1 Hedonic Regressions: A Review of Some Unresolved Issues Erwin Diewer 1, Deparmen of Economics, Universiy of Briish Columbia, Vancouver, B.C., Canada, V6T 1Z1. email: diewer@econ.ubc.ca January 2, 2003. 1. Inroducion Three recen publicaions have revived ineres in he opic of hedonic regressions. The firs publicaion is Pakes (2001) who proposed a somewha conroversial view of he opic. 2 The second publicaion is Chaper 4 in Schulze and Mackie (2002), where a raher cauious approach o he use of hedonic regressions was advocaed due o he fac ha many issues had no ye been compleely resolved. A hird paper by Heravi and Silver (2002) also raised quesions abou he usefulness of hedonic regressions since his paper presened several alernaive hedonic regression mehodologies and obained differen empirical resuls using he alernaive models. 3 Some of he more imporan issues ha need o be resolved before hedonic regressions can be rouinely applied by saisical agencies include: Should he dependen variable be ransformed or no? Should separae hedonic regressions be run for each of he comparison periods or should we use he dummy variable adjacen year regression echnique iniially suggesed by Cour (1939; 109-11) and used by Bernd, Griliches and Rappapor (1995; 260) and many ohers? Should regression coefficiens be sign resriced or no? Should he hedonic regressions be weighed or unweighed? If hey should be weighed, should quaniy or expendiure weighs be used? 4 How should ouliers in he regressions be reaed? Can influence analysis be used? The presen paper akes a sysemaic look a he above quesions. Single period hedonic regression issues are addressed in secions 2 o 5 while wo year ime dummy variable regression issues are addressed in secions 6 and 7. Some of he more echnical maerial relaing o secion 7 is in an Appendix, which examines he properies of bilaeral weighed hedonic regressions. Secion 8 discusses he reamen of ouliers and 1 The auhor is indebed o Erns Bernd and Alice Nakamura for helpful commens. 2 See Hulen (2002) for a nice review of he issues raised in Pakes paper. 3 The observaion ha differen varians of hedonic regression echniques can generae quie differen answers empirically daes back o Triple and McDonald (1977; 150) a leas. 4 Diewer (2002b) recenly looked a hese weighing issues in he conex of a simplified adjacen year hedonic regression model where he only characerisics were dummy variables.
2 influenial observaions and secion 9 addresses he issue of wheher he signs of hedonic regression coefficiens should be resriced. Secion 10 concludes. 2. To Log or No o Log We suppose ha price daa have been colleced on K models or varieies of a commodiy over T+1 periods. 5 Thus p k is he price of model k in period for = 0,1,...,T and k S() where S() is he se of models ha are acually sold in period. For k S(), denoe he number of hese ype k models sold during period by q k. 6 We suppose also ha informaion is available on N relevan characerisics of each model. The amoun of characerisic n ha model k possesses in period is denoed as z kn for = 0,1,...,T, n = 1,...,N and k S(). Define he N dimensional vecor of characerisics for model k in period as z k [z k1,z k2,...,z kn ] for = 0,1,...,T and k S(). We shall consider only linear hedonic regressions in his review. Hence, he unweighed linear hedonic regression for period has he following form: 7 (1) f(p k ) = β 0 + n=1 N f n (z kn )β n + ε k ; = 0,1,...,T; k S() where ε k is an independenly disribued error erm wih mean 0 and variance σ 2, f(x) is eiher he ideniy funcion f(x) x or he naural logarihm funcion f(x) ln x and he funcions of one variable f n are eiher he ideniy funcion, he logarihm funcion or a dummy variable which akes on he value 1 if he characerisic n is presen in model k or 0 oherwise. We are resricing he f and f n in his way since he ideniy, log and dummy variable funcions are by far he mos commonly used ransformaion funcions used in hedonic regressions. Recall ha he period characerisics vecor for model k was defined as z k [z k1,z k2,...,z kn ]. We define also he period vecor of he β s as β [β 0,β 1,...,β N ]. Using hese definiions, we simplify he noaion on he righ hand side of (1) by defining: (2) h (z k,β ) β 0 + n=1 N f n (z kn )β n = 0,1,...,T; k S(). The quesion we now wan o address is: should he dependen variable f(p k ) on he lef hand side of (1) be p k or lnp k ; i.e., should f be he ideniy funcion or he log funcion? 8 We also would like o know if he choice of ideniy or log for he funcion f should affec our choice of ideniy or log for he f n ha correspond o he coninuous (i.e., non dummy variable) characerisics. 5 Models sold in differen oules can be regarded as separae varieies or no, depending on he conex. 6 If a paricular model k is sold a various prices during period, hen we inerpre q k as he oal quaniy of model k ha is sold in period and p k as he corresponding average price or uni value. 7 Noe ha he linear regression model defined by (1) can only provide a firs order approximaion o a general hedonic funcion. Diewer (2001) made a case for considering second order approximaions bu in his paper, we will follow curren pracice and consider only linear approximaions. 8 Griliches (1971a; 58) noed ha an advanage of he log formulaion is ha β n would provide an esimae of he percenage change in price due o a one uni change in z n, provided ha f n was he ideniy funcion. Cour (1939; 111) implicily noed his advanage of he log formulaion.
3 Suppose ha we choose f o be he ideniy funcion. Suppose furher ha here is only one coninuous characerisic so ha N = 1. In his siuaion, he hedonic regression is essenially a regression of price on package size and so if we wan o have as a special case, ha price per uni of useful characerisic is a consan, hen we should se f 1 (z 1 ) = z 1. 9 Under hese condiions, he model defined by (1) and f(p) = p will be consisen wih he consan per uni price hypohesis if β 0 = 0. In he case of N coninuous characerisics, a generalizaion of he consan per uni characerisic price hypohesis is he hypohesis of consan reurns o scale in he vecor of characerisics, so ha if all characerisics are doubled, hen he resuling model price is doubled. If our period model is defined by (1) and f(p) = p, hen h mus saisfy he following propery: (3) β 0 + n=1 N f n (λz kn )β n = λ[β 0 + n=1 N f n (z kn )β n ] for all λ > 0. In order o saisfy (3), we mus choose β 0 = 0 and he f n o be ideniy funcions. Thus if f is chosen o be he ideniy funcion, hen i is naural o choose he f n ha correspond o coninuous characerisics o be ideniy funcions as well. 10 Now suppose ha we choose f o be he log funcion. Suppose again ha here is only one coninuous characerisic so ha N = 1. In his siuaion, again he hedonic regression is essenially a regression of price on package size and so if we wan o have as a special case, ha price per uni of useful characerisic is a consan, hen we need o se f 1 (z 1 ) = lnz 1 and β 1 = 1. Under hese condiions, he model defined by (1) and f(p) = lnp will be consisen wih he consan per uni price hypohesis. In he case of N coninuous characerisics, a generalizaion of he consan per uni price hypohesis is he hypohesis of consan reurns o scale in he vecor of characerisics. If our period model is defined by (1) and f(p) = lnp, hen h mus saisfy he following propery: (4) β 0 + n=1 N f n (λz kn )β n = lnλ + β 0 + n=1 N f n (z kn )β n for all λ > 0. In order o saisfy (4), we mus choose he f n (z n ) o be log funcions 11 and he β n mus saisfy he following linear resricion: (5) n=1 N β n = 1. Thus if f is chosen o be he log funcion, hen i is naural o choose he f n ha correspond o coninuous characerisics o be log funcions as well. 9 We are no arguing ha his consan reurns o scale hypohesis mus necessarily hold (usually, i will no hold); we are jus arguing ha i is useful for he hedonic regression model o be able o model his siuaion as a special case. The consan reurns o scale hypohesis is required in some hedonic models; e.g., see Muellbauer (1974; 988) and Pollak s (1983) L Characerisics model, which is also used by Triple (1983). 10 If we change he unis of measuremen for he coninuous characerisics, hen he linear hedonic regression model will be unaffeced by his change in he unis; i.e., he change in he unis for he nh characerisic can be absorbed ino he regression coefficien β n. 11 Noe ha all of he coninuous characerisics mus be measured in posiive unis in his case.
4 An exremely imporan propery ha a hedonic regression model should possess is ha he model be invarian o changes in he unis of measuremen of he coninuous characerisics. Thus suppose ha we have only coninuous characerisics and he period model is defined by (1) wih f arbirary and he f n (z n ) = lnz n. Suppose furher ha new unis of measuremen for he N characerisics are chosen, say Z n, where (6) Z n z n /c n ; n = 1,...,N where he c n are posiive consans. The invariance propery requires ha we can find new regression coefficiens, β n *, such ha he following equaion can be saisfied idenically: (7) β 0 + n=1 N (lnz n )β n = β 0 * + n=1 N (lnz n )β n * = β * 0 + N * n=1 (lnz n /c n )β n = β * 0 N n=1 (lnc n )β * n + N n=1 (lnz n )β * n. using (6) Hence o saisfy (7) idenically, we need only se β n * = β n for n = 1,...,N and se β 0 * = β 0 n=1 N (lnc n )β n. Thus in paricular, he hedonic regression model where f and he f n are all log funcions will saisfy he imporan invariance o changes in he unis of measuremen of he coninuous characerisics propery, provided ha he regression has a consan erm in i. 12 We now address he following quesion: should he dependen variable f(p k ) on he lef hand side of (1) be p k or lnp k? If f is he ideniy funcion, hen using definiions (2), equaions (1) can be rewrien as follows: (8) p k = h (z k,β ) + ε k ; = 0,1,...,T; k S() where ε k is an independenly disribued error erm wih mean 0 and variance σ 2. On he oher hand, if f is he logarihm funcion, hen equaions (1) are equivalen o he following equaions: (9) p k = exp[h (z k,β )]exp[ε k ] ; = 0,1,...,T; k S() = exp[h (z k,β )]η k ; where η k is an independenly disribued error erm wih mean 1 and consan variance. Which is more plausible: he model specified by (8) or he model specified by (9)? We argue ha i is more likely ha he errors in (9) are homoskedasic compared o he errors in (8) since models wih very large characerisic vecors z k will have high prices p k and 12 Noe ha he above argumen is independen of he funcional form for f; i.e., if he f n for he coninuous characerisics are log funcions, hen for any f, he hedonic regression mus include a consan erm o be invarian o changes in he unis of hese coninuous characerisics.
5 are very likely o have relaively large error erms. On he oher hand, models wih very small amouns of characerisics will have small prices and small means and he deviaion of a model price from is mean will be necessarily small. In oher words, i is more plausible o assume ha he raio of model price o is mean price is randomly disribued wih mean 1 and consan variance han o assume ha he difference beween model price and is mean is randomly disribued wih mean 0 and consan variance. Hence, from an a priori poin of view, we would favor he logarihmic regression model (9) (or (1) wih f(p) lnp) over is linear counerpar (8). The regression models considered in his secion were unweighed models and could be esimaed wihou a knowledge of he amouns sold for each model in each period. In he following secion, we assume ha model quaniy informaion q k is available and we consider how his exra informaion could be used. 3. Quaniy Weighs versus Expendiure Weighs Usually, discussions of how o use quaniy or expendiure weighs in a hedonic regression are cenered around discussions on how o reduce he heeroskedasiciy of error erms. In his secion, we aemp a somewha differen approach based on he idea ha he regression model should be represenaive. In oher words, if model k sold q k imes in period, hen perhaps model k should be repeaed in he period hedonic regression q k imes so ha he period regression is represenaive of he sales ha acually occurred during he period. 13 To illusrae his idea, suppose ha in period, only hree models were sold and here is only one coninuous characerisic. Le he period price of he hree models be p 1, p 2 and p 3 and suppose ha he hree models have he amouns z 11, z 21 and z 31 of he single characerisic respecively. Then he period unweighed regression model (1) has only he following 3 observaions and 2 unknown parameers, β 0 and β 1 : (10) f(p 1 ) = β 0 + f 1 (z 11 )β 1 + ε 1 ; f(p 2 ) = β 0 + f 1 (z 21 )β 1 + ε 2 ; f(p 3 ) = β 0 + f 1 (z 31 )β 1 + ε 3. Noe ha each of he 3 observaions ges an equal weigh in he period hedonic regression model defined by (10). However, if say models 1 and 2 are vasly more popular han model 3, hen i does no seem o be appropriae ha model 3 ges he same imporance as models 1 and 2. 13 Thus our represenaive approach follows along he lines of Theil s (1967; 136-138) sochasic approach o index number heory, which is also pursued by Rao (2002). The use of weighs ha reflec he economic imporance of models was recommended by Griliches (1971b; 8): Bu even here, we should use a weighed regression approach, since we are ineresed in an esimae of a weighed average of he pure price change, raher han jus an unweighed average over all possible models, no maer how peculiar or rare. However, he did no make any explici weighing suggesions.
6 Suppose ha he inegers q 1, q 2 and q 3 are he amouns sold in period of models 1,2 and 3 respecively. Then one way of consrucing a hedonic regression ha weighs models according o heir economic imporance is o repea each model observaion according o he number of imes i sold in he period. This leads o he following more represenaive hedonic regression model, where he error erms have been omied: (11) 1 1 f(p 1 ) = 1 1 β 0 + 1 1 f 1 (z 11 )β 1 ; 1 2 f(p 2 ) = 1 2 β 0 + 1 2 f 1 (z 21 )β 1 ; 1 3 f(p 3 ) = 1 3 β 0 + 1 3 f 1 (z 31 )β 1 where 1 k is a vecor of ones of dimension q k for k = 1,2,3. Now consider he following quaniy ransformaion of he original unweighed hedonic regression model (10): (12) (q 1 ) 1/2 f(p 1 ) = (q 1 ) 1/2 β 0 + (q 1 ) 1/2 f 1 (z 11 )β 1 + ε 1 * ; (q 2 ) 1/2 f(p 2 ) = (q 2 ) 1/2 β 0 + (q 2 ) 1/2 f 1 (z 21 )β 1 + ε 2 * ; (q 3 ) 1/2 f(p 3 ) = (q 3 ) 1/2 β 0 + (q 3 ) 1/2 f 1 (z 31 )β 1 + ε 3 *. Comparing (10) and (12), i can be seen ha he observaions in (12) are equal o he corresponding observaions in (10), excep ha he dependen and independen variables in observaion k of (10) have been muliplied by he square roo of he quaniy sold of model k in period for k = 1,2,3 in order o obain he observaions in (12). A sampling framework for (12) is available if we assume ha he ransformed residuals ε k * are independenly normally disribued wih mean zero and consan variance. Le b 0 and b 1 denoe he leas squares esimaors for he parameers β 0 and β 1 in (11) and le b 0 * and b 1 * denoe he leas squares esimaors for he parameers β 0 and β 1 in (12). Then i is sraighforward o show ha hese wo ses of leas squares esimaors are he same 14 ; i.e., we have: (13) [b 0,b 1 ] = [b 0 *,b 1 * ]. Thus a shorcu mehod for obaining he leas squares esimaors for he unknown parameers, β 0 and β 1, which occur in he represenaive model (11) is o obain he leas squares esimaors for he ransformed model (12). This equivalence beween he wo models provides a jusificaion for using he weighed model (12) in place of he original model (10). The advanage in using he ransformed model (12) over he represenaive model (11) is ha we can develop a sampling framework for (12) bu no 14 See, for example, Greene (1993; 277-279). However, he numerical equivalence of he leas squares esimaes obained by repeaing muliple observaions or by he square roo of he weigh ransformaion was noiced long ago as he following quoaion indicaes: I is eviden ha an observaion of weigh w eners ino he equaions exacly as if i were w separae observaions each of weigh uniy. The bes pracical mehod of accouning for he weigh is, however, o prepare he equaions of condiion by muliplying each equaion hroughou by he square roo of is weigh. E. T. Whiaker and G. Robinson (1940; 224).
7 for (11), since he (omied) error erms in (11) canno be assumed o be disribued independenly of each oher. However, in view of he equivalence beween he leas squares esimaors for models (11) and (12), we can now be comforable ha he regression model (12) weighs observaions according o heir quaniaive imporance in period. Hence, we definiely recommend he use of he weighed hedonic regression model (12) over is unweighed counerpar (10). However, raher han weighing models by heir quaniy sold in each period, i is possible o weigh each model according o he value of is sales in each period. Thus define he value of sales of model k in period o be: (14) v k p k q k ; = 0,1,...,T ; k S(). Now consider again he simple unweighed hedonic regression model defined by (10) above and round off he sales of each of he 3 models o he neares dollar (or penny). Le 1 k* be a vecor of ones of dimension v k for k = 1,2,3. Repeaing each model in (10) according o he value of is sales in period leads o he following more represenaive period hedonic regression model (where he errors have been omied): (15) 1 1* f(p 1 ) = 1 1* β 0 + 1 1* f 1 (z 11 )β 1 ; 1 2* f(p 2 ) = 1 2* β 0 + 1 2* f 1 (z 21 )β 1 ; 1 3* f(p 3 ) = 1 3* β 0 + 1 3* f 1 (z 31 )β 1. Now consider he following value ransformaion of he original unweighed hedonic regression model (10): (16) (v 1 ) 1/2 f(p 1 ) = (v 1 ) 1/2 β 0 + (v 1 ) 1/2 f 1 (z 11 )β 1 + ε 1 ** ; (v 2 ) 1/2 f(p 2 ) = (v 2 ) 1/2 β 0 + (v 2 ) 1/2 f 1 (z 21 )β 1 + ε 2 ** ; (v 3 ) 1/2 f(p 3 ) = (v 3 ) 1/2 β 0 + (v 3 ) 1/2 f 1 (z 31 )β 1 + ε 3 **. Comparing (10) and (16), i can be seen ha he observaions in (12) are equal o he corresponding observaions in (10), excep ha he dependen and independen variables in observaion k of (10) have been muliplied by he square roo of he value sold of model k in period for k = 1,2,3 in order o obain he observaions in (16). Again, a sampling framework for (16) is available if we assume ha he ransformed residuals ε k ** are independenly disribued normal random variables wih mean zero and consan variance. Again, i is sraighforward o show ha he leas squares esimaors for he parameers β 0 and β 1 in (15) and (16) are he same. Thus a shorcu mehod for obaining he leas squares esimaors for he unknown parameers, β 0 and β 1, which occur in he value weighs represenaive model (15) is o obain he leas squares esimaors for he ransformed model (16). This equivalence beween he wo models provides a jusificaion for using he value weighed model (16) in place of he original model (10). As before, he advanage in using he ransformed model (16) over he value weighs represenaive model (15) is ha we can develop a sampling framework for (16) bu no
8 for (15), since he (omied) error erms in (15) canno be assumed o be disribued independenly of each oher. I seems o us ha he quaniy weighed and value weighed models are clear improvemens over he original unweighed model (10). Our reasoning here is similar o ha used by Fisher (1922; Chaper III) in developing bilaeral index number heory, who argued ha prices needed o be weighed according o heir quaniaive or value imporance in he wo periods being compared. 15 In he presen conex, we have a weighing problem ha involves only one period so ha our weighing problems are acually much simpler han hose considered by Fisher: we need only choose beween quaniy or value weighs! Bu which sysem of weighing is beer in our presen conex: quaniy or value weighing? The problem wih quaniy weighing is his: i will end o give oo lile weigh o models ha have high prices and oo much weigh o cheap models ha have low amouns of useful characerisics. Hence i appears o us ha value weighing is clearly preferable. Thus we are aking he poin of view ha he main purpose of he period hedonic regression is o enable us o decompose he marke value of each model sold, p k q k, ino he produc of a period price for a qualiy adjused uni of he hedonic commodiy, say P, imes a consan uiliy oal quaniy for model k, Q k. Hence observaion k in period should have he represenaive weigh Q k in consan uiliy unis ha are comparable across models. Bu Q k is equal o p k q k /P, which in urn is equal o v k /P, which in urn is proporional o v k. Thus weighing by he values v k seems o be he mos appropriae form of weighing. Our conclusions abou single period hedonic regressions a his poin can be summarized as follows: Wih respec o aking ransformaions of he dependen variable in a period hedonic regression, aking of logarihms of he model prices is our preferred ransformaion. 15 I has already been obser ved ha he purpose of any index number is o srike a fair average of he price movemens or movemens of oher groups of magniudes. A firs a simple average seemed fair, jus because i reaed all erms alike. And, in he absence of any knowledge of he relaive imporance of he various commodiies included in he average, he simple average is fair. Bu i was early recognized ha here are enormous differences in imporance. Everyone knows ha pork is more imporan han coffee and whea han quinine. Thus he ques for fairness led o he inroducion of weighing. Irving Fisher (1922; 43). Bu on wha principle shall we weigh he erms? Arhur Young s guess and oher guesses a weighing represen, consciously or un consciously, he idea ha relaive money values of he various commodiies should deermine heir weighs. A value is, of course, he produc of a price per uni, muliplied by he number of unis aken. Such values afford he only common measure for comparing he sreams of commodiies produced, exchanged, or consumed, and afford almos he only basis of weighing which has ever been seriously proposed. Irving Fisher (1922; 45).
9 If informaion on he number of models sold in each period is available, hen weighing each observaion by he square roo of he value of model sales is our preferred mehod of weighing. If he log ransformaion is chosen for he dependen variable, hen we have a mild preference for ransforming he coninuous characerisics by he logarihm ransformaion as well. If he coninuous characerisics are ransformed by he logarihmic ransformaion, hen he regression mus have a consan erm o ensure ha he resuls of he regression are invarian o he choice of unis for he characerisics. If he dependen variable is simply he model price, hen we have a mild preference for no ransforming he coninuous characerisics as well. Wih he above general consideraions in mind, we now urn o a discussion of how single period hedonic regressions can be used by saisical agencies in a sampling conex. 4. The Use of Single Period Hedonic Regressions in a Replacemen Sampling Conex In his secion, we consider he use of single period hedonic regressions in he conex of saisical agency sampling procedures where a sampled model ha was available in period s is no available in a laer period and is replaced wih a new model ha is available in period. We assume ha s < and ha model 1 is available in period s (wih price p s 1 and characerisics vecor z s 1 ) bu is no available in period. We furher assume ha model 1 is replaced by model 2 in period, wih price p 2 and characerisics vecor z 2. The problem is o somehow adjus he price relaive p 2 /p s 1 so ha he adjused price relaive can be averaged wih oher price relaives of he form p k /p s k ha correspond o models k ha are presen in boh periods s and in order o form an overall price relaive for he iem level, going from period s o. If he iem level index is a chain ype index, hen s will be equal o 1 and if he iem level index is a fixed base ype index, hen s will be equal o he base period 0. Recall he family of single period hedonic regressions defined in secion 2 above by equaions (1). If we use definiions (2) and assume ha he funcion of one variable f(x) has an inverse funcion f 1, hen we may rewrie equaions (1) as follows: (17) p k = f 1 [h (z k,β ) + ε k ] ; = 0,1,...,T; k S(). Assume ha we have a vecor of esimaes b for he period vecor of parameers β and define he model k sample residuals for period, e k, as follows: 16 (18) e k f(p k ) h (z k,β ) ; = 0,1,...,T; k S(). 16 Definiions (18) need o be modified if weighed regressions are run insead of unweighed regressions.
10 Thus he sample counerpars o equaions (17) are he following equaions: (19) p k = f 1 [h (z k,b ) + e k ] ; = 0,1,...,T; k S(). Now suppose ha he period s hedonic regression is available o he saisical agency. Thus equaion (19) for period s and model 1 is: (20) p 1 s = f 1 [h s (z 1 s,b s ) + e 1 s ]. Recall ha model 2, he replacemen for model 1 in period, has he vecor of characerisics z 2. Hence, using he period s hedonic regression, a comparable price for model 2 in period s is f 1 [h s (z 2,b s )], he prediced period s price using he period hedonic regression for a model wih he vecor of characerisics z 2. Thus our firs esimaor for an adjused price relaive for models 1 and 2 going from period s o is: (21) r(1) p 2 /f 1 [h s (z 2,b s )]. However, here is a problem wih he use of (21) as an adjused price relaive. The problem will become apparen if z 2 = z 1 s, so ha he wo models are in fac idenical. In his case, we wan our price relaive o equal he acual price raio: (22) p 2 /p s 1 = p 2 /f 1 [h s (z s 1,b s ) + e s 1 ] using (20) p 2 /f 1 [h s (z s 1,b s )] if e s 1 0. Hence if he regression residual for model 1 in period s, e 1 s, is no equal o zero, hen r(1) defined by (21) will no be an appropriae adjused price relaive. In order o compare like wih like, we mus muliply r(1) by an adjusmen facor equal o (23) f 1 [h s (z 1 s,b s )]/p 1 s = f 1 [h s (z 1 s,b s )]/f 1 [h s (z 1 s,b s ) + e 1 s ]. Thus our second esimaor r(2) for an adjused price relaive is r(1) defined by (21) imes he adjusmen facor defined by (23), which adjuss he period s observed price for model 1, p 1 s, ono he period s hedonic regression surface: 17 (24) r(2) {p 2 /f 1 [h s (z 2,b s )]}{f 1 [h s (z 1 s,b s )]/p 1 s } = {p 2 /f 1 [h s (z 2,b s )]}/{p 1 s /f 1 [h s (z 1 s,b s )]}. The second expression for r(2) in (24) is insrucive. We can inerpre p 2 /f 1 [h s (z 2,b s )] as he period price for model 2 expressed in consan qualiy uiliy unis, using he period s hedonic regression as he qualiy adjusmen mechanism. Similarly, we can inerpre p 1 s /f 1 [h s (z 1 s,b s )] as he period s price for model 1 expressed in consan qualiy uiliy unis, using he period s hedonic regression as he qualiy adjuser. Thus he price relaive defined by (24) compares he price of model 2 in period o he price of model 1 17 If e 1 s = 0, hen r(1) will equal r(2).
11 in period s in consan uiliy quaniy unis. Hence, he period s hedonic regression may be used o express model prices in homogeneous qualiy adjused unis. 18 Obviously, if he saisical agency has he period hedonic regression available o i, hen he above analysis can be repeaed, wih some modificaions. In his case, equaion (19) for period and model 2 is: (25) p 2 = f 1 [h (z 2,b ) + e 2 ]. Recall ha model 1 has he vecor of characerisics z 1 s. Hence, using he period hedonic regression, a comparable price for model 1 in period is f 1 [h (z 1 s,b )], he prediced period price using he period hedonic regression for a model wih he vecor of characerisics z 1 s. Thus our hird esimaor for an adjused price relaive for models 1 and 2 going from period s o is: (26) r(3) f 1 [h (z 1 s,b )]/p 1 s. However, again, here is a problem wih he use of (26) as an adjused price relaive. As above, he problem becomes apparen if z 2 = z s 1, so ha he wo models are in fac idenical. In his case, we wan our price relaive o equal he acual price raio: (27) p 2 /p s 1 = f 1 [h (z 2,b ) + e s 2 ]/p 1 f 1 [h (z 2,b s )]/p 1 using (25) if e 2 0. Hence if he regression residual for model 2 in period, e 2, is no equal o zero, hen r(3) defined by (26) will no be an appropriae adjused price relaive. In order o compare like wih like, we mus muliply r(3) by an adjusmen facor equal o (28) p 2 /f 1 [h (z 2,b )] = f 1 [h (z 2,b ) + e 2 ]/f 1 [h (z 2,b )]. Thus our fourh esimaor r(4) for an adjused price relaive is r(3) defined by (26) imes he adjusmen facor defined by (28), which adjuss he period observed price for model 2, p 2, ono he period hedonic regression surface: 19 (29) r(4) {f 1 [h (z 1 s,b )]/p 1 s }{p 2 /f 1 [h (z 2,b )]} = {p 2 /f 1 [h (z 2,b )]}/{p 1 s /f 1 [h (z 1 s,b )]}. The second expression for r(4) in (29) is again insrucive. We can inerpre p 2 /f 1 [h (z 2,b )] as he period price for model 2 expressed in consan qualiy uiliy unis, using he period hedonic regression as he qualiy adjusmen mechanism. Similarly, we can inerpre p 1 s /f 1 [h (z 1 s,b )] as he period s price for model 1 expressed in 18 This basic idea can be raced back o Cour (1939; 108) as his hedonic suggesion number one. The idea was explicily laid ou in Griliches (1971a; 59-60) (1971b; 6) and Dhrymes (1971; 111-112). I was implemened in a saisical agency sampling conex by Triple and McDonald (1977; 144). 19 Of course, if e 2 = 0, hen r(3) will equal r(4).
12 consan qualiy uiliy unis, using he period hedonic regression as he qualiy adjuser. Thus he price relaive defined by (29) compares he price of model 2 in period o he price of model 1 in period s in consan uiliy quaniy unis, using he period hedonic regression o do he qualiy adjusmen. If he period s and hedonic regressions are boh available o he saisical agency, hen i is bes o make use of boh of he adjused price relaives r(2) and r(4) and generae a final adjused price relaive ha is a symmeric average of he wo esimaes. 20 Thus define our final preferred adjused price relaive r(5) as he geomeric mean of r(2) and r(4): (30) r(5) [r(2)r(4)] 1/2. We chose he geomeric mean in (30) over oher simple symmeric means like he arihmeic average because he use of he geomeric average leads o an adjused price relaive ha will saisfy he ime reversal es. 21 Finally, suppose ha period s and hedonic regressions are no available o he saisical agency bu a base period hedonic regression is available. In his case, he obvious adjused replacemen price raio is: (31) r(6) {p 2 /f 1 [h 0 (z 2,b 0 )]}/{p 1 s /f 1 [h 0 (z 1 s,b 0 )]}. Thus he price relaive defined by (31) compares he price of model 2 in period o he price of model 1 in period s in consan uiliy quaniy unis, using he period 0 hedonic regression o do he qualiy adjusmen. Obviously, he adjused price relaive r(5) would generally be preferable o he price relaive defined by r(6), since he period 0 hedonic regression may be quie ou of dae if period 0 is disan from periods s and. 22 Similar consideraions sugges ha more reliable resuls will be obained if he chain principle is used in forming he adjused price relaives defined by (5); i.e., he gap beween he equally valid r(2) and r(4) is likely o be minimized if period s is chosen o be period 1. 23 20 Griliches (1971a; 59) noed he exisence of hese wo equally valid esimaes. Griliches (1971b; 7) also suggesed aking an average of he wo esimaes and, as an alernaive mehod of averaging or smoohing, he suggesed using adjacen year regressions, which will be sudied in secions 7 and 8 below. 21 See Diewer (1997; 138) for an argumen along hese lines. 22 Tases will probably change over ime and he characerisics domain of definiion for models ha exis in period 0 may be quie differen from he domains of definiion for he models ha exis in periods s and ; i.e., he z region spanned by he period 0 hedonic regression may be quie ou of dae for he laer periods. 23 Our advocacy of he chain principle and of averaging equally valid resuls seems o be consisen wih he posiion advocaed by Griliches (1971b; 6-7): This approach cal ls for relaively recen and ofen changing price weighs. Since such saisics come o us in discree inervals, we are also faced wih he usual Laspeyres-Paasche problem. The ofener we can change such weighs [i.e., run a new hedonic regression], he less of a problem i will be. In pracice, while one may wan o use he mos recen cross secion o derive he relevan price weighs, such esimaes may flucuae oo much for comfor as he resul of mulicollineariy and sampling flucuaions. They should be smoohed in some way, eiher by choosing w i = (1/2)[w i () + w i (+1)], or by using adjacen year regressions in esimaing hese weighs.
13 In he following secion, we shall assume ha he saisical agency has esimaed single period hedonic regressions as in his secion bu in addiion, we assume ha informaion on quaniies sold of each model is available. Hence, Paasche, Laspeyres and superlaive indexes of he ype advocaed by Silver and Heravi (2001) (2002a) (2002b) and Pakes (2001) can be calculaed. 5. Single Period Hedonic Regressions in he Scanner Daa Conex In his secion, we assume ha he saisical agency has boh price and quaniy (or value) daa for he subse of he K models ha are available in each period. As in he previous period, we will assume ha he saisical agency has run single period hedonic regressions for periods s and. 24 The hedonic regression of period s can be used in order o calculae he following Paasche ype index going from period s o : 25 (32) P P (s,) k S() p k q k /{ k [S() S(s)] p k s q k + k [S() S(s)] f 1 [h s (z k,b s )]q k }. The summaion in he numeraor of he righ hand side of (32) is simply he sum of price p k imes quaniy q k over all of he models k sold during period, which is he se of indexes k represened by S(). The firs summaion in he denominaor of he righ hand side of (32) is he produc of he period s model k price, p s k, over all models ha are presen in boh periods s and while he second se of erms uses he period s esimaed hedonic price of a model k ha is sold in period (which has characerisics defined by he vecor z k ) bu is no sold in period s, f 1 [h s (z k,b s )], imes he period quaniy sold for his model, q k. If we make he srong assumpions on demander s period s preferences 26 ha are lised in Diewer (2001), hen we can inerpre f 1 [h s (z k,b s )] as an approximae Hicksian (1940; 114) reservaion price for model k ha is sold in period bu no in period s; i.e., if price is above his limiing price, hen purchasers will no wan o buy any unis of i in period s. Thus under appropriae assumpions on consumer s preferences, he Paasche index defined by (32) will be an approximae lower bound o a heoreical Paasche-Konüs cos of living index; see Diewer (1993; 80). 27 Thus he esimaed period s hedonic regression enables us o calculae a mached model ype Paasche index beween 24 Wih he availabiliy of quaniy informaion on he models sold, value weighed hedonic regressions of he ype recommended in secion 4 can be run for each period. 25 This is Pakes (2001; 22) Paasche complee hedonic hybrid price index. Excep for error erms, i is also equal o one of Silver and Heravi s (2001) Paasche ype lower bounding indexes for a rue cos of living index. 26 A sronger bu simpler se of assumpions han hose of Diewer (2001) are ha all period s demanders of he hedonic commodiy evaluae he uiliy of a model wih characerisics vecor z according o he magniude g s (z), where g s (z) is a separable (cardinal) uiliy funcion. Under hese assumpions, he equilibrium price of a model wih characerisics vecor z should have he period s hedonic price funcion equal o g s (z) imes a consan. If f 1 [h s (z,β s )] can approximae his rue period s hedonic price funcion and if he fi of he period s hedonic regression is good so ha b s is close o β s, hen f 1 [h s (z k,b s )] will be an approximae Hicksian reservaion price for model k ha is sold in period bu no in period s. 27 See Diewer (1993; 103-104) for an exposiion of he use of Hicksian reservaion prices for new and disappearing commodiies in he conex of Paasche and Laspeyres indexes.
14 periods s and, where he prices for he models ha were sold in period bu no period s are filled in using he period s hedonic regression. In a similar manner, we can use he hedonic regression for period o fill in he missing reservaion prices for models ha were sold in period s bu no and we can calculae he following Laspeyres ype index going from period s o : 28 (33) P L (s,) { k [S(s) S()] p k q k s + k [S(s) S()] f 1 [h (z k s,b )]q k s }/{ k S(s) p k s q k s }. The summaion in he denominaor of he righ hand side of (33) is simply he sum of price p k s imes quaniy q k s over all of he models k sold during period s, which is he se of indexes k represened by S(s). The firs summaion in he numeraor of he righ hand side of (33) is he produc of he period model k price, p k, over all models ha are presen in boh periods s and while he second se of erms uses he period esimaed hedonic price of a model k ha is sold in period s (which has characerisics defined by he vecor z k s ) bu is no sold in period, f 1 [h (z k s,b )], imes he period s quaniy sold for his model, q k s. Under appropriae assumpions on consumer s prefer ences, he Laspeyres index defined by (33) will be an approximae upper bound o a heoreical Laspeyres-Konüs cos of living index; see Diewer (1993; 80). Thus he esimaed period hedonic regression enables us o calculae a mached model ype Laspeyres index beween periods s and, where he prices for he models ha were sold in period s bu no period are filled in using he period hedonic regression. If boh period s and hedonic regressions are available o he saisical agency, hen since he Paasche and Laspeyres measures of price change beween periods s and are equally valid, i is appropriae o ake a symmeric average of hese wo esimaors of price change as a final esimaor of price change beween he periods. 29 As usual, we chose he geomeric mean of P L and P P over oher simple symmeric means like he arihmeic average because he use of he geomeric average leads o an index ha will saisfy he ime reversal es. 30 Hence, define he Fisher (1922) index beween periods s and as: (34) P F (s,) [P L (s,) P P (s,)] 1/2 where P P and P L are defined by (32) and (33). 31 28 Excep for error erms, i is equal o one of Silver and Heravi s (2001) Laspeyres ype upper bounding indexes for a rue cos of living index. 29 If all models are presen in boh periods, hen he Laspeyres ype index defined by (33) reduces o an ordinary Laspeyres index beween periods s and and he Paasche ype index defined by (32) reduces o an ordinary Paasche index. I can be seen ha he weighs for each of hese indexes is no represenaive of boh periods and hence each of he indexes (32) and (33) will be subjec o subsiuion or represenaiviy bias; see Diewer (2002a; 45) on he concep of represenaiviy bias. Hence, o eliminae his bias, i is necessary o ake an average of he wo indexes defined by (32) and (33). 30 See Diewer (1997; 138). 31 An argumen due originally o Konüs (1924) can be used o prove ha a heoreical cos of living index lies beween he Paasche and Laspeyres indexes; see also Diewer (1993; 81). However, his argumen will only go hrough for he case where all of he characerisics are of he coninuous ype.
15 I is of some ineres o compue P P, P L and P F defined by (32)-(34) above for he case where here are only wo models: model 1, which is available in period s bu no period, and model 2, which is available in period bu no period s; i.e., we are revisiing he sampling model ha was sudied in secion 4 above. Under hese condiions, P P defined by (32) simplifies o he following expression: (35) P P (s,) p 2 q 2 /f 1 [h s (z 2,b s )]q 2 = p 2 /f 1 [h s (z 2,b s )] = r(1) where r(1) was defined in secion 4 by (21). Similarly, P L defined by (33) simplifies o he following expression: (36) P L (s,) f 1 [h (z 1 s,b )]q 1 s /p 1 s q 1 s = f 1 [h (z 1 s,b )]/p 1 s = r(3) where r(3) was defined in secion 4 by (26). Recall ha our preferred replacemen price raios obained in secion 4 were r(2) and r(4) raher han r(1) and r(3). Hence he resuls obained in his secion seem o be slighly inconsisen wih he resuls obained in secion 4. 32 This sligh inconsisency can be resolved if we make srong assumpions abou he preferences of purchasers of he hedonic commodiies. Suppose all purchasers of he hedonic commodiy evaluae he relaive uiliy of each model in period s according o he cardinal uiliy funcion g s (z) so ha he relaive value o purchasers of a model wih characerisics vecor z 1 versus a model wih characerisics vecor z 2 is g s (z 1 )/g s (z 2 ). Then in equilibrium, he period s relaive price of he wo models should also be g s (z 1 )/g s (z 2 ). Thus he period s price of a model wih characerisics vecor z should be proporional o g s (z). Finally, suppose ha he period s economerically esimaed hedonic funcion, f 1 [h s (z,b s )], can provide an adequae approximaion o he heoreical hedonic funcion, ρ s g s (z), where ρ s is a posiive consan. Under hese srong assumpions, he oal marke uiliy for period s ha is provided by purchases of he hedonic commodiies is equal o: (37) Q s k S(s) ρ s g s (z k s )q k s k S(s) f 1 [h s (z k s,b s )]q k s where we have approximaed he uiliy o purchasers of model k in period s, ρ s g s (z s k ), by he period s hedonic regression esimaed value, f 1 [h s (z s k,b s )]. Thus Q s can be inerpreed as he aggregae quaniy of all of he models purchased in period s, where each model has been qualiy adjused ino consan uiliy unis using he period s hedonic aggregaor funcion, g s (z). In wha follows, we will neglec he approximaion error beween lines 1 and 2 of (37) so ha we idenify he period s aggregae quaniy purchased of he hedonic commodiy, Q s (s), using he period s hedonic regression o do he qualiy adjusmen, as follows: 32 We say slighly inconsisen because usually he hedonic regression observed errors e 1 s and e 2 will be small and hence he differences beween r(1) and r(2) and r(3) and r(4) will also be small.
16 (38) Q s (s) k S(s) f 1 [h s (z k s,b s )]q k s. For each period, we can define he value of all models purchased as: (39) V k S() p k q k ; = 0,1,...,T. For laer reference, we also define he period expendiure share of model k as follows: (40) s k p k q k / i S() p i q i ; = 0,1,...,T; k S(). Corresponding o he period s quaniy aggregae defined by (38), we can define an aggregae period s price level, P s (s), by dividing Q s (s) ino he period s value aggregae, V s : (41) P s (s) V s /Q s (s) = V s / k S(s) f 1 [h s (z s k,b s s )]q k using (38) = 1/[ k S(s) {f 1 [h s (z s k,b s )]/p s k }p s k q s k /V s ] = 1/[ k S(s) {f 1 [h s (z s k,b s )]/p s k }s s k ] using (40) for = s = [ k S(s) s s k {p s k /f 1 [h s (z s k,b s )]} 1 ] 1. Thus he aggregae period s price level using he period s hedonic regression, P s (s), is equal o a period s share weighed harmonic mean of he period s acual model prices, p s k, relaive o he corresponding prediced period s model prices using he period s hedonic regression, f 1 [h s (z s k,b s )]. 33 Since p s k = f 1 [h s (z s k,b s )+e s k ] where e s k is he regression residual for model k in period s 34 and hese residuals are ypically close o 0 and randomly disribued around 0, i can be seen ha under normal condiions, P s (s) defined by (41) will be close o 1. Now le us use he period s hedonic regression o form a consan uiliy quaniy aggregae for he models sold in period. Thus model k in period, using he esimaed hedonic valuaion funcion of period s, will have he consan uiliy value f 1 [h s (z k,b s )]. Hence, he period aggregae quaniy purchased of he hedonic commodiy, Q (s), using he period s hedonic regression o do he qualiy adjusmen ino consan uiliy unis, can be defined as follows: (42) Q (s) k S() f 1 [h s (z k,b s )]q k. 33 I can be seen ha he expression on he righ hand side of (41) is a ype of Paasche price index, where he price and quaniy daa of period s, p k s and q k s for k S(s), ac as he comparison period daa and he hedonic regression period s prediced prices, f 1 [h s (z k s,b s )] for k S(s), ac as base period prices. 34 Our algebra here assumes ha unweighed hedonic regressions have been run. If a value weighed hedonic regression has been run for period s, hen he equaion p k s = f 1 [h s (z k s,b s )+e k s ] mus be replaced by p k s = f 1 [h s (z k s,b s )+ (v k s ) (1/2) e k s ] where he e k s are he residuals for he ransformed period s hedonic regression.
17 Corresponding o he period quaniy aggregae defined by (42), we can define an aggregae period price level using he preferences of period s o do he qualiy adjusmen, P (s), by dividing Q (s) ino he period value aggregae, V : (43) P (s) V /Q (s) = V / k S() f 1 [h s (z k,b s )]q k using (42) = 1/[ k S() {f 1 [h s (z k,b s )]/p k }p k q k /V ] = 1/[ k S() {f 1 [h s (z k,b s )]/p k }s k ] using definiions (40) = [ k S() s k {p k /f 1 [h s (z k,b s )]} 1 ] 1. Thus he aggregae period price level using he period s hedonic regression, P (s), is equal o a period share weighed harmonic mean of he period acual model prices, p k, relaive o he corresponding prediced period s model prices using he period s hedonic regression, f 1 [h s (z k,b s )]. 35 Having defined he period s price level P s (s) by (41) and he period price level P (s) by (43) using he hedonic regression of period s o do he consan uiliy qualiy adjusmen, we can ake he raio of hese wo price levels o form a Paasche ype price index going from period s o, using he hedonic regression of period s, as follows: (44) P s (s) P (s)/p s (s) = [ k S() s k {p k /f 1 [h s (z k,b s )]} 1 ] 1 /[ k S(s) s k s {p k s /f 1 [h s (z k s,b s )]} 1 ] 1. The above Paasche ype index can be compared wih our earlier Paasche ype index defined by (32): (45) P P (s,) k S() p k q k /{ k [S() S(s)] p s k q k + k [S() S(s)] f 1 [h s (z k,b s )]q k } = V /{ k [S() S(s)] p s k q k + k [S() S(s)] f 1 [h s (z k,b s )]q k } using definiion (39) = 1/{ k [S() S(s)] [p s k /p k ]p k q k + k [S() S(s)] (f 1 [h s (z k,b s )]/p k )p k q k }/V = 1/{ k [S() S(s)] [p s k /p k ]s k + k [S() S(s)] (f 1 [h s (z k,b s )]/p k )s k } using (40) = { k [S() S(s)] s k [p k /p s k ] 1 + k [S() S(s)] s k [p k /f 1 [h s (z k,b s )]] 1 } 1 = { k [S() S(s)] s k (p k /f 1 [h s (z k,b s )+e s k ]) 1 + k [S() S(s)] s k [p k /f 1 [h s (z k,b s )]] 1 } 1 since p s k = f 1 [h s (z k,b s )+e s k ] for k S() S(s) { k [S() S(s)] s k (p k /f 1 [h s (z k,b s )]) 1 + k [S() S(s)] s k (p k /f 1 [h s (z k,b s )]) 1 } 1 neglecing he regression residuals e s k for k S() S(s) = { k S() s k (p k /f 1 [h s (z k,b s )]) 1 } 1 = P (s) using (43). Thus our old Paasche ype index P P (s,) is approximaely equal o he numeraor of our new Paasche ype index P s (s). However, as we menioned before, he denominaor of P s (s), P s (s), will be approximaely equal o 1, and hence, our new Paasche ype index will be approximaely equal o our old Paasche ype index; i.e., we have 35 I can be seen ha he expression on he righ hand side of (43) is a Paasche price index, where he price and quaniy daa of period, p k and q k for k S(), ac as he comparison period daa and he hedonic regression period s prediced prices, f 1 [h s (z k,b s )] for k S(), ac as base period prices.
18 (46) P s (s) P P (s,). Now consider our new Paasche ype index for he case where here are only wo models: model 1, which is available in period s bu no period, and model 2, which is available in period bu no period s so ha we are revisiing he sampling model ha was sudied in secion 4 above. Under hese condiions, P s (s) defined by (44) simplifies o r(2) defined in secion 4 by (24). Hence our new Paasche ype index is perfecly consisen wih he hedonically adjused sampling price raio r(2) defined earlier in secion 4. Obviously, he above analysis can be repeaed excep ha he hedonic regression for period is used o do he qualiy adjusmen raher han he period s hedonic regression. Thus, we now suppose ha all purchasers of he hedonic commodiy evaluae he relaive uiliy of each model in period according o he cardinal uiliy funcion g (z). Then in equilibrium, he period price of a model wih characerisics vecor z should be proporional o g (z). Suppose ha he period economerically esimaed hedonic funcion, f 1 [h (z,b )], can provide an adequae approximaion o he period heoreical hedonic funcion, ρ g (z), where ρ is a posiive consan. Under hese srong assumpions, he oal marke uiliy for period ha is provided by purchases of he hedonic commodiies is equal o: (47) Q k S() ρ g (z k )q k k S() f 1 [h (z k,b )]q k where we have approximaed he uiliy o purchasers of model k in period, ρ g (z k ), by he period hedonic regression esimaed value, f 1 [h (z k,b )]. Thus Q can be inerpreed as he aggregae quaniy of all of he models purchased in period, where each model has been qualiy adjused ino consan uiliy unis using he period hedonic aggregaor funcion, g (z). In wha follows, we will again neglec he approximaion error beween lines 1 and 2 of (47) so ha we idenify he period aggregae quaniy purchased of he hedonic commodiy, Q (), using he period hedonic regression o do he qualiy adjusmen, as follows: (48) Q () k S() f 1 [h (z k,b )]q k. Corresponding o he period quaniy aggregae defined by (48), we can define an aggregae period price level, P (), by dividing Q () ino he period value aggregae, V : (49) P () V /Q () = V / k S() f 1 [h (z k,b )]q k using (48) = 1/[ k S() {f 1 [h (z k,b )]/p k }p k q k /V ] = 1/[ k S() {f 1 [h (z k,b )]/p k }s k ] using definiions (40) = [ k S() s k {p k /f 1 [h (z k,b )]} 1 ] 1.
19 Thus he aggregae period price level using he period hedonic regression, P (), is equal o a period share weighed harmonic mean of he period acual model prices, p k, relaive o he corresponding prediced period model prices using he period hedonic regression, f 1 [h (z k,b )]. 36 Since p k = f 1 [h (z k,b )+e k ] where e k is he regression residual for model k in period and hese residuals are ypically close o 0 and randomly disribued around 0, i can be seen ha under normal condiions, P () defined by (49) will be close o 1. Now use he period hedonic regression o form a consan uiliy quaniy aggregae for he models sold in period s. Thus model k in period s, using he esimaed hedonic valuaion funcion of period, will have he consan uiliy value f 1 [h (z k s,b )]. Hence, he period s aggregae quaniy purchased of he hedonic commodiy, Q s (), using he period hedonic regression o do he qualiy adjusmen ino consan uiliy unis, can be defined as follows: (50) Q s () k S(s) f 1 [h (z k s,b )]q k s. Corresponding o he period s quaniy aggregae defined by (50), we can define an aggregae period s price level using he preferences of period o do he qualiy adjusmen, P s (), by dividing Q s () ino he period s value aggregae, V s : (51) P s () V s /Q s () = V s / k S(s) f 1 [h (z s k,b s )]q k using (50) = 1/[ k S(s) {f 1 [h (z s k,b )]/p s k }p s k q s k /V s ] = 1/[ k S(s) {f 1 [h (z s k,b )]/p s k }s s k ] using definiions (40) = [ k S(s) s s k {f 1 [h (z s k,b )]/p s k }] 1. Thus he aggregae period s price level using he period hedonic regression, P s (), is equal o he reciprocal of a period s share weighed arihmeic mean of he prediced period s model prices in period using he period hedonic regression, f 1 [h (z k s,b )], relaive o he period s acual model prices, p k s. 37 Having defined he period s price level P s () by (51) and he corresponding period price level P () by (49) using he hedonic regression of period o do he consan uiliy qualiy adjusmen, we can ake he raio of hese wo price levels o form a Laspeyres ype price index going from period s o, using he hedonic regression of period, as follows: (52) P s () P ()/P s () 36 I can be seen ha he expression on he righ hand side of (49) is a ype of Paasche price index, where he price and quaniy daa of period, p k and q k for k S(), ac as he comparison period daa and he hedonic regression period prediced prices, f 1 [h (z k,b )] for k S(), ac as base period prices. 37 I can be seen ha he expression on he righ hand side of (51) is he reciprocal of a kind of Laspeyres price index, where he price and quaniy daa of period s, p s k and q s k for k S(s), ac as he base period price and quaniy daa and he hedonic regression period prediced prices, f 1 [h (z s k,b )] for k S(s), ac as comparison period prices.