he aedix o his glossary gives he mai aggregae umber formulae used for cosumer rice (CI) uroses ad also exlais he ierrelaioshis bewee hem. Acquisiios aroach Addiiviy Aggregae Aggregaio Axiomaic, or es aroach Base eriod Baske Baske rice Bias Boucig A aroach o CIs i which cosumio is ideified wih he cosumio goods ad services acquired by a household i some eriod (as disic from hose wholly or arially used u for uroses of cosumio). Deedig o he ieded scoe of he CI, acquisiios may iclude o oly goods ad services urchased, bu also hose acquired by ow-accou roducio or as social rasfers i kid from goverme or o-rofi isiuios. A curre rices, he value of a aggregae is ideical o he sum of he values of is comoes. Addiiviy requires his ideiy o be reserved for he exraolaed values of he aggregae ad is comoes whe heir curre values i some eriod are exraolaed usig a se of ierrelaed quaiy idices; or, aleraively, whe he curre values of a aggregae ad is comoes i some eriod are deflaed usig a se of ierrelaed rice idices. A se of rasacios relaig o a secified flow of goods ad services, such as he oal urchases made by reside households o cosumer goods ad services i some eriod. he erm aggregae is also used o mea he value of he desigaed se of rasacios. he rocess of combiig, or addig, differe ses of rasacios o obai larger ses of rasacios. he larger se is described as havig a higher level of aggregaio ha he ses of which i is comosed. he erm aggregaio is also used o mea he rocess of addig he values of he lower-level aggregaes o obai higher-level aggregaes. I he case of rice idices, i meas he rocess by which rice idices for lower-level aggregaes are averaged, or oherwise combied, o obai rice idices for higher-level aggregaes. he aroach o umber heory ha deermies he choice of umber formula o he basis of is mahemaical roeries. A lis of ess is draw u, each es requirig a o ossess a cerai roery or saisfy a cerai axiom. A umber may he be chose o he basis of he umber of ess saisfied. No all ess may be cosidered o be equally imora ad he failure o saisfy oe or wo key ess may be cosidered sufficie grouds for rejecig a. he base eriod is usually udersood o mea he eriod wih which all he oher eriods are comared. he erm may, however, have differe meaigs i differe coexs. hree yes of base eriod may be disiguished: he rice referece eriod he eriod ha rovides he rices o which he rices i oher eriods are comared. he rices of he rice referece eriod aear i he deomiaors of he rice relaives, or rice raios, used o calculae he. he rice referece eriod is yically desigaed as eriod ; he weigh referece eriod he eriod, usually oe or more years, of which he exediures serve as weighs for he. Whe he exediures are hybrid (i.e., he quaiies of oe eriod are valued a he rices of some oher eriod), he weigh referece eriod is he eriod o which he quaiies refer. he weigh referece eriod is yically desigaed as eriod b i his maual; he referece eriod he eriod for which he value of he is se equal o 1. I should be oed ha, i racice, he duraio of he weigh referece eriod for a CI is yically a year, or eve wo or more years, whereas he CI is calculaed mohly or quarerly, he duraio of he rice referece eriod beig a moh or quarer. hus, he weigh ad rice referece eriods seldom coicide i racice, a leas whe a CI is firs calculaed, alhough he rice ad referece eriods frequely coicide. A secified se of quaiies of goods ad services. I a CI coex, he se may comrise he acual quaiies of cosumio goods or services acquired or used by households i some eriod, or may be made u of hyoheical quaiies. A rice ha measures he roorioae chage bewee eriods ad i he oal value of a secified baske of goods ad services: ha is, q/ q, where he erms q are he secified quaiies. See Lowe. A sysemaic edecy for he calculaed CI o diverge from some ideal or referred, resulig from he mehod of daa collecio or rocessig, or he formula used. See Cos of livig bias ad Rereseaiviy bias. A siuaio i which he se of rices for he secod eriod is simly a reorderig of he se of rices for he firs eriod, he rice relaives hus beig obaied by machig each rice i he firs eriod wih aoher rice from he same se of rices. 443
Carli rice Carry forward Ceral roduc classificaio (CC) Chai Characerisics Circulariy (rasiiviy) Collecive cosumio Commesurabiliy Commodiy reversal es Comoe Codiioal cos of livig Cosisecy i aggregaio Cosumer rice (CI) Cosumers Cosumio Cosumio of ow roducio Coiuiy Cos of livig bias Cos of livig (COLI) Coverage A elemeary rice defied as a simle, or uweighed, arihmeic average of he samle rice relaives. A siuaio i which a missig rice i some eriod is imued as beig equal o he las rice observed for ha iem. A ieraioally agreed classificaio of goods ad services based o he hysical characerisics of goods or o he aure of he services redered. A umber series for a log sequece of eriods obaied by likig ogeher umbers saig shorer sequeces of eriods. See Likig; see also equaio (6) of he Aedix. he hysical ad ecoomic aribues of a good or service ha serve o ideify i ad eable i o be classified. A umber roery such ha, if j I k deoes a aricular kid of rice ha measures he chage bewee eriods j ad k, he j I l : j I k k I l where he idices j I l ad k I l are of he same ye. Whe a is rasiive, he ha comares eriods j ad l idirecly hrough eriod k is ideical wih he ha comares j ad l direcly. Oe es ha migh be required uder he axiomaic aroach is ha he umber should be rasiive. Goods ad services ha are cosumed simulaeously by a grou of cosumers or by he commuiy as a whole; for examle, defece services rovided by he Sae. See Ivariace o chages i he uis of measureme es. A es ha migh be used uder he axiomaic aroach, which requires ha, for a give se of roducs, he rice should remai uchaged whe he orderig of he roducs is chaged. A subse of he goods ad services ha make u some defied aggregae. A codiioal cos of livig measures he chage i he cos of maiaiig a give uiliy level, or sadard of livig, o he assumio ha all he facors, exce he rices covered by he, ha ifluece he cosumer s uiliy or welfare (e.g., he sae of he hysical evirome) remai cosa. See Cos of livig. A is said o be cosise i aggregaio whe he for some aggregae has he same value wheher i is calculaed direcly i a sigle oeraio, wihou disiguishig is comoes, or wheher i is calculaed i wo or more ses by firs calculaig searae idices, or sub-idices, for is comoes, or sub-comoes, ad he aggregaig hem, he same formula beig used a each se. A mohly or quarerly rice comiled ad ublished by a official saisical agecy ha measures chages i he rices of cosumio goods ad services acquired or used by households. Is exac defiiio may vary from coury o coury. Idividual ersos or grous of ersos livig ogeher as households. here are several yes of cosumio: iermediae cosumio cosiss of he goods ad services used by eerrises as ius io heir rocesses of roducio; i is excluded from CIs; collecive cosumio cosiss maily of he collecive services rovided by govermes o he commuiy as a whole; i is excluded from CIs; fial idividual cosumio cosiss of goods ad services ha idividual households may acquire i order o saisfy heir ow eeds ad was. See also Households cosumio exediures. Goods or services ha are cosumed by he same household ha roduces hem. he housig services cosumed by ower-occuiers fall wihi his caegory. If goods ad services roduced ad cosumed wihi he same household are o be icluded i CIs, rices mus be imued for hem. heir iclusio or exclusio deeds o he ieded scoe of he CI. he roery whereby he rice is a coiuous fucio of is rice ad quaiy vecors. A aleraive erm used o describe Subsiuio bias. A ha measures he chage bewee wo eriods i he miimum exediures ha would be icurred by a uiliy-maximizig cosumer, whose refereces or ases remai uchaged, i order o maiai a give level of uiliy (or sadard of livig or welfare). As cosumers may be execed o chage he quaiies hey cosume i resose o chages i relaive rices (see Subsiuio effec), he COLI is o a baske. he exediures i oe or oher, or ossibly boh, eriods cao usually be observed. COLIs cao be direcly calculaed bu may be aroximaed by suerlaive idices. See Codiioal cos of livig. he se of goods ad services of which he rices are acually icluded i he. For racical reasos, coverage may have o be less ha he ideal scoe of he, ha is, he se of goods ad services ha he comilers of he would refer o iclude if i were feasible. 444
Curre eriod, or comariso eriod Curre rices Curre value Cu-off samlig Deflaio Democraic Discou Divisia Domai Drif Drobisch rice Durable cosumio good Duo Ecoomic aroach Edgeworh rice Ediig Elemeary aggregae Elemeary rice Exediure weighs Exlici qualiy adjusme I ricile, he curre eriod should refer o he mos rece eriod for which he has bee comiled or is beig comiled. he erm is widely used, however, o mea he comariso eriod; ha is, he eriod ha is comared wih he base eriod, usually he rice referece or referece eriod. I is also widely used simly o mea he laer of he wo eriods beig comared. he exac meaig is usually clear i he coex. he acual rices revailig i he eriod i quesio. he acual value of some aggregae i he eriod i quesio: he quaiies i he eriod mulilied by he rices of he same eriod. A samlig rocedure i which a redeermied hreshold is esablished wih all uis i he uiverse a or above he hreshold beig icluded i he samle, ad all uis below he hreshold beig excluded. he hreshold is usually secified i erms of he size of some releva variable, he larges samlig uis beig icluded ad he res give a zero chace of iclusio. I he case of reail oules, size may be defied i erms of sales. he divisio of he curre value of some aggregae by a rice (described as a deflaor) i order o revalue is quaiies a he rices of he rice referece eriod. A form of CI i which each household is give equal weigh i he calculaio of he, irresecively of he size of is exediures. A deducio from he lis or adverised rice of a good or a service ha is available o secific cusomers uder secific codiios. Examles iclude cash discous, rom ayme discous, volume discous, rade discous ad adverisig discous. A rice or quaiy ha reas boh rices ad quaiies as coiuous fucios of ime. By differeiaig wih resec o ime, he rae of chage i he value of he aggregae i quesio is ariioed io wo comoes, oe of which is he rice ad he oher he quaiy. I racice, he idices cao be calculaed direcly, bu i may be ossible o aroximae hem by chai idices i which idices measurig he chages bewee cosecuive eriods are liked ogeher. A aleraive erm for he scoe of a. A chai is said o drif if i does o reur o uiy whe rices i he curre eriod reur o heir levels i he base eriod. Chai idices are liable o drif whe rices flucuae over he eriods hey cover. he arihmeic average of he Laseyres rice ad he aasche rice. A cosumio good ha ca be used reeaedly or coiuously for uroses of cosumio over a log eriod of ime, yically several years. A elemeary rice defied as he raio of he uweighed arihmeic averages of he rices i he wo eriods comared. he ecoomic aroach o umber heory assumes ha he quaiies are fucios of he rices, he observed daa beig geeraed as soluios o various ecoomic oimizaio roblems. I he CI coex, he ecoomic aroach usually requires he CI o be some kid of cos of livig. A baske rice i which he quaiies i he baske are simle arihmeic averages of he quaiies cosumed i he wo eriods. he rocess of scruiizig ad checkig he rices reored by rice collecors. Some checks may be carried ou by comuers usig saisical rograms wrie for he urose. he smalles aggregae for which exediure daa are available ad used for CI uroses. he values of he elemeary aggregaes are used o weigh he rice idices for elemeary aggregaes o obai higher-level idices. he rage of goods ad services covered by a elemeary aggregae should be relaively arrow, ad may be furher arrowed by cofiig he goods ad services o hose sold i aricular yes of oule or i aricular locaios. Elemeary aggregaes also serve as sraa for he samlig of rices. A elemeary is a rice for a elemeary aggregae. Exediure weighs cao usually be assiged o he rice relaives for he samled roducs wihi a elemeary aggregae, alhough oher kids of weighig may be exlicily or imlicily iroduced io he calculaio of elemeary idices. hree examles of elemeary umber formulae are he Carli, he Duo ad he Jevos. See Weighs. A direc esimae of how much of he chage i he rice of a roduc is aribuable o chages i is hysical or ecoomic characerisics. I requires a evaluaio of he coribuios of he differeces i aricular characerisics o he differeces i he observed rices of wo roducs. I icludes qualiy adjusmes based o hedoic mehods. See also Imlici qualiy adjusme. 445
Facor reversal es Fisher rice Fixed baske idices Fixed weigh idices Geomeric Laseyres Goods Hedoic mehod Suose he rices ad quaiies i a rice are ierchaged o yield a quaiy of exacly he same fucioal form as he rice. Uder he axiomaic aroach, he facor reversal es requires ha he roduc of his quaiy ad he origial rice should be ideical o he roorioae chage i he value of he aggregae i quesio. he geomeric average of he Laseyres rice ad he aasche rice. I is a symmeric ad a suerlaive. A ime series of baske idices ha all use he same baske; see equaio (4) of he Aedix. Ia CI coex, he fixed baske usually cosiss of he oal quaiies cosumed by he desigaed se of households over a eriod of a year or more. A abbreviaed descriio for a series of weighed arihmeic averages of rice relaives ha all use he same weighs; see equaio (13) of he Aedix. he weighs are usually eiher acual or hybrid exediure shares. A weighed geomeric average of he rice relaives usig he exediure shares of he rice referece eriod as weighs. Also called Logarihmic Laseyres. hysical objecs for which a demad exiss, over which owershi righs ca be esablished ad for which owershi ca be rasferred bewee uis by egagig i rasacios o he marke. A regressio model i which he marke rices of differe roducs are exressed as a fucio of heir characerisics. No-umerical characerisics are rereseed by dummy variables. Each regressio coefficie is reaed as a esimae of he margial coribuio of ha characerisic o he oal rice. he esimaes may be used o redic he rice of a ew roduc for which he mix of characerisics is differe from ha of ay roduc already o he marke. he hedoic mehod ca herefore be used o esimae he effecs of qualiy chages o rices. Higher-level A aggregae as disic from a elemeary. Household budge surveys (HBSs) Households Households cosumio exediures Hybrid values or exediures Hybrid weighs Ideiy es Imlici qualiy adjusme Imued rice Idexaio Idex referece eriod Isiuioal ui Ivariace o chages i he uis of measureme es Samle surveys of households i which he households are asked o rovide daa o, or esimaes of, he amous hey sed o cosumio goods ad services, ad for oher uroses over a give eriod of ime. Households may be eiher idividual ersos livig aloe or grous of ersos livig ogeher who make commo rovisio for food or oher esseials for livig. Mos couries choose o exclude grous of ersos livig i large isiuioal households (barracks, reireme homes, ec.) from he scoe of heir CIs. Exediures o fial cosumio goods ad services icurred by idividual households o heir ow behalf. hey exclude exediures icurred by govermes or o-rofi isiuios o goods or services rovided o households as free social rasfers i kid. Hyoheical values, or exediures, i which he quaiies are valued a a differe se of rices from hose a which hey were acually bough or sold: for examle, whe he quaiies urchased i a earlier eriod, such as b, are valued a he rices revailig i a laer eriod, such as. Weighs defied as hybrid value, or hybrid exediure, shares. A es uder he axiomaic aroach ha requires ha, if each ad every rice remais uchaged bewee he wo eriods, he rice mus equal uiy. Iferrig idirecly he chage i he qualiy of a roduc of which he characerisics chage over ime by esimaig, or assumig, he ure rice chage ha has occurred. For examle, if he ure rice chage is assumed o be equal o he average for some oher grou of roducs, he imlied chage i qualiy is equal o he acual observed rice chage divided by he assumed ure rice chage. If he whole of he observed rice chage is assumed o be ure rice chage, here is assumed o be o chage i qualiy. See also Exlici qualiy adjusme. he rice assiged o a iem for which he rice is missig i a aricular eriod. he erm imued rice may also refer o he rice assiged o a iem ha is o sold o he marke, such as a good or service roduced for ow cosumio, icludig housig services roduced by oweroccuiers, or oe received as ayme i kid or as a free rasfer from a goverme or o-rofi isiuio. he eriodic adjusme of he moey values of some regular scheduled aymes based o he moveme of he CI or some oher rice. he aymes may be wages or salaries, social securiy or oher esios, oher social securiy beefis, res, ieres aymes, ec. he eriod for which he value of he is se a 1. A aioal accous coce defied as a ecoomic eiy ha is caable, i is ow righ, of owig asses, icurrig liabiliies ad egagig i ecoomic aciviies ad rasacios wih oher eiies. Households are isiuioal uis. Oher kids of uis iclude eerrises ad govermes. A es uder he axiomaic aroach ha requires ha he rice does o chage whe he uis of quaiy o which he rices refer are chaged: for examle, whe he rice of some drik is quoed er lire raher ha er i. his es is also described as he commesurabiliy es. 446
Ivariace o roorioal chage i curre or base quaiies es Iverse roorioaliy i base year rices es Iem Iem or roduc roaio Jevos rice Laseyres rice Likig Lowe Lower-level Mached roducs or models Mea value es for rices No-robabiliy samlig Oe hoss shay Oulier Ower-occuied housig aasche rice rice referece eriod rice relaive A es uder he axiomaic aroach ha requires ha he rice remais uchaged whe all he base eriod quaiies, or all he curre eriod quaiies, are mulilied by a osiive scalar. A es which may be ivoked uder he axiomaic aroach ha requires ha, if all he base eriod rices are mulilied by he osiive scalar l, he ew rice is 1/l imes he old rice. A idividual good or service i he samle of roducs seleced for ricig. he deliberae relaceme of a samled iem, or roduc, for which rices are beig colleced, by aoher roduc before he relaced roduc has disaeared from he marke or idividual oule. I is desiged o kee he samle of roducs u o dae ad reduce he eed for forced relacemes caused by he disaearace of roducs. A elemeary rice defied as he uweighed geomeric average of he samle rice relaives. A baske i which he baske is comosed of he acual quaiies of goods ad services i he earlier of he wo eriods comared, he rice referece eriod; see equaio (3) of he Aedix. I ca also be exressed as a weighed arihmeic average of he rice relaives ha uses he exediure shares i he earlier eriod as weighs; see equaios (7) o (1) of he Aedix. he earlier eriod serves as boh he weigh referece eriod ad he rice referece eriod. Slicig ogeher wo cosecuive sequeces of rice observaios, or rice idices, ha overla i oe or more eriods. Whe he wo sequeces overla by a sigle eriod, he usual rocedure is simly o rescale oe or oher sequece so ha he value i he overla eriod is he same i boh sequeces ad he sliced sequeces form oe coiuous series. See equaio (6) of he Aedix. A rice ha measures he roorioae chage bewee eriods ad i he oal value of a secified baske of goods ad services; ha is, q/ q, where he erms q are he secified quaiies. he baske does o ecessarily have o cosis of he acual quaiies i some eriod. See Aedix. his ye of is described i he maual as a Lowe afer he umber ioeer who firs roosed his geeral ye of. he class of idices covered by his defiiio is very broad ad icludes, by aroriae secificaio of he erms q, he Laseyres, aasche, Edgeworh ad Walsh idices, for examle. Lowe idices are widely used for CI uroses, he quaiies i he baske yically beig hose of some weigh referece eriod b, which recedes he rice referece eriod. A elemeary as disic from a aggregae. he racice of ricig exacly he same roduc i wo or more cosecuive eriods. I is desiged o esure ha he observed rice chages are o affeced by qualiy chage. he chage i rice bewee wo erfecly mached roducs is described as a ure rice chage. A es uder he axiomaic aroach, which requires ha he rice should lie bewee he smalles rice relaive ad he larges rice relaive. he deliberae, i.e. o-radom, selecio of a samle of oules ad roducs o he basis of he kowledge or judgeme of he erso resosible. Also kow as urosive samlig ad judgemeal samlig. A model of dereciaio, i which he durable delivers he same services for each viage: a chair is a chair, o maer wha is age (uil i falls o ieces ad is scraed). Also kow as he ligh bulb model of dereciaio. A erm ha is geerally used o describe ay exreme value i a se of survey daa. I a CI coex, i is used for a exreme value of rice or rice relaive ha requires furher ivesigaio or ha has bee verified as beig correc. Dwelligs owed by he households ha live i hem. he dwelligs are fixed asses ha heir owers use o roduce housig services for heir ow cosumio, hese services beig usually icluded wihi he scoe of he CI. he res may be imued by he res ayable o he marke for equivale accommodaio or by user coss. See Real equivalece ad User cos. A baske i which he baske is comosed of he acual quaiies of goods ad services i he laer of he wo eriods comared. he laer eriod serves as he weigh referece eriod ad he earlier eriod as he rice referece eriod. he aasche ca also be exressed as a weighed harmoic average of he rice relaives ha uses he acual exediure shares i he laer eriod as weighs. See equaios (7) o (11) of he Aedix. he eriod of which he rices aear i he deomiaors of he rice relaives. See also Base eriod. he raio of he rice of a idividual roduc i oe eriod o he rice of ha same roduc i some oher eriod. 447
rice udaig robabiliy roorioal o size samlig (S) robabiliy samlig roducs roorioaliy i curre rices es urchaser s rice ure rice chage Qualiy adjusme Quaiy relaive Quaiy weighs Rebasig Referece oulaio Real equivalece Relaceme roduc Rereseaive roduc Rereseaiviy bias Reweighig Samle augmeaio Samled rice Samled roduc Samle roaio A rocedure whereby he quaiies i a earlier eriod are revalued a he rices of a laer eriod. he resulig exediures are hybrid. I racice, he rice-udaed exediures may be obaied by mulilyig he origial exediures by rice relaives or rice idices. A samlig rocedure whereby each ui i he uiverse has a robabiliy of selecio roorioal o he size of some kow variable, such as he value of he sales of a oule. he radom selecio of a samle of uis, such as oules or roducs, i such a way ha each ui i he uiverse has a kow o-zero robabiliy of selecio. A geeric erm used o mea a good or a service. Idividual samled roducs seleced for ricig are ofe described as iems. A es uder he axiomaic aroach ha requires ha, if all curre eriod rices are mulilied by he osiive scalar l, he ew rice is l imes he old rice. he amou ayable by he urchaser o acquire a good or service. he urchaser s rice icludes ay charges icurred i order o ake delivery a he ime ad lace required by he urchaser. he chage i he rice of a good or service of which he characerisics are uchaged; or he chage i he rice afer adjusig for ay chage i qualiy. A adjusme o he chage i he rice of a roduc of which he characerisics chage over ime ha is desiged o remove he coribuio of he chage i he characerisics o he observed rice chage. I a CI coex, he adjusme is eeded whe he rice of a relaceme roduc has o be comared wih he rice of he roduc i relaces. I racice, he required adjusme ca oly be esimaed. Differe mehods of esimaio, icludig hedoic mehods, may be used i differe circumsaces. See Exlici qualiy adjusme ad Imlici qualiy adjusme. he raio of he quaiy of a roduc i oe eriod o he quaiy of ha same roduc i some oher eriod. A erm someimes used o describe he quaiies i he baske. However, exediures raher ha quaiies ac as weighs for rice relaives. See Weighs. Rebasig may have differe meaigs i differe coexs. I may mea: chagig he weighs used for a series of idices; or chagig he rice referece eriod used for a series of idices; or chagig he referece eriod for a series of idices. he weighs, rice referece eriod ad referece eriod may be chaged searaely or a he same ime. he se of households icluded wihi he scoe of he. he esimaio of he imued res ayable by ower-occuiers o he basis of he res ayable o he marke for accommodaio of he same ye. A roduc chose o relace a roduc for which rices have bee colleced reviously, eiher because he revious roduc has disaeared alogeher or because i accous for a dimiishig share of he sales of he oule, or he exediures wihi he elemeary aggregae. A roduc, or caegory of roducs, ha accous for a sigifica roorio of he oal exediures wihi a elemeary aggregae, ad/or for which he average rice chage is execed o be close o he average for all roducs wihi he aggregae. Bias i a baske ha resuls from he use of quaiies ha are o rereseaive of he wo eriods comared; ha is, ha sysemaically diverge from he average quaiies cosumed i he wo eriods. For examle, rereseaiviy bias may resul from he use of a old, ou-of-dae baske which deviaes sysemaically from he baskes i boh he eriods comared. I racice, rereseaiviy bias eds o be similar o subsiuio bias, as i is aribuable o he same ecoomic facors. Relacig he weighs used i a by a ew se of weighs. Maiaiig ad addig o he samle of oules i he survey ael o esure ha hey coiue o be rereseaive of he oulaio of oules. A fixed samle of oules eds o be deleed over ime, as oules cease radig or so resodig. Icludig ew oules also eds o faciliae he iclusio of ew roducs i he CI. he rice colleced for a samled roduc, someimes described as a rice quoe. A idividual roduc ha is icluded i he samle seleced for ricig wihi a elemeary aggregae. Limiig he legh of ime ha oules ad/or roducs are icluded i he rice surveys by droig a roorio of hem, or ossibly all of hem, afer a cerai eriod of ime ad selecig a ew samle of oules ad/or roducs. Roaio is desiged o kee he samle u o dae. 448
Samlig frame Scaer daa Scoe Seasoal roducs Secificaio Sochasic aroach Subsiue Subsiuio Subsiuio bias Subsiuio effec Suerlaive Symmeric Sysem of Naioal Accous (SNA) ime reversal o rqvis rice A lis of he uis i he uiverse from which a samle of uis ca be seleced. he lis may coai iformaio abou he uis, which may be used for S samlig. Examles of liss ha may be used for reail oules are busiess regisers, elehoe direcories ( yellow ages ), local auhoriy records, rade direcories, ec. Such liss may o cover all he uis i he desigaed uiverse ad may also iclude uis ha do o form ar of ha uiverse. Deailed daa o sales of cosumer goods obaied by scaig he bar codes for idividual roducs a elecroic ois of sale i reail oules. he daa ca rovide deailed iformaio abou quaiies, characerisics ad values of goods sold, as well as heir rices. Scaer daa cosiue a raidly exadig source of daa wih cosiderable oeial for CI uroses. hey are icreasigly used for uroses of hedoic aalysis. he se of roducs for which he is ieded o measure he rice chages. he scoe of a CI will geerally be defied i erms of a desigaed se of cosumio goods ad services urchased by a desigaed se of households. I racice, cerai goods ad services or households may have o be excluded because i is oo difficul, ime-cosumig or cosly o collec he releva daa o exediures or rices: for examle, illegal exediures. he coverage of a deoes he acual se of roducs icluded, as disic from he ieded scoe of he. Seasoal roducs are roducs ha eiher are o available o he marke durig cerai seasos or eriods of he year, or are available hroughou he year bu wih regular flucuaios i heir quaiies ad rices ha are liked o he seaso or ime of he year. A descriio or lis of he characerisics ha ca be used o ideify a idividual samled roduc o be riced. A igh secificaio is a fairly recise descriio of a iem ieded o arrow he rage of iems from which a rice collecor migh choose, ossibly reducig i o a uique iem, such as a aricular brad of elevisio se ideified by a secific code umber. A loose secificaio is a geeric descriio of a rage of iems ha allows he rice collecor some discreio as o which aricular iem or model o selec for ricig, such as colour elevisio ses of a aricular size. he aroach o umber heory ha reas he observed rice relaives as if hey were a radom samle draw from a defied uiverse for which he mea ca be ierreed as he geeral rae of iflaio. he samle mea rovides a esimae of he rae of iflaio. A roduc of which he characerisics are similar o hose of aoher roduc ad ha ca be used o mee he same kids of cosumer eeds or was. he relaceme of roducs by subsiues, yically i resose o chages i relaive rices. Raioal uiliy-maximizig cosumers, as rice akers, yically reac o chages i relaive rices by reducig, a leas margially, heir cosumio of goods ad services ha have become relaively dearer ad icreasig heir cosumio of subsiues ha have become relaively cheaer. Subsiuio resuls i a egaive correlaio bewee he quaiy ad rice relaives. his is geerally udersood o be he bias ha resuls whe a baske is used o esimae a cos of livig, because a baske cao ake accou of he effecs o he cos of livig of he subsiuios made by cosumers i resose o chages i relaive rices. I geeral, he earlier he eriod of which he baske is used, he greaer he uward bias i he ; see also Rereseaiviy bias. he effec of subsiuio o he value of a. A ye of formula ha ca be execed o aroximae o he cos of livig. A is said o be exac whe i equals he rue cos of livig for cosumers whose refereces ca be rereseed by a aricular fucioal form. A suerlaive is he defied as a ha is exac for a flexible fucioal form ha ca rovide a secod-order aroximaio o oher wice-differeiable fucios aroud he same oi. he Fisher, he o rqvis ad he Walsh rice idices are examles of suerlaive idices. Suerlaive idices are geerally symmeric idices. A ha reas boh eriods symmerically by aachig equal imorace o he rice ad exediure daa i boh eriods. he rice ad exediure daa for boh eriods eer io he formula i a symmeric way. A cohere, cosise ad iegraed se of macroecoomic accous, balace shees ad ables based o a se of ieraioally agreed coces, defiiios, classificaios ad accouig rules. Household icome ad cosumio exediure accous form ar of he SNA. he exediure daa are oe of he sources ha are used o esimae exediure weighs for CI uroses. A umber roery such ha, if j I k deoes a aricular kid of rice formula ha measures he chage from eriod j o eriod k, he j I k :1/ k I j where k I j measures he chage from eriod k o eriod j. Whe a has his roery, he chage is he same wheher i is measured forwards from he firs o he secod eriod or backwards from he secod o he firs eriod. A may be required o saisfy he ime reversal es uder he axiomaic aroach. A symmeric defied as he weighed geomeric average of he rice relaives i which he weighs are simle arihmeic averages of he exediure shares i he wo eriods. I is a suerlaive. Also kow as he o rqvis heil rice. 449
rasiiviy Ui value or average value User cos Uses aroach Value Walsh rice Weigh referece eriod Weighed arihmeic average Weighs Youg See Circulariy. he ui value of a se of homogeeous roducs is he oal value of he urchases/sales divided by he sum of he quaiies. I is herefore a quaiy-weighed average of he differe rices a which he roduc is urchased/sold. Ui values may chage over ime as a resul of a chage i he mix of he roducs sold a differe rices, eve if he rices do o chage. he cos icurred over a eriod of ime by he ower of a fixed asse or cosumer durable as a cosequece of usig i o rovide a flow of caial or cosumio services. User cos cosiss maily of he dereciaio of he asse or durable (measured a curre rices ad o a hisoric cos) lus he caial, or ieres, cos. A aroach o CIs i which he cosumio i some eriod is ideified wih he cosumio goods ad services acually used u by a household o saisfy heir eeds ad was (as disic from he cosumio goods ad services acquired). I his aroach, he cosumio of cosumer durables i a give eriod is measured by he values of he flows of services rovided by he socks of durables owed by households. hese values may be esimaed by he user coss. rice imes quaiy. he value of he exediures o a se of homogeeous roducs ca be facored uiquely io is rice ad quaiy comoes. Similarly, he chage over ime i he value of a se of homogeeous roducs ca be facored uiquely io he chage i he ui value ad he chage i he oal quaiies. here are, however, may differe ways of facorig he chage over ime i he value of a se of heerogeeous roducs io is rice ad quaiy comoes, a heomeo ha gives rise o he umber roblem. A baske i which he quaiies are geomeric averages of he quaiies i he wo eriods; see he Aedix. I is a symmeric ad a suerlaive. he eriod of which he exediure shares serve as he weighs for a Youg, or of which he quaiies make u he baske for a Lowe. here may be o weigh referece eriod whe he exediure shares for he wo eriods are averaged, as i he o rqvis, or whe he quaiies are averaged, as i he Walsh. See also Base eriod. A defied as a weighed arihmeic average of he rice relaives: amely, w( / ), where he weighs w sum o uiy. A se of umbers summig o uiy ha are used o calculae averages. I a CI coex, he weighs are geerally acual or hybrid exediure shares ha sum o uiy by defiiio. hey are used o average rice relaives, or elemeary rice idices; see he Aedix. Quaiies of differe kids of roducs are o commesurae ad o addiive. hey cao serve as weighs. he quaiies ha make u a baske should herefore o be described as quaiy weighs. A Youg is a weighed arihmeic average of he rice relaives, w( / ), i which he erms w refer o he acual exediure shares of eriod b, he weigh referece eriod; ha is, w=s b = b q b / b q b. I is a weighed versio of he Carli. 45
Aedix o he glossary. Some basic umber formulae ad ermiology 1. hroughou his Aedix, he sums are udersood o be ruig over all iems. A baske rice is a of he form q q (A:1) which comares he rices of eriod wih hose of (a earlier) rice referece eriod, usig a cerai secified quaiy baske. he baske does o have o cosis of he acual quaiies i ay aricular eriod. his geeral ye of is called a Lowe rice afer he umber ioeer who firs roosed his geeral ye of. he family of Lowe idices icludes some well-kow idices as secial cases: whe q =q, we ge he Laseyres ; whe q =q, we ge he aasche ; whe q =(q +q1 )=2, we ge he Marshall Edgeworh ; ad whe q =(q q )1=2, we ge he Walsh. I racice, saisical offices frequely work wih a Lowe i which q =q b, where b deoes some weigh referece eriod ha is yically rior o. 2. A useful feaure of a Lowe for eriod relaive o eriod is ha i ca be decomosed, or facored, io he roduc of wo or more idices of he same ye: for isace, as he roduc of a for eriod 1 relaive o eriod ad a for eriod relaive o eriod 1. Formally, q 1 = q q q q (A:2) 1 q I aricular, whe q =q, exressio (2) urs io q 1 q = q q q 1 q (A:3) he lef-had side of exressio (3) is a direc Laseyres. Noe ha oly he firs of he idices ha make u he righhad side is iself a Laseyres, he secod beig a Lowe for eriod relaive o eriod 1 ha uses he quaiy baske of eriod (o 1). Some saisical offices describe he o he righ-had side of exressio (3) as a modified Laseyres. 3. I a ime series coex, say whe rus from 1 o, he series 1 q 2, q,..., q q q q (A:4) is ermed a series of fixed baske rice idices. I aricular, whe q =q, we ge a series of Laseyres idices. 4. A eriod oe could chage o a ew quaiy baske q, ad calculae from his eriod owards +1 q +2 q +3 q, q q, q ;...: (A:5) o relae he rices of eriods +1, +2, +3,... o hose of eriod, chai likig ca be used o rasform he series (5) io a series of he form q +1 q q q, q +2 q q q, q +3 q q q,...: (A:6) his could be ermed a series of chai-liked fixed baske rice idices. I aricular, whe q =q ad q ¼ q, we ge a series of chai-liked Laseyres idices. Sice he baske is chaged a eriod, he adjecive fixed alies lierally oly over a cerai umber of ime iervals. he baske is fixed from eriod 1 o eriod, ad is agai fixed from eriod +1 owards. Whe he ime iervals durig which he baske is ke fixed are of he same legh, such as oe, wo or five years, his feaure ca be made exlici by describig he as a aual, bi-aual or five-yearly chai-liked fixed baske rice. 5. A weighed arihmeic-average rice (so-called o disiguish i from a geomeric or oher kid of average) is a of he form w ( = ) (A:7) which comares he rices of eriod wih hose of eriod, usig a cerai se of weighs addig u o 1. I aricular, whe he weighs are he eriod b value shares w =s b b qb = b qb (A:8) we obai he Youg. Noe ha ay baske rice (1) ca be exressed i he form (7), sice q = q q q : (A:9) Whe he weighs are he eriod value shares, w =s q = q exressio (7) urs io he Laseyres. Whe (A:1) w = q = q (A:11) ha is, hybrid eriod (, ) value shares, we ge he aasche. Oe could also hik of seig w =s b ( =b )= s b ( =b )= qb = qb (A:12) ha is, rice-udaed eriod b value shares. Noe ha hybrid value shares, such as give by exressios (11) or (12), are o observable bu mus be cosruced. 6. I a ime series coex, whe rus from 1 o, he series w ( 1 = ), w ( 2 = ),...::, w ( = ) (A:13) is ermed a series of fixed weighed arihmeic-average rice idices. I aricular, whe he weighs are equal o he eriod exediure shares, we ge a series of Laseyres idices, ad whe he weighs are equal o he rice-udaed eriod b exediure shares, we ge a series of Lowe idices i which he quaiies i he baske are hose of eriod b. 7. I eriod oe could chage o a ew se of weighs w, ad calculae from his eriod owards w ( +1 = ), w ( +2 = ), w ( +3 = ),...:: (A:14) 451
or, usig chai-likig o relae he rices of eriods +1, +2, +3,... o hose of eriod, w ( = ) w ( +1 = ), w ( = ) w ( +2 = ),...: (A:15) his could be ermed a series of chai-liked fixed weighed arihmeic-average rice idices. I aricular, whe w =s ad w =s, we ge a series of chai-liked Laseyres idices. Whe w =s b ( =b )= s b ( =b ) ad w =sb ( =b )= s b ( =b ) for some laer eriod b, we ge a series of chai-liked Lowe idices. 8. Agai, sice he weighs are chaged a eriod, he adjecive fixed alies lierally oly over a cerai umber of ime iervals. he weighs are fixed from eriod 1 o eriod, ad are agai fixed from eriod +1 owards. Whe he ime iervals durig which he weighs are ke fixed are of he same legh, his feaure ca be made exlici by addig a emoral adjecive, such as aual, bi-aual or five-yearly. 452