Model Quality Report in Business Statistics

Transcription

1 Moel Quali Repor in Buine Saiic Ma Bergal, Ole Blac, Ruell Boaer, Ra Camber, Pam Davie, Davi Draper, Eva Elver, Suan Full, Davi Holme, Pär Lunqvi, Sien Lunröm, Lennar Norberg, Jon Perr, Mar Pon, Mie Preoo, Ian Ricaron, Cri Sinner, Paul Smi, Ceri Uneroo, Mar William General Eior: Pam Davie, Paul Smi Volume I Teor an Meo for Quali Evaluaion

2 Preface Te Moel Quali Repor in Buine Saiic proec a e up o evelop a eaile ecripion of e meo for aeing e quali of urve, i paricular applicaion in e cone of buine urve, an en o appl ee meo in ome eample urve o evaluae eir quali. Te or a pecifie an iniiae b Euroa folloing on from e Woring Group on Quali of Buine Saiic. I a fune b Euroa uner SUP-COM 997, lo 6, an a been uneraen b a conorium of e UK Office for Naional Saiic, Saiic Seen, e Univeri of Souampon an e Univeri of Ba, i e Office for Naional Saiic managing e conrac. Te repor i ivie ino four volume, of ic i i e fir. Ti volume eal i e eor an meo for aeing quali in buine urve in nine caper folloing e urve proce roug i variou age in orer. Tee fall ino ree par, one ealing i ampling error, one i a varie of non-ampling error, an one covering coerence an comparabili of aiic. Oer volume of e repor conain: a comparion of e ofare meo an pacage available for variance eimaion in ample urve (volume II); eample aemen of quali for an annual an a monl buine urve from Seen an e UK (volume III); guieline for an eperience of implemening e meo (volume IV). An ouline of e caper in e repor i given on e folloing page. Acnolegemen Apar from e auor, everal oer people ave mae large conribuion iou ic i repor oul no ave reace i curren form. In paricular e oul lie o menion Tim Jone, Ania Ullberg, Jeff Evan, Trevor Fenon, Jonaan Goug, Dan Helin, Sue Hibbi an Seve Jame, an e oul alo lie o an all e oer people o ave been o elpful an uneraning ile our aenion a been focue on i proec! Ouline of Moel Quali Repor Volume Volume I. Meoolog overvie an inroucion Par : Sampling error. Probabili ampling: baic meo 3. Probabili ampling: eenion 4. Sampling error uner non-probabili ampling Par : Non-ampling error

3 5. Frame error 6. Meauremen error 7. Proceing error 8. Non-repone error 9. Moel aumpion error Par 3: Oer apec of quali 0. Comparabili an coerence Par 4: Concluion an Reference. Concluion. Reference Volume II. Inroucion. Evaluaion of variance eimaion ofare 3. Simulaion u of alernaive variance eimaion meo 4. Variance in STATA/SUDAAN compare i analic variance 5. Reference Volume III. Inroucion Par : Te rucural urve. Quali aemen of e 995 Sei Annual Proucion Volume Ine 3. Quali aemen of e 996 UK Annual Proucion an Conrucion Inquirie Par : Sor-erm aiic 4. Quali aemen of e Sei Sor-erm Proucion Volume Ine 5. Quali aemen of e UK Ine of Proucion 6. Quali aemen of e UK Monl Proucion Inquir Par 3: Te UK Sampling Frame 7. Sampling frame for e UK Volume IV. Inroucion. Guieline on implemenaion 3. Implemenaion repor for Seen 4. Implemenaion repor for e UK ii

4 Conen Meoolog overvie an inroucion.... General rucure.... A guie o e conen..... Toal urve error..... Sampling error Non-ampling error Comparabili an coerence Concluing remar...6 Par : Sampling Error Probabili ampling: baic meo...7. Baic concep Targe populaion an ample populaion Sample frame an auiliar informaion Probabili ampling...8. Saiical founaion Y an X variable Finie populaion parameer Populaion moel Sample error an ample error iribuion Te repeae ampling iribuion v. e uperpopulaion iribuion Bia, variance an mean quare error....3 Eimae relae o populaion oal Te eign-bae approac Sample incluion probabiliie Te Horviz-Tompon eimae Deign-bae eor for e Horviz-Tompon eimae Deign-bae eor for fie ample ize eign Approimaing econ orer incluion probabiliie Problem i e eign-bae approac Te ue of moel for eimaing a populaion oal Te uperpopulaion moel Te omogeneou raa moel Te imple linear regreion moel Te general linear regreion moel Te cluer moel Ignorable ampling Bia, variance an mean quare error uner e moel-bae approac Weanee of e moel-bae approac Linear preicion Robu preicion variance eimaion Te moel-aie approac Te GREG an GRAT eimae for a populaion oal Variance eimae for e GREG an GRAT Calibraion eiging Meo for nonlinear funcion of e populaion value Variance eimaion via Talor erie lineariaion Differeniable funcion of populaion oal Funcion efine a oluion of eimaing equaion Replicaion-bae meo for variance eimaion Ranom group eimae of variance Jacnife eimae of variance Te linearie acnife Boorapping Concluion Probabili ampling: eenion Domain eimaion Deign-bae inference for omain Deign-bae inference uner SRSWOR Moel-bae inference en N i unnon...4 iii

5 3..4 Moel-bae inference en N i non Moel-bae inference uiliing auiliar informaion An eample Domain eimaion uing a linear eige eimae Moel-aie omain inference Eimaion of cange Linear eimaion Eimae of cange for funcion of populaion oal Eimae of cange in omain quaniie Oulier robu eimaion Oulier robu moel-bae eimaion Winoriaion-bae eimaion Variance eimaion for inice Concluion Sampling error uner non-probabili ampling Inroucion Volunar ampling Quoa ampling Jugemenal ampling Proucer price ine conrucion in e EU Te UK eperience Cu-off ampling Variaion : Ignore e cu-off uni Variaion : Moel e cu-off uni Concluion...80 Par : Non-ampling error 5 Frame error Inroucion A Buine Regier an i ue a a frame Uni, elineaion, an variable Upaing e BR uing everal ource Te BR a a frame uni, variable an reference ime Frame an arge populaion Targe populaion Frame, an frame populaion Difference beeen e frame populaion an e arge populaion Uner- an over-coverage of e populaion Difference iin e populaion Some commen on frame error Defining a Buine Regier covering a ime perio Te arge populaion: eimaion an inaccurac Eimaion proceure an informaion neee Uing e frame populaion onl Upaing e ample onl Uiliing laer BR informaion on e populaion Uiliing a BR covering e reference perio Some commen on e BR an effec of coverage eficiencie Illuraion aminiraive aa an buine emograp Illuraion ime ela an aing frame Te UK Buine Regier Te Sei Buine Regier Some comparion beeen UK an Seen Illuraion cange beeen frame an eir effec Difference beeen UK curren an frozen claificaion Difference iin e Sei populaion one ear apar Difference for e populaion a a ole; Seen A fe ummariing concluion Meauremen error Naure of meauremen error True value...04 iv

6 6.. Source of meauremen error Tpe an moel of meauremen error Te conribuion of meauremen error o oal urve error Toal urve error Bia Variance inflaion Diorion of eimae b gro error Deecing meauremen error Comparion a aggregae level i eernal aa ource Comparion a uni level i eernal aa ource Inernal comparion an eiing Follo-up Embee eperimen an obervaional aa Quali meauremen Quali inicaor Aeing e bia impac of meauremen error Aeing e variance impac of meauremen error Proceing error Inroucion o proceing error Sem error Meauring em error Sem error: o eample Sampling in e ONS Variable forma in compuer program Minimiing em error Daa anling error Daa ranmiion Daa capure Daa eing from pencil an paper queionnaire Meauring error occurring uring aa eing Minimiing error occurring uring aa eing Daa capure uing canning an auomae aa recogniion Meauring error aociae i canning an auomae aa recogniion Minimiing error aociae i canning an auomae aa recogniion Coing error Meauring coing error Conienc Accurac Te impac of coer error on e variance of urve eimae Te ri of coer error inroucing bia in urve eimae Minimiing coing error Daa eiing Meauring e impac of eiing on aa quali Minimiing error inrouce b eiing An eample of error a e publicaion age Nonrepone error Inroucion Tpe of nonrepone Paern of miing aa Miing aa mecanim Problem caue b nonrepone A baic eing Bia Variance inflaion Effec of confuing uni ouie e populaion i nonrepone Effec of nonrepone on coerence Quali meauremen Repone rae Meaure bae on follo-up aa Comparion i eernal aa ource an bencmar...3 v

7 8.4.4 Comparion of alernaive aue poin eimae Weiging aumen Te baic meo Ue of auiliar informaion Poraificaion Regreion eimaion an calibraion Weiging an nonrepone error Variance eimaion Impuaion Ue Deucive impuaion an eiing La value impuaion Raio an regreion impuaion Donor meo Socaic meo Impuaion an nonrepone error Variance eimaion Moel Aumpion Error Inroucion Ine number Bencmaring Seaonal aumen Cu-off ampling Small omain of eimaion Non-ignorable nonrepone Selecion moel for coninuou oucome Paern-miure moel for caegorical oucome Concluion...60 Par 3: Oer Apec of Quali 0 Comparabili an coerence Inroucion Coerence empaiing e uer perpecive Definiion in eor Definiion in pracice Accurac an conien eimae Comparabili over ime Inernaional comparabili Some uer-bae concluion Proucer apec on coerence, incluing comparabili Definiion in eor Definiion in pracice Accurac an conien eimae Some commen on meoolog, epeciall bencmaring Comparabili over ime Inernaional comparabili Some proucer-bae concluing commen Some illuraion of coerence an co-orinaion...74 Par 4: Concluion an Reference Concluing remar Meoolog for quali aemen Recommenaion for quali aemen...79 Reference Ine...89 vi

8 Meoolog overvie an inroucion. General rucure Paul Smi, Office for Naional Saiic Ti volume cover e eor an meo for aeing quali in buine urve uner eig main eaing. Te main bo of e repor i ivie ino nine caper, i e probabili ampling main eaing pli ino o caper. Te non-ampling error ecion follo e claificaion of e Euroa oring group on Quali of Buine Saiic. Te caper are. Probabili ampling: baic meo 3. Probabili ampling: eenion 4. Sampling error uner non-probabili ampling 5. Frame error 6. Meauremen error 7. Proceing error 8. Nonrepone error 9. Moel aumpion error 0. Comparabili an coerence Tee fall ino ree par, i caper -4 ealing i ampling error (par ), caper 5-9 i variou apec of non-ampling error (par ) an caper 0 forming a par on i on (par 3). Te coverage of eac caper i ecribe in ummar in ecion., an e iea are neie an line o e Moel Quali Repor in e final caper, caper. Reference o oer or menione in i volume appear a e en, an e noaion generall follo Särnal, Senon & Wreman (99) ecep ere furer noaion i require, in ic cae i i efine.. A guie o e conen.. Toal urve error I i enible o r o lin e meo in ee ampling an non-ampling error caper ino a common frameor (a) a a guie o a i of mo inere an relevance an ic ource of error i liel o be mo imporan in a given cone, an (b) o elp in navigaion roug e opic conaine in e variou caper. Ti i epeciall imporan in ome of e non-ampling error caper ere opic ill ofen fi comforabl uner more an one eaing, an i ma no be immeiael obviou ere o loo for informaion on a paricular opic. Te be concep for proviing a unifing frameor i e concep of oal urve error (Grove 989), ic emboie e ifference beeen e urve eimae an e concepual real or rue value. In buine urve e real value (oal ale b manufacuring inurie, for eample) mol a a founaion in reali if i ere poible o loo a ever manufacuring buine ale an recor em accurael, e coul arrive a

9 e real value. For oer aiic uc a e average price movemen e rue value i no ell-efine an i conruc brea on. So, auming a e real value i ell-efine, e can imagine a e an o meaure e ifference beeen our urve eimae an e rue value. Conier e problem of eimaing a oal pical eimaor ae e form i i U i of a variable acro a populaion U. Te, ere i i e urve eig, i i e repore value of i an e um i over e ample. Te oal urve error i en oal urve error i i U i an i ma be broen on ino o componen (ee Grove, 989, p.): error from obervaion error from non - obervaion i i i i U i i i ( ) Te fir (obervaion error) componen reflec meauremen error, a ell a proceing, coing an impuaion error an oul iappear if e recore value i ere equal o e rue value i. Te econ (non-obervaion error) componen reflec ampling error, frame error an nonrepone error an oul iappear if e uni upon ic e eimae i bae comprie preciel e arge populaion U. Te oal urve error provie an overall meaure of quali. Te problem i o o ae i magniue. To meaure e ampling error i i uual o e up a moel for e iribuion of e ampling error an en o eimae e caraceriic of i iribuion. Uuall, i i aume (e aumpion being bae on ampoic eor) a i ampling iribuion i approimael normal an cenre a zero o a e onl a i o eimae e variance of e iribuion. To een i iea o oal urve error i i necear o e up a moel for e iribuion of e oer componen of error. Toal urve error can be coniere in a ifferen a oo broen on ino o componen, a ifference ic i approimael invarian over repeiion of e urve, e bia, an a ifference ic varie i ifferen repeiion of e urve, e variance. Te repeiion ue in i efiniion are ofen poeical, a i e urve i no acuall repeae. We eplore ee o pe of error in more eail belo. Te bia an variance ogeer conribue o a meaure of e oal urve error, calle e mean quare error (me), uc a me bia variance alo omeime epree a i quare roo, e roo mean quare error (rme). Bo e bia an e variance are mae up of everal componen erm correponing o paricular pe of error. In e cae of e bia ome of ee componen ill almo cerainl cancel eac oer ou (e a a ere are poiive an negaive biae), o a e overall bia ill be i i i

10 e ne of ee effec. Variance are ala non-negaive an o ill cumulae over componen. If all e relevan biae an variance are inclue in calculaing e me, i ill be a goo eimaor of e oal urve error. Ti give u o broa approace o man error. We can rea e repone of a given uni a fie for an occaion en i i inclue in e ample (a in of eerminiic approac). Ta i, if a buine i inclue in e ample, e aume a i ala mae e ame repone/nonrepone eciion, ala give e ame aner on e queionnaire, an o on. Ti almo ala lea u o eimae biae. Alernaivel e can conier a a buine repone/nonrepone eciion arie from ome probabili iribuion, an a i aner alo come from ome iribuion, in ic cae mo of e error ill aiionall ave a variance componen. Ti laer approac i ain o e moel-bae ampling approac (ecion.3.), a e aume a uperpopulaion of poible oucome i e ampling forming onl one componen of eermining ic oucome e acuall oberve in e urve. We ill ue i iincion in approac beeen eerminiic an uperpopulaion moel in icuing e error ic mae up oal urve error... Sampling error Cerain aumpion an moel are require o eimae e componen of oal urve error, an e begin b coniering ranom ampling mecanim; in i ecion e aume a all urve age afer ampling are error-free. Wen a urve i o be conuce, e ample can be elece accoring o ome probabili mecanim. A lea concepuall e can elec more en one ample uing e ame probabili mecanim (b running e elecion proce everal ime), an eac ample oul reul in a ifferen eimae if e urve ere acuall run, impl becaue ifferen uni oul be inclue in e ample. Eac of ee poenial eimae oul in general be ifferen from e rue oal. We ave ere e iuaion a e urve eimae are ifferen b repeiion over ifferen ample, an e can meaure o muc ee eimae iffer from eir mean on average, uing e average iance of e ample elemen from eir mean o eimae e average iance of populaion elemen from e mean. Ti give u a variance, e ampling variance. Over all poible ifferen ample, e mean of e eimae i e ame a e rue value (ill auming no oer error); in pracice e normall ave onl one ample, an ave o ue e mean of a ample o approimae e rue populaion mean. Effecivel, a menione in ecion.., e aume a e ampling error i cenre aroun e eimae e o ave. Caper cover e eor an meo ic give rie o ampling error an ampling error eimae uing firl e eign-bae an moel-aie approace, uner ic ifferen moel of e relaionip beeen a urve repone an non auiliar value are ue o improve e eimaion. Tee approace baicall involve accouning for e elecion probabiliie from e ampling in all e eimaion an variance calculaion in an appropriae a. Ti caper alo inrouce e moel-bae approac, ic aume a Unle eimae b a variance componen moel; if a negaive variance i obaine i probabl inicae a e moel i inappropriae. 3

11 e urve repone are realiaion from an poeical infinie populaion of poible oucome. In i cae, i an appropriae moel e elecion probabiliie are ignorable, a i e ave no effec on e eimaion or variance eimaion an o no nee o be inclue eplicil. Caper 3 ae ee o approace an een em from raigforar eimaion meo o more complicae aiic, incluing eimaion of cange, eimaion for omain (ube of e populaion) an eimaion in e preence of oulier. Tere i alo a ummar of ome or on e variabili of a muliource inicaor, ic conier e effec of e variabili of ifferen erie ic go o mae up an ine on i oal variance. Conier no ample elecion mecanim ic are no bae on probabili. In ee cae e pe of error e obain epen on e acual mecanim of elecion. If repeiion a no effec on e ample compoiion (a i, e ame ample elemen are coen ever ime), en e ifference beeen e urve eimae an e rue value i conan over repeiion: i i a (pure) bia. If e ample can be ifferen over repeiion, en ere ill be a range of poenial eimae, an ere ill be a variance componen an a bia. In pracice e o effec ma no be eparael eimable, or even eimable a all if e rue value i unnon (ic i picall e cae). Ti ubec i aree in caper 4 (nonprobabili ampling), concenraing paricularl on cu-off ampling an volunar ampling (ample obaine from volunar urve), bu alo menioning quoa ampling an ugemenal ampling...3 Non-ampling error No relaing e aumpion from ecion.. a evering ele apar from ampling i perfec, le u conier e oer poible error. Tee are arrange o follo approimael e orer of proceing in a buine urve. Frame error conribuing mainl o e bia componen of e oal error are icue in caper 5. Tee error generall em from ifference beeen frame- an arge populaion. Hence problem of uner- an over-coverage are imporan. Since buine populaion uuall cange rapil, e upaing of uni an of variable aace o ee uni become imporan. Delineaion of buinee ino ifferen pe of uni (local uni, in-of-acivi uni ec) i anoer acivi i a large impac on frame quali. All of ee iue are eal i in caper 5. Meauremen error are error ic are inrouce en ring o ge e eire informaion from conribuor. In caper 6, e loo a a meauremen error moel for o aner var over ifferen (concepual) repeae queioning, an i conribue o e variabili of e eimae b giving a variable meauremen for a ingle reponen. Meauremen error are liel o conribue o bo componen bia an variance of e oal error bu e are ofen ifficul or epenive o ae, epeciall in cae ere folloup uie become necear. Ye meauremen error ma ofen ave a large influence on accurac in buine urve. Approace o eecion an aemen of meauremen error are icue in caper 6. 4

12 Proceing error are icue in caper 7. Tee are error connece i aa anling aciviie eer manual or auomae uc a aa ranmiion, aa capure, coing an aa eiing. A paricular form of proceing error, calle em error in caper 7, are error ariing from ofare an arare. I i ifficul o enviage a probabili mecanim i a real inerpreaion for em error, an in fac e are ver ifficul o meaure a all. Proceing error in general ma conribue o bo componen e bia an e variance of e oal error aloug e bia i liel o be e more imporan one. Nonrepone, reae in caper 8, arie en a ample uni fail o provie complee repone o all queion ae in a urve. Tere are o a of coniering nonrepone in a fie ample. Te eerminiic approac aume a fie bu unnon repone inicaor value ( if value i recore, 0 if value i miing) for ever uni in e ample. Te ocaic approac rea e repone inicaor variable a oucome of ranom variable. Te naure of error ariing from nonrepone en epen on aumpion abou i ranom mecanim. Te ocaic approac i e one folloe in caper 8. Meo o meaure or inicae e impac of nonrepone on accurac are reae. Ti caper alo rea implicaion of nonrepone uc a bia, variance inflaion an effec of confuing nonrepone i over-coverage. Re-eiging an impuaion meo o compenae for bia caue b nonrepone are icue. Caper 9 icue error an inaccurac caue b uing moel aumpion concenraing on eimaion problem an pe of moel ic are no menione eleere. Te aim of inroucing a moel ma be o reuce variance an/or o reuce bia, bu ere i alo a ri of inroucing bia if e moel i no ell coen. Small area eimaion i one par of e urve proce ere moel are imporan, bencmaring anoer (noe a calibraion belong o ampling error; e iea i imilar bu e ecnique ifferen). Non-ignorable nonrepone i icue ere, aloug i a rong lin o e non-repone meo in caper 8. Te icuion of cu-off ampling a are in caper 4, non-probabili ampling, an i i coninue ere, empaiing e ue of moel o eimae for e par of e populaion a a cu off. Anoer reaon for uing moel i o elp o compenae for a lac of up-o-ae informaion, for eample on eig in caine price inice, a problem ic i inrouce in i caper. Seaonal aumen i alo ecribe, incluing commen on e ofare in ue; aemen of e reuling accurac i a ifficul maer...4 Comparabili an coerence Ti i an area ic oe no fi uner e uual efiniion of oal urve error, becaue i oe no eal i e error in a ingle urve, bu inea conier o ell o or more e of aiic can be ue ogeer. Ti caper cover efiniion in eor an in pracice, accurac, ifferen co-orinaion aciviie, an comparabili of urve over ime an naional bounarie. Bo uer an proucer perpecive are coniere, an illuraion are given. 5

13 ..5 Concluing remar Te final caper in i volume, caper, lin e concep ecribe in i inroucion an ra ou e imporan eme for aeing oal urve error in ome given cone. I alo correpon i caper of e Implemenaion Guieline (volume IV), ic provie a ummar of e meo ecribe in i volume a e are applie in e Moel Quali Repor. Tere i an eample running roug e ampling error caper ( an 3), an ic alo appear in caper 4, 8 an 9, ic correpon rongl i e Annual Buine Inquir in e UK, ic i e annual rucural urve eample from e UK in e Moel Quali Repor (volume III, caper 3). 6

14 Par : Sampling Error Probabili ampling: baic meo. Baic concep Ra Camber, Univeri of Souampon Man cienific an ocial iue revolve aroun e iribuion of ome pe of caraceriic over a populaion of inere. Tu e number of unemploe people in a counr labour force an e average annual profi mae b buinee in e privae ecor of a counr econom are o e inicaor of a counr economic ell-being. Te fir of ee number epen on e iribuion of labour force ae among e iniviual maing up e counr labour force ile e econ i eermine b e iribuion of annual profi acieve b e counr buinee. Bo ee number are picall meaure b ample urve. Ta i, a ample of iniviual belonging o e counr labour force i urvee an eir emplomen/unemplomen aue eermine. Similarl a ample of privae ecor buinee i urvee an eir annual profi meaure. In bo cae e informaion obaine from e urve can be ue o infer e unnon correponing value (unemplomen oal or average profi) for e counr... Targe populaion an ample populaion Since in general i i meaningle o al abou a ample iou referring o a i i a ample of, e concep of a populaion from ic a ample i aen i baic o ample urve eor. In e eample above ere are o populaion e populaion of iniviual maing up e labour force of e counr, an e populaion of buinee maing up e privae ecor econom of e counr. In general, oever, e populaion from ic a ample i aen, an e populaion of inere can an o iffer. Te arge populaion of a urve i e populaion a ic e urve i aime, a i e populaion of inere. Hoever, a arge populaion i no necearil a populaion a can be urvee. Te acual populaion from ic e urve ample i ran i calle e urve populaion. A baic meaure of e overall quali of a ample urve i e coverage of e urve populaion, or e egree o ic arge an ample populaion overlap. Aemen of i quali i coniere in Caper 5. Here e all aume ere i no ifference beeen e arge an urve populaion. Ta i, e ave complee coverage. From no on e ill u refer o e populaion... Sample frame an auiliar informaion A anar meo of ampling i o elec e ample from a li (or erie of li) ic enumerae e uni (iniviual, buinee, ec) maing up e ample populaion. Ti li i calle e (ample) frame for e urve. Eience of a ample frame i necear for e ue of man ampling meo. Furermore, applicaion of ee meo ofen require a 7

15 a frame conain more an u ienifier (for eample, name an aree) for e uni maing up a ample populaion. For eample, raifie ampling require e frame o conain enoug ienifing informaion abou eac populaion uni for i raum memberip o be eermine. In general, e refer o i informaion a auiliar informaion. Tpicall, i auiliar informaion inclue caraceriic of e urve populaion a are relae o e variable meaure in e urve. Tee inclue raum ienifier an meaure of ize. For economic populaion, e laer correpon o value for eac uni in e populaion ic caracerie e level of economic acivi b e uni. Te een o ic e ample frame enumerae e ample populaion i anoer e meaure of ample urve quali. Ti iue i coniere in Caper 5. In a follo oever e all aume a ample frame ei an i perfec. Ta i, i li ever uni in e populaion once an onl once, an ere i a non number N of uc uni...3 Probabili ampling A probabili ampling meo i one a ue a ranomiaion evice o ecie ic uni on e ample frame are in e ample. Wi i pe of elecion meo, i i no poible o pecif in avance preciel ic uni on e frame mae up e ample. Conequenl uc ample are free of e (ofen ien) biae a can occur i ampling meo a are no probabili-bae. In a follo e mae e baic aumpion a e probabili ampling meo ue i uc a ever uni on e frame a a non-zero probabili of elecion ino e ample. Ti aumpion i necear for valii of e eign-bae approac o urve eimaion an inference ecribe in ecion.3. belo. Some relevan eor for e cae ere a non-probabili ampling meo i ue i e ou in Caper 4.. Saiical founaion A noe earlier, e baic aim of a ample urve i o allo inference abou one or more caraceriic of e populaion. Suc caraceriic are picall efine b e value of one or more populaion variable. A populaion variable i a quani a i efine for ever uni in e populaion, an i obervable en a uni i inclue in e ample. In general, urve are concerne i man populaion variable. Hoever, mo of e eor for ample urve a been evelope for e cae of a mall number of variable, picall one or o. In a follo e aop e ame implificaion. Iue ariing ou of e nee o meaure man variable imulaneoul in a ample urve are coniere in ecion Y an X variable Aociae i eac uni in e populaion i a e of value for e populaion variable. Some of ee are recore on e frame, an o are non for ever uni in e populaion. We refer o ee auiliar variable a X-variable. Te oer coniue e variable of inere (e u variable) for e urve. Tee are no non. Hoever e aume a eir value are meaure for e ample uni, or can be erive from ample aa. We uuall refer o ee variable a Y-variable. 8

16 For eample, e quarerl urve of capial epeniure (CAPEX) carrie ou b e U.K. Office for Naional Saiic (ONS) a everal u (Y) variable, e mo imporan being acquiiion, ipoal an e ifference beeen acquiiion an ipoal, e ne capial epeniure. Te frame for i urve i erive from e Iner-Deparmenal Buine Regier (IDBR) of e ONS. Tere are a number of X-variable on e urve frame, e mo imporan of ic are e inur claificaion of a buine (Sanar Inur Claificaion), e number of emploee of e buine an e oal VAT urnover of e buine in e preceing ear... Finie populaion parameer Te populaion caraceriic a are e focu of ample urve are omeime referre o a i arge of inference. In general, ee arge are ell-efine funcion of e populaion value of Y-variable, picall referre o a parameer of e populaion. An populaion covere b a frame-bae urve i necearil finie in erm of e number of uni i conain. Suc a parameer ill be referre o a a finie populaion parameer (FPP) in a follo in orer o iingui i from e parameer a caracerie e infinie populaion ue in anar aiical moelling. Some common eample of FPP are: - e populaion oal an average of a Y-variable; - e raio of e populaion average of o Y-variable; - e populaion variance of a Y-variable; - e populaion meian of a Y-variable...3 Populaion moel A populaion of Y-value a an one poin in ime repreen e oucome of man cance occurrence. Hoever, i oe no mean a ee value are compleel arbirar. Tere i picall a rucure ineren in a e of populaion value a can be caracerie in erm of a moel. Suc moel are uuall bae on pa epoure o aa from oer populaion ver muc lie e one of inere, or ubec maer nolege abou o e populaion value oug o be iribue. Conequenl i moel i no caual i oe no a o ee Y-value came o be bu ecripive, in e ene a i i a maemaical ecripion of eir iribuion. In man cae i moel i ielf efine in erm of parameer ic capure ee iribuional caraceriic. A anar a of pecifing uc a aiical moel i in erm of an unerling ocaic proce. Ta i, e N value coniuing e finie populaion of inere are aume o be realiaion of N ranom variable oe oin iribuion i ecribe b e moel. If i approac i aen, en e moel ielf i referre o a a uperpopulaion moel for e finie populaion of inere. Te parameer a caracerie i moel are picall unnon, an are referre o a e uperpopulaion parameer for e populaion. Unlie FPP, uperpopulaion parameer are no real e can never be non preciel, even if e uperpopulaion moel i an accurae epicion of o e finie populaion value are iribue an ever populaion value i non. Some eample of uc uperpopulaion parameer are momen (mean, variance, covariance) of e oin iribuion of e Y- 9

17 variable efining e populaion value an relae quaniie (for eample regreion coefficien)...4 Sample error an ample error iribuion Once a ample a been elece, an ample value of Y-variable obaine, e are in a poiion o calculae e value of variou quaniie bae on ee aa. Tee quaniie are picall referre o a aiic. Te aim of ample urve eor i o efine o pe of aiic: (i) eimae of e FPP of inere; (ii) quali meaure for e eimae in (i). In i repor e ill be mainl concerne i e econ pe of aiic above, a i aiic meauring e quali of e eimae. Hoever, before e can ecribe o uc aiic can be erive, e nee o icu e concep of ample error an ample error iribuion. Te ample error of a urve eimae i u e ifference beeen i oberve value an e unnon value of e FPP of ic i i an eimae. Clearl one oul epec a ig quali urve eimae o ave a mall ample error. Hoever, ince e acual value of e FPP being eimae i unnon, e ample error of i eimae i alo unnon. Bu i oe no mean a ere i noing e can a abou i error. Te meo b ic e ample i coen, an e uperpopulaion moel for e populaion, allo u o pecif a varie of iribuion for e ample error. In urn, i allo u o ue aiical meo o meaure e quali of e urve eimae in erm of e caraceriic of ee iribuion. Before going on o ecribe o ee iribuion are erive an inerpree, i i imporan o noe a i quali meauremen relae o a quani (e ample error) ic aume a ere are no oer ource of error in e urve. In reali, ere are man oer ource of error (frame error, nonrepone error, meauremen error, moel pecificaion error, proceing error) in a urve. Meo for aeing ee are icue in Par of i repor...5 Te repeae ampling iribuion v. e uperpopulaion iribuion Tere are o anar a of efining a iribuion for a ample error. One i i repeae ampling iribuion. Ti i e iribuion of poible value i error can ae uner repeiion of e ampling meo. Concepuall, i correpon o repeaing e ampling proce, elecing ample afer ample from e populaion, calculaing e value of e eimae for eac ample, generaing a (poeniall) ifferen ample error eac ime an ence a iribuion for ee error. Te oer a of efining a iribuion for a ample error i in erm of e uperpopulaion iribuion. Uner i iribuion e ample eimae a ell a e FPP are bo bae on realiaion of e Y-variable a efine e populaion value. Conequenl e ample error i alo a ranom variable i a iribuion efine b e uperpopulaion moel. 0

18 Operaionall i iribuion correpon o e range of poenial value e ample error can ae given e range of poenial value for e populaion Y-variable uner i moel. Tere are funamenal ifference beeen ee iribuion. Te repeae ampling iribuion rea e populaion value a fie. Conequenl e ource of variabili unerling i iribuion i e ample elecion meo. Sample elecion meo a are no probabili bae are erefore no uie o evaluaion uner i iribuion. In conra, e uperpopulaion iribuion rea e ample a fie. Ta i, e unerling variabili in i cae arie from e uncerain abou e iribuion of Y-value for e ample uni an non-ample uni, bu e ample/non-ample iincion i fie accoring o a acuall oberve. To iingui beeen ee o iribuion, e ue a ubcrip of p in a follo o enoe epecaion, variance, ec, aen i repec o e repeae ampling iribuion, an a ubcrip of o enoe correponing quaniie aen i repec o e uperpopulaion iribuion. Tere are aiical argumen for an again e ue of ee o iribuion for e ample error en e an o caracerie e quali of e acual ample eimae. Baicall, e repeae ampling (or ranomiaion) iribuion of e ample error i viee a appropriae for meauring e quali of a urve eign, a i e meo ue o elec e ample. Ti i becaue i reflec our uncerain abou ic ample ill be coen prior o e acual coice of ample. Hoever, bo meo ave been ue o caracerie uncerain abou e ize of e ample error afer e ample aa are obaine. Te argumen for uing e ranomiaion iribuion involve e aumpion a ee aa o noing o cange e ource of our uncerain, e u provie u i a mean o meaure i. We ill caracerie uncerain b e iribuion of ample error aociae i ample a mig ave been coen bu ere no. In conra, ue of e uperpopulaion iribuion eeniall come on o aing a e populaion Y-value, being unnon, repreen e rue ource of uncerain a far a urve inference i concerne. In paricular, afer e ample aa are obaine e ave no uncerain abou ic ample a elece, bu e ill ave uncerain abou e populaion Y-value efining e FPP of inere. In i repor e ill evelop meaure bae on bo iribuion, inicaing eir reng an eanee ere appropriae...6 Bia, variance an mean quare error In orer o ue a iribuion for e ample error o meaure e quali aociae i e acual ample eimae, e nee o pecif e caraceriic of i iribuion a are appropriae for i purpoe. Saiical pracice eeniall focue on o uc caraceriic e cenral locaion of e iribuion, a efine b i mean or epecaion, an e prea of i iribuion aroun i mean, a efine b i variance. Ofen bo are combine in e mean quare error, ic i e variance plu e quare mean. Te mean of e ample error iribuion i picall referre o a e bia of e eimaion meo, o e mean quare error become variance plu quare bia.

19 A ig quali eimae ill be aociae i a ample error iribuion a a bia cloe o or equal o zero an lo variance. In i cae e can be ure a e oberve value of e eimae ill, i ig probabili, be cloe o e unnon FPP being eimae. Conequenl e focu on e bia an variance of e ample error iribuion a e e quali meaure of a ample eimaion meo. In e ne ecion e evelop epreion for ee quaniie, ogeer i relevan meo for eimaing em from e ample aa. In oing o e focu on one FPP a i of paricular inere in man urve ampling iuaion. Ti i e FPP efine b e oal of e value aen b a ingle Y-variable..3 Eimae relae o populaion oal Le U enoe e finie populaion of inere, an le U enoe e N uni maing up i populaion. For eac uni e aume a a Y-variable i efine, i e realie (bu unnon) value of i variable for e uni enoe b. Te oal of e N value of i Y-variable in e populaion ill be enoe. Folloing common pracice e o no iingui beeen a a realiaion (a i a number) an a e ranom variable a le o a realiaion. I ill be clear from e cone a paricular inerpreaion oul be place on i quani. Similarl, e ill no iingui beeen an eimae (a realie value) an an eimaor (e proceure a le o e realie value)..3. Te eign-bae approac Ti approac, ofen referre o a eign-bae eor, evaluae an eimae of in erm of e repeae ampling iribuion of i ample error. Ta i, a goo eimae for i efine a one for ic e aociae ample error i non o be a ra from a repeae ampling iribuion a a eier zero bia or bia a i approimael zero an a mall variance. A ill become clear belo, e uefulne of i approac epen on eer or no a ranom meo i non ample incluion probabiliie i emploe for ample elecion..3.. Sample incluion probabiliie In orer o generae i repeae ampling iribuion e nee o inrouce e concep of a ample incluion inicaor. Ti i a binar value ranom variable a ae e value if a uni i inclue in ample an i zero oerie. We enoe i b I in a follo. Clearl e iribuion of I epen on e proce ue o cooe e ample. Suppoe no a i proce i ranom in ome a. Ten e can pu π Pr p (I ) Pr(uni i inclue in ample given fie populaion value for Y an e auiliar variable X). Since e aume a ever uni in e populaion a a non-zero probabili of incluion in e ample, e mu ave π > 0 for all U. Noe a e o NOT aume a e I are inepenen ranom variable. Te properie of e oin iribuion of an ube of ee ranom variable ill epen on e acual ampling meo emploe. Te imple oin iribuion i of o incluion variable, I an I, ere. In i cae e pu π Pr p (I, I ) Pr(uni an are bo

20 inclue in ample given fie populaion value for Y an X). I i anar o refer o π a e incluion probabili for uni, an π a e oin incluion probabili for uni an..3.. Te Horviz-Tompon eimae Suppoe no a e value π are non for eac uni in e populaion. Ten, irrepecive of ic ample i acuall coen, e can efine an eimae of of e form HT π. Te noaion mean a e ummaion above i rerice o e ample uni, ile e ubcrip HT refer o e fac a i eimae a fir pu forar in Horviz & Tompon (95) Deign-bae eor for e Horviz-Tompon eimae I i raigforar o o a e repeae ampling iribuion of e ample error of e HTE (Horviz-Tompon eimae) a mean zero. An equivalen a of aing i i o a a HT i unbiae uner repeae ampling, or, more commonl, a i i eign unbiae a i, unbiae i repec o repeae ampling uner e probabili ampling eign. Te mean an variance of e repeae ampling iribuion of (e ample error efine b) HT are eail obaine. I u require one o noice a e onl ranom variable conribuing o i iribuion are e ample incluion variable I efine above. All oer quaniie (an in paricular e value of Y) are el fie a eir populaion value. Conequenl, ince E p (I ) π, E p ( ) HT E E p p U E U p π I ( I π π ) U U U 0. Ta i, e ample error iribuion of e HTE a zero bia uner repeae ampling. Noe a i proof i epenen on ever uni in e populaion aving a non-zero probabili of incluion in ample. Te eign variance of HT (a i e variance of e repeae ampling iribuion of e ample error efine b HT ) i obaine roug a ver imilar argumen. Since i coniere fie in i cae, i variance i given b 3

21 4 ( ) ( ). ) ( ), ( C ) ( V V V V U U U U U p U p U p J p HT p I I I I π π π π π π π π π π π π Wiou lo of generali e efine π π. Ten e above variance i ( ) ( ) U U HT p π π π π π V. Noe a i variance i a FPP. Conequenl e can ue e argumen a o HT i eign unbiae o obain an eimae of i variance a i alo eign unbiae. Ti i e o-calle HT eimae of variance ( ) ( ) HT HT p π π π π π π V Deign-bae eor for fie ample ize eign An imporan cla of ample eign ave fie ample ize. For uc eign e um of an realiaion of e N ample incluion inicaor equal a fie number n (e ample ize). I immeiael follo a for fie ample ize eign e um of e populaion value of π mu alo equal n. Furermore, en U U U n I n I I I I I π π ) ( ) (. Tee equaliie allo u o epre e eign variance of HT a lile ifferenl. Ta i, en a fie ample ize eign i ue i variance i ( ) ( ) U U HT p V π π π π π. A eign unbiae eimae of i variance i eail een o be ( ) HT SYG p V π π π π π π. Te upercrip SYG above an for Sen-Yae-Grun, e original eveloper of i paricular variance eimae (Yae & Grun, 953; Sen, 953).

22 Te HT variance eimae can ae negaive value en ample uni ave ig incluion probabiliie. Similarl, e SYG variance eimae can be negaive if π π < π for ome. Since in mo pracical cae i coniion oe no ol, e SYG eimae i uuall preferre for eimaing e eign variance of e HTE Approimaing econ orer incluion probabiliie An imporan pracical problem unerling bo variance eimae above i a e require e urve anal o no e oin incluion probabiliie π. In e cae of imple ranom ampling, or raifie ranom ampling, ee probabiliie are non. For eample, uner raifie ranom ampling π n ( n N ( N nng N N g ) ) if, are bo in raum ; if i in raum an i in raum g. For oer meo of ampling, oever, e oin incluion probabiliie are rarel non. In uc cae, one can approimae ee probabiliie o a, iin raa, e are a lea correc for imple ranom ampling. Ta i, e pu π N n ( n ) ( N ) π π en an are in e ame raum. Obvioul, en an are in ifferen raa e ave π π π. In e pecial cae of probabili proporional o ize (PPS) ampling Berger (998) a propoe an alernaive approimaion. Ti i bae on e folloing approimaion o e variance of e HTE (Hae, 964): ( ) V ~ π H ( N p HT ) ( π ) G ( π ) N U π ere an ( π ) π ( π ) U G( π ) ( π ) U ( π ). Berger variance eimae replace e populaion quaniie in Hae approimaion b eign-unbiae eimae, leaing o e variance eimae 5

23 V B p ( ) HT n ( π ) ( n ) ( π ) ( π ) G ( π ) π ere ( π ) ( ) π an ( π ) ( π ) G ( π ) U π. I oul be empaie a i variance eimaor i onl uiable for PPS eign. I can give erioul mileaing reul if ue i general unequal probabili eign. For eample, if ue i raifie ranom ampling i a a large poiive bia. Coniion for B V are e ou in Berger (998). applicabili of ( ) p HT.3..6 Problem i e eign-bae approac Te main reng of eign-bae eor i a i mae no aumpion abou e populaion value being ample. Hoever i i alo i eane, ince ere i noing in e approac o inicae o o mae efficien inference. In paricular, e HTE can be quie inefficien. Uner e eign-bae approac o ample urve inference, eign unbiaene i a e meaure of quali for a urve eimae. A ill be clear from e evelopmen above, i proper a noing o o i e acual value of e ample error of i eimae. I i a proper of e probabili ampling meo. On average, over repeae ampling from e fie finie populaion of Y-value acuall ou ere, i error i zero. Bu e ize of e acual error ma be far from zero. If e variance of e repeae ampling iribuion i alo mall, en i error ill be mall i ig probabili. Sanar probabili eor aure u a i ill be e cae provie e ample ize i large. Hoever, ere i lile o guie one on a large mean ere, ince e coniion require for i eor o ol epen on e (unnon) caraceriic of e populaion. Furermore, in man pracical iuaion ample ize are NOT large, an eign-unbiaene i of limie uefulne. Tee commen appl equall ell o a eign-unbiae eimae of e eign variance of an eimae. Wen a ample i no large e accurac of i eimae of variance (a i e ifference beeen i an e rue ampling variance of e eimae) uffer from e ame problem a e acual ample error ielf e canno a o mall (or o large) i acuall i. All e can a i a e proceure ue o calculae i eimae ill on average prouce an eimae a i e rig value. A furer problem relae o e ue of e eign variance a e meaure of e error of a paricular ample eimae. Ti quani i no e acual value of i error. In fac, e eign variance remain e ame irrepecive of e ize of i error. Ti invariance a 6

24 been criicie (Roall, 98). Furermore, e anar eimae of i eign variance (ic, ince e var from ample o ample, DO var i e acual error) ave been criicie a being mileaing. In paricular, in ome circumance ee variance eimae can be negaivel correlae i e acual error, leaing o mileaing quali aemen for e urve eimae. See Roall & Cumberlan (98). Bo e above problem (efficien eimae an meaningful variance eimae) can be reolve if one aop a moel-bae approac o ample urve inference. Hoever, i i no free of co. One en a o rel on e aequac of one moel for e uperpopulaion iribuion of e Y-variable of inere. Since all moel are, o a greaer or leer een, incorrec i mean a one oul aop robu moel-bae meo, a i meo a o no erioul loe efficienc uner moo eviaion from aumpion. Ti iue i aen up in more eail in Belo e evelop e baic eor unerlining e moelbae approac..3. Te ue of moel for eimaing a populaion oal A on above, e eign variance of e HTE epen on e acual populaion value of Y. Conequenl, iou ome a of moelling e iribuion of ee populaion Y- value, ere i lile one can a abou e properie of e HTE. Over e la 5 ear a conierable bo of eor a erefore evelope ic aemp o uilie nolege abou e probable iribuion of populaion value for Y in orer o improve eimaion of a FPP. Tpicall, i informaion i caracerie in erm of a ocaic moel for i iribuion. Tere are o baic a uc a moel can be ue. Te moel-aie approac eeniall ue i o improve eimaion of e FPP iin e eign-bae frameor. Ta i, e moel i ue o moivae an eimae i goo moel-bae properie. Hoever, i eimae i ill aee in erm of eirable eign-bae properie lie eign unbiaene an lo eign variance. Furermore, e e quali meaure of an eimae uner i approac remain i eimae eign variance. Te oer baic approac i full moel-bae. Here e rericion of eign unbiaene an lo eign variance are ipene i, being replace b moel unbiaene an lo moel variance. Belo e ecribe e baic of e moel-bae approac. Correponing evelopmen of e moel-aie approac i e ou in ecion Te uperpopulaion moel In orer o ecribe i approac, e inrouce e iea of a uperpopulaion moel. Ti i a moel for e oin iribuion of e N ranom variable Y, U oe realiaion correpon o e populaion Y-value, given e value of e auiliar variable X. Tpicall uc a moel pecifie e fir an econ orer momen of i oin iribuion raer an e complee iribuion. Tu e can rie 7

25 C E ( ) µ ( ; ω ) V ( ) σ ( ; ω ) (, ) 0 for ere µ an σ are pecifie funcion of oe value epen on ω, a picall unnon parameer. Noe a e aumpion a iinc populaion uni are uncorrelae given X ma eem rericive, bu i anar for urve of economic uni ere X can be quie informaive abou Y. In oueol urve X ma provie ver lile informaion abou Y, in ic cae i i anar o allo uni a group ogeer (for eample iniviual in oueol) o be correlae. See ecion.3..5 belo..3.. Te omogeneou raa moel Ti moel i iel ue in buine urve pracice. Here, e populaion i pli ino raa an i i aume a e mean an variance of e populaion Y-variable are e ame for all uni iin a raum, bu ifferen acro raa. In i cae X i a raum inicaor. Auming e raa are inee b,,, H, en for in raum e ave ( ; ω ) µ an σ ( ω ) σ µ beeen e raum mean an variance Te imple linear regreion moel ;. Noe a i moel oe no aume an relaionip Anoer commonl ue moel i ere i a meaure of e ize of e populaion uni, an i i reaonable o aume a linear relaionip beeen Y an X. Tpicall i linear relaionip i couple i eeroeaici in X, in e ene a e variabili in Y en o increae i increaing X. A pecificaion a allo for i beaviour for poiive γ value X i µ ( ; ω ) α β an ( ) σ ; ω ψ φ. In man economic populaion e regreion of Y on X goe roug e origin, an i moel reuce o e imple raio form efine b α ψ Te general linear regreion moel Bo e omogeneou raa moel an e imple linear regreion moel are pecial cae of a moel ere e auiliar informaion correponing o X conain a mi of raum ienifier an ize variable. We enoe i mulivariae auiliar variable b X. Ten T ( ω ) β µ. I i anar in i cae o epre e eeroeaici in Y in erm of a ; ingle auiliar variable Z, ic can be one of e auiliar ize variable in X, or ome poiive value funcion of e componen of i vecor (for eample a poer ranformaion lie γ above). In eier cae e pu σ ( ; ω ) σ z. I i imporan o noe a e pecificaion of X i quie general. In mo applicaion i vecor conain onl main effec, bu concepuall ere i noing o op i conaining an funcion (incluing ineracion erm) efine b e auiliar informaion on e ample frame. 8

26 .3..5 Te cluer moel A common feaure of e moel e ou above i a e aume iniviual populaion uni are uncorrelae, irrepecive of eir iance from oer populaion uni. Ta i, afer coniioning on e auiliar informaion in X, ere i no reaon o epec populaion uni a are coniguou in ome ene o be more alie i repec o eir value of Y an uni a are no coniguou. Anoer a of epreing i i a ee moel aume e oberve imilari in Y value for coniguou uni i compleel eplaine b eir imilar value of X. Wen e eplanaor poer of X i ea, a i e cae in mo uman populaion, i aumpion of lac of correlaion canno be uaine. In uc cae i i uual o epan e moel in.3.. o allo correlaion beeen coniguou uni. In paricular, a ierarcical rucure for e populaion i ofen aume, i iniviual groupe ogeer ino mall non-overlapping cluer (for eample oueol). All cluer are aume o be more or le imilar in ize, an eeniall imilar in erm of e range of Y value e conain. Hoever, iniviual from e ame cluer are aume o be more alie an iniviual from ifferen cluer. Tpicall i i moelle b an unobervable cluer effec variable ic a a iribuion acro e cluer maing up e populaion. Te effec of i variable i o inuce a iin cluer correlaion for Y. Since e focu of i repor i quali meaure for buine urve, an cluer pe moel are rarel ue o moel buine populaion, e ill no purue i iue an furer. See Roall (986) for furer icuion of moel-bae eimaion uner a cluer pecificaion Ignorable ampling An imporan aumpion a i picall mae a i age i a e oin iribuion of e ample value of Y can be euce from e aume uperpopulaion moel. In paricular, i i ofen aume a if uni i in ample, en e mean an variance of are e ame a pecifie b e moel. Ta i, e fac a a uni i elece in e ample a no impac on our uncerain abou e iribuion of poenial value aociae i i correponing Y- value. Ti i e o-calle ignorable ampling aumpion. I i aifie b an meo of probabili ampling a epen a mo on non populaion auiliar informaion. We all aume ignorable ampling in a follo, ince i i a i one in pracice. An inveigaion of non-ignorable ampling i e ou in Caper Bia, variance an mean quare error uner e moel-bae approac Uner e moel-bae approac e oal of e populaion value of Y i a ranom variable, o e problem of eimaing i FPP i acuall a preicion problem. An eimae of e populaion oal of Y i a funcion of e ample Y-value, eac one of ic i a realiaion of a ranom variable uner e aume uperpopulaion moel. Conequenl i alo e realiaion of a ranom variable. Te ample error i a preicion error uner i approac. Te moel bia of an eimae of i en e epece value of i ample 9

27 error uner e moel, a i ( ) moel bia i zero, a i E ( ) 0 E. Ti eimae i ai o be moel unbiae if i. Te moel mean quare error of i e um of i moel variance an e quare of i moel bia E ( ) V ( ) [ E ( ) ]. Noe a bo bia an mean quare error above ill epen on ω. Provie i parameer can be eimae from e ample aa, a b ω, en e can eimae e moel mean quare error of b replacing ω b ω in e variance an bia erm above. Suc a plugin eimae ma ielf be biae, oever. Bia correcion can be conruce, epening on e acual populaion moel aume Weanee of e moel-bae approac I i imporan o realie a e moel-bae properie of an eimae are a conequence of e uperpopulaion moel aume. Since e correcne of i aumpion i eeniall unverifiable (aloug e ample aa can ro lig on i appropriaene) ere a been criicim of i approac a being moel epenen. A crucial quali requiremen of a moel-bae approac erefore i robune o pecificaion of e uperpopulaion moel. Tere are o baic a uc robune can be acieve. Te fir (an mo effecive) i o eign e ample o a e urve eimae i in fac moel unbiae i repec o bo e uperpopulaion moel oug o be mo appropriae for e populaion value a ell a i repec o a large cla of alernaive uperpopulaion moel a coul poeniall unerlie ee value. Te econ (an picall le effecive) i o ue a ver general moel, picall one a i overpecifie, a e eimaion age of e urve. Ta i, e replace e original urve eimae (ic a eigne o be unbiae i repec o a muc maller moel) b e eimae uggee b i eene moel. See ecion.3.4. Probabili ampling i a e elemen of a robu ample eign raeg. Ti i becaue probabili ampling can provie average robune b elecing ample ere e bia ue o mipecificaion of e uperpopulaion moel i mall. Hoever, i i uuall avie a one oul no rel enirel on probabili ampling in i regar, effecivel leaving robune o cance, bu a one oul alo implemen robu ample eign raegie lie ize raificaion an orere emaic ampling iin raa. Tee raegie effecivel prea e ample acro e populaion in uc a a a mipecificaion bia i conierabl reuce. For furer icuion of i iue ee Roall & Heron (973) Linear preicion A iel ue cla of eimae of i linear in e ample Y-value. Ta i, e eimae i of e form. L 0

28 In general, e eig above ill epen on an ill be ample epenen, in e ene a i ill alo epen on e X-value of all e ample uni. Hoever, i i no a funcion of e ample Y-value, an ence i a fie quani uner e populaion moel. Ti i in conra o e eign-bae approac, ic oul rea i eig a a ranom variable in i cae. For a large number of commonl ue uperpopulaion moel i i poible o conruc eig a enure e linear eimae L above i moel unbiae an a minimum preicion variance. Suc eig are picall referre o a Be Linear Unbiae (BLU) ample eig, an e eimae L i en e Be Linear Unbiae Preicor (BLUP) of uner e moel. Since ee eig epen on e acual uperpopulaion moel, e ill var accoring o o i moel i pecifie. To illurae, e omogeneou raa moel an e linear regreion moel of.3.. an.3..3 are ofen merge o give a moel ere e populaion i pariione ino raa i a eparae regreion relaionip beeen e u variable Y an e auiliar variable X in eac raum. If, in aiion, bo e linear regreion of Y on X in eac raum, an e variaion of Y abou i regreion line, are ricl proporional o X (a i e regreion line goe roug e origin, i reiual variance proporional o X) en e general moel in.3.. become C E ( ) β for V ( ) σ for (, ) 0 for raum raum Uner i moel e BLUP of i e eparae raio eimaor R, ep N b N ( ) ere b i e Be Linear Unbiae Eimae (BLUE) of β, efine a e raio of e ample mean of Y in raum o e correponing ample mean of X, an i e populaion mean of X in raum. Noe a i eimaor i a paricular cae of L, i ( N ) ( n ) for ample uni in raum. Uner e uperpopulaion moel e ou in.3.., e moel bia of L i eail een o be Since µ ( ;ω ) E ( L ) µ ( ; ω ) µ ( ω ) ; U i O(), i immeiael follo a mu be O(N/n) if L unbiae. Furermore, e preicion variance of L uner e moel i V ( L ) ( ) σ ( ; ω) σ ω) ( ;. i o be moel

29 Given an eimae ω of ω calculae from e ample aa, a imple plug-in eimae of i preicion variance i ( L ) ( ) σ ( ; ω) σ ( ; ω ) L V Since i O(N/n), i i clear a e leaing erm in i eimae variance i e fir (ample) erm on e rig an ie above. Te valii of i eimae erefore re on e accurac of σ ( ;ω ) a a pecificaion for e variance of. Reurning o e cae of e eparae raio eimaor efine above, one can o a i eimae preicion variance en become V L ( ) ( N ) n R, ep σ n N ere σ. n ( b ).3..0 Robu preicion variance eimaion A more robu eimae of e preicion variance of L can be efine b replacing i leaing erm b one oe valii onl epen on e uperpopulaion moel being correc µ µ ; ω i an unbiae o fir, raer an econ, orer. In paricular, uppoe ( ) eimae of µ ( ;ω ) uner e uperpopulaion moel. Ten E ( ) V ( ) O( n ) µ irrepecive of e acual rue pecificaion of e uperpopulaion variance of. Conequenl e alernaive preicion variance eimae for L R ( L ) ( ) ( µ ) σ ( ; ω ) V ill be vali even en e econ orer momen in e uperpopulaion moel are incorrecl pecifie. In pracice, ligl moifie verion of i robu variance eimae are uuall emploe, picall i e quare reiual above muliplie b an O() aumen, u enuring i i alo en an unbiae eimae of e variance of a pecifie b e uperpopulaion moel. To illurae i approac, conier e cae ere i i convenien o aume a all uni in ome pecifie par of e populaion (for eample a raum) ave e ame mean value, a µ, an e ame variance σ. For convenience e all aume a i ubpopulaion i e onl one e are ineree in, an o e rea i a e arge populaion. Suppoe alo a

30 3 ample eig are available, an i i propoe o eimae uing e linear preicor L ecribe in Uner i moel ( ) N E L µ o e ample eig ave o um o e populaion ize N for i eimae o be unbiae. We aume i. Te preicion variance of L i en ( ) ( ) ) ( V n N L σ. An unbiae eimae of µ i e eige average N µ. Alo, an unbiae eimae of σ i en ( ) N N n σ µ o an unbiae eimae of e preicion variance of L uner i moel i ( ) ( ) ) ( V n N L σ. Unforunael, i eimae ill be biae if e aumpion of conan variance for e i incorrec. In paricular, uppoe a e uni in e populaion ave poeniall ifferen (an unnon) variance, a σ. To iingui i cae from e conan variance moel aume o far, e ue a ubcrip of η belo. Te rue preicion variance of L ill en be ( ) ( ) L V σ σ η. Te robu variance eimae R V i en ( ) ( ) ( ) ) ( V L R n N σ µ I i ea o ee a i robu variance eimae ill no be eacl unbiae uner. Hoever, e ligl moifie alernaive D V belo i: ( ) ( ) ( ) ) ( V L D n N N N σ µ.

31 4 Eenion of i robu approac o preicion variance eimaion for e eparae raio eimaor inrouce in.3..9 i icue in Roall & Cumberlan (98). Ti lea o e variance eimae ( ) ( ) loer orer erm. ) ( ) ( ) ( V, ep R D n b n n N n b n N n n N A e all ee in.3.3. belo, i urn ou a e leaing erm above in i robu moelbae variance eimae i eeniall ienical o a eign-bae variance eimae for e eparae regreion eimae a arie uner e moel-aie approac o ample urve inference..3.3 Te moel-aie approac An alernaive approac o incorporaing uperpopulaion moel informaion ino urve eimaion i o ue e moel o ugge improvemen o e anar HTE, bu o coninue o bae all inference on e eign-bae properie of e reuling eimae. Ti approac i commonl referre o a moel-aie. See Särnal e al. (99) Te GREG an GRAT eimae for a populaion oal Given a uperpopulaion moel of e form e ou in.3.., ere are o anar a e HTE i picall improve upon. Ti i via generalie regreion eimaion (GREG) or via generalie raio eimaion (GRAT). In orer o moivae ee approace, conier e folloing equivalen a of reriing e populaion oal of Y, ere ( ) ω µ µ ;, ( ) U U U U e µ µ µ an U U U U R µ µ µ. An improve eimae of bae on e fir ecompoiion above can en be efine b replacing e unnon µ b a uiabl coen plug-in eimae, an e populaion oal of e e b i HTE. Ti lea o e GREG eenion of e HTE: U GREG e π µ ere ) ; ( p ω µ µ, e µ an p ω i a eign conien eimae of e parameer ω efine b e uperpopulaion moel. Tpicall, e la coniion i equivalen

32 o requiring a in large populaion an ample, ω a a eign bia of ( n ) p O en ue a an eimae of a FPP ω N, ic i ielf a moel unbiae eimae of ω bae on e full populaion. An alernaive improve eimae of can be bae on e econ ecompoiion above. Ti i e GRAT eenion of e HTE: µ GRAT µ R p µ π π U U. Clearl e eign unbiaene of e HTE, couple i e eign conienc of ω p, enure a bo e GREG an e GRAT are approimael eign unbiae in large ample Variance eimae for e GREG an GRAT Eac epreion for e eign variance of e GREG an GRAT eimae are unavailable in general. Hoever, i i relaivel raigforar o rie on fir orer approimaion. In e cae of e GREG, one can noe a e eign conienc of ω p implie a e leaing erm in e eign variance of i eimae i e eign variance of e generalie ifference eimae GDIFF, ic i u e GREG eimae bu i ω p replace b ω N. Te HT eimae of variance for i generalie ifference eimae i V HT p ( GDIFF ) ( µ ( ; ω N ))( µ ( ; ω N )) ( π π π π ) π π On e oer an, if a fie ample ize eign a been ue, e SYG variance eimae can be calculae. V SYG p ~ ( GDIFF ) π π π π µ ( π ; ω N ) µ ( π ; ω N ). A fir orer eimae of e eign variance of e GREG i en obaine b ubiuing for ω N in eier of e above variance eimae. For eample e SYG eimae of e eign variance of e GREG i ω p V p ( GREG ) π π π π e π e π. A imilar leaing erm approimaion o e eign variance of e GRAT can be evelope. We again replace ω p b ω N in e pecificaion of i eimae an en ue a fir orer 5

33 6 Talor Serie approimaion o e variance of e raio erm in e reuling eimae o ge e approimaion ( p C enoe eign-bae covariance) N p N N p N p GRAT p R R π ω µ π ω µ π π ) ; ( V ) ; (, C V ) ~ ( V ere U N U N R ) ; µ( ω. Auming a fie ample ize eign an ubiuing SYG eimae for e variance an covariance on e rig an ie of i epreion, replacing N ω b p ω, N R b p R an collecing erm lea o e folloing fir orer eimae for e eign variance of e GRAT p p GRAT p R R ) ~ ( V π µ π µ π π π π. Noe a i eimae i imilar o, bu no e ame a, e variance eimae for e GREG. If e mean funcion µ(; ω) i linear in, an e eimae p ω i a moel-unbiae linear funcion of e ample Y-value, en bo e GREG an GRAT eimaor are alo moelunbiae linear funcion of e ample Y-value. Ta i, e can be rien in e form L inrouce in In uc a cae e can erive an alernaive variance eimae for e GREG/GRAT ic i cloel relae o e robu moel-bae preicion variance eimae ecribe in To ar, pu ( ) N ω µ µ ; ~ an e µ ~ ~. Le enoe e ample eig of e ample uni in e linear repreenaion of e GREG. Since e mean funcion i linear in, an e GREG i moel-unbiae, i immeiael follo a U µ µ ~ ~. Conequenl e GREG can be equivalenl rien U U GREG e g e π µ µ ~ ~ ~ ~ ere g π i e g-eig aociae i e GREG. I immeiael follo a ( ) ( ) U U p GREG p e g e g e g π π π π π π ~ ~ ~ V V

34 an e can ue anar eign-bae eor o rie on an eimae of i variance, ubiuing e µ for e unnon e~. For eample, e HT eimae of variance ariing from i repreenaion i ile e SYG verion i V HT p ( ) GREG ( π π π ) π π π g e g e V SYG p ( ~ ) GREG π π π π π π π π g e g e π π ( e e ). Equivalen variance eimae for e GRAT are eail evelope. We illurae e preceing eor b reurning o e cae of e eparae raio eimae inrouce in.3..9, auming in aiion a e ampling meo iin a raum i imple ranom ampling. Since ample incluion probabiliie iin a raum are conan uner i eign, an recollecing a e efiniion of b enure e um of reiual iin a raum i zero, e can repreen i eimae in e form b R ep b b,. π U U Ta i, e eparae raio eimae i a GREG eimae, i g-eig g / for ample uni in raum. Furermore, uner imple ranom ampling iin a raum e HT an SYG variance eimae are ienical, an o e eor above lea o e variance eimae V HT p ( ) R, ep N n ( π π π ) ( b )( b ) n π π ( b ). A noe a e en of.3..0, i i eeniall e leaing erm in e robu moel-bae variance eimae for e eparae raio eimae..3.4 Calibraion eiging Ti i an area of urve eimaion a a een conierable evelopmen over e la five ear. I i alo an area ere bo eign-bae an moel-bae iea are relevan. Baicall, calibraion i e proce b ic a e of urve eig (eier moel-bae BLU eig or eign-bae invere π-eig) are moifie in a minimal a o a π 7

35 en ee moifie eig are applie o pecifie conrol variable, non populaion oal for ee variable are recovere from e urve aa. Deign-bae uificaion for calibraion i mainl euriic. Te iea i a ince e calibrae eig recover populaion conrol oal, e oul alo be goo for oer urve variable. Calibraion mae more ene from a moel-aie viepoin, ince i cerain pe of calibraion (eeniall bae on a minimum ci-quare crierion for e iance beeen e original uncalibrae eig an e calibrae eig), calibraion i equivalen o GREG eimaion bae on a uperpopulaion moel a i linear in e variable efining e conrol oal. From a moel-bae viepoin minimum ci-quare calibraion i raigforar. I eeniall correpon o moifing e iniial e of ample eig o a e final calibrae eimae i moel unbiae uner i linear uperpopulaion moel. Oer pe of iance crieria can be imilarl moel-moivae. In e moel-bae frameor calibraion i a naural a o generalie ample eig o e are vali uner larger moel (pecifie b e conrol oal) an oe a ere originall oug o be appropriae for e populaion. In i ene calibraion i alo a raeg for ealing i a mulipurpoe urve, paricularl one i man Y variable eac one folloing perap a ifferen uperpopulaion moel pecifie b ifferen X-variable. B calibraing o e conrol oal of eac of ee poenial covariae, one can efine a ingle ample eig a oul lea o unbiae eimae for an paricular Y variable. Since coice of calibraion conrol oal i equivalen o coice of a uperpopulaion moel, all e problem aociae i uner- an over-pecificaion of uc moel flo roug o calibraion eiging. Tu calibraing on oo large a range of conrol oal i analogou o moel overpecificaion an en o reul in inefficien eimae an igl variable eig. In paricular, uner minimum ci-quare calibraion one can obain eig a are negaive or large poiive in uc cae. On e oer an, miing ou a e calibraion conrain i equivalen o leaving a e eplanaor facor ou of a moel, an can lea o ubanial moel bia in e urve eimae. Since aeing e quali of urve eiging meoolog i no e primar focu of i repor, e o no purue i iue furer. Ineree reaer are referre o Camber (997)..4 Meo for nonlinear funcion of e populaion value Aloug eimaion of populaion oal i a e obecive of man buine urve, i i alo imporan o be able o conruc eimae of FPP a are nonlinear funcion of e populaion value. For eample, raio of populaion oal are ofen of inere, a are finie populaion quanile. 8

36 9.4. Variance eimaion via Talor erie lineariaion.4.. Differeniable funcion of populaion oal In general, le ( ) m f,,, θ enoe a iffereniable funcion of e populaion oal of m Y-variable. Furermore, le m,,, enoe eimae of ee oal. A naural eimae of θ i en e plug-in eimae ( ) m f,,, θ. If e componen eimae m,,, are unbiae, en θ ill be approimael unbiae in large ample. A fir orer approimaion o e ample error of θ i ( ) ( ) ( ) m a a a a m m f f f,,, θ θ ere a f enoe e parial erivaive of f i repec o i a argumen, evaluae a m,,,. Conequenl, uner eier e eign-bae or moel-bae approace, a fir orer approimaion o e variance of i ample error i ( ) ( ) m a m b b b a a b a f f, C V θ θ. Here V enoe variance an C enoe covariance. I immeiael follo a an eimae of i fir orer approimaion i ( ) ( ) m a m b b b a a b a f f, Ĉ V θ θ ere Ĉ enoe an eimae covariance an a f enoe e parial erivaive of f i repec o i a argumen, evaluae a m,,,. Noe a Ĉ can be calculae uing an of e ifferen variance eimaion meo ecribe in ecion.3. An imporan pecial cae i ere e eimae m,,, all ave e linear form icue in.3..9, ic inclue e HTE, linear preicion eimaion an calibraion eimaion. Ten raigforar algebra can be ue o o ( ) ( ) z z V V θ θ ere z i e populaion oal of e linearie variable m a a a f z an z i e linear eige eimae of i oal. Ta i, m a a a z f z.

37 Noe a a enoe e value of e variable efining a for e populaion uni. In principle a fir orer approimaion o e variance of θ can en be compue a e eimae variance of e ample error of z. In pracice e o no no e value of e parial erivaive efining z ince e are evaluae a e unnon a, a,,, m. Hoever ee value can be replace b e eimae,,, m, o give an eimae ẑ ic replace z in e formula for z above an i en reae a a anar Y-variable. Ti approac a fir uggee b Wooruff (97)..4.. Funcion efine a oluion of eimaing equaion No all FPP of inere can be epree a moo funcion of e populaion oal of iinc Y-variable, for eample e finie populaion meian. A ier cla of FPP i erefore obaine b coniering oe a can be efine a oluion o populaion level eimaing equaion. In general, θ i efine b a populaion level eimaing equaion if i i a oluion o H ( θ ) f (,, ; θ ) 0 U m ere f i picall aume o be a iffereniable funcion of θ. A linear eimae of θ i θ, ere ( ) f (,, ; θ ) 0 H θ m. Talor erie lineariaion can be ue eimae e variance of θ. We rie 0 H H ( θ ) H ( θ ) ( θ θ ) H ( θ ) ( θ θ ) θ from ic e obain e fir orer approimaion ( θ θ ) (,, ; θ ) ( H ( θ )) f (,, ; θ ) θ m (,, θ ) m f m ; V θ θ f V J. Te o-calle anic eimae of variance i obaine b evaluaing e parial erivaive above a θ, an replacing e variance erm in e mile b an appropriae plug-in eimae. For arbirar θ ere z ( θ ) f (, ;θ ) ( H ( θ )) V f (, ; θ ) V z ( θ ) V m i u anoer populaion Y-variable. Conequenl e m variance on e rig an ie above i e variance of a linear eimae of e populaion oal of i erive variable, an e can ue e eor evelope in e previou ecion o 30

38 eimae i. Plugging in θ for θ in i variance eimae give an eimae of i variance en θ i replace b θ. We enoe i eimae b V ( H ( θ ). Te final anic eimae of variance for θ i en V ( θ θ ) f (, ; θ ) θ m ( V ( H ( θ ) f.4. Replicaion-bae meo for variance eimaion (, ; θ ) Aloug mo FPP of inere can be efine in erm of a moo funcion of populaion oal, or a e oluion of a populaion eimaing equaion, ere remain iuaion ere e efiniion of e FPP i o comple a applicaion of Talor erie lineariaion meo for variance eimaion i ifficul. In uc cae e can ue alernaive variance eimaion meo a are imple o implemen, bu are picall numericall inenive. Te bai for all ee meo i e iea a one can imulae e variance of a aiic b (i) maing repeae ra from a iribuion oe variance i relae in a imple (an non) a o e variance of inere; (ii) empiricall eimaing e variance of i econar iribuion, an (iii) auing i variance eimae o a i i an eimae of e variance of inere..4.. Ranom group eimae of variance Te imple a of implemening e above iea i roug e ue of inerpeneraing ample, ee Maalonobi (946), Deming (956). Here e acual ample elece i mae up of G inepenen replicae or inerpeneraing ubample, eac one of ic i repreenaive of e populaion, being ran accoring o e ame eign an i e ame ample ize n/g. Le θ g enoe e eimae of e FPP θ bae on e g replicae ample. Te overall eimae of i quani i e average θ of ee θ g. B conrucion, e e of replicae eimae {, g, G} g, θ m θ are inepenen an ienicall iribue. Conequenl, e can eimae e variance of eir (common) iribuion b eir empirical variance aroun eir average, e overall eimae θ. Furermore e variance of θ i u i replicae variance ivie b e number of replicae, G. Conequenl e can eimae e variance of θ b impl iviing i empirical variance b G, leaing o e eimae V R ( ) θ. ( G ) G g G ( θ g θ ) In fac, e above iea ill or even if e replicae eimae are no ienicall iribue. All a i require i a e are inepenen of one anoer, an eac i unbiae for e FPP θ. Sraigforar algebra can en be ue o o. 3

39 G ( ( θ ) V( θ ) V( θ ) E V R G g g o e replicae variance eimae i ill unbiae for e variance of e average of e replicae eimae. In pracice replicae ample eign a ecribe above are rare. Hoever, e iea of replicaion-bae variance eimaion i ill applicable. Wa i one in ee cae i o conruc e replicae afer e ample i elece, b ranoml allocaing ample uni o G group in uc a a a eac group i a lea approimael inepenen of e oer group. Wi raifie eign uc po-ample ranom grouping can be accomplie b ranom grouping iin e raa, provie ere i ufficien ample ize iin eac raum o carr i ou. If i i no e cae, en ranom grouping can be applie o e ample a a ole, preerving e raa en pliing e ample beeen e group. In e cae of muliage eign, pliing i picall carrie ou a PSU (primar ampling uni) level. In aiion, e average eimae θ in e variance formula above i ofen replace b e full ample eimae of i quani. Finall, i oul be poine ou a e replicaion variance eimae i an eimae of e variance (eier eign-bae or moel-bae) of θ, no e variance of e ample error θ θ. A conequence i a i variance eimae oe no go o zero a e ample ize approace e populaion ize. Ti i of no grea concern en ample ize iin raa are mall compare o raum populaion ize. Hoever, in man buine urve, ample ize iin raa are a ubanial fracion of e raum populaion. In uc cae, i i anar o mulipl e raum level replicae group variance eimae b appropriae finie populaion correcion facor..4.. Jacnife eimae of variance A problem i e replicaion-bae approac o variance eimaion i e abili of ee eimae. Clearl, e more group ere are, e more able ee variance eimae are. Hoever, e more group ere are, e arer i i o ranoml group e ample. A meoolog a circumven i problem, bu a e co of ropping e proper of inepenen ubgroup eimae, i o ue overlapping group. Tere are eeniall o approace o uing overlapping group. Te fir i via Balance Repeae Replicaion (BRR) ere e group are forme uing eperimenal eign precep o a covariance inuce b e ame uni belonging o ifferen group cancel ou in e (non-overlapping) ranom group variance formula above. Ti can be quie ifficul o accompli in general, an o i meo i picall rerice o cerain pe of muliage eign a are rarel ue in buine urve. See Woler (985) an Sao & Tu (995). Te econ, an more common meo, i o compue a acnife variance eimae. 3

40 Uner e acnife approac, e ample i again ivie ino G group, bu i ime G eimae are compue b ropping ou eac of e G group from e ample in urn. Te variabili beeen ee epenen eimae i en ue o eimae e variabili of e overall eimae of θ. Le θ ( g ) enoe e eimae of θ bae on e ample ecluing group g. Te acnife eimae of variance i V J G ( G ) ( θ θ ) θ ( g ). G A i e replicae group variance eimae, ere are o form of e acnife variance eimae. Te fir, ic e refer o a e Tpe acnife, efine θ a e average of e θ. Te econ, e Tpe acnife, efine θ a e full ample eimae of θ. Since ( g ) G g g G G G ( θ ( g ) θ ) θ ( g) θ ( ) Gθ n g e Tpe acnife ill be more conervaive an e Tpe acnife. Unbiaene of e acnife variance eimae oe no follo a eail a unbiaene of e replicae group variance eimae. For e Tpe acnife, ufficien coniion for unbiaene are C ( G V θ ) ( ( g ) V θ ) G ( G ) ( G ) θ, θ ( g ) ( ) ( G ) V( θ ). For e Tpe acnife e econ coniion above oul be replace b ( θ, θ ) V( θ ) C ( g ). A i e ranom group variance eimae, e acnife variance eimae i picall compue a PSU level in muliage ample. Ta i, e G group are efine a group of PSU. Furermore, e mo common pe of acnife i en G i equal o e number of PSU in ample, a i one PSU i roppe from e ample eac ime a value of θ ( g ) i calculae. Tere i empirical evience a, provie e arge parameer θ i ufficienl moo, i coice of G minimie e variance of e eimae of variance (Sao & Tu, 995; eample..4). Finall, one can noe a, lie e ranom group variance eimae, e acnife variance eimae oe no inclue a finie populaion correcion. Ti nee o be applie eparael Te linearie acnife Te compuaional eman of e acnife en G n (e number of ample PSU) a le o reearc ino a of approimaing i o a i can be compue in one pa of e n θ ( ) 33

41 ample aa. If θ i a moo funcion of e ample aa, i can be accomplie b eeniall replacing V J ( θ ) b a fir orer Talor erie approimaion o i. In a follo e aume ingle age ampling. Furermore, e aume e eience of a uperpopulaion moel uner ic ( ) µ ample epece value. We can en approimae θ b θ θ E for. Le µ enoe e n-vecor of ee ( ) θ µ ( µ ) ere θ ( µ ) enoe e value of θ en e ample Y-value are replace b µ an e parial erivaive in e econ erm on e rig an ie are evaluae a µ a ell. Similarl, le µ () enoe µ i e epece value for elee, an pu θ ( ) equal o e eimae bae on e ample ecluing. Te correponing approimaion o θ ( ) i en ( ) θ θ ( ) ( ) ( ) µ ( ) ( ) µ θ µ ( ) ( µ ) ere θ ( ) µ ( ) enoe θ ( ) evaluae a µ (). We no mae o era aumpion: () ( ) ( ) ( ) ( µ ) θ 0 θ µ θ ; () θ µ n θ n ( ) µ ( ). Te fir of ee aumpion i unconroverial, ince i eeniall correpon o e requiremen a e rop ou an full ample eimae are eimaing e ame ing. Te econ aumpion i reaonable en θ i linear in Y, bu ma no be reaonable in oer cae. Wi ee aumpion e can replace e approimaion o θ ( ) above b θ n ( ) θ θ ( µ ) n θ. n 0 Subiuing i approimaion ino e Tpe acnife variance eimae lea o e linearie verion of i eimae () ( ) n ( ) ( ) θ θ V JL θ µ µ n n µ µ ere µ enoe e full ample eimae of µ. Te correponing linearie Tpe acnife i obaine imilarl, afer replacing θ 0 b θ. I i µ 34

42 V n ( ) JL ( θ ) ( ) µ θ n n( n ) θ µ n 3n Comparing e preceing o epreion one can eail ee a e linearie Tpe acnife variance eimae ill ala be greaer an e linearie Tpe acnife variance eimae, a proper a i generall oberve for Tpe acnife variance eimae. Noe a e linearie acnife i eeniall a moel-bae variance eimaion proceure, ince i require pecificaion an eimaion of µ. Furermore, i i unclear eer i lea o aning ubaniall ifferen from uing e Talor approimaion approac iin a moel-bae frameor for variance eimaion. For eample, e linearie Tpe acnife eimae of e variance of e linear eimaor L efine in.3..9, () n V JL ( L ) ( µ ) ( µ ) n n i (o a fir orer approimaion) equivalen o e robu moel-bae variance eimaor ( ) R V L ecribe in Boorapping Bo e ranom group an e acnife meo reul in eimae of variance for a aiic a i an eimae of a FPP. In general, oever, our inere in uc eimae i bae on e eire o compue inerval eimae (for eample confience inerval) for i FPP. Suc quaniie are efine in erm of e properie of e ampling iribuion of e eimae. For large ample, e cenral limi eorem picall applie, an i ampling iribuion can be ell approimae b a normal iribuion. In uc cae i i ufficien (provie e eimae i ampoicall unbiae for e FPP) o eimae e variance of e ampling iribuion in orer o rie on confience inerval for i FPP. Hoever, for man ampling eign e level a ic variance are calculae can be quie eaile (for eample fine raa or omain conaining relaivel fe uni). Here an aumpion of cenral limi beaviour ma be quie inappropriae, in e ene a e ampling iribuion (eier eign-bae or moel-bae) ma be quie non-normal. In ee cae e ma an o compue an eimae of e ampling iribuion irecl. Te boorapping iea provie a a b ic i obecive can be acieve. To ar, e ecribe a moel-bae boorap, ince i i relaivel raigforar. In paricular, e aume a e FPP of inere i efine in erm of e populaion value of a ingle Y-variable oe uperpopulaion iribuion i pecifie b e moel in.3.., an a moel-unbiae eimae ω of e parameer ω in i moel can be calculae from e ample aa.. 35

43 Le { r ; }, enoe e e of uenie reiual generae b e ample aa uner i moel. Ta i, ee reiual epen on ω an aif E (, ) 0 V, B ampling a ranom i replacemen from { r ; } * boorap reiual { ; U} r value of Y, efine b r r. an ( ), e can en generae a e of N an conequenl a boorap realiaion of e populaion ( ω ) σ ( ; ω ) r * * ; µ. Given i boorap realiaion, e can compue a boorap eimae of θ bae on e * ;, ic e enoe b θ *, ogeer i e acual value of θ for e value { } * boorap populaion, ic e enoe b θ. Te boorap realiaion of e ample error i * * en θ θ. Ti proce i no repeae a large number of ime, leaing o a iribuion of uc boorap ample error. We enoe e mean of i boorap iribuion b * * ( * * * E θ θ ), an i variance b V ( θ * θ ). * * Te boorap eimae of θ i en θ E ( * B θ θ θ ) * * eimae i omeime aen a V ( θ * θ ). Te boorap variance of i. Hoever, i ill picall be an unereimae ince i oe no ae accoun of e error in eimaion of ω in e above proce. Conequenl i i uuall beer o recale e boorap ample error iribuion o a i variance i e larger of i iniial variance or an eimae of e variance ic allo for error in eimaion of ω (for eample, a acnife eimae). If i i alo believe a θ repreen a be eimae of θ, en e boorap ample error iribuion can be cenre a zero prior o i recaling. In an cae, afer recenering an recaling, i i imple o rea off a 00(α)% confience inerval for θ from e boorap ample error iribuion. Eeniall uc a confience inerval i efine b θ B Q α, θ B Q * α * ere Q * (γ) enoe e γ- quanile of i iribuion. One problem i e boorap proceure efine above i a i epen on correc pecificaion of e eeroeaici funcion σ(; ω). A eeroeaici-robu moelbae boorap i eail efine, oever. Eeniall, all one nee o o i o replace e uenie reiual unerpinning e boorap proceure b ra reiual r µ ; ω. Te remaining ep in e boorap proceure are uncange. See ra, ( ) Camber & Dorfman (994). Boorapping e eign-bae iribuion of e ample error i alo poible, bu can be quie complicae epening on e acual urve eign ue. Ti i becaue one nee o 36

44 ample i replacemen from e ample Y-value in uc a a a o a lea preerve e fir an econ orer incluion probabiliie of e eign. Conequenl, a e ime of riing, a number of boorap-pe meo for eimaing e eign variance ave been uggee (Sao & Tu, 995, Caper 6), i no obviou preferre meo. Te imple of ee a preen i e boorap proceure ecribe b Can & Davion (997). We ecribe i in e cone of eimaion of e variance of e linear eimae L efine in.3..9, ere e ample eig are calibrae o e populaion oal of an auiliar variable X. Ta i, en e eimae L i calculae i e ample Y-value replace b ample X-value, e non populaion oal of X i obaine. A boorap replicaion ere coni of e folloing ep: () elec a imple ranom ample of n label from i replacemen. Le i ine e n * ra maing up i boorap ample. Tu i enoe e value of Y correponing o e ample label elece a e i * ra, i enoe e ample eig aociae * i i value, an i enoe e correponing value of e auiliar variable; () recalibrae e eig aociae i e boorap ample. Le * i enoe e recalibrae eig aociae i e i boorap ample Y-value; (3) recompue e boorap realiaion of L. Auming L i a GREG eimae, i ill be of e form: n n n * * * * i i i i β i i i * L * i * i * ere β enoe e eimae of e regreion of Y on X bae on e boorap ample. Repeaing e above proceure a large number of ime en generae e boorap iribuion of L. A uual e enoe e mean an variance of i boorap iribuion (a i, coniional on e ample Y-value) b E* an V* repecivel. Te boorap * variance eimae i e empirical variance of e boorap value L over ee replicaion. Aloug eac epreion for e momen of e above boorap iribuion are generall unavailable, goo approimaion are eail ore ou. For an paricular boorap * replicaion, efine I i a one if e ample uni a elece a e i ra maing up e boorap ample elece a a replicaion, an a zero oerie. Ten I * n i I * i enoe e number of ime e ample uni conribue o i boorap ample. I follo ( I ), V ( ) * I * * * * ( n ) n an C ( I, I ) n E * *. Furermore, ince e can rie 37

45 * * I β * L * e can approimae i boorap realiaion of L b replacing β b e coefficien β of e full ample regreion of Y on X. Wi i approimaion i i ea o ee a E * *, ile ( L ) L V * ( * * ) V ( β ) L n V () ( JL L ). n * ( * * ) ( ) ( )( * * * β V I β β ) C ( I, I ) ( β ) ( β ) I n Ta i, i fir orer approimaion o e Can-Davion boorap variance eimae i (n )/n ime e linearie Tpe acnife variance eimae. Clearl, i approimaion i eacl e acnife variance eimae provie e moif e boorap proceure above o elec n raer an n ample label a eac replicaion..5 Concluion Te purpoe of i caper a been o e ou e baic eor for ampling error relae bia an variance aemen of anar urve eimae. Ti eor a eier epene on, or require, e ue of ome form of probabili ampling meo. To baic paraigm for efining bia an variance ave been preene: e eign-bae approac ic meaure ee quaniie relaive o e uncerain aociae i e ifferen ample a coul ave been elece uner e meo ue; an e moel-bae approac ic meaure e uncerain in erm of e poible value a e urve variable can ae in e arge populaion. Bo approace ave reng an eanee, an ee ave been poine ou. In e en, i eem clear a robu moel-bae/moel-aie meo an enibl coniione eign-bae meo for aeing bia an variance en o lea o imilar concluion, an o i caper a aempe, ere poible, o inicae e connecion beeen e o. From e poin of vie of be pracice a far a minimiaion of ampling bia an aemen of ampling variance are concerne, e ugge e folloing poin be ep in min: robu probabili ampling meo oul be ue erever poible. Tee are eign ic blen ranomiaion an moelling iea in orer o enure a e ample a are finall elece are no onl ranom bu alo repreenaive of e full range of poenial Y-value uner a carefull pecifie moel for e arge populaion. Suc I * 38

46 ample are necear if e ize of e ampling error i o be ep iin accepable boun; robu meo of ampling variance eimaion oul be ue if a all poible. Given e repreenaive balance ample a arie uner e preceing recommenaion, ee meo provie able an accurae aemen of e poenial ize of e ample error. Hoever, i oul alo be ep in min a ee meo are no guaranee o or if e ample i unrepreenaive. Eeniall all robu meo for eimaing ample error variabili aume a e variabili in e ample value i repreenaive of a in e arge populaion. Ti i no e cae if e ample i unrepreenaive; for comple FPP one a a coice beeen plug-in meo bae on Talor erie lineariaion argumen or a varie of replicaion or reampling meo. Te former are le compuer inenive bu (omeime) require conierable analic ill o evelop an program. Te laer are generall ea o program bu are picall igl compuer inenive. Te coice beeen ee meo epen on e reource a an. Some appreciaion for e ifferen operaing caraceriic of ee meo can be obaine b reaing e volume of i repor ealing i aemen of ifferen compuer ofare for urve inference. I uffice o poin ou a generall, becaue of eir plug-in naure, Talor erie lineariaion meo en o unereimae ampling variabili, ile replicaion/reampling meo en o overeimae i. In meium o large ample, oever, ere i lile o cooe beeen ee meo ince all are eeniall fir orer equivalen. 39

47 3 Probabili ampling: eenion 3. Domain eimaion Ra Camber, Univeri of Souampon A common problem in urve inference i eimaion of e populaion oal of a urve variable Y for a omain of inere. For eample, in man buine urve e ample frame i ou of ae, o e inur an ize claificaion of man uni on e frame o no agree i eir curren inur an ize claificaion. Afer e urve i carrie ou, eimae are require for e curren inur b ize clae. Tee clae en correpon o omain of inere a far a e urve i concerne. In general, a omain i ome ubgroup of e ample populaion. Ofen omain cu acro raum bounarie an are referre o a cro-clae. A baic aumpion in omain eimaion i a omain memberip i obervable on e ample. Ta i, one can efine a omain memberip variable D i value for populaion uni, uc a if uni i in e omain an i zero oerie, an e value of D are obervable for e ample uni. Te number of populaion uni in e omain i u e populaion um of D an i enoe b N. B conrucion, e populaion oal for e omain i. U 3.. Deign-bae inference for omain Wiin e eign-bae frameor, omain eimaion poe no pecial problem. I i ufficien o noe a e omain oal i u e populaion oal of e variable DY. Conequenl e HTE for i u HT π i eign variance V p ( ) HT ( π π π ) U U π π Te SYG eimae of e variance of i eimae i V SYG p ( ) HT π π π π π π. 3.. Deign-bae inference uner SRSWOR Te cae of imple ranom ampling iou replacemen (SRSWOR) i inrucive, ince i i e one iuaion ere moel-bae inference an eign-bae inference come ogeer. In i cae 40

48 HT N n Np ere i e ample average of Y for uni in e omain, an p i e ample proporion of uni in e omain. Ti eimaor i inuiivel reaonable. One moifie an eimae of e populaion oal a effecivel rea all populaion uni a belonging o e omain b an eimae of e proporion of populaion uni a acuall belong o e omain. Te eign variance of i eimaor i (afer ome algebra) V p ( N n ) [ p p ( p ) ] HT n N ere p N N i e proporion of populaion uni a are in e omain, i e average value of Y in e omain an i e anar eviaion of e Y-value in e omain. Ignoring ( ) O erm, e SYG eimae of i variance i N ( N n ) [ p p ( p ) ] SYG V p HT n N ere i e anar eviaion of e ample Y-value in e omain Moel-bae inference en N i unnon Moel-bae inference for a omain oal epen on a one no abou e omain, an in paricular on eer one no o man populaion uni are in e omain. Ta i, i epen on eer one no e value of N. I alo epen on eer e meo of ample elecion epen on omain incluion or no (remember e are auming a e ampling meo i uninformaive a far a Y i concerne). To ar, e conier e mo common iuaion, ere e value of N i unnon. To illurae e moel-bae approac, conier e cae ere e eimaor of coice i e HTE efine in 3... A uual, e le a ubcrip of enoe quaniie efine i repec o a uperpopulaion moel. Te paricular moel e aume i ver imple an i pecifie b C E ( ) µ E ( ) θ V ( ) σ V ( ) θ ( θ ) (,, ) 0 C (, ) 0 Ta i, omain memberip in e populaion i moelle a e oucome of a Bernoulli proce i fie ucce probabili θ, an coniional on omain memberip e populaion value of Y are uncorrelae i conan mean an variance. A i e moel-bae approac in general, ere i an implici aumpion a ample incluion i inepenen of e value of e variable of inere. In i cone, i require 4

49 4 a ample incluion an omain memberip be inepenen of one anoer. Ti aumpion i vali if e ample i coen via imple ranom ampling. Uner e above moel i i ea o ee a ( ) ( ) ( ) ( ) 0, C V E θ θ µ θ σ θ µ o e Be Linear Unbiae Preicor (BLUP) for i u e HTE. Furermore e moel variance of e HTE/BLUP i ( ) ( ) [ ] V HT N n n N µ θ θ σ θ o e SYG variance eimae in 3.. i alo an unbiae eimae of i moel variance. For i cae, moel-bae an eign-bae inference coincie Moel-bae inference en N i non Here one i lea o inference a coniion on i non value of N. To illurae, e conier e ame iuaion a in In i cae, oever, e nee o moif e moel coniere ere o ae accoun of e era informaion provie b nolege of N. Le C,,V E enoe epecaion, variance an covariance coniional on noing N. A before e pu N N p. Ten, ince ( ) N N E an ( ) 0 V N, mmer-bae argumen can be ue o o a ( ) ( ) ( ) ( ) ( ) ( ), C V E N p p p p p Furermore, if e aume a Y i inepenen of N coniional on D (a i, noing N ell u noing era abou an noing e value of ), an e coniional momen of Y given D are a pecifie in 3..3, en e folloing reul ol ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), C, C ) (, C V E N p p p p N p p p p p p µ µ µ µ σ µ From e fir ree ieniie above e ee a, i repec o i coniional iribuion, e erive ranom variable DY a a mean an variance a i e ame for all populaion uni. Furermore, e covariance beeen an o populaion value of DY i conan. I i

50 43 raigforar o o a e BLUP efine in erm of i erive variable i en ill e HTE. In fac, e ave ( ) ( ) ( ) [ ] ( )., C V V HT p p N N p N n n N N n n N µ σ Hoever, in i iuaion ere eem no rong reaon one oul reric aenion o eimae a are linear in DY. An obviou alernaive i e nonlinear raio-pe eimae R N N Ti eimae i approimael moel-unbiae in large ample. Furermore, e variance of i eimae can be approimae uing a anar Talor erie argumen. In fac, one can o ( ). V V U R p N n n N N n n N σ µ Comparing i variance i e variance of e HTE, e ee a ere ill picall be large efficienc gain from ue of e raio-pe eimae. Tere i a funamenal principle omeime invoe in moel-bae inference calle e coniionali principle (Co & Hinle, 974). Ti ae a one oul ala coniion on ancillar variable in inference. An ancillar variable i one oe iribuion epen on parameer a are iinc from oe aociae i e iribuion of e variable of inere. In e cone of omain anali, i can be argue a e parameer() aociae i e iribuion of e omain incluion variable D are iinc from oe aociae i e iribuion of e urve variable Y. Conequenl, one oul coniion on D in inference. Ti i equivalen o coniioning on bo e populaion coun N of e number of uni in e omain, an e correponing ample coun n. If one coniion in i a i i raigforar o o a e raio-pe eimae above i e BLUP for (efine in erm of Y) an a moel variance ( ), V R N n n N N n σ Ti i omeime referre o a e variance of e poraifie eimae for e omain oal.

51 44 Wic of e o immeiael preceing variance for e raio-pe eimae i correc i e ubec of ebae. Clearl, plug in eimae for bo ill be ifferen in general, i equali onl if e populaion ampling fracion equal e omain ampling fracion. An argumen again e poraifie approac i bae on e fac a e iribuion of e populaion parameer epen on e parameer of Y a ell a e parameer of D. Conequenl i i a cae ere e ancillari principle i no applicable. Raie again i, oever, i e argumen a, unlie e coniional variance, e poraifie variance i zero if N n, en e no a e raio-pe eimae a zero error. Hoever, ofen one ill ave N >> n an o a cauiou approac oul be o eimae e variance of e raio-pe eimae b e maimum of e o variance eimae Moel-bae inference uiliing auiliar informaion We reurn o e cae ere e omain coun N i unnon. Hoever, e een e moel for Y o e one coniere in.3... Ta i, e aume ( ) ( ) ( ) ( ) ( ). for 0,, C ; V ; E ω σ ω µ We coninue o aume a omain memberip i efine b a equence of inepenen an ienicall iribue Bernoulli rial, inepenenl of e value of Y. Hoever, omain memberip can epen on X, o ( ) ( ) ( ) ( ) ( ) [ ] ( ). 0, C ; ; V ; E γ θ γ θ γ θ Wi i e-up e ave ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ). 0, C ; ; ; ; ; V ; ; E γ θ γ θ ω µ γ θ ω σ γ θ ω µ Given probabili ampling, conien eimae for e parameer ω an γ above can be obaine from e ample aa. A plug in moel-bae eimae of i en ( ) ( ) γ θ ω µ ; ; ere a a enoe a ample eimae. Clearl i eimae ill alo be conien. Te moel-variance of i eimae can be rien ( ) ( ) ( ) ( ) γ θ ω µ V V V ; ; V V

52 45 Te leaing (bigge) erm in i variance i V. I can be eimae uing compuer inenive meo lie e acnife or boorap. For eample, e rop-ou Tpe acnife eimae of i quani i () ( ) ( ) ( ) ( ) ( ) ( ) J n n ; ; ; ; V γ θ ω µ γ θ ω µ ere ( ) ω enoe e ample eimae of ω bae on e ample uni ecluing uni, an ( ) γ i efine imilarl. Tpicall ( ) ω i u ω for all ample uni no in e omain, o ome implificaion of e above formula i poible. Alernaivel, a Talor erie lineariaion approac can be ue o conruc a irec eimae of V. Ti i bae on e approimaion ( ) ( ) ( ) ( ) ; ; ; ; V V ω ω µ γ θ ω γ γ θ ω µ γ ere 0 γ an 0 ω are e rue value of 0 γ an 0 ω, an e parial erivaive are evaluae a ee rue value. Depening on e pecificaion of e funcion µ an θ, eimae of e variance of ω an γ an eir covariance can be eimae from e ample aa. Uing a o enoe ee eimae in e uual a, i ugge a Talor erie eimae of V of e form ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ω ω µ γ θ γ γ θ ω µ ω γ ω ω µ γ θ ω γ γ θ ω µ γ ; ; ; ;, Ĉ ; ; V ; ; V V Te econ erm V in e variance formula a a imple plug-in eimae bae on e moel pecificaion above. Ti i ( ) ( ) ( ) ( ) ( ) [ ] ( ) γ θ γ θ ω µ γ θ ω σ ; ; ; ; ; V An eample A imple eample illuraing e above eor i ere e populaion i raifie an e regreion of Y on X i linear an roug e origin for uni in e omain, bu e lope of i regreion line varie from raum o raum. Furermore, e proporion of e populaion in e omain varie ignificanl from raum o raum. Here e pu θ equal o e probabili a a populaion uni in raum lie in e omain an β equal o e lope

53 of e regreion line for omain uni in raum. Our eimae of e omain oal of Y for e populaion i en p ( N n ) Here inee e raa, p enoe e ample proporion of raum uni in e omain, β enoe e raum eimae for e lope of e regreion of Y on X in e omain, enoe e raum average for X an i e ample average for X in raum. Te Talor erie eimae of e leaing erm in e moel-variance of i eimae i ere ( ) p V β ( N n ) ( V ( p ) β V ( β ) p ) V enoe e eimae variance of p an ( ) β V enoe e eimae variance of β. Noe a inepenence of D an Y iin a raum caue e covariance erm in i eimae o iappear. Tpicall ( p ) n p ( p ) V an, if e alo aume a e reiual variance for e regreion of Y on X i proporional o X iin a raum b omain cell, en V ( β ) ( n ) σ ere σ i e uual eimae of e reiual cale parameer for i regreion, n i e number of omain uni in ample in raum an i eir average X-value. Subiuing ee eimae an aing on V for i cae lea o a variance eimae of e form V ere V V p β σ n ( p ) ( N n ) ( N n ) enoe e average of X in raum, an n p N ( N n ) n i e correponing ample quani. In e pecial cae ere X i i raigforar o ee a i epreion reuce o e raifie ranom ampling verion of e SYG variance eimae ecribe in Domain eimaion uing a linear eige eimae Mo compuing pacage for urve eimaion ic ue a linear eimae of e form L ecribe in.3..9 carr ou omain eimaion b impl replacing e in i eimae b. Ta i, e calculae e linear eige omain eimae 46

54 47 L. Uner e general omain moel of 3..5 e moel-bia of i eimae i ( ) ( ) ( ) ( ) ( ) U L γ θ ω µ γ θ ω µ ; ; ; ; E. Tere i no paricular reaon for i moel-bia o be zero, or even cloe o zero. To illurae, uppoe (a i ofen e cae) a e regreion of Y on X in e populaion i linear in X an e eig are calibrae on X. Ti i ufficien o enure moelunbiaene of L. Suppoe alo a e regreion of Y on X in e omain i linear in X, bu i e aiion of a omain if erm. Ta i ( ) η β ω µ T ;. Ten ( ) ( ) ( ) ( ) ( ). ; ; ; ; E T T U U L γ θ γ θ η β γ θ γ θ Unle e omain incluion probabili oe no epen on X, i i clear a bo erm in i bia ill be nonzero in general, irrepecive of e calibrae naure of e eig. One iuaion ere e econ erm in e above bia iappear i ere X inclue raum inicaor, o e calibrae eig um o e raum populaion coun iin a raum, an ere omain incluion probabiliie are conan iin a raum. In i cae ( enoe e raum ubample, U enoe e raum populaion) ( ) z U L z z β θ T T E ere z enoe i raum inicaor remove, an β z i e correponing componen of β. Clearl i remaining moel-bia ill vani if e eig are acuall calibrae on X iin raa, ic i equivalen o requiring moel-unbiaene for L in e cae ere e linear regreion moel for Y inclue ineracion beeen e raum inicaor componen of X an e remaining componen of i auiliar variable. In principle, one can eimae e moel-bia of e linear eige omain eimae via ( ) ( ) ( ) ( ) ( ) U L B γ θ ω µ γ θ ω µ ; ; ; ; an ence correc i eimae for i moel-bia. For eample, in e raifie cae icue above i bia eimae i ( ) ( ) z L z z p N B β T T

55 ere z i e eige average of e ample z in raum, an z i e acual raum average for i auiliar variable. Te aiical properie of i bia correce eimae are unnon a e ime of riing Moel-aie omain inference We focu on e eenion of e GREG iea o omain eimaion. Te correponing moificaion o e GRAT iea i raigforar. Tu, appling e GREG iea uner e general moel of 3..5 lea o e eimae ere ~ GREG U µ ( ; ω ) θ ( ; γ ) p p µ ( ; ω ) θ ( ; γ ) ω p i a eign conien eimae of ω, an γ p i efine imilarl. Defining reiual e µ ( ; ω ) θ ( ; γ ) p p, a fir orer approimaion o e SYG eimae of e leaing erm in e eign variance of i eimae i en eail een o be π p p V p ( ~ ) GREG π π π π θ π θ π. Noe a e GREG eimae above i no e ame in general a e eimae obaine b ubiuing for in a anar GREG eimae for a populaion oal. Ti imple ubiuion eimae i moel-biae, a on in 3..7 above. 3. Eimaion of cange Mo buine urve are coninuing urve. Ta i, e urve i repeae monl, quarerl, annuall or i ome oer fie frequenc. An imporan reaon for oing i i o eimae e cange in populaion quaniie from one urve perio o e ne. Ti eimaion oul be relaivel raigforar if e arge populaion an e urve ample remaine e ame from one perio o e ne. Unforunael, i i almo never e cae. Meo for coping i e complicaion caue b ample an populaion cange over ime are icue belo. To eep noaional complei o a minimum e reric ourelve o cange in a finie populaion oal beeen o ime poin. Le enoe e populaion oal of a urve variable Y a ime T an le enoe e correponing oal a ime T. Te value of Y a ime T ill be enoe an e value of Y a ime T ill be enoe. Te aim i o eimae eier e abolue cange δ or e relaive cange φ ( ). Real populaion are rarel aic. Tu, e uni maing up e populaion conribuing o ill be ifferen from oe maing up e populaion conribuing o. We pu N u, u, equal o e number of uni in e populaion a ime T u. In man cae ere ill be conierable overlap beeen e populaion a e o ime poin. We pu C ( coninuing ) equal o e e of populaion uni common o bo ime poin. Te e of populaion uni 48

56 conribuing o an no ill be enoe D ( ea ) ile e e conribuing o an no ill be enoe B ( bir ). Le N C, N D an N B enoe e number of uni in ee e repecivel. Ten N N C N D an N N C N B. Te oal populaion ill be enoe a e e of uni conribuing o eier or or bo. Clearl i conain N C N D N B uni. A imilar ecompoiion of e ample a ime T an T can be efine. Tu i e ample a ime T, i e ample a ime T, c i e ample common o bo ime, i e e of ample uni unique o ime T an b enoe e ample uni unique o ime T. Noe a uni in c mu, b efiniion, be in C, bu uni in o no ave o be in D, an imilarl uni in b o no ave o be in B. We pu D equal o oe uni in an D, i C D. Similarl, e pu bb equal o oe uni in b an B, i bc b bb. 3.. Linear eimaion Suppoe ome form of eige linear eimae of e populaion oal of Y i compue a eac ime perio. Tee are eimae of e form (u, ) u u ere e L an ubcrip ave been roppe for e ae of clari. Te eig u are aume o be calibrae i repec o non populaion caraceriic a ime T u. An obviou eimae of δ i en e ifference δ. Provie e level eimae u i unbiae for u, i i clear a δ ill alo be unbiae for δ. A correponing eimae for φ i en φ. Developmen of eign variance for ee eimae i complicae b e nee o evaluae e eign covariance beeen an. To illurae, uppoe bo an are HTE, an le e inicaor I u enoing ample incluion/ecluion a ime T u, o e probabili of incluion in ample of populaion uni a ime T u i equal o π u. Le U enoe e populaion a T an U enoe e populaion a T. Ten V p V ( δ δ ) p B I I V p U π U π I I I π C π π ic can onl be epane furer provie e oin iribuion of I an I can be pecifie for all pair of uni in e oal populaion. Ti i rivial if inepenen ample are elece a eac ime perio. Hoever, i i far more common a ome form of conrolle ample roaion ceme i ue. In uc cae calculaion of i variance can be raer comple. For eample, Norberg (998) e ou e eor for eimaion of e eign variance of bo δ an φ uner e SAMU ample co-orinaion em ue a Saiic u u D I π 49

57 50 Seen for e paricular cae ere imple ranom ampling iin raa i emploe a eac ime perio. Ti approac coniion on e realie ample ize efine b e ranom e, b, c, D an bb, an o i eeniall equivalen o e moel-bae approac ouline belo. A moel-bae approac o meauring e variance of δ i reaonabl raigforar o evelop, oug noaionall cumberome. We ave ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) ( )( ) ( ) { } ( ) ( ) ( ) { } { } ( ) D C bc c bb C D c bc bb D C c bc bb D C B C D B \ \ \ V V σ σ σ σ ρ σ σ σ σ σ ρ σ σ σ σ σ ρ σ σ σ σ ρ σ σ δ δ ere σ u enoe e moel anar eviaion of u, an ρ enoe e moel correlaion beeen an. Noe a B\ bb enoe all elemen of B a are no in bb, an o on. Provie uni belonging o e variou ample componen in e above variance are ienifiable, e can eimae e moel-variance of δ uing plug in eimae for e variou moel parameer in i epreion. A eeroeaici robu eimae of e moel-variance of δ can be rien on uing e eor e ou in Define u µ a e moel epecaion of u, i unbiae eimae u µ. Suppoe furer a for ome non conan u e ave ( ) [ ] ( ) E E u u u u u µ µ. Ten e can eimae e moel-variance of δ b ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) { } ( ) ( ) { } ( ) D C bc c bb D C c bc bb D C B D \ \ \ V σ σ χ σ σ µ µ µ µ µ µ µ µ δ δ

58 5 ere u σ, u, an χ are moel-bae eimae of V ( u ) an C (, ) repecivel. Tu, for e iuaion coniere in.3..0, e ave ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( )( ) V C bc c C D D bb B D n n n N n N n N D C c bc bb σ χ σ σ σ µ µ µ µ µ µ µ µ δ δ ere ( ) u u u u u u u u u u u u u u u u u n N N N µ σ µ an ( )( ) c c c c c c u u u u uc u u u uc c c c c c n µ µ µ χ Turning no o φ, e noe a a fir orer approimaion o i moel-variance can be rien on uing a Talor erie argumen. Ti i ( ) ( ) ( ) ( ) V, C V V φ φ ere ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) bc c bb c bc bb D D bc c C D c C B D C \ \ \ \ V, V σ σ σ σ σ σ σ σ σ σ an

59 5 ( ) ( ) ( ) ( )( ) ( ) C bc c c C bc C \, C σ σ ρ σ σ ρ σ σ ρ σ σ ρ We can eimae e componen of i variance uing e eeroeaici robu variance eimaion eor e ou in Deail are omie, bu follo e correponing evelopmen for δ cloel. 3.. Eimae of cange for funcion of populaion oal Te Talor erie lineariaion meo ecribe in.4. can alo be ue o eimae e variance of e eimae of cange in a funcion of e populaion oal a eac ime poin. To illurae, conier e cae ere e are ineree in e cange in e raio of e populaion oal of o variable, a Y a an Y b. Ti cange i efine a U b U a U b U a R R R δ Suppoe furer a ee oal are eimae via unbiae linear eige eimae a eac ime poin. A conien eimae of δ R i en b a b a R R R δ. Te approac ecribe in.4.. can be ue o linearie e eimae of e raio a eac ime poin. Tu u u u u z R ere u u u U bu U au u U bu bu u au u R R z µ µ µ ~ ~ an µ au, µ bu repreen e epece value of au an bu repecivel. Conequenl R z z δ

60 an e can appl e reul in 3.. above o eimae e variance of δ R, replacing in e variance eimae formula ere b au R u bu z u. u Alernaivel, eier e boorap or acnife approace o variance eimaion can be ue. In eier cae, e ample unerling e proceure i e union c b of e ample an. Tu e rop ou acnife in i cae procee b eleing one uni a a ime from. See Can & Davion (997) for an applicaion of e boorapping iea o eimaion of cange Eimae of cange in omain quaniie Given linear eige eimaion a eac ime perio, an eimae of e cange in omain oal beeen an, δ U ere u enoe e value of e omain inicaor a ime T u for uni, i δ. A noe in 3..7, e level eimae componen of δ ma be biae, an o i eimae of cange ma be biae a ell. To illurae, conier e raum moel of 3..7 i auiliar variable X efine b raum inicaor plu a ize variable Z, an i calibrae eig. Aume furer a e coefficien for Z in e regreion of Y on X i e ame a bo ime perio. Te bia in δ i en u bu U E ( δ δ ) θ ( ) c z T b z T z T B z T D z T β z ere a ubcrip of enoe rericion o raum. Ti bia vanie if θ i e ame for all (e coniion for e omain eimae a eac ime perio o be unbiae). In general, oever, ere i lile e can a abou i bia. One ecepion i ere e bir an ea iin a raum ave approimael e ame iribuion for Z, in ic cae e ir erm in brace above oul be mall. Similarl, if e eig for e common ample iin a raum remain approimael e ame from T o T, an e incoming ample a ime T i coen o a i repreen e ame proporion of e raum oal of Z a e ougoing ample from T, en e fir an econ erm in brace ill alo be cloe o zero an o e bia in δ ill be mall. Variance eimaion for a linear eige δ i raigforar. We replace in e variance formulae in e preceing ecion b. Noe a a correponing moificaion o e eimae of e epece value µ of i variable i alo require en compuing reiual for ue in e variance eimae. Furermore, ince e omain incluion variable D an e urve variable Y are 53

61 54 uncorrelae uner (a i, given e value of e auiliar variable), ( ) ( ) [ ] E V µ. Appling e moel-robu variance eimae evelope in 3.. en lea o ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) { } ( ) ( ) { } ( ) D C bc c bb D C c bc bb D C B D v v v v v \ \ \ V µ µ µ µ µ µ µ µ δ δ ere e eimae variance an covariance conribuing o e econ orer (uneige) erm in i variance eimae are given b ( ) v θ θ µ θ σ an [ ] v θ θ θ µ µ θ σ. Here σ i e covariance beeen an, an θ i e probabili of omain incluion a bo T an T. Bo ee quaniie nee o be moelle uing aa from e common ample c. 3.3 Oulier robu eimaion Oulier are a common problem in ample urve, an paricularl in buine urve. Given a moel for a urve variable Y, an oulier i a value for i variable, ic i eeniall impoible uner. An oulier i erefore an inicaion of a breaon in e moel pecificaion for Y. Oulier can be bo ample an non-ample value. In e laer cae, oever, e are no oberve, an o i mipecificaion i never ienifie. In a follo erefore e confine aenion o ample oulier. We alo aume uc oulier are repreenaive, in e ene a e are no caue b error in aa collecion or proceing. Ta i ee value are real e are u no a all lie e re of e ample value. B efiniion, oulier are rare. Conequenl, aloug eir preence in e ample ell u a i mipecifie, ere are o fe of em a ere i no enoug informaion o moif e efiniion of in orer o accommoae em. For eample, oulier ofen arie becaue inur an ize caraceriic ue o efine raa are ou of ae, an o a raum en up conaining uni oe curren caraceriic (an reuling economic performance) are quie unrelae o a of e maori of uni in e raum. If ere i a ubanial proporion of ee incorrecl claifie uni, en raum level eimae can be replace b omain eimae. Hoever, picall ere are onl a fe uc oulier, an omain eimaion proceure bae on ee are igl unable.

62 Tere are ree baic approace o ealing i ample oulier. Te fir i e mo common in pracice an e lea efenible. Ti i o elee e oulier from e ample. Ti canno be uifie unle ere i rong evience a e ample oulier are unrepreenaive, being ue o incorrec aa collecion meo or error inrouce in ample proceing. Te econ i o eep e oulier in ample, bu o give em eig equal o one. Ti correpon o e aumpion a e oulier are unique, an a ere are no remaining oulier in e nonample par of e populaion. Ti aumpion abilie e overall ample eimae, bu a e co of a poeniall large bia. Te eoreicall mo accepable opion i o eep e oulier in e ample, bu o moif em o a eir impac on e ample eimae i ep mall. In effec, e normal ample eig a oul be aociae i e oulier i ep, bu e oulier value i moifie o omeing le ereme. In e folloing o ub-ecion e icu approace o i value moificaion. B efiniion ee are moel-bae. Sricl peaing, oulier are irrelevan from e eignbae poin of vie ince i eor mae no aumpion abou eer a realie ample value i conien i an aume uperpopulaion moel for e populaion aa. We alo reric ourelve o a are omeime referre o a Y-oulier, a i ere e problem i in e realie Y-value of cerain ample uni. Anoer cla of oulier occur ere e X value of a fe ample uni are ver ian from e X-value of e oer ample uni. Tee are X-oulier, an e can ave a ubanial impac on e abili of e overall ample eimae becaue of eir o-calle leverage. Ti i picall manifee in ouling ample eig, raer an ouling ample value. Tere are meo for ealing i uc ouling eig (ee Camber, 997), bu ince e primaril relae o efficien eiging meo raer an o bia an variance iue uner probabili ampling, e are no icue furer in i repor Oulier robu moel-bae eimaion Robu moel-bae meo for urve eimaion are reviee in Camber & Koic (993). See alo Lee (995). We aume a e non-oulier ample value follo e uperpopulaion moel pecifie in.3.., a i ere C E ( ) µ ( ; ω ) V ( ) σ ( ; ω ) (, ) 0 for Hoever, e ample aa conain a fe value a are inconien i i moel. If e ignore ee inconiencie (a i, inclue e aa a normal), our eimae of e populaion oal of Y i picall of e form µ ( ; ω ) ere ω i an eimae of ω bae on e ample aa. Tpicall, in e inere of efficienc, i eimae i bae on e applicaion of nonrobu eimaion meo lie lea 55

63 quare or maimum lielioo. Te preence of ample oulier can erioul eabilie i eimae oever. Oulier in e populaion can be moelle b auming a e populaion i in fac a miure of oulier an non-oulier. Ta i, e rue uperpopulaion moel for Y i mae up of value ran from an value ran from an oulier moel η. Ti can be repreene a ( µ ( ; ω ) ε ) ( δ )( ν ( γ ) ε ) δ ; ere δ i an inicaor ranom variable ic eermine eer a value i an oulier ( δ 0 ) or a non-oulier ( δ ), an ε an ε η are zero mean ranom variable uc a ( ) σ ( ω ) an V η ( ) τ ( ; γ ), i τ ( ; γ ) σ ( ; ω ) V ; ε ε η aume a e ranom variable δ an rue populaion moel i uc a E ε, η >>. If e furer ε η are inepenen of one anoer, en e ( ) µ ( ; ω ) ( π )( ν ( ; γ ) µ ( ; ω ) an V ( ) π σ ( ; ω ) ( π ) τ ( ; γ ) π ( π )( µ ( ; ω ) ν ( ; γ )) ere π Pr(δ ). Te bia in i erefore ( ) E{ µ ( ; ω ) µ ( ; ω )} ( π ){ ν ( ; γ ) µ ( ; ω )} E. Te fir erm on e rig an ie above ill be eeniall zero provie e meo for calculaing ω can be mae oulier robu (for eample if e ample oulier ave lile or no influence on i value). Ti lea o e eimae ere * µ ( ; ω ) ω robu i e oulier robu eimae of ω (i ma be impl e eimae of ω obaine afer oulier are elee from e ample). In an cae e all aume a o e bia of * become E robu { ( ; ω ) µ ( ; ω )} 0 µ robu ( ) ( π ){ ν ( ; γ ) µ ( ; ω )} E *. Ti bia can ill be ubanial. Conequenl, i i generall inufficien o replace nonrobu parameer eimae b robu parameer eimae en ealing i oulier in ample urve aa. Hoever, ince 56

64 E { ( )} µ ; ω ( π ){ ν ( ; γ ) µ ( ; ω )}, one can ee a i bia can be eimae b eimaing e nonample oal of e reiual generae b. I follo * can be correce b ubracing i eimae bia. One problem i i eimae bia correcion i a e preence of ample oulier can mae i ver unable. Camber (986) erefore uggee a i correcion be robuifie a ell, leaing o e moifie eimae ** µ ( ; ω robu ) m σ ( ; ω robu ) ψ σ µ ( ; ω robu ) ( ; ω ) ere m i a uiable coen eig of orer O((N n)/n) an ψ i a boune emmeric funcion ic eermine e influence of e ample reiual on e bia correcion. In e cae ere i a general linear eige eimae, efine b ample eig { }, ** i given b ( ; ω robu ) ( ) σ ( ; ω robu ) ψ σ robu µ ( ; ω robu ) ( ; ω ) ** µ. robu A GREG verion of ere of ω. ** π ** can alo be rien on. Ti i U µ ( ; ω ) π robu σ ( ; ω π robu) µ ( ; ω π robu) ψ π σ ( ; ω ) π robu ω π robu enoe a eign conien eimae of a FPP ic i ielf a robu eimae Coice of e influence funcion ψ i picall lef o e uer. A ie varie of uc funcion are available in e aiic lieraure (Huber, 98). In general a afe coice eem o be e Huber influence funcion ψ() gn() min(ab(), c), i c no oo mall, a c 6. Ti allo e ample oulier o ave ome a in e bia correcion erm, bu no enoug o eabilie i compleel. In general, none of e above verion of e robu eimae ** i unbiae. Hoever, eir mean quare error properie are picall uperior o bo an e naive robu eimae Variance eimaion for ** i complicae b i bia proper, a ell a b e inrinic nonlineari of e eimae. Camber & Dorfman (994) repor on e ue of e boorap o eimae confience inerval for robu eimae lie **. In general, e foun a e *. 57

65 boorap variance eimae coul no anle e bia, leaing o acual confience inerval coverage a a le an nominal coverage. Te eimae ** i moivae b a i omeime referre o a a gro error moel for e populaion oulier. Ti moel i queionable en e oulier are e conequence of a long aile error iribuion for Y raer an conaminaion. Here e oulier arie becaue of mipecificaion of. Camber e al. (993) uggee a in i cae one oul a a nonparameric bia correcion erm o. Uner long-aile alernaive o, i i ie o robuif i nonparameric correcion erm o a, lie e parameric correcion erm ue in **, i i relaivel unaffece b a fe ver ereme ample value. Ti lea o e eimae B ere B i e fie value a of a robu nonparameric moo of e ample reiual r µ ( ;ω ),. In e empirical u repore in Camber & Dorfman (994), i eimae, bae on a Huber-pe local linear mooer, performe eremel ell, recoring bo a lo bia an a lo mean quare error. Boorap confience inerval bae on alo a e be coverage properie of all e robu eimae coniere in a reference Winoriaion-bae eimaion A a been noe a number of ime before, e ue of ample eige eimae i common in buine urve. Conequenl, ere i a eman for robu eimaion meo a can (a lea nominall) fi ino i linear eimaion frameor. Te moel-bae robu eimaion meo ecribe above are no eail compue in i a. An alernaive meo a fi naurall ino i frameor an a goo oulier robune properie i e o-calle inoriaion approac. Uner i meo, ouling ample Y-value are moifie o e are no longer ouling, an e linear eige eimae i en calculae uing ee moifie value. More preciel, ince an linear eige eimae of a populaion oal can be epree a ( ), L inoriaion procee b replacing an ouling value in e econ erm on e rig an ie above b a le ouling value. In paricular, e inorie eimae can be rien * L ( )[ I ( L U ) L I ( < L ) U I ( > U )] ere I () enoe an inicaor funcion ic ae e value if i argumen i rue an i zero oerie an L, U are loer an upper boun for e Y-value of populaion uni. 58

66 Deerminaion of ee boun epen on e uperpopulaion moel for Y. A uual e aume e general uperpopulaion moel of.3... Ta i, e aume e mean an variance of uner are given b µ ( ;ω ) an σ ( ;ω ) repecivel, ere ω i an unnon parameer. In man buine urve applicaion, Y i inrinicall poiive, an o L i e o zero. Ti i referre o a one-ie inoriaion. For i cae Koic & Smi (999a) parameerie e upper boun U in erm of a ingle parameer U, via ere µ ( ; ω ) U µ ( ; ω ) U i an unbiae eimae of e epece value of uner. Te en evelop proceure for cooing U in orer o minimie e mean quare error of L * uner. Tee proceure require acce o iorical urve informaion in orer o eimae ω. Empirical reul quoe in eir paper inicae ubanial gain from inoriaion in urve of oulier prone populaion. A problem i one-ie inoriaion i a, b conrucion, e reuling eimae a a negaive bia. Tpicall, eimaion i carrie ou eparael in variou raa an ee eimae are en ae o give an overall populaion eimae. If inoriaion i applie iin eac raum (a i U above i eermine eparael for eac raum in orer o minimie mean quare error a raum level), en e overall populaion eimae ma ave a ubanial negaive bia caue b ummaion of e iniviual raum biae. Tu, aloug e iniviual raum level eimae are ell beave, e overall eimae ma ave an unaccepable level of error. On e oer an, if U i eermine a populaion level (a i, e ame U in all raa), en i ma lea o raum level eimae a are unaccepable. In a ubequen paper (Koic & Smi, 999b) ave eene eir meoolog o a bo loer an upper boun are eermine in uc a a a o enure a e inorie eimae a minimum variance uner ubec o i being (approimael) unbiae uner i moel. Teir parameeriaion for U an L in i cae are U ( ; ~ ω ) σ ( ; ω )U µ robu an L ( ; ~ ω ) σ ( ; ω )L µ robu ere ω ~ i an inepenen unbiae eimae of ω (for eample obaine from iorical urve aa) an σ ( ω ) ; i an oulier robu eimae of e anar eviaion of robu uner. Te cu-off parameer U an L are en coen in orer o minimie e moel variance of L * ubec o i aving zero moel bia. I urn ou a ee opimal value epen on oluion of o ifferenial equaion efine b e common iribuion F of e r µ ω σ ; ω. Tee are anarie reiual ( ( ) ( )) ; 59

67 F ( L) L F( U ) ( )U an U L ( U L) { f ( U ) f ( L) L U} ere f i e eni correponing o F. Empirical reul repore in Koic & Smi (999b) inicae a i o-ie inorie eimae overcome e cumulaive bia problem ecribe above for one-ie inoriaion, ile ill reaining e oulier robune properie aociae i e inoriaion iea. Provie ~ ω (an ence U an L ) i bae on inepenen iorical informaion, variance eimaion for L * i raigforar, ince e meo ecribe in previou ecion of i repor can be applie, i replace b i inorie value * ( L U ) L I( < L ) U I ( U ) I > Wen iorical aa are no available, i i unclear o one can procee o eermine L an U above. One poibili i o ue cro-valiaion, uing par of e ample o eermine ω ~ an e re o eermine L an U, an en repeaing i proce for a e of nonoverlapping ubample ic eeniall cover e original ample. Te final value of L an U are en obaine a average of ee ubample-bae eimae. Te properie of i approac are unnon a e ime of riing. 3.4 Variance eimaion for inice Man e official aiic are preene in e form of inice, emelve calculae uing eimae erive from a number of ource, bo urve an aminiraive em. Te purpoe of i ecion i o briefl ouline meoolog for variance eimaion for uc aiic. To provie a focu for i icuion, e cae of variance eimaion for e UK Ine of Proucion (IoP) ill be coniere. For a more compreenive aemen, ee Koic (998). Te IoP i an economic inicaor prouce b Unie Kingom Office for Naional Saiic (ONS). I i a monl ine of e oal volume of inurial oupu (or proucion). I cover e Mining, Manufacuring an Agriculural ecor of e UK econom an i currenl bae o 990 price. I i one of e main inicaor of economic gro iin e UK. Te IoP i obaine b combining everal ifferen ource of aa. B far e mo ignifican ource i ONS urve. Tee inclue e Monl Proucion Inquir (MPI), Proucer Price Ine (PPI) an e Quarerl Soc Inquir (QSI). Oer aa ue in i conrucion inclue e Epor Price Deflaor (EPD), ic i currenl erive from a combinaion of aa collece b ONS an b UK Cuom an Ecie, an aiional aa on e oil, ga, elecrici an mining inurie from e UK Deparmen of Trae an Inur, an on foo proucion from e UK Minir of Agriculure, Fierie an Foo. 60

68 Te IoP i fir conruce iin inur group a e 4-igi anar inurial claificaion (SIC) level (Cenral Saiical Office, 99). Le I 0T be e IoP eimae for ime perio T relaive o a reference perio 0 in inur group. Higer level eimae are prouce b aing eige average of ee IoP eimae, ere e eig are eermine b e value ae in e bae ear (eimae from e Annual Buine Inquir urve). Tu e overall ine I 0T i given b I 0T I 0T0 0 ere 0 i a value ae eig for inur. Te relaive cange in e IoP beeen ime perio T an T ma be rien a I T I T I 0T 0 0 From no on, ecep ere necear for clari, e all onl mae reference o one bae ear, a ingle reference perio T an one 4 igi inur, an o for implici e ubcrip 0, T an ill be roppe. Te proce of ine conrucion can be broen on ino a number of iinc ep. Sep : Conrucion of e combine price eflaor. A price eflaor for ome (a i, omeic) ale i eimae from PPI aa, an anoer for epor ale i eimae from EPD aa. Te invere of ee eflaor eimae e average price increae from e bae ear for commoiie prouce an ol b all member of a given inur. Te combine eflaor i a armonic mean of ee ome an epor price eflaor, eige b oal ome ale an oal epor ale, bo eimae from MPI aa. I i efine b 0. S D D ome ome S D epor epor S ere D ome i e eimae ome price eflaor (from PPI), D epor i e eimae epor price eflaor (from EPD), epor ale (from MPI) an Ŝ ome i eimae ome ale (from MPI), Ŝ epor i eimae S S S. ome epor Sep : Conrucion of a eflae eige ale ine. Ti ine repreen e relaive increae in real erm of ale in e curren mon compare o e bae ear. For i purpoe ale are pli beeen mercane goo an non-mercane goo. Mercane goo are oe prouc ol on b a buine iou being ubece o a manufacuring proce. Te ine i efine b I ale S S S m S S G G m m S m S S G m m D 6

69 ere Ŝ m i e eimae of ale of mercane goo (from MPI), Ĝ m i e eimae of monl average ale of ee goo in e bae ear, an Ĝ i e correponing eimae of monl average ale of all goo in e bae ear. Sep 3: Creaion of a bencmar ale ine. Ti ine i calculae b a linear ranformaion of e eflae eige ale ine. A muliplicaive aumen i ue o enure a e ine mee cerain (eernall impoe) conrain for publicaion, an aiive uning conan are ue for minor aumen ere e ine value oe no follo paern epece in e relevan inur. Te ine value a i prouce i erefore c I I ale a ere c i e conraining facor, i e monl average of e eflae eige ale ine in e bae ear an a i e uning conan. Te final value of e IoP i obaine afer carring ou a furer aiive oc aumen o e bencmar ale ine above. Ti i en eaonall aue before publicaion, uing X-ARIMA. Talor erie lineariaion an boorap meo for eimaing e variance of e noneaonall aue IoP are icue in Koic (998). Bo are bae on e aumpion a i approimael one an o S S S m G G G S S S ome epor I ale. GD G Dome Depor m I follo a V ( I ) c V( I ) ale ere V(I ale ) can be eimae via Talor erie lineariaion or boorapping. In e former cae i lea o e eimae V ( ) ( ) ( ) ( ) S S ome epor V G S V V ome epor D S D I ome epor ale G D ome V D ( S ) Ĉ( S ome, S epor ) V ( S epor ) ome ome D epor D G ome D epor D 4 ome D epor D 4 epor 6

70 ere a a, a uual, enoe an eimae, an e ave ue e fac a Ĝ, an (, S ) D ome, D epor S ome epor are uncorrelae eimae, being bae on aa collece a o ifferen ime poin an from ree ifferen ource (PPI, EPD an MPI). A parameric boorap eimae of e variance of I ale i alo eail compue. Ti involve ampling i replacemen from e large ample approimae oin iribuion of Ĝ, D, D an ( S, S ) epor ome epor. Uing a ubcrip of b o enoe uc a ra, e ave ome I b ci ale, b ere I ale, b G b S D ome, b ome, b S D epor, b epor, b an D S S D ome, b epor, b ome, b epor, b G b IID ~ IID ~ IID ~ IID ~ N N N N ( D ( D ome, V ome ) ( D ( D epor, V epor ) S ( S ) ( S S ) ome V ome Ĉ,, S Ĉ( S, S ome epor ) V ( S epor ome epor epor ) ( G, V ( G ) ere IID ~ enoe a ranom ra from e inicae iribuion. Given B imulae value I b generae accoring o i moel, e boorap variance eimae for I i erefore B B V boorap ( I ) I b B I b. B b b In e imulaion u repore in Koic (998), i approac an Talor erie lineariaion le o comparable variance eimae. 3.5 Concluion Ti caper a eene e eor for eimaion an ample error variance eimaion inrouce in e previou caper o four imporan pecial cae a occur ofen in buine urve. Tee are eimaion for omain, eimaion of cange, eimaion in e preence of ample oulier an eimaion of inice. All four iuaion require careful applicaion of e eor evelope in caper, i an empai perap on e ue of moel-bae iea o iglig iue relaing o e overall quali of e eimae prouce. Hig quali omain eimaion i a funamenal obecive of mo buine urve. For eample, i i a baic requiremen for an urve ere e inur an ize claificaion on 63

71 e frame i ou of ae. In ecion 3., erefore, e e ou e relevan eor for i obecive. I i inereing o noe a if one rea omain memberip on e ame bai a an oer urve variable, en anar eign-bae an moel-bae eimaion meo eeniall reul in e ame inference. Hoever, e inroucion of era informaion abou e omain (for eample eir ize) can onl be eail accommoae from a moel-bae perpecive, oug even ere ere i ome ebae abou eacl o i oul be one. Conequenl e recommen a en meoolog for omain eimaion i ue in a urve, careful aenion i pai o informing e uer of ee eimae abou e meo of compuaion, plu e bai of e ampling variance calculaion (a i, eer e are coniional on omain memberip in e ample or no). Eimaion of cange bae on aa obaine from picall overlapping ample i anoer common feaure of buine urve. One coul in fac claim a uc a meaure of cange i in fac e e obecive of mo uc urve. In i cone e ave inicae e manner in ic variance eimae bo for abolue a ell a relaive cange nee o be calculae. Of necei, ee calculaion are raer comple involving e inegraion of urve aa from o (an omeime more) ource. A preen e are no aare of an ofare a can auomaicall carr ou ee calculaion, o e appropriae meoolog nee o be cuom programme ino a urve aa anali em. Te eor e ou in ecion 3. oul be elpful in i regar. Sample oulier are a perennial problem in buine urve an form e focu of e icuion in ecion 3.3. Here i uffice o noe a a conenu on ealing i ee uni a e o be reace, in large par ue o e fac a e concep of a coniue an oulier remain e obec of ebae. Te inoriaion meo icue in ecion 3.3. offer conierable promie an are e ubec of curren reearc. Again, ue of ee meo ill generall abilie e eimae variance of a urve eimae, bu a e co of ome increae in bia. Ti rae-off i picall avanageou if one main concern i racing e beaviour of e non-volaile par of e arge populaion. In oing o, one oul ae care, oever, o enure a ample uni ienifie an oneige a oulier oul be inveigae an e reaon for eir ouling value eablie. A e en of e a e preence of oulier i a mpom of a bal pecifie moel for e populaion, an o e informaion e provie nee o be ue o upae an improve ample eimaion an inference proceure. Finall, in ecion 3.4 e acle e iue of variance eimaion for an ine calculae on e bai of coninuing urve aa. Becaue of e ie varie of uc inice in ue, e ave coen o confron i problem via icuion of one paricular ine, e UK Ine of Proucion, an o o o e comple aiic meoolog icue in ecion.4 can be aape o e problem of eimaing e ampling variabili of i ine. Te meo (Talor erie lineariaion, boorapping) e ecribe are generall applicable o an ine, oever. 64

72 4 Sampling error uner non-probabili ampling 4. Inroucion Davi Draper an Ruell Boaer, Univeri of Ba In Caper e eamine ampling error ariing from probabili ampling. In e ranomampling approac o urve an auming (a e i in Caper ) (a) a e arge an urve populaion coincie, o a one ma pea iou confuion impl abou e populaion, an (b) a e available frame i perfec e ampling meo i aume o rea e N populaion uni in uc a a a ever uni a a non-zero probabili of incluion in e ample. Coninuing e noaion in Caper, le be an oucome variable of inere an efine e ample incluion inicaor I if uni i in e ample an 0 oerie. Probabili ampling mae e I ranom variable, o a i i meaningful o pea of e incluion probabili for uni, Pp ( I ) π P ( I, I ) p π, an e oin incluion probabili for uni an,. Here, a in Caper, e ubcrip p enoe probabili a efine b e (eign-bae) poeical proce of repeae ranom ampling. A Särnal e al. (99) noe, a probabili ampling eign for ic e folloing o properie ol, π π > 0 for all > 0 for all N N, (4.) an for ic all of e π an π coniion in (4.) (ogeer i e ipulaion a e are non, i calle meaurable. Te fir of e π are non) i necear an ufficien for obaining a eign-unbiae eimaor of e populaion oal an e econ coniion permi e calculaion of a (nearl) eign-unbiae eimae of e variance of e ample error iribuion for eimaor of. From e eign-bae poin of vie, meaurable probabili ampling eign are u clearl eirable (Neman 934, Cocran 977), an a noe in Caper probabili ampling alo provie an imporan egree of robune from e moel-bae perpecive. Depie i, non-meaurable ampling i frequenl emploe in ome fiel even oa: ample of convenience, in ic e π are unnon becaue no aemp a mae o cooe e ample ranoml, are ubiquiou in meicine an e ocial cience (Draper 995b give everal eample of uc ample), an probabili-ampling eign in ic N We are graeful o Ra Camber (Univeri of Souampon), Eva Elver (Saiic Seen) an Paul Smi (UK Office for Naional Saiic) for commen an reference, an o Paul Smi for ome uggee e fragmen. Memberip on i li oe no impl agreemen i e iea epree ere, nor are an of ee people reponible for an error or omiion a ma be preen. 65

73 ome of e π are zero (uc a raifie ranom ampling i onl one ample uni in one or more raa) can occur in pracice. Non-probabili ampling i alo omeime ue in buine urve (ee Särnal e al. 99 an Leler & Kalbee 99 for eample). A noe in Euroa (996:04), i can occur en ere i no reail available ampling frame, or en e urve i volunar. In i caper in Secion e conier eac of e four leaing poenial inance of nonprobabili ampling in buine urve - volunar ampling, quoa ampling, ugemenal ampling, an cu-off ampling. In Secion 4.6 e provie ome concluion, incluing brief recommenaion on be pracice an eir implicaion for moel quali repor. I i perap or empaiing a e oue (a) a one of e main problem poe b nonprobabili ampling i bia (a efine in Caper ), an (b) a bia i qualiaivel ifferen from e in of error a can arie i (mall) ranom ample. In e laer cae (eign) unbiaene i guaranee, in e uual long-run-average ene (ee caper an 3), b e ranomiaion, an e ave onl o ae larger ample o imini e liel amoun b ic our eimae ill iffer from eir rue value. Bia i more iniiou: i ill no go aa i increaing ample ize, becaue repeaing a biae meo of aa collecion on a larger cale merel perpeuae e bia. Tu ere i a maor buren on anone o ie o ue a non-probabili ampling meo, namel emonraing a an bia inuce b e ampling meo can be largel iminie b aumen uc a poraificaion (o be ecribe in Secion 4.). Even if bia i largel conrolle, e unavailabili (or non-poiivi) of e accurae uncerain aemen. 4. Volunar ampling π an/or π ma creae eriou problem for Volunar ampling arie en, for eample, buinee are requee, bu no require, o ae par in a urve, an e urve reul are bae u on e aa receive from e companie o cooe o repon. Te coice of eer o paricipae u mae e ample non-probabili-bae: even if one ie o acnolege uncerain, before e aa arrive, abou ic companie ill repon b regaring e ample incluion inicaor I a ranom, e incluion probabiliie π are renere unnon b e coice mecanim. A i quoa ampling (Secion 4.3), e reul can range from igl accurae o igl inaccurae, epening on e (poibl unnon) egree o ic e voluneer uni repreen e populaion in all relevan repec. An bia a arie from failure of e volunar ample o mac e populaion in i a i an eample of elecion bia (ee Freeman e al. 998 for a icuion), in ic e elf-elecion mecanim i correlae i e oucome of inere an ome or all of i mo imporan preicor. An eample of volunar ampling i provie b e Soc Inquir buine urve, conuce b e UK Office for Naional Saiic (ONS). Ti urve a bo a monl volunar componen an a quarerl componen bae on probabili ampling: ranom ample of companie are (a) coen, (b) require o provie quarerl aa, an (c) requee 66

74 (in aiion) o provie monl aa, o a e companie proviing volunar monl informaion form a elf-elece ube of e probabili ample. In pracice abou 30% of e ample companie cooe o uppl e volunar aa. Noe a i pe of ample coul equall ell be ecribe a a probabili ample (a) i a volunar ub-ample or (b) i a ig egree of (almo cerainl) non-ignorable non-repone (ee caper 8 an ecion 9.7). Perio Inur Inur Inur 3 P V B P V B P V B 97/Q 3,40 5,45,005 38,0 38, ,67 6,534 34,97 97/Q 3,456 6,48,69 40,50 43,7,769 7,439 6,990 35,55 97/Q3 3,455 6,008,553 36,940 44,70 7,30 6,059 59,93 33,87 Table 4. Eimae bae on e Probabili (P) an volunar (V) ample, b inur an perio, for or-in-progre Opening oc. All figure are in 000. B eimae bia. Available variable in e anali e preen ere inclue inur group number (four-igi SIC9; e focu ere on onl 3 inurie, coe -3); perio of reurn from 0/997 o 09/997; regier emplomen an (VAT) urnover (in 000) bae on aa gaere rougl 3 mon previoul; an e Opening an Cloing oc (in 000) for eac of ree caegorie: maerial, or in progre, an finie goo. Te number of companie involve in e volunar an probabili ample in i perio ere an 6-75, repecivel, varing a bi from quarer o quarer. We concenrae ere on e or-inprogre oc (reul for e oer o caegorie ere imilar). For eae of epoiion (a) e preen reul onl on e 77 an 6 companie in e volunar an probabili ample i complee aa a all ime poin relevan o e anale belo, an (b) e anale e aa a if e probabili ample a been a imple ranom ample (in fac i a a raifie ranom ample; e poin e i o mae in i ecion come roug more clearl iou e era iue of re-eiging e probabili ample bac o e populaion). Perio Inur Inur Inur 3 P V B P V B P V B 97/Q 3,456 6,48,69 40,50 43,7,769 7,439 6,990 35,55 97/Q 3,455 6,008,553 36,940 44,70 7,30 6,059 59,93 33,87 97/Q3 3,898 7,88 3,930 39,356 49,605 0,49 4,67 56,638 3,0 Table 4. Eimae bae on e Probabili (P) an volunar (V) ample, b inur an perio, for or-in-progre Cloing oc. All figure are in 000. B eimae bia. Some inicaion of e biae a coul arie from baing inference on e volunar monl ample i provie b a irec comparion beeen e monl an quarerl aa in eac of e ree perio 0-03/97, 04-06/97, an 07-09/97 a ere common o bo urve (for comparabili beeen e monl an quarerl erie, e opening an cloing 67

75 of e fir quarer of 997 ere aen o be 0/97 an 03/97 for e volunar erie, an analogoul for e oer quarer). Table 4.-Table 4.3 preen ample eimae b inur an perio for or-in-progre Opening, Cloing, an (Cloing Opening) oc in eac of ee ree quarer. Wiin eac inur coe, probabili (P) an volunar (V) eimae are given, an ince e are aing e probabili-ampling reul o be (eign) unbiae e eimae bia B V P from e volunar aa ma alo be calculae. I i evien from ee able (a) a e volunar reul for bo opening an cloing oc are enormoul biae on e ig ie, an (b) a muc oug b no mean all of i bia cancel in e ubracion en proucing e (Cloing Opening) oc eimae, ic are e principal oucome of inere. Perio Inur Inur Inur 3 P V B P V B P V B 97/Q ,49 4,366,875 8, /Q , ,46 -,380-3,059 -,679 97/Q3 443,80,377,46 5,435 3,09 -,43-3,93 -,86 Table 4.3 Eimae bae on e Probabili (P) an volunar (V) ample, b inur an perio, for or-in-progre (Cloing Opening) oc. All figure are in 000. B eimae bia. Te leaing meo for bia reucion i volunar ample i poraificaion (for eample, Hol & Smi 979, Jager 986, Smi 99, Lile 993). Taing for implici e cae of a ingle oucome of inere, o ingreien are require for i meo: (i) a li, preferabl (cloe o) eauive, of covariae liel o be (igl) correlae i e oucome; an (ii) e abili o gaer aa on ee covariae bo in e volunar ample an in e populaion ielf. Diviing eac covariae ino raa an cro-abulaing e reuling caegorical variable, poraificaion involve (a) eimaing bo populaion an volunar ample prevalence in e cell of i raificaion gri, an (b) re-eiging e volunar ample o mac e eimae populaion prevalence. Ieall e abili of i meo oul be cece b eniivi anali (ee Draper e al. 993a for eample), varing e covariae ue an e cu-poin efining eir raa acro plauible range an eeing eer e bia-aue eimae are imilar. Te (approimae) ucce of i meo re on e aumpion a (mo or all of) e imporan covariae ave been correcl ienifie, meaure, an aue for. Variable Probabili Sample Volunar Sample Inur Inur Inur 3 Inur Inur Inur 3 Regier emplomen Regier urnover,949 7,7 9,775 5,06 6,45 8,388 68

76 Table 4.4 Comparion of probabili an volunar ample on meian regier emplomen (number of people) an urnover ( 000), b inur, in e fir quarer of 997 (reul for e oer o quarer ere imilar). In i eample e onl available covariae are regier emplomen (E) an urnover (T), ic are fairl igl correlae in bo e P an V ample (for eample, e correlaion, i bo variable on e log cale, i 0.74 in e volunar ample). Table 4.4 o a a lea ome of e icrepanc beeen e probabili an volunar ample oul inee be eplainable on e bai of E an/or T: e 30% of e quarerl probabili ample a coe o voluneer monl aa eavil over-repreene large companie. To avoi reunanc e preen poraificaion reul ere onl for one inur (reul ere imilar i e oer o inurie). Wi onl 7 companie per quarer in i inur in e volunar ample, bivariae raificaion on bo E an T oul leave emp cell, ic oe no permi re-eiging, o in e or preene ere e fir raifie onl on regier urnover (in an cae e ig correlaion beeen E an T inicae a ere i no muc informaion in E afer T a been accoune for). We coe four raa, i e malle cupoin elece o a e loe raum a a lea one compan in bo ample, an i e oer o cupoin coen o prea e re of e iribuion ou approimael evenl. Table 4.5 inicae o e probabili an volunar ample in inur ere iribue acro raa bae on regier urnover. Ti provie anoer vie of o arpl e large companie ere over-ample in e volunar urve, for eample, 43% of e probabiliample companie ere in e malle regier-urnover raum, veru 6% in e volunar ample. Te eig ue in e poraificaion are alo given in i able; for eample, e volunar-ample compan in e loe raum a given eig ( 30 70) ( 7) 7. 9, erea e 6 volunar companie in e ige raum receive eig ( 70) ( 6 7) Regier urnover inerval ( 000) P V Weig [0-8,455] (8,455-4,784] (4,784-84,657] (84,657-,84,4] Toal 70 7 Table 4.5 Frequenc iribuion of probabili (P) an volunar (V) ample, acro e four regier urnover raa, ogeer i e poraificaion eig. Table 4.6 preen e reul of e bia reucion ariing from poraificaion on regier urnover. Separael for eac of e oc caegorie {Opening, Cloing, an (Cloing Opening)}, e P column give e probabili-ample eimae (repore previoul in Table 4.-Table 4.3), e PV column i e volunar-ample eimae re-eige b e 69

77 poraificaion on regier urnover, B PV P i e eimae bia afer poraificaion, an R i e percenage (relaive) reucion in eimae bia iele b e poraificaion. For eample, in 997/Q e ra volunar-ample eimae for Perio Opening Cloing P PV B R (%) P PV B R (%) Q 3,40 3, ,456 3, Q 3,456 3, ,455 3, Q3 3,455 3, ,898 4, Perio Cloing Opening P PV B R (%) Q Q Q3 443, Table 4.6 Reul, b perio, from poraifing on regier urnover. In eac of e oc caegorie {Opening, Cloing, an (Cloing Opening)}, P i e probabili-ample eimae, PV i e poraifie volunar ample eimae, B PV P i e eimae bia afer poraificaion, an R i e percenage reucion in eimae bia ariing from e poraificaion. inur a 6,48, giving an eimae bia of,69 (Table 4.); afer re-eiging e ne volunar-ample eimae i 3,556, i an eimae bia of 00 (Table 4.6); an iminiing e eimae bia from,69 o 00 repreen an eimae bia reucion of (,69 00) %. Poraificaion a reule in maive eimae bia reucion ranging from 8% o 97% for e opening an cloing oc, bu a prouce a more moe eimae improvemen in e crucial ifference (Cloing Opening), i gain from 53% o 90%. Perio Opening Cloing P PV B R (%) P PV B R (%) Q 3,40 3, ,456 3, Q 3,456 3, ,455 3, Q3 3,455 3, ,898 4, Perio Cloing Opening P PV B R (%) Q Q Q Table 4.7 Reul, b perio, from poraifing on regier emplomen. In eac of e oc caegorie {Opening, Cloing, an (Cloing Opening)}, P i e probabili-ample 70

78 eimae, PV i e poraifie volunar ample eimae, B PV P i e eimae bia afer poraificaion, an R i e percenage reucion in eimae bia ariing from e poraificaion. Seniivi anali on e poraificaion proce i raigforar. For eample, baing e raa on regier emplomen an uing ree raa inea of four (i raum efiniion [0-5], (5-449], an (449-,378]), coen o creae approimael equalize group in e volunar ample, iele e reul in Table 4.7. Te o approace o poraificaion ave in i cae le o imilar amoun of bia reucion, aloug i nee no ala be rue. In pracice, en a gol-anar (uc a e probabili-ample reul ere) i no available, an ifference reveale b a comparion of i pe ma inicae a oer variable oul ieall ave been par of e raum efiniion, a i a poraificaion ma no ave been enirel ucceful in removing e elecion bia preen in e volunar ample. 4.3 Quoa ampling For a raigforar efiniion of quoa ampling e urn o Särnal e al. (99, p 530) Quoa ampling i ofen ue in mare reearc. Te baic principle i a e ample conain a fie number of elemen in pecifie populaion cell. Suppoe a e populaion i ivie accoring o ree conrol: e, age group, an geograpic area. Wi o ee, four age group, an i area, e ge a oal of populaion cell. In eac cell, e inveigaor fie a number (a quoa ) of elemen o be inclue in e ample. No e inervieer impl fill e quoa, a i, inervie e preeermine number of peron in eac of e quoa cell. Tee ma be e fir peron encounere, or i ma be lef o e inervieer o eercie ugemen in e quoa elecion. Te meo reemble raificaion, bu e elecion iin raa i non-probabiliic [empai ae]. Becaue a elecion i non-probabiliic, ere i neier an unbiae poin eimae nor a vali variance eimae iin e cell. (Alo ee Deville 99 for one aemp a eabliing a eoreical bai for quoa ampling.) In pracice quoa ampler ofen impl aume a e populaion uni ic en up in eac of e cell are lie a one oul ave obaine i imple ranom ampling iin eac cell, bo for an of aning beer o aume an becaue i aumpion urn quoa ampling ino raifie ranom ampling (SRS) an e uual eimae of error (for eample, Cocran 977) are en available. Inee, a Särnal e al. (99) noe, aoping a moel-bae approac in ic e are aume o be ranom variable i E ( ) µ, V ( ) σ, ere inee e cell in e quoa-ampling gri in ic i oberve, iel preciel e ame eimae of e populaion oal a i SRS, ere H i e number of cell in e gri an H N (4.) N an are e populaion ize an ample mean in cell, repecivel. Moreover, e uual SRS eimae variance of i eimaor, 7

79 ere n i e ample ize in cell an H f V () N ( ) (4.3) n n n f, i unbiae uner i moel. Tu vali N inerval eimae for uc quaniie a e populaion oal or mean an raum (cell) mean are available, uner e aumpion a e moel i correc (ee alo caper 9). In Baeian reamen of ample urve i or of aumpion oul be ecribe a a ugemen a e ample an unample uni in eac of e populaion cell are ecangeable (ee Draper e al. 993b for icuion), ic u mean a one preicive uncerain for bo e ample an unample uni before an aa are gaere oul be e ame. If aiional relevan raifier (a Särnal e al. (99) calle conrol in e quoe above) are available in e quoa ample an populaion prevalence are non, poraificaion (a in Secion 4.) iin eac cell can be emploe o au for poible elecion bia ariing from e apazar coice mecanim (ee Smi 983, 993 for eample). Quoa ampling oe no eem o be muc in ue in European rucural an or-erm buine urve a preen, aloug a in of quoa ampling a coul alo be erme ugemenal ampling (ee Secion 4.4 belo) i emploe b man EU Member Sae in e compilaion of price aiic (Euroa 998:07). 4.4 Jugemenal ampling A noe b Euroa (996:04), everal [EU] counrie ue uiciou ample bae on a ig coverage of relevan caraceriic (for eample, proucion, emplomen, an urnover). Ti mainl concern proucion an oupu price inice for ic ere i no regier of prouc. In effec, uc ample are bae on eper ugemen a o repreenaivene raer an full probabili ampling. An eample of o i ma arie occur in one or more age of e ampling proce upporing e creaion of proucer price inice. For inance, Euroa (998:07, abbreviae E98) conain an eenive icuion of meoological apec of eimaing proucer price on e epor mare; mo of e maerial in i ecion i bae on i ocumen Proucer price ine conrucion in e EU Bacgroun on e problem aree b epor-mare proucer price inice i a follo. Proucer price inice in general oul cover e price of all commoiie prouce in a given counr in orer o be conien i [e counr' overall ine of proucion].... Wile oal proucer price inice (PPI) o e evoluion of price for goo prouce on e omeic mare, irrepecive of eer e are ol on e omeic mare or abroa, proucer price on e epor mare (PPI ) onl ae ino accoun e price for oe commoiie ic are ol abroa.... Te main purpoe of e PPI i o provie rapi informaion on buine ccle movemen, a i, o erve a an economic 7

80 inicaor. Furermore PPI alo erve a a eflaor for foreign rae aa an for naional accoun.... [Te] PPI for a given inur group oul be calculae a a eige average of commoi price inice, bae on a ample of enerprie an ample of repreenaive commoiie. Tu e fir ep in e compilaion of a PPI i e elecion of a bae of repreenaive goo, a i, eaing a a given level of a prouc nomenclaure (uc a PRODCOM or HS). In accorance i e elece goo, enerprie ave o be coen ic prouce ee goo on a regular bai eine o be ol abroa. Te la ep coni in efining in eac enerprie e prouc repreening ee goo, for ic price ill en be repore eac mon. [E98, pp. 45] In oer or, e creaion of a PPI picall involve ree age of ampling: (i) cooing a in of mare-bae of goo, (ii) elecing enerprie (companie) maing oe goo, an (iii) aing a ample of acual prouc repreening e goo mae b e enerprie. In pracice eac age of elecion in i ierarc ma ue one or more ampling meo in a more or le formal a, for eample, raificaion, probabili proporional o ize, cu-off ampling, an/or eper ugemen. Here are o eample from pecific EU Member Sae:. In e Neerlan, Te elecion of prouc an reporing uni i bae on eaile bae ear proucion an conumpion aa from ifferen aiical ource, uc a proucion aiic an foreign rae aiic.... In orer o guaranee a minimum quali of price inice, e folloing rule applie: per commoi group e elece reporing uni oul on average cover 80% of ale (cu-off meo). If for a paricular commoi more an 5 reporing uni are require in orer o aain 80% coverage, a ranom ampling meo can be applie.... Once e reporing uni ave been elece, e ne ep i o elec for eac reporing uni cerain prouc iin a pecifie commoi group. Te price aiician no for a in of prouc e an o gaer price from e reporing uni. So, i e elp of a fiel urveor, a vii i mae o e reporing uni. Te reporing uni i ae o pecif e price of a prouc, iin e commoi group, ic i repreenaive for e epor. A lea one, bu normall o or more, price are ae for.... A preen abou 7,000 epor price quoaion are collece a frequen inerval from abou 5,500 reporing uni. [E98, pp. 34]. In Seen, Te ample of repreenaive iem i revie annuall an i mae in four ep: (i) Inurial aciviie (a pecifie b [e Sei verion of] SIC9) are ample b cu-off accoring o epor value. Wiin eac acivi (ii) commoiie (a pecifie b HS) are en alo ample b cu-off accoring o Foreign Trae Saiic ic ave been procee for e naional accoun. (iii) Proucer of elece commoiie are en ample b cu-off from e Foreign Trae Saiic regier of eporer. (iv) Finall, repreenaive iem are elece [ugemenall] in conulaion i e reponen (proucer). Te are elece i preference o prouc i ig ale value, ic coul be epece o be ol uring all mon, an if poible are repreenaive of price movemen iin e commoi group (HS number). [E98, p. 44] 73

81 A ee ecerp emonrae, e coice of eaile commoi pecificaion i liel o involve icuion i eac enerprie a a bai for eper ugemen. Te Sei eample o a ee commoiie are picall coen o be repreenaive of price cange, an o be ol bo frequenl (o a monl aa are available) an for a long perio of ime. I i imporan o ae e accurac of e pe of ample u menione. For eample, if prouc are coen becaue e ave enoe frequen ale, i ma be ue o lo price, an oe price ma, uring perio of riing inflaion, increae more an oer. I oe no appear a man EU Member Sae are aemping a preen o ae e bia or ampling variabili i ic eir PPI are eimae. Te effec of ugemenal ampling are normall ifficul o quanif, bu ere are everal approace ic can be aope, ome of ic rel on e eience of oer informaion, an ome of ic are onl available roug aiional uie. We conclue i ecion i a icuion of ome meo currenl in ue in e UK Te UK eperience Te fir poin o noe, in e cone of price inice, i a ere i rarel a frame i prouc informaion from ic commoiie can be elece. A menione above, i mean a ampling i uuall rerice o cooing an enerprie, an en ienifing a repreenaive prouc on a ugemenal bai. Tere a been a enenc in e UK PPI o obain more an one quoe from buinee for imilar prouc, ic in pracice give lile aiional informaion, ince buinee uuall ave conien pricing policie; i oul be beer o obain quoe for ifferen prouc, or o ample a ne buine. Ti i epeciall imporan if e ample ize in erm of number of price quoe i fie or conraine. Small-cale uie of e effec of i ampling can be mae b enumeraing e prouc manufacure b a buine, elecing a probabili-bae ample, an en looing a e price movemen over a or perio in comparion o e eiing ugemenal ample. Ti approac i epenive in collecing aiional informaion an forming e prouc li o ample from. Te UK i in e proce of raniion from a ugemenal ample o a ample bae on i concep. Li of prouc ale a e eaile (8-igi) level of e PRODCOM claificaion are obaine a par of e PRODCOM urve for a (probabili) ample of buinee from e IDBR. Tee ill en be ue o form a frame from ic ampling of 8-igi prouc can ae place accoring o a probabili mecanim in e PPI, giving a o-pae eign. Tere i ill an iue of ic prouc o cooe iin an 8-igi eaing, bu a lea e buine-prouc pair ill be elece b a probabili mecanim from e PRODCOM ampling, an appropriae eiging can be ue o give a eign-unbiae eimaor of e populaion PPI. Te fir age in e inroucion of i eign i unera in e UK, an reul comparing e curren ugemenal em (ic alo ineri man caraceriic of a previou volunar urve) an e ne probabili-bae em are epece aroun April

82 Tere are paricular problem i e prouc of ome inurie ic ma mae ugemenal elecion of a repreenaive prouc eremel ifficul. In e cloing inur, for inance, iem an faion cange on a eaonal bai, an geing a coninuou price quoe for a ranien line i impoible. Tu ere ill be a enenc o elec coninuoul-prouce prouc, even en ee o no accurael repreen e overall price movemen uner e appropriae eaing. In a imilar a i mig be epece a pical raer an repreenaive prouc are ienifie, an a for i reaon minori proucion (ic mig ave a more volaile price) ma be mie. Ti i ver ifficul o ae: e informaion require i abou e proporion of ereme price movemen, ic require a large ample for eimaion. Hoever, in cae ere prouc ienificaion inrucion ra aenion o i problem, i oul be noe a par of e quali aemen a i ma be an iue. Some aemen of e quali of a ugemenal ample can alo be mae uing e moelbae approac b invoing e ignorable ampling aumpion (ee Caper an 9). If e aume (probabl falel) a e ugemenal ample i approimael repreenaive, en e can calculae e variabili of price in prouc caegorie (cooing a iger or loer level epening on e ample ize available o a o obain a reaonable eimae). Ti elp o ae e ampling variabili of e ugemenal ample, an b reallocaing e ample uing a Neman-pe allocaion an calculaing e epece variance (noing a e epece variance i maller an a ill be acieve in pracice becaue i ue e ame aa for allocaion an ampling variance eimaion), e o can be compare. Ti approac a been aope in e opimiaion of e UK CPI, ere for eample e number of quoe for poaoe a increae becaue of e variabili inuce b e ig price of impore ne poaoe a cerain ime of e ear. 4.5 Cu-off ampling Once again Särnal e al. (99) i a goo ource for a imple ecripion of cu-off ampling. A in Secion 4. le e N uni in e populaion U be inee b, an efine π a e probabili a uni i coen in e ample. Probabili ampling require a π > 0 for all U. Tere are ampling meo in curren ue a emplo probabili elecion i π > 0 for par of e populaion U, erea π 0 for e remainer of U. Suc meo ae an inermeiae poiion beeen probabili ampling an non-probabiliic elecion i π a are unnon rougou e populaion. One of ee ecnique i cuoff ampling. In cu-off ampling ere i a uuall eliberae ecluion of par of e arge populaion from ample elecion. Ti proceure, ic lea o biae eimae, i uifie b e folloing argumen: (i) a i oul co oo muc, in relaion o a mall gain in accurac, o conruc an mainain a reliable frame for e enire populaion; an (ii) a e bia caue b e cu-off i eeme negligible. In 75

83 paricular, e proceure i ue en e iribuion of e value,, i N igl ee, an no reliable frame ei for e mall elemen. Suc populaion are ofen foun in buine urve. A conierable porion of e populaion ma coni of mall buine enerprie oe conribuion o e oal of a variable of inere (for eample, ale) i moe or negligible. A e oer ereme, uc a populaion ofen conain ome gian enerprie oe incluion in e ample i viruall manaor in orer no o ri large error in an eimae oal. One ma ecie in uc a cae o cu off (eclue from e frame, u from ample elecion) e enerprie i fe emploee, a five or le. Te proceure i no recommene if a goo frame for e ole populaion can be conruce iou eceive co. (See Sugen & Smi (984) an Haan, Opperoe & Scu (997) for more on cu-off ampling.) A an illuraion of e in of aa for ic cu-off ampling mig be ue, conier e annual UK Annual Buine Inquir (ABI) urve, ic eimae curren emplomen, urnover, an value ae bae on a ample coen i e ai of regier emplomen an urnover (Table 4.8; e regier conain informaion from 3-6 mon before e urve). ABI raifie on inur (b 3-igi SIC), region ( caegorie) an regier emplomen, over-ampling large companie (compare e ra-mean an eige-mean column in Table 4.8 o ee o arp e over-ampling i). Te ample eig require o compenae for i varie in 996 from o 7.9 i a mean of We can ue e ample of ize,737 an,453 in 995/96 a e bai of an eercie in ic (a) imulae populaion are creae an (b) cu-off ample are coen from ee populaion, o eplore e biae a reul from ignoring or moelling e malle companie. Variable Ra mean Weige mean Regier emplomen Reurne emplomen Regier urnover ( 000) 33,49.4,307. Reurne urnover ( 000) 3, ,757.5 Table 4.8 Variable available in e anali of e UK ABI urve preene ere (value are from e 996 ample). Reurning o e quoe from Särnal e al. (99), Le U c enoe e cu-off porion of e populaion an le U 0 be e re of e populaion, from ic e aume a a probabili ample i elece in e normal a. Te ole populaion i u U U 0 U c. Eac elemen in e cu-off porion a zero incluion probabili; a i, π 0 for all U c. Le 0 be an 76

84 eimaor of e ole oal 0 U, for eample, 0 0 U 0 π. Ho can i be acieve?. Bu e nee an eimaor of Te o poible coure of acion in i iuaion are evienl o ignore e cu-off uni alogeer or o r o eimae eir conribuion o e oal. In e ne o ubecion e conier eac of ee poibiliie in urn Variaion : Ignore e cu-off uni A Särnal e al. (99) noe, in i variaion, ic i equivalen o eimaing e oal acro e cu-off uni a zero, Te aiician ma be illing o aume a T c U i a negligible porion c of e ole oal. If 0 b ielf i ue o eimae, e relaive bia i U ( ) E 0 c 0, (4.4) ic i negaive bu negligible uner e aumpion. We aume a i an ala poiive variable. Coninuing e ABI eample above, conier a given inur, i an oucome variable uc a urnover, an uing a pro variable for urnover uc a number of emploee. One a o efine e cu-off uni U c i b (a) oring all companie in e regier on emploee number, obaining ( ),..., ( N ), ere ( ) i e malle number of emploee; (b) calculaing e cumulaive um of emploee number from e malle o e large N companie, obaining S (),..., S J S N ( ),..., ( ); an (c) cuing off all e companie for S ( ε ) S, for ome mall ε uc a Here i i a oug e N populaion of inere i efine o be u e op 00( ε )% companie in emploee number. Probabili ampling from e reuling e U 0 of non-cu-off companie coul no be uneraen, a Särnal e al. (99) menion, or complee enumeraion of e value in U 0 coul occur. A raeg relae o e one u ouline oul be o impl efine e populaion of inere o be all companie i (a) 5 or more emploee, ample from e companie i (a) 5-00 emploee, an aemp a full enumeraion of e companie i more an 00 emploee. Here one poin of ignoring e in companie b efiniion i a la prevening e governmenal urve buren on mall companie from being oo grea ma mae i impracical or impoible o ge aa from em in an cae. Hoever, b cooing ε appropriael an over-ampling i ufficien vigour on e large companie (efine b emploee number), i approac i een o be a cloe approimaion of e meo in e previou paragrap, on ic e focu belo. 77

85 To eimae e bia ariing from variaion of cu-off ampling, for eac of everal value of ε e repeael (00 ime) (a) re a ample of ize,453 (e 996 ABI ample ize) i replacemen from e ABI aa bu i unequal elecion probabiliie eermine b e ampling eig, o creae a peuo-populaion reflecing e acual iribuion of UK companie (i i a in of eige boorap; ee Efron & Tibirani 993), (b) ue e regier emplomen variable in i populaion o cu off e loer 00ε% of e companie (b cumulaive emploee number, a ecribe above), an (c) eimae e oal reurne urnover b e oal acro e companie no cu off. To focu on bia iue e are u emploing e raeg of full enumeraion iin U 0. ε Relaive bia, in % (SE) Maimum bia, in % Average emplomen of buinee cu-off (SE) % of buinee cu-off (0.6) (0.6) 75.8 (0.) (0.) (0.3) 66.7 (0.) (0.08) (0.) 53. (0.) (0.04) (0.) 3.5 (0.) Table 4.9 Simulaion reul from cu-off ampling e 996 ABI aa, bae on 00 imulaion repeiion (SE Mone Carlo anar error). Te ample ize in eac cae a,453. Table 4.9 preen a ummar of i imulaion eercie. (Reul i larger ample ize of 5,000 an 0,000 ere viruall ienical.) To inerpre e reul in e able, conier e ro for ε 0.0 (a i, uing a 0% cu-off). Acro e 00 imulaion replicaion, e average amoun b ic e cu-off eimae fell or of e oal acro all,453 companie a.5% of e rue oal, an e maimum uc relaive bia acro e 00 replicaion a 6.6%. On average e cu-off companie a abou 54 emploee or le, an uc companie mae up abou 76% of all companie. I can be een from e ε 0.05 ro in e able a, i aa of i pe, cuing off e 5% malle companie (in erm of oal emploee in e regier) lea o a onar bia of abou 3% in oal urnover, ile alloing e ampling proce o ignore abou a ir of e companie. Weer a bia of i magniue i accepable epen on e cone. In pracice e ucce of i variaion of cu-off ampling varie rongl i ε, in a populaion- an problem-pecific manner. For inance, e icuion u far a empaie e eimaion of e level of, for eample, urnover a one poin in ime raer an e cange in urnover level over ime. Wen e main aim i o eimae cange, e proporion of cu-off uni in e populaion ma be aen o be iger (for a given bia olerance) an in e cae of a level, becaue ome of e bia oul cancel in e ubracion unerling e cange eimae. To illurae i poin, e replicae e anali of Table 4.9 on bo e 995 an 996 ABI ample, repeael (00 ime) creaing 78

86 peuo-populaion for eac ear an recoring e abolue an relaive biae from ignoring e cu-off uni in 995, 996, an e cange from 995 o 996. Table 4.0 preen e reul of i econ imulaion. Column 6 an 7 (couning from e lef) in e able eibi e epece bia cancellaion, in abolue an relaive erm, in eimaing e cange from 995 o 996; for eample, a ε 0.5, biae of 7-9% in e Bia 995 Bia 996 Bia ( ) ε Abolue Relaive (%) Abolue Relaive (%) Abolue Relaive (%) 0.0-4, ,53 -.3, , , , , Table 4.0 Abolue (in M) an relaive (in %) bia reul from ignoring e cu-off uni in eimaing e 995 an 996 oal urnover value, an e cange from 995 o 996, in e UK ABI urve. Te 996 reul iffer a bi from oe in Table 4.9 becaue a ifferen ranom number ee a ue in eac cae. iniviual ear are reuce o 5% en e cange from ear o ear i e quani of principal inere Variaion : Moel e cu-off uni Te oer leaing approac o eimaing populaion oal i cu-off ampling i o r o eimae e conribuion o e oal provie b e cu-off porion of e populaion U c. A Särnal e al. (99) pu i, A econ approac i o ue a raio aumen for e cu-off. Le be an auiliar variable, for eample, e variable of inere meaure for e enire populaion a an earlier ae, or ome oer non variable rougl proporional o e curren variable of inere. Le an le U 0 R U (4.5) 0 U 0 R U 0 S S 0 0 π π (4.6) 79

87 be e [eign]-conien eimaor of R U 0, bae on e probabili ample from U 0. To een e concluion o e ole populaion, an unverifiable aumpion i necear. Aume a RU 0 R U ell, an b raio aumen e arrive a cu U U. Ten RU can erve o eimae R 0 U a off RU (4.7) 0 U a an eimaor of e ole curren oal U, auming U or a cloe eimae of i i available. Te relaive bia i approimael ( ) E cuoff U 0 R R U, (4.8) ic can be poiive or negaive. I i zero if e aumpion R U R U 0 ol. Ti aumpion i one a e aiician ma be more incline o mae an e aumpion in e fir approac a c i negligible. Ti raeg i bae on raio eimaion, bu i i no e onl opion: raio eimaion i equivalen o regreion eimaion i e preume regreion line going roug e origin (ee Cocran 977), an one ma ue regreion eimaion iou e inercep being u rerice. Moreover, a e ill ee in Caper 9, e regreion eimaion coul occur eier on e ra cale, for bo e an variable, or on e log cale. Euroa (997:04) conain anoer eample of Variaion : Wen uni are elece i cerain folloing a rucural auiliar variable, uc a earl value ae, a more opiicae inicaor coul be buil uing an economeric moel in orer o eimae e effec of enerprie no elece. Te approac in i variaion i no aen in mo or all EU regulaion involving cu-off ampling (Euroa 997:06, 997:07). Becaue of i epenence on moelling aumpion e popone furer icuion of i variaion o Caper 9. In bo variaion, one problem i a legilaion ma a one oul obain aa from e companie proviing e op (a) 95% of curren emplomen, bu in fac pa emplomen i picall ue (aever i e mo curren figure, ic ma be anere from 3-6 mon o - ear ou of ae, epening on EU Member Sae) a a pro. Te erioune of i problem naurall gro i e gap in ime beeen curren an regier emplomen. 4.6 Concluion We conclue i caper i a e of recommenaion for eac of e non-probabiliampling iuaion eamine in e ecion above. 80

88 Recommenaion: Moel reporing in buine urve involving volunar ampling oul Acnolege eplicil a volunar ampling a been ue; an Preen eimae an uncerain aemen bo i an iou poraificaion on e mo imporan available covariae, o a conumer of e anali can ee bo (a) eer e agree a all relevan covariae ave been accoune for an (b) e irecion an magniue of e bia aumen. Recommenaion: Moel reporing in buine urve involving quoa ampling oul Acnolege eplicil a quoa ampling a been ue; Preen proviional eimae an uncerain aemen a if e aa a been gaere uing raifie ranom ampling, i e ame raificaion gri a a ue o efine e quoa; an Preen evience, if available, emonraing a e quoa ample iin e cell of e gri provie approimael unbiae eimae of e populaion mean in oe cell. Ti evience coul ae e form of eniivi anale oing a e reul of principal inere are lile cange en raificaion i repec o aiional plauibl relevan variable i uneraen. If no uc evience i available, e quoa ampling eimae an uncerain aemen oul be preene i an eplici aemen a e unbiaene of e eimae cell mean a no been concluivel eablie. Recommenaion: Moel reporing in buine urve involving ugemenal ampling, for eample, in e creaion of proucer price inice, oul Rouinel ee an preen evience a ugemenall pical prouc are in fac repreenaive of acual price movemen, an Perioicall calculae e variabili of price in prouc caegorie bae on an aumpion a e ugemenal ample i approimael repreenaive. Recommenaion: Moel reporing in buine urve involving cu-off ampling iou an aemp o eimae e conribuion of e cu-off populaion uni (variaion in Secion 4.5.) oul Provie evience, of a imulaion naure or oerie, a e percenage of populaion uni cu off an ignore lea o accepabl lo bia i problem an populaion imilar o oe currenl uner u. 8

89 Par : Non-ampling error 5 Frame error 5. Inroucion Eva Elver 3, Saiic Seen Among e non-ampling error a conribue o e overall inaccurac are frame error, o be ecribe in i caper. Te conrucion of a frame i one of e fir ep in e proucion proce an eenial for e ep o follo. Te frame mu, of coure, be efine i regar o e final goal, e reuling aiic. Tee are eimae of finie populaion parameer (FPP). Ingreien in uc parameer are aiical meaure (oal, mean, meian, ec); variable (proucion, number of our ore, ec); uni (enerprie, in-of-acivi uni, ec); omain (ub-populaion, for eample efine b a anar claificaion lie NACE Rev. ); reference ime; bo uni an variable value relae o pecific ime. Te reference ime are mol ime inerval, lie a calenar ear, a quarer, or a mon. Hoever, ome variable ma refer o a poin in ime, for eample e aring poin of e perio. Uuall reference ime agree for all variable an uni in a FPP. Ti mean, for eample, for monl aiic a e elineaion of uni oul refer o e curren mon. I follo from e above a uni, claificaion, oer auiliar variable, an reference ime are eenial o aiic an o alo o e frame. Te empai ere i o be on e aemen of quali, bu ome bacgroun i necear. Secion 5. eal i a Buine Regier an i ue a frame a founaion iou ic e aiic can arl be buil. Secion 5.3 ecribe frame an arge populaion. Te accurac o be meaure epen on e frame bu alo on eimaion proceure; Secion 5.4 ecribe ome iuaion. Secion illurae; oing aminiraive ource, ime ela an frame conrucion, an frame ifference an quali aemen meaure. Tere are ome ummariing concluion in Secion A Buine Regier an i ue a a frame 5.. Uni, elineaion, an variable Te abbreviaion SIC ill be ue for convenience for Sanar Inurial Claificaion, meaning NACE Rev. an ofen referring o e primar acivi. 3 Man peron ave conribue i aa, eample, an commen, epeciall Pär Lunqvi a Saiic Seen, an Ole Blac, Jon Perr, Ian Ricaron, an Mar William a ONS, UK. 8

90 A Buine Regier i ere in agreemen i e Council regulaion No 86/93 on raing up buine regier for aiical purpoe, an, ence, alo in agreemen i e Council regulaion No 696/93 on aiical uni regare a a aabae i a e of uni; a lea enerprie, legal uni, an local uni; a e of variable o eac uni; uc a SIC coe an ize, for eample e number of emploee; a e of ime amp (eplici or implici); a lea e ime of regiraion for upae; lin beeen uni, i ime amp. Te BR buil on aminiraive informaion, inveigaion of i on, an informaion from aiical urve. Noe a urve feebac a o be ue i care en ampling i co-orinaion over ime in orer no o ior e ranomne of e ample, ee Olon (995). Te informaion in e BR i a recen a poible. Ti goe bo for eac variable an for e elineaion of uni. Te elineaion refer no onl o ingle uni bu alo o informaion on lin beeen uni, for eample lin beeen legal uni an enerprie. Ho recen e BR informaion i varie beeen variable an alo beeen uni, epening on upaing proceure. Te BR o eac uni i i SIC coe, ize meaure, lin o oer uni ec. Variable on a iger level in e ierarc of uni are in man cae erive b aggregaion from a loer level, for eample number of emploee an SIC coe. Some variable ma, oever, no be available on a lo level, for eample urnover connece o VAT (value-ae a). Te coice of ic variable o pu on e BR oul conier bo e uni level an e uefulne a auiliar variable in ifferen proceure (for upaing, creaing frame, eimaion ec). 5.. Upaing e BR uing everal ource Some pical eample of upae are a follo. Informaion on bir an ea arrive from aminiraive ource regularl i non frequenc. Te ime-lag beeen an acual even an en i i recore ma be ifferen for bir an ea. For eample, e imelag from e fir paing of VAT o bir in e BR ma be or in comparion i e imelag from ceae acivi o regiraion of ea in e BR if a i bae on a e-regiraion a fical auoriie. A urve ma eec e no-acivi ae muc quicer an e BR e ifficul for e urve ma be o iingui beeen i ae an nonrepone. Te informaion available on an enerprie a i bir in e BR ma be fairl limie, an i uuall ae ome ime before i a an aequae ize an SIC coe. For cerain aiic, for eample on invemen in fie ae (invemen for or in e folloing), an earl eecion of ne acivi i imporan. A e ime e invemen i mae, ere are liel o be fe emploee an e uni ma no e ave urnover. Hence, i i eirable o fin aiional ource of informaion ic o uc aciviie a an earl age. I i imporan a ee ource are conien over ime an pace. 83

91 Te ource of e BR for bir, ea, an upae coul be PAYE (i abbreviaion ill be ue in e folloing for aminiraive informaion from e collecion of ae on earning, ic inclue emplomen) an VAT. Te BR ma ave a urve of i on, for eample concerning uni, lin beeen uni, an claificaion. Wen an upae i mae, no onl a cange of e value i mae, bu ere i alo a noaion a o ime. Te imple ing i o noe e ime of regiraion. Tere oul preferabl be alo a ime of occurrence. A ne SIC coe ma for eample be regiere in Februar 998 bu be vali from Januar 996. Te ime i poibl non implicil from e ource. Time amp a o e informaion an e are valuable in emograpic uie, bu e alo mae e anling more comple. Te ue of everal ource mae i necear o ave ome ienificaion. Tere ma, for eample, be an ienificaion number (i.nr) for legal uni ue b fical auoriie, a i, e BR obain VAT aa b legal uni i.nr. Some ienificaion i necear no onl o upae bu alo o merge informaion from ifferen ource. Suc merging i imple if ere i a unique ienificaion number common o all ource. Ti i, oever, rarel e cae. For eample, ere are ifferen i.nr in German an Irelan for e o aminiraive aa e regaring VAT an PAYE, maing i necear o merge e informaion b name an are. In Seen, ere i a inge number for a legal uni, bu an enerprie coniing of everal legal uni ma cooe o repor VAT an PAYE aa for one an e ame acivi a belonging o ifferen legal uni. Ti mean a e legal uni number are no ienificaion number in e ene of buine acivi. Te UK eperience i a i a foun buine rucure o be comple an bae on aminiraive proceure a are no ala uiable for aiical inquirie. Te VAT uni i ere o faciliae e collecion of VAT, an i ma no be able o provie e urve informaion require. Alo ome emploer mainain eparae PAYE em for alarie an non-alarie orer, giving o aminiraive uni an maing i necear o merge informaion from e o em en upaing e frame. Te above eample o a uplicae can eail arie on e BR unle couner-acion are aen ince a ingle acivi ma lea o everal bir roug ifferen aminiraive ource Te BR a a frame uni, variable an reference ime Conier fir uni for ifferen purpoe in ifferen par of e proucion proce. Sampling i performe in one or poibl more age i a ampling uni a eac age (for eample a ingle age i enerprie a e ampling uni). Te aa collecion i aree o e reporing uni (for eample e enerprie roug a queionnaire) or more generall o e ource of informaion (ic coul be an aminiraive regier). Te obervaion of e aiical urve are ie o e obervaion uni. Te reporing uni can be equal o e 84

92 obervaion uni or be ifferen: an enerprie a e reporing uni an a in-of-acivi uni a e obervaion uni provie an eample of e laer cae. Noe: Te erminolog i no unique; collecion uni i omeime ue for reporing uni, an reporing uni i omeime ue for obervaion uni. I i imporan o conier e omain of eimaion en cooing uni. Te obervaion uni oul no cu acro everal omain; for eample an enerprie coniing of everal in-of-acivi uni oul no be e obervaion uni for aiic a are bae on inof-acivi uni, o-calle funcional aiic. Here, e empai i on Srucural Buine Saiic (SBS) an Sor-Term Saiic (STS), i uni of e FPP being: enerprie for SBS, enerprie for par of e STS, inof-acivi uni (KAU) for par of e STS, an en poibl alo legal uni, local uni, an local in-of-acivi uni. Te BR (a efine ere) i uc a ere i an agreemen beeen e regier uni an e uni o be ue in buine aiic. Te ep from e BR an i uni o a frame populaion i en principall or an imple. I involve maing a li of uni i regar o SIC coe an poibl alo ize; variable a are available in e BR. Te mo pronounce principal ifficul ma be e in-of-acivi uni, epening on eer i i inclue in e BR or no. Ti uni coul alernaivel be creae a e aa collecion age (KAU from enerprie, an local KAU from local uni). Srui & Willeboore (995) icu uni an cange of uni. 5.3 Frame an arge populaion 5.3. Targe populaion A ae, e arge parameer ave e reference ime for bo uni an variable equal o e curren mon/quarer/ear. Te arge populaion coul for eample be all enerprie or all in-of-acivi uni in e manufacuring inur ic are acive in e curren perio Frame, an frame populaion Ieall ere i a perfec frame ic li ever uni in e arge populaion once an onl once ogeer i baic eign variable. In reali e frame i affece b variou imperfecion for everal reaon, for eample ime ela an coing miae. For buine aiic, lie SBS an STS, e frame i normall bae on a BR. Te frame populaion for a paricular urve i bae on e arge populaion of a urve. I i normall epree in e ame a a e arge populaion, a i, in erm of uni, SIC coe, an poibl ize; for eample all enerprie in e manufacuring inur. I ue e informaion available in e BR, an i ma pu on rericion, for eample a e enerprie inclue are acive en e frame i conruce. An annual urve collec aa afer e reference ear, an a or-erm urve collec aa uring e ear (orl afer eac mon/quarer). If e frame i conruce orl before 85

93 ening ou e queionnaire, a ime i a e en of e ear for e annual urve, an orl before e reference ear for e or-erm urve. Te laer ma ae furer ample uring e ear. Ano, e frame error are ifferen for ee o e of aiic unle e annual aiic eliberael ue e ame frame a e or-erm aiic for e ae of agreemen, compare Caper 0. Te frame populaion i bae on e informaion a i available a a ime. For or-erm aiic regaring ear, e SIC coe refer o ear ( ) a be more liel o ear ( ) or poibl even earlier, epening on e proucion ime of e aiic ue an e frequenc of upaing. In e cae of e manufacuring inur i normall epen on en PRODCOM informaion become available. Noe: PRODCOM i or for e Frenc or Proucion communauaire meaning Communi proucion Difference beeen e frame populaion an e arge populaion Tere are o pe of ifference beeen e frame an arge populaion: ifference for e populaion a a ole; ifference iin e populaion, affecing omain (ub-populaion). Anoer a of epreing i i e claificaion of for eample an enerprie ino urve or ino omain iin a urve. (Ti coul be manufacuring veru ervice inurie, an inurie iin e manufacuring inur, repecivel.) Toe o cae ill be eal i in Secion an 5.3.5, repecivel. A par of e arge populaion ma eliberael be lef ou of e urve, for eample enerprie belo a cerain ize ma be cu off. Te eimaion for i par of e populaion a o be bae on moel aumpion, ee Caper 4 an 9. Aminiraive aa ma be ueful, epeciall if ere are variable rongl relae o oe of e aiic. A ifferen claificaion of frame error i i repec o e ime i ae unil e are correce. Some are impl ue o ime ela in e informaion from ifferen ource. Suc error can be evaluae afer upae. Oer error are eier eece in pecial circumance lie a urve or a cange incluing a informaion or (more or le) never eece. Toe error can arl be uie; a e lea e require pecial inveigaion. Small uni epeciall ma be ubec o an error for a long ime. Te upaing proceure ma omeime be el bac eliberael, a menione above in Secion 5.3. for coerence beeen or-erm an annual aiic in ome Member Sae. Anoer eample i for or-erm aiic uing e ame e of claificaion an ize meaure uring e ear, ue in e UK in orer no o a e effec of re-claificaion o e iin-ear-cange. Bo raum an omain are frozen, ee furer Secion 5.6. an

94 5.3.4 Uner- an over-coverage of e populaion Tere are o pe of eviaion beeen e frame populaion an e arge populaion: uner-coverage: uni belonging o e arge populaion bu no o e frame populaion over-coverage: uni belonging o e frame populaion bu no o e arge populaion Tere i an ammer beeen e o. A conequence of uner-coverage i a obervaion are no collece for a par of e arge populaion. Ti ma impl a bia in e aiic. Over-coverage mean a reource are ue on uninereing uni. Te overcoverage ma be regare a an era omain of eimaion, an one of e reul (in comparion i no over-coverage) i an increae in uncerain en eimaing e regular omain. If e uni memberip of e arge populaion i no cece, ere ma be a bia. For bo uner- an over-coverage, e reuling inaccurac epen on e amoun of e coverage eficiencie, e abili o eec em, an e couner-acion aen in e eimaion proceure. Furermore, ere ma be pracical ifficulie in iinguiing over-coverage an uni nonrepone. A uni ouie e arge populaion a receive a queionnaire ma be more or le incline o reurn i an a uni belonging o e arge i i ea o reurn, bu on e oer an ere eem o be no reaon o fill in e queionnaire. Some queionnaire ma be reurne b e poal auoriie becaue e are i no longer vali a oul, of coure, be folloe up. See Caper Difference iin e populaion Te reaoning a a ue in e previou ecion for e ole populaion i o ome een alo vali for eac ub-populaion. Hoever, uner-coverage of one omain i overcoverage for anoer. Tere are ome ifferen poibiliie ere for coverage eficiencie: remain uneece (for eample an erroneou SIC coe remain) eece for e ample (or more accurael for e reponing uni; for eample e number of emploee in e queionnaire) eece on e populaion level (for eample a general upae of SIC coe beeen ampling an eimaion) Again, e reuling inaccurac epen on e amoun of e coverage eficiencie, e abili o eec em, an e couner-acion aen in e eimaion proceure Some commen on frame error Even if e conrucion of a frame populaion i ea in principle, ere i muc or in pracice i e BR an e frame i regar o bir, ea, organiaional cange, conraicor piece of informaion, uplicae, miae, ienificaion problem, ime ela, ec. Ienificaion i imporan, for eample o eliminae uplicae ue o ifferen ource. Arcer (995) ecribe e mainenance of buine regier, incluing ome 87

95 eample from Ne Zealan. One aemen mae i a ienifing bir picall involve a quarer of e oal reource neee. A cloe co-operaion beeen e BR an e aiical urve uing i a a frame i imporan. Ti inclue an uneraning on bo ie of e ifferen ue. I alo mean a lo of or on ingle cae o anle em correcl bo over ime an in ifferen urve, for eample in cae of reclaificaion an reorganiaion. Paricular care i neee i large enerprie group ic ave comple rucure an pan everal ifferen aciviie. Suc eniie ma cu acro ifferen urve, an e rucure are ubec o cange. I i imporan a e are moniore cloel o a cange can be pice up quicl an anle conienl. In e UK ere i a Comple Buine Uni o i en. A number of oer counrie ave a imilar organiaion, ome of em alo being reponible for all urve aa collecion. In e icuion of quali aurance for buine urve b Griffi & Linacre (995), frame creaion, mainenance, an monioring i an imporan par, incluing illuraion of bir, ea, an ime lag. Te erm frame error i no ala a correc ecripion coverage eficienc i ofen more aequae, oing e conequence an no u blaming e frame, for eample for no aving inclue merger in Januar 998 in a frame conruce a e en of Defining a Buine Regier covering a ime perio Te arge populaion a reference ime for e uni a equal oe of e variable, a menione above. Ti mean, for enerprie an annual aiic for eample, a e enerprie inclue oul no be oe a are acive a e ime of e frame conrucion bu all enerprie a are acive uring e ear, eer acive e ole ear or uring a par of e ear onl. If e frame i conruce a e en of e ear (ee icuion in Secion 5.3. an ), e enerprie miing in e frame are earl ea an lae bir, a i broal oe a are (i) no longer acive accoring o e BR bu ave been acive previoul in e ear, an (ii) no acive in e BR bu acive laer in e ear. Moreover, i SIC coe referring o a ifferen perio an e arge calenar ear, ere ill be miclaificaion. Ti o e frame eficiencie affecing aiic unle acion are aen. A pecial BR i e purpoe of uc acion i inrouce belo. A ome poin afer e calenar ear i i poible a lea in principle an if e informaion neee a been ep o combine informaion from e BR incluing ime amp, an poibl alo from oer ource, o erive a ne Buine Regier a refer o e calenar ear. In e cae of enerprie, i inclue all enerprie a ave been acive a ome ime uring e calenar ear. Te value of e variable alo refer o e full ear. If e baic value ave reference ime a are poin in ime, ome proceure i neee, perap a uiabl coen average of value before/uring/afer e ear. Te ame i poible 88

96 for a ifferen perio, lie a quarer, bu ue o e ime ela, uc a regier i le liel o be ueful. Seen a ome eperience of a BR covering a calenar ear an i ue, illurae in Secion I i en regare a e be nolege aaine. Saiic bae on i BR an anoer, previou verion are compare. Ti i one a o evaluae effec of frame error. Furermore, e improvemen of e accurac roug uing i BR oul be coniere ogeer i e effor involve, o ee if e effor i co-effecive. An orinar BR o e iuaion a ome poin in ime, lie a napo. Hoever, coniering a e rae of upaing varie beeen variable an uni, i i raer a miure of napo of e uni i regar o elineaion, SIC coe, ize meaure ec. 5.4 Te arge populaion: eimaion an inaccurac 5.4. Eimaion proceure an informaion neee A ae everal ime, e arge populaion a reference ime of baic variable lie SIC coe a are equal o oe of e aiic. For eample, bo annual an or-erm aiic referring o ear oul be bae on elineaion of uni an SIC coe of a ear. Te frame i bae on a BR a a ime oo earl o acieve i. Tere are everal poibiliie a e eimaion age, i ifferen ambiion for upaing e informaion an, a e ame ime, i ifferen reul a o accurac i repec o frame error (coverage eficiencie). Waever e proceure coen, e reuling (in)accurac nee o be meaure. A pical iuaion i a eign i raificaion b inur an ize. A ranom ample i ran for eac raum. Te greae ize raa ave e elecion probabili equal o one. Te raificaion ino e of SIC coe correpon o e omain, eac raum being equal o (or more eaile an) a omain. Size i ue in e raificaion o improve accurac. Te baic eimaor of e oal value of proucion, a, for a paricular inur i en impl a um over e ize group for a inur. Te variance of e eimaor i alo compue b umming over ee raa. Te eimaion proceure can ue a Horviz- Tompon eimaor, epaning ample value b invere probabiliie of elecion (in e cae of full repone), ee furer Caper. Ti i o for e ampling uni an i omain a given b e frame. Wi a ifferen obervaion uni, e conribuion o a paricular inur ill alo come from oer raa, for eample if enerprie are ample an eir in-ofacivi uni are e obervaion uni. Tere are furer poible eimaor, epening on a informaion i available in aiion o a in e frame. Tere are o main reaon o ue furer informaion: o reuce bia b incluing correcion an upae; o reuce variance roug uiliing auiliar informaion. Te amoun of furer informaion ma var: i can be limie o e ample or i can be available for e populaion, for eample in erm of furer variable or an upae BR. 89

97 Some iuaion are ecribe belo in Secion For eimaion proceure, ee Caper -3 or e lieraure, for eample for calibrae eig in generalie regreion eimaor ee Deville & Särnal (99) Uing e frame populaion onl Te imple eimaion proceure i o eep o e frame populaion, a i, eac uni eep i omain of eimaion a on e frame. A ecribe above, eac pair of poin eimae an anar error i compue b umming over e correponing raa. Ti proceure can be ue no onl for claificaion bu alo for uni a are in fac ea or oerie no belonging o e arge populaion, b reaing em lie nonrepone. If ere i no reneal of e ample, uc an eimaion proceure can be regare a incluing a moel aumpion on e relaionip of uner- an over-coverage: a e are equal in ize. Tere i bia ue o uner- an over-coverage for e populaion a a ole an for eac omain, unle e aumpion i rue. Wen e bir rae i ig compare o e ea rae, ere i uner-eimaion an vice vera. Care nee o be aen in uing implifie aumpion. Invemen provie a paricular callenge. Ne uni an one ic are groing are liel o be rong inveor. Converel uni ic are ruggling an, a a reul, iminiing in ize ill ave lile opporuni o bu ne ae. Elver (993) icue i for a urve bae on a cu-off ample i e rericion 0 emploee or more. An alernaive leaving e frame informaion o ome een i o ienif e overcoverage an pu variable value equal o zero for ee uni. If ere i no reneal of e ample, ere i en an imbalance, ince over-coverage bu no uner-coverage i aen ino accoun. Illuraion: Table 5.3-Table 5.4 in Secion o an eample of over- an unercoverage i a cu-off urve. Te bia ue o an ol SIC-coe i on for an eample in Figure 5. in Secion Upaing e ample onl If e uni in e ample ave eir omain cece in e urve, inerior movemen an correcion can be aen ino accoun b aigning eac ample uni o i proper omain of eimaion. Ti implie a e bia from i error ource i eliminae. Tere i, oever, an increae variance ue o incluing i informaion ic ma be a rare caraceriic bae on ample informaion onl. Caper 3 provie formula in i ecion on omain eimaion, for eample a imple cae in Secion 3... Tere ma, in fac, be quie a ifference in going from (i) e variance coming from a mall e of ailor-mae raa a inicae in Secion , o (ii) e variance erive from ee raa an ome furer raa ere a fe uni i acual value conribue o e variance ogeer i a large number of nil value. Ti i a conequence of frame eficienc. 90

98 Tere are alo eerior movemen/correcion, uni leaving an enering e populaion. An upae in e fir repec mean for eample ienifing over-coverage an giving i a nil value. Tere i en an ammer if no acion i aen for e uner-coverage, a ae in Secion Eier aiional ampling or moel aumpion are neee o eimae for uni no in e populaion originall ample, e frame populaion. A ver imple moel i o aume equal effec beeen over- an uner-coverage, bu i aumpion i onl liel o be realiic en e econom i able an no ala even en. For uni inclue i probabili one, cange can be mae iou affecing e variance, for eample reorganiaion can be aen ino accoun an claificaion upae can be mae, a long a eac uc uni repreen ielf onl. Hoever, care mu be aen if urve ino ifferen ecor are run inepenenl. For eample, if uc a uni i reclaifie from reailing o manufacuring, i coul be remove from e reailing urve. A econ acion nee o be aen a e ame ime o enure i i inclue in e manufacuring urve. Tere ma be ifficulie in oing i in pracice. Illuraion: Te increae in variance (or raer i quare roo) en upaing an ol SICcoe bae on ample informaion i on for an eample in Figure 5. in Secion Uiliing laer BR informaion on e populaion A iuaion i even more informaion i ere ere i a furer variable for all uni, no ue in e eign, or ere ere i renee informaion on e original eign variable. One eimaion meo i o-calle poraificaion, ere a raificaion variable i ae a e eimaion age. Te calibraion ecnique i an eample of incluing uc auiliar informaion (poibl quaniaive) o improve e eimaion. Ti ma lea o a reucion of bo bia an variance. I i a moel-aie eimaion meo a i ue for e urvee par of e populaion. Movemen of uni ino e populaion are no inclue in e proceure u menione. Te require moel-bae proceure i aumpion abou ee uni. Again, ere i an ammer o be overcome. Tere are illuraion of cange in SIC coe an number of emploee from one ear o e ne in Table 5. an Figure 5., repecivel. Table 5. a SIC coe for a orer perio Uiliing a BR covering e reference perio Te ecnique of conrucing a BR covering a perio a ecribe above in Secion Te arge populaion i ere coniere full non. Ti i, of coure, a implificaion, ince ome error ill remain. Ti BR i, oever, a conierable improvemen over e verion a e ime of frame conrucion. From e eimaion poin of vie, e iuaion i i BR covering e reference perio i rougl e ame a a in Secion in erm of meo an aumpion. Ti mean for eample a poraificaion an calibraion meo are available for inerior movemen. 9

99 Movemen ou of e populaion are ienifie, a i, e over-coverage i non. Te uner-coverage i alo ienifie. Te eimaion a o be moel-bae for oe uni (unle ere i ime for furer queionnaire), uing for eample imilar uni in e urvee par of e populaion an/or aminiraive aa. Again, e reaoning i bae on i lae BR covering a perio oing e ru; in pracice ere are, of coure, remaining eficiencie. In Secion 5.7.3, Table 5.3-Table 5.4 illurae over- an uner-coverage i a cu-off urve, an ere i informaion on e era uni provie b e BR covering e calenar ear Some commen on e BR an effec of coverage eficiencie Dicuion on e opic of quali of a BR are going on a e EU level (Euroa 998a). Te connecion beeen Buine Regier an e aiic uing em are geing ronger. Tere i an increaing inere in buine emograp, an regular or on quali aemen of buine regier i aing place a ome aiical office. See alo Srui & Willeboore (995), Arcer (995), an Griffi & Linacre (995), alrea menione, an illuraion belo. Te meauremen of inaccurac caue b coverage eficiencie ma be uneraen in ree ifferen a: ) Revie upaing proceure of e BR o loo a ime ela. Ti ill provie a broa inicaor onl, bu i i available a e ime en e frame i conruce. ) Compare uni on an upae BR i e BR ue. Coun can be mae of e number of uni erroneoul inclue or eclue. Lieie e number of uni claifie o e rong omain of eimaion can be evaluae. 3) Compue approimael e level of inaccurac. Eimae can be mae for e frame populaion an for e eimae arge populaion, uing a variable a i available a e populaion level (for eample urnover from VAT, or alarie an age or number of emploee from PAYE). Wil i meo provie e mo informaion i i e mo emaning an reource inenive. Te illuraion in Secion 5.5 are ie o e BR, an Secion provie a range of illuraion for frame, aloug nearl rerice o e UK an Seen. Mo illuraion in Secion 5.6 an 5.7 belong o e fir an econ of e above meo. Tere are, oever, a fe eample on accurac meaure in Secion 5.7 belonging o e ir meo. Ti i e preferable one, ince a quali aemen oul aim a e effec of frame error (coverage eficiencie). 5.5 Illuraion aminiraive aa an buine emograp Buine Regier are epenen on aminiraive aa an influence b aminiraive rule, ic ma var over ime an, of coure, beeen counrie. A an eample, a bir in e BR can ave ifferen caue: ere are pure bir in e ene of ne acivi, an ere are ne regiraion ue o a ne legal form or an enerprie 9

100 reorganiaion ino everal par ec. Accoring o a urve on ifferen caraceriic of ne Sei enerprie, abou 54 % of e 997 ne BR enerprie ere purel ne, ee SOS (998); e figure for e previou ear a 60 %. (Tee figure refer o enerprie i more an SEK (approimael ECU) in annual urnover, bu e urve alo cover maller enerprie.) Saiic Finlan (996) give imilar reul. Te percenage epen on e BR em, of coure, an i varie beeen counrie an over ime. Anoer a o u buine emograp i o uilie iniviual emplomen aa ogeer i e BR. A ecripion for Seen i given in Saiic Seen (995); e meo i ae o be a ranformaion of original iea from Denmar. Te epenence on aminiraive rule i illurae in o able. Te fir one o e number of uni in e Sei BR b ear, i ome commen on conierable cange. Year Number of acive Cange in Ta an VAT-rule in Seen legal uni From 990 inclue uni iou acivi coe Cange in VAT-rule Ne in of a (ome influence on 993 alo) Te ne able i a relae one from e UK. Te bai of e aa collecion b e ONS i e Iner-Deparmenal Buine Regier (IDBR), ic a inrouce in 994 an became full operaional in 995. Te IDBR combine informaion on VAT raer an PAYE emploer in a aiical regier compriing million enerprie, repreening nearl 99% of economic acivi. Te regier comprie companie, parnerip, ole proprieor, public auoriie, cenral governmen eparmen, local auoriie an non-profi maing boie. Te main aminiraive ource for e IDBR are HM Cuom an Ecie, for VAT informaion (pae o e ONS uner e Value Ae Ta Ac 994) an Inlan Revenue for PAYE informaion (ranferre uner e Finance Ac 969). Oer informaion i ae o e regier if require for ONS aiical purpoe. Ti able inclue informaion onl on VAT-bae enerprie. Noe: Te coun of buinee belo e VAT reol repreening volunar regiraion an i zero urnover are inclue in e o fir par of e able ( an ). Figure for e fir par are coun of iniviual legal uni. Coun for e econ par o VAT-bae enerprie coniing of one or more legal uni. Te ir par ( ) eclue uni i zero VAT urnover an all enerprie iou a VAT bai. Te GBP i currenl aroun.4 ecu. 93

101 Year Number of legal uni / enerprie Percenage cange in number Cange in VATregiraion ae an reol value in GBP % % % % % % % % % % % a above % % % Illuraion ime ela an aing frame 5.6. Te UK Buine Regier Te UK regier ol o claificaion an o meaure of ize. A curren value o e lae poiion an i ue o form e frame for e annual inquirie. A frozen value (upae onl a e ar of e ear, before Januar elecion, from e curren value a a ime) i aen roug e ear o enure conienc rougou e ear for ub-annual inquirie. Tu e annual frame relae o a laer perio an e or-erm frame, e UK concenraing on accurac for rucural aiic in preference o congruence i or-erm urve. Te regier i upae from a number of ource uring e ear: i. PAYE Upae. Tape are receive from e a auori ever quarer giving eail of ne uni, cloure an cange of rucure. ii. VAT Upae. A eel ape i receive from HM Cuom an Ecie conaining eail of bir (ne regiraion), ea (eregiraion) an amenmen. Enerprie i no local uni or PAYE uni ave an emplomen impue from e VAT uni urnover uing e urnover per ea figure appropriae o e claificaion. iii. Surve Informaion. Size an claificaion aa upae onl e curren claificaion. iv. Vii b e Comple Buine Uni (ee Secion 5.3.6). Tee are upplemene b e profiling iin e Buine Regier Uni. Te bir, ea, an rerucuring pice up from ee ource are acione immeiael. Claificaion an ize amenmen affec onl curren value unle a unle a buine i in e proce of being profile or a ignifican error i foun. Upaing of e regier ae place roug e ear from quarerl ource uc a PRODCOM, bu e main upae i in Augu from e Annual Emplomen Surve (o be 94

102 incorporae ino e annual rucural urve from 998). Te reul of e upae ill rive elecion for e ub-annual inquirie for e folloing ear. Te ource of informaion ue o upae e IDBR are lie b variable belo: I. Turnover. Te VAT aminiraive em i e main ource. Surve aa are ue from e iribuion ( rae ) an ervice ecor bu rarel from eleere. Enerprie i no VAT or urve informaion ave a urnover value impue from emplomen informaion. II. Emplomen. Te preferre ource i e Annual Emplomen Surve (o be incorporae ino a ne annual rucural urve from 998). Emplomen informaion come from e PAYE ( Pa-a-ou-earn ) a aminiraive em if Annual Emplomen Surve aa are no available. Enerprie i no emplomen informaion (eier from PAYE or from AES) ave emplomen impue from urnover. III. Claificaion informaion come from a varie of ource. Te folloing priori applie: A. Comple Buine Uni B. PRODCOM/Reail Inquir/Financial inquirie C. Annual Regier Inquir D. Sor Perio Turnover Inquir E. Oer buine urve F. Builer' Are file G. VAT H. PAYE Te annual regier inquir i a ne urve ic ill replace o-calle regier proving from 999. Te Builer Are File conain informaion on conrucion buinee from e Deparmen of e Environmen, Tranpor an e Region (DETR) conrucion inur urve. Care mu be aen en uing o aminiraive ource uc a PAYE an VAT o upae e BR o enure a erroneou informaion i no aen on an ue in proucing eimae. Wen a ne PAYE uni i ienifie i 0 or more emploee, an aemp i mae o mac i i a VAT uni or a local uni eleere on e regier. If no correponing uni i foun, e uni i en a regier proving form an eclue from all eimae unil i valii i confirme. Lieie a ne VAT uni oul be mace i PAYE, an proving uneraen if no correponing uni can be foun. Eenive macing i carrie ou for uni i feer an 0 emploee, bu ere i no proving for ee uni ue o reource an compliance conrain. Small unmace PAYE uni in VAT eemp inurie an corporae PAYE uni are ae o e regier iou proving. Te annual rucural urve ample are ran a e en of Ocober eac ear. Te orerm urve are ran namicall eac mon or quarer. Sample from e or-erm inquirie are ran from e frozen fiel il e annual inquirie elec from e curren fiel. Sample are raifie b inur an ize. Te meaure i uuall emplomen. Te 95

103 ize group for e annual rucural ample for e proucion inurie an for e Monl Proucion Inquir are on belo. Annual Proucion Monl Proucion Te Sei Buine Regier Ti ecripion refer o e mile of e 990, mainl before e EU Regulaion came ino Sei ue (Seen became a Member Sae in 995). Te Sei BR obain informaion on bir an ea from e Naional Ta Boar ever econ ee. Te number of emploee i upae roug everal ource. Te o main one are e Ta Paroll an a pecial queionnaire o muliple-locaion enerprie, bo once a ear. Tere i alo informaion from e urve of Saiic Seen. For Diviion 0-37 of e Sei SIC 99 a i armonie i NACE Rev. a e four igi level, ere i an annual urve rougl a e local uni level a i an imporan ource for e SIC coe (uing oupu informaion, incluing aa on commoiie). Tere i a moifie verion of e BR, calle e Saiical Regier (SR), ic i ue a e frame for buine urve. Some uni on e SR coni of a e of legal uni. Te are e malle one for ic balance ee an profi an lo aa can be obaine. Te are eenial o e Financial Accoun Surve, an e are inclue in oer frame for coerence. Tere are abou 60 uc large aiical uni, coniing of more an 400 legal uni. In e folloing, e erm enerprie ill be ue o mean uc uni enever e occur an legal uni oerie. (Ti enerprie efiniion i omea ifferen from e EU one. An enerprie inclue more legal uni in ome cae, an feer in oer cae; ere oul be furer enerprie i everal legal uni. Te number of uc enerprie a, oever, increae recenl.) In e ampling em, mo ample are ran in November (an ome in Ma). Te SR can en be epece o ecribe e iuaion a e en of Sepember a o acive enerprie an local uni. Te number of emploee refer o e pring i ear,, for muliplelocaion enerprie (BR queionnaire) an o December la ear, ear ( ), for inglelocaion enerprie (PAYE informaion). Single-locaion enerprie born in ear normall ave 0 emploee in e BR a ear. Hence urve a require a minimum of for eample 0 or 0 emploee o no cover bir in ear. Te ample obaine are ue for a ear b annual urve an for e ne ear b orerm urve (ome ampling i alo mae in Ma). All urve ue inur (e SIC coe) for raificaion. Mo urve alo raif b ize, an e ize meaure i mol e number of emploee. Te ize group in e urve ere are i, i e o op one 96

104 oall enumerae (a i 00 emploee or more). Te are bae on enerprie in Diviion 0-37 i a lea 0 emploee. Te alo inclue (rougl) local uni i a lea 0 emploee in Diviion 0-37 belonging o enerprie in oer Diviion for funcional aiic, bu a par of e populaion i iregare ere for implici. number of emploee: Some comparion beeen UK an Seen UK an Seen ave imilar rouine in everal repec, for eample in uing bo PAYE an VAT a ource, an b puing era empai on e BR quali aroun Ocober i regar o frame. Sample for annual urve epen on a frame, an o largel o orperio ample. Sraificaion b inur an ize i ue. Tere are alo ifference, for eample UK ue namic ampling for or-perio inquirie an Seen run urve i a cu-off limi. Tere are ifference beeen uni, for eample e enerprie concep an e een of appling e in-of-acivi uni. UK a a pecial eam for comple buinee, an Seen a a pecial BR covering a calenar ear. 5.7 Illuraion cange beeen frame an eir effec 5.7. Difference beeen UK curren an frozen claificaion Te mari in Table 5. o for e UK o enerprie are claifie on e BR in relaion o curren an frozen SIC claificaion. I reveal e een o ic e frozen claificaion i rong a one poin in ime (Sepember 998, folloing e ae-on of e 997 Annual Emplomen Surve (AES) informaion). I oul be remembere a or-erm inquirie elec from e frozen fiel for purpoe of conienc uring e ear. Te mari oul be inerpree in e folloing a: Ro: e figure a e en of e ro o e percenage of buinee a ave remaine in e iviion of eir frozen claificaion folloing e AES upae (an an oer informaion (for eample from PRODCOM) receive uring e ear). I alo o e een o ic buinee ill be reclaifie ou from an inur. Column: e figure a e boom of e column o e percenage of buinee currenl claifie o a cerain iviion ic ere claifie o a iviion in e frozen fiel alo. I alo o e een o ic buinee ill be reclaifie in o an inur. Te mari reveal a relaivel mall amoun of reclaificaion in erm of number of buinee i reclaificaion in or ou of le an 3 % for nearl all inurie. I oul be inereing o ee e anali carrie ou on emplomen oo. (Noe: I oul alo be more inereing o ave a full ear mari, bu i i no poible for 997 or earlier.) Te inur i e ige percenage of inar reclaificaion ore up i iviion 3. Here, 95. % of e enerprie in e curren fiel are alo in e frozen fiel, o 4.8 % (308 buinee) ill be ae en e curren fiel i copie over ino e frozen fiel. Te inurie ic provie e mo enerprie are iviion 3 an 33. Converel,. % of 97

105 Curren Sic9 % on Frozen Sic Toal iagonal Toal % on iagonal Table 5. Comparion of frozen an curren SIC-coe in 998 on o igi level 98

106 e enerprie i e frozen claificaion in iviion 3 ill be leaving e inur. Diviion 3 an 33 feaure again, no a e main einaion inurie. Te inurie i e large amoun (in percenage erm) of ouar reclaificaion aaie are iviion 40 an 4;.6 % an 4. % of buinee repecivel ill be leaving e inur. Due o e fac a ere are no a man buinee operaing in ee inurie i oe no repreen man buinee (37 an 5 repecivel) Difference iin e Sei populaion one ear apar Tere are four main aa ource ic can be ue o u cange among enerprie in Diviion 0-37 i a lea 0 emploee. Fir, e BR covering a calenar ear, ere 995. Secon, e frame for e or-erm urve, bo in November 994 an in November 995. Te frame for i urve i eeniall e ame a a of e annual urve, bu ue one ear earlier. Tir, e regier ere obervaion an impuaion from e annual urve 995 ave been ae for comparaive purpoe. Four, ere i aminiraive aa, PAYE an VAT. Te main file ue in i Secion are e frame from November 994 an 995, an e inclue enerprie i a lea 0 emploee in Diviion Hence, ifference beeen iuaion one ear apar are on. Te correpon o e frame for e orerm an annual aiic for 995. Te or-erm aiic largel eep eir claificaion, bu e annual aiic mae ne one, o e ifference in inur in e aiic ill be bae on aa o ear apar. Te file are a e enerprie level. Te SIC coe a five igi, e fif being a Sei aiion, ic i rarel ifferen from zero. Cange are for convenience uie b uing all five igi, iou regar o leer, maing ifference bae on e fir o igi a bi unequal. Table 5. o b ro o ic igi e SIC coe agree for enerprie in 994 an 995. Te column o ize group in 995. Nearl 500 uni ave a cange in SIC coe. Tere are no conierable ifference beeen ize group a o percenage of cange. Number of equal igi in SIC coe Size group for 995 (number of emploee) Toal Toal Table 5. Comparion of SIC-coe 994 an 995 i regar o ize 995. In eac cell, e upper figure o e frequenc, an e loer figure o e column percen.

107 Te cange ere are coniere fairl normal. (Tere i an ecepion for e iviion, i conierable cange i e ne SIC coe. If cange from 995 o 996 a been coen o overcome e SIC coe effec, ere oul ave been a greaer influence from collecing commoi aa in a ne nomenclaure an in a ne a.) Cange for large enerprie ill ave a conierable effec on iniuional (enerpriebae) aiic. Te effec for or-erm funcional aiic ma be fairl mall: if ere are o in-of-acivi uni i rougl e ame ize, e cange in primar acivi of e enerprie ma be caue b mall cange in relaive ize beeen e o in-of-acivi uni. Te effec on e o-igi-level of iniuional aiic are on in erm of abolue number on e verical ai of Figure 5., uing e neer number of emploee (from e 995 frame). Tere are 7 omain of eimaion. Si of ee omain are unaffece. To of em are affece b more an 5 %. A more eaile level i, of coure, more eniive. Legen: A ob, B ob, ec. abol. bia 000 A 500 A 000 A A 500 A A AA A A B A A 0 A B A B A Šƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ quare roo of e increae of e variance NOTE: 6 ob a miing value. Figure 5. Comparion of abolue bia ue o e ol SIC coe in e frame an e quare roo of e increae of e variance en upaing e ample onl.

108 Te orizonal ai of Figure 5. o e increae in e quare roo of e variance in uing e SIC coe of e ample inea of e SIC coe of e frame. Te ample ize a been erive b a imple Neman allocaion in e 994 frame i e preciion crierion % for e number of emploee. Te eail are lef ou, a e aim i u a imple illuraion. Te line in Figure 5. correpon o e o mean quare error being equal. Poin above a line (8 in number) correpon o inurie a ge a maller mean quare error if e bia i eliminae b upaing e SIC coe for e ample. For poin belo e line (3 in number) e increae in variance en ere are conribuion no onl from e ailor-mae raa i o large a e reuling mean quare error i iger an e original one. Legen: A ob, B ob, ec. No 95 B A B A A 00 A A A AE A A AA AB A A B A AA A A A AAA CEBA 80 A A A AAABAA A AAAAEGBA A A A AA B AABA A 60 A AABAA AB CACICAA B A A A A BAAB EHAA A AA A AAABAABDDA A 40 A BAAB CBBABAGCCA A A A B BAA ACABCIFC B AAA A AA BAABAAAACDIDAA A 0 BABAAAAABACDHDA A A A A A CEABABHEC AAA A A A A B A AAA A BADCGCCAA 00 AAACBAADIJGDAAA A A A A A BDADDEJEQQFB AA A A BAA B BCCDDHLOIBD AB A A 80 A AABDDEELGTQJC A A AA BA AAAEGFFJGHYNJBA CA A B B ACB GIGKPYKD CBAA 60 B BBDIGTRZZLIDCB B A AACFHRSTLZZMCA A A A CADBDKWZZZWMBCDB A A 40 A DAKMNZZZZZJAFA A AA A BBFQZZZZZZOBA A A ENZZZZZZKD 0 ZZZZZZBHD A ZZZVFEAB 0 Šƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ No 94 NOTE: 844 ob ien. 47 ob ou of range. Figure 5. Number of emploee in 995 veru a in 994, accoring o frame. 3

109 No o ize an cange in ize, fir illurae in a imple plo, Figure 5.. Te figure i rerice o e number of emploee being a mo equal o 00, a i i o e ample par of e populaion. A greaer prea of a urve variable iin raa can be epece en e raificaion i bae on e ize of e 994 frame an oul ave been e cae i e ize of e 995 frame. Te plo inicae a ome uni ill ave muc iger or muc loer value an e raificaion inicae. Te accurac of e eimae ill be loer an e oul ave been i a more up-o-ae ize. A relae illuraion i a cro-claificaion of e ize group in e o frame. Te reuling able (no inclue ere) o a omea more an 0% of e enerprie move upar or onar b one or poibl o ize clae. Te ear 994 an 995 ere uc a e movemen upar ominae over oe onar. To remain in e ame ize cla i, of coure, b far e mo frequen cae, een in omea le an 90 % of e enerprie Difference for e populaion a a ole; Seen Te aa file ue o u ifference for e populaion a a ole are oe menione in e previou ecion. Te BR covering 995 i in i cone coniere a e final reul. Te number of emploee i a convenien meaure of e effec. Tere are o iavanage, oever: ere i no conribuion from enerprie i no emploee an ere i a full conribuion from enerprie ic ere acive for onl par of e ear. A coun of e number of emploee in mall enerprie (le an 0 emploee) in e BR covering 995 o a 7.4 % of e oal number of emploee i ere, maing 55 ouan emploee belo e cu-off. Tere are 679 ouan emploee above e cu-off. Accoring o e o frame, e number are 639 an 657 ouan emploee, repecivel, e ifference being ue o bo ifference in uni an ifference in reference ime. Te over- an uner-coverage of eac of e o frame are on in Table 5.3 on e fir ro (bol ialic) an e fir column (bol), repecivel, in erm of number of enerprie. Te group belo inclue bo mall an non-acive enerprie. I oul be noe a e figure given are mainl e reul of a blin mac-merging. Enerprie a belong o e oall enumerae group on one occaion an e belo group on anoer are liel o ave gone roug ome re-organiaion, aen ino accoun b e urve. Looing a e annual urve, ere e BR covering a calenar ear i ue o prouce e aiic, e percenage of aiion relaive o e 995 frame are of e ame overall orer. Tere are ome ifference in proceure. Te uner-coverage foun in a urve i cece o avoi ouble couning. On e oer an, enerprie ma falel be roppe a an earl age a over-coverage an en reurn a uner-coverage. A e of enerprie of ancillar caracer i pice up from e Financial Accoun Surve. Ou of e 567 enerprie a ere in e BR covering 995 bu no in e 995 frame, ere ell above e cu-off bu in oer inurie, 09 ere no acive, an 358 ere belo e cu-off (6 of ee iou emploee). Conier no e ample par onl, bu in more eail. Fir in Table 5.4, over-coverage i 4

110 group in e group in e frame 994 group in e frame 995 BR cov. 995 belo ample o.enum. belo ample o.enum. belo ample o.enum Table 5.3 Over- an uner-coverage of e frame 994 an 995 on i number of uni an e number of emploee in ouan a meaure b e frame an b e BR covering 995. Ten e uner-coverage i on i number of uni, number of emploee accoring o e BR covering 995, an in relaive erm ummarie for ree variable: number of emploee, alarie an age, an urnover from VAT. Te figure ere refer o e ole of Diviion Te relaive effec on an inur level ma, of coure, be ifferen, larger or maller. over-coverage uner-coverage _ uni empl. 000 uni empl. 000 ree variable 994 frame 5 en en. 5 aroun 3.0 o 3.7 % 995 frame 76 en en. aroun 0.8 o.7 % Table 5.4 Over- an uner-coverage of e ample par, frame 994 an A fe ummariing concluion Te BR an e frame erive from i provie a funamenal bai o e aiic. Te frame populaion oul be efine i regar o e arge populaion, an e uni of e BR oul correpon o e aiical uni. Ti i in line i e EU regulaion. Obvioul, correc elineaion an claificaion of uni are imporan for e omain of eimaion. Size informaion i ofen ue o improve accurac; eficiencie in ize informaion ill mae e eimaion proceure le efficien an caue rouble i oulier ec. Te iincion beeen frame error an oer non-ampling error i no ala clearcu a meauremen error ma be relae o unclearl or erroneoul pecifie uni, an nonrepone an over-coverage are no ala ea o iingui. I i no onl e frame e BR a e ime en e frame i conruce ic i imporan, bu alo o a een e eimaion proceure ae laer informaion ino accoun. Ti i o bo for uni a repreen onl emelve an uni a repreen oer a ell. I i normall e cae a e incluion of ne informaion for e ample par of e populaion implie a iger variance in comparion i e ieal iuaion i a perfec frame, bu o iregar e informaion normall implie a bia. Wen aeing e quali of e aiic, e reuling accurac i e main aim. Time ela for ne uni an upae are inicaor, bu inicaor onl (Secion 5.4.6). 5

111 6 Meauremen error 6. Naure of meauremen error 6.. True value Cri Sinner, Univeri of Souampon Meauremen error i efine relaive o e value of a given variable (a i a queion) repore b a given reponen. Te baic aumpion i a ere ei a rue value of i variable for i uni, o a ere i no ambigui in e efiniion of e variable. Given i aumpion, e meauremen error i efine a e ifference beeen e repore value an e rue value. Ti i no an operaional efiniion, of coure. Even if i i accepe a ere can be no ambigui in e efiniion of e rue value, ere ma be no operaional a for an agenc o obain e rue value i cerain. Inea, variou inirec meo ma be ue o eec meauremen error a ecribe in i caper. 6.. Source of meauremen error In i repor meauremen error ill be equae i repone error, a i error ariing becaue e reponen fail for ome reaon o provie e rue value eire. Error on e par of e aa collecion agenc, for eample falel rancribing value from queionnaire or mirecoring value repore b elepone, ill be reae a proceing error (ee caper 7). Error in auiliar variable recore on a buine regier ill, furermore, be reae a frame error (ee caper 5). Tee error ma be aribuable impl o ou-of-ae informaion on regier variable bu ma alo arie for imilar reaon o repone error, a i becaue a buine fail for ome reaon o provie e rue value of e variable require. Repone error ma arie from ree ource. True value unnon or ifficul o obain Someime e buine ma eep informaion accoring o ifferen efiniion, for eample man buinee mainain accoun accoring o ifferen financial ear an i ma be ifficul o repor value i repec o a ifferen ime perio, for eample a calenar ear, requee b e agenc. In uc circumance e buine ma repor e value of e variable accoring o e cloe efiniion available, for eample e buine financial ear. Someime e buine ma no eep e informaion require, for eample bo e value an quani of ga or elecrici purcae, a ae in ONS Annual Buine Inquir. Alernaivel, e buine ma be unilling o go o e effor require o rerieve e informaion. In uc cae e value ma be guee or e queion lef blan. Te occurrence of uc meauremen error ma erefore be inicae b ig rae of iem nonrepone on a queion. Suc error ma ave a paricular effec on oer caegorie. For eample, e ONS ABI require a epeniure in ifferen area oul um o e oal epeniure repore. One 6

112 of e la epeniure queion i for oer ervice purcae. I i poible a i i ue a a balancing bo, accoring o ic buinee impl or ou a epeniure for e ear a no alrea been accoune for. Miuneraning of queion or oer lip Inrucion on queionnaire ma be miuneroo or impl no rea. A common eample of an error i e reporing of a value in e rong uni. For eample, a queion ma a for a value o be repore in uni of ouan of poun. A rue value of,488,500 oul erefore be repore a,489. A buine ma, oever, erroneoul repor e figure a,488,500. Some form inclue boe iin ic igi oul be recore for canning an buinee ma complee ee rongl, for eample riing NIL roug e boe. Te queion emelve ma alo be funamenall miuneroo. For eample, a conrucion firm mig recor e value of reail urnover on e ABI a e firm epeniure on conrucion of reail oule, erea e rue value oul be zero. Error in informaion ue b e reponen Finall, i i poible a e informaion ue b e reponen, for eample from a buine informaion em, i ielf ubec o error Tpe an moel of meauremen error Four in of meauremen error ma be iinguie. Coninuou variable: maor occaional error Eample of maor occaional error are e occaional reporing of value in e rong uni (for eample in ingle currenc uni raer an 000 currenc uni) or e occaional recoring of epeniure uner e rong eaing (o a epeniure uner one eaing i greal reuce an epeniure uner anoer eaing i greal increae). Tee error ill ofen be ienifiable uner cloe inpecion a oulier (Lee, 995). Tee are oulier ic arie from error raer an oulier ic are unuual bu correc. If poible e oul be eece an reae a par of e eiing proce (ee ecion 6.3.3). A ocaic moel for uc error in a meaure variable Y oul be a Y equal e rue value i probabili -ε an i ran from a ver ifferen iribuion i probabili ε, ere ε i a mall number, for eample 0.0. Coninuou variable: mireporing of zero A pecific inance of maor error i e mireporing of zero. One eample i e eing above ere epeniure i recore uner e rong eaing o a epeniure uner e correc eaing ma be erroneoul zero erea epeniure uner anoer eaing ma be erroneoul non-zero. Suc error ma cancel ou uner aggregaion of eaing. Oer erroneou reporing of zero ma arie en informaion i unavailable or ifficul o obain, a queion i lef blan an en impue a zero. In i cae, meauremen error i cloel relae o iem nonrepone (ee Cae Su in Secion 6.3.). 7

113 Coninuou variable: oer error Gueing of value an error ue o minor ifference in reference perio mig be epece no o lea o maor error bu raer o error ic mig be repreene b e claical error moel Y e (6.) ere Y i e repore value, e rue value an e i e meauremen error ran from a coninuou probabili iribuion. Someime e iribuion of e error mig reaonabl be uppoe o be cenre abou zero, for eample uner one gueing b an eperience reporer, o a e meauremen error ma be viee a approimael unbiae. Someime, bia ma be epece. Caegorical variable: miclaificaion Meauremen error in caegorical variable involve miclaificaion. Te baic moel in i cae involve a miclaificaion mari i elemen q i, e probabili of claifing caegor i a caegor. Te iagonal elemen of i mari oul be cloe o one an e off-iagonal elemen mall. 6. Te conribuion of meauremen error o oal urve error 6.. Toal urve error Le Y be e repore value for e value, aume o be ell-efine. Ten Y ample uni an le be e correponing rue i e meauremen error for ample uni an e conribuion of meauremen error for all ample uni o a eige eimae Y i given b ( Y ). Ti conribuion o oal urve error reflec no onl meauremen error bu alo proceing, coing an impuaion error. In orer o ae e magniue of e conribuion of ( Y ) o oal urve error (ee Secion..), i i necear o concepualie e iribuion of i erm an o eimae e caraceriic of i iribuion. Te iribuion of Y uuall involve e pecificaion of a meauremen error moel a in (6.). Te meauremen error iribuion in uc moel mig be conceive of in erm of poeical repeae meauremen (Grove, 989, p.5). For eample, a reponen mig provie ifferen guee value if ae (poeicall) e ame queion repeael, or ifferen iniviual mig complee a form ifferenl uner (poeical) repeae mailing o a firm. Te iribuion mig alo be conceive of in erm of e iribuion of error acro buinee. For eample, an error ariing becaue a reponen refer o e buine financial ear raer an a calenar ear ma no cange uner repeae queioning, bu i ma be poible o inerpre e iribuion of error e in e moel in (6.) a reflecing e iribuion of financial ear (in eir impac on e urve variable) acro buinee. 8

114 Given a meauremen error moel, e iribuion of e oal urve error can be conceive of a reflecing e oin iribuion ariing from meauremen error, ampling an nonrepone. If E enoe epecaion i repec o e oin iribuion, e bia an variance ariing from meauremen error (an aociae proceing, coing an impuaion error) ma be epree a ( ( Y ) Bia E ) (6.) ( ( Y )) Variance E (6.3) Te aemen of ee i coniere in Secion 6.4 belo. For e purpoe of quali meauremen, e primar inere ill be in oal urve error an an overall meaure of quali i 6.. Bia E[ Y Y ] Mean quare oal urve error. Te bia in (6.) ma arie from all in of meauremen error. For eample, a emaic enenc o unerrepor cerain micellaneou co ma lea o onar bia in e eimaion of oal micellaneou co. A enenc o repor accoring o an earlier financial ear raer an a requee calenar ear ma lea o onar bia for variable ic eibi upar ren over e ime perio concerne Variance inflaion Te variance inflaing impac of meauremen error i liel o be mo imporan for e large buinee in e compleel enumerae raa. Suc buinee o no conribue a all o e ampling variance, bu ranom error in eir repore value ma ave a ignifican impac on e oal variance of e urve eimae. Ti i coniere furer in Secion Diorion of eimae b gro error Uuall, i i aume a e oal urve error an i componen are normall iribue o a e iribuion can be ummarie b bia an variance. An ecepion ma arie i gro error ic are no eece or reae. Gro error for iniviual buinee ma erioul ior eimae, epeciall eimae for omain bae on mall number of obervaion, one (or more) of ic i ubec o gro error. 6.3 Deecing meauremen error 6.3. Comparion a aggregae level i eernal aa ource Surve eimae ma be compare i aggregae figure from anoer ource, uc a anoer urve, an aminiraive ource or rae organiaion aa. Suc a comparion ma reveal bia from meauremen error, aloug i ma be ifficul o ienangle meauremen error bia from nonrepone bia an i ma be ifficul o eermine o a een e P 9

115 ifference beeen eimae i aribuable o error in e urve of inere or error in e oer aa ource. Cae Su. Comparion of mail urve i inervie urve In e 980 Saiic Seen conuce an annual urve on forer (logging) among privae oner (a oppoe o large corporaion, e governmen or e Curc). Te privae oner mae up abou 50% of all forer in Seen. Ti urve a one b a convenional mail queionnaire eign an involve a ample of 7,000 uc oner (oning le an,000 ecare eac). Te aim a o eimae a e naional level, among oer quaniie, e oal volume (in million cubic meer) logge b final felling (a i a ole area i cu on), inning (elece ree onl) an micellaneou felling (in ice, uner poer line ec). Becaue of concern abou quali, i a ecie in 988 o ivie e urve ino o par on an eperimenal bai: a mail queionnaire a iribue o abou 4,500 oner ile abou,500 oner ere inclue in an inervie urve, abou 00 local forer eper performing e inervie. Te reul are given in e folloing able. π-eige eimae of proporion of oner oing acivi Eimae volume (million cubic meer) Mail Inervie Mail Inervie Final felling: 0% % Tinning: 3% 39% Micellaneou: 8% 38% Toal logging: Te eimae volume for e mail urve en o be le an for e inervie urve, epeciall for e micellaneou caegor. Ti ma be eplaine b e muc greaer number of zero (oner no uneraing e acivi) in e mail urve, epeciall for e micellaneou caegor. Man of ee zero repreen eier meauremen error (e failure o repor acual acivi) or iem nonrepone (a blan reurn ere an acual reurn ma be ifficul). Final felling i ea o ienif an quanif (for eample lo of paperor i involve o ge a permi), ile inning an paricularl micellaneou logging are arer o ienif, quanif an remember. I a conclue a e quali of e reul from e mail queionnaire a unaccepable an e urve a cange o an inervieer moe from Comparion a uni level i eernal aa ource A more ueful comparion i poible if e reponen recor can be mace o recor from anoer ource uc a a a regier, conaining relae variable. Suc comparion 0

116 mig onl be mae i a ube of ample recor, for eample e repone of u e buinee in e compleel enumerae raum mig be compare i informaion in publicl available annual repor. Gro error mig be eece in value ic o no follo e normal relaionip i variable in e eernal ource. Difference in efiniion beeen e o ource, in paricular ifference in reference perio, ill ofen complicae uc comparion, oever. I ma alo be a e eernal ource, for eample an auie e of compan accoun, onl become available afer e urve eimae ave been publie, o a meauremen error eimae can onl be mae reropecivel. Cae Su. Comparion of queionnaire repone i value on VAT regier Te urve on omeic rae in e ervice ecor a Saiic Seen aim o eimae quarerl urnover b inur (4 igi NACE) in e ervice ecor. A probabili ample of legal uni i ran from e Buine Regier (BR) an a queionnaire i maile o ee uni. In 997 a u a mae o fin ou eer e mail queionnaire coul be replace b aa aen irecl from e VAT regier. Suc a if oul reuce co conierabl, for Saiic Seen a ell a for reponen, an a e ame ime mae i poible o if from a ample of abou 4,500 o a oal enumeraion of abou 0,000 enerprie. To eimae of urnover b 4-igi NACE ere compare. Te fir a a π-eige eimae from e original urve obervaion. Te econ a a moifie eimae, i e queionnaire obervaion replace b e correponing VAT obervaion (ecep in e ae-all raa). Difference beeen e eimae ere reaonabl mall in mo NACE group compare i e ranom variaion in e urve. Hoever, in ome NACE group e ifference ere muc larger an one oul epec from e ranom variaion. For 4 legal uni e π-eige ifference beeen queionnaire an VAT aa eceee 50 million SEK. Abou one ir (37) of ee ere elece for a elepone inervie o fin ou e reaon for e icrepancie. For pracical reaon e inervie a o be one uring e olia eaon in e ummer, an onl inervie ere complee. Neverele, a lo a learne from ee inervie: ) In 0 cae (legal uni) e large icrepancie ere ue o e coice of uni. Tee legal uni urne ou o be par of muli-legal uni enerprie. Te urnover in e ample cae ma be repore o e VAT regier from anoer legal uni iin e ame muli-uni enerprie, an i VAT-reporing uni ma even be an ou-of-cope uni, for inance a manufacuring uni. In ome cae e elece uni repore zero urnover ile e correponing VAT urnover a ubanial. In ome cae i a agree (i e reponen) a e queionnaire urnover a inee e correc one ile e VAT urne ou o be e correc figure in oer cae. ) In 3 cae e reponen a b miae given e rong number (urnover) on e queionnaire. Ti a been correce uring e icuion, maing queionnairean VAT aa coincie.

117 3) To cae ere ue o aa enr error mae b Saiic Seen bu no eece b eiing. 4) To cae ere ue o error in NACE claificaion in e BR. Te reponen a repore Manufacuring inea of e ervice ecor coe foun in e BR. Tee uni a been claifie a over-coverage in e urve an given value of urnover equal o zero. 5) One cae, a oleale rae agen (NACE 5.) a inclue a urnover e ole rae urnover inea of onl i on urnover a requee in e queionnaire. 6) To cae ere race o miuneraning of e queionnaire. 7) One cae a ue o reference perio problem. Ti enerprie a involve in a 6 mon long proec. Te VAT pamen ere ivie ino i monl equal um ile acual pamen oo place on one or o occaion. I o appene a e queionnaire-urnover a aribue o anoer quarer an e one in e u ile e VAT aa eeme o be ver conien from mon o mon. I i clear a uc comparion i eernal ource can reveal man ource of error in aiion o meauremen error. In paricular e mo riing aiional pe of error in i u coni of frame error ariing from problem in elineaing uni. Suc comparion ma alo ugge meo for improving quali. Ti u ugge, for eample, a VAT aa ma be ueful for eiing. A large ifference beeen queionnaire repone an VAT urnover oul be a goo reaon for a elepone conac Inernal comparion an eiing A impler approac i o eamine e inernal conienc of e value repore in e urve a par of e uual eiing proce (Hiiroglou & Berelo, 986; Pierzcala 990; Granqui an Kovar, 997). Tu, one ma cec accouning ieniie, for eample ere componen um o a oal, an inequaliie, for eample a ome variable are poiive. Comparion ma be mae i value repore in previou urve b e ame reponen. For eample, a variable i mon o mon variaion normall no in ece of 5%, ic uenl cange b 000% i a liel cae of gro meauremen error. See caper 7, on proceing error, for furer icuion Follo-up Wen ei conrain are faile, ere are generall o opion. Fir, e repore value ma be moifie o a e o obe e conrain, for eample folloing e proceure of Fellegi an Hol (976). Secon, e reponen ma be folloe up in orer o clarif e reaon for e faile ei conrain an ence o eabli, if necear, a value i reuce meauremen error. Suc follo-up ma be epece o provie more informaion abou e naure an ize of e meauremen error. I ma be elecive, a i onl value coniere liel o ave a non-negligible effec on e aiical eimae mig be folloe up. Follo-up can range from a imple elepone call o cec a ingle value roug o a more eaile reinervie, aime a eabliing e ource of informaion ue a ell a e

118 reponen uneraning of queion an inrucion. Dippo, Cun & Saner (995, p.95) refer o i a a repone anali urve. Suc a urve ma reveal meauremen error irecl, for eample roug miuneraning iplae, or ma ugge ubgroup for ic e quali of e aa ma be or. For eample, reponen mig be ae eer eir repone ere bae on memor or involve reference o appropriae informaion ource. Te proporion of reponen uing memor mig be aen a an inicaor of poor aa quali an mig be compare beeen ifferen ubgroup of buinee. Reinervie appear o be relaivel uncommon in European buine urve. An illuraion of repone variabili i provie b a u of Friberg (99) in ic reinervie aroe b accien! He repor on a Saiic Seen urve on environmenal invemen an co in Seen. A reminer a iribue a ome poin o oe enerprie a a no e repone. Five enerprie among oe receiving e reminer a in fac en in eir queionnaire u one of o a before. I o appene in oe five cae a a ifferen peron a e enerprie an e one o a alrea repone (an en poibl gone on olia - i appene in e ummer) fille in e queionnaire. Ti mae i poible for Saiic Seen o compare e o verion from eac of e five enerprie. Ver large ifference ere foun beeen e repone of e pair of reponen from eac of e five enerprie. Ti eem o reflec e large egree of error in meauring a variable uc a environmenal invemen, ic i ifficul o efine an quanif Embee eperimen an obervaional aa Ranomie eperimenal eign ma, in principle, be ue o eec meauremen error bia b comparing alernaive meauring inrumen (Biemer & Feco, 995, p.68). For eample, ifferen form eign or ifferen moe (for eample mail veru elepone) mig be aigne ranoml beeen ifferen reponen. See Cae Su in Secion 6.3. for an eample. Ranomie aignmen ma ofen be ifficul o implemen in pracice. For eample, aloug an agenc ma reque a a form be anere b a paricular caegor of aff, i ma be ifficul in pracice o enforce i. I mig erefore be ifficul o implemen a ranomie eperimen comparing e effec of uing, for eample, managemen veru clerical aff a reponen. I ma, oever, be poible o recor obervaional aa on e caegor of aff reponing in an ongoing urve. Te fac a e allocaion of aff i no eperimenall aigne mae e inerpreaion of ifference in e urve oucome beeen ifferen caegorie of aff more ifficul, becaue of poenial confouning i oer variable, bu no impoible (Biemer & Feco, 995, p. 69). 6.4 Quali meauremen 6.4. Quali inicaor Tere are everal a a problem in e quali of repone o a paricular queion ma be reveale: 3

119 a) ig rae of failure of ifferen ei conrain involving e variable; b) ig rae of iem nonrepone ma inicae ifficulie in anering e queion an poenial meauremen error; c) uneplaine large variaion beeen urve occaion; ) ponaneou repor on ifficulie from reponen; e) a repone anali urve (Dippo e al., 995) ma reveal miuneraning or e frequen ue of memor in anering a queion; f) ubec maer uneraning of e naure of e queion, for eample invemen are arer o quanif an e number of people emploe. Tere are alo everal inicaor for problem i e ole queionnaire: a) repone buren in erm of ime an effor; b) number of people involve in reponing o e urve; c) cange of peron reponible for filling in e queionnaire; ) proporion of lae/elae repone. Quali inicaor erive from uc ource ma be ueful for monioring quali an for comparing quali beeen queion an beeen urve. Te ma ugge poible irecion of bia bu are unliel o provie muc elp in e aemen of e magniue of e bia or variance of oal urve error Aeing e bia impac of meauremen error Were pecific ource of meauremen error are concerne, bia ma be aee b moelling e mecanim leaing o error. For eample, e effec of buinee uing eir on financial ear raer an e requee calenar ear mig be aue for b appling a ren moel iin inurial caegorie o e ample buinee ic o no ue e calenar ear. Or e impac of buinee allocaing acivi o erroneou eaing mig be aee b eimaing e probabili of miclaificaion beeen eaing. Someime i ma be poible o conuc eperimen (ee ecion 6.3.5) o ae e bia impac of alernaive meauring inrumen, for eample ifferen form eign or mail urve veru elepone urve. Difference beeen meauring inrumen onl reflec ifferen biae, oever, an o no necearil provie accurae eimae of abolue biae. Anoer approac o bia aemen i roug comparion i eernal ource (ee ecion 6.3.). Again, i ma no necearil be poible o ecie ic ource i lea biae an, moreover, meauremen error biae ill generall be confoune i oer ource of bia, uc a nonrepone. Te ieal a o ae bia i o conuc reinervie i e ample o eabli e rue value. Suc an eercie face, of coure, man pracical obacle (Biemer & Feco, 995, p.70). 4

120 6.4.3 Aeing e variance impac of meauremen error Variance eimaor eigne o eimae e ampling variance (ee caper ) ma alo be epece o capure an imporan componen of e variance of oal urve error aribuable o meauremen error. Conier, for eample, e claical error moel in (6.), ere e repore value Y i eermine from e rue value b Y e ere e i e meauremen error. Conier a ingle raum, iin ic e rue value an error are inepenenl iribue i common variance σ an σ e repecivel. Le Y be e mean of e meaure variable Y for e n ample uni in e raum an le µ be e mean of e rue value for N populaion uni in e raum. Tu Y i e urve eimaor of µ. Auming imple ranom ampling iin e raum, e variance of e oal urve error Y µ acro bo e ampling an meauremen error iribuion i obaine a var( Y µ ) σ σ n N e / n, Te uual eimaor v ( Y ) of e ampling variance of Y i e ample variance of Y iin e raum muliplie b ( n ) an i a epecaion E N [ v ( Y )] ( σ var( Y σ e ) n µ ) σ N Te eimaor i erefore biae onar, failing o capure e componen ariing from meauremen error, bu e bia ill be mall if N i large. A conervaive approac i o remove finie populaion correcion from e variance eimaor (a i replace ( n ) N b n ). Ti i liel o be oo conervaive, oever, epeciall for compleel enumerae raa. To obain an improve variance eimaor i i necear o eimae e variance σ e of e meauremen error. Ti mig be aempe via a reinervie urve (Biemer & Feco, 995, p.65). If no, ome in of eniivi anali i liel o be necear. Te conribuion σ e n of e meauremen error o e variance above aume inepenen meauremen error. If meauremen error for ifferen buinee are poiivel correlae en i ill en o inflae e variance. I i imporan erefore a e variance eimaor i bae on reporing uni beeen ic inepenen reporing i a reaonable aumpion. If, for eample, a ingle reponen provie repone for everal enerprie, e meauremen error coul be correlae beeen ee repone an o e e of enerprie oul be reae a a ingle reporing uni for e purpoe of variance eimaion. e / N σ e N 5

121 7 Proceing error 7. Inroucion o proceing error Pam Davie, Office for Naional Saiic Once urve aa ave been collece from reponen, e pa roug a range of procee before e final eimae are prouce. Tee po-collecion operaion can ave effec on e quali of e urve eimae. Error inrouce a i age are calle proceing error. Proceing error can be ivie ino o caegorie: em error an aa anling error. Te opic of proceing error i u one componen of non-ampling error. Non-ampling error, incluing proceing error, affec no onl aa from ample urve bu alo aminiraive an cenu aa. Proceing error, along i oer nonampling error, ma lea o biae an increae in e variance. Ti caper concenrae on ecribing e variou componen of proceing error in e cone of buine urve. Some uggeion are mae for reucing e effec of proceing error on aa quali. Te repor i illurae b eample, from e UK an Seen, of reearc o meaure an minimie proceing error. 7. Sem error Sem error are error in e pecificaion or implemenaion of em ue o carr ou urve an proce reul. One ource of em error i auomae aa capure. Sem error picall affec eier all or paricular clae of eimae. Te impac of em error on aa quali i influence b en e error are icovere. Te impac of e error on aa quali nee o be evaluae an compare i e co of correcing e error, bo in erm of uman reource an a poible ela in e releae of e aa, before maing a eciion eer o correc e error. Sem error ic are icovere before e beginning of aa collecion are more eail correce an error ic are ienifie in e coure of e urve. Wi e ue of compuer aie aa collecion, omeime program error are no eece unil afer aa collecion a are. Sem error laer in aa proceing omeime are no eece unil laer on, or, a or, unil afer reul are publie, leaing o e nee o publi correcion. Clearl uc error are poeniall ver eriou. 7.. Meauring em error In orer o meaure e effec of a em error, e par of e em a are incorrec nee o be correce. Te eimae nee o be prouce on bo e incorrec an correc em, an e ifference in e reul from e o em nee o be compare. 6

122 7.. Sem error: o eample 7... Sampling in e ONS From abou 99 o 994, probabili proporional o ize (PPS) ampling a ue in ome buine urve run b e UK Cenral Saiical Office (CSO), noabl in e quarerl capial epeniure an quarerl ocbuiling inquirie. Te em of ample elecion a implemene on CSO buine regier em, an a pecifie o a a ranom number a generae, an eac buine a repreene b a par of e ranom number range proporional o i ize, an a elece if e generae number fell ino i inerval. Te coing in e program i no, oever, follo i proceure eacl, an in 994 i a icovere a e elecion probabiliie ere no a inene. Te uggee oluion a o or ou a elecion probabiliie ere implie b e elecion proceure, an o ue oe o prouce an unbiae (bu poibl raer variable) e of urve eimae. Te reul of i epioe a a general iru of PPS ampling for buine urve, an, aloug a correce elecion algorim i available, e meo a been moballe in ONS (e ucceor o CSO) ince en Variable forma in compuer program Wen a compuer program i being rien, variable ma be allocae cerain fie forma, an a for a paricular variable e forma i efine o be an ineger i o igi. A e ime a value above 99 i coniere impoible. In ime, value above 99 become poible an occur, bu nobo amen e forma. Te em cop value o ore em iin e ae forma, an oe o iou arning, for eample 3 impl become 3. Te aiic en o no move a epece. Afer a ile omebo realie e caue! 7..3 Minimiing em error Sem error are minimie b e ue of quali aurance an eing proceure a e em i rien. Were appropriae, e ue of armonie meo acro urve enable e ame ell-evelope an ee program coe o be ue for proceing aa in all e urve. Ti reuce e cope for programmer error b reucing e amoun of coe o be rien, an free up reource for eveloping an eing oer par of e em. 7.3 Daa anling error Poenial ource of aa anling error range from procee ue o capure an clean e aa o ecnique ue for e final proucion of eimae an e anali of e aa. Te main ource of aa anling error are: aa ranmiion: i cover error ariing in e ranmiion of informaion from e fiel, ere aa are collece, o e office ere e aa are ubece o furer proceing; aa capure: e pae of e urve ere informaion recore on a queionnaire i convere o a forma ic can be inerpree b a compuer ; 7

123 coing: e proce of claifing open-ene repone ino preeermine caegorie (Kaprz & Kalon, 998); aa eiing: a proceure for ienifing error creae roug aa collecion or aa enr uing eablie ei rule (Kaprz & Kalon, 998). Daa eiing alo refer o e auomaic correcion of cerain error ere e error i (apparenl) ienifiable or ere e co of cecing i manuall ecee e benefi over auomaic correcion; an proce a i applie o e aa, from e ienificaion of oulier o e eaonal aumen proceure, can inrouce proceing error. Ti proceing error i no caue b e meo ielf, bu b e incorrec applicaion of e proce. Ti repor icue error inrouce a e aa ranmiion, capure, coing an eiing pae of e urve. 7.4 Daa ranmiion For mo buine urve, aa ranmiion from e fiel i via poal queionnaire. In i cae, ranmiion error are unliel o caue a ignifican problem becaue e aa oul arrive inac. In ome inance, aa ma be fae or given over e elepone an in ee cae e cope for error increae. Fae informaion ma be illegible, an informaion given over e elepone ma be miuneroo, or recore rongl b e urve orer. In bo ee cae, if ere i an oub, e recore value oul be cece i e reponen before i i capure. A relaivel ne evelopmen, a lea for ONS buine urve, i e ue of ouc-one, raer an mailing, for aa ranmiion. Clearl ere i cope for reponen o eier fail o operae e em correcl, or o pre an incorrec buon. To minimie e ri of error, e em oul be eigne o a reponen are require o confirm eir reurn. Gro error are eece in e eiing pae, bu maller error ma oerie pa uneece. 7.5 Daa capure A varie of meo ma be ue o capure aa. Tee inclue: eing repone from pencil an paper queionnaire; uing canning o capure image folloe b auomae aa recogniion o ranlae oe image ino aa recor; eing b inervieer of repone uring compuer aie inervie; an ee are icue in urn belo Daa eing from pencil an paper queionnaire Te raiional meo of aa capure for buine urve i e eing of repone from pencil an paper queionnaire ono compuer b a cenrall locae aa enr eam. Ti i a ver labour inenive a, ic a no been replace on man urve b more moern ecnologie. Some moe of aa collecion uc a compuer aie peronal inervieing (CAPI) an compuer-aie elepone inervieing (CATI) ener e aa ono compuer in e coure of e inervie. 8

124 Daa eing i ue on man poal urve ere pencil an paper queionnaire are e imple a o collec informaion. ONS i inveigaing e poenial of oer meo of aa capure incluing canning an auomae aa recogniion o reuce e number of urve ere aa are capure in i a Meauring error occurring uring aa eing Te accurac of aa eing can be meaure b eier comparing a bac of enere aa i e original queionnaire or more commonl b re-enering e bac an comparing e o e of aa. Lberg & Kaprz (997) give a range of eample i error rae varing from 0.% o.6%. An ne meo of aa capure mu ave error rae a lea a goo a ee o mainain e quali of urve aa Minimiing error occurring uring aa eing Meo of minimiing error uring aa eing inclue: cecing regular bace of queionnaire for eing error; in-buil ei in compuer aie ranmiion can ienif eing error; cecing all aa enr or of ne aff unil e reac an accepable level of accurac Daa capure uing canning an auomae aa recogniion Te poenial co aving offere b e ue of canning an auomae aa recogniion over raiional aa eing a le o increaing inere in i ecnolog. In ONS canning i being ue for ome buine urve. For eample, e la Cenu of Emplomen carrie ou in e UK ue canning equipmen o capure all e aa reuling in quicer proceing an a loer co for a ver large urve. Oer organiaion o ave inveigae e ue of canning an auomae aa recogniion for aa capure inclue Saiic Seen (Blom & Friberg, 995), an Saiic Canaa (Vezina, 996). Te age in e aa capure proce are: Scanning Te queionnaire are eparae ino ingle ee an fe ino e canner ic ore e image of eac page a a TIF file. Te preparaion of queionnaire for e canner can be fairl labour inenive (Eler & McAleee 996) ince an aple nee o be remove an e queionnaire correcl aligne. Te orage of image of queionnaire a e aiional avanage of proviing rapi acce o queionnaire if an querie arie an reucing e nee for orage of large volume of paper queionnaire. Form Ou In man aa recogniion em e image of e original prine queionnaire i remove elecronicall from e image of e aa fille in b e reponen. Ti reuce e compuer memor neee o ore e image of e aa an clarifie e image for auomae aa recogniion. 9

125 Auomae aa recogniion Differen meo are ue o erac e aa from e image epening on e pe of informaion being capure. Tee inclue: Bar coe recogniion (BCR). Ue o rea bar coe, for eample erial number on paper queionnaire. Ver accurae. Opical Mar Recogniion (OMR). Ue o rea repone in ic boe. Over 99% of iem are (preumabl correcl) recognie b e em. Opical Caracer Recogniion (OCR). Ue o rea macine-prine e. Over 99% of iem are (preumabl correcl) recognie b e em. Inelligen Caracer Recogniion (ICR). Ue o rea an-rien caracer. For an rien numerical informaion 65%-90% of queion repone ere recognie. Ti figure i loer for an rien e informaion; a a reul ICR i rarel ue for collecing uc informaion. Te recogniion figure quoe above are from Saiic Seen eperience of auomae aa recogniion a repore in Blom & Friberg (995). I mu be empaie a ecnolog i eveloping quicl in i area o a e accurac of auomae aa recogniion em can be epece o improve Meauring error aociae i canning an auomae aa recogniion Auomae aa recogniion ma inrouce error ino aa en caracer are incorrecl recognie; for eample e number 3 an 8 ma be confue, a ma e number an 7. If e em i more liel o confue a 3 for an 8 an vice vera, an imilarl for e number an 7, en ee error coul caue an upar bia in e urve eimae. Some of ee error ma be eece a e eiing age bu ome inaccuracie ma lip roug. Te accurac of auomae aa recogniion ma be compare i ee aa enr b proceing a bac of form in bo a an comparing e reuling aa. Eler & McAleee (996) repor e reul of uc a comparion ere e foun a for ome queionnaire e accurac acieve b e auomae recogniion em a a lea a ig a a acieve b e ee aa enr proce Minimiing error aociae i canning an auomae aa recogniion Te mo effecive a o enure ig quali aa capure uing auomae aa recogniion i o eign form a are eail canne an inerpree b e aa recogniion proce. Vezina (996) provie a ueful icuion of apec of form eign a influence aa quali. Tee inclue: e caraceriic of e paper i nee o fee eail ino e canner; e colour of e in canner pic up ome colour beer an oer an i can be ue o enance e image of e aa; page ienifier; regiraion poin mar on e form ic enable e em o align e canne image i a i epecing; 0

126 efiniion of zone of aa o be capure i i paricularl imporan for par of e form ere e reponen i ae o rie in number or leer. Te proviion of boe encourage e reponen o prin caracer in capial a are eaier for e em o recognie an manucrip; an o ee e coul a one ic Vezina oe no menion: inrucion aing e conribuor o provie aa in e require forma. 7.6 Coing error Te aim of coing i o ranform open-ene, eual informaion ino caegorie a can be ue in aa anali. In e buine urve fiel, e commonl ue coing claificaion i NACE Rev., bu in e UK i i replace i e comparable Sanar Inurial Claificaion 99 (CSO, 99). A maor ue of coing in buine urve i on e buine regier. In e UK, buinee provie a ecripion of eir acivi, ic nee o be coe accoring o e Sanar Inurial Claificaion. In ome buine urve open-ene ecripion, for eample of commoiie, are require a nee o be coe accoring o a prouc claificaion. Te accurac of coing i eavil epenen on e ill of coer, o ere i e poenial for inroucing bo bia an variance uring e coing proce. Coing a o age: e evelopmen of a claificaion or coing frame. Ti coing frame i non a a nomenclaure or icionar an i accompanie b a e of coing inrucion. Nomenclaure nee o be frequenl revie o a e repreen e full range of poible caegorie; rien or verbal repone o urve queion are coe ino caegorie. Ti coing ma be: ricl manual ere e uman coer loo up e coe in e icionar; compuer aie ere repone are available in elecronic form or pe ino a compuer an ome purpoe-rien ofare ugge a range of poible coe. Te uman coer eier elec one of ee coe or ei e verbal ecripion an a e compuer o ugge furer poible coe; compleel auomae. In compleel auomae coing e urve repone are available in elecronic form or enere ino a compuer an e compuer ofare allocae e coe Meauring coing error Te impac of ifferen coer on aa quali can be aee in erm of conienc (or reliabili) an accurac compare o a anar.

127 7.6.. Conienc A conien coing em ill give e ame coe for iem in e ame caegor. Compuer auomae em are b efiniion compleel conien ince given e ame ecripion of a caegor e ill allocae e ame coe. Differen uman coer implemen coing rule ifferenl, eer concioul or ubconcioul, o e ma allocae ifferen coe o e ame ob ecripion. Te conienc of coing em can be meaure b aing a e of ifferen coer o coe a common li of ob ecripion an calculaing e proporion of all paire comparion of coe ere e coer agree (Kalon & Soell, 979) Accurac Aloug auomae em are compleel conien e ave anoer le eirable feaure: e ma no allocae e be coe o a ecripion, a i, e coe ma no be an accurae one. Auomae coing em rel on e macing of e ring; if e macing i no eac en e aignmen of coe ma no be accurae. Te accurac of coe can be meaure b comparing coe allocae b anar coer i oe allocae b an eper coer, o i preume o be infallible Te impac of coer error on e variance of urve eimae In manual coing an compuer aie coing ifferen coer ma allocae ifferen coe o e ame ecripion. In paricular eac iniviual coer ma unconcioul over-allocae buinee o ome coe an uner-allocae em o oer. Ti i non a correlae coer error. Te error in e coe allocae b a paricular coer ma lea o bia in e eimae of e proporion of buinee in a given inur group for inurie coe b a coer. Hoever ince for man urve coing i are over a number of coer, if e error mae b coer are ifferen e impac of ee iniviual biae on e final urve eimae ma cancel ou. In i cae aloug e final urve eimae ma no be biae e variance of e eimae ill be increae. Te overall bia i reuce a e number of ifferen coer increae, o in ome urve e coe li i provie i or a par of e queionnaire, o a eac reponen coe eir on aner. Ti minimie correlae coer error a e epene of a poenial increae in meauremen error (ee caper 6) Te ri of coer error inroucing bia in urve eimae Bia ill be inrouce ino urve eimae if a lea ome coer emaicall aign incorrec coe o cerain occupaion. One cenario ere i ma occur i in compuer aie coing ere e compuer ugge a preferre coe ic e coer ma accep or reec. If ere i a enenc for coer o accep e uggee coe even en i i incorrec en e coing error ma inrouce bia ino e urve eimae (Bunell 996) Minimiing coing error Te impac of coer error on aa quali can be minimie b: e effecive raining of coer in uing e coing em;

128 ell eigne, up-o-ae coing em; in manual an compuer-aie coing em, coer nee o be upervie an e quali of eir coing cece regularl. In ome cae coer ma be unure ic coe o allocae an ee querie ill nee o be referre o upervior an in ome cae reearcer for reconciliaion; ome urve (or more localie eperimen) coe informaion more an once uing ifferen coer an compare e reuling claificaion o elp reolve cae ere ere i ome oub a o e rue coe. Ueful reference on coer error inclue Lberg & Kaprz (997). 7.7 Daa eiing Granqui (984) ecribe eiing a aving ree goal: o provie informaion abou aa quali; o provie informaion o elp bring abou fuure improvemen in e urve proce; an o clean up poibl erroneou aa. Cec ue o ienif upiciou aa iem are calle ei rule. Tee inclue: range or valii cec i e aa iem in e vali range for e aa? conienc cec i e aa iem conien i oer aa provie b e reponen eier in a inervie/queionnaire or on a previou occaion? rouing cec a e reponen anere e correc queion? Ti form a large par of eiing cec for pencil an paper queionnaire. Compuer program are ue o implemen ee ei rule eier on-line uring e aa enr proce (inegrae eiing) or in a bac proce ic prouce a li of upec aa iem for manual revie. Supiciou aa iem can be claifie ino faal ei or quer ei. Faal ei ienif clearl erroneou aa erea quer ei ienif aa a are implauible. In aiion o ifferen pe of ei rule ere i a varie of ifferen approace o eiing: eiing can compare ifferen iem of aa for a given iniviual (i i iem conien i e oer iem?) or compare e ame iem for ifferen iniviual (i i aa iem muc iger an e oer?); eiing can be conuce on aggregae (o e ummar aiic or eimae for i bac of aa loo upiciou?) or on iniviual aa. Supiciou bace of aa can en be ubivie an e aggregae eiing proce repeae unil e error() are narroe on o iniviual aa ; eiing can be manual, b inpecion of paper form before or uring aa enr, or auomae. For general icuion on eiing ee Granqui (995), Lberg & Kaprz (997), an Granqui & Kovar (997). 3

129 7.7. Meauring e impac of eiing on aa quali Differen organiaion, an inee iniviual iin organiaion, ave ifferen eiing policie. Tere i conenu on e imporance of correcing faal error ere aa are clearl erroneou. Hoever ome argue a urve, paricularl buine urve, are overeie, an a muc of e eiing conuce o reolve quer ei a lile impac on e quali of eimae an erefore oul be reuce. Ti oul ave a large impac on e co of running urve: eiing can aborb a muc a 0-40% of oal urve buge (Granqui & Kovar 997). If e reource evoe o eiing ere reuce i oul free aff o concenrae on minimiing oer ource of urve error ic mig ave a greaer poenial impac on aa quali. Oer argue a ince i i impoible o pre-pecif all e ue o ic aa ill be pu, e poenial impac of inconiencie in e aa on eimae canno be aee. Daa oul erefore be eie unil e are inernall conien, paricularl if one oupu of a urve i a aa e o be ore a an eernal arcive a ma be ue b econar anal Minimiing error inrouce b eiing Eiing can inrouce bia ino urve eimae if i i bae on pre-conceive iea of a e aa oug o loo lie ic urn ou in pracice o be unrue. Eiing ma alo arificiall reuce e variance of urve eimae if real ereme value are incorrecl aue oar e mean of e iribuion. Ti can reul in over-opimiic claim abou e preciion of urve eimae. Sraegie o minimie error inrouce b eiing inclue: involving ubec maer peciali in e eiing proce o a ei are appropriae for e aa; uing anarie eiing coe for queion a are ue on a range of urve; eing program coe ue in eiing b eamining a appen o buinee i paricular combinaion of aa value; feeing bac informaion abou aa quali o e urve, queionnaire an ei eign age o a poible amenmen o queionnaire, fiel proceure an ei rule a oul improve aa quali can be icue. 7.8 An eample of error a e publicaion age Proucion of man ifferen official aiic, an in paricular monl aiic, i ofen ubec o ig ime conrain. All age of e proucion proce are en carrie ou i no ime o pare. One of e ep o be aen quicl i moving a able ino e pre releae. In comparion i e previou able a ne mon i ae, an previou mon ma be revie. In Seen recenl, a ne mon a ae o a able in a pre releae an e reviion for e previou mon a overlooe. Several earlier mon ere on in bol a reviion. Hence, e earlier figure for e previou mon ma be rea a confirme, an i i le accurae an i oul be. Te leon i a e le manual ping of figure e beer; able oul be move a a ole, or an auomaic proceure for generaing em from e final aa oul be ue. 4

130 8 Nonrepone error 8. Inroucion Cri Sinner, Univeri of Souampon Nonrepone arie en a ample uni fail o provie complee repone o all queion ae in e urve. Error ariing from nonrepone ma be coniere a an eenion of error ariing from volunar ampling, a icue in Secion 4., ince e failure o voluneer informaion ma be viee a a form of nonrepone. Nonrepone error are reae ere a iinc from frame error, a icue in Caper 5. In paricular, ample uni ic fail o repon bu are ouie e arge populaion (ineligible) are reae a frame error. In aiion, noncoverage (a i uni in e arge populaion bu ouie e ample urve populaion) i reae a a frame error. 8. Tpe of nonrepone 8.. Paern of miing aa Uni nonrepone arie en a uni fail o provie an aa for a given roun of a urve. Tere are o broa reaon for uc nonrepone: (i) nonconac e form ma no reac an appropriae reponen for variou reaon, for eample cange of are, failure of e poal em, failure o forar from iin e buine; (ii) refual e form oe reac an appropriae reponen bu e reponen oe no reurn e form. Uni nonreponer ma be claifie ino o pe accoring o e informaion available abou e uni o e agenc: uni ic ave never previoul repone (ee ill coni primaril of maller uni ic are ample afre a eac urve occaion, or oe nel recruie o e ample in roaing ceme) for uc uni e onl informaion available ma be a recore on e frame; uni ic ave previoul repone (ave nonrepone) ee uni ill uuall coni eier of compleel enumerae uni ic are ample on ever occaion or ele larger uni ic are ample over everal occaion in a roaion eign paern of nonrepone over e roun of e urve mig be enoe XXOXOOXX, for eample, ere X enoe repone an O nonrepone an e mo recen roun of e urve i on e rig. Iem nonrepone arie en a form i reurne from e uni bu repone o ome queion are miing. Suc miing aa ma arie, for eample, becaue queion ere overlooe or becaue e informaion require o aner e queion a no available o e reponen. A paricular problem in buine urve i e eparaion of iem nonrepone from zero. Reponen ill ofen leave blan aner o queion abou amoun, for eample e value of proucion in a cerain caegor, en e aner i zero. 5

131 8.. Miing aa mecanim In orer o ae e error ic ma arie from nonrepone i i necear o eabli a aiical frameor iin ic e mecanim of nonrepone ma be coniere. Formall, nonrepone ma be repreene b 0- repone inicaor variable of e form R 0 if value i recore (repone) if value i miing (nonrepone) Uni nonrepone ma be repreene b a erie of inicaor variable R, efine for eac uni in e ample. Ti efiniion ma be eene in variou a. To allo for repeae roun of a urve, one ma efine variable R for occaion an uni. Iem nonrepone ma be repreene b a erie of repone inicaor, one for eac variable for ic miing value ma occur. Tere i a number of alernaive aiical frameor iin ic e nonrepone mecanim ma be repreene. See Leler & Kalbee (99, Caper 7) for a lieraure revie. Te eerminiic approac aume a repone inicaor variable R are efine for all uni in e populaion an a eir value are fie. Tu, in e cae of uni nonrepone, i i uppoe a e populaion i ivie ino o raa : e reponen o ala repon an e nonreponen o never repon. Te naure of e error ariing from nonrepone ill epen on o ell e eimaion meo ue o anle nonrepone compenae for ifference beeen ee o raa. Te ocaic approac rea e repone inicaor variable R a oucome of ranom variable. A number of ifferen ocaic frameor i poible. In e cae of uni nonrepone, one approac i o rea e e of reponen (oe ample uni for ic R ) a a ranom ubample of e elece ample obaine roug a proce analogou o o-pae ampling (Särnal & Senon, 987). Te naure of error ariing from nonrepone en epen on aumpion abou o e ubampling occur. In e remainer of i repor a ocaic approac i aope, correponing o moern aiical moelling. Bo e repone inicaor R an e urve variable are conceive of a oucome of ranom variable an aumpion abou e miing aa mecanim are repreene roug aumpion abou e oin iribuion of e R an e. Ti approac i paricularl fleible for anling ifferen in of nonrepone, for eample bo uni nonrepone an iem nonrepone, an for eening o an inegrae frameor ic allo for bo nonrepone an meauremen error. Te above frameor i ver general an in orer o mae ueful progre in aeing nonrepone error or in auing for nonrepone i i necear o mae more pecific aumpion abou e naure of e miing aa mecanim. Tree erm ill be ueful for ecribing uc mecanim. Miingne i ai o occur compleel a ranom if R i ocaicall inepenen of e relevan urve variable. For eample, if uni nonrepone in a urve of proucion i being coniere, i coniion oul impl a buinee i lo level of proucion oul 6

132 be a liel o repon a buinee i ig level of proucion. Ti coniion i a ver rong one an ma arie onl rarel in pracice. Miingne i ai o occur a ranom given an auiliar variable (or variable) if R i coniionall inepenen of relevan urve variable given e value of. Suppoe, for eample, a i a meaure of ize, uc a emplomen or urnover, available on e frame. In a urve of proucion, nonrepone oul occur a ranom given e ize variable if nonrepone i unrelae o proucion among firm of an given ize. Te iribuion of nonrepone coul var, oever, beeen firm of ifferen ize. Ti aumpion i generall le ringen an e aumpion a aa are miing compleel a ranom. I i alo an aumpion ic unerlie man aumen meo b uiciou coice of meaure auiliar variable. A miing aa mecanim ic oe no occur a ranom given available auiliar variable i ai o be informaive or non-ignorable in relaion o e relevan urve variable. Conier, for eample, iem nonrepone on a comple variable, for ic e iger e value of e variable, e more or ill en o be require of a buine of a given ize o rerieve e informaion. In uc circumance, i ma be a even afer conrolling for meaurable facor, uc a ize of e buine, e rae of iem nonrepone en o increae a e value of e variable increae. Iem nonrepone on i variable oul erefore be informaive in relaion o i variable. 8.3 Problem caue b nonrepone 8.3. A baic eing Te problem caue b nonrepone ill clearl epen on e a nonrepone i reae. For convenience of epoiion, a imple buine urve eing i coniere ere raifie imple ranom ampling i emploe an ere, in e abence of nonrepone, e populaion oal of a urve variable i eimae b e epanion eimaor H N. Here, i e ample mean in raum, N i e number of buinee on e frame in raum an H i e number of raa. Perap e imple a of reaing bo uni nonrepone an iem nonrepone i o emplo e ame eimaor i replace b e mean acro all reponing uni in raum ic provie repone o i variable. Te laer mean i enoe r, ere e ubcrip r inicae a i eimaor i bae upon reponen aa. Te eimaor of e oal i en N. r H r 7

133 8.3. Bia Wiin e eing in Secion 8.3., e epecaion of r ma be epree a E H ( ) N µ r,r ere µ,r i e mean of e urve variable in raum among oe o repon (R), an i epreion ma be compare i e epecaion of in e abence of nonrepone. E H () N µ ere µ i e mean of e urve variable in raum. Te ifference beeen ee o epecaion eermine e bia ariing from nonrepone. H ( r ) N ( µ,r µ ). bia Wriing µ,r0 a e mean of e urve variable in raum among oe o o no repon an R a e rae of repone in raum e ma rie µ ( R ), 0 R µ, R µ R an u an alernaive epreion for e bia i H ( r ) N ( R )(, R, R 0 ) bia µ µ (8.) Tu no bia arie if eier ere i no nonrepone ( R ) or if e reponen an nonreponen are e ame mean value of e urve variable iin raa, ic occur en miingne i ranom iin raa, a i en nonrepone i inepenen of e urve variable iin raa. In general, oever, i coniion ill no ol an nonrepone ill lea o biae eimaion of oal a ell a of oer populaion parameer Variance inflaion Wiin e eing again of Secion 8.3., e variance of r ill epen again on aumpion abou e miing aa mecanim. One imple aumpion, ic illurae e variance impac of nonrepone, i a e reponen iin raum form a imple ranom ubample of ize m among e n uni of e elece ample. In i cae e variance before an afer nonrepone repecivel are 8

134 var var () H H ( r ) N N n N m N S n S m (8.) ere S i e populaion variance in raum. Auming an approimael uniform repone rae acro raa an ignoring e finie populaion correcion, e variance ill be inflae b a facor rougl equal o e reciprocal of e repone rae. Nonrepone in oe raa i a ig ampling fracion an epeciall in compleel enumerae raa ill en o inflae e variance furer Effec of confuing uni ouie e populaion i nonrepone I ill ofen be ifficul o iingui uni nonrepone from a uni ic i ouie e arge populaion, for eample becaue i a ceae o be acive. If uc a uni i reae a nonrepone en bia ill uuall arie. Wen eimaing oal of variable uc a proucion, a value of zero oul be ue erea e reamen of nonrepone ecribe in ecion 8.3. ill effecivel ae e value a e raum mean, biaing e eimae upar. On e oer an, if a uni in e arge populaion fail o repon an i rongl reae a ouie e arge populaion en i ill en o lea o onar bia Effec of nonrepone on coerence Man variable appearing in buine urve are ubec o arimeic conrain. For eample, queion mig be ae on capial epeniure uner ree eaing a ell a on oal capial epeniure. Tere ma be inere no onl in e populaion oal A, B an C of e ree pecific pe of capial epeniure bu alo in D ABC, e oal capial epeniure overall. Hoever, iem nonrepone ma occur on ifferen buinee, for ifferen variable an o, if nonrepone i reae variable b variable a in 8.3., i i poible a e reuling eimae Â, B, Ĉ an D are no coeren, a i D Â B Ĉ. Man agencie ma vie uc incoerence a uneirable, in paricular becaue i ma confue uer. Impuaion provie one approac o ealing i i problem (ee Secion 8.6). 8.4 Quali meauremen 8.4. Repone rae Tere are man repone rae ic ma be calculae. Uni repone rae ma be calculae b ize raum an b inur raum an ma be eige ogeer acro raa. Cumulaive uni repone rae ma be calculae accoring o o man reminer ave been iue. Uni nonrepone rae ma be iaggregae b reaon for nonrepone, nonconac, refual ec. Iem repone rae ma alo be calculae for eac urve variable. Te baic efiniion of a repone rae i 9

135 number of reponing uni number of eligible ample uni ere an eligible ample uni i one ic i in e arge populaion. Te numeraor i uuall reail available. Tere ma, oever, be ifficulie in eermining e enominaor becaue i ma be ifficul o ecie eer ample uni ic o no repon are eligible. Some eimaion of i number ill generall be necear, bae for eample on pa eimae of ea rae of buinee. Repone rae ave ifferen ue, upon ic e coice of rae ill epen. One ue i o monior problem in aa collecion. For i purpoe, i ma be ueful, for eample, o recor cumulaive repone rae over ime folloing e iniial iue of form. Suc evience ma be relevan, for eample, o eciion abou e iming an number of reminer. Te principal concern ere i i e ue of repone rae for quali meauremen. A baic problem i a e repone rae i no irecl relae o e principal problem caue b nonrepone bia. I i, in principle, poible for nonrepone rae o be lo an bia o be ig an vice vera. Neverele, equaion (8.) oe emonrae an inirec relaion beeen repone rae an bia. If e repone rae R iin raa are ig en e nonreponen nee o be muc more ifferen from e reponen o acieve e ame level of bia a en e repone rae R are muc loer. Hig repone rae mig erefore be viee a a form of proecion again bia. Comparing uni repone rae beeen inur an ize raa ma be informaive for quali conrol of aa collecion bu ee rae nee ummariing if an overall inicaor of quali i o be eermine. Te a in ic ee rae oul be ummarie epen on e impac of nonrepone. A imple aumpion i o uppoe a e componen ( µ µ ) of bia in (8.) i proporional o e mean,r,r 0 of a given auiliar variable, uc a emplomen, iin raum. If i i alo aume for implici a e mean of e urve variable i proporional o iin raa e ma approimae e relaive bia b ere bia ( ) r r N ( R ) N i e raum oal of e auiliar variable. Uner ee aumpion i eem appropriae o eig e raum repone rae R b e raum oal if an overall meaure of quali relae o nonrepone bia i require. Te eige rae erefore ae e form N R 30

136 H R eige repone rae, (8.3) H ere R i e repone rae in raum an i e raum oal of an auiliar variable uge o be proporional o e principal urve variable of inere. For eample, if e auiliar variable i emplomen en i meaure ma be inerpree a e epece proporion of oal emplomen in buinee ic repon. In orer o reuce nonrepone bia i i common pracice o evoe greaer reource o repone caing i e larger buinee. For eample, in e Annual Buine Inquir, buinee of 00 emplomen are argee eavil. A a reul e repone rae R i iger in e larger ize raa an e eige repone rae ill be greaer an an uneige rae. Te ample verion of formula (8.3) can be epree a ere e um i over ample uni an e eig R eige repone rae (8.4) n, ere n i e ample ize in raum an eig. Generaliing e formula, e ma ae for a ample buine in raum i N n i e epanion eig for ample buine in eige repone rae in (8.3) eimaion eig for buine ize meaure for buine Suc a eige repone rae reflec e relaive imporance of ifferen ample uni roug bo eir eig in eimaion an eir ize, aume rougl proporional o e urve variable Meaure bae on follo-up aa Repone rae are, oever, unaifacor a meaure of quali. Even if a loer repone rae inicae e poibili of greaer bia, repone rae provie no informaion on o large a bia ma be. One approac o eimaing nonrepone bia i o follo up nonreponen (eier uni or iem nonreponen) an collec e urve informaion from ee buinee. To ource of bia can be aree in i a. Te mo imporan ource arie impl from e value miing ue o nonrepone. Tee are collece in e follo-up urve. A econ ource arie becaue ome aume nonreponing uni ma in fac be ineligible an vice vera. Follo-up enable ee o poibiliie o be iinguie. Of coure, 3

137 complee repone in e follo-up ill rarel be acieve in pracice an o e eimae of bia ariing from follo-up aa ill emelve be ubec o ome error. Mo buine urve are ubec o preure for e earl releae of reul. Someime i mean a preliminar eimae are eermine afer an iniial ime perio an final eimae are obaine afer a longer perio incluing perap furer reminer. An eimae of e bia in e preliminar eimae i obaine impl from a ifference beeen ee eimae an e final eimae. Ti iea ma be eene b collecing furer aa beon e perio upon ic e final eimae are bae. In i a e bia in e final eimae ariing from nonrepone can be eimae. In aiion o eening e perio available for aa collecion, oer more inenive meo of follo-up can be ue, in paricular i ifferen moe of aa collecion, uc a e elepone an peronal inervie. Recogniing e fac a full ucceful follo-up i no onl impracical bu col, elecive follo-up raegie ma be coniere, focue oar larger uni ic ma be epece o mae a greaer conribuion o e bia Comparion i eernal aa ource an bencmar An alernaive approac o eimaing nonrepone bia i o mae comparion i eernal ource, uc a oer urve, aminiraive ource or rae organiaion aa. Naional accoun ource ma alo provie bencmar for comparion. To in of comparion are poible. Fir, comparion beeen overall eimae ma be mae. In i cae ifference beeen eimae ma reflec no onl nonrepone bia bu alo oer ource of bia uc a meauremen error, an i ma be ifficul o ienangle ee ifferen ource. Moreover, ifference beeen eimae ma reflec bia in eier e eimae of inere or in e comparaive ource an again i ma omeime be ifficul o eparae ee effec. See e caper on meauremen error (caper 6) for an eample of a comparion beeen a mail urve an an inervieer urve, ere ifferen rae of nonrepone arie. A econ in of comparion ma be uneraen en e urve reponen (an ieall nonreponen) ma be mace o recor in e eernal ource. Te mo obviou eample i ere e eernal ource i e buine regier from ic e ample a ran. In i cae comparion ma be mae beeen reponen an oer uni in e eernal ource i repec o variable available in a ource. Anoer eample i e comparion of urve repone i auie accoun, aloug ee ma onl become available ome ime afer e urve. Suc comparion ma ill be ueful for aeing nonrepone bia even if e variable in e eernal ource are ubec o meauremen error, o long a e are ufficienl correlae i e urve variable of inere Comparion of alernaive aue poin eimae In ecion 8.5 an 8.6 e conier eiging an impuaion meo aime a auing for nonrepone bia. Tee aumen meo are bae upon rong aumpion, in paricular a nonrepone occur a ranom given value of cerain auiliar variable (ee ecion 3

138 8.. for e efiniion of miing a ranom ). Deparure from ee aumpion ma be epece o lea o biae in e aue eimaor. Some aemen of bia ma be mae b comparing eimaor bae upon ifferen aumpion, pecificall uing ifferen coice of auiliar variable. In aiion, e poibili of informaive (non-ignorable) nonrepone (ee ecion 8..) ma be coniere. Alernaive plauible moel for informaive nonrepone mecanim mig be pecifie an en e impac on eimaion coniere. Wa in ic i mig be one are icue furer in e caper on moel aumpion error (caper 9). I ma be poible o evelop pecial eimaion proceure uner e pecifie informaive nonrepone mecanim a Copa & Li (997) ave one for cerain moelling purpoe. Alernaivel, imulaion-bae proceure mig be emploe. Perap e mo raigforar approac i o ae a complee e of recor from e ample aa an rea i a if i i an arificial ample. Ne, miing value ma be creae in i arificial ample accoring o aume nonrepone mecanim (ic ma emelve ave been arrive a b fiing moel o e original aa ubec o nonrepone). Eimae ma be compue from e ne aa accoring o e anar proceure emploe in e urve an ee eimae ma be compare i eimae obaine from e full arificial ample. Te proce of creaing miing value oul preferabl be repeae an e bia an variance of e eimaor uner e pecifie nonrepone mecanim eimae a in an imulaion u. 8.5 Weiging aumen 8.5. Te baic meo Te populaion oal of a urve variable i eimae b, ere e um i acro reponen. Te baic iea i a eac reponing uni repreen populaion uni. Te eig ma be epree a nr, ere i e ampling eig an nr e nonrepone eig. Variou meo ma be ue o conruc e eig. In pracice a ingle e of eig ill uuall be ue for all urve variable. Ti i eirable no onl for implici of compuaion bu alo o enure a arimeic relaionip beeen variable (for eample oal capial epeniure i e um of e componen of capial epeniure) are preerve in e eimae. For i reaon eiging, i e anar proceure ue o au for uni nonrepone (ic applie o all variable in a uniform a) bu i uuall unuiable for iem nonrepone, ince ifferen eig ill be necear for variable for ic value are miing for ifferen uni Ue of auiliar informaion In orer o reuce nonrepone bia i i necear o ue auiliar informaion abou uni ic are no reponen. To broa in of informaion ma be ue. Fir, cerain informaion ma be available on nonreponen in e ample bu no for oer populaion uni. One eample arie in a monl buine urve en e ample coni of e ame 33

139 buinee eac mon. In i cae informaion ma be available on ample buinee in Februar, a, ic ma be ue o eig for nonrepone in Marc. Suc eiging i calle ample-bae eiging. Quaniaive informaion on nonreponen, uc a repore value from e previou mon in a monl urve, i more liel o be ue for impuaion an for eiging. Caegorical informaion, uc a an inurial claificaion, mig be ue o efine repone omogenei group iin ic e nonrepone eig ma be eermine b e invere repone rae. Te econ broa in of informaion i a available on e ole populaion, mo obvioul informaion recore on e buine regier. Weiging meo bae on uc informaion are calle populaion-bae eiging. Te folloing o ecion concern ifferen meo of uc populaion-bae eiging Poraificaion Ti meo i applicable en a claificaion of buine i available ic a no ue for ampling. Te claificaion pariion buinee ino poraa g, ere e number of buinee N g iin poraum g i non. An eample arie en e claificaion of buinee b inur or ize i upae an coniere o be more accurae an e original claificaion ue for ampling (Hiiroglou e al., 995). Te poraifie eimaor of a oal ae e eige form nr in ecion 8.5., ere e nonrepone eig for all uni in poraum g i nr N g N g, an N g i obaine b umming e ample eig acro reponing uni in poraum g Regreion eimaion an calibraion Poraificaion i a pecial cae of regreion eimaion ic ielf i a pecial cae of calibraion eimaion (Deville & Särnal, 99; Lunröm, 997). Meo of raio eimaion ue iel for buine urve are alo pecial cae. Te imple approac o anling uni nonrepone in ee meo i o rea e reponen a e acieve ample i incluion probabiliie proporional o e ample incluion probabiliie. If e regreion relaionip beeen e urve variable an e auiliar variable i e ame for reponen an nonreponen, e correponing regreion (or calibraion) eimaor ill remove bia ue o nonrepone (Hiiroglou e al., 995, p.49). Ti i eeniall e miing a ranom coniion referre o earlier. Uner eparure from i aumpion, regreion eimaion ma ill be ueful for reucing nonrepone bia. A more comple approac involve fir auing e ample incluion probabiliie b eimae nonrepone probabiliie. Beleem (988) argue a i aumen ma be epece o reuce bia Weiging an nonrepone error Weiging ma be epece o affec bo e bia an e variance ariing from nonrepone. Te aim i o remove nonrepone bia aloug, in pracice, i i unliel o be full 34

140 acieve. A comparion of alernaive eige eimaor provie ome iea of o bia ma var accoring o ifferen aumpion. Tee aumpion ill be of e form miing a ranom given meaure auiliar variable. Tee auiliar variable mig, for eample, be oe ue o efine repone omogenei group in e ample, or o efine poraa for populaion eiging. A comparion of eige eimaor erefore repreen a eniivi anali i repec o a limie e of aumpion. Weiging ill alo generall affec e variance of e oal urve error in o a. Fir, poraificaion an more generall calibraion eiging can ac o reuce e variance if e auiliar variable ue elp o preic e urve variable iin raa. Secon, variabili in e eig can inflae e variance an i variance inflaion en o increae a e amoun of auiliar informaion increae (Nacimeno Silva & Sinner, 997) Variance eimaion Tere ei a number of variance eimaor in e preence of nonrepone. Te imple i o rea e nonrepone eig a fie quaniie for ic variaion beeen eig inflae e variance. Ti approac fail o allo for e reucion of variance acieve b populaion eiging. Ti variance reucion i alloe for b anar variance eimaor for calibraion eimaion (for eample Deville & Särnal, 99). More complicaion arie if ample-bae eiging i alo involve. In i cae, more complicae variance eimaor are require, ic inclue componen bo a e ample level an a e reponen level (Särnal & Senon, 987; Lunröm, 997). All of ee eimaor effecivel mae a miing a ranom aumpion an u o no allo for e poibili of informaive nonrepone. See e caper on moel aumpion error (ecion 9.7) for furer icuion of i cae. 8.6 Impuaion 8.6. Ue Impuaion i ue generall for iem nonrepone an, in paricular, for allocaing acivi beeen e componen, for eample local uni, of an enerprie en onl aggregae value are repore. Impuaion ma alo be ue for uni nonrepone, epeciall for buinee in e compleel enumerae raum ere previoul repore value ma be poerful preicor of miing value Deucive impuaion an eiing Te imple form of impuaion involve e ue of logical relaionip beeen variable an i uuall performe a par of e eiing proce (Hiiroglou & Berelo, 986). For eample if e oal of non-negaive variable i recore a zero, en e value of ee variable can be impue a zero La value impuaion For frequen (for eample monl) urve a ver imple impuaion meo i o ue e mo recenl repore value. 35

141 8.6.4 Raio an regreion impuaion A imple moificaion of la value impuaion i o cale i value i a raio of eimae bae on e curren an previou value. Tu, if i e repore value of uni a mon en, e value miing a mon, mig be impue b r. r Here e mean r an r of repore value a mon an repecivel mig be obaine from buinee in a imilar inurial claificaion an ize. Ereme value mig be rimme en calculaing ee mean, o avoi oulier aving eceive influence. Ti approac i paricularl uie o variable ic o no var greal over ime. More generall, a linear regreion moel β mig be fie o e urve variable p p i miing value, i e covariae p incluing previou value of e urve variable a ell oer variable, for eample oe on e buine regier. Te impue value ma en be aen a e uual preicor p β p, ere β p i e lea-quare eimaor of β p. Buine urve en o be ell uie o uc meo ince rong correlaion beeen variable are common Donor meo Raio an regreion meo mae efficien ue of auiliar informaion bu are no uie o ever applicaion. Te are ifficul o appl o miing value in caegorical variable an, ince e are uuall applie variable b variable, e ma no preerve relaionip beeen variable. In uc circumance, onor meo uc a o ec impuaion ma be ueful. A onor uni i elece ic i a imilar a poible o e uni i miing value an e value from e onor are ue o impue one or more miing value. Similari ma be meaure for eample accoring o ize an inurial claificaion of e uni (Kovar & Wirige, 995) Socaic meo A furer problem i raio an regreion meo i a e en o reuce e variaion in e variable impue. Ofen onl naional oal are of inere an i enenc ill no be of concern. Hoever, if iribuional quaniie are of inere, bia ma arie. For eample, if e proporion of buinee performing poorl accoring o ome crierion i of inere, an e impue value en oar e cenre of e performance iribuion, i proporion ma be unereimae. Ti problem ma be aree roug e ue of ocaic meo of impuaion (Kovar & Wirige, 995). For eample, e regreion impuaion p β p in ecion mig be replace b e ocaic regreion 36

142 impuaion p β p e, ere e i a ranom reiual, obaine b raing a reiual a ranom from oe ariing in e regreion anali ue o obain e β p Impuaion an nonrepone error Lie eiging, impuaion ma be epece o affec bo e bia an variance ariing from nonrepone. Regaring bia, ere are o broa conieraion. Te fir one i e mo obviou an applie equall o eiging. Te ucce of impuaion in removing bia for e eimaion of caraceriic of a given urve variable ill epen on o ell e impuaion moel capure e iribuion of e miing value. Comparing e reul of ifferen impuaion meo ill provie ome evience on e ize of uc bia. A econ, more uble conieraion i a impuaion can inrouce bia in eimae ic epen on more an one variable if ee variable are no full conrolle for in e impuaion. Conier, for eample, a variable ic ae e folloing value for a buine December 000 Repore Januar 050 Nonrepone (000 impue) Februar 00 Nonrepone (000 impue) Marc 50 Repore Suppoe a bo e Januar an Februar value are miing an are eac impue b e la value 000 (ee ecion 8.6.3). Suppoe a an eimae i require for e number of buinee ic ave cange b over 00 from Februar o Marc. Te above buine ill be erroneoul claifie in i caegor an impuaion ma lea o an upar bia in e eimaion of i number. Ti coul, in principle, ave been avoie if e Marc figure a been ue alo o impue e Februar figure bu, in pracice, uc reviion are ofen viee a uneirable. Impuaion ma alo be epece o ave an impac on e variance of e eimaor. In general, e ma epec e variance o become V amp Vimp, ere V amp i e ampling variance ic oul ave arien in e abence of nonrepone an V imp i e aiional variance ariing from impuaion. Te ize of i erm ill epen on e form of impuaion. Te erm V imp ill en o be maller for meo of raio or regreion impuaion ic are bae on moel i ig preicive poer. Te erm V imp ill en o be larger for meo ic ave le preicive poer, for eample la value impuaion, an for ocaic meo. Kovar & Wirige (995) ugge a impuaion can inflae e variance b o 0 percen in e cae of a 5 percen nonrepone rae or b 0 o 50 percen in e cae of 30 percen nonrepone. 37

143 8.6.8 Variance eimaion An imporan problem for quali meauremen i a e variance impac of impuaion i muc arer o eimae an a of eiging. Te imple approac i o rea impue value a real value an o ue e uual eimaor of ampling variance. Unforunael, i ill uuall unereimae e variance becaue no accoun i aen of V imp, e aiional uncerain ariing from e fac a e impue value ill, in pracice, no equal e rue value. Te egree of unereimaion ma be evere in buine urve, in paricular becaue e uual eimaor of ampling variance ae no accoun of impuaion error among large buinee in e compleel enumerae raa. Conier, for eample, e ue of a eparae raio eimaor. Te convenional variance eimaor, reaing e impue value a real, ae e form H b analog i epreion (), ere N ( m N ) m (incluing impue value) are available,, (5) m i e number of uni in raum for ic aa N i e correponing populaion ize an i e ample variance of e reiual (reaing e impue value a real). Auming raio impuaion i emploe uing e ame auiliar variable a in e raio eimaor, e acual variance oul be: H N * * ( m N ) S m (6) * ere m i e number of obervaion in raum ecluing impue value an S i e variance of e reiual in e abence of iem nonrepone. Auming raio impuaion a above, eac of e reiual in S correponing o an impue value ill be zero an ( m * ) S ( m ). Tu e erm ( m N ) m * * correponing erm ( m N ) S m in (6) b a facor * * ( m N )( m ) m * [( m N )( m ) m ] in (5) en o unereimae e ( m N ) * ( m N ) Te amoun of unereimaion ill en o be large if eier e ampling fracion m N i large, epeciall for compleel enumerae raa i m, or if e fracion of impue value ( m * ) m m m N i large. A imple aue variance eimaor ae e form H N * ( m N )( m ) * * [ m ( m ) ] an involve appling a correcion o e anar variance eimaor iin eac raum. Ti eimaor aume a e ame auiliar variable i ue for impuaion a for * 38

144 eimaion. Ti ill ofen no be e cae. An alernaive approac o aumen i o replace e impue value b aue impue value for e purpoe of variance eimaion. Suppoe, for eample, a impue value are of e form * β, ere i a previou value recore for buine an β i a raio. Ten, for e purpoe of variance eimaion * mig be replace b ** * ε, ere ε i a ranoml generae value from a normal iribuion i mean zero an variance σ. Te problem en i o cooe e in uc a a a e anar variance eimaor (reaing e * a real value) i approimael unbiae for e oal variance V amp Vimp. One approac i icue b Rao (996) in e cone of acnife variance eimaion. Särnal (99) ecribe an approac ic involve eimaing e componen V amp an V imp eparael. A furer approac i muliple impuaion ic involve creaing muliple aae i impue value an comparing e eimae obaine from eac (Rubin 996, Fa 996). None of ee meo eem o ave e foun eir a ino buine urve pracice in Europe, oever, an e evelopmen an implemenaion of pracical variance eimaion meo remain an ouaning reearc problem. σ 39

145 9 Moel Aumpion Error 9. Inroucion Davi Draper & Ruell Boaer 4, Univeri of Ba Te original goal of eign-bae anali meo in urve ampling a e evelopmen of a ampling eor a i moel-free (Cocran 977). Even iin claical eign-bae meo, oever, e incorporaion of auiliar informaion roug uc ecnique a raio an regreion eimaion i eeniall (if perap omea coverl) moel-bae. Toa overl moel-bae meo are commonl emploe in buine aiic, in e calculaion of ine formulae, in e ue of bencmaring an eaonal aumen (ere moel-bae oulier eecion an correcion are crucial), an in eimaion en no aa for a ub-populaion are available (for eample, enerprie a fall belo a ize reol, a in cu-off ampling, or mall-area eimaion from aggregae aa). Moel are u ubiquiou in e anali of buine urve aa (ee, for eample, Särnal e al. 99), an e aumpion e mae mu be criicall reviee i an ee o quanifing moel aumpion error. We ave alrea encounere e ue of moel in everal previou caper; in paricular, in ecion.3. e eamine e iea of reaing e populaion from ic e ample a an a ran a ielf a ample from a uperpopulaion pecifie b a moel. An eample of i iea a i relevan o moel aumpion error came up in e icuion of quoa ampling in ecion 4.3: if e populaion value in e cell of e quoa-ampling gri are aume o be ranom variable i E ( ) µ, an V ( ) σ, ere inee e cell in e gri in ic i oberve, en moel-unbiae eimae bo of e populaion oal (, a) an e variance of are available an coincie i e uual eign-unbiae eimae from raifie ampling. Hoever, i i equivalen o e moelling aumpion a e oberve value in e quoa ample are ocaicall iniinguiable from a one oul obain i imple ranom ampling (iou replacemen) from e cell in e gri, an ere i no a o compleel verif i aumpion from e aa. Error in i moel aumpion coul lea o a bia in e eimae of oe magniue an even irecion are ar o quanif. In e folloing ecion e eamine in urn e five leaing area in ic moel aumpion error appear crucial in buine urve: ine formulae, bencmaring, eaonal aumen, cu-off ampling, an coping i non-ignorable nonrepone. In e final ecion e offer ome recommenaion on be pracice in e reporing of poible moel aumpion error in buine urve. 4 We are graeful o Ra Camber (Univeri of Souampon), Eva Elver (Saiic Seen) an Paul Smi (UK Office for Naional Saiic) for commen an reference, an o Paul Smi for ome uggee e fragmen. Memberip on i li oe no impl agreemen i e iea epree ere, nor are an of ee people reponible for an error or omiion a ma be preen. 40

146 9. Ine number A noe b Jazairi (98), an ine number i a meaure of e magniue of a variable a one poin relaive o i value a anoer poin. Te variable in queion i ofen eier e price or e (ale) quani (or volume) of a commoi. Te poin in queion ma be ifferen ime, or locaion, or group of oueol; e ill focu ere on ime, meaure in mon. In e imple form of i iea ere are onl o poin in ime being compare; one, a (ofen e earlier ime-poin), i elece a e reference or bae mon, an e oer, a, i e curren mon. Conier a e or mare bae, C, of commoiie c,,c oberve a n ime, an le m pi an q i be e price an volume, repecivel, of commoi c i a ime. Te mone value of ci a ime i b efiniion impl e prouc vi piqi. Te raio p i pi of e price of commoi c i a ime o i price a ime i e price raio; e correponing fracion q i qi i e volume raio. In aemping o meaure o muc e price of e mare bae C a cange over ime, an ol (8 cenur) iea a impl o form e average m pi of e price raio; in e 9 cenur e German economi Lapère an m i pi Paace inrouce a refinemen of i iea ic i ill ue oa. Te Lapère price an volume inice, repecivel, are raio of eige um of e form m pi qi qi pi LP i i, LV m m ; (9.) p q q p i i i for eample, e Lapère price ine repreen e raio of e co of e bae mon mare bae a e curren mon price o i co a e price of e bae mon. Similarl e Paace price an volume inice, repecivel, are m m i pi qi qi pi PP i i, PV m m ; (9.) p q q p i i i u e Paace inice are imilar o oe of Lapère ecep a in Lapère eige um e eig are meaure in e bae mon an Paace' eig are oe in e curren mon. Wi an given mare bae, an bae an curren mon, e Lapère an Paace price inice ill picall no agree (eeniall for e ame reaon a e relaive cange of a quani q from ime o, (( q q ) q ), oe no coincie i e relaive cange from o, (( q ) q ) q ); e Fier ieal ine, e geomeric mean of e Lapère an Paace formulae, i frequenl ue a a compromie. Tere are man m i i i i i 4

147 variaion on e iea illurae ere; Jazairi (98) li no le an 4 pe of alernaive ine number. A imple eample of e role of moel aumpion in e creaion of ine number arie from reriing e Lapère price ine a LP m m m i p i q pi qi v i i i p i i i m m m i p i q i p i v i i p p v i i i, (9.3) ereb epreing i ine a a eige average of price raio, uing e value a ime a e eig. To prouce LP for ime, price raio an value for ime are neee; in pracice e value (ofen eimae from naional accoun) mig, for eample, refer o e previou ear an e price raio mig compare e curren mon i e previou December. A e ime en e ine i o be prouce, reliable value for ime are ofen no e available. I i en necear o mae an approimaion, for eample, o ae value referring o an earlier ear forar on e bai of ome aumpion on gro rae. An uc aumpion ill be moel-bae, eier implicil or eplicil, an e poibili of error in e moel aumpion ieall nee o be eplore. An eample of an eplicil moel-bae approac o e conrucion of price an volume inice i given b e erivaion of be linear inice. Teil (960), e originaor of i meo, aume a e price of e m commoiie move proporionael, apar from ranom flucuaion. A noe b Fi (977), one a o epre i aumpion i roug e moel m m pi i vi pi e, (9.4) p i i i in ic picall i pi vi i e average mone value recore a pen b a ample group of oueol on commoi i in ime perio, an p i pi i e price raio for commoi i obaine from an inepenen ource, uuall a urve of price in reail oule. Here e i reae a a ocaic error erm aume o ave mean zero, aloug Fi noe a in pracice non-ampling error ma prove more imporan an ampling error an e ma conain a bia componen ic i no necearil conan for all pair ( ),. To conruc e price an volume inice for m commoiie over n ime perio one ma form e n m price an quani marice P an Q, efine e mone value mari T M PQ, an obain e be linear price an volume inice p an q b uneige lea quare, a e vecor a minimie e um of quare of e elemen of e reiual mari T R M pq. In Secion 9.8 e icu o o ae e effec of error in e aumpion unerling moel uc a (9.4). 4

148 9.3 Bencmaring A goo efiniion of i opic i given b Colee & Dagum (994): Bencmaring iuaion arie enever o (or more) ource of aa are available for e ame arge variable i ifferen frequencie, for eample, monl veru annuall, or monl veru quarerl. Generall, e o ource of aa o no agree; for eample, e annual um of monl meauremen of a variable are no equal o e correponing annual meauremen. Furermore, one ource of aa, picall e le frequen, i more reliable an e oer, becaue i originae from a cenu, eauive aminiraion recor, or a larger ample. Te more reliable meauremen are coniere a bencmar. Traiionall, bencmaring a conie of auing e le reliable erie o mae i conien i e bencmar. Bencmaring, oever, can be efine more broal a e proce of opimall combining o ource of meauremen, in orer o acieve improve eimae of e erie uner inveigaion. Uner uc a efiniion, benc-mar are reae a auiliar obervaion. A pical eample of bencmaring i e folloing. In Saiic Canaa, e monl eimae of age an alarie originae from e Surve of Emplomen, Paroll, an Hour, erea e annual bencmar meauremen of e ame variable originae from eauive aminiraive recor, namel e income a form file b Canaian an compile b Revenue Canaa. Bencmaring au e monl aa o a e conform o e bencmar an preerve e original mon-o-mon movemen a muc a poible. Coninuing e cone of e la paragrap in i quoe, in i ecion e ae e lefrequen erie e bencmar o be annual an e more frequen erie o be monl, an e ue age an alarie a e oucome variable of inere. For mo of e pa 5 ear, mo aiical agencie orlie ave performe bencmaring uing one variaion or anoer of a meo propoe b Denon (97). In i meo, ic a no originall bae on a aiical moel for e o ime erie, e monl erie i require o eacl mac e bencmar, ic are regare a bining, bu a muc a poible of e mon-o-mon movemen of e original le-reliable erie i preerve. More recenl, eplicil moel-bae meo ave emerge for eample, oe of Colee & Dagum (994, ereafer CD) an Durbin & Quenneville (997), folloing on from or of Hillmer & Trabeli (987) ic aemp o generalize e Denon approac o increae e realim of e bencmaring. CD oberve a urve error (of e pe liel o affec e monl aa) are ofen eeroeaic an auocorrelae, an e urve ma be biae ue o facor uc a nonignorable nonrepone (ee ecion 9.7) an frame eerioraion over ime. Te propoe an improvemen o e Denon meo bae on a regreion moel a ae accoun of ee facor. Teir moel i a θ e, z m m θ m,,, T; m,, M. (9.5) 43

149 i e erie of monl meauremen, ecompoe ino e um of (i) a Here {, T} bia erm a; (ii) e unerling rue (unoberve) value of e age an alarie erie θ ; an (iii) urve error e, aume o aif E ( e ) E( ee ) 0 for all an. { m M} z m, i e erie of annual bencmar, poeniall ubec o e error m ic aif E ( m ) E( mm ) 0 for all m an (e e an m are aen o be muuall inepenen). If e bencmar are oug no o be ubec o error en e ma all be aen o be zero; in i cae e z m erie i bining. Moel (9.5) ma be rien in e familiar regreion form T ( u) 0, E( uu ) V z Xβ u, E, (9.6) ere e β vecor inclue bo a an e vecor θ of rue value. Auocorrelae error e in e monl erie can be accommoae in i meo b auming a e e follo a aionar ARMA(p, q) moel an compuing e (eimae) covariance mari V in (9.6) in erm of e eimae parameer of e ARMA moel. Weige lea quare, aing e T T reuling mari V a non, en prouce e uual eimae β ( X V X) X V z m, from ic complicae mari epreion for â an θ (ic e omi) ma be euce. * Heeroceaici in e e ma alo be anle b riing e e, ere e, e anar eviaion of e monl erie error, are alloe o var over ime, an leing e * e (no e e ) follo an ARMA moel. CD o a Denon-pe meo for bencmaring are a pecial cae of i regreion frameor, an e alo emonrae a eir approac i more efficien an Denon aumen uner a varie of ime erie moel for e e. Durbin & Quenneville (997, ereafer DQ) ae a ifferen approac o e conrucion of opimal bencmaring eimae, bae on ae-pace rucural ime erie moel. Teir approac aume an aiive error rucure for e annual erie, bu can anle eier aiive or muliplicaive beaviour of e monl erie. In e cae of aiive monl error, for eample, DQ aume a e monl erie follo e moel η u,,, T, (9.7) ere e η are unerling rue value, e are anar eviaion of e urve error, an e u are aen o be realiaion of a uni-variance aionar ARMA(p,q) moel. p, q, an e are aume non from ubanive nolege of e urve. Te furer aume a e annual bencmar ( z z ) T e regreion moel z are relae o η ( η η )T roug,, M,, T z Lη e e ~ N( 0, Σ ), (9.8), e 44

150 ere e marice L an Σ e are again aume non. A i e approac of Colee & Dagum (994), en e error vecor e i aume o be zero e bencmar are bining. Te ae-pace caracer of DQ' moel ener roug e aumpion a η µ γ δ ε, ε ~ N( 0, ), (9.9) ere µ accoun for an ren a ma be preen, γ moel e eaonal componen (if an), an δ i e raing-a aumen. Man moel are available for e ren an eaonal componen (for eample, Harve 989); DQ ue µ µ γ γ µ ω, ζ, ζ ~ N ω ~ N ( 0, σ ζ ) ( 0, σ ) ω σ ε (9.0) Te fir of ee equaion iel a conan linear ren if σ 0 bu oerie aap o a ime-varing lope; e econ force a conan eaonal paern if σ ω 0 bu permi i paern o var over ime oerie. DQ' moel i complee i e aumpion a e coefficien in e raing-a aumen follo e relaion δ δ ς, ς ~ N ( 0, σ ); (9.), ς ere once again, conan coefficien are obaine b eing σ 0, bu ime-varing ζ ς coefficien ma be accommoae oerie. All of e error erie ε, ζ, ω an ς are aume o be oinl inepenen of eac oer an of u. DQ ue anar Kalman filering an mooing (KFS) meo (ee, for eample, Harve 989) o fi i moel. I i clear from equaion (9.5-9.) a bencmaring meo in curren ue or recenl propoe are bae on moel i rong rucural an iribuional aumpion. Effec of error in moelling aumpion lie oe in bencmaring are icue in ecion Seaonal aumen Man buine ime erie eibi eaonal variaion, picall annual in paern en e erie i oberve monl. Harve (989) efine eaonal ren a a par of e erie ic, en erapolae, repea ielf over an one-ear ime perio an average ou o zero over uc a ime perio. Since uc ren conain no informaion on e general irecion of e erie, eier in e long run or e or run, eaonali i uuall eal i b eimaing i, ubracing ou e eimae, an uing e properie of e reuling eaonall-aue erie. Simple a oc eimae can reail be conceive for eample, a Cafiel (996) noe, For erie oing lile ren, i i uuall aequae o eimae e eaonal effec for a paricular perio (for eample, Januar) b fining e average of eac Januar obervaion [in e oberve ime erie] minu e correponing earl 45

151 average en e eaonal componen i oug o be a lea rougl aiive in caracer bu in pracice more complicae moel-bae meo are picall emploe. For eample, e UK Office for Naional Saiic (ONS) ue e compuer program X- ARIMA for almo all of i eaonal aumen. X-ARIMA involve e coice of an appropriae auoregreive inegrae moving average (ARIMA) moel (for eample, Bo, Jenin & Reinel 994) for forecaing obervaion a bo en of a finie erie, an i augmene erie i en pae roug a erie of Heneron filer (for eample, Kenn & Durbin 98) involve in (a) oulier eecion an removal/on-eiging, (b) coice of an appropriae filer for eaonal aumen (generall bae on e irregular-o-cclic (IC) raio) an (c) e aumen ielf. (Heneron filer are mooing ecnique bae on moving average ic aim o follo a cubic polnomial ren iou iorion (Cafiel 996)). Reearcer a e US Cenu Bureau (Finle e al. 998) ave recenl releae X- ARIMA, a upere of X-ARIMA bae on regarima moeling an inene o improve on e oler ofare in a number of a. A noe b ee auor, Te baic eaonal-aumen proceure of X-ARIMA an [i preeceor] X- ecompoe a monl or quarerl ime erie ino a prouc of (eimae of) ree componen: a ren componen, a eaonal componen, an a reiual componen calle e irregular componen. Suc a muliplicaive ecompoiion i uuall appropriae for erie of poiive value (ale, ipmen, epor, ec) in ic e ize of e eaonal componen increae i e level of e erie, a caraceriic of mo eaonal macroeconomic ime erie. Uner e muliplicaive ecompoiion, e eaonall aue erie i obaine b iviing e original erie b e eimae eaonal componen.... Given a ime erie Y o be moele, i i ofen necear o ae a nonlinear ranformaion of e erie, f ( Y ), o obain a erie a can be aequael fi b a regarima moel. For eample, if Y i a poiive-value erie i eaonal movemen proporional o e level of e erie, one oul uuall ae logarim or, more generall, [or i] Y log logy log, (9.) ere i ome appropriae equence of ivior.... Le B enoe e bacif operaor, B. X-ARIMA can eimae regarima p,, q P, D, Q for. Tee are moel of e form moel of orer ( )( ) φ p r, (9.3) i D ( B) Φ p ( B )( B) ( B ) β i i θ q ( B) ΘQ ( B ) a ere i e leng of e eaonal perio [(picall )]. 46

152 Here a i a ie-noie IID erie i mean 0 an anar eviaion σ a, e i are r ime erie oug o be preicive of, an φ p ( z), Φ P( z), θ q ( z) an Θ Q ( z) are polnomial of egree p, P, q, an Q, repecivel. In e uual a i ARIMA moel, p an P are e orer of e auoregreive par of e non-eaonal an eaonal moel, q an Q are e orer of e moving average par, an an D are e number of ime e non-eaonal an eaonal par of e erie nee o be ifference o acieve aionari. Te ame efiniion appl o X-ARIMA. Te efaul ARIMA moel ue b e ONS i ( 0,,)( 0,,) (e fir moel ee b X- ARIMA, an accepe in e maori of cae, aloug oer moel are ue oo). A ifferen efaul moel, ( 0,,)( 0,,) i ue in ren eimaion. Te elecion of a Heneron filer for e main eaonal aumen par i auomaic bae on e IC raio, i coice beeen a 3-erm an 3-erm moving average for monl erie. Tere are everal level of aumen for more or le evere oulier removal, in eac cae i e mo apical obervaion compleel replace b an eimae conien i e moel, an i e eig of oer oulier reuce in e eaonal aumen. Te anal can cooe cerain aa poin o be mare manuall a oulier, bu i i more ofen one roug prior aumen in ic e reaon for an unuual obervaion i noe (for eample, a rie acion). Tee prior aumen can be emporar (unuual reiual a fee roug o eaonall aue erie) or permanen (aue aa alo ue in oupu). Te principal aumpion ic affec e ONS eaonal aumen meo an ence e final aa can u be ummarie a follo: (i) ue of X-ARIMA over an oer eaonal aumen ofare, i implici reliance on Heneron filer in all cae (raer an, a, Kalman filer (for eample, Abraam & Leoler 983; ee belo) or raigforar Bo-Jenin-le ARIMA moelling); (ii) coice of level of oulier eecion/reamen; (iii) ue an een of permanen/emporar prior aumen; an (iv) e eail of e ARIMA moel ue for forecaing beon e en of e finie inpu erie. Te effec of error in moelling aumpion uc a ee ill be coniere in ecion 9.8. Te problem i paricularl ifficul becaue eaonal aumen i an aemp o eimae a couner-facual oucome namel, a value e ime erie unergoing eaonal aumen oul ave eibie a ere been no eaonal effec o a no gol anar (rue) value are available for comparion. A leaing alernaive o e X(X)-ARIMA approac o eaonal aumen i foun in e program TRAMO an SEATS evelope b Gomez & Maravall (994a, b) a e Ban of Spain an no in ieprea ue rougou Europe. TRAMO (Time erie Regreion i ARIMA noie, Miing Obervaion, an Oulier) i a regarima moel-bae meo ic eimae miing aa, ienifie an oneig four in of oulier, an cope 47

153 i pecial circumance uc a olia an calenar effec. TRAMO can be ue a a pre-proceor o SEATS (Signal Eracion in ARIMA Time Serie), ic ue minimum mean-quare error meo o ecompoe e erie ino ren, eaonali, ccle, an irregular componen. Finle e al. (998) oberve a e TRAMO-SEATS proceure i equivalen o e moifie Kalman-filer of Kon & Anle (986), ic een e approac propoe b Jone (980) o e cae of moel i ifferencing an miing aa in e fir D ime poin. Tee auor foun in a comparion of X-ARIMA an TRAMO-SEATS on aa in ic obervaion a been eliberael e aie an mare miing a e eimae of e miing value from bo proceure ere ala cloe o eac oer, an ere alo uuall quie cloe o e value of e elee aum (< % error). Maravall (998), in i icuion of e Finle e al. paper, aer a TRAMO-SEATS i uperior o X-ARIMA in ome repec, bu Finle e al. emonrae in eir reoiner a e o approace ofen prouce imilar reul (ee Euroa 998b for aiional comparion). 9.5 Cu-off ampling Coninuing e icuion of cu-off ampling in caper 4, conier a populaion of N companie an le be regier emplomen a ome fie ime poin of inere, ore from large o malle, i e correponing urnover value. Te oal urnover N i o be eimae. Le. N In cu-off ampling ill be non, bu onl (a mo) e fir of e ill be oberve, ere (in one leaing applicaion of e meo) i e malle ineger uc a ( ε ) (9.4) for ome 0 < ε < (picall on e orer of ). Wi i approac complee enumeraion of all of e {, } ma be uneraen, or a ample ma be coen; e focu ere on e former cae. Having ienifie, i i ueful o efine N * *, ere an o ecompoe ino e um. In ecion 4.5. e eamine e approac o eimaing bae on ignoring * (in effec, eimaing i b 0); ere e conier e effec of moel aumpion error on aemp o eimae *. Perap e imple eimae i obaine b efining (unverifiable) aumpion a N * an maing e 48

154 * * ( N ) ( N ) N N. (9.5) If (9.5) ere rue en coul be eimae b * ( ) *, raio ic i recogniable a a imple raio eimaor. * (9.6) For eample, in one of e imulae populaion bae on e 996 ABI (Annual Buine Inquir) aa ecribe in Secion 4.5, ere ere N,453 companie in e populaion, i oal regier emplomen acro all N companie of 69, 03 an rue oal urnover,739,96. Uing an ε value of 0. (cuing off 0% of e emploee, o o pea) iel 699 companie in e ample; for ee companie 35, 4 an 8,884,96. In i cae, ignoring * alogeer oul iel an eimae of, ic i biae lo b ( ) %,739,96 8,884,96,739, If inea aumpion (9.5) ere mae, e reuling raio eimae oul be 8,884,96 ( ) 69,03 3,599, 904, ic i biae ig b 8.6%. raio 35,4 In i eample e raio eimaor acieve a bia reucion of (. 8.6) 3. 35% 3 larger improvemen are poible. To ee require moivaing raio eimaion from a moel-bae perpecive an looing for moel aumpion error. I can be on (ee Cocran 977 or Särnal e al. 99) a if e N populaion value (, ) are emelve aume o be a ranom ample from a uperpopulaion in ic, bu β e, (9.7) ere e e are inepenen of e an aif E( e ) 0, V( e ) σ, en ( ) raio i be linear unbiae for i an ample, ranom or no, elece olel accoring o e value of e. Tu, in i paricular ene, e moel unerling ( (or, a lea, a ) raio ) raio leaing iuaion in ic ( oul be epece o perform ell) i a linear regreion roug e origin of e on e, in ic e variance of i proporional o. 49

155 50 Regier Emplomen Reurne Turnover *0^5 4*0^5 6*0^5 8*0^5 0^6.*0^6 Regier Emplomen Reiual *0^5 0 *0^5 4*0^5 6*0^5 8*0^5 0^6.*0^6 Figure 9. Scaerplo (lef panel) an reiual plo (rig panel) from fiing moel (9.7) o e ABI imulae populaion. Sanar aiical/economeric moel-cecing meo uc a caerplo an reiual plo are elpful in evaluaing e fi of moel (9.7). Te lef panel of Figure 9. i a caerplo of reurne urnover again regier emplomen for e 699 ample companie in e ABI eample above, i e fie line from e raio eimaion moel uperimpoe. I i evien from e arpl non-ellipical ape of ee plo a lea quare even eige lea quare i no maing e be ue of e bivariae aa ( ),, an i i alo clear a e eimae lope i quie poibl being riven b a mall number of poin i large regier emplomen value. A anar reme for i i o rim a mall fracion, a 00γ %, of poin i e large before eimaing e lope, ere γ i perap in e range Denoe e reuling populaion oal eimae b ( ) rim raio. Anoer anar approac o eimaing * arie from relaing e aumpion of a zero inercep in fiing a linear moel o e ( ), pair. Figure 9. oe appear o inicae ome or of eeroeaici (a i, ( ) V i no conan i varing ), bu e

156 rong cluering of e poin near e origin mae i ifficul o ee a form e variance funcion oul ae. Auming conan variance a a aring poin amoun o fiing e moel ( e ) 0, ( e ) σ 0 β e E V β (9.8) b orinar (uneige) lea quare (OLS), leaing o e folloing eimae of oal urnover: N ( ) ( β ) reg β (9.9) 0. A i raio eimaion, i i enible o rim e 00γ % of poin i e large before eimaing e lope, ieling e eimaor (. Wi or iou rimming, regreion (raer an raio) eimaion ma be a poor coice in cu-off ampling, leaing o even more biae eimae an oe prouce b e unrimme raio meo: i e eample aa β, β 805.7, 5., leaing o ( ) 8,03, 374, ic ) rim reg given above, for inance, ( ) ( ) 0 i biae ig b 8.9%. Wa i ore, i meo oe no even guaranee poiive preice urnover value (aemping o repon o an eeroceaici a ma be preen, b uing eige lea quare i a free inercep parameer an i on e ra cale, ma alo fall vicim o negaive oal urnover eimae). Te naural aa-analic oluion o ee problem i o fin a cale for e (, ) value on ic OLS perform ell (an on ic e eimae oal urnover canno be negaive). Te vigorou buncing up of e poin in e loer lef corner of e caerplo ugge a logarimic ranformaion for bo variable. So le log( ) an log( ), an regre on uing OLS, obaining inercep an lope value (on e log cale) β 0 an β, repecivel. Ten olving e equaion ( ) ( log ) β log( ) log 0 β reg (9.0) for ŷ iel a log-log regreion eimae of oal urnover: ( ) reg ( log) N e β 0 β. (9.) (Ti eimae i biae ue o e nonlinear naure of e log ranformaion an coul poeniall benefi from bia aumen, bu e bia i mall in i eample, a Table 9. ill emonrae.) Figure 9. parallel Figure 9., i ime on e log-log cale. Wi i ranformaion e poin-clou i nicel ellipical (ecep for e lef-runcaion caue b cuing off e malle companie), an OLS oul perform efficienl. Wi e eample aa given ere, 5

157 5 log( Regier Emplomen ) log( Reurne Turnover ) log( Regier Emplomen ) Reiual Figure 9. Scaerplo (lef panel) an reiual plo (rig panel) from fiing a linear moel on e log-log cale o e ABI imulae populaion. e reul are ( ) ( ) ,, 0 β β an ( ) ( ) 0,885,766 log reg, ic iffer from e rue populaion value b onl 3.9% on e lo ie. On e log-log cale regier emplomen an reurne urnover ave a ample correlaion of 0.84 (e correponing figure on e ra-ra cale i 0.56), an regreion eimaion on i cale can mae effecive ue of i relaionip. Tere i no nee o rim an poin i i approac, becaue e log ranformaion a remove e ig-leverage naure of e companie i large regier emplomen. Table 9. preen e reul of a imulaion comparing e ree oal urnover eimaor ( ) raio, ( ) ( ) ra reg an ( ) ( ) log reg. A in Secion 4.5 e repeael (00 ime) (a) re a ample of ize,453 (e ABI erac ample ize in 995) i replacemen from e ABI aa bu i unequal elecion probabiliie eermine b e ampling eig, o creae a peuo-populaion reflecing e acual iribuion of UK companie, an (b) ue e regier emplomen variable in i populaion o cu off e loer 00ε% of e companie (b cumulaive emploee number); bu i ime e (c) eimae e oal reurne urnover i eac of e ree eimaor uie ere, varing e rimming fracion γ in e cae of e fir o eimaor from 0.0 o 0.0.

158 Mean relaive bia (%) Opimal rim fracion ε Raio Raio rimme Regreion ra Regreion ra rimme Regreion log Raio Regreion ra Table 9. Simulaion reul i e 996 ABI aa. Te mean value of e rue populaion oal urnover acro e 00 imulae replicaion a 6,650,30. From e able i i evien a i i pe of populaion e unrimme raio eimaor i biae ig b an unaccepabl large margin for all bu e malle value of ε in fac, comparing ee reul i oe in Table 4.9, raio eimaion iou rimming i almo a ba a ignoring e cu-off uni alogeer. Hoever, afer rimming, e raio eimaion approac perform ver ell, i relaive error of le an 0-4%; unrimme regreion eimaion on e ra cale i even ore an unrimme raio eimaion, overaing e rue populaion oal b up o 33% a a funcion of cu-off fracion. Trimming elp, bu no enoug o mae e meo viable i ABI-pe aa; an regreion eimaion on e log-log cale (iou an nee o earc for an opimal rimming fracion) perform ver ell, ieling icrepancie beeen eimae an rue oal on e orer of onl -3% of e ru. Ti eample a illurae e value of bo (a) anar aiical moel-cecing meo uc a e eaminaion of reiual an (b) eniivi anali, eploring a varie of moel (in i cae, (9.7), (9.8), an e log-log moel leaing o (9.)) o oberve e effec of moel aumpion on e quaniie of irec inere. 9.6 Small omain of eimaion Mo of e icuion in i repor a o far focue on e eimaion of a oal or mean for e enire populaion of inere (an ecepion i ecion 3.). In man buine urve, oever, ere i alo inere in eimaing e oal or mean in ube, or omain, of e populaion. Someime ee omain are efine b variable along ic raificaion a aen place in e urve eign. In uc cae i i ofen poible o over-ample rare ubgroup (or mall omain) o obain accurae omain-pecific eimae, iou acrificing muc accurac in e overall populaion eimae (ee, for eample, Cocran 977). In oer cae, oever, e omain are oo numerou for i raeg o or effecivel. A claic eample occur en a urve i carrie ou fairl parel over a ie 53

159 geograpic region, bu i i ill eire o mae eimae a e level of mall area iin e region. Becaue of e frequenc i ic i eample arie in pracice, mallomain eimaion i ofen referre o a mall-area eimaion, even en e omain are no efine geograpicall. In ecribing ere e maor moelling iue a arie in mallarea eimaion e follo cloel e noaion an piri of Camber (997); oer ueful reference on e ubec inclue Go & Rao (994) an Draper e al. (993). Conier a coninuou urve variable efine on a populaion U i non overall ize N. A ample of ize n uni i ran ranoml from U, i e main arge being e populaion oal or mean of. Le r an for e unample uni in e populaion, an le U be ivie ino mall area a,, A, i non ize N a. Afer e ample i ran one can ivie i up ino area-pecific ubample, of ize n a, an a econar goal i e eimaion of e area oal a or mean a. In man cae i canno be one iou e ai of a moel a ugge o informaion oul be combine acro e area, o improve e eimaion iin a given area (anoer name for i iea i borroing reng from all e area o eimae a an a for eac area a). Given a vecor of p covariae ic are relae o an available on eac uni in U, a pical moel-bae approac o mall-area eimaion oul aume a linear moel of e form () e 0, V() e V Xβ e, E σ. (9.) Here i e vecor of leng N conaining all e populaion value of, X i e N p mari of populaion covariae value, β i a p-vecor of regreion coefficien, e i an unobervable vecor of error, an e covariance mari V of e i aume non (an ofen iagonal). Uner (9.) a moel-unbiae eimae of e populaion mean i ere T β, (9.3) N r β T T ( X V X ) X V an e ubcrip in e equaion belo (9.3) refer o e ub-vecor an ub-marice coniing onl of e ample uni. Ti i a pical preicion-le eimaor of ŷ (ee Caper ): e ample value of in e populaion are ue irecl in e eimaion of e oal, an e unample value of are preice b e moel. Probabl e mo iel ue meo for eimaing e mean a of e mall area in e cone of a moel uc a e one above i neic eimaion. Te e aumpion in i approac i a e ame linear moel (9.) ol in eac mall area, a i, e relaionip beeen an i conan acro omain. Uner i aumpion i i enible o eimae β from e enire ample, bu en mimic e fir line of (9.3) in eac area eparael: 54

160 T a β, (9.4) N a a r a ere a an r a are e ample an unample uni in area a. Hoever, e omogenei aumpion unerling (9.4) ma ell no be rue. Maing i aumpion creae an eimae of a i relaivel lo variance (becaue e eimae of β borro reng acro e ole ample) bu poeniall large bia in an given area (if e conan-β aumpion i far from rue). One a o avoi e area-level bia poeniall ineren in (9.4) i o eimae e relaionip beeen an eparael in eac omain, b fiing e moel a ( e ) 0, V( e ) σ V. T β e, E (9.5) a a a a a a a Te irec eimae of a uggee b i moel i en a T β N a a r a a (9.6) ere β a T T ( X V ) a X a X a V a a a a i, impl eimae eparae regreion in eac area. Ti eimaor i moel-unbiae in eac omain bu ill picall ave large variance, ince e omain-level ample ize are uuall mall. Tu eac of e neic an irec eimae a poenial fla, in e irecion of large bia an large variance, repecivel, ic ugge earcing for a compromie eimaor. Te anar coice i bae on an epanion of moel (9.), T a ( a) e, β I α (9.7) in ic I ( p) i if propoiion p i rue an 0 oerie, an e (unobervable) α a are referre o a area effec (moel (9.) u correpon o e pecial cae a all of e area effec are zero). If e α a are regare a IID ranom variable i mean 0 an variance σ α, e reuling pecificaion i a ranom-effec moel; if e α a are impl parameer (repreening area-pecific eviaion from a common inercep) umming o 0 acro e omain, e reul i a fie-effec moel. Uner eier pecificaion e compromie eimaor ae e form a a ( ) T β α a a, (9.8) N a a r a 55

161 in ic eimae of β a an e α a are obainable from anar muli-level moelling ofare uc a MLiN (Golein e al. 997). Cooing beeen fie- an ranom-effec formulaion epen on e number of mall area an e relaive magniue of e iin- an beeen-area variaion in oucome of inere: for eample, i a large number of omain an a fairl large egree of beeen-area omogenei, a ranom-effec moel oul be inicae. Wen e omain are in fac geograpic area an ere i reaon o believe a aacen area oul eibi imilar repone, variaion on (9.7) bae on paial mooing are poible; ee Coling e al. (996). Oer refinemen of e meo ecribe ere inclue empirical Bae mooing of irec eimae (ee, for eample, Draper e al. 993) an mall-area eimaion of coun raer an coninuou oucome, bae on SPREE eimae (for eample, Purcell & Ki 980). A an eample of ee iea in acion, e UK Office for Naional Saiic (ONS) a in e pa ue a verion of irec eimaion in e Annual Buine Inquir (ABI; ee Secion 9.5 for anali of ome ABI aa). Te baic iea a a a raio-pe eimaor (bae on regreion roug e origin) a fie for eac urve variable i a ifferen parameer in eac raum, an en bae on auiliar aa from e regier a complee urve recor a mae for eac non-ample buine in e populaion o upplemen e ample repone (i i an applicaion of equaion (9.6)). Ti alloe cro-abulaion of reul for ver mall omain in more or le an conceivable combinaion, bu i no mae an commen abou e quali of e aa. In effec an eimae for a region (no par of e urve raificaion) a mae up of an eimae of eac cell in e region b raum croabulaion, i e appropriae irec eimaor ae o give an overall eimae. Ti relie eavil on e aumpion a e raa efine all e variabili in e aa, an a e ample are repreenaive. In curren pracice a ONS, mo mall-area eimae are for omain ic coincie i raa, an ence are no ubec o e uncerain ariing from aving o eimae e omain ize. To urve (one ean an monl, e oer planne an annual) mae omain-pe eimae for region from aa ic are no raifie b region, an en conrain ee eimae oal (a) o reprouce non auiliar variable oal an (b) o um o e ame overall eimae for e UK (a in of bencmaring; ee ecion 9.3). Some ONS urve (normall in ic e ample ize per raum i mall) ue combine raio eimaion, ic i bae on e aumpion of a conan raio (or regreion lope) over e ize raa; i i imilar o e neic eimaion meo (9.4) above. ONS oe no a preen ue muli-level moel, of e pe leaing o eimaor uc a (9.8), in buine urve, bu plan o eplore eir ue in e fuure. Te effec of moel aumpion error imilar o oe in mall-area eimaion ill be eplore in Secion

162 9.7 Non-ignorable nonrepone In caper 8 e icue e effec of nonrepone error on buine urve. Tree pe of aa miingne a e uni level ere efine in a caper: leing R be if ample uni repon an 0 if no, miingne compleel a ranom occur en R i ocaicall inepenen of e oucome(); miingne a ranom given an auiliar variable X occur if inepenen of given ; an 57 R i coniionall informaive or non-ignorable miingne occur en R an remain epenen even afer coniioning on (auing for) all available auiliar variable. Te o verion of miingne a ranom in i li are referre o a ignorable nonrepone, becaue in oe cae no bia in eimaing apec of e populaion iribuion of i inuce b e miingne (aloug, a caper 8 poin ou, miingne a ranom ill inflae oberve ampling variance, becaue e obaine ample ize i maller an e planne ample ize). Te main problem creae b non-ignorable nonrepone (NINR) i a, en i occur, eimae bae onl on e reponen ill be biae. In e eing of raifie ranom ampling eamine in ecion 8.3, for eample, equaion () of a ecion ummarize e bia in eimaing e populaion oal of a ere H bia H ( r ) N ( R )(, R µ, R 0 ), µ (9.9) N i e eimae of bae onl on e reponing uni an in ic r µ, R an, R 0 r µ are e mean in raum for e reponen an nonreponen, repecivel. NINR implie a ee o mean are no equal, an e greaer e ipari beeen em, e larger e overall bia in (9.9). Tu i NINR i i no necearil aequae impl o ac a if e reponen aa e, of oal ample ize H n r n r, i equivalen o a one oul ave obaine i an inene ample of ize n r a a no miingne. Tere i an operaional problem i i concluion, oug: o can one uge eer e miingne i ignorable, en b efiniion e value are no oberve for e uni i R 0? One approac o anering i queion in longiuinal urve i o conul e frame for variable a are goo proie for, for eample, in perio ma be rongl correlae i in perio (), an ma ell be available for man of e uni for ic R 0 a ime ; or one ma be able o compare ample reponen an nonreponen a ime i repec o eir value on auiliar variable, ic ave in e pa been rongl correlae i, a ime ().

163 An even greaer ifficul i a o o abou NINR en i i upece. Auming NINR, e onl a forar i evienl roug e pecificaion of a moel ic preic a e oberve value oul ave been for e uni for ic R 0. Tere appear o ave been lile or no emaic aemp in e lieraure o ailor e conrucion of uc moel o e buine aiic frameor (for eample, e UK Office for Naional Saiic mae no ue of NINR moel in e anali of an of i buine urve reul a preen). Aemp ave been mae in oer eing, oever, an in e re of i ecion e revie o leaing meo a appear of poenial relevance o buine aiic Selecion moel for coninuou oucome Copa & Li (997) anale aa from a local ill aui conuce a a ample urve in Covenr, UK, in 988. In one anali of n 435 aul non o be in full-ime emplomen (an aume o be ranoml ample from e populaion of uc aul in Covenr), e oucome of inere a income (poun per ee), i gener an age a e principal auiliar () variable. Tere a no miingne on e -value, bu 8% of e aul refue o provie income informaion, ieling a complee-cae ample ize of 33. A repone rae of 9% ma eem amirabl ig, bu ere a goo reaon o believe a e probabili of nonrepone a a funcion of income. Copa & Li ue elecion moel, an approac aing bac o e 970 in e economeric lieraure (ee, for eample, Hecman 979), o quanif e poible effec of NINR in i problem. Along i e oberve an value, ere i in general a vecor, e baic iea of ee moel i o poi e eience of an unoberve, or laen, variable z ic repreen e propeni o repon in e urve, an o relae (,, z) b e pair of regreion equaion z T β, (9.30) T σe γ ε in ic e pair ( e, ε ) i aen o be bivariae normal i E ( ) E( ) 0 V ( e ) V( ε ), an corr ( e, ε ) ρ e ε,. Te fir equaion in (9.30) mig be erme e obervaion equaion, e econ e elecion equaion, an applicaion o miing aa in urve arie b auming a i onl oberve if e laen variable z i poiive. Te correlaion beeen e error erm in e o equaion capure e premie a (i) e i a in of place-oler for a e of unoberve auiliar variable a oul elp o preic if e a been oberve, (ii) ε imilarl conain anoer e of unoberve auiliar variable z a oul elp o eplain e propeni o repon if e a been meaure, an (iii) e o e of variable in an z are liel o overlap, inucing a correlaion beeen e an ε. If ρ 0 ere i no informaion in e elecion equaion for preicing, ic implie ignorable nonrepone, bu if ρ 0 en i ubec o NINR. 58

164 Copa & Li fi moel (9.30) b profile maimum lielioo (ee Draper & Ceal 998 for a Baeian anali) an eamine e eniivi of reul o e poibili of NINR b calculaing eimae of e populaion mean for, an anar error of oe eimae, a a funcion of ρ. Te noe a For a ell-eigne an ell-eecue urve uc a [e Covenr ill aui] i i implauible a ρ oul be ver large. Wi an overall [nonrepone] rae of 8%, a fairl ereme poibili mig be a e probabili of miing aa a e loer quarile of [e iribuion of] i 4% erea a e upper quarile i i % (ree ime a large). Ti give a plauible range for ρ beeen 0.40 an 0.40, leaing o bia-aue populaion mean eimae in e range (38, 48) poun per ee a compare i e unaue eimae of 4. Tu i a nonrepone rae of onl 8%, e bia correcion o au for NINR in i eample i onl abou 34% of e unaue eimae, bu i i of e ame orer of magniue a e anar error of, o a (ince i i no clear eer e bia i poiive or negaive) e era uncerain [aace o ariing from e poibili of NINR] coul be oug of a oubling e [variance] of eimaion. Ti provie a concree ummar of e poible effec of NINR an (ieall) a o o abou ee effec: en uni-level nonrepone occur in a urve, if bo e irecion an e magniue of biae inrouce b e nonrepone can be quanifie, bae on reaonable moelling an pa eperience, en bia aumen oul be uneraen; an if e irecion an magniue are ar o pin on, en e anar uncerain ban bae onl on e oberve aa oul ien o acnolege e poibili of non-ignorable nonrepone Paern-miure moel for caegorical oucome Forer & Smi (998) eamine aa from e 99 Brii general elecion panel urve o quanif e effec of poible NINR on eimae of voing inenion (ic a caegorical a four level). In eir ranom ample of 4 iniviual e available auiliar variable ere gener an ocial cla (caegorical a five level), ic ere non for all ample people, bu 375 (30%) of e ample iniviual refue o mae eir voing inenion non. Denoing e vecor of auiliar variable b an e repone inicaor b R, e problem (a above) i o conruc oin probabili moel for (, R), a ill permi impuaion of a e oberve voing inenion oul ave been for oe people for om R 0. Maimum lielioo eimaion of voing inen bae olel on e oberve -value in e urve iele (Conervaive, Labour, Liberal Democra, Oer) (C, L, LD, O) (45.6%, 34.3%, 7.%, 3.0%). Uing e noaion of coniional inepenence evelope b Dai (979), e aumpion of miingne compleel a ranom correpon o R {, } (a i, R i inepenen of (, )), erea miingne a ranom given i epree a R. All oer moel aume NINR in one form or anoer. Differen moelling raegie correpon o ifferen facoriaion of e oin iribuion p (, R,), for eample, e facoriaion 59

165 (, R, ) p( ) p( R ) p( R) p, reuce, uner e aumpion of miingne a ranom, o e moel (, R, ) p( ) p( R ) p( ) p (9.3) for e full oberve aa, ic i in e cla of ecompoable grapical log-linear moel (ee for eample, Dai & Laurizen 993). Ti moel, approace in a Baeian a bu i prior iribuion on e parameer i lile informaion conen, iele i e above urve reul a i mu reul in cloe agreemen i e maimum lielioo eimae: (C, L, LD, O) (44.8%, 35.0%, 7.%, 3.%), i 95% uncerain ban [(4.3, 48.3), (3.6, 38.5), (4.5, 9.7), (.0, 4.5) ]. In eir cenral NINR moelling Forer & Smi emplo e facoriaion (, R, ) p( R, ) p( R, ) p, (9.3) a paern-miure pecificaion (for eample, Glnn e al. 986). Forer & Smi' main approac i a follo: A e are onl coniering non-repone on, ( R, ) again ] an (,, R ) require for inference abou p ( R, ) an (, R ) miing en R 0 an o... an inference for (, ) informaion concerning (, R 0) n [e cro-abulaion of R n are full oberve. Hence, e ave all e informaion p. Hoever, i compleel p require ome in of prior p. Ti prior iribuion oug perap o be referre o a e ubecive iribuion, a i remain unalere in e lig of e oberve aa.... An inuiivel aracive an compuaionall raigforar approac R, p, R an θ,, i o conier e parameer p ( ), ( ), [ere i e number of iinc value aen b ]. Te parameer θ repreen e een of prior belief in non-ignorabili. If θ 0 en i correpon o ignorabili of nonrepone for raum, an if all θ 0 en R an non-repone i [miing a ranom given ].... Hence, e θ are ea o inerpre an prior informaion regaring ignorabili ma be raigforarl incorporae ino e moel via a prior iribuion.... We cooe o ue mulivariae normal iribuion for θ, i mean µ an variance σ eermine b e prior belief concerning e een an rucure of non-ignorabili. Te parameer θ in i formulaion pla e role of e correlaion parameer ρ in e Copa & Li approac in ecion Tere a evience from e lieraure a nonreponen o poll in Brii general elecion prior o 99 ere more eavil pro-conervaive an reponen. Uing a reaonable prior pecificaion bae on i evience, Forer & Smi obaine aue eimae of (C, L, LD, O) (47.6, 33.0, 6.5,.9), i 95% inerval [(4., 53.0), (8.7, 37.6), (3.6, 9.7), (.9, 4.)]. In comparion i e reul above bae onl on e 60

166 reponen, e bia aumen ere on e orer of -3 percenage poin for e o large poliical parie, increaing e eimae lea of e Conervaive over Labour b 5 percenage poin (a large ifference in pracical erm), an e 95% uncerain ban ere on average 34% ier afer e poibili of NINR a accoune for. Forer & Smi alo provie a ueful formula for ample ize calculaion a eign ime o anicipae poible NINR: in eir frameor, e effec of alloing for non-ignorabili i n R, X in raum o o reuce e effecive oberve ample ize ( ) n n( R, ) ( R, ) σ [ n( R 0, ) n( )], (9.33) [ere i e number of oberve level of ]. Te proporion of reponen an nonreponen in eac raum ill no be non in avance an a prior eimae ill be require, a ill a prior pecificaion of σ, e amoun of uncerain abou o rong e NINR ill be in raum. Tee ing ma no be ea o pecif a eign ime, bu a i pical of urve eign, an in an cae (9.33) can erve a e bai of a eniivi anali. Effec of error in moelling aumpion imilar o oe ariing from aemp o cope i non-ignorable nonrepone ill be coniere in e ne an final ecion of i caper. 9.8 Concluion We ave een in e previou ecion a moel are ubiquiou in e anali of buine urve. Since a aiical moel i noing more (or le) an a collecion of aumpion abou e relaionip beeen oberve an unoberve aa, an ince b eir naure ome of ee aumpion are no non o be vali i cerain, aeing e impac of error in moelling aumpion i evienl crucial o e ucce of buine urve a emplo em. Tree eample of i ariing from Secion 9.3, 9.6 an 9.7 are a follo. On e opic of moel for bencmaring, Colee & Dagum (994) ami a In real cae, e gain in efficienc from e regreion meo [for bencmaring ic e avocae] ill epen on o ell e ARMA moel [for e monl erie o be bencmare] are ienifie an eimae. In mall-area eimaion Camber (997) conclue a A e ime of riing a general conenu on an appropriae robu meoolog for meauring e overall reliabili of mall-area eimae a no been reace, ic i one a of aing a moel aumpion error in mall-area eimaion ma ell ominae oer ource of error. Wi regar o non-ignorable nonrepone, Forer & Smi (998) repor on e reul of a follo-up urve of e 4 original paricipan in e 99 Brii general elecion panel urve: iniviual i no repon an 86 claime no o ave voe. Of e remaining 35, 44.% [repore voing] Conervaive, 3.% Labour,.0% Liberal Democra, an.8% oer. Of ee, 37 ere nonreponen o e original urve, for 6

167 om e correponing proporion ere 4.0%, 5.6%, 30.0%, an 3.5%. Tu in e en e original nonreponen repore voing in a a a a oll unanicipae far more rongl for e Liberal Democra (LD) an an eper oul ave preice ieling an overall percenage for e LD a fell ouie e 95% inerval from e paern miure moelling (even i i muc ier uncerain ban). Ti iglig e fac a even en reaonable moelling aumpion are emploe bae on eper nolege, occurrence ouie e realm of plauible prior epecaion can be lef unanicipae b e moelling. I oul appear a be pracice in ealing i moel aumpion error in buine aiic mace e iuaion in aiical moelling quie generall, in a o main ool are available: Te enible ue of moel iagnoic (ee, for eample, Coo & Weiberg 98); an A illingne o emplo eniivi anali (ee, for eample, Sene e al. 986): varing e moelling aumpion acro plauible range o icover eir effec on e eimae of e quaniie of principal inere. Ti ill ofen involve imulaion uie (ee, for eample, Hammerle & Hancomb 979). In e cla of linear moel, for eample, a uggeive (bu no eauive) li of caegorie of moelling aumpion or eploring mig inclue e folloing: Tranformaion of oucome an one or more preicor ; Coice of e funcional form b ic an e are relae; Aumpion abou e variance rucure an iribuion of e error erm in e moel; Coice of preicor variable from among a poeniall large e of ; an Coice of oulier reamen meo. Bo of ee approace, incluing a number of e moel aumpion caegorie lie ere, ere illurae in Secion 9.5 on cu-off ampling. Figure 9. give a caerplo of reurne urnover again regier emplomen in a imulae populaion bae on e 995 UK ABI urve, an a reiual plo obaine from fiing a regreion roug e origin i bo variable on e ra cale. Bo plo o (a) a number of ig-leverage poin (Weiberg 985) companie eering a large influence on e eimae lope, ic can ramaicall if e raio eimaor bae on e regreion moel (9.7) an (b) a rong buncing up of poin near e origin, ic implie a e eige lea quare meo ue o eimae e lope ma no be maing e mo efficien ue of e aa. Eac of ee problem ugge alernaive moelling aumpion. Difficul (a) i a robune problem (Huber 98), perap mo impl olve b mean of rimme regreion: e aie a mall proporion of e companie i e ige regier emplomen, an fi moel (9.7) o e remaining aa. Difficul (b) ugge a aaanalic oluion (ee, for eample, Moeller & Tue 977) bae on variable ranformaion: inea of regreing on, regre ( ). Ti line of reaoning iel ree main cu-off eimaor, bae on ree ifferen moel: (i) regreion log on log ( ) 6

168 roug e origin on e ra cale, emploing all of e aa; (ii) regreion roug e origin on e ra cale, rimming e ig-leverage companie; an (iii) orinar leaquare regreion uing all of e aa on e log-log cale. Evaluaing e quali of ee eimaor i an eercie in eniivi anali bae on imulaion: one ma (A) repeael generae imulae populaion imilar o e reali in ic e coen cu-off eimaor ill be emploe, compuing e rue populaion oal urnover in eac imulaion repeiion; (B) compue eac of e ree cu-off eimae for eac imulae populaion; an (C) evaluae e eimaion meo in erm of uc ummarie a relaive bia an/or roo mean quare error. Te reul, in Table 9., o clearl a for populaion lie e ABI aa rimme raio eimaion on e ra cale an regreion eimaion on e log-log cale perform ell. Ti oe no prove a ee meo oul or equall ell on oer populaion; imulaion-bae eniivi anali of i pe mu be emploe on a ie varie of populaion pe o ra uc a concluion, an an ineracion beeen populaion pe an eimaion meo ma ell be foun: meo (ii) or be i populaion pe I, meo (iii) or be i pe II, an o on. Tere i anoer varie of eniivi anali or menioning a ell: eamining e effec of moel aumpion on a ingle (real) ample raer an acro a number of imulae populaion an ample. In i approac one mae a li { A,, } A of moelling aumpion a all eem o be plauible for e given ample, bae on eper ugemen an moel iagnoic, an en one compue e correponing concluion {, } C,C reuling from e e of aumpion. Te reul of i pe of eniivi anali ma be ummarie eier qualiaivel or quaniaivel, a follo. Qualiaive ummar. Te iea i impl o ee if all reaonable roa lea o Rome, a i, o ee if acro e pan of plauible { A,, } e reuling {, } A C,C largel agree i regar o e quaniie of principal inere. If e o, en confience increae a moel aumpion error o no pla a large par in e rea o e urve' valii. If e o no, en i approac i more problemaic; one i lef i a qualiaive ummar of e form If aumpion A en concluion C, if A en C, (9.34) ic ma ell no be aifacor a a bai for eciion-maing bae on e urve. Quaniaive ummar. To go beon (9.34) one mu be illing o place eig on e relaive plauibili (a i, probabiliie) of e aumpion A i, o prouce a compoie ummar a reflec bo iin-moel an beeen-moel uncerain. Tere i no a ell-evelope Baeian approac o oing i (for eample, Draper 995): i a an oucome o be preice, moel i (bae on aumpion A i ) given probabili leaing o preicive iribuion for i mean an anar eviaion (SD) repecivel, p i an µ i an σ i, 63

169 ere V [ ] ( ) V Ê( ) i p i ( µ µ ) beeen - moel variance Ê i ( ) E Ê( ) Ê i [ V ( )] p σ i i iin - moel variance [ ] µ µ p i i i σ (9.35). (9.36) Tu e overall preicive uncerain abou ecompoe ino e um of {e uncerain coniional on a given e of moelling aumpion} an {e uncerain abou e moelling aumpion emelve}. Tere ma be ubanive an ecnical ifficulie in implemening i approac in pracice, oever, an i a no e been aempe i buine urve aa; i pe of moel uncerain aui i in e caegor of poible fuure be pracice in buine urve. We conclue i ecion, an e caper, b ummarizing e above icuion. Recommenaion: Be-pracice reporing in buine urve involving moel-bae meo oul Ue a blen of moel iagnoic, imulaion uie, an qualiaive eniivi anale o mae conumer of e urve aare of (a) e plauibili of e principal aumpion mae b e moel emploe an (b) e effec of varing ee aumpion, acro reaonable alernaive pecificaion, on e ummar eimae of principal inere. 64

170 Par 3: Oer Apec of Quali 0 Comparabili an coerence 0. Inroucion Eva Elver 5, Saiic Seen Coerence relae o e of aiic an ae ino accoun o ell e aiic can be ue ogeer. Saiic are eimae of finie populaion parameer (FPP), a ecribe in previou caper an in e ne ecion. Te arge i rarel acieve for man reaon. Te maller e icrepanc beeen e value of a aiic an i arge, e more accurae i e aiic. A aiic can be coniere a coniing of e um of e FPP an an eimaion error. Tere are o principal error par, emaic error (a ma lea o a bia) an ranom error. Te proucer normall aim a e bia being nil or negligible, an alo a ranom error being mall (cloe o zero in abolue or relaive erm). One a of ecribing e inaccurac i roug e roo mean quare error, anoer i an uncerain inerval. Te inerval coul be mmeric aroun e poin-eimae. Te uer a a e of FPP in min a e/e ie o u. Ten ere ma be aiic publie a ui ee ie off-e-elf bu ofen i i necear o ue everal e of aiic. Suc a uage ma inclue combinaion of everal FPP ino ne one. Te uer nee o no if ere are aiic i arge FPP a are equal or a lea cloe o i/er ieal. Coerence i a more general concep an comparabili. Queion on coerence arie for eample en proucion aiic an foreign rae aiic are ue ogeer, or proucion aiic an emplomen aiic, or annual aiic an or-erm aiic. In quali repor o Euroa, comparabili an coerence are o quali componen. Since ee componen ave muc in common e former being a pecial cae of e laer e are ere ecribe an icue in a ingle caper. Obvioul, comparabili beeen Member Sae (MS) i imporan o Euroa, an alo comparabili beeen counrie in general. Comparabili over ime i anoer comparabili apec. A preen Euroa oe uuall no inclue comparion beeen non-geograpical omain in e comparabili componen. Coerence apec are icue belo fir i empai on e uer in Secion 0. an en i empai on e proucer in Secion 0.3. Te rucure i largel e ame in bo cae, uing i ub-eaing, mainl a belo. efiniion in eor. efiniion in pracice 3. accurac an conien eimae 5 Several peron ave conribue i commen an eample, epeciall Ole Blac an Mar William a ONS 65

171 4. comparabili over ime 5. inernaional comparabili 6. concluing commen Te eample all refer o buine aiic, bu e eor i general for official aiic. Secion 0.4 i more illuraive, bae on ome naional iuaion. Summarie an concluion are given along i e e, largel in Secion 0..6 an Coerence empaiing e uer perpecive 0.. Definiion in eor A ae previoul, aiic are eimae of finie populaion parameer (FPP). Ingreien in uc a parameer are aiical meaure (oal, mean, meian, ec); variable (proucion, number of our ore, ec); uni (enerprie, in-of-acivi-uni, ec); omain (ub-populaion, for eample efine b a anar claificaion lie NACE Rev. ); reference ime; bo uni an variable value relae o pecific ime. Te reference ime are mol ime inerval, lie a calenar ear, a quarer, or a mon. (Hoever, ome variable ma refer o a poin in ime, for eample e aring poin of e perio.) Uuall reference ime agree for all variable an uni in a FPP. Ti mean for eample for monl aiic a e elineaion of uni oul refer o e curren mon. I follo from e above a uni, claificaion, oer auiliar variable, an reference ime are eenial o conier enever uing aiic. In a oin ue of everal e of aiic, e uer ie o eep ome of e ingreien of e FPP conan an var one or more of e oer. Some pical eample, i empai on a i varie: comparion over ime: reference ime, for eample ever mon from a given one onar; comparion of counrie: omain are Member Sae or oer counrie; comparion beeen non-geograpical group: omain lie inurie are varie; ne aiic uing everal urve: combining aiic from ifferen buine urve (proucion & emplomen, annual & or-erm) for furer anali of inurie for eample. A imple eample of a comple eing i: fir aing raio beeen proucion an number of our ore uing o urve an en comparing oe relaive quaniie over ifferen apec of pace: geograpical area, inurie, ize group ec. To i en, e urve oul be equal in eir uni, omain, an reference ime. Te omain are efine b for eample an inurial claificaion a nee o be e ame for all urve. Wen a uer i uging coerence, efiniion of e arge finie populaion parameer (regaring uni, populaion an omain elineaion, variable, an reference ime) pla a 66

172 primar role. Accurac i imporan, bu i pla a econar an ifferen role. Te more accurae e aiic, e maller e iurbance; e u i more eail performe, an e concluion ran are uuall ronger. 0.. Definiion in pracice A ecribe in e previou ecion, oin ue of e of aiic buil on ome ingreien of e arge aiic being e ame. Te ifficulie meeing e uer ofen epen rongl on e iance beeen e aiic ue oinl. I ma no be rivial even iin a ingle urve, ince efiniion can var (for eample for proucion an emplomen, reference ime coul be a perio for one an a poin in ime for e oer). Normall, oever, e problem increae conierabl en uing everal urve. Even if efiniion are e ame in principle a far a e uer can ee e ma iffer in pracice. One urve ma ave e reference ime of e omain equal o a of e variable an e oer ue a of e frame (ic e quali repor oul o). A furer eample i e enerprie uni; i a o be efine an applie in e ame a in bo urve. In a comparion beeen MS, e enerprie efiniion ma var a lo, in pie of ere being a Regulaion on aiical uni. In pracice ere i an influence from e meoolog ue for eample in aa collecion an eimaion. Hence, e uer nee informaion alo on uc influenial facor Accurac an conien eimae Accurac a, of coure, o be coniere en uing for eample o e raio beeen proucion an our ore varie over inurie, o a ifference a can be ue onl o noie are no ae o be ignifican. Te uer nee a meaure of e overall accurac in e oin ue. Ti mean an aemen of inaccurac from all ource, no onl ue o aing a ample. I i imporan a e meaure i realiic. If ere i a relaionip beeen e FPP involve, man uer fin i convenien if e eimae alo fulfil i relaionip. To imple eample: (i) Te number of emploee in o ifferen urve (on emplomen an proucion) i efiniion uc a e FPP are equal. (ii) Monl an annual proucion aiic i efiniion uc a e um over e elve calenar mon equal e annual value. Te epreion conien eimae ill be ue ere o empaie a e eimaion proceure ave force e eimae o ave e ame relaionip a e FPP, ee Secion for ome eail. Obvioul, aiic can be coeren iou giving conien eimae. Ti i normall e cae i preliminar an efiniive aiic. Noe a e concep of conien eimae i ifferen from conienc in ampoic eor. If a uer a o aiic a e/e believe eimae e ame FPP an ee eimae iffer more an epece, from e inaccurac meaure given, e uer oul upec 67

173 eficiencie in coerence. A imple eample i a follo. Wiou going ino ecnical eail, aume a uncerain inerval are given. ) Te figure are 750 ± 5 an 705 ± 0 Tee are no coeren from a can be een. Ti ignal a ere are ifference in efiniion a ave no been ae or e uer a no oberve. Anoer poibili i a one or o of e inerval i oo or. ) Te figure are 700 ± 5 an 705 ± 0 Tee are coeren from a can be een. I oul be more convenien for e uer o ave a ingle figure (conien eimae), a 704 ± 9 Te icuion in i ecion a empaie e ranom par of e eimaion error. Tere ma alo be emaic error o ae ino accoun en uing aiic. Suc error coul be caue for eample b e aa collecion. Te iincion beeen efiniion an emaic eviaion i no ala clear-cu, oug, ince efiniion in pracice are influence b man facor in, for eample, aa collecion an eimaion Comparabili over ime Comparion over ime are frequen. Tere are ofen o conflicing uer inere a o e aiic o be prouce: abili of efiniion o compare e preen i e previou for a pecial iue; e curren ae oul be ell ecribe. Te fir one or in e irecion of comparabili, erea e econ one goe in e oppoie irecion. Ti ma be a caue of enion in aiical em. Wen a cange i mae, pecial acion are ofen aen o improve e comparabili, for eample b proucing aiic in bo a on one occaion or even re-eimaing a par of e ol erie in erm of e ne efiniion. Tere ma be ifferen opinion a o eer i i more imporan o eimae e level or e cange accurael ifferen aiic ma ave ifferen prioriie. Sor-erm aiic ofen empaie cange. To mae a poible, comparabili i neee over e ime perio a e cange refer o. Uer of annual aiic ma fin e level o be more imporan. Te Naional Accoun nee o ecribe bo level an cange. A furer apec of comparabili over ime i a cerain uer (for eample uing economic aiic inicaing or erm cange) are aniou o be able o eparae for eample ren an regular eaonal variaion. Tecnical mean for i purpoe are eaonal aumen an calenar aumen. To inclue uc parameer i an enricmen of e aiic. 68

174 0..5 Inernaional comparabili A paricular, imporan apec i comparabili beeen Member Sae, oer counrie an geograpical area in general. Ti involve no onl ifferen proucer of e aiic bu alo furer ifference ue o ineren iimilariie beeen counrie: labour mare rule, economic pracice, a rule, ec. Aemp o reuce ifference o increae comparabili for e benefi of e uer b uing imilar concep an efiniion ave been going on inernaionall for a long ime; e are ime-conuming a. Tere are man aciviie for armoniaion in buine aiic in Euroa an oer inernaional auoriie, ee Secion Some uer-bae concluion In ummar, comparabili an coerence iin an beeen e of aiic require ome efiniion o be e ame, for eample uni, variable, or reference ime, epening on e paricular oin ue. Te uer nee informaion on ifference an eir conequence from e proucer. Te quali repor for a cerain e of aiic oul provie uc informaion i regar o comparabili over ime an coerence i oer e of aiic. I i no poible o inclue all oer e bu eperience oul be ue o li ue a are frequen an ere uer are liel o nee elp. Comparabili an coerence in general epen on efiniion. Accurac pla a ifferen role. Tere i, oever, no ala a clear-cu iincion. Definiion ma eem clear an unambiguou in eor bu ill var in pracical or. Tere ma be a enenc no o inclue uc eviaion en meauring accurac, aloug a oul be one. If, for eample, ere i an uneclare emaic eviaion in one urve bu no anoer, ere ill be eficiencie in coerence beeen e o e of aiic. A a conequence of e above, comparabili an coerence epen on e iance beeen proucer; e eficiencie mol increae in e folloing orer: par of a ingle urve, ifferen urve a e ame agenc, ifferen organiaion in e ame counr, aiical office in ifferen counrie. I i imporan for e uer o ave accurac meaure en uing aiic ogeer. I i convenien if e oin ue a been foreeen an prepare, for eample o a eimae are conien. Eplanaor commen in cae of ifference are elpful, for inance en ere are ubanial reviion. 0.3 Proucer apec on coerence, incluing comparabili 0.3. Definiion in eor Te mean of e proucer o acieve coerence are everal. To ue e ame efiniion i, of coure, one of em o be conien iin e auori an i inernaional anar. 69

175 Tere are man armoniaion aciviie inernaionall, an ifferen aciviie ave gone on for a long ime in ifferen fora. Tere i, for eample, muc effor a e EU level, performe b Euroa an oer auoriie. Tere are aiical anar claificaion, lie NACE for aciviie. Tere are alo claificaion for prouc. Furermore, ere are regulaion on Buine Regier (BR), an on aiic, lie Srucural Buine Saiic (SBS) an Sor-Term Saiic (STS). Tere i a Regulaion on aiical uni for e obervaion an anali of e proucion em in e Communi. Uni elineaion an e BR ogeer form an imporan par of e bai of e aiic. Te Naional Accoun are a e op, builing on a lo of oer aiic an being one reaon for coerence among em. Even if ere i a conierable e of efiniion a ave been agree upon, i oe no mean a ere i full armoniaion. Inerpreaion an pracice ma ill iffer beeen counrie Definiion in pracice Tere are man apec o conier in e efiniion of a variable, bo o acieve coerence beeen urve an i inernaional guieline, an o mae e meauremen an aa collecion proceure ea an accurae. Reponen mol provie informaion from eir accouning em, ic avocae a coice of efiniion in agreemen i accouning em in general ue. Buine organiaion a o be coniere carefull en efining an elineaing bo uni an variable. An eample ere i o o anle proucion b boug-in emplomen. Ieall ere oul be co-orinaion aciviie beeen aiical urve, for eample in queionnaire, inrucion o reponen, an aa eiing. Ti ma be more raigforar iin a Naional Saiical Iniue (NSI) an beeen organiaion. Te aciviie iin an NSI ma inclue e baic: uni, elineaion of populaion an omain, variable, aiical meaure, an reference ime, an alo proceure lie aa collecion an eimaion. Uing e ame BR a a frame, conrucing e frame a e ame ime, upaing e uni in e ame a a e ame ime (i regar o buine rucure, claificaion ec), areing queionnaire o e ame uni, ec are furer acion influencing coerence an accurac. Tere ma alo be aciviie beeen organiaion an ifferen counrie. Foreign rae aiic i a clear-cu eample ere inveigaion are poible roug o-calle mirror aiic; e epor of counr A o counr B oul equal e impor of counr B from counr A. Tere are ifference o be uie, largel ue o inaccurac, for eample meauremen error, bu alo ue o ifference in efiniion beeen counrie. Overall, ere are everal principle ic can be ue o acieve comparabili an coerence in general, bo iin an beeen naion, more or le far-reacing. Te European Saiical Sem goe for e ubiiari approac, ere eac Member Sae ma implemen urve in i on a, ogeer i quali aemen. Ti i preferre 70

176 o aemp o armonie proucion an o ocumenaion of ifference, ofen leaing o able i lo of foonoe Accurac an conien eimae A ae in Secion 0..3, i i imporan for e uer o ave appropriae accurac meaure in i/er oin ue of aiic. Te accurac ma be ifficul o quanif. Tere coul for inance be meauremen error for uni ample i probabili one (a o no conribue o e ampling error). If uc a uni a ifferen reponen in ifferen urve, an one of e reponen onl inclue one of everal brance, i i a meauremen error i evere conequence, if uneece. Te raio beeen proucion an our ore b inur ma be affece if e miing par i large, an ere i clearl a ri of e accurac meaure no incluing i error full. Hence, ere ma be a fale concluion. I ma be regare a ue o a non-ampling error; i coul alo be viee a an unereimaion of inaccurac. Te eample ma eem eaggerae, bu uc ing appen. Te folloing overall, an vague, aemen eem reaonable (an in line i e previou ecion): e furer apar e urve are, e greaer e ri of ifference beeen em ifference a affec e accurac, ofen in a a a i no ea o ae. Te oin ue of aiic i inaccurac of ifferen caracer i more ifficul an o ue aiic from e ame urve ere e error are relae, perap becaue ere are emaic eviaion a cancel a lea parl in comparaive uie or becaue e ranom error are correlae. In line i e above, incluing e eample in Secion 0..3, conier aiic a coniing of e arge parameer an an eimaion error, an aume e imple cae i a mmeric uncerain inerval aroun e poin eimae. Te orer e leng of e inerval, e more accurae e aiic, an e ronger e aiical inference in e oin ue of aiic, for eample comparion. A u icue above, ere i a ri of proucing oo or an inerval, no aing all e error ource ino accoun. Te coerence concep i ie o e arge. In oin ue of aiic, cerain par of e arge involve nee o be equal, a Secion 0.. illurae. Conier e raio beeen oal proucion an oal number of our ore i bo aiic bae on a ample urve. If e emanae from e ame urve, e are bae on obervaion on e ame e of uni. So, if a ample appen o conain mol mall uni, i i o for bo numeraor an enominaor. Te raio oe no var o muc aroun e populaion value a i oul i o ifferen ample. A maller variaion i obaine no onl i repec o e ampling error, bu i can be epece o ol for everal furer error ource, for eample meauremen error. Hence, comparabili an coerence apec in general mae i eirable o co-orinae e proucion of aiic a are ue oinl. Te eimaion proceure ma be co-orinae beeen urve. Ti can be one a ifferen age, i ifferen reng, an i ifferen aim. 7

177 Te aim coul be o give e uer a imple an coeren a meage a poible, a i o ave a ig egree of co-orinaion of e oupu from ifferen urve. Ti i ifferen from anling eac urve on i on an from uing auiliar informaion i e ingle aim of improving accurac Some commen on meoolog, epeciall bencmaring One meo of co-orinaing aiical oupu i o-calle bencmaring, ere one e of eimae i force o agree i anoer. Ti i a pecial cae of conien eimae inrouce in Secion Tpicall, or-erm aiic coul be bencmare on annual aiic, if e former (afer aggregaion o e calenar ear) are an inicaor of e laer. One reaon coul be o implif for e uer b unifing e o ime erie (enuring a e monl erie a e ame annual um a e annual erie), anoer o improve e accurac of e or-erm aiic. For i o be meaningful, e o e of aiic oul ave e ame arge parameer for e calenar ear. Te ue of proceure o mae eimae conien ma influence no onl one bu bo e of aiic. Te implemenaion of bencmaring of, a, or-erm aiic on annual aiic, involve comparion. Tee ma conier no onl e macro level, bu alo e micro level. Te evaluaion performe ma impl furer ei for bo or-erm an annual aiic. In cae lie bencmaring or-erm aiic on annual aiic, e former ave been publie en e laer appear. Ta mean a reviion, ma be one or o ear afer e fir publicaion (longer for Januar an for December), or even more. Man uer ill reac bal o reviion in eir ime erie. Avanage an iavanage ave o be balance. Tere are everal meo for bencmaring, bae on ifferen approace o e o ime erie a o a i fie an a i ranom variaion, ee for eample Colee & Dagum (994) i empai on urve error, Durbin & Quenneville (997) i empai on ocaic ime erie moel (an alo reference erein), an e ver recen Dagum, Colee & Cen (998). Tere i a recen uggeion on co-orinaion a e eimaion age b Renen & Nieuenbroe (997), o call eir proceure aligning eimae. Surve i variable in common variable a are oberve in ee urve an ave unnon populaion oal are poole an e common variable are ue a regreor (in aiion o variable i non oal). Ten e eimae obaine i ue a auiliar informaion in e iniviual urve. Te proceure i inereing from bo coerence an accurac poin of vie. Furermore, aiic ma be relae, aloug iou clear connecion in erm of, for eample, uni. Labour mare aiic bae on buine urve an on oueol urve provie an eample, ee alo Secion

178 0.3.4 Comparabili over ime Tere i uuall inere bo in recen aiic an in long ime erie. Accurac of cange i ofen a lea a imporan a accurac of level. Sabili of efiniion i imporan, bu cange in rucure oul alo be aen ino accoun. For eample, e ue of cain-line inice a increae, an an ine i a fie bae i recommene o be rebae fairl frequenl, a lea ever fif ear. I ma be necear o cange variable o be in line i accouning pracice if ee cange. Ne aminiraive rule ma influence e BR in a a a carrie over o e aiic. Comparabili over ime oul be aen ino accoun en cooing variable: curren price are ofen complemene i conan price or volume meaure. Te meoolog ue a an influence on comparabili, an ere a o be a compromie beeen inroucing for eample improve eimaion meo an eeping e ol a i regar o ime erie. Tere i ofen a ump in a ime erie en a cange i mae. Hence, care i neee en inroucing cange in meo. I ma be ie o ave a ouble-run perio, a i o run e o meo in parallel o meaure e effec an poibl lin e o ime erie. A a minimum, eplanaion oul be provie o e uer. Wen comparing or-erm aiic, calenar an eaonal aumen are imporan ool, i regar o correponing perio in ifferen ear an aacen perio. Tere are ifferen meo of aumen, builing on ifferen aumpion, lie aiive or muliplicaive componen. Te appropriaene of a meo i no necearil e ame in all counrie. Sill, for comparabili reaon ere oul be ome armoniaion of e aumen of ime erie Inernaional comparabili A alrea inicae above in Secion 0.3., ere are man inernaional armoniaion aciviie o improve comparabili beeen counrie. Sanar claificaion i a pical eample, i for eample NACE Rev. for claificaion of economic aciviie. Tere i a Regulaion on Srucural Buine Saiic a inclue efiniion of variable. A Regulaion on Sor-Term Saiic a become la uring 998. Tere i a Regulaion on aiical uni lie enerprie, in-of-acivi uni an local uni an alo one on Buine Regier. Tee regulaion aim a increaing e comparabili roug maing baic efiniion equal an alo e applicaion imilar b proviing no onl eor bu alo manual i eample. Hoever, e ubiiari approac mean a eac Member Sae ma implemen urve in i on a, even en ere i a regulaion uc a oe menione. Similarl, regulaion on for eample aiical uni ma be inerpree an applie omea ifferenl beeen MS ue o ifferen raiion, prerequiie, ec. Tere are ineren culural ifference, lie e number of oring our per full-ime an par-ime emploee, e iribuion of oring our over e ear an over e ee, aaion rule ec. Te variable invemen in fie ae provie an eample ere e precie efiniion of e variable ma var 73

179 beeen counrie, a lea before regulaion ave come ino ue. In uc a cae, i ma be poible o mae ome in of eimae a o e effec of a ifferen naional efiniion in comparion i e European concep. Ta i an aemp o overcome e lac in comparabili, bu o meaure e ifference i a ifficul a. Among eample of meo in e irecion of comparabili, anariaion of ea rae in populaion emograp i an ol an illuraive one. Depouo & Aronel (997) icu buine aiic, an e avocae economeric moel. Dalén (998) preen ource of non-comparabili in a general approac o e cae of conumer price inice, an e preen empirical anale of e effec of ifferen concepual an ecnical ifference bae on Sei an Finni aa Some proucer-bae concluing commen Te icuion above an belo o naional an inernaional acion o improve coerence incluing comparabili, bu alo eample ere eficiencie ill remain. Man claificaion em an regulaion or in e irecion of coerence beeen aiic from ifferen urve. Sill, ere are everal claificaion em. Ti mean for inance a aiic on proucion of commoiie an foreign rae aiic are ifficul o ue ogeer en e former i bae on PRODCOM an e laer on CN8. Ti influence for eample e Proucer Price Ine (roug e eig ue for price inice for e omeic mare, epor ec) an e Naional Accoun. Te SBS an STS Regulaion ave muc in common. Tere ma ill be eficiencie in coerence beeen annual an or-erm aiic. One reaon for ifference i a ee aiic parl buil on ifferen uni, enerprie for annual aiic an in-of-acivi uni for or-erm. Moreover, e laer ue in-of-acivi uni for eample for manufacuring bu, a lea a preen, enerprie for cerain inurie, for eample ervice. Te populaion i no clearl epree for e STS, an e miure of uni eem o involve ifferen pracice, leaing o furer coerence an comparabili eficiencie, i manufacuring in-of-acivi uni iin non-manufacuring enerprie an vice vera. Anoer reaon for ifference beeen e o e of aiic i ifferen ime ceme of proucion for e aiic for a given reference ear. Te annual aiic are collece afer e ear, ile e or-erm aiic are collece uring e ear. Te populaion being urvee cange uring e ear; bir an ea, merger an brea-up ec. Suc cange are beer non en proucing e annual an e or-erm aiic. Hence, even if e arge populaion are e ame, e frame an e nolege available ma be ifferen for e o urve. Ta ma impl ifference perap above all for e accurac a e proucer oul inform e uer abou. Alernaivel, e proucer ma eier revie e or-erm aiic or refrain from uing ne populaion informaion for e annual aiic. Ti i an eample of ifferen pracice in ifferen Member Sae. See alo Caper 5. 74

180 Te Naional Accoun buil fir on e or-erm aiic. Laer, en e annual aiic are available, bo annual an or-erm aiic are inegrae ino e accoun. Te annual an e quarerl accoun are o be conien, an o are e ifferen accoun, lie proucion an ue. Te Naional Accoun are o cover e ole econom. Te inegraion ma impl coerence eficiencie i bo e annual an e or-erm aiic, an, above all, inconien eimae. A furer eample ere coerence i inereing i beeen official or-erm aiic an relae aiic from oer, poibl privae, iniue; e laer ma be qualiaive, a baromeer urve or buine enenc urve. A ae, i i imporan for e uer o no if efiniion are equal, or if e are no a e ifference are. Te ifference oul preferabl be epree in erm of effec on e aiic. Te more accurae e aiic, e beer in e oin ue. Accurae aiic canno, oever, overcome ifferen efiniion. Te uer ma fin i convenien if eimaion proceure are uc a conien eimae are obaine. Te proucer oul conier ee apec en proucing an preening e aiic. 0.4 Some illuraion of coerence an co-orinaion A ae everal ime above, efiniion are funamenal for coerence, incluing comparabili. Accurac i imporan, bu in a ifferen imenion. Te more accurae e aiic, e ronger e inference ic can be mae in e oin ue. Ranom variaion (for eample ue o ampling) i ofen eaier o meaure an ae ino accoun an emaic eviaion (for eample ue o nonrepone) a i feare o be ere, aloug ifficul o quanif. If ere are emaic eviaion, i i eaier o mae comparion if e eviaion ave a paern a i able. In general, e cloer e urve, e le e problem i eficiencie in coerence. I i, oever, neier poible nor eirable o ave u one or a fe urve. Tere i a balance beeen irece urve i fe variable on e one an an urve i a broa cope an man variable on e oer. Te former a ma allo comparaivel mall ample, bu i ma be convenien o inclue ome variable lie e number of emploee in eac urve. Ta mean a e ame variable value i repore man ime. Ti increae e repone buren. Te em coen oul inclue illingne o repon an r o eep repone buren lo an prea ou. Co-orinaion aciviie are imporan en everal urve are equal or a lea imilar. German i a noable eample. Man urve are performe on a ub-naional level in 6 Buneläner (region) an i i imporan o co-orinae ee urve o obain aiic no onl for eac Bunelan bu alo on e feeral level. Tere ave o be compromie ince opimal oluion are ifferen epening on e level. For eample, a goo ample allocaion for German ma be quie ifferen o a of iniviual Buneläner. Tere i muc o co-orinae: variable, inrucion, queionnaire, eiing ec. Spain provie a imilar eample; 50 Province perform e iniial aa collecion an eiing. 75

181 Tere are relae coerence problem in mo counrie. A urve ma comprie all inurie, or i ma be more convenien o perform urve for manufacuring an ervice inurie eparael. Annual an or-erm urve ma be more or le co-orinae a o variable. Ofen e annual urve i more eaile. A variable lie urnover, or alarie an age, ma be inclue in bo cae. Tere i an argumen a for uni in bo ample i i unnecear o a for e um of elve value alrea collece, even if ome of ee are impue. On e oer an, if monl an annual aa are collece, e annual urve ill ave problem i inconienc if ere a been ome miing perio, bu if impuaion i reaonable i inconienc ma be mall. Moreover, a maor aim of or-erm urve i o prouce eimae quicl. If reponen o no ave final reul available for e mon, e ma be encourage o provie eimae (eir informe ugemen being beer an impuing for nonrepone). Te ource of e aa for or-perio an rucural urve ma be ifferen. Te former ma emanae from managemen or operaional accoun. Te laer are liel o be prouce from e final auie accoun for e ear an ma inclue ome aumen ic are mae a e en of e ear. On balance mo counrie ee rong argumen for eparae annual an or-perio aa collecion. Several counrie no ave one BR a i ue a frame for all, or a lea mo, urve. If all urve ue a alo for upae, a ill mae e oin ue of e aiic eaier. Several counrie ave inrouce co-orinaion of e ampling. Tere ma be one or more aim: poiive co-orinaion o improve accurac over ime or beeen urve an negaive co-orinaion (roaion i a poibili) mainl o prea e repone buren. Tere i a enion beeen annual aiic being a accurae a poible an being coeren i or-erm aiic. Unil recenl in e UK coerence a e main riving force i annual panel elece o be conien i or-erm aiic. Hoever, e empai a no ice o accurac i e aim a e rucural urve oul ue e mo up o ae informaion available on uni an claificaion. Ti cange in polic mean eimae cloer o e arge bu larger reviion en bencmaring or-erm aiic on annual aiic. Suc a pracice a a longer ior of ue in Seen. Tere ma be co-orinaion beeen urve o enure a e final aiic agree. Tere are ifferen ecnique epening on informaion an cloene of urve. In Seen, for eample, e or-erm ine of proucion for e manufacuring inurie i bencmare on e annual ine in pie of ere being ome ifference in efiniion; e or-erm ine being regare a an inicaor of e annual ine, ee e Sei Moel Quali Repor for ecripion an figure 6. In e UK e or-erm proucion ine i no currenl bencmare o e annual urve (bu bencmaring i uneraen eleere). Hoever, e UK raeg for e longer erm i o move o cain lining uppore b annual inpu-oupu able. A conequence i a e value ae from e annual urve ill replace eimae of gro oupu ue in e or erm. In i approimaion a necear aumpion i a e raio of gro o ne oupu i conan over ime. Ta poei i 6 Ti bencmaring a been ebae in Saiic Seen, an in lae 998 i a ecie o iconinue. 76

182 liel o become rece, paricularl a loer level of e SIC, e furer one move from e bae ear. In man cae, ifferen efiniion ma be foun impoible o overcome an imporan o ue for eac of e ingle urve. Tere ma for eample be ifferen e of emplomen aiic an of aiic on alarie an age, coming from labour force urve an buine urve, an from buine urve an aminiraive aa ie o emploer eclaraion. A UK eperience i a one a of elping uer o uneran e ifference beeen e labour force an e emplomen urve i o empaie e ifference beeen people an ob, ee Peae (997). Maing i iincion clear a elpe o preven uer from focuing on e ifference beeen e eimae ic, en ampling error are aen ino accoun, are relaivel mall. Similarl, conierable reource ave been ue in e Neerlan on aiical inegraion for e labour mare, i aiic bae on eablimen urve, oueol urve, an cenral regier, Leuni & Alena (996). Te co-orinaion ma be on e macro level, a u menione, an/or on e micro level. Tere ma for eample be an ecange of figure for iniviual uni beeen urve perap o enure a an enerprie a i comple an/or re-organiing i full inclue or a a par of e eiing em. Tere are uc pracice in German an Denmar. Similarl, aff in e UK generall or on more an one inquir. In e proucion ecor e ame aa collecor ill or on Soc, Capial Epeniure, Monl Turnover an e Annual Srucural Surve. Tu comparion of aa a conribuor level can eail be mae an acion aen o reconcile ifference. Member Sae ofen mae cange o eir inquir em o improve e meoolog an acieve greaer conienc i oer urve, claificaion or European regulaion. Aloug ee evelopmen ma increae coerence beeen urve an counrie, e inrouce iconinuiie en e cange are mae ioring e comparabili over ime. Specific eample of cange a influence efiniion an/or accurac inclue: (a) cange of aminiraive rule or aa, for eample aa ue for upae, regional bounar cange (b) conrucion of a ne regier or frame (c) ne ampling eign () cange in eimaion meoolog (e) ne oulier reamen (f) move o NACE Rev. (g) move o ESA (European Sem of Accoun) In orer o calculae a lin beeen e o ime erie, i i necear o ave aiic on bo e ol an e ne bai. Tere i analical or an ofen era aa collecion. Noneele, e or i vial ince e lin facor are ofen large even for cange ic ma eem o be lig. For eample in e UK cange in eimaion meo ave a ime alere inur oal a cla level b over en per cen. Te lin ma be calculae for a 77

183 mon, a quarer, or a number of perio. Were lin are large an coul var from perio o perio i ma be be o loo a ome average lin over a perio of ime o enure abili. An cae ere e facor i urpriingl large or mall oul be folloe up. Lin can be applie o eier e ol or e ne erie. 78

184 Par 4: Concluion an Reference Concluing remar. Meoolog for quali aemen Paul Smi, Office for Naional Saiic Ti volume conain a lo of informaion on e eor an meo bein e aemen of quali in buine urve, covering a uge range of ecnique. In man urve iuaion i ill be pracical onl o ue a mall number of ee o ae e quali of e urve reul, becaue of e limiaion of ime, mone an available informaion. A naural coice i o aim for a balance beeen oe meo ic are ea o appl an evaluaion of e quali componen ic are e mo imporan one. Some accurac meaure ave a long raiion, for eample e ampling error en e ample eign i probabili-bae. Ofen ee meaure are oe mo amenable o eoreical reamen. Sofare for aeing e ampling error i reviee in Volume II, an e properie of ampling error are alo inveigae ere. Non-ampling error an non-probabili ampling ceme are acceible o inveigaion b ree main general meo: inicaor; follo-up uie; an eniivi anali. Inicaor are aiic, normall available a b-prouc of e urve proceing, ic are oug o be (rongl) correlae i e quali of e eimae, bu ic o no irecl meaure a quali. Te are e eaie aiic on quali o calculae, an e preominae in e moel quali repor (volume III), aloug e precie eail iffer accoring o a nee o be eimae. Bo follo-up uie an eniivi anali are limie b e aa ic are available (or obainable); follo-up uie are picall igco (for NSI an conribuor) bu aim o ge cloer o e rue value an e original urve i, uuall for a ube of e original obervaion. Seniivi anale rel on e aa alrea available (bo urve an auiliar aa) o ugge plauible moel, an inicae o e eimae cange i ifferen moel (or ifferen aumpion). In a mall number of cae NSI obain follo-up aa a par of e urve proce, an nee onl iner ome era orage or unerae ome aiional or o ue i in paricular proceing error an coing error (ere all e original repone are available (if e are ore) an can be re-evaluae) an nonrepone error (ere e cange of repone i ime give ome iea of e caraceriic of nonreponen). In general oever, follo-up uie are eaile an ver epenive, an are uneraen rarel an on a mall cale. Seniivi anali i ceaper a i ue onl e aa alrea collece an require onl e reproceing of i aa uner ifferen cenario. I give an inicaion of o e eimae 79

185 i affece b cerain moel an aumpion, bu oe no a o cloe ee eimae are o e rue value, aloug ere i an implici aumpion a if all e cenario inveigae ave imilar oucome, en ee oucome ill be cloe o e rue value. Deucing ic componen of oal urve error (ee ecion..) ave e bigge conribuion i muc more problemaic, ince in ifferen urve e aner ma ell be ifferen, an ere i onl a mall number of uie ic inveigae everal error in a ingle urve in a comparaive a. I i perceive iom a non-ampling error ma oueig ampling error ubaniall, bu ere i lile evience of e relaive imporance of error in pracice. Muc of e meoolog bein urve eimaion involve on e one an removing bia a muc a poible an acceping an increae in variance (for eample in compenaing for nonrepone, caper 8), an on e oer an inroucing bia in a rucure a o reuce e variabili of urve eimae (for eample roug poraificaion or oulier aumen), o i i meauremen of ee biae an variance ic ill lea o e oal urve error.. Recommenaion for quali aemen Clearl i i inappropriae o unerae an in-ep u of all e biae an variance componen of a urve on ever occaion a i i run. Hoever, i i alo clear a i or of u i e onl a in ic a complee evaluaion of e urve quali can be mae. Ti lea u o ugge a ree-pronge approac o evaluaing quali: (a) Inicaor oul be inclue a par of urve proceing em, an oul be prouce eac ime e urve i run. Te no onl inicae e quali, bu alo o ere urve procee are failing. Tee oul inclue for inance eige an uneige repone rae, rae of ienificaion of miclaificaion an ea uni, an aa ei failure rae. (b) Quali meaure oul be prouce perioicall (a lea annuall) ere e are clearl efine. Tee oul inclue ampling error. (c) Tere oul be a rolling programme of evaluaion of e overall quali of e urve, covering ome opic eac ear. Ti oul involve e ue of follo-up inervie an oer eaile uie, in orer o eimae e rue oal urve error. Te eac li of componen o be inclue oul nee o be ecie; ieall all componen oul be meaure. Some of e buren of meauremen coul be move aa from e urve b, for eample, uneraing an evaluaion of e frame quali, a e frame i ue for man urve. In aiion o ee ree, a ueful qualiaive meaure of urve quali i o ave e meo full ocumene, an o ave e quali aemen pracice rien on, muc a in e Moel Quali Repor. Te ac of proucing ee repor ill force e meo of e urve o be coniere criicall, an i ill influence e quali. Te Moel Quali Repor (volume III) inclue bo imple inicaor an more ambiiou meaure lie eniivi anale, bu no in-ep uie. Te Implemenaion Repor an e Guieline on implemenaion (volume IV) inclue icuion of balancing iue. 80

186 Reference ABRAHAM, B. & LEDOLTER, J. (983) Saiical meo for forecaing. Ne Yor: Wile. ARCHER, D. (995) Mainenance of buine regier. In Buine urve meo (e. B.G. Co, D.A. Biner, B.N. Cinnappa, A. Criianon, M.J. College & P.S. Ko), pp Ne Yor: Wile. BETHLEHEM, J.G. (988) Reucion of nonrepone bia roug regreion eimaion. Journal of Official Saiic, 4, BERGER, Y.G. (998) Rae of convergence o ampoic variance for e Horviz-Tompon eimaor. Journal of Saiical Planning an Inference, 7, BIEMER, P.P. & FECSO, R.S. (995) Evaluaing an conrolling meauremen error in buine urve. In Buine urve meo (e. B.G. Co, D.A. Biner, B.N. Cinnappa, A. Criianon, M.J. College & P.S. Ko), pp Ne Yor: Wile. BLOM, E. & FRIBERG, R. (995) Te ue of canning a Saiic Seen. Proceeing of e Inernaional Conference on Surve Meauremen an Proce Quali, Conribue paper, pp Virginia: American Saiical Aociaion. BOX, G.E.P., JENKINS, G.M. & REINSEL, G.C. (994) Time erie anali, forecaing, an conrol, ir eiion. Engleoo Cliff, NJ: Prenice-Hall. BUSHNELL, D. (996) Compuer-aie occupaion coing. In Proceeing of e Secon ASC Inernaional Conference (e. R. Ban, J. Fairgrieve, L. Gerrar, T. Orcar, C. Pane, & A. Welae), pp Ceam: Aociaion for Surve Compuing. CANTY, A.J. & DAVISON, A.C. (997) Variance eimaion for e Labour Force Surve. Repor prepare uner conrac o e Univeri of Ee, on bealf of e UK Office for Naional Saiic. CENTRAL STATISTICAL OFFICE (99) Sanar inurial claificaion of economic aciviie 99. Nepor: CSO. CHAMBERS, R.L. (986) Oulier robu finie populaion eimaion. Journal of e American Saiical Aociaion, 8, CHAMBERS, R.L. (997) Weiging an calibraion in ample urve eimaion. In Proceeing of e Conference on Saiical Science Honouring e Bicenennial of Sefano Francini Bir (e. C. Malaguerra, S. Morgenaler & E. Roncei), Mone Verià, Sizerlan, Bael: Biräuer Verlag. CHAMBERS, R.L. (997) Small-area eimaion: a urve ampler perpecive. Tecnical repor, Deparmen of Social Saiic, Univeri of Souampon, UK (preene a a meeing organize b e Small Area Heal Saiic Uni of Imperial College, UK, Ma 997). 8

187 CHAMBERS, R.L. & DORFMAN, A.H. (994) Robu ample urve inference via boorapping an bia correcion: e cae of e raio eimaor. Invie Paper, Join Saiical Meeing of e ASA, IMS an e Canaian Saiical Socie, Torono, Augu 3-8, 995. CHAMBERS, R.L., DORFMAN, A.H. & WEHRLY, T.E. (993) Bia robu eimaion in finie populaion uing nonparameric calibraion. Journal of e American Saiical Aociaion, 88, CHAMBERS, R.L. & KOKIC, P.N. (993) Oulier robu ample urve inference. Invie Paper, Proceeing of e 49 Seion of e Inernaional Saiical Iniue, Firenze, Augu 5-Sepember, 993. CHATFIELD, C. (996) Te anali of ime erie: an inroucion. Lonon: Capman & Hall. CHOLETTE, P.A. & DAGUM, E.B. (994) Bencmaring ime erie i auocorrelae urve error. Inernaional Saiical Revie, 6, COCHRAN, W.G. (977) Sampling ecnique, ir eiion. Ne Yor: Wile COOK, R.D. & WEISBERG, S. (98) Reiual an influence in regreion. Lonon: Capman & Hall. COPAS, J.B. & LI, H.G. (997) Inference for non-ranom ample (i icuion). Journal of e Roal Saiical Socie, Serie B, 59, COWLING, A., CHAMBERS, R., LINDSAY, R. & PARAMESWARAN, B. (996) Applicaion of paial mooing o urve aa. Surve Meoolog,, COX, D.R. & HINKLEY, D.V. (974) Teoreical aiic. Lonon: Capman & Hall. DAGUM, E.B., CHOLETTE, P.A. & CHEN, Z.-G. (998) A unifie vie of ignal eracion, bencmaring, inerpolaion an erapolaion in ime erie. Inernaional Saiical Revie, 66, DALÉN, J. (998) Suie on e comparabili of conumer price inice. Inernaional Saiical Revie, 66, DAWID, A.P. (979) Coniional inepenence in aiical eor (i icuion). Journal of e Roal Saiical Socie, Serie B, 4, -3. DAWID, A.P. & LAURITZEN, S.L. (993) Hper Marov la in e aiical anali of ecompoable grapical moel. Annal of Saiic,, DEMING, W.E. (956) On implificaion of ample eign roug replicaion i equal probabiliie an iou age. Journal of e American Saiical Aociaion, 5, DENTON, F.T. (97) Aumen of monl or quarerl erie o annual oal: An approac bae on quaraic minimizaion. Journal of e American Saiical Aociaion, 66,

188 DEPOUTOT, R. & ARONDEL, PH. (997) Inernaional comparabili an quali of aiic. Preene o CAED97, Inernaional Conference on Comparabili Anali of Enerprie (micro)daa, 5-7 December 997, Bergamo, Ial. DEVILLE, J.-C. (99) A eor of quoa urve. Surve Meoolog, 7, DEVILLE, J.-C. & SÄRNDAL, C.-E. (99) Calibraion eimaor in urve ampling. Journal of e American Saiical Aociaion, 87, DEVILLE, J.-C. & SÄRNDAL, C.-E. (994) Variance eimaion for e regreion impue Horviz-Tompon eimaor. Journal of Official Saiic, 0, DIPPO, C.S., CHUN, Y.I. & SANDER, J. (995) Deigning e aa collecion proce. In Buine urve meo (e. B.G. Co, D.A. Biner, B.N. Cinnappa, A. Criianon, M.J. College & P.S. Ko), pp Ne Yor: Wile. DRAPER, D. (995a) Aemen an propagaion of moel uncerain (i icuion). Journal of e Roal Saiical Socie, Serie B, 57, DRAPER, D. (995b) Inference an ierarcical moelling in e ocial cience (i icuion). Journal of Eucaional an Beavioral Saiic, 0, 5-47, DRAPER, D. & CHEAL, R. (998) Caual inference via Marov cain Mone Carlo. Tecnical repor, Saiic Group, Deparmen of Maemaical Science, Univeri of Ba, UK. DRAPER, D., GAVER, D., GOEL, P., GREENHOUSE, J., HEDGES, L., MORRIS, C., TUCKER, J. & WATERNAUX, C. (993a) Combining informaion: aiical iue an opporuniie for reearc. Conemporar Saiic Serie, No.. Aleanria VA: American Saiical Aociaion. DRAPER, D., HODGES, J., MALLOWS, C. & PREGIBON, D. (993b) Ecangeabili an aa anali (i icuion). Journal of e Roal Saiical Socie, Serie A, 56, DURBIN, J. & QUENNEVILLE, B. (997) Bencmaring b ae pace moel. Inernaional Saiical Revie, 65, EFRON, B. & TIBSHIRANI, R.J. (993) An inroucion o e boorap. Lonon: Capman & Hall. ELDER, S. & MCALEESE, I. (996) Applicaion of ocumen canning, auomae aa recogniion an image rerieval o paper elf-compleion queionnaire. In Surve an Saiical Compuing 996. [E noe: full ref pleae] ELVERS, E. (993) A ne Sei buine regier covering a calenar ear an eample of i ue for eimaion. Proceeing of e Inernaional Conference on Eablimen Surve. American Saiical Aociaion, pp EUROSTAT (996:04) Propoal for a quali repor on rucural buine inicaor Euroa/D3/Quali/96/04. Luembourg: Euroa. 83

189 EUROSTAT (997:04) Propoal for a quali repor on or-erm inicaor. Euroa/A4/Quali/97/04. Luembourg: Euroa. EUROSTAT (997:06) Variance eimaion of aic aiic. Par : Overvie}. Euroa/A4/Quali/97/06. Luembourg: Euroa. EUROSTAT (997:07) Variance eimaion for namic aiic. A imulaion u (raf). Euroa/A4/Quali/97/07. Luembourg: Euroa. EUROSTAT (998:07) Meoological apec of proucer price on e epor mare. Euroa/4E/Energ an Inur/98/07. Luembourg: Euroa. EUROSTAT (998a) Quali of Buine regier. Euroa/D/BRSU/98- (Woring group meeing 9-30 June 998). EUROSTAT (998b) Seaonal aumen meo: a comparion. Saiical Documen, Teme 4E, 998. Luembourg: Euroa. FAY, R.E. (996) Alernaive paraigm for e anali of impue urve aa. Journal of e American Saiical Aociaion, 9, FELLEGI, I.P. & HOLT, D. (976) Semaic approac o auomaic ei an impuaion. Journal of e American Saiical Aociaion, 7, 7-35 FINDLEY, D.F., MONSELL, B.C., BELL, W.R., OTTO, M.C. & CHEN B.-C. (998) Ne capabiliie an meo of e X-ARIMA eaonal aumen program (i icuion). Journal of Buine an Economic Saiic, 6, FISK, P.R. (977) Some approimaion o an ieal ine number. Journal of e Roal Saiical Socie, Serie A, 40, 73. FORSTER, J.J. & SMITH, P.W.F. (998) Moel-bae inference for caegorical urve aa ubec o non-ignorable nonrepone (i icuion). Journal of e Roal Saiical Socie, Serie B, 60, 57-70, FREEDMAN, D., PISANI, R. & PURVES, R. (998) Saiic, ir eiion. Ne Yor: Noron. FRIBERG (99) Surve on environmenal invemen an co in Sei inur. Saiical Journal of e UN Economic Commiion for Europe, 9, 0-0. GHOSH, M. & RAO, J.N.K. (994) Small-area eimaion: an appraial (i icuion). Saiical Science, 9, GLYNN, R.J., LAIRD, N.M. & RUBIN, D.B. (986) Selecion moeling veru miure moeling i non-ignorable nonrepone. In Draing inference from elf-elece ample (e. H. Wainer), pp Ne Yor: Springer. GOLDSTEIN, H., RASBASH, J., PLEWIS, I., DRAPER, D., BROWNE, B., YANG, M. & WOODHOUSE, G. (997) A Uer' Guie o MLn for Wino (MLiN), Verion.0b, November 997. Lonon: Iniue of Eucaion. 84

190 GOMEZ, V. & MARAVALL, A. (994a) Program SEATS (Signal Eracion in ARIMA Time Serie): Inrucion for e Uer. Woring Paper ECO 94/8, European Univeri Iniue, Florence. GOMEZ, V. & MARAVALL, A. (994b) Program TRAMO (Time erie Regreion i ARIMA noie, Miing Obervaion, an Oulier): Inrucion for e Uer. Woring Paper ECO 94/3, European Univeri Iniue, Florence. GRANQUIST, L. (984) On e role of eiing. Saiical Revie,, GRANQUIST, L. (995) Improving e raiional eiing proce. In Buine urve meo (e. B.G. Co, D.A. Biner, B.N. Cinnappa, A. Criianon, M.J. College & P.S. Ko), pp Ne Yor: Wile. GRANQUIST, L. & KOVAR, J.G. (997) Eiing of urve aa: o muc i enoug? In Surve Meauremen an Proce Quali (e L. Lberg, P. Biemer, M. Collin, E. e Leeu, C. Dippo, N. Scarz & D. Trein), pp Ne Yor: Wile. GRIFFITHS, G. & LINACRE, S. (995) Quali aurance for buine urve.. In Buine urve meo (e. B.G. Co, D.A. Biner, B.N. Cinnappa, A. Criianon, M.J. College & P.S. Ko), pp Ne Yor: Wile. GROVES, R.M. (989) Surve error an urve co. Ne Yor: Wile. HAAN, J. DE, OPPERDOES, E. & SCHUT, C. (997) Iem ampling in e conumer price ine: a cae u uing canner aa. Paper ubmie o e Join ECE/ILO Meeing on Conumer Price Inice (Geneva, 4-7 November 997). HÀJEK, J. (964) Ampoic eor of reecive ampling i varing probabiliie from a finie populaion. Annal of Maemaical Saiic, 35, HAMMERSLEY, J.M. & HANDSCOMB, D.C. (979) Mone Carlo meo. Lonon: Capman & Hall. HARVEY, A.C. (989) Forecaing, rucural ime erie moel, an e Kalman filer. Cambrige: Cambrige Univeri Pre. HECKMAN, J.J. (979) Sample elecion bia a a pecificaion error. Economerica, 47, HIDIROGLOU, M.A. & BERTHELOT, J.M. (986) Saiical eiing an impuaion for perioic buine urve. Surve Meoolog,, HIDIROGLOU, M.A., SÄRNDAL, C.-E. & BINDER, D.A. (995) Weiging an eimaion in buine urve. In Buine urve meo (e. B.G. Co, D.A. Biner, B.N. Cinnappa, A. Criianon, M.J. College & P.S. Ko), pp Ne Yor: Wile. HILLMER, S.C. & TRABELSI, A. (987) Bencmaring of economic ime erie. Journal of e American Saiical Aociaion, 8,

191 HOLT, D. & SMITH, T.M.F. (979) Po-raificaion. Journal of e Roal Saiical Socie, Serie A, 4, HORVITZ, D.G. & THOMPSON, D.J. (95) A generalizaion of ampling iou replacemen from a finie univere. Journal of e American Saiical Aociaion, 47, HUBER, P.J. (98) Robu aiic. Ne Yor: Wile. JAGERS, P. (986) Po-raificaion again bia in ampling. Inernaional Saiical Revie, 54, JAZAIRI, N.T. (98) Ine number. Enr in Encclopeia of Saiical Science, 4 (e. N.L. Jonon & S. Koz). Ne Yor: Wile. JONES, R.H. (980) Maimum lielioo fiing of ARMA moel o ime erie i miing obervaion. Tecnomeric,, KALTON, G. & STOWELL, R. (979) A u of coer variabili. Journal of e Roal Saiical Socie, Serie C, 8, KASPRZYK, D. & KALTON, G. (998) Meauring an reporing e quali of urve aa. Smpoium 97, Ne Direcion in Surve an Cenue: proceeing pp Oaa: Saiic Canaa. KENNY, P.B. & DURBIN, J. (98) Local ren eimaion an eaonal aumen of economic an ocial ime erie (i icuion). Journal of e Roal Saiical Socie, Serie A, 45, -45. KOHN, R. & ANSLEY, C.F. (986) Eimaion, preicion, an inerpolaion for ARIMA moel i miing obervaion. Journal of e American Saiical Aociaion, 8, KOKIC, P.N. (998) Eimaing e ampling variance of e UK Ine of Proucion. Journal of Official Saiic, 4, KOKIC, P.N. & SMITH, P.A. (999a) Winoriaion of oulier in buine urve. Submie o Journal of e Roal Saiical Socie, Serie D. KOKIC, P.N. & SMITH, P.A. (999b) Oulier-robu eimaion in ample urve uing oie inoriaion. Submie o Journal of e American Saiical Aociaion. KOVAR, J.G. &WHITRIDGE, P.J. (995) Impuaion of buine urve aa. In Buine urve meo (e. B.G. Co, D.A. Biner, B.N. Cinnappa, A. Criianon, M.J. College & P.S. Ko), pp Ne Yor: Wile. LEE, H. (995) Oulier in buine urve. In Buine urve meo (e. B.G. Co, D.A. Biner, B.N. Cinnappa, A. Criianon, M.J. College & P.S. Ko), pp Ne Yor: Wile. LESSLER, J.T. & KALSBEEK, W.D. (99) Non-ampling error in urve. Ne Yor: Wile. 86

192 LEUNIS, W.P. & ALTENA, J.W. (996) Labour accoun in e Neerlan, Ho o cope i fragmene macro aa in official aiic. Inernaional Saiical Revie, 64, -. LITTLE, R.J.A. (993) Po-raificaion: a moeller perpecive. Journal of e American Saiical Aociaion, 88, LUNDSTRÖM, S. (997) Calibraion a a anar meo for reamen of nonrepone. Docoral ieraion, Deparmen of Saiic, Univeri of Socolm. LYBERG, L. & KASPRZYK, D. (997) Some apec of po-urve proceing. In Surve Meauremen an Proce Quali (e L. Lberg, P. Biemer, M. Collin, E. e Leeu, C. Dippo, N. Scarz & D. Trein), pp Ne Yor: Wile. MAHALONOBIS, P.C. (946) Recen eperimen in aiical ampling in e Inian Saiical Iniue. Journal of e Roal Saiical Socie, 09, MARAVALL, A. (998) Commen on Ne capabiliie an meo of e X-ARIMA eaonal aumen program, b Finle, D.F., Monell, B.C., Bell, W.R., Oo, M.C., Cen, B.-C.. Journal of Buine an Economic Saiic, 6, MOSTELLER, F. & TUKEY, J.W. (977) Daa anali an regreion. Reaing, MA: Aion-Wele. NASCIMENTO SILVA, P.L.D. & SKINNER, C.J. (997) Variable elecion for regreion eimaion in finie populaion. Surve Meoolog, 3, 3-3. NEYMAN, J. (934) On e o ifferen apec of e repreenaive meo: e meo of raifie ampling an e meo of purpoive elecion. Journal of e Roal Saiical Socie, 97, NORDBERG, L. (998) On variance eimaion for meaure of cange en ample are coorinae b a permanen ranom number ecnique. R&D Repor 998:6, Saiic Seen. OHLSSON, E. (995) Coorinaion of ample uing permanen ranom number. In Buine urve meo (e. B.G. Co, D.A. Biner, B.N. Cinnappa, A. Criianon, M.J. College & P.S. Ko), pp Ne Yor: Wile. PEASE, P. (997) Comparion of ource of emplomen aa. Labour Mare Tren, December 997. Lonon: Office for Naional Saiic. PIERZCHALA, M. (990) A revie of e ae of e ar in auomae aa eiing an impuaion. Journal of Official Saiic, 6, PURCELL, N.I. & KISH, L. (980) Po-cenal eimae for local area (or omain). Inernaional Saiical Revie, 48, 3-8. RAO, J.N.K. (996) On variance eimaion i impue urve aa. Journal of e American Saiical Aociaion, 9,

193 RENSSEN, R.H. & NIEUWENBROEK, N.C. (997) Aligning eimae for common variable in o or more ample urve. Journal of e American Saiical Aociaion, 9, ROYALL, R.M. (98) Finie populaion, Sampling from. Enr in e Encclopeia of Saiical Science (e. N.L. Jonon & S. Koz). Ne Yor: Wile. ROYALL, R.M. (986) Te preicion approac o robu variance eimaion in o-age cluer ampling. Journal of e American Saiical Aociaion, 8, 9-3. ROYALL, R.M. & CUMBERLAND, W.G. (98) An empirical u of e raio eimaor an eimaor of i variance. Journal of e American Saiical Aociaion, 76, ROYALL, R.M. & HERSON, J. (973) Robu eimaion in finie populaion I. Journal of e American Saiical Aociaion, 68, RUBIN, D.B. (986) Baic iea of muliple impuaion for nonrepone. Surve Meoolog,, RUBIN, D.B. (996) Muliple impuaion afer 8 ear. Journal of e American Saiical Aociaion, 9, SÄRNDAL, C.-E. (99) Meo for eimaing e preciion of urve eimae en impuaion a been ue. Surve Meoolog, 8, 4-5. SÄRNDAL, C.-E. & SWENSSON, B. (987) A general revie of eimaion for o pae of elecion i applicaion o o-pae ampling an non-repone. Inernaional Saiical Revie, 55, SÄRNDAL C.-E., SWENSSON B. & WRETMAN, J. (99) Moel-aie urve ampling. Ne Yor: Springer-Verlag. SEN, A.R. (953) On e eimaion of e variance in ampling i varing probabiliie. Journal of e Inian Socie of Agriculural Saiic, 5, 9-7. SHAO, J. & TU, D. (995) Te acnife an boorap. Ne Yor: Springer-Verlag. SKENE, A.M., SHAW, J.E.H. & LEE, T.D. (986) Baeian moeling an eniivi anali. Te Saiician, 35, SMITH T.M.F. (983) On e valii of inference from non-ranom ample. Journal of e Roal Saiical Socie, Serie A, 46, SMITH T.M.F. (99) Po-raificaion. Te Saiician, 40, SMITH T.M.F. (993) Populaion an elecion - limiaion of aiic. Journal of e Roal Saiical Socie, Serie A, 56, SOS (998) Ne are enerprie in Seen 996 an 997. Saiical Repor Nv SM 980 in e erie Official Saiic of Seen. Örebro, Seen: Saiic Seen. STATISTICS FINLAND (996) Progre Repor. Conribuion b T. Viiaaru an A. Heinonen o e 0 Inernaional Rounable on Buine Surve Frame. 88

194 STATISTICS SWEDEN (995) Demograp of enerprie an eablimen in Seen. An emplomen approac o meauring e namic among uni. Conribuion b B. Tegö o e 9 Inernaional Rounable on Buine Surve Frame. SCB, Örebro, pp STRUIJS, P. & WILLEBOORDSE, A. (995) Cange in populaion of aiical uni. In Buine urve meo (e. B.G. Co, D.A. Biner, B.N. Cinnappa, A. Criianon, M.J. College & P.S. Ko), pp Ne Yor: Wile. SUGDEN R.A. (993). Parial ecangeabili an urve ampling inference. Biomeria, 80, THEIL, H. (960) Be linear ine number of price an quaniie. Economerica, 8, VEZINA, S. (996) Saiic Canaa eperience i auomae aa enr. In Proceeing of Saiic Canaa Smpoium 96, Oaa. WEISBERG, S. (985) Applie regreion anali, econ eiion. Ne Yor: Wile. WOLTER, K.M. (985) Inroucion o variance eimaion. Ne Yor: Springer-Verlag. WOODRUFF, R.S. (97) A imple meo for approimaing e variance of a complicae eimae. Journal of e American Saiical Aociaion 66, YATES, F. & GRUNDY, P.M. (953) Selecion iou replacemen from iin raa i probabili proporional o ize. Journal of e Roal Saiical Socie, Serie B, 5,

195 3 Ine ABI...See Annual Buine Inquir aminiraive aa...83, 9 PAYE...95, 97 VAT...95, 97 ancillar variable...43 Annual Buine Inquir...6, 76, 77 ARIMA...45 auomae aa capure...4 auomae aa recogniion...7 auiliar informaion... 8, 8, 44, 3, 34, 70 auiliar variable...8, 68, 56 balance repeae replicaion...3 bar coe recogniion...8 bencmaring... 30, 4, 54, 59, 70 be linear inice...40 be linear unbiae preicor..., 43 bia..., 53, 66, 78, 07, 63, moel...9 aeing... effec of impuaion...35 nonrepone... 6, 8, 3 bir...83, 9 BLUP...See be linear unbiae preicor boorap variance eimaor... 35, 53, 58, 6, eign-bae...36, moel-bae...35 BRR...See balance repeae replicaion buine regier , 30, feebac o...83, for calenar ear...88, upaing...83, 86 calibraion... 7, 9, 3, 33 coice of conrol oal...8 cange, eimaion cange, eimaion of...78, 66 cluer moel...9 coing...6, 9 auomae...9 compuer aie...9 conienc...0 coing error See alo claificaion error coerence... 7, 63, 64, 66, 67, 69, 7, 74 comparabili...66, 67 inernaional...67, 7 ime...7, 75 coniionali principle...43 conienc inernal...0 conien eimae...65, 70 aligning eimae...70 conrol oal...see populaion oal, non co-orinaion...83 ample...74 urve... 68, 70, 73, 75 co-orinaion, ample...49 correlae coer error...0 correlaion, moel...50 covariance eign...49 covariae...see auiliar variable coverage eficienc...88, 9 cu-off ampling...75, 8, 86, 46 bia...5 ignore cu-off uni...77 moel e cu-off...79 aa capure...5 aa eiing... See eiing aa anling error...5 aa ranmiion...5, 6 ea...83 eign conienc...5 eign unbiaene...6, 5 eign-bae approac..., 38, 38, problem i...6 omain eimaion...40 iagnoic...60 irec eimae...53 iconinuiie...75 omain...40, 8, 89, 64, 68 omain eimaion , 54, 90, 5 cange omain memberip...40, 4, 44, 53, 85 onor impuaion...34 uplicaion...84 eiing...05, 0, 6,, 68 over-eiing... enerprie group...88 error, maor occaional...05 eimaing equaion...30 ecangeabili...7 eperimen ranomie... eernal aa ource...08 eernal aa ource...07, 30 finie populaion correcion...3, 33, 3 finie populaion parameer...9, 30, 35, 8, 63, 65 moel-bae approac...9 Fier ieal ine...39 fie ample ize eign...4 fie-effec moel...53 follo-up of reponen...0 u...77 urve...9 FPP...See finie populaion parameer frame...7, 74, 8, 85, 7 frame error...8, 88, 89, 0 frame populaion...85, 90, 03 frozen aa...94, 98 90

196 general linear regreion moel...8 generalie ifference eimaion...5 generalie raio eimaion...4 eign variance...5 generalie regreion eimaion...4, 8, 57 eign variance...5 GRAT...See generalie raio eimaion GREG...See generalie regreion eimaion gro error...07, 09 gro error moel...58 armoniaion... 67, 68, 7 Heneron filer...44 eeroeaici...36, 48 omogeneou raa moel...8, Horviz-Tompon eimae...3, 89 eign-bae eor for...3 o ec impuaion...34 Huber funcion...57 IDBR...See Iner-Deparmenal Buine Regier ignorable ampling...9, 75 impuaion ocaic...34 incluion probabili...3, oin...3 approimaing...5 ine number...39 ine of proucion...60 inicaor of quali...8 of queionnaire quali... of repone quali... informaive nonrepone...33, 55 inelligen caracer recogniion...8 Iner-Deparmenal Buine Regier...93 inerpeneraing ample...3 IoP...See Ine of proucion iem nonrepone...04, 33 acnife variance eimaor... 3, 36, 45, 53 linearie...33 ugemenal ampling...7, 8 Kalman filer...43 eing...6 Lapère ine...39 laen variable...56 level eimae...49 level eimaion...66 linear eimaion...49, 58 linear preicion...0 linear regreion moel... mean quare error..., moel...0 meauremen error... 04, 06,, 30, 69 meian...30 miclaificaion...06 miclaificaion mari...06 miing a ranom...5, 3, 33, 55, 57 miing compleel a ranom...4, 55, 57 miure...56 moel...7 for caraceriing populaion...9 meauremen error...07 moel aumpion error...38, 47, 59 moel aumpion...86 moel epenence...0 moel mipecificaion...54 moel-aie approac...7, 4, 8, 9 moel-bae approac...3, 7, 8, 38, 63, 7, 47, problem i...0 omain eimaion...4 oulier...55 muli-level moelling...54 mulipurpoe urve...8 muliage ample...33 NACE...8 NINR... See non-ignorable nonrepone nonconac...3 non-ignorable nonrepone5, 3, 55, 56, 57, 59 non-obervaion error... non-probabili ampling...66 nonrepone...3 informaive...5 iem...3 uni...3 ave...3 eiging...3 non-ampling error...40 obervaion equaion...56 obervaion error... obervaion uni...84, 89 opical caracer recogniion...8 opical mar recogniion...8 oulier...54, 44 repreenaive...54 overcoverage...87, 0 overlap...48 Paace ine...39 populaion oal, non...8, 4, 7 eimaion... poraificaion...68, 9, 3, 33 poraifie eimaor omain oal, for...43 PPI... See proucer price ine PPS amplingsee probabili proporional o ize ampling preicion variance, robu eimaion..., unbiae robu eimaion...3 prior aumen...45 probabili proporional o ize ampling...5, 5

197 probabili ampling... 8, 0, 38, 44, 65 proceing error...4 PRODCOM... 74, 86, 94, 7 proucer price ine...6, 7, 7 quali aurance...5 quali meaure...0 quoa ampling...7, 8 ranom error...63 correlae...69 ranom group variance eimaor...3 ranom-effec moel...53 ranomiaion iribuionsee repeae ampling iribuion raio eimaion...3, 47, eparae..., 7 combine...54 for cu-off...80 reference ime... 8, 88, 64, 68 refual...3 regiraion...84 regreion eimaion...3, 49 reinervie...0, 3 repeae ampling iribuion...0,,, compare o uperpopulaion iribuion... reporing uni...84 repreenaive...75 reiual...48 repone anali urve... repone error...04 repone omogenei group...3 repone inicaor...4 repone rae...7, 8 eige...8 robune... 0, 4, 60 oulier robu eimaion roaion...49, 74 ample error...0, ample frame...40 ample incluion inicaor... ample of convenience...65 ampling uni...84 ampling variance...3, 6, 3 canning...6, 7 eaonal aumen...43, 66 elecion equaion...56 elecion moel...56 eniivi anali... 68, 7, 33, 60, 6, 77 Sen-Yae-Grun variance eimaor...4 omain eimaion...4 imple linear regreion moel...8 ae-pace moel...4 aiic...0 raifie ampling...8, 5 ubiiari...7 uperpopulaion...3 uperpopulaion iribuion...0, compare o repeae ampling iribuion... uperpopulaion moel9, 7,, 8, 34, 55, 59, 38 urve populaion...7 neic eimaion...5 emaic error...63 em error...4 arge populaion7, 85, 86, 87, 89, 9, 03, 7, 8, 7 arge of inference.see finie populaion parameer Talor erie lineariaion...9, 34, 43, 45, 5, 6 oal... See populaion oal oal urve error..., 06 ouc-one...6 ranformaion...49, 60 rue value..., 04 unercoverage...87, 0 uni...8, 09, 64, 68 uni nonrepone...33 upaing e ample onl...90 variance, boorap eimaor of...35, acnife eimaor of...3, moel...0, preicion..., ranom group eimaor of...3, replicaion eimaor of...3, anic eimaor of...30, Sen-Yae-Grun eimaor of...4 effec of correlae coer error...0 effec of eiing... effec of impuaion...35, effec of meauremen error...3 effec of non-ignorable nonrepone...57 effec of nonrepone...6, 3 inflaion...07, 7 of a iribuion... of an ine...60 volunar ampling...66, 8, 3 eiging for nonrepone...3 populaion-bae...3 ample-bae...3 inoriaion one-ie...59 o-ie...59 X-ARIMA