Keywords: Hedonic regressions; hedonic indexes; consumer price indexes; superlative indexes.

Transcription

1 Hedonc Imuaon versus Tme Dummy Hedonc Indexes Erwn Dewer, Saeed Herav and Mck Slver December 5, 27 (wh a commenary by Jan de Haan) Dscusson Paer 7-7, Dearmen of Economcs, Unversy of Brsh Columba, Eas Mall, Vancouver, BC V6T Z Canada. Emal: dewer@econ.ubc.ca Absrac: Sascal offces ry o mach em models when measurng nflaon beween wo erods. However, for roduc areas wh a hgh urnover of dfferenaed models, he use of hedonc ndexes s more arorae snce hey nclude he rces and quanes of unmached new and old models. The wo man aroaches o hedonc ndexes are hedonc muaon (HI) ndexes and dummy me hedonc (HD) ndexes. Ths sudy rovdes a formal analyss of he dfference beween he wo aroaches for alernave mlemenaons of an ndex ha uses weghng ha s comarable o he weghng used by he Törnqvs suerlave ndex n sandard ndex number heory. Ths sudy shows exacly why he resuls may dffer and dscusses he ssue of choce beween hese aroaches. An llusrave sudy for desko PCs s rovded. Keywords: Hedonc regressons; hedonc ndexes; consumer rce ndexes; suerlave ndexes. JEL classfcaon: C43, C82, E3.. Inroducon The urose of hs aer s o comare wo man and que dsnc aroaches o he measuremen of hedonc rce ndexes: me dummy hedonc ndexes and hedonc muaon ndexes. Boh aroaches no only correc rce changes for changes n he qualy of ems urchased, bu also allow he ndexes o ncororae mached and unmached models. They rovde a means by whch rce change can be measured n roduc markes where here s a rad urnover of dfferenaed models. However, hey can yeld que dfferen resuls. Ths aer rovdes a formal exoson of he facors underlyng such dfferences and he mlcaons for choce of mehod. We consder boh weghed and unweghed hedonc regresson models. Unweghed hedonc regresson models wll be consdered n secons 2 and 3 below. These models are of course useful n a samlng conex where nformaon on he quany or value of sales (or urchases) s unavalable. Weghed hedonc regresson models are consdered n secons 4 and 5 below. The weghng s chosen so ha f we are acually n a mached model suaon for he wo erods beng consdered, hen he resulng hedonc regresson measures of rce change resemble sandard suerlave ndex number formulae. The sandard way rce changes are measured by naonal sascal offces s hrough he use of a mached models mehodology. Usng hs mehodology, he deals and rces of a reresenave selecon of ems are colleced n a base reference erod and her mached rces colleced n successve erods so ha he rces of lke are comared wh lke. However, f here s a rad urnover of avalable models, hen he samle of roduc rces used o measure rce changes Our hanks o Erns Bernd, John Greenlees, Jan de Haan, Alce Nakamura and Jack Trle for helful commens. The auhors are a he Unversy of Brsh Columba, he Unversy of Cardff and he Inernaonal Moneary Fund resecvely. The aer s forhcomng n Prce Index Conces and Measuremen, W.E. Dewer, J. Greenlees and C. Hulen, Unversy of Chcago Press.

2 2 becomes unreresenave of he caegory as a whole. Ths s as a resul of boh new unmached models beng nroduced (bu no ncluded n he samle), and older unmached models beng rered (and hus drong ou of he samle). Hedonc ndexes use mached and unmached models and n dong so u an end o he mached models samle selecon bas. 2 The need for hedonc ndexes can be seen n he conex of he need o reduce bas n he measuremen of he U.S. consumer rce ndex (CPI), whch has been he subjec of hree major reors: he Sgler (96) Commee Reor, he Boskn (996) Commsson Reor and he Schulze and Macke (22) Commee on Naonal Sascs Panel Reor. Each found he nably o roerly remove he effec on rce changes of changes n qualy o be a major source of bas. Hedonc regressons were consdered o be he mos romsng aroach o conrol for such qualy changes, hough he Schulze anel cauoned for he need for furher research on mehodology: Hedonc echnques currenly offer he mos romsng aroach for exlcly adjusng observed rces o accoun for changng roduc qualy. Bu our analyss suggess ha here are sll subsanal unresolved economerc, daa, and oher measuremen ssues ha need furher aenon. C.L. Schulze and C. Macke (22; 6). A frs sgh he wo aroaches o hedonc ndexes aear que smlar. Boh rely on hedonc regresson equaons o remove he effecs on rce of qualy changes. They can also ncororae a range of weghng sysems and can be formulaed as a geomerc, harmonc or arhmec aggregaor funcon of qualy adjused rces, and as chaned or drec, fxed base comarsons. Ye hey can gve que dfferen resuls, even when usng comarable weghs, funconal forms and he same mehod of makng comarsons over erods. Ths s due o he fac ha hey work on dfferen averagng rncles. The dummy varable mehod consrans hedonc regresson arameers o be he same over me. A hedonc muaon ndex conversely allows he qualy adjusmen arameers o change n each erod and underakes wo ses of qualy adjusmens o rces for each comarson of rces beween wo erods and hen averages over hese wo comarsons. There has been some valuable research on he wo aroaches 3 hough no formal analyss has been resened, o he auhors knowledge wh a few exceons 4, of he facors governng he dfferences beween he aroaches. Bernd and Raaor (2) and Pakes (23) have hghlghed he fac ha he wo aroaches can gve dfferen resuls and boh of hese aers advse he use of hedonc muaon ndexes when arameers are unsable, a roosal whch wll be consdered n secons 5 and 7 below. Secon 2 below looks a a smle unweghed wo erod me dummy varable hedonc regresson model. We focus on he esmaon of he me dummy esmae of he change n log rces gong from erod o bu we reresen hs measure of overall log rce change as a dfference n log rce levels for he wo erods. In secon 3, we ake he same unweghed model bu run searae hedonc regressons for boh erods and use hese regresson arameers o form wo mued measures of consan qualy log rce change. These wo measures are hen averaged o oban an overall mued measure of log rce change. 5 An exac exresson for he dfference n consan qualy log rce change beween he me dummy and muaon measures s also develoed n secon 3. I s found ha n order for hese wo overall measures o dffer, we requre: 2 See for examle, Cole, Chen, Barqun-Solleman, Dulberger, Helvacan and Hodge (986), Slver and Herav (23) (25), Pakes (23) and Trle (24). 3 See Bernd, Grlches and Raaor (995), Bernd and Raaor (2), Dewer (23b), Slver and Herav (23), Pakes (23), Haan (23) (24) and Trle (24). 4 The frs exceon s Slver and Herav (27a) who consdered he case of one characersc and used a raher dfferen mehodologcal aroach based on he bas generaed by omed varables n regresson models. The second exceon s he commen by Jan de Haan whch follows hs aer, who develoed an exresson for he dfference based on a framework oulned n Trle and MacDonald (977). The ons made n Haan s commenary are develoed more fully n Haan (27). 5 An alernave nerreaon of hs measure of rce change s derved n Aendx.

3 3 dfferences n he wo varance covarance marces eranng o he model characerscs n each erod; dfferences n average amouns of model characerscs resen n each erod 6 and dfferences n esmaed hedonc coeffcens for he wo searae hedonc regressons. The analyss n secons 2 and 3 s reeaed n he weghed conex n secons 4 and 5. Secon 6 rovdes an emrcal sudy for desko PCs and secon 7 concludes by dscussng he ssue of choce beween he aroaches n lgh of he heorecal and emrcal fndngs. Aendx consders wo alernave mehodologes for consrucng measures of overall log rce change usng he hedonc muaon mehodology where wo searae hedonc regressons are esmaed for he wo erod under consderaon. The frs mehodologcal aroach s due o Cour (939; 8) where ndvdual rces n each erod are qualy adjused usng her characerscs vecors and he characerscs rces obaned from one of he wo hedonc regressons and hen he resulng qualy adjused rces are comared across he wo erods. Fnally, he resulng wo measures of qualy adjused overall log rce change are averaged. In he second mehodologcal aroach o hedonc muaon ndexes, due orgnally o Grlches (967), he mean vecor of characerscs ha erans o he models observed n erod s calculaed and hen he dsance beween he wo hedonc regressons a hs mean characerscs on s calculaed, whch generaes a frs measure of overall rce change (he Laseyres measure of log rce change). The Paasche measure of overall log rce change s calculaed usng he mean vecor of characerscs ha erans o he models observed n erod and hen he dsance beween he wo hedonc regressons a hs mean characerscs on s calculaed. Fnally he wo esmaes of overall log rce change are averaged. Aendx shows ha hese wo mehodologcal aroaches o hedonc muaon ndexes lead o exacly he same numercal esmaes of overall rce change. I s ofen hough ha a major advanage of he me dummy varable mehod for obanng measures of overall log rce change s ha a sandard error for he log rce change s obaned. In Aendx 2, a mehod for obanng aroxmae sandard errors for he Laseyres and Paasche hedonc muaon measures of log rce change s derved. 2. Unweghed Tme Dummy Hedonc Regressons We begn by consderng a smle unweghed wo erod me dummy varable hedonc regresson model. We assume ha here are N() observaons on he rces, n, of varous models n n erod for =,. Observaon n n erod has a vecor of K characerscs assocaed wh, say [z n,z n2,...,z nk ] for =, and n =,2,...,N(). The me dummy regresson model has he followng form: () ln n y n = α + k= K z nk γ k + ε n ; =, ; n =,2,...,N() where he ε n are ndeendenly dsrbued normal varables wh mean and consan varance and α, α, γ,...,γ K are arameers o be esmaed. The arameers α and α are measures of he average level of consan qualy rces of he ems n erod and resecvely and he γ,...,γ K are qualy adjusmen facors for he K characerscs;.e., γ k s he conrbuon o he log rce of he roduc of addng an exra un of characersc k. Noe ha we have arameerzed he me dummy hedonc regresson model n a slghly dfferen way o he way s usually done snce we do no have an overall consan erm n he regresson lus a me dummy varable n erod ; nsead 6 If he models are exacly he same n he wo erods beng consdered, hen hs se of dfferences wll be zero and he characerscs varance covarance marces wll also be dencal. Hence he hedonc me dummy and hedonc muaon esmaes of rce change wll be dencal under hese condons. Thus he wo mehods wll gve rse o subsanal conflcng esmaes only n markes where here are many new models beng nroduced no he markelace or many dsaearng models (or boh).

4 4 we have searae consan erms for each erod. The overall measure of logarhmc rce change gong from erod o erod s α α. 7 Le and be vecors of ones and zeros of dmenson N(), le y and y be he N() and N() dmensonal vecors of erod and logarhms of roduc rces, le ε and ε be he N() and N() dmensonal vecors of erod and sochasc dsurbances and le Z and Z be marces of he roduc characerscs n erods and resecvely. Then he model defned by () can be wren n marx noaon as follows: (2) y = α + α + Z γ + ε ; (3) y = α + α + Z γ + ε. Le α *,α *,γ *,...,γ K * be he maxmum lkelhood or leas squares esmaors for he arameers ha aear n (2) and (3). Then leng e and e be he vecors of leas squares resduals for equaons (2) and (3) resecvely, he followng equaons wll be sasfed by he arameer esmaes and he daa: (4) y = α * + α * + Z γ * + e ; (5) y = α * + α * + Z γ * + e. Le y [y T,y T ] T and e [e T,e T ] T and defne φ * [α *,α *,γ *,...,γ K * ] T. Now rewre (4) and (5) as (6) y = Xφ * + e. I s well known ha he columns of he X marx are orhogonal o he vecor e of leas squares resduals;.e., we have (7) X T e = X T [y Xφ * ] = 2+K. The frs wo equaons n (7) are equvalen o he followng wo equaons: 8 (8) T y = N()α * + T Z γ * ; (9) T y = N()α * + T Z γ *. Equaons (8) and (9) can be used o solve for he followng erod and consan qualy log rce levels: () α * = { T y /N()} { T Z γ * /N()} = T [y Z γ * ]/N() ; () α * = { T y /N()} { T Z γ * /N()} = T [y Z γ * ]/N(). Noe ha T y /N() s he arhmec average of he log rces n erod for =,. Furhermore, noe ha T Z /N() s he arhmec average of he amouns of each characersc ha are resen n he erod models and T Z /N() s he corresondng arhmec average amoun of each characersc ha s resen n he erod models. Thus each α * s equal o he average of he log rces for he models resen n erod less a qualy adjusmen conssng of he nner roduc of he characersc rces γ * wh he average amoun of each characersc across he models ha are resen n erod. Alernavely, he second se of equales n equaons () and () shows ha each α * s equal o he arhmec average of he qualy adjused log rces, y Z γ *, for he models 7 Our mehod of arameerzaon s equvalen o he sandard mehod for arameerzng a me dummy hedonc regresson model whch s o have a common consan erm for he wo erods and a me dummy varable for he second erod regresson. 8 Dewer (23a; 335) (23b; 39) and Slver and Herav (25) used hs orhogonaly mehod of roof o rovde an nerreaon of he hedonc me dummy n erms of qualy adjused rces.

5 5 resen n ha erod. In any case, he (unweghed) hedonc me dummy esmae of he change n log rces gong from erod o, LP HD, s he followng dfference n he log rce levels: 9 (2) LP HD α * α *. For laer reference, we work ou an exresson for he esmaed characersc rces, γ *. Recall equaon (6), y = Xφ * + e, whch defned he N()+N() by 2+K marx X. We rewre X as follows: (3) X = [V,Z] where Z T [Z T,Z T ] and V s an N()+N() by 2 marx whch has he frs column equal o [ T, T ] T and second column equal o [ T, T ] T. Now solve he leas squares mnmzaon roblem ha corresonds o (6) n wo sages. In he frs sage, we condon on γ and mnmze wh resec o he comonens of α [α,α ] T. The resulng condonal leas squares esmaor for α s: (4) α(γ) (V T V) V T [y Zγ]. The second sage mnmzaon roblem s he roblem of mnmzng f(γ) wh resec o he comonens of γ where f s defned as follows: (5) f(γ) [y Zγ Vα(γ)] T [y Zγ Vα(γ)] = [y Zγ V(V T V) V T (y Zγ)] T [y Zγ V(V T V) V T (y Zγ)] usng (4) = [My MZγ] T [My MZγ] = [y Zγ] T M T M[y Zγ] = [y Zγ] T M[y Zγ] snce M = M T and M 2 = M where he rojecon marx M s defned as follows: (6) M I V(V T V) V T. A smle way o solve he roblem of mnmzng f(γ) wh resec o γ s o make use of he hrd equaly n (5);.e., defne he rojecons of y and Z ono M as follows: (7) y * My ; Z * MZ. Usng defnons (7), can be seen ha (8) f(γ) = [y * Z * γ] T [y * Z * γ]. Thus he soluon o he second sage leas squares mnmzaon roblem s: (9) γ * (Z *T Z * ) Z *T y *. Once γ * has been deermned by (9), hen we can use (4) or (8) and (9) o deermne he leas squares esmaors for α * and α *. 9 Ths mehodology can be raced back o Cour (939; 9-) as hs hedonc suggeson number wo. Noe also ha f he models are he same n he wo erods beng consdered, hen N() equals N() (equals N say) and Z equals Z, so ha he wo characerscs marces are dencal and hus α * α * = T [y Z γ * ]/N() T [y Z γ * ]/N() = T y /N T y /N, whch s he arhmec mean of he erod log rces less he arhmec mean of he erod log rces. Thus under hese condons, here s no need o run a hedonc regresson; he usual mached model mehodology can be used.

6 6 Usng he defnon of V, can be shown ha he rojecon marx M defned by (6) s block dagonal, wh he wo man dagonal blocks M and M defned as follows: (2) M I T /N() ; M I T /N() where I and I are deny marces of dmenson N() and N() resecvely. Usng (2), we can deermne more recsely wha he vecor y * equal o My and he marx Z * equal o MZ look lke. Le y *T = [y *T,y *T ] and Z *T = [Z *T,Z *T ]. Then usng (2), we have: (2) y * = y T y /N() ; =, ; (22) Z * = Z T Z /N() ; =,. Thus each rojeced vecor y * s equal o he corresondng erod log rce vecor y less a vecor of ones mes he average of he log rces for erod, n= N() y n /N(), and each rojeced marx Z * s equal o he corresondng erod characerscs marx Z less a column vecor of ones mes a row vecor equal o he average of he characerscs n each model for erod, [ n= N() z n /N(), n= N() z n2 /N(),..., n= N() z nk /N()]. Thus y * and Z * are smly he corresondng y and Z wh he erod means subraced from each comonen. Usng he block dagonal srucure of M, can be verfed ha we have he followng alernave reresenaon for he leas squares characerscs rces γ * defned by (9): (23) γ * (Z *T Z * + Z *T Z * ) [Z *T y * + Z *T y * ]. We now urn our aenon o hedonc muaon ndexes. 3. Unweghed Hedonc Imuaon Indexes Insead of runnng one hedonc regresson where he same characerscs rces are used o qualy adjus rces n each erod, we can run wo enrely searae hedonc regressons wh searae characerscs rces, γ n erod and γ n erod. Thus usng he same noaon as n secon 2, our models now are: (24) y = β + Z γ + η ; (25) y = β + Z γ + η where η and η are ndeendenly dsrbued normal random varables wh means zero and consan varance whn each erod. Le β *,γ *,...,γ K * be he maxmum lkelhood or leas squares esmaors for he arameers ha aear n (24) and le β *,γ *,...,γ K * be he maxmum lkelhood or leas squares esmaors for he arameers ha aear n (25). Then leng u and u be he vecors of leas squares resduals for equaons (24) and (25) resecvely, he followng equaons wll be sasfed by he arameer esmaes and he daa: (26) y = β * + Z γ * + u ; (27) y = β * + Z γ * + u. The counerars o equaons (8) and (9) n he resen conex are: (28) T y = N()β * + T Z γ * ; (29) T y = N()β * + T Z γ *. Equaons (28) and (29) lead o he followng counerars o equaons (8) and (9):

7 7 (3) β * = { T y /N()} { T Z γ * /N()} = T [y Z γ * ]/N() ; (3) β * = { T y /N()} { T Z γ * /N()} = T [y Z γ * ]/N(). Recall ha he hedonc me dummy esmae of he change n log rces gong from erod o, LP HD, was defned by (2) as he dfference n he log rce levels, α * α *. In he resen conex, we canno smly ake he dfference beween β * and β * as a measure of consan qualy log rce change beween erods and, because he qualy adjusmen arameers, γ * and γ *, are dfferen beween he wo erods. However, we can use he erod arameers, γ *, o form esmaes of qualy adjused log rces for he models resen n erod and hen ake he average of he resulng qualy adjused log rces, whch we denoe by δ * : (32) δ * { T y /N()} { T Z γ * /N()} = T [y Z γ * ]/N(). Noe ha he above esmae of a erod log rce level s analogous o β * defned by (3) exce ha he erod hedonc qualy adjusmen facors, γ *, are used n (32) whereas he erod hedonc qualy adjusmen facors, γ *, were used n (3). Snce he erod and esmaed rce levels, β * and δ *, use he same qualy adjusmen facors γ * n order o form consan qualy log rces n each erod, we can ake he dfference δ * less β * as a measure of qualy adjused log rce change beween erods and. We call hs hedonc muaon measure of log rce change a Laseyres ye measure of rce change and denoe by φ * L δ * β *. Ths measure of overall log rce change deends asymmercally on he characerscs rce vecor γ * ha was obaned from he erod hedonc regresson. I can be seen ha we can oban an alernave measure of log rce change beween he erods usng he erod hedonc regresson characerscs rce vecor γ *. Thus use he erod characerscs rce vecor, γ *, o form esmaes of qualy adjused log rces for he models resen n erod and hen ake he average of he resulng qualy adjused log rces, whch we denoe by δ * : (33) δ * { T y /N()} { T Z γ * /N()} = T [y Z γ * ]/N(). Noe ha he above esmae of a erod log rce level s analogous o β * defned by (3) exce ha he erod hedonc qualy adjusmen facors, γ *, are used n (33) whereas he erod hedonc qualy adjusmen facors, γ *, were used n (3). Snce he erod and esmaed rce levels, δ * and β * use he same qualy adjusmen facors γ * n order o form consan qualy log rces n each erod, we can ake he dfference β * less δ * as a second measure of log rce change beween erods and. We call hs hedonc muaon measure of log rce change a Paasche ye measure of rce change and denoe by φ * P β * δ *. Followng Grlches (97b; 7) and Dewer (23b; 2), seems referable o ake a symmerc average of he above wo esmaes of log rce change over he wo erods. We choose he arhmec mean 2 as our symmerc average and defne he (unweghed) hedonc muaon esmae of he change n log rces gong from erod o, LP HI, as follows: 3 Ths basc dea can be raced back o Cour (939; 8) as hs hedonc suggeson number one. Hs suggeson was followed u by Grlches (97a; 59-6) (97b; 6) and Trle and McDonald (977; 44). In Aendx 2, we develo a smle mehod for obanng aroxmae sandard errors for he hedonc muaon Laseyres and Paasche measures of log rce change beween he wo erods. 2 If we chose o measure rce change nsead of log rce change, hen he arhmec mean esmaor of log rce change convers no a geomerc mean of he wo measures of level rce change, ex[δ * β * ] and ex[β * δ * ]. 3 In Aendx, we show ha he measure of qualy adjused change n log rces defned by (34), whch followed he mehodology due orgnally o Cour (939; 8), can also be nerreed as a measure of he dsance beween he wo hedonc regressons. The rncles of such measures were dscussed n Grlches (967) and Dhrymes (97; -2) and furher develoed by Feensra (995) and Dewer (23a; ). Emrcal sudes usng hs aroach nclude

8 8 (34) LP HI (½)φ L * + (½)φ P * = ½[δ * β * ] + ½[β * δ * ] = ½{ T [y Z γ * ]/N() T [y Z γ * ]/N() + T [y Z γ * ]/N() T [y Z γ * ]/N()} usng (3)-(33) = T {y Z [½γ * +½γ * ]}/N() T {y Z [½γ * +½γ * ]}/N(). Recall ha n he hedonc me dummy mehod for qualy adjusng log rces y for each erod, we used he characerscs qualy adjusmens defned by Z γ *, where γ * was a consan across erods vecor of qualy adjusmen facors. Lookng a he rgh hand sde of (34), can be seen ha he hedonc muaon mehod for qualy adjusng log rces n each erod s smlar bu now he erod vecor of qualy adjusmens s Z [½γ * +½γ * ] nsead of Z γ *. Thus for he hedonc muaon mehod of qualy adjusng rces, he average of he wo searae hedonc regresson esmaed qualy adjusmen facors, ½γ * +½γ *, relaces he sngle regresson esmaed qualy adjusmen facor vecor γ * ha was used n he hedonc me dummy mehod for qualy adjusmen. Noe also ha f he models are he same n he wo erods beng consdered, hen N() equals N() (equals N say) and Z equals Z, so ha he wo characerscs marces are dencal, hen LP HI defned by (34) collases o T y /N T y /N, whch s he arhmec mean of he erod log rces less he arhmec mean of he erod log rces. Thus under hese condons, here s no need o run hedonc regressons; he usual mached model mehodology can be used. Usng (2) and (34), we can form he followng exresson for he dfference n he overall log rce change usng he wo mehods for qualy adjusmen: (35) LP HD LP HI = T Z γ * /N() + T Z γ * /N() + T Z [½γ * +½γ * ]/N() T Z [½γ * +½γ * ]/N() = T Z [½γ * +½γ * γ * ]/N() T Z [½γ * +½γ * γ * ]/N() = {[ T Z /N()] [ T Z /N()]}{½γ * +½γ * γ * }. Thus he hedonc me dummy and hedonc muaon measures of log rce change wll be dencal f eher of he followng wo condons are sasfed: (36) T Z /N() = T Z /N() ; (37) γ * = ½γ * +½γ *. Condon (36) says ha he average amoun of each characersc for he models resen n erod equals he corresondng average amoun of each characersc for he models resen n erod. Condon (37) says ha he me dummy vecor of qualy adjusmen facors, γ *, s equal o he arhmec average of he wo searae hedonc regresson esmaes for he qualy adjusmen facors, ½γ * +½γ *. Condon (36) s somewha unancaed. I ells us ha f he average amoun of characerscs resen n he models n each erod does no change much, hen he hedonc me dummy and hedonc muaon esmaes of qualy adjused rce change wll be much he same, even f characersc valuaons change over he wo erods. Condon (37) can be refned. Recall (23) n he revous secon, whch rovded a formula for he hedonc me dummy vecor of qualy adjusmen facors, γ *. The echnques ha were used o Bernd, Grlches and Raaor (995), Ioannds and Slver (999), Bernd and Raaor (2; 27), Koskmäk and Vara (2) and Slver and Herav (23) (27b).

9 9 esablsh (23) can be used n order o esablsh he followng exressons for he erod and leas squares esmaes γ * and γ * ha aear n (26) and (27): (38) γ * = (Z *T Z * ) Z *T y * ; γ * = (Z *T Z * ) Z *T y * where he y * and Z * are he demeaned y and Z as n he revous secon. 4 Now remully boh sdes of (23) by he marx Z *T Z * + Z *T Z * and we oban he followng equaon: 5 (39) [Z *T Z * + Z *T Z * ]γ * = Z *T y * + Z *T y * = Z *T Z * [Z *T Z * ] Z *T y * + Z *T Z * [Z *T Z * ] Z *T y * = Z *T Z * γ * + Z *T Z * γ * usng (38). Equaon (39) ells us ha f γ * equals γ *, hen γ * s necessarly equal o hs common vecor. We now use equaon (39) n order o evaluae he followng exresson: (4) 2[Z *T Z * + Z *T Z * ][½γ * +½γ * γ * ] = [Z *T Z * + Z *T Z * ][γ * + γ * ] 2[Z *T Z * γ * + Z *T Z * γ * ] usng (39) = [Z *T Z * ][γ * γ * ] [Z *T Z * ][γ * γ * ] = [Z *T Z * Z *T Z * ][γ * γ * ]. Now remully boh sdes of (4) by (/2)[Z *T Z * + Z *T Z * ] and subsue he resulng exresson for ½γ * +½γ * γ * no equaon (35) n order o oban he followng exresson for he dfference beween he hedonc dummy esmae of consan qualy rce change and he corresondng symmerc hedonc muaon esmae: (4) LP HD LP HI = (/2){[ T Z /N()] [ T Z /N()]}[Z *T Z * + Z *T Z * ] [Z *T Z * Z *T Z * ][γ * γ * ]. Usng (4), can be seen ha he hedonc me dummy and hedonc muaon measures of log rce change wll be dencal f any of he followng hree condons are sasfed: T Z /N() equals T Z /N() so ha he average amoun of each characersc across models n each erod says he same or Z *T Z * equals Z *T Z * so ha he model characerscs oal varance covarance marx s he same across erods 6 or γ * equals γ * so ha searae (unweghed) hedonc regressons n each erod gve rse o he same characerscs qualy adjusmen facors. 7 In he followng secons, we wll ada he above maeral o cover he case of weghed hedonc regressons. 4. Weghed Tme Dummy Hedonc Regressons 4 Noe ha Z *T y * /N() can be nerreed as a vecor of samle covarances beween he log rces n erod and he amouns of he characerscs resen n he erod models whle Z *T Z * /N() can be nerreed as a samle varance covarance marx for he model characerscs n erod. In he man ex, we refer o Z *T Z * as a characerscs oal varance covarance marx. 5 In he sngle characersc case, (4) ells us ha γ * s a weghed average of γ * and γ *. 6 Ths condon for equaly s also somewha unancaed. 7 Usng (38), we can oban a more fundamenal condon n erms of varance covarance marces for he equaly of γ * o γ * ; namely he equaly of (Z *T Z * ) Z *T y * o (Z *T Z * ) Z *T y *.

10 We now consder a weghed wo erod me dummy varable hedonc regresson model. We agan assume ha here are N() observaons on he rces, n, of varous models n n erod for =, bu we now assume ha he quanes urchased for each model n n erod, q n, are also observable. Model n n erod agan has he vecor of K characerscs assocaed wh, [z n,z n2,...,z nk ] for =, and n =,2,...,N(). The exendure share of model n n erod s (42) s n n q n / = N() q ; =, ; n =,2,...,N(). Le s [s,...,s N() ] T denoe he erod exendure share vecor for =, and le S denoe he dagonal erod share marx ha has he elemens of he erod exendure share vecor s runnng down he man dagonal for =,. The marx (S ) /2 s he square roo marx of S ;.e., he osve square roos of he elemens of he erod exendure vecor s are he dagonal elemens of hs dagonal marx for =,. As n he revous secons, y s he vecor of erod log rce changes and Z s he erod model characerscs marx for =,. Wh he above noaonal relmnares ou of he way, he weghed me dummy regresson model ha s he counerar o he unweghed model defned earler by (2) and (3) s defned as follows: 8 (43) (S ) /2 y = (S ) /2 [ α + α + Z γ] + ε ; (44) (S ) /2 y = (S ) /2 [ α + α + Z γ] + ε where he ε vecors have elemens ε n ha are ndeendenly dsrbued normal varables wh zero means and consan varances. Le α *,α *,γ *,...,γ K * be he maxmum lkelhood or leas squares esmaors for he arameers ha aear n (43) and (44). Then leng e and e be he vecors of leas squares resduals for equaons (43) and (44) resecvely, he followng equaons wll be sasfed by he arameer esmaes and he daa: (45) (S ) /2 y = (S ) /2 [ α * + α * + Z γ * ] + e ; (46) (S ) /2 y = (S ) /2 [ α * + α * + Z γ * ] + e The counerars o equaons () and () are now: (47) α * = s T y s T Z γ * = s T [y Z γ * ] ; (48) α * = s T y s T Z γ * = s T [y Z γ * ]. Noe ha s T y = n= N() s n ln n s he erod exendure share weghed average of he log rces n erod for =,. Furhermore, noe ha s T Z s a by K vecor whose kh elemen s equal o he erod exendure share weghed average n= N() s n z nk of he amouns of characersc k ha are resen n he erod models for =, and k =,...,K. Thus each α * s equal o he exendure share weghed average of he log rces for he models resen n erod less a qualy adjusmen conssng of he nner roduc of he characersc rces γ * wh an exendure share weghed average amoun of each characersc across he models ha are resen n erod. Alernavely, he second se of equales n equaons (47) and (48) shows ha each α * s equal o he erod 8 Dewer (23b; 26) exlaned he logc behnd he weghng scheme n he regresson model defned by (43) and (44). For addonal maeral on weghng n hedonc regressons, see Dewer (25; 563) (26; 3). Bascally, he form of weghng ha s used n (43) and (44) leads o measures of rce change ha are comarable (n he case where he characerscs of he models can be defned by dummy varables) o he bes measures of rce change n blaeral ndex number heory. I should be noed ha he resen weghed model wll be equvalen o he revous unweghed model (4) and (5) only f he number of observaons n each erod are equal, so ha N() equals N(), and each share comonen s n s equal o a common value.

11 exendure share weghed average of he qualy adjused log rces, y Z γ *, for he models resen n ha erod. Now use (47) and (48) n order o defne he followng weghed hedonc me dummy esmae of he change n log rces gong from erod o : LP WHD, s he followng dfference n he log rce levels: (49) LP WHD α * α * = s T [y Z γ * ] s T [y Z γ * ]. Thus he weghed hedonc me dummy esmae of he change n log rces s equal o a erod exendure share weghed average of he qualy adjused log rces, y Z γ *, less a erod exendure share weghed average of he qualy adjused log rces, y Z γ *. For laer reference, we can ada he mehodology resened a he end of secon 2 n order o work ou an exlc exresson for he esmaed characersc rces, γ *. Recall he defnons of he dagonal marces S and S. Defne S as a block dagonal marx ha has he blocks S and S on he man dagonals and le S /2 be he corresondng square roo marx;.e., hs marx has he elemens of he erod exendure share s and he erod exendure share vecor s runnng down he man dagonal. Adang he analyss n secon, we need o relace he y vecor n ha secon by S /2 y and he Z marx n ha secon by S /2 Z, he V marx by S /2 V and defne he counerar M o he rojecon marx M defned by (6) as follows: (5) M I S /2 V(V T SV) V T S /2 = I S /2 VV T S /2 where he second equaly n (5) follows from he fac ha V T SV equals I 2, a wo by wo deny marx. Now defne y and Z n erms of M and he orgnal y vecor and Z marx as follows: (5) y M S /2 y = [I S /2 VV T S /2 ]S /2 y usng (5) = S /2 [I VV T S]y ; (52) Z M S /2 Z = [I S /2 VV T S /2 ]S /2 Z usng (5) = S /2 [I VV T S]Z. The new vecor of me dummy qualy adjusmen facors γ * can now be defned as he followng counerar o (9): (53) γ * (Z T Z ) Z T y. Once γ * has been deermned by (53), hen we can use (47) and (48) o deermne he weghed leas squares esmaors for α * and α *. I s ossble o exress he γ * defned by (53) n a more ransaren way usng our defnons of he marces Z, V and S. Recall ha n secon 2, he demeaned log rce change vecors y * defned by (2) and he demeaned characerscs marces Z * defned by (2) roved o be useful. In hose defnons, we used smle unweghed means. In he resen conex, we use exendure share weghed average means as follows: (54) y * y s T y ; =, ; (55) Z * Z s T Z ; =,.

12 2 Le he nh comonen of he vecor y * be y n * and le he nh row of he N() by K dmensonal marx Z * be z n * for =, and n =,2,...,N(). Then can be shown ha he vecor of characerscs rces γ * defned by (53) can be wren n erms of he comonens of he demeaned log rce change vecors y *, he comonens of he exendure share vecors s and he rows of he demeaned characerscs marces Z * as follows: (56) γ * = [ n= N() s n z n *T z n * + n= N() s n z n *T z n * ] [ n= N() s n z n *T y n * + n= N() s n z n *T y n * ]. Noe ha n= N() s n z n *T z n * can be nerreed as a erod exendure share weghed samle varance covarance marx 9 for he model characerscs resen n erod and n= N() s n z n *T y n * can be nerreed as a erod exendure share weghed samle covarance marx beween he erod log rces and he characerscs of he models resen n erod. We now generalze he analyss on unweghed hedonc muaon ndexes resened n secon 3 o he weghed case. 5. Weghed Hedonc Imuaon Indexes Usng he noaon exlaned n he revous secon, he wo searae weghed hedonc regressons ha are counerars o he searae unweghed regressons (24) and (25) are now (57) and (58) below: (57) (S ) /2 y = (S ) /2 [ β + Z γ ] + η ; (58) (S ) /2 y = (S ) /2 [ β + Z γ ] + η where η and η are ndeendenly dsrbued normal random varables wh means zero and consan varance whn each erod. Le β *,γ *,...,γ K * be he maxmum lkelhood or leas squares esmaors for he arameers ha aear n (57) and le β *,γ *,...,γ K * be he maxmum lkelhood or leas squares esmaors for he arameers ha aear n (58). Then leng u and u be he vecors of leas squares resduals for equaons (57) and (58) resecvely, he followng equaons wll be sasfed by he arameer esmaes and he daa: (59) (S ) /2 y = (S ) /2 [ β * + Z γ * ] + u ; (6) (S ) /2 y = (S ) /2 [ β * + Z γ * ] + u. The weghed counerars o equaons (3) and (3) are: (6) β * = s T y s T Z γ * = s T [y Z γ * ] ; (62) β * = s T y s T Z γ * = s T [y Z γ * ] As n secon 3, we canno smly ake he dfference beween β * and β * as a measure of consan qualy log rce change beween erods and, because he qualy adjusmen arameers, γ * and γ *, are agan dfferen beween he wo erods. As n secon 3, we can use he erod arameers, γ *, o form esmaes of qualy adjused log rces for he models resen n erod and hen ake he erod weghed average of he resulng qualy adjused log rces, whch we denoe by δ * : (63) δ * s T y s T Z γ * = s T [y Z γ * ]. 9 Noe ha for he resen weghed model, n= N() s n z n *T z n * s a rue erod exendure share weghed samle varance covarance marx, whereas for he earler unweghed model, he counerar o hs samle varance covarance marx was Z *T Z *, whch was he erod oal varance covarance marx, equal o N() mes he samle varance covarance marx for he characerscs resen n erod models.

13 3 Noe ha he above esmae of a erod log rce level s analogous o β * defned by (62) exce ha he erod hedonc qualy adjusmen facors, γ *, are used n (63) whereas he erod hedonc qualy adjusmen facors, γ *, were used n (62). Snce he erod and esmaed rce levels, β * and δ *, use he same qualy adjusmen facors γ * n order o form consan qualy log rces n each erod, we can agan ake he dfference δ * less β * as a measure of qualy adjused log rce change beween erods and. 2 Ths measure of overall log rce change deends asymmercally on he characerscs rce vecor γ * ha was obaned from he erod hedonc regresson. I can be seen ha we can oban an alernave measure of log rce change beween he erods usng he erod hedonc regresson characerscs rce vecor γ *. Thus use he erod characerscs rce vecor, γ *, o form esmaes of qualy adjused log rces for he models resen n erod and hen ake he erod weghed average of he resulng qualy adjused log rces, whch we denoe by δ * : (64) δ * s T y s T Z γ * = s T [y Z γ * ]. Noe ha he above esmae of a erod log rce level s analogous o β * defned by (6) exce ha he erod hedonc qualy adjusmen facors, γ *, are used n (64) whereas he erod hedonc qualy adjusmen facors, γ *, were used n (6). Snce he erod and esmaed rce levels, δ * and β * use he same qualy adjusmen facors γ * n order o form consan qualy log rces n each erod, we can ake he dfference β * less δ * as a second measure of log rce change beween erods and. 2 Agan followng Dewer (23b; 2) and Haan (23; 4) (24), seems referable o ake a symmerc average of he above wo esmaes of log rce change over he wo erods. We agan choose he arhmec mean as our symmerc average and defne he weghed hedonc muaon esmae of he change n log rces gong from erod o, LP WHI, as follows: 22 (65) LP WHI ½[δ * β * ] + ½[β * δ * ]. = ½{s T [y Z γ * ] s T [y Z γ * ] + s T [y Z γ * ] s T [y Z γ * ]} = s T {y Z [½γ * +½γ * ]} s T {y Z [½γ * +½γ * ]}. Recall ha n he weghed hedonc me dummy mehod for qualy adjusng log rces y for each erod, we used he characerscs qualy adjusmens defned by Z γ *, where γ * was a consan across erods vecor of qualy adjusmen facors. Lookng a he rgh hand sde of (65), can be seen ha he weghed hedonc muaon mehod for qualy adjusng log rces n each erod s smlar bu now he erod vecor of qualy adjusmens s Z [½γ * +½γ * ] nsead of Z γ *. Usng (49) and (65), we can form he followng exresson for he dfference n he overall log rce change usng he wo weghed mehods for qualy adjusmen: (66) LP WHD LP WHI = s T [y Z γ * ] s T [y Z γ * ] (s T {y Z [½γ * +½γ * ]} s T {y Z [½γ * +½γ * ]}) 2 Haan (23; 2) defned he exonenal of hs measure of log rce change as he geomerc Laseyres hedonc muaon ndex of rce change, exce ha for he mached models n boh erods, he used acual rces raher han redced rces. 2 Haan (23; 3) defned he exonenal of hs measure of log rce change as he geomerc Paasche hedonc muaon ndex of rce change, exce ha for he mached models n boh erods, he used acual rces raher han redced rces. Dewer (23b; 3-4) consdered smlar hedonc muaon ndexes exce ha he worked wh ordnary Paasche and Laseyres ye ndexes raher han geomerc Paasche and Laseyres ye ndexes. 22 The exonenal of hs measure of rce change s aroxmaely equal o Haan s (23; 4) geomerc mean of hs geomerc Paasche and Laseyres hedonc muaon ndexes, whch he regarded as an aroxmaon o Törnqvs hedonc muaon ndex.

14 = {s T Z s T Z }{½γ * +½γ * γ * }. 4 Thus he hedonc me dummy and hedonc muaon measures of log rce change wll be dencal f eher of he followng wo condons are sasfed: (67) s T Z = s T Z ; (68) γ * = ½γ * +½γ *. Condon (67) says ha he exendure share weghed average amoun of each characersc for he models resen n erod equals he corresondng exendure share weghed average amoun of each characersc for he models resen n erod. Condon (68) says ha he me dummy vecor of qualy adjusmen facors, γ *, s equal o he arhmec average of he wo searae weghed hedonc regresson esmaes for he qualy adjusmen facors, ½γ * +½γ *. As n secon 3, condon (68) can be srenghened. Recall (56) n he revous secon, whch rovded a formula for he hedonc me dummy vecor of qualy adjusmen facors, γ *. Usng he same noaon o ha used n he revous secon, we can esablsh he followng exressons for he erod and weghed leas squares esmaes γ * and γ * ha aear n (59) and (6): (69) γ * = [ n= N() s n z n *T z n * ] [ n= N() s n z n *T y n * ] ; (7) γ * = [ n= N() s n z n *T z n * ] [ n= N() s n z n *T y n * ]. where he y * and Z * are he demeaned y and Z as n he revous secon. Now remully boh sdes of (56) by he marx [ n= N() s n z n *T z n * + n= N() s n z n *T z n * ] and we oban he followng equaon whch s he weghed counerar o (4): (7) [ n= N() s n z n *T z n * + n= N() s n z n *T z n * ]γ * = n= N() s n z n *T y n * + n= N() s n z n *T y n * = [ n= N() s n z n *T z n * ][ n= N() s n z n *T z n * ] [ n= N() s n z n *T y n * ] + [ n= N() s n z n *T z n * ][ n= N() s n z n *T z n * ] [ n= N() s n z n *T y n * ] = [ n= N() s n z n *T z n * ]γ * + [ n= N() s n z n *T z n * ]γ * usng (69) and (7). Equaon (7) ells us f γ * equals γ *, hen γ * s necessarly equal o hs common vecor. We now use equaon (7) n order o evaluae he followng exresson: (72) 2[ n= N() s n z n *T z n * + n= N() s n z n *T z n * ][½γ * +½γ * γ * ] = [ n= N() s n z n *T z n * + n= N() s n z n *T z n * ][γ * + γ * ] 2{[ n= N() s n z n *T z n * ]γ * + [ n= N() s n z n *T z n * ]γ * } usng (7) = [ n= N() s n z n *T z n * ][γ * γ * ] [ n= N() s n z n *T z n * ][γ * γ * ] = [ n= N() s n z n *T z n * n= N() s n z n *T z n * ][γ * γ * ]. Now remully boh sdes of (72) by (/2)[ n= N() s n z n *T z n * + n= N() s n z n *T z n * ] and subsue he resulng exresson for ½γ * +½γ * γ * no equaon (66) n order o oban he followng exresson for he dfference beween he hedonc dummy esmae of consan qualy rce chan`ge and he corresondng symmerc hedonc muaon esmae usng weghed hedonc regressons n boh cases: (73) LP WHD LP WHI = [s T Z s T Z ][½γ * +½γ * γ * ] = ½[s T Z s T Z ][ n= N() s n z n *T z n * + n= N() s n z n *T z n * ] [ n= N() s n z n *T z n * n= N() s n z n *T z n * ][γ * γ * ].

15 5 Usng (73), can be seen ha he weghed hedonc me dummy and he weghed hedonc muaon measures of log rce change wll be dencal f any of he followng hree condons are sasfed: s T Z equals s T Z so ha he erod exendure share weghed amoun of each characersc across models n each erod says he same or n= N() s n z n *T z n * equals n= N() s n z n *T z n * so ha he exendure share weghed model characerscs varance covarance marx s he same n he wo erods or γ * equals γ * so ha searae (weghed) hedonc regressons n each erod gve rse o he same characerscs qualy adjusmen facors. Whch weghed mehod of qualy adjusmen s bes? If eher he weghed average amouns of each characersc are much he same n he wo erods beng consdered so ha s T Z s close o s T Z, or f he exendure share weghed model characerscs varance covarance marces are smlar across erods, or f he searae weghed hedonc regresson qualy adjusmen facors do no change much across he wo erods, hen wll no maer much whch mehod s used, whch s he new resul ha s demonsraed n hs aer. If however, s T Z s no close o s T Z, he exendure share weghed model characerscs varance covarance marces are dfferen across erods, and he wo searae weghed hedonc regressons (57) and (58) generae very dfferen esmaes for he qualy adjusmen facors, γ * and γ *, hen he mehod of qualy adjusmen could maer. If (57) and (58) are run ogeher as a ooled regresson model and an F es rejecs he equaly of γ and γ, hen seems sensble o use he weghed hedonc muaon mehod whch does no deend on havng γ equal o γ as does he hedonc me dummy mehod. If he F es does no rejec he equaly of γ and γ and here are a large number of characerscs n he model, hen valuable degrees of freedom wll be saved f he weghed me dummy hedonc regresson model s used. However, n hs case, snce γ and γ are necessarly close, should no maer much whch mehod s used. Thus seems ha he hedonc muaon mehods robably gve rse o beer qualy adjusmens han dummy varable mehods. We wll revs hs dscusson n secon 7 below. 6. Emrcal llusraon: desko ersonal comuers (PCs) The emrcal sudy s of he measuremen of changes n he qualy-adjused monhly rces of Brsh desko PCs n 998. The daa are monhly scanner daa from he bar-code readers of PC realers. The daa amouned o 7,387 observaons (a arcular make and model of a PC sold n a gven monh n an eher secalzed or non-secalzed PC sore-ye) reresenng a sales volume of.5 mllon models worh.57 bllon. Table shows ha for he January o February rce comarson here were 584 mached models avalable n boh monhs for he rce comarson. However, for he January o December rce comarson only 6 mached models were avalable wh 59 unmached old models (avalable n January, bu unmached n December) and 436 unmached new models (avalable n December bu unavalable n January for machng). 23 For roduc markes where here are a hgh rooron of unmached models, Slver and Herav (25) demonsraed why mached model ndexes suffer from samle selecvy bas and why hedonc ndexes should be used nsead. The calculaon of hedonc ndexes requres he esmaon of hedonc regresson equaons. To smlfy he llusraon we frs nclude only a sngle exlanaory varable n he hedonc (rce) regressons, he seed n MHz. The regressons were run searaely for each monh for he hedonc muaon ndexes, and over January and he curren monh, ncludng a dummy varable for he 23 Bear n mnd ha some of he ndexes esmaed n hs aer are also weghed by shares of sales values and ha he fall off n he coverage of he mached samle by sales s even more dramac: for he January o December comarson mached models made u only 7% of he January sales value and a mere 2% of he December sales value.

16 6 laer, for he hedonc dummy ndexes. The esmaed coeffcens for seed n he hedonc regressons were sascally sgnfcan coeffcens wh he execed osve sgns. 24 Table 2 columns () and (2) show falls n he unweghed hedonc dummy and hedonc muaon ndexes of 73.2 and 76.4 ercen resecvely, a dfference of 3.2 ercenage ons. Columns (4)-(6) show he consuen elemens of equaon (4) ha make u hs dfference: column (4) s he change n mean characerscs, [ T Z /N()] [ T Z /N()] average seed ncreased by 27 MHz. over he year; column (5) s he change n he (oal) varance-covarance characerscs marces relave o her sum n he wo erods, [Z *T Z * + Z *T Z * ] [Z *T Z * Z *T Z * ] wh one characersc n hs llusraon s he relave change n he varance of seed, fallng n some early monhs bu ncreasng hereafer; and column (6) s he change n he characersc arameer esmaes, [γ * γ * ] he esmaed arameers decreased over me. The decomoson of he dfference beween he hedonc dummy and muaon ndexes gven n column (3) s exac as demonsraed n column (7) he dfference s equal o one-half of he roduc of columns (4), (5) and (6), followng equaon (4). We sress ha he decomoson s based on he roduc of hese consuen ars. If eher of hese changes s zero, hen here wll be no dfference beween he ndexes. Whle s clear ha here were large ncreases n he mean seed of PCs over he year, column (4), hey dd no maeralze n subsanal dfference beween he formulas, beng emered by he smaller changes n columns (5) and (6). The formulaon also rovdes nsghs no he facors behnd any dfference n he resuls from hese mehods. For examle, n Seember and December he changes n he esmaed arameers were abou he same, ye he dfference beween he hedonc dummy and muaon ndexes n column (3) was hgher n December han n Seember, drven by he larger change n mean characerscs n December. The ndexes consdered n Table 2 were unweghed and hus unreresenave f models dffer n her oulary. Convenonal ndex number heory requres ha rce changes should be weghed by relave exendures shares and he same requremen should aly o hedonc ndexes. In Secons 4 and 5 above weghed hedonc dummy and hedonc muaon ndexes were formulaed and equaon (73) rovded a decomoson of he dfference beween hem. The resuls for PCs for weghed hedonc ndexes, her dfference, and he consuen elemens underlyng he dfference are gven n Table 3. Agan he dfference deends on he roduc of hree erms: he change n he exendure share weghed mean of each characersc, [s T Z s T Z ]; he change n he exendure share weghed characerscs varance-covarance marx, [ N() n= s n z *T n z * n + N() n= s n z *T n z * n ] mes [ N() n= s n z *T n z * n N() n= s n z *T n z * n ]; and he change n he arameers esmaes, [γ * γ * ] from he searae (weghed) hedonc regressons n each erod. Table 3 shows resecve falls for he weghed hedonc dummy and weghed hedonc muaon ndexes of 43.and 5.6 ercen (muaon) over he year, comared wh falls n he corresondng unweghed ndexes of 73.2 and 76.4 ercen n Table 2: hus weghng maers. The decomoson n Table 3 follows equaon (73) and s exac wh (one-half of) he roduc of columns (4) o (6), n column (7), equalng he dfference beween he formulas n column (3). The dfferences are a her hghes n Seember and December a over 7 ercenage ons, n ar due o he relavely hgh arameer changes. Agan comarng Seember wh December, he arameer changes n column (6) are smlarly hgh a.39. Bu agan he change n he mean seed n column (4) a 39.6 MHz. n December s much hgher han s Seember fgure of MHz. 25 Ths should ranslae no a very much lower dfference beween he formulas n Seember 24 The F-sascs for he null hyohess of coeffcens beng equal o zero averaged 34.2 for hedonc muaon ndexes and 53.4 for hedonc dummy ndexes, conssenly rejecng he null a a.% level and lower. The exlanaory ower of he esmaed equaons were naurally low for hs secfcaon wh a sngle exlanaory varable, esecally snce hey dd no nclude dummy varables on brand. Deals of esmaes from a fully secfed model are avalable from he auhors. 25 The mullcaon s of he change n average seed by he arameer esmae for seed. The resul s nvaran o he uns of measuremen used for seed snce an accordngly lower coeffcen would resul f he uns of measuremen for seed were say doubled.

17 7 comared wh December, bu for he weghed resuls does no. Ths s because he much hgher December change n characerscs, column (4), s largely offse by he much lower change n he relave varance n column (5). The emrcal examle above was lmed o a sngle rce-deermnng qualy characersc varable for llusrave uroses. The decomoson for more han one characersc s smlar n rncle o he case of a sngle exlanaory varable, bu he consuen ems of equaon (73) are marces and s he roduc of hese marces ha s requred o accoun for he dfference beween he formulas. The regresson esmaes are now based on hree qualy characerscs, seed n MHz., he hard dsk caacy (CAP), and random access memory (RAM), boh n MB. The esmaed coeffcens for he hree varables n he hedonc regressons were sascally sgnfcan wh he execed osve sgns. 26 The resul of he decomoson of he dfference beween weghed hedonc dummy and weghed hedonc muaon ndexes based on mulle characerscs for he January wh December comarson only (for ease of exoson) s resened below n he marx forma of equaon (73) and as Table 4. The weghed hedonc dummy ndex n December comared wh January, based on he exended secfcaon, fell by 5.5 ercen, comared wh he fall n he weghed hedonc muaon ndex of 63.3, a szable dfference of.8 ercenage ons. From Tables 2 and 3 we saw ha weghng maers. Here we denfy he morance of a fuller secfcaon of he hedonc regresson used and s effec on he magnude of he ndex change and he sread beween he wo esmaes. The fuller secfcaon has led o an ncrease n he sread beween he wo ndexes from 7.5 o.8 ercenage ons. Prces are esmaed o have fallen furher usng he exended varable se: a fall for he weghed hedonc dummy hedonc of 5.5 comared wh 43. and for he hedonc muaon ndex of 63.3 comared wh 5.6 ercen. Addonal exlanaory varables n a qualy adjused hedonc regresson based ndex are robably referable. 27 Ye n se of hs llusraon, a fuller secfcaon of he hedonc regresson need no necessarly lead o an ncrease n he dfference beween he formulas. If any comonen of an addonal varable s change n relave dserson, covarance, arameer, or mean value s neglgble, hen wll have lle effec on he dfference. Indeed, he overall mac of addonal varables n he roduc of marces n equaon (73) may ake a dfferen sgn and reduce he dscreancy. However, n general, roduc develomen n hgh echnology roducs such as PCs akes he form of ncreased roduc dfferenaon (dserson of characerscs values), mrovemens n many roduc dmensons a he same me (whch wll generally change means and covarances), and decreasng characersc roducon coss and margnal ules, as consumers realgn her references o he new sandards ( whch wll generally lead o changes n arameer esmaes). Thus our execaon s ha he HI and HD aroaches o measurng rce change may frequenly gve dfferen esmaes of overall rce change n dynamc markes. The four erms n equaon (73) are reroduced below as marces, and as Table 4, for he December wh January comarson, he roduc of whch s equal o he dfference beween he formulas,.e. ½[s T Z s T Z ][ n= N() s n z n *T z n * + n= N() s n z n *T z n * ] [ n= N() s n z n *T z n * ' 6,245.2 = ½[39.593, 8.4, 4.45] % % 5,422.8 % , , ,626. n= N() s n z n *T z n * ][γ * γ * ] $ 3,626. " ",3." 26 The adjused-r 2 were.4 and.3 for he hedonc regresson equaons n January and November resecvely. The secfcaon could be exended o nclude many more varables ncludng dummy varables for brands. The exoson here was smlfed o llusrae he decomoson for more han one exlanaory varable. 27 There are sascal caveas o hs saemen,.e., may no be useful o nclude addonal exlanaory varables f here are nsuffcen degrees of freedom or f mulcollneary beween ncluded and omed varables s srong.