Erwin Diewert Department of Economics University of British Columbia. August Discussion Paper No.: 05-11

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1 ADJACENT PERIOD DUMMY VARIABLE HEDONIC REGRESSIONS AND BILATERAL INDEX NUMBER THEORY by Erwin Diewer Deparmen of Economics Universiy of Briish Columbia Augus 2005 Discussion Paper No.: DEPARTMENT OF ECONOMICS THE UNIVERSITY OF BRITISH COLUMBIA VANCOUVER, CANADA V6T 1Z1 hp://

2 1 Adjacen Period Dummy Variable Hedonic Regressions and Bilaeral Index Number Theory Erwin Diewer 1, Augus 5, Discussion Paper No , Deparmen of Economics, Universiy of Briish Columbia, Vancouver, B.C., Canada, V6T 1Z1. Absrac A hedonic regression regresses he price of various models of a produc (or service) on he characerisics ha describe he produc. The issue ha is addressed in his paper is he following one: if informaion on he prices and quaniies purchased of he various models is available, hen how should his exra informaion be used? The paper suggess various mehods of weighing ha migh be used in an adjacen period dummy ime variable hedonic regression framework. Some of he ideas ha are presen in he es approach o index number heory are used in an aemp o cas some ligh on he consequences of differen ypes of weighing. Secion 2 weighing in a single period hedonic regression framework. Secion 3 discusses he adjacen period hedonic regression model wihou weighing. Secion 4 inroduces weighing ino he adjacen period framework and he ideas developed in he earlier secions are applied. Secion 5 specializes he general model of secion 4 o he pure dummy variable approach o he specificaion of characerisics of models and secion 6 concludes. Key Words Hedonic regression, es approach o index number heory, consumer heory, characerisics, qualiy change, mached models, consumer price index. Journal of Economic Lieraure Classificaion Sysem Numbers C23, C43, C51, D11, D12, E31 1 The auhor is indebed o Ana Aizcorbe, Erns Bernd, Carol Corrado, Melvyn Fuss, Rober Gordon, Jacques Mairesse, Alice Nakamura, Mick Silver and Manuel Trajenberg for helpful commens and o he SSHRC of Canada for financial suppor. None of he above are responsible for he views expressed in his chaper. Secions 2-4 and he Appendix in his paper originally appeared in Diewer (2003b). This paper will appear in Essays in honor of Zvi Griliches, J. Mairesse and M. Trajenberg (eds.), forhcoming in a Special issue of he Annales d'economie e Saisique, 2005.

3 2 1. Inroducion Some recen publicaions have revived ineres in he opic of hedonic regressions. The firs 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 second 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. 2 Finally, he comprehensive monograph by Triple (2004) argued srongly in favor of he use of hedonic regression mehods o adjus prices for qualiy changes over oher mehods ha have been suggesed. One imporan problem area associaed wih he use of hedonic regressions is he issue of wheher he regressions should be weighed or no. This is he issue ha we will address in his chaper. 3 Our approach o answering his quesion will be somewha novel: we will ry o use ideas ha occur in he index number lieraure on weighing o help guide us in evaluaing alernaive approaches o he weighing quesion in he hedonic regression conex. Thus consider he problem of consrucing a Consumer Price Index ha compares prices beween wo periods. Index number heory approaches o his problem end up consrucing a CPI beween he wo periods as a share weighed average of he price relaives for all he commodiies in he domain of definiion of he index ha can be mached beween he wo periods. Hedonic regressions are used o exend his framework o siuaions where he qualiy of he commodiies may change over ime (and so complee maching across he wo periods is no possible) bu he overall goal is he same as wih a sandard CPI: we wan a single number ha describes he average price change beween he wo periods. Thus in his chaper, we will focus on he dummy variable adjacen year hedonic regression echnique iniially suggesed by Cour (1939; ) and used by Bernd, Griliches and Rappapor (1995; 260) and many ohers, since his hedonic regression framework gives us an unambiguous measure of price change beween he wo periods under consideraion. 4 Our general sraegy will be o sugges various mehods of weighing ha migh be used in his adjacen period dummy ime variable hedonic regression framework, look a he resuling measures of overall price change bu hen specialize he model o he mached model conex so ha he resuling measure of price change can be compared o more radiional index number measures of overall price change. Thus we se he sage for his mehodology by firs discussing weighing in a single period hedonic regression framework in secion 2 and hen in secion 3, we discuss he adjacen period hedonic regression model wihou weighing. In secion 4, we inroduce weighing ino he adjacen period framework and apply our index number ype mehodology. Secion 5 2 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. 3 The only exensive discussion of his issue ha I am aware of is by Triple (2004; ). 4 When we esimae separae hedonic regressions for boh periods, here are many ways ha his informaion from he wo regressions can be used in order o obain a single overall measure of price change; see Silver and Heravi (2002) (2003) (2004) (2005) and Triple (2004) for discussions and comparisons of hese various mehods.

4 3 specializes he general model of secion 4 o an ineresing case considered by Aizcorbe, Corrado and Doms (2000), who inroduced a pure dummy variable approach o he specificaion of characerisics of models in addiion o he usual ime dummies. Secion 6 concludes and an Appendix presens proofs of he various resuls ha are used in secions Quaniy Weighs versus Expendiure Weighs As an inroducion o our main opic, we discuss alernaive mehods of weighing model prices in a single equaion hedonic regression. Thus if informaion on model prices, characerisics and sales o households is available o a saisical agency producing a Consumer Price Index, hen how exacly could he exra informaion on sales be used in running a single period hedonic regression? We inroduce some noaion a his poin. We suppose ha price daa have been colleced on K models or varieies of a commodiy for some period. 5 Thus p k is he price of model k in period and k S() where S() is he se of models ha are acually purchased by households 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 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 k S(). We shall consider only linear hedonic regressions in his chaper. 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 ; k S() where he β n are unknown parameers o be esimaed, he ε k are independenly disribued error erms wih mean 0 and variance σ 2 and he funcions f and f n are known funcions ha are used o ransform he daa (ypically, hey 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). Usually, economeric 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 an idea aken from index number heory namely 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 5 Models purchased in differen oules can be regarded as separae varieies or no, depending on he conex. 6 If a paricular model k is purchased 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 (2003a) made a case for considering second order approximaions bu in his chaper, we will follow curren pracice and consider only linear approximaions.

5 4 he period regression is represenaive of he sales ha acually occurred during he period. 8 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 : (2) 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 (2). 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 in he regression as models 1 and 2. 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: (3) 1 1 f(p 1 ) = 1 1 β f 1 (z 11 )β 1 ; 1 2 f(p 2 ) = 1 2 β f 1 (z 21 )β 1 ; 1 3 f(p 3 ) = 1 3 β 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 (2): (4) (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 (2) and (4), i can be seen ha he observaions in (4) are equal o he corresponding observaions in (2), excep ha he dependen and independen variables in 8 Thus our represenaive approach follows along he lines of Theil s (1967; ) weighed sochasic approach o index number heory, which is also pursued by Clemens and Izan (1981), Selvanahan and Rao (1994), Rao (2002) and Diewer (1995) (2004). 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 5 observaion k of (2) 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 (4). A sampling framework for (4) 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 (3) and le b 0 * and b 1 * denoe he leas squares esimaors for he parameers β 0 and β 1 in (4). Then i is sraighforward o show ha hese wo ses of leas squares esimaors are he same 9 ; i.e., we have: (5) [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 (3) is o obain he leas squares esimaors for he ransformed model (4). This equivalence beween he wo models provides a jusificaion for using he weighed model (4) in place of he original model (2). The advanage in using he ransformed model (4) over he represenaive model (3) is ha we can develop a sampling framework for (4) bu no for (3), since he (omied) error erms in (3) canno be assumed o be disribued independenly of each oher. 10 However, in view of he equivalence beween he leas squares esimaors for models (3) and (4), we can now be comforable ha he regression model (4) weighs observaions according o heir quaniaive imporance in period. Hence if we ake he poin of view ha regards weighing according o economic imporance as fundamenal, hen we can recommend he use of he weighed hedonic regression model (4) over is unweighed counerpar (2). 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: (6) v k p k q k ; k S(). Now consider again he simple unweighed hedonic regression model defined by (2) 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 (2) 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): 9 See, for example, Greene (1993; ). 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). 10 I is possible o develop a descripive saisics inerpreaion for b 0 and b 1, he leas squares esimaors for he β 0 and β 1 parameers in (3); see secion 8 in Diewer (2004).

7 6 (7) 1 1* f(p 1 ) = 1 1* β * f 1 (z 11 )β 1 ; 1 2* f(p 2 ) = 1 2* β * f 1 (z 21 )β 1 ; 1 3* f(p 3 ) = 1 3* β * f 1 (z 31 )β 1. Now consider he following value ransformaion of he original unweighed hedonic regression model (2): (8) (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 (2) and (8), i can be seen ha he observaions in (8) are equal o he corresponding observaions in (2), excep ha he dependen and independen variables in observaion k of (2) 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 lef hand side variables in (8). Again, a sampling framework for (8) 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 (7) and (8) 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 (7) is o obain he leas squares esimaors for he ransformed model (8). This equivalence beween he wo models provides a jusificaion for using he value weighed model (8) in place of he value weighs represenaive model (7). As before, he advanage in using he ransformed model (8) over he value weighs represenaive model (7) is ha we can develop a sampling framework for (8) bu no for (7), since he (omied) error erms in (7) canno be assumed o be disribued independenly of each oher. From he viewpoin of index number heory, i seems o us ha he quaniy weighed and value weighed models are clear improvemens over he original unweighed model (2). 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. 11 In he 11 I has already been observed 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).

8 7 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 in he single period conex, i appears o us ha value weighing is clearly preferable. Thus we are aking he poin of view ha he main purpose of a 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. We will draw on he maerial in his secion in secion 4 below. However, in he following secion, we provide an inroducion o he heory of weighed adjacen period hedonic regressions by considering he unweighed case firs. 3. Unweighed Bilaeral Hedonic Regressions wih Time as a Dummy Variable We now consider he following hedonic regression model, which uilizes he daa of periods s and : (9) f(p s k ) = β 0 + N n=1 f n (z s kn )β n + ε s k ; k S(s); (10) f(p k ) = γ s + β 0 + N n=1 f n (z kn )β n + ε k ; k S(); where he variables in (9) and (10) are defined in he same manner as in equaion (1) above. In paricular, ε s k and ε k are independenly disribued error erms wih mean 0 and variance σ 2. Noe ha he β regression coefficiens in (9) are consrained o be he same as he corresponding β coefficiens in (10). Noe also ha equaions (10) have added a ime dummy variable, γ s, and his coefficien will summarize he overall price change in he various models going from period s o This wo period ime dummy variable hedonic regression (and is exension o many periods) was firs considered explicily by Cour (1939; ) as his hedonic suggesion number wo. Cour (1939; 110) chose o ransform he prices by he log ransformaion on empirical grounds: Prices were included in he form of heir logarihms, since preliminary analysis indicaed ha his gave more nearly linear and higher simple correlaions. Cour (1939; 111) hen used adjacen period ime dummy hedonic regressions as links in a longer chain of comparisons exending from 1920 o 1939 for US auomobiles: The ne regressions on ime shown above are in effec price link relaives for cars of consan specificaions. By joining hese ogeher, a coninuous index is secured. If he wo periods being compared are consecuive periods, Griliches (1971b; 7) coined he erm adjacen year regression o describe his dummy variable hedonic regression model.

9 8 Before proceeding furher, we briefly discuss some of he advanages and disadvanages of he dummy variable model defined by (9) and (10) versus running separae single period regressions of he ype defined by (1) for periods s and and hen using hese separae regressions o form wo separae esimaes of qualiy adjused prices which would be averaged in some way in order o form an overall measure of price change beween periods s and. The main advanage of he laer mehod is ha i is more flexible; i.e., changes in ases beween periods can readily be accommodaed. However, his mehod has he disadvanage ha wo disinc esimaes of period s o price change will be generaed by he mehod (one using he regression for period s and he oher using he regression for period s) and i is somewha arbirary how hese wo esimaes are o be averaged o form a single esimae of price change. 13 The main advanages of he dummy variable mehod are ha i conserves degrees of freedom and is less subjec o mulicollineariy problems 14 and here is no ambiguiy abou he measure of overall price change beween periods s and. 15 We have considered only he case of wo periods since his is he case of mos ineres o saisical agencies who mus provide measures of price change beween wo periods. However, he bilaeral model defined by (9) and (10) can encompass boh he fixed base siuaion (where s will equal he base period 0) or he chained siuaion where s will equal 1. I is also of ineres o consider he wo period case because in his siuaion, we can draw on many of he ideas ha have been inroduced ino bilaeral index number heory, which also deals wih he problem of measuring price change beween wo periods. We firs consider he case where f is he ideniy ransformaion. 16 Le us esimae he unknown parameers in (9) and (10) by leas squares regression and denoe he esimaes for he β n by b n for n = 0,1,...,N and he esimae for γ s by c s. Denoe he leas squares s residuals for equaions (9) and (10) wih f defined o be he ideniy ransformaion by e k and e k respecively. Then we have he following equaions, which relae he model prices in he wo periods o heir prediced values and he sample residuals: (11) p k s = b 0 + n=1 N f n (z kn s )b n + e k s ; k S(s); 13 For reamens of he issues involved in averaging he resuls of wo period specific hedonic regressions o obain measures of overall price change, see Koskimäki and Varia (2001) for he unweighed case and Silver and Heravi (2002) (2003) (2004) (2005), Diewer (2003b) and Triple (2004) for he weighed case. 14 This advanage was noed by Griliches (1971b; 8): The ime dummy approach does have he advanage, if he comparabiliy problem can be solved, of allowing us o ignore he ever presen problem of mulicollineariy among he various dimensions. 15 Griliches (1971b; 7) has he following very nice summary jusificaion for he use of he ime dummy variable mehod: The jusificaion for his [mehod] is very simple and appealing: we allow as bes we can for all of he major differences in specificaions by holding hem consan hrough regression echniques. Tha par of he average price change which is no accouned for by any of he included specificaions will be refleced in he coefficien of he ime dummy and represens our bes esimae of he unexplained-byspecificaion-change average price change. 16 This model is no wihou ineres. Suppose here is a minimal base period model bu various addiional amouns of useful characerisics can be purchased a consan prices in each period. Then i would be naural o se he f n o be equal o ideniy funcions as well as f and wihin each period, he linear funcional form for he hedonic regression would be quie appropriae.

10 9 (12) p k = c s + b 0 + n=1 N f n (z kn )b n + e k ; k S(). Now consider a hypoheical siuaion where he models sold during periods s and are exacly he same so ha here are say K common models peraining o he wo periods. Suppose furher ha he model prices in period are all exacly λ imes greaer han he corresponding model prices in period s, where λ is a posiive consan. Under hese condiions, i seems reasonable o ask ha he regression prediced values for he period models be exacly equal o λ imes he regression prediced values for he same models in period s; i.e., we wan he following equaions o be saisfied: 17 (13) c s + b 0 + n=1 N f n (z kn )b n = λ[ b 0 + n=1 N f n (z kn )b n ] ; k = 1,...,K. In general, if K > N+2 and λ 1, i can be seen ha equaions (13) canno be solved for any coefficiens c s, b 0, b 1,...,b N. Hence, our conclusion is ha he linear ime dummy hedonic regression model defined by (11) and (12) is no a very good one, since i will no give us he righ answer in a simple siuaion where all model prices are proporional for he wo periods. 18 Of course, his homogeneiy problem wih he linear dummy variable regression model can be solved if we replace equaions (12) by he following equaions: (14) p k = c s [b 0 + n=1 N f n (z kn )b n ] + e k ; k S(). In equaions (14), he ime dummy variable, c s, now appears in a muliplicaive fashion. Thus, he problem wih he esimaing equaions (11) and (14) is ha we no longer have a linear regression model; nonlinear esimaion echniques would have o be used. This is our firs example of how he es approach ha is commonly used in bilaeral index number heory can be adaped o he adjacen period or ime dummy hedonic regression conex in order o obain useful resricions on he form of he hedonic regression. Many addiional examples will be presened in wha follows. 19 Since nonlinear regression models are more difficul o esimae and may suffer from reproducibiliy problems, we will urn our aenion o he second se of bilaeral hedonic regression models, where f is he log ransformaion. In his case, he counerpars o equaions (11) and (12) are he following equaions: (15) ln p s k = b 0 + N n=1 f n (z s kn )b n + e s k ; k S(s); (16) ln p k = c s + b 0 + N n=1 f n (z kn )b n + e k ; k S(). 17 Le z kn z s kn = z k denoe he common amoun of characerisic n ha he idenical model k has in each period. 18 Diewer (2003a) also argued on heoreical grounds ha dummy variable hedonic regression models ha used unransformed prices as dependen variables did no have good properies. 19 The es approach o bilaeral index number heory is reviewed in Diewer (1992), Balk (1995) and he ILO (2004; ).

11 10 Exponeniaing boh sides of (15) and (16) leads o he following equaions ha will be saisfied by he daa and he leas squares esimaors for (15) and (16): (17) p s k = exp[b 0 + N n=1 f n (z s kn )b n ]exp[e s k ] k S(s); (18) p k = exp[c s ]exp[b 0 + N n=1 f n (z kn )b n ]exp[e k ] ; k S(). Again consider a hypoheical siuaion where he models sold during periods s and are exacly he same so ha here are K common models peraining o he wo periods. Again suppose ha he model prices in period are all exacly λ imes greaer han he corresponding model prices in period s, where λ is a posiive consan. Again we ask ha he regression prediced values for he period models be exacly equal o λ imes he regression prediced values for he same models in period s; i.e., we wan he following equaions o be saisfied: (19) exp[c s ]exp[b 0 + n=1 N f n (z kn )b n ] = λ{exp[b 0 + n=1 N f n (z kn )b n ]} ; k = 1,...,K. I can be seen ha if we choose c s = ln λ, hen we can saisfy equaions (19). Hence we conclude (from a es approach perspecive) ha if we wan o use linear regression echniques o esimae he parameers of he hedonic regression, hen i is preferable o run linear bilaeral dummy variable hedonic regressions using he log ransformaion for he dependen variable raher han leaving he model prices unransformed. 20 The bilaeral log hedonic regression model is defined by (9) and (10) where f is he log ransformaion. I can be seen ha in his case, he heoreical index of price change going from period s o is exp[γ s ] and a sample esimaor of his populaion measure is: (20) P(s,) exp[c s ] where c s is he leas squares esimaor for he shif parameer γ s. Noe ha we pu he shif parameer in equaions (10) raher han in equaions (9). The choice of base period should no maer so le us consider he following bilaeral log regression model which pus he shif parameer γ s in he period s equaions raher han in he period equaions: (21) ln p s k = γ s + β * 0 + N n=1 f n (z s kn )β * n + ε s k ; k S(s); (22) ln p k = β * 0 + N n=1 f n (z kn )β * n + ε k ; k S(). Denoe he leas squares esimaes for β n * by b n * for n = 0,1,...,N and he esimae for γ s by c s. For he regression model defined by (21) and (22), i can be seen ha he heoreical index of price change going from period o s is exp[γ s ] and he sample esimaor of his populaion measure is: (23) P(,s) exp[c s ]. 20 However, recall our earlier qualificaion which noed ha if addiional amouns of all characerisics can be purchased a consan prices in each period, hen a nonlinear regression model wih he f and f n se equal o ideniy funcions is preferable.

12 11 The quesion now is: how does P(s,) defined by (20) relae o P(,s) defined by (23)? Ideally, we would like hese wo esimaors of price change o saisfy he following ime reversal es: (24) P(,s) =1/P(s,). If we compare he original log linear regression model defined by (9) and (10) (wih f being he log ransformaion) wih he new model defined by (21) and (22), i can be seen ha he righ hand side exogenous variables are idenical excep ha γ s appears in he firs se of equaions in (21) and (22) while γ s appears in he second se of equaions in (9) and (10). The ranspose of he column in he X marix ha corresponds o γ s in (21) and (22) is equal o [1 1 T,0 2 T ] where 1 1 is a column vecor of ones of dimension equal o he number of models in he se S(s) and 0 2 is a column vecor of zeros of dimension equal o he number of models in he se S(). The ranspose of he column in he X marix ha corresponds o γ s in (9) and (10) is equal o [0 1 T,1 2 T ] where 0 1 is a column vecor of zeros of dimension equal o he number of models in he se S(s) and 1 2 is a column vecor of ones of dimension equal o he number of models in he se S(). However, noe ha boh models have he consan erm β 0 (or β 0 * ) in every equaion and he ranspose of he column in he X marix ha corresponds o his consan erm is equal o [1 1 T,1 2 T ] in boh models. I can be seen ha he subspace spanned by he X columns corresponding o β 0 and γ s in (9) and (10) is equal o he subspace spanned by he X columns corresponding o β 0 * and γ s in (21) and (22) and he wo ses of parameers are relaed by he following equaions: (25) [0 1 T,1 2 T ] γ s + [1 1 T,1 2 T ] β 0 = [1 1 T,0 2 T ] γ s + [1 1 T,1 2 T ] β 0 *. Equaions (25) are equivalen o he following 2 equaions in he four variables γ s, β 0, γ s and β 0 * : (26) 0 γ s + 1 β 0 = 1 γ s + 1 β 0 * ; 1 γ s + 1 β 0 = 0 γ s + 1 β 0 *. Thus given γ s and β 0, he corresponding γ s and β 0 * can be obained using equaions (26) as: (27) γ s = γ s ; β 0 * = γ s + β 0. Equaions (27) also hold for he leas squares esimaors for he wo hedonic regression models. In paricular, we have: (28) c s = c s. Hence, exponeniaing boh sides of (28) gives us exp[c s ] = 1/exp[c s ] and his equaion is equivalen o (45) using definiions (20) and (23). Thus we have shown ha he esimaor

13 12 of price change P(s,) defined by (20) (which corresponds o he leas squares esimaors of he iniial log hedonic regression model defined by (9) and (10) wih f(p) ln p) is equal o he reciprocal of he esimaor of price change P(,s) defined by (23) (which corresponds o he second log hedonic regression model defined by (21) and (22) so ha he wo bilaeral dummy variable hedonic regressions saisfy he ime reversal es (24). If i is desired o avoid he use of nonlinear regression echniques, hen he resuls in his secion suppor he use of he logarihms of model prices as he dependen variables in an unweighed bilaeral hedonic regression model wih a ime dummy variable. In he following secion, we will sudy he properies of weighed bilaeral hedonic regression models. 4. Weighed Bilaeral Hedonic Regressions wih Time as a Dummy Variable Given he resuls in he previous secion, we consider only weighed bilaeral hedonic regressions ha use he log of model prices as he dependen variable, before weighing he equaions. We also draw on he resuls in secion 2 and consider only value weighing. Thus we now consider he following value weighed hedonic regression model, which uilizes he daa of periods s and : (29) (v s k ) 1/2 ln p s k = (v s k ) 1/2 [β 0 + N n=1 f n (z s kn )β n ] + ε s k ; k S(s); (30) (v k ) 1/2 ln p k = (v k ) 1/2 [γ s + β 0 + N n=1 f n (z kn )β n ] + ε k ; k S(); where he model sales values for period, v k, were defined by (6) and ε s k and ε k independenly disribued error erms wih mean 0 and variance σ 2. are The weighed model defined by (29) and (30) is he bilaeral counerpar o our single equaion weighed hedonic regression model ha was sudied in secion 2 above. However, in he presen bilaeral conex, we now encouner a problem ha was absen in he single equaion conex. The problem is his: if here is high inflaion going from period s o, hen he period model sales values v k can be very much bigger han he corresponding period s model sales values v s k due o his general inflaion. Hence, he assumpion of homoskedasic residuals beween equaions (29) and (30) is unlikely o be saisfied. 21 Hence, i is necessary o pick new weighs ha will eliminae his problem. In order o address he above problem, we firs define he period expendiure share of model k as follows: (31) s k p k q k / i S() p i q i ; k S(). Our iniial soluion o he above problem caused by general inflaion beween he wo periods is o use he model expendiure shares, s k s and s k as he weighs in (29) and (30) 21 From he viewpoin of he descripive saisics approach, if we wan he weighs for each period o be equally imporan or represenaive in he regression, hen i is naural o require ha he weighs o sum o one for each period.

14 13 in place of he model expendiures, v k s and v k. Thus we recommend he use of he following expendiure share weighed hedonic regression model, which uilizes he daa of periods s and : 22 (32) (s s k ) 1/2 ln p s k = (s s k ) 1/2 [β 0 + N n=1 f n (z s kn )β n ] + ε s k ; k S(s); (33) (s k ) 1/2 ln p k = (s k ) 1/2 [γ s + β 0 + N n=1 f n (z kn )β n ] + ε k ; k S(); where ε k s and ε k are independenly disribued error erms wih mean 0 and variance σ 2. Denoe he leas squares esimaes for β n by b n for n = 0,1,...,N and he esimae for γ s by c s. For he regression model defined by (32) and (33), i can be seen ha he heoreical index of price change going from period o s is exp[γ s ] and he sample esimaor of his populaion measure is: (34) P 1 (s,) exp[c s ]. I can be shown ha P 1 (s,) defined by (34) in his secion has he same desirable propery ha P(s,) defined by (20) in he previous secion had: namely, if he models are idenical in he wo periods (and he model expendiure shares are idenical for he wo periods) and he model prices in period are all exacly λ imes greaer han he corresponding model prices in period s, hen P 1 (s,) is exacly equal o λ. 23 The resricion ha he expendiure shares be idenical in he wo periods in he idenical model case is a bi unrealisic. Moreover, in he idenical models case, i would be nice if P 1 (s,) defined by (34) urned ou o equal he Törnqvis price index, since his index is a preferred one from he viewpoins of boh he sochasic and economic approaches o index number heory. 24 Hence in place of he model defined by (32) and (33), when a model is presen in boh periods, le us use he average sales share for ha model, (1/2)(s s k +s k ), as he weigh for ha model in boh periods. In his revised weighing scheme, he old period s equaions (32) are replaced by he following wo ses of equaions: (35) (s k s ) 1/2 ln p k s = (s k s ) 1/2 [β 0 + n=1 N f n (z kn s )β n ] + ε k s ; k [S(s) S()]; (36) [(1/2)(s k s +s k )] 1/2 ln p k s = [(1/2)(s k s +s k )] 1/2 [β 0 + n=1 N f n (z kn s )β n ] + ε k s ; k S(s) S(). Thus if a model k is presen in period s bu no presen in period, hen we use he square roo of he period s sales share for ha model, (s k s ) 1/2, as he weigh, which means his model is included in equaions (35). On he oher hand, if model k is presen in boh periods,, hen we use he square roo of he arihmeic average of he period s and sales shares for ha model, [(1/2)(s k s +s k )] 1/2, as he weigh, which means his model is 22 Diewer (2005) considered a model similar o (32) and (33) excep ha all of he explanaory variables were dummy variables and he showed ha weighing by he square roos of expendiure shares led o a very reasonable index number formula o measure he price change beween he wo periods. 23 See Proposiion 1 in he Appendix. 24 See Diewer (2002; ) and he ILO (2004; and ).

15 14 included in equaions (36). Similarly, he old period s equaions (33) are replaced by he following wo ses of equaions: 25 (37) (s k ) 1/2 ln p k = (s k ) 1/2 [γ s + β 0 + n=1 N f n (z kn )β n ] + ε k ; k [S() S(s)]; (38) [(1/2)(s k s +s k )] 1/2 ln p k = [(1/2)(s k s +s k )] 1/2 [γ s + β 0 + n=1 N f n (z kn )β n ] + ε k ; k S(s) S(). Thus if a model k is presen in period bu no presen in period s, hen we use he square roo of he period sales share for ha model, (s k ) 1/2, as he weigh, which means his model is included in equaions (37). On he oher hand, if model k is presen in boh periods, hen we use he square roo of he arihmeic average of he period s and sales shares for ha model, [(1/2)(s s k +s k )] 1/2, as he weigh, which means his model is included in equaions (38). As usual, we assume ha ε s k and ε k are independenly disribued error erms wih mean 0 and variance σ 2. Denoe he leas squares esimaes for β n by b n for n = 0,1,...,N and he esimae for γ s by c s. For he regression model defined by (35)-(38), i can be seen ha he heoreical index of price change going from period o s is exp[γ s ] and he sample esimaor of his populaion measure is: (39) P 2 (s,) exp[c s ]. I can be shown ha P 2 (s,) defined by (39) has he following desirable propery: if he models are idenical in he wo periods, hen P 2 (s,) is equal o he Törnqvis price index beween he wo periods. 26 Hence i appears ha he weighed hedonic regression model defined by (35)-(38) is a naural weighed hedonic regression model ha provides a generalizaion of he Törnqvis price index o cover he case where he models are no mached. If here are no models in common for he wo periods under consideraion, hen he model defined by (35)-(38) becomes a special case of our earlier model defined by (32)-(33). As in he previous secion, i is somewha arbirary wheher we pu he ime dummy variable in he period equaions or wheher we pu i in he period s equaions. If we pu he ime dummy in he period s equaions as he parameer γ s and obain a weighed leas squares esimae c s for his populaion parameer, he heoreical index of price change going from period o s is exp[γ s ] and he sample esimaor of his populaion measure is: (40) P * (,s) exp[c s ]. As in he previous secion, we would like P * (,s) o equal he reciprocal of P(s,). I urns ou ha his propery is rue for he weighed hedonic regressions defined by (32) and 25 Noe ha he mixed period s share weighs used in (35) and (36) and he mixed period share weighs used in (37) and (38) do no necessarily sum o one whereas he period s and weighs used in (32) and (33) respecively did sum o one for each period. 26 This follows from Corollary 5.3 in he Appendix.

16 15 (33) and (35)-(38) in his secion as well as for he unweighed ones defined in he previous secion; see Proposiion 4 in he Appendix. Hence i does no maer wheher we pu he ime dummy variable in period s or : our measure of overall price change beween he wo periods will be invarian o his choice for he wo weighed hedonic regressions considered in his secion. Using he resuls in he Appendix, we can also show ha P 1 (s,) and P 2 (s,) boh saisfy he ideniy es (A6), he homogeneiy ess (A4) and (A5) and he ime reversal es (A7) as we have already indicaed. Thus boh of hese hedonic price indexes have some good axiomaic properies. Which bilaeral weighed hedonic index is bes? From he viewpoin of represenaiviy, P 1 (s,) seems bes: he models presen in each period are weighed by expendiure shares ha perain o ha period. However, he loss of represenaiviy for P 2 (s,) is probably no large in mos applicaions and P 2 (s,) has he advanage of being consisen wih he use of a Törnqvis price index in he mached models case. Thus eiher index can be jusified. 5. The Pure Dummy Variable Adjacen Period Hedonic Regression Model In his secion, we specialize he resuls in he previous secion in order o provide a generalizaion (o he weighed case) of a model due o Aizcorbe, Corrado and Doms (2000), which inroduced a dummy variable for each model in addiion o he usual ime dummies. Thus heir model had no oher characerisics oher han hese model specific dummy variables. 27 We can use he resuls in he previous secion o see how he weighed ACD model works in he case of wo periods. We will work wih he firs share weighed model defined by (32) and (33) in he previous secion, excep ha he N characerisics funcions f n are now assumed o be dummy variables. 28 Thus we assume ha here are a oal of N differen models sold in periods s and. The old period s equaions (32) are broken up ino wo ses of equaions, (41) and (42), where he models k which appear in (41) are presen in boh periods s and and he models m which appear in (42) are presen in period s and no period : (41) (s s k ) 1/2 ln p s k = (s s k ) 1/2 β k + ε s k ; k S(s) S(); (42) (s s m ) 1/2 ln p s m = (s s m ) 1/2 β m + ε s m ; m [S(s) S()]. Similarly, he old period equaions (33) are broken up ino wo ses of equaions, (43) and (44), where he models k which appear in (43) are presen in boh periods s and and he models n which appear in (44) are presen in period and no period s: 27 Their model is equivalen o he Counry Produc Dummy model used in making inernaional comparisons of prices ha was pioneered by Summers (1973). For exensions of his model o he weighed case, see Diewer (2004) (2005). 28 We also need o se β 0, he consan erm in he regression, equal o zero in order o idenify all of he parameers in his pure dummy variable hedonic regression.

17 16 (43) (s k ) 1/2 ln p k = (s k ) 1/2 [γ s + β k ] + ε k ; k S(s) S(); (44) (s n ) 1/2 ln p n = (s n ) 1/2 [γ s + β n ] + ε n ; n [S() S(s)]. Now run a leas squares regression on he model defined by (41)-(44). Denoe he leas squares esimaes for β n by b n for n = 1,...,N and he esimae for γ s by c s. Use he firs order necessary (and sufficien) condiions for he leas squares minimizaion problem for he b n o solve for each b n in erms of c s and hen subsiue hese expressions for he b n ino he firs order condiion for he c s parameer. The resuling equaion simplifies o: (45) c s = W 1 k S(s) S() [s k s + s k ] 1 s k s s k ln[p k /p k s ] where (46) W k S(s) S() [s k s + s k ] 1 s k s s k. Thus c s, he log of he hedonic price index going from period s o, is equal o a weighed average (where he weighs are posiive and sum o one) of he log price raios, ln[p k /p k s ], over all of he models k ha are presen in boh periods. Noe ha his pure dummy variable hedonic model boils down o a (weighed) mached model price index. If we drop he share weighs in (41)-(44) and simply run an unweighed regression model, hen we can use he above algebra by simply seing each share equal o 1 and we find ha (47) c s = (1/M) ln[p k /p k s ] where M is he number of models ha are presen in boh periods s and. This capures he original ACD model for he case of only wo periods. Thus for he case of only wo ime periods, he Aizcorbe, Corrado and Doms (2000) model of hedonic price change reduces o a saisical agency mached model esimae of price change; i.e., heir hedonic esimae of he price change going from period s o is equal o he equally weighed geomeric mean of he price relaives of he models ha are presen in boh periods. The exac equivalence of he ACD measure of price change in he wo period case o he equally weighed geomeric mean of he price relaives of he models ha are presen in boh periods does no carry over o he case where here are more han 2 periods. However, i is likely ha even in he many period case, he ACD measures of price change will have a endency o follow he mached model resuls. In any case, he ACD measures of price change can be quie differen from wha a hedonic model wih coninuous characerisics would yield Conclusion 29 See Triple and McDonald (1977), Triple (2004) and Silver and Heravi (2002) (2003) (2004) (2005) for some empirical evidence and explanaion on his poin.

18 17 We conclude his chaper by noing ha we have no resolved all of he issues surrounding he quesion as o wheher hedonic regressions should be weighed according o heir economic imporance or no. There is a ension beween he index number approach o hedonic regressions and he economeric approach. This ension is bes described by Triple (2004), who wroe he mos sysemaic discussion of he weighing issue o dae. 30 I is worh quoing par of Triple s conclusion on he weighing issue: Dickens (1990) conends ha when weighed and unweighed regression esimaes differ, i is a sign of specificaion error in our conex, an example would be a hedonic funcion in which crucial characerisics variables were missing. Missing informaion on sofware characerisics and some characerisics of hardware are common in hedonic invesigaions so hedonic funcion sensiiviy o weighing in he presence of missing variables is consisen wih Dickens conenion. On Dickens analysis, in a properly specified hedonic funcion, weighed and unweighed regressions should no differ. Jack E. Triple (2004; ). Thus from he economerics poin of view, he emphasis is on he saisical model: is accurae specificaion and is mos efficien esimaion. However, from he viewpoin of he price saisician, he model is no he mos imporan consideraion: he mos imporan consideraion is o obain an overall measure of price change over wo periods over some domain of definiion of admissible prices and ransacions involving hose prices. Thus he price saisician akes a descripive saisics perspecive whereas he economerician akes a saisical model and esimaion perspecive. 31 The difference beween he wo approaches can be illusraed in he mached model conex. In his conex, he economerician running a hedonic regression model in his conex would probably follow Dickens advice and focus on a simple unweighed model of he ype defined by (9) and (10) in secion 3 and would end up wih he equally weighed geomeric mean of he price relaives of he mached models (he Jevons index) as he measure of overall price change beween he wo periods. However, he descripive saisics price saisician migh run he weighed model defined by (35)-(38) in secion 4 and would end up wih he Törnqvis price index as he measure of overall price change. 32 In many cases, he wo esimaes of overall price change could be quie differen bu from he viewpoin of sandard index number heory, he Törnqvis index is clearly preferable o he Jevons index. The heory of hedonic regressions leaves a grea deal of leeway open o he empirical invesigaor wih respec o he deails of implemenaion of he models. Our sraegy in his chaper has been o use some of he ideas ha are presen in he es approach o 30 Silver and Heravi (2004) also discuss alernaive approaches o weighing in hedonic regressions in a sysemaic manner. 31 Triple (2004; 190) noes ha he economeric approach can be used in a weighed conex so he wo perspecives can be complemenary: However, he dummy variable mehod is only one mehod for esimaing hedonic price indexes and i is he only one where weighs for he index and weighs for he hedonic funcion imply he same quesions. Chaper 3 describes hree oher mehods for esimaing hedonic price indexes he characerisics price index mehod, he hedonic impuaion mehod and he hedonic qualiy adjusmen mehod in all of which a weighed index number can be produced using an unweighed hedonic funcion. 32 See Diewer (2004; secion 8) for a mehod for convering weighed models like (32)-(33) or (35)-(38) ino descripive saisics models similar o Theil s (1967; 138) so ha c * s can be inerpreed as a descripive saisics measure of overall log pure price change beween he wo periods.

19 18 index number heory in an aemp o work ou some of he axiomaic properies of adjacen period ime dummy hedonic regression models in an aemp o cas some ligh on he issue of weighing. Our resuls are no definiive bu perhaps hey can cas some ligh on he consequences of differen ypes of weighing. 33 We conclude by noing ha he cauious aiude owards he use of hedonic regressions expressed by Schulze and Mackie (2002) echoes he following commens made by Bean in his discussion of Cour s (1939) pioneering paper on hedonic regressions: Mr. Cour s ineresing work should be carried much furher, as he suggess. We should, however, no be disappoined if neiher public agencies nor rade associaions adop he policy of publishing prices, values and index numbers based on he relaively ricky resuls ha one is sure o ge by applying he device of muliple correlaion. The only group who would sponsor such a procedure would be he non-exisen Naional Associaion of Expers in Muliple Correlaion, he demand for whose services would be enormously increased. Louis H. Bean (1939; 119). Hopefully, in he nex few years, as users form a consensus on wha he bes procedures are, hen he use of hedonic regressions by saisical agencies will become much more widespread and rouine. 34 Appendix: Properies of Bilaeral Weighed Hedonic Regressions We consider some of he mahemaical properies of a sligh generalizaion of he share weighed bilaeral hedonic regression model defined by (32) and (33) in secion 4. The generalizaion is ha we do no resric he weighs o sum up o 1 in each period. Thus, we replace he period s share weighs s k s in (32) and he period share weighs s k in (33) by he posiive weighs w k s and w k respecively, where hese weighs do no necessarily sum o 1 in each period. We assume ha hese weigh funcions are known funcions of he price and quaniy daa peraining o periods s and ; i.e., we have for some funcions, g k s and g k : (A1) w k s = g k s (p s,p,q s,q ) for k S(s) ; w k = g k (p s,p,q s,q ) for k S() where p s and p are price vecors of he model prices for periods s and respecively and q s and q are he corresponding period s and quaniy vecors of he models sold in periods s and. In he Proposiions below, we will place furher resricions on he weighing funcions g k s and g k as hey are needed. The weighed leas squares esimaors for γ s, β 0, β 1,...,β N for his new model are he soluions c * s, b * 0, b * * 1,..., b N o he following quadraic weighed leas squares minimizaion problem: I is encouraging o he auhor ha he earlier version of his paper, Diewer (2003b), has been exensively used by Silver and Heravi (2004) (2005) and Haan (2003) and he work of hese auhors has influenced saisical agency pracice in he UK and he Neherlands. 34 The exensive work by Jack Triple (2004) and Silver and Heravi (2001) (2002) (2003) (2004) (2005) should help form his consensus.