INVESTIGATION OF ALTERNATIVE REPLICATE VARIANCE ESTIMATORS FOR THE ANNUAL CAPITAL EXPENDITURES SURVEY

003 Jon Sascal Meengs - Secon on Survey Researc Meods INVESTIGATION OF ALTERNATIVE REPLICATE VARIANCE ESTIMATORS FOR THE ANNUAL CAPITAL EPENDITURES SURVEY aerne J. Tompson Rcard S. Sgman Roger L. Goodwn U.S. Bureau o e Census. INTRODUCTION Te Annual Capal Ependures Survey ACES collecs normaon abou e naure and level o capal ependures n non-arm companes organzaons and assocaons n e Uned Saes. ACES uses a one-sage sraed smple random sample wou replacemen SRS- OR desgn and samplng racons n several sraa are larger an 0.0. ACES perorms weg adusmen or un non-response and does no perorm mpuaon or em nonresponse. Secon descrbes e ACES sample desgn and esmaon meodology. From e collecon year 000 daa onward ACES began usng e U.S. Census Bureau s Sandard Economc Processng Sysem SEPS as s pos-daa-collecon sysem Amed and Tasy 000. le e esng SEPS esmaon module soware easly accommodaed e ACES esmaors varance esmaon enancemens were requred. In pror collecons perods ACES used a samplng eory ormula S varance esmaor w non-response adused wegs n place o samplng wegs so a e raoadused wegs were reaed as consans n e producon varance esmaes. SEPS s a generalzed sysem wc lends sel more o replcaon varance esmaon meods. Tus e prmary purpose o s sudy was o deermne weer replcae varance esmaon could be used o esmae ACES varances. By 000 SEPS ncluded varance esmaon soware or e meod o random groups. Ts varance esmaon meod s que popular w many o e U.S. Census busness surveys or eorecal and or operaonal reasons. Teorecally random group varance esmaes o epanson esmaors are nearly unbased or sraed SRS-OR samples w small samplng racons [e mos-commonly used desgn or our non-manuacurng busness surveys]. Operaonally surveys a ncorporae brs new busnesses no ongong samples can easly and correcly nclude e new uns n e varance esmaons by assgnng new uns o random groups as ey are seleced. Moreover random group esmaon requres ewer Te auors an llam C. Dave Jr. Jaclyn Jonas Andromace Howe and Quaraca llams or er useul commens on earler versons o s manuscrp. Ts paper repors e resuls o researc and analyss underaen by e U.S. Census Bureau sa. I as undergone a Census Bureau revew more lmed n scope an a gven o ocal Census Bureau publcaons. Ts repor s released o norm neresed pares o ongong researc and o encourage dscusson o wor n progress. compuer resources an oer more popular meods suc as e sraed acne: replcae esmaes or replcae wegs were s e number o random groups versus one replcae esmae or replcae weg per responden or e sraed acne. Random group varance esmaon as wo drawbacs. Frs can be unpredcable wen appled o SRS-OR samples because e random group esmaor ends o esmae e varance as e sample were seleced w replacemen oler 985 p.43. Te second drawbac s e nsably o e random group varance esmaes especally wen e number o sampled observaons n eac random group s small or wen ere s a g rae o un non-response. Because o ese drawbacs we nvesgaed e delee-a-group acne varance esmaor. Ts meod can be appled o e same ypes o sample desgns as e random group meod and sould yeld more sable varance esmaes snce replcaes are consruced rom more sample uns. Moreover snce SEPS already ad random group esmaon capably we new a a e number o delee-a-group replcaes or replcae wegs would no pose an operaonal problem. o 00 and 998 repors ecellen resuls usng e delee-a-group acne or several o e Naonal Agrculural Sascs Servce NASS programs w a varey o sample desgns ncludng sraed SRS-OR or epanson rao and resrced regresson esmaors. Sm 00 also repors some success w e delee-a-group acne varance esmaor or New Zealand s labor orce survey. Our nvesgaon speccally eamned ow o mody random group and delee-a-group acne varance esmaors or wou replacemen samples w nonneglgble samplng racons and weer e non-response adusmen procedure sould be repeaed n eac replcae. Te rs ssue s dscussed n Secons 3 and 6 and e second ssue s dscussed n Secons 4 and 5. Secon 4 provdes our emprcal resuls or ree ey capal ependures caracerscs usng 999 ACES daa. Te resuls rom s emprcal esmaon movaed e smulaon sudy descrbed n Secon 5. Secon 6 descrbes an approprae way o use ese replcaon meods o oban varance esmaes or combned rao or rend esmaes or survey desgns w non-neglgble samplng racons. Secon 7 provdes our conclusons.. ACES SAMPLE DESIGN AND ESTIMATION METHODOLOGY Te ACES unverse conans wo sub-populaons: employer companes and non-employer companes. Deren orms are maled o sample uns dependng on weer ey are 46

003 Jon Sascal Meengs - Secon on Survey Researc Meods employer ACE- companes or non-employer ACE- companes. New ACE- and ACE- samples are seleced eac year bo w sraed SRS-OR desgns. Te ACE- sample comprses appromaely seveny-ve percen o e ACES sample rougly 45000 companes seleced per year or ACE- and 5000 seleced per year or ACE-. Te ACE- rame s sraed rs by prmary ndusry acvy ound n e Census Bureau s Busness Regser. Fve separae sraa are ormed wn ndusry: one cerany sraum conssng o companes w 500 or more employees and our non-cerany sraa deermned usng a moded Lavallee-Hdrglou meod w payroll as a measure o sze Slana and renze 996. Samplng racons n e noncerany ACE- sraa can be que g: n e 999 desgn 3 o e 54 sraa ad samplng racons greaer an 0.0. Unle e ACE- desgn samplng racons n all ACE- sraa are que low all less an 0.0 n e 999 desgn. Two o e ACE- sraa are pos-sraed usng updaed normaon rom e Busness Regser aer daa collecon. Te ACE- and ACE- non-response weg procedures ollow e adusmen-o-sample models descrbed n alon and asprzy 986.e. all samplng wegs n a wegng class l are mulpled by a acor derved rom daa correspondng o sample uns. Te ACE- non-response adusmen procedure conrols samplng wegs o ndependenly obaned esmaes o payroll; a s e nonresponse wegng adusmen acor or a wegng cell l s e sum o e sample-weged payroll esmaes or uns n e wegng cell dvded by e sum o e sampleweged payroll esmaes or all respondng uns n e wegng cell. Under complee non-response n a cerany sraum or complee non-response n e large company sraum e wo sraa are combned no one wegng cell wn e sample ndusry. Presenly ere s no collapsng procedure n place or complee non-response n e ree remanng non-cerany sraa. Te pos-sraed ACE- esmaes are conrolled o sample couns wn sraa; a s e non-response wegng adusmen acor or a wegng cell l s calculaed as e number o sampled uns n e wegng cell dvded by e number o respondng uns n e wegng cell [Noe: snce ACE- perorms non-response adusmen wn sraa e sample wegs cancel ou]. Tus e nal weg or survey v adv s gven by S l ad v or all l Rl p l Sl n l w l ACE l ACE prw l Rl r l l ACE l ACE were ndees e sraum v ndees e survey ndees e sample un p s e esmaed payroll n sraum n s e sample sze o sraum N /n e samplng weg and r s a varable ndcang e response saus o sample un. See Caldwell 999a or more deals on e ACE- and ACE- non-response wegng adusmen procedures ACES publses epanson esmaes or all caracerscs. Tecncally ese esmaes are non-lnear because o e rao non-response weg adusmen procedure. Addonally ACES publses year-o-year rend esmaes. 3. VARIANCE ESTIMATION METHODOLOGY Ts secon descrbes our consdered varance esmaon procedures. Our non-replcae varance esmaor or caracersc s an appromae v ad v samplng-ormula S varance gven by vâr vâr 3. s v v s NSR Sraa v ACE v vâr ACE N ad v s s v v 3. were N n e appromaed populaon sze o sraum and n r r r s oerwse r r r See Caldwell 999b. Usng o esmae s wen a sraum conans one responden allows e sraum o conrbue o e varance compuaon owever poorly. Is use s no eorecally used. Noce a ormula 3. drecly ncorporaes e ne populaon correcon pc. oler 985 C. provdes modcaons or random group and delee-a-group acne esmaors or sraed samples w nonneglgble samplng racons speccally suggesng usng * n place o were s e sraum nal weg and n /N s e sraum samplng racon. e used s adusmen n all o e replcae procedures descrbed below. Te random group meod begns by splng e noncerany poron o e survey sample no random groups 47

003 Jon Sascal Meengs - Secon on Survey Researc Meods usng e survey s samplng meodology oler 985 pp. 3-3. Eac random group s sample s en reweged o represen e ull sample eer by smply mulplyng e random group esmae by smple rewegng or by developng replcae wegng acors wn eac sraum sraa-specc rewegng. Te sraa-specc replcae acors or sraum and random group are n / m were m s e number o sample uns n sraum assgned o random group. Developng sraa-specc replcae wegng acors yelds replcae esmaes a are condonally unbased. Suc replcae wegng acors may owever ncrease e varance o e esmaed varances snce ey der by sraa. Cerany uns are ncluded n eac random group. Tese cases or er assocaed replcae wegs are no mulpled by or any adusmen acor. Tus replcae wegs are assgned o eac sample un. I un s n a non-cerany sraum e replcae weg s zero unless un s n random group. In a cerany sraum all replcae wegs are equal o e samplng or nal weg. Te ull sample esmaon procedure s en appled o eac o e replcae wegs e.g. non-response adusmens pos-sracaon or o e replcae esmaes. Te random group varance or any esmae s RG 0 vâr RG 3.3 were RG s e random group replcae esmae and 0 s e ull-sample esmae. I e replcae wegs used ncorporae e pc adusmens descrbed above en and e abulaed ull-sample esmae wll no be equvalen. For delee-a-group acne varance esmaon agan e non-cerany poron o survey sample s dvded no random groups. However e delee-a-group acne replcae esmae s compued or eac replcae by removng e random group rom e ull sample. Replcaes are obaned eer by mulplyng eac replcae esmae by /- or by developng sraum-specc replcae wegs. Te sraa-specc replcae acors or sraum and delee-a-group acne replcae are n / m m. Cerany uns are ncluded n eac delee-agroup acne replcae esmae. Tus or delee-a-group acne replcaon replcae wegs are assgned o eac sample un. I un s n a non-cerany sraum e replcae weg s zero wen un s n random group. In a cerany sraum all replcae wegs are equal o e samplng or nal weg. 0 Snce acne replcae sample szes are larger an e correspondng random group replcae sample szes deleea-group acne varance esmaes are oen more sable a leas or smoo sascs. Te delee-a-group acne varance or an esmae s vârdag DAG 0 3.4 were DAG s e replcae delee-a-group acne esmae. As saed earler we also waned o nvesgae e sascal properes o replcang e non-response adusmen procedure.e. ndependenly perormng e non-response adusmen procedure on eac se o replcae wegs. Ts can be que me-consumng and compuer resourcenensve so we consdered a sorcu approac usng e ull sample non-response adused wegs n all replcaes. Insuonal nuon eld a bo e samplng ormula varance procedure and sorcu approac would underesmae e rue varance by alng o eplcly accoun or non-response varance. Ts nuon s somewa suppored n e leraure: or eample Cany and Davson 999 ound replcang e calbraed wegng procedure reduced e degree o relave bas n er sraed acne varance esmaes rom ose usng e sorcu approac or a smlar desgn. On e oer and oler 985 pp. 83-84 ces resuls rom wo sudes a sowed e slg mprovemens n random group varance esmaes usng ull replcae rewegng versus e sorcu approac dd no ose e addonal compung coss. In a smlar ven Scndler 00 ound rval derences beween e varance compued w a ully-reweged sraed acne procedure versus ose obaned w a smple acne a used nal wegs n all esmaes sorcu procedure or seleced dual sysem esmaes rom e Census 000 Accuracy and Coverage Enumeraon Survey. e consdered ree deren replcae wegng varaons per replcaon meod: Smple Consruc replcae wegs rom e ull sample s non-response adused wegs e sorcu. Random group esmaon uses as e replcae adusmen acor; delee-a-group acne esmaon uses /-. Smple Reweged Consruc replcae wegs rom e sample wegs en perorm e non-response adusmen procedure on eac se o replcae wegs. Random group esmaon uses as e replcae adusmen acor; delee-a-group acne esmaon uses /-. Sraed Reweged Consruc replcae wegs rom e sample wegs were non-cerany uns adused wegs n a gven replcae are mulpled by sraaspecc replcae acors. Perorm non-response adusmen on eac se o replcae wegs. In subsequen secons we use RG o ndcae random group and DAG o ndcae delee-a-group acne 48

003 Jon Sascal Meengs - Secon on Survey Researc Meods combned w S smple SR smple reweged and STR sraed reweged. Cerany cases are ecluded rom all o e dscussed replcae varance esmaes va e pc-adusmen all cerany cases ave replcae wegs o zero. None o e replcae varance meods descrbed accoun or e varance conrbuon due o non-respondng cerany uns. Ts s conssen w ACES curren producon meod. e used 5 random groups n all applcaons. For a sraed SRS-OR desgn o 00 proves a e delee-a-group acne varance esmaor s appromaely unbased or e rue varance wen e sample sze n eac sraum s larger an e number o random groups and all samplng racons are neglgble less an or equal o /5 and s based upwards oerwse. Tus e 999 desgn ACE- delee-a-group acne varance esmaes are appromaely unbased. Ts s no e case w ACE-: ere were welve o 54 ACE- sraa a dd no ave sample n all een random groups. Moreover e proporon o sraa a are no represened n any random group s acually ger due o un non-response. Consequenly e ACE- TR esmaes are based upwards. Te bound on s bas gven by o 00 a s 4/5mn {/n -} s probably no applcable because suc a g proporon o e ACE- samplng racons are que large. 4. EMPIRICAL DATA RESULTS Inally we compared e s replcae varance esmaors o e samplng ormula appromaon or ree capal ependure sascs Toal Capal Ependures; Capal Ependures on Srucures; Capal Ependures on Equpmen; usng 999 ACES daa. e ound several neresng paerns. Frs e samplng ormula S sandard errors were generally larger an correspondng replcae esmaes. e ound s perpleng avng assumed a e S and smple replcae varance meods would conssenly underesmae e varance snce ey do no eplcly accoun or e nonresponse adusmen. Second perormng non-response adusmen n eac delee-a-group acne replcae e smple reweged SR or sraed reweged STR meods -- usually reduced e esmaed sandard error rom e correspondng smple replcaon esmae agan a couner-nuve resul. Tere s no conssen paern w e random group esmaes. Fnally e smple reweged acne esmaes were less an or equal o e correspondng sraed reweged acne esmaes. Ts was reasonable snce e varable replcae acors used or sraed rewegng sould ncrease e varably among e replcae esmaes. In conras usng sraaspecc random group adusmen acors reduced e esmaed sandard errors or all ACE- caracerscs and or ree ACE- caracerscs. Ts dd no seem reasonable. Ts las derence could parally accoun or e varably n replcae acors or e RGSTR and TR meods. As Table sows e sraa-level random group adusmen acors were que varable and were on e average que deren rom er epeced value unle e correspondng delee-a-group acne adusmen acors. Table : Sraa-Level Adusmen Facors or 999 ACES Daa Frame Meod Epeced Mean Sample Mean Sandard Devaon Mnmum Mamum ACE- RGSTR 5.00 5.59 3.38.00 9.00 TR.07.07 0.03.03.00 ACE- RGSTR 5.00 5..4.5 0.00 TR.07.07 0.0.05.09 Te nconssen emprcal resuls or e ACE- random group esmaes are parally eplaned by e replcaed non-response adusmen procedures. Table presens e number o weg adusmen cells w complee nonresponse by replcaon meod or ACE- [Noe: e resuls are equvalen or e smple and sraed reweged meods]. Toal A reers o e large-sze non-cerany wn-ndusry sraa wc are collapsed w ndusry cerany sraa under complee non-response or wegng adusmens. In all oer ACE- sraa e wegng cell l s equvalen o e sraum. e random group meods an overly g proporon o ACE- sraa ave no respondens n a replcae wegng cell. Ts poses wo problems. Te rs s mecancal: ecep or ACE- Sraum A ACES does no ave a collapsng mecansm n place or non-cerany sraa. Second and ar more mporan e random group varance esmaon meod s no mmcng e ull-sample esmaon procedure. In conras n all bu one replcae e delee-a-group acne replcaes use e same wegng cells as e ull sample. Table : ACE- egng Cells w Complee Non-Response Meod Full Replcae Sample 3 4 5 6 7 8 9 0 3 4 5 RG Toal A 8 0 9 4 8 5 3 3 7 Oer 0 40 38 4 37 4 40 39 34 33 35 30 37 39 34 38 DAG Toal A Oer 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Snce appromaely weny-percen o e ACE- sraa samplng raes are larger an 0.0 we were concerned a no ncorporang e pc no e replcae varance esmaes could lead o subsanal overesmaes o varance. Table 3 presens raos o unadused o pc-adused replcae sandard errors or e same ree ACE- caracerscs [Noe: ACE- samplng racons are all less an 0.0 so all sandard error raos are as epeced]. Table 3: ACE- Sandard Error Raos ou FPC/ FPC RGS RGSR RGSTR R TR Toal.0.0.0.0.0.03 Srucures.00.00.00.00.0.0 Equpmen.06.06.06.06.06.07 49

003 Jon Sascal Meengs - Secon on Survey Researc Meods Falng o accoun or e non-neglgble pcs n e varance esmaes leads o a s-percen overesmae o sandard error or capal ependures on equpmen and conssenly overesmaes e SE or oal capal ependures e prmary sasc o neres by appromaely wo-percen. Tere s no real derence beween sandard error esmaes or capal ependures on srucures bu s s a arly small caracersc. Te overesmaon or e oer wo caracerscs could aec coverage n parcular or capal ependures on equpmen us usyng e need or ncorporang pcs n e replcaon. Te overly varable replcae acors or e sraed reweged random group meod concerned us. Te unrealsc non-response adusmen collapsng paern w e wo reweged meods convnced us no o urer pursue random group varance esmaon meods w ACES daa and o nsead concenrae on delee-a-group acne varance esmaon meods. O course our emprcal resuls sll le us w our deren ses o varance esmaes and no gold sandard agans wc o measure em. So we conduced a Mone Carlo smulaon sudy o evaluae e properes o ese our deren varance esmaors. 5. SIMULATION STUDY 5. Creaon o e Frame and Sample Selecon Capal ependures daa are dcul o model. Frs ey are oen poorly correlaed w aulary daa suc as payroll or employmen especally or small companes. Second purcasng paerns are no necessarly conssen wn an ndusry. For eample n some ndusres capal ependures on srucures and equpmens are negavely correlaed or small companes and posvely correlaed or large companes. Te mulvarae correlaon srucure becomes more complcaed wen capal ependures daa s urer cross-classed by new or used saus. Consequenly we developed models only or non-cerany employer companes usng e acual repored sample daa or cerany companes n eg sample NAICS ndusres provded by ACES meodologss. Tese ndusres encompassed a varey o proessonal secors: Ules; Manuacures; olesale Trade; Real Trade; Inormaon; Proessonal Scenc and Tecncal Servces; and Admnsrave Suppor ase Managemen and Remedaon Servces. Ts smulaon sudy dd no nclude e ACE- rame daa wc represens appromaely 5% o e oal ACES sample unverse bu only appromaely seven percen o e oal esmaed capal ependures. In general eac sample ndusry requred ree separae ses o models: one or uns a repored all capal ependures on equpmen; one or uns a repored all capal ependures on srucures; and one or uns a repored spendng on bo. In e laer case we modeled wo o e ree caracerscs eplcly dervng e remanng caracersc as e derence o e oer caracerscs. In eac sample ndusry we randomly appled e ree ses o models o e rame daa n e same proporons as n e responden sample daa: a s rs we smulaed oal capal ependures daa n e same proporon as repored n e ndusry en we appled our ree deren models o e uns a ad nonzero smulaed capal ependures daa. e sraed s complee rame daa usng e ACES producon programs. Aer sracaon we used a mssng-a-random model o assgn response saus n wc e probables o non-response maced e ACES nonresponse raes by sraum. Tus we assumed a ed se o non-respondens on e rame addng bas o e esmaor bu allowng or deren response paerns by sample. Fnally we seleced 5000 sraed SRS-OR samples rom s smulaed populaon usng e sraa samplng raes rom e ACES sracaon and allocaon programs. In 000 o e 5000 samples we assgned sample uns o 5 random groups. Le e emprcal sudy no all sraa conan all random groups: one o e 3 non-cerany sraa conans en sample uns. 5. Evaluaon Crera To eamne e sascal properes o e our deren varance esmaon meods we used our 5000 sraed random samples o consruc e emprcal varance o eac caracersc n sample ndusry u as 5000 r ru u V u 5. 5000 were s e esmae o caracersc n ndusry u n sample r and u s e mean o e ru. ru Ne we calculaed our varance esmaes v me per caracersc n ndusry u rom 000 o e 5000 samples. e compared ese varance esmaes n erms o 000 v me 000 r Relave Bas V c.v.v me 000 000 r u ru [ v V ] me V Relave bas s a measure o e bas o e varance esmae as a proporon o e rue varance. Te coecen o varaon c.v. measures e varance o e varance esmaes; s sasc s called e sably n oer publcaons e.g. Rao and Sao 996. an opmal varance esmaor bo e relave bas and e c.v. wll be near zero. 5.3 Resuls Table 4 presens e relave bases o e our varance esmaon meods. Sascally sgncan conrass beween bases α 0.05 are saded. To compare conrass n relave bases beween varance esmaon meods u ru u 430

003 Jon Sascal Meengs - Secon on Survey Researc Meods wn ndusry or eac caracersc we used an ANOVA approac w e repeaed measures model v me ru µ me u + ε me ru. For eac caracersc/ndusry we rs esed e omnbus ypoess H 0 : µ Su µ u µ Ru µ TRu. Snce we reeced H 0 or all caracerscs n all ndusres we esed e conrass n varance esmae means. Parwse derences beween relave bases are sascally deren wen e conrass beween e correspondng varance esmae means are sgncanly deren.e p 0.05. Table 4: Relave Bases o ance Esmaors Ind. Relave Bas Conrass Relave Bases Toal S R TR B S - B B S - B R B S - B TR B - B R B R - B TR 3-0.07-0.03-0.0-0.0-0.04-0.05-0.05-0.0 0.00 353 0.03 0.04 0.06 0. -0.0-0.03-0.08-0.0-0.05 3369-0.0-0.07 0.06 0. -0.3-0.6-0.3-0.3-0.06 40-0.05-0.03-0.0-0.0-0.0-0.04-0.04-0.0 0.00 4480-0.04 0.05 0.07 0.08-0.09-0. -0. -0.0-0.0 5-0.0 0.00 0.5 0.0-0.0-0.35-0.40-0.5-0.05 545-0. -0. -0.0-0.09 0.00-0.0-0.03-0.0-0.0 569-0.03 0.03 0.06 0.07-0.06-0.09-0.0-0.03-0.0 Srucures 3 0.00 0.0 0.03 0.03-0.0-0.03-0.03-0.0 0.00 353 0.07 0.05 0.0 0. 0.0-0.03-0.04-0.05-0.0 3369-0. -0.7-0.09-0.06-0.04-0. -0.5-0.08-0.03 40-0.0-0.0-0.0-0.0 0.0 0.00 0.00-0.0 0.00 4480-0.06-0.06-0.04-0.04 0.00-0.0-0.0-0.0 0.00 5-0.06-0.0 0.05 0.08-0.04-0. -0.4-0.07-0.03 545-0. -0. -0. -0.0 0.00-0.0-0.0-0.0-0.0 569-0.04-0.04-0.0-0.0 0.00-0.0-0.0-0.0 0.00 Equpmen 3-0.4-0. -0.09-0.09-0.03-0.05-0.05-0.0 0.00 353 0.04 0.04 0.06 0. 0.00-0.0-0.08-0.0-0.06 3369-0.6-0.03 0.09 0.6-0.3-0.5-0.3-0. -0.07 40-0. -0.07-0.05-0.03-0.04-0.06-0.08-0.0-0.0 4480-0.0 0.07 0.09 0.0-0.09-0. -0. -0.0-0.0 5-0.0 0.00 0.5 0.9-0.0-0.35-0.39-0.5-0.04 545-0. -0. -0.0-0.09-0.0-0.0-0.03-0.0-0.0 569-0.0 0.04 0.07 0.07-0.06-0.09-0.09-0.03 0.00 Tere s very lle evdence o derence beween e wo reweged delee-a-group acne varance esmaes R and TR. Oerwse e maory o conrass are sgncanly deren. Te relave bas resuls can be summarzed as ollows:! S relave bases are negave or all caracerscs n all bu one ndusry. On e average s varance esmaon meod underesmaes e rue varance;! For caracerscs n ndusres w sgncan derences beween e S and relave bases e meod generally yelds varance esmaes wose relave bas s closer o zero 4 o 6 or oal capal ependures; 3 o 3 or srucures; 4 o 6 or equpmen;! R relave bases are always larger an correspondng relave bases. Some o s bas ncrease beween and R could be caused by avng one sraum a s no represened n all random groups. To summarze Table 4 sows clear gans n relave bas usng eer e or R meod over e S meod bu does no deny a clearly superor meod n erms o bas. Table 5 presens e c.v.s o e varance esmaes or eac caracersc or eac varance esmaon meod. Table 5: C.V.s o e Four ance Esmaon Meods Indusry S R TR Toal Srucures Equpmen 3.37.43.45.45 353 0.93.04.08.5 3369 0.79.09.50.75 40..5.7.9 4480 0.76 0.89 0.93 0.99 5 0.65 0.85.5.33 545.4.6.6.7 569 0.93.0.4.7 3.70.75.79.79 353.67.7.85.88 3369.59.8.09.0 40.70.67.69.69 4480.67.64.67.68 5.05.5.45.5 545.6.8.8.3 569 3.55 3.4 3.45 3.45 3.09.6.9.9 353 0.97.09.3.9 3369 0.93.9.70.93 40 0.53 0.6 0.66 0.98 4480 0.86.0.05.07 5 0.66 0.86.5.33 545..4.4.8 569 0.9.08.3.5 Across e board e S varance esmaes are e leas varable. O course e S esmaor does no eplcly accoun or e varance componen due o un non-response and does no ncur an addonal resamplng varance componen. Reasonably e varably o varance esmaes ncreases w replcaon: all o e delee-agroup acne c.v.s are ger an e correspondng S c.v.s. Te varably o e varance esmaes urer ncreases wen replcang e non-response adusmen c.. e o e R and TR sables especally n e ransporaon manuacurng 3369 and soware publsers 5 ndusres. Fnally usng e sraa-specc adusmen acors TR nsead o consan adusmen acors R ncreases e varance o e varance. Consequenly w our daa ses ere s no advanage o e TR meod: yelds overly varable varance esmaes s varance esmaes are no sascally deren rom e R meod and requres e mos compuer resources. ese resuls we ave wo almos equally good replcae varance esmaors or ACES: and R. Bo ave very smlar sascal properes. Moreover n mos samples e wo ses o varance esmaes were very close. e epeced e esmaes o conssenly underesmae e rue varance. Ts dd no appen n our 43

sample daa: 49-percen o e varance esmaes are larger an e R esmaes or Toal Capal Ependures; 37-percen or Capal Ependures on Srucures; and 49-percen or Capal Ependures on Equpmen. Ts paern s conssen w our emprcal resuls. Here e slg relave bas mprovemens o e R meod over e meod do no compleely ose e worsenng sably measures.e. ncreased cvv me o e R meod. Moreover e meod s muc aser and s less compuer-resource nensve an e R meod. Fnally as menoned n Secon 4 e ACE- non-response adusmen procedures only provde collapsng creron or wo o e ve wn-ndusry sraa. By desgn complee non-response n e remanng ree sraa s gly unlely. However we canno guaranee a wll always appen we use R varance esmaon or ACES and we wan o avod e ype o collapsng problem seen n Secon 4 w random group esmaon. For ese reasons ACES meodologss eleced o use e meod. 6. COMPUTING VARIANCES OF COMBINED RATIO AND TREND ESTIMATORS Te leraure suppors usng acne-ype varance esmaes or rao esmaors wen samplng racons can be gnored e.g. o 00 Rao and Sao 996. en samplng racons are large drecly replcang e varance o a combned rao esmae can yeld large overesmaes. To see s consder e smple eample o a SRS-OR desgn. For any esmaor SRS-OR - SRS-R 6. 6. Our esmaor mulples eac replcae weg by e square-roo o eac un s pc. For epanson esmaors e esmae s R SRS DAG > sraa s pc-adusmen o e replcae wegs gves unbased varance esmaes or lnear esmaors see o 00 pp. 53-54 replacng w. * For rao esmaes owever e esmae rom a SRS-OR desgn s Y y y y y Y R SRS DAG Obvously suc cancellaon does no occur w a sraed SRS-OR desgn unless s sel-wegng. By eenson oug our replcae wegng procedure wll overesmae e varance o combned rao esmaes. Furermore can be sown a ere are only wo survey desgns or wc applyng e square-roo-pc correcon o only e numeraor replcae wegs wll yeld correc combned rao esmaes: SRS-OR and sel-wegng sraed SRS-OR. Ts ecnque wll provde correcly adused esmaes or separae rao esmaors under unresrced sraed random samplng. Te ACES publses year-o-year rend esmaes. Le combned rao esmaors rend esmaors use esmaes consruced rom e ull sample n bo e numeraor and e denomnaor a rend esmae s e derence o e curren and pror perod esmaes o caracersc dvded by e pror perod esmae o caracersc. Drec replcaon usng square-roo-pc adused replcae wegs s napproprae as llusraed a me or a SRS-OR desgn: were s e samplng racon or e curren sample and s e samplng racon or e pror sample. Ts s an obvously poor appromaon: < e rao o e wo pc s s larger an one and consequenly e esmaed SRS-OR varance s larger an e SRS-R wc sould be mpossble. Moreover even w s smple desgn s dcul o come up w a sraegy a appropraely combnes e wo pc adusmens e geomerc mean mg be an opon. To avod s problem we use Taylor Seres meods o esmae rend varances oler 985 C. 6. Te Taylor Seres appromaon or e varance o e rend esmaor s gven by + TAYLOR Cov were e varance and covarance esmaes are appropraely adused varance esmaes or e epanson esmaes [Noe: e coce o replcae meod s no parcularly mporan]. ACES e covarance 003 Jon Sascal Meengs - Secon on Survey Researc Meods 43

003 Jon Sascal Meengs - Secon on Survey Researc Meods erm s zero because o e ndependenly-seleced samples Foreman 99 p.49. For non-ndependen samples e covarance erm can be obaned va subracon. 7. CONCLUSION e presen resuls o a sudy comparng e sascal properes o wo deren replcae varance esmaors random group and delee-a-group acne or a survey a uses a one-sage SRS-OR desgn w non-neglgble samplng racons n several sraa. e also eamne e eecs o ully replcang e non-response adusmen procedure versus a sorcu approac o usng e ullsample s non-response adused wegs o consruc eac replcae. Muc o our sudy ocuses on lnear esmaors aloug we dscuss applcaons o our meods o combned rao and rend esmaors. Our emprcal daa comparsons led us o elmnae e random group varance esmaor rom consderaon or ACES. Our smulaon sudy ocused on e benes o replcang e non-response adusmen procedure n delee-a-group acne replcaes. Te smulaon sudy resuls demonsraed some sascal advanages o bo e smple delee-a-group-acne and e smple reweged delee-a-group acne R meods over e appromae samplng ormula meod ormerly used by ACES. Tey also provded evdence agans usng sraa-specc replcae wegng acors recommended by o 00: ere were ew any relave bas mprovemens w s meod over e oers coupled w ncreased varably o e varance esmaes. Ulmaely e coce beween e and R meods or ACES was no obvous n erms o e suded sascal properes. Tus admnsrave consderaons suc as compuer resources and producon run me were e decdng acors leadng us o recommend usng e smple delee-a-group acne varance esmaor or ACES a leas nally. Aer coosng our varance esmaor we eamned ow o calculae replcae varances o non-lnear esmaors suc as combned rao esmaors or year-o-year rend esmaors or sraed SRS-OR desgns w non-neglgble samplng racons. For ese esmaors we sow a usng replcae wegs a ncorporae e pc o consruc replcae esmaes en drecly replcang combned rao or rend varance ends o overesmae e varance [Noe: s also apples o random group esmaon under e same condons]. Usng Taylor lnearzed varance esmaes reduces s overesmaon. Pror o s sudy we assumed a drecly replcang e non-response adusmen procedure was sascally preerable. Our resuls dd no suppor s ypoess. e used one samplng desgn and one non-response adusmen meodology and we suded a survey a radonally as a very g un response rae appromaely 75%. le our resuls suppor conclusons ced n oler 985 and Scndler 00 more varaons on bo sample desgn and weg adusmen meodology are requred beore mang any general recommendaons. Delee-a-group acne varance esmaon s one o a varey o acne esmaors. Ts parcular esmaor was appealng or anecdoal and producon reasons. Eamnng alernave acne esmaors suc as e sraed acne or surveys w smlar desgns s an area o uure sudy. REFERENCES Amed S.A. and Tasy D.L. 000. An Overvew o e Sandard Economc Processng Sysem. Proceedngs o e Inernaonal Conerence on Esablsmen Surveys II Aleandra VA: Amercan Sascal Assocaon pp. 633-64. Caldwell C. 999a. Non-response egng or e 999 Annual Capal Ependures Survey ACES. Unpublsed nernal memorandum. asngon DC: U.S. Bureau o e Census. Caldwell C. 999b. Esmaon and ance Esmaon or e 999 Annual Capal Ependures Survey ACES. Unpublsed nernal memorandum. asngon DC: U.S. Bureau o e Census. Cany A.J. and Davson A.C. 999. Resamplng-based ance Esmaon or Labour Force Surveys. Te Sascan 48 pp. 379-39. Fay R.E. 995. VPL: ance Esmaon or Comple Surveys. Unpublsed program documenaon. asngon DC: U.S. Bureau o e Census. Foreman E.. 99. Survey Samplng Prncples. New Yor: Marcel Decer Inc. alon G. and asprzy 986. Te Treamen o Mssng Survey Daa. Survey Meodology pp. 6. o P. 998. Usng e Delee-A-Group Jacne ance Esmaor n NASS Surveys. RD Researc Repor Number RD-98-0. asngon DC: Naonal Agrculural Sascs Servce. o P. 00. Te Delee-A-Group Jacne. Journal o Ocal Sascs 7 pp. 5-56. Rao J.N.. and Sao J. 996. Some Recen or on Resamplng Meods or Comple Surveys. Survey Meodology 8 pp. 09-7. Scndler E. 00. Smpled ance Esmaon or Comple Surveys. Proceedngs o e Secon on Survey Researc Meods Amercan Sascal Assocaon orcomng. Slana J.G. and renze T.R. 996. Applyng e Lavallee and Hdroglou Meod o Oban Sracaon Boundares or e Census Bureau s Annual Capal Ependures Survey. Survey Meodology pp.65-76. Sm H. 00. Invesgaon o e Delee-a-Group Jacne ance Esmaor or e HLFS. Researc repor Sascs New Zealand Paper 3 Te Tar Taau. ellngon New Zealand. oler r M. 985. Inroducon o ance Esmaon. New Yor: Sprnger-Verlag. 433