How to better measure hedonic residential property price indexes

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1 WP/YY/XX How o beer measure hedonc resdenal propery prce ndexes by Mc Slver IMF Worng Papers descrbe research n progress by he auhor(s) and are publshed o elc commens and o encourage debae. The vews expressed n IMF Worng Papers are hose of he auhor(s) and do no necessarly represen he vews of he IMF, s Execuve Board, or IMF managemen.

2 YYYY Inernaonal Moneary Fund WP/[Paper o. ex. 4/xx] IMF Worng Paper Sascs Deparmen How o beer measure hedonc resdenal propery prce ndexes Prepared by Mc Slver Auhorzed for dsrbuon by Clauda Dzobe May 26 IMF Worng Papers descrbe research n progress by he auhor(s) and are publshed o elc commens and o encourage debae. The vews expressed n IMF Worng Papers are hose of he auhor(s) and do no necessarly represen he vews of he IMF, s Execuve Board, or IMF managemen. Absrac Hedonc regressons are used for propery prce ndex measuremen o conrol for changes n he qualy-mx of properes ransaced. The paper consoldaes he hedonc me dummy approach, characerscs approach, and mpuaon approaches. A praccal hedonc mehodology s proposed ha () s weghed a a basc level; () has a new (quas-) superlave form and hus mgaes subsuon bas; () s suable for sparse daa n hn mares; and (v) only requres he perodc esmaon of hedonc regressons for reference perods and s no subjec o he vagrances of msspecfcaon and esmaon ssues. JEL Classfcaon umbers: C43, E3, E3, R3. eywords: Hedonc Regressons; Resdenal Propery Prce Index; Commercal Propery Prce Index; House Prce Index; Superlave Index umber.] Auhor s E-Mal Address: mslver@mf.org mcslver@gmal.org

3 3 Conens Page I. Inroducon...5 II. Measures of hedonc consan-qualy propery prce change...5 A. Hedonc regressons...5 Funconal forms of he hedonc regresson: a lnear form...6 Funconal forms of he hedonc regresson: a log(arhmc)-lnear form...7 B. The me dummy varable approach....9 The mehod...9 Feaures of he mehod...2 Channg, rollng wndows and smoohng...4 C. The characerscs approach...7 Wha averages of characersc values o use? Means, medan, and represenave characersc values...8 Hedonc characerscs ndexes: a lnear funconal form...9 Types of hedonc characerscs ndexes: log-lnear funconal form...22 Dual mpuaons...24 D. The mpuaon approach...25 E. An ndrec approach o hedonc prce ndexes...28 F. Arhmec versus geomerc aggregaon: how much does maer?...3 On he mporance of a geomerc versus an arhmec hedonc formulaon.3 Duo s falure of he uns of measuremen (commensurably) es...32 So wha deermnes he dfference beween hedonc Duo and Jevons and when wll be mnmal?...34 III. Some equvalences...36 When mpuaon ndex equals characerscs ndex...36 Addvy...38 IV. Weghs and superlave hedonc prce ndexes...4 A. Lower-level weghs for a lnear/arhmec hedonc formulaons...42 Use of acual prces as weghs...45 Quas-superlave ndexes: Fsher ndexes...46 B. log-lnear hedonc model...47 Applcaon of explc reference and curren perod weghs: a hedonc quas- Törnqvs prce ndex...48 C. The naure of subsuon bas for a hedonc prce ndex...5 D. Hedonc superlave ndexes and sample selecon bas...5 E. Hedonc superlave prce ndex number formulas: Hll and Melser (28)...53 F. Weghs for he me dummy approach...56 Leverage effecs and he need for ouler deecon and robus esmaors...57 G. Soc weghs...6

4 4 V. Hedonc propery prce ndexes seres: perodc rebasng, channg and rollng wndows.62 On perodc lnng and channg...63 VI. A praccal choce of formula: equvalences, nfrequen hedonc esmaon, weghng, hn mares, and he ndrec approach...65 A. Praccal problem of approprae hedonc formulas for hn mares Arhmec mplc weghs and quas-fsher ndexes...69 Geomerc explc weghs and quas-törnqvs ndexes...7 VII. Summary...76 Annex A: dfference beween hedonc arhmec and geomerc mean propery prce ndexes Annex B: Oulers and leverage effecs on coeffcen esmaes....8 Annex C: Equang an esmaed coeffcen on a me dummy from a log-lnear hedonc model o he geomerc mean of he prce changes References...85

5 5 I. ITRODUCTIO Ths paper examnes, consoldaes, and provdes mproved praccal mehods for he mely esmaon of hedonc resdenal propery prce ndexes (RPPIs), hough he proposed mehods apply equally o commercal propery prce ndexes (CPPIs). Hedonc regressons are he man mechansm recommended for and used by counres for a crucal aspec of RPPI esmaon prevenng changes n he qualy-mx of properes ransaced ranslang o prce changes. RPPIs and CPPIs are hard o measure. Houses, never mnd commercal properes, are nfrequenly raded and heerogeneous. Average house prces may ncrease over me, bu hs may n par be due o a change n he qualy-mx of he houses ransaced; for example, more 4- bedroom houses n a beer (more expensve) pos-code ransaced n he curren perod compared wh he prevous or some dsan reference perod would bas upwards a measure of change n average prces. A purpose and crucal challenge of RPPIs and CPPIs s o preven changes n he qualy-mx of properes ransaced ranslang o measured prce changes. The need s o measure consan-qualy propery prce changes and whle here are alernave approaches, he concern of hs paper s wh he hedonc approach as a recommended wdely used mehodology for hs. 2 Alernaves nclude repea sales, mx-adjusmen by weghng more-homogeneous sraa, and he sales prce apprasal rao (SPAR) are furher alernaves. Each s used o form consan qualy prce ndexes, SPAR by usng he relaonshp beween apprasal and ransacon prces where such daa coexs and predcng ransacon prces. Repea sales only uses ha par of he daase where here s more han one ransacon over a gven perod. Deals of all hese mehods are gven n Eurosa e al. (23); see also Hll (23) for a survey of hedonc mehods for resdenal propery prce ndexes; Slver and Herav (27a) and Dewer, Herav, and Slver (29) on hedonc mehods; Dewer and Shmzu (23b) and Shmzu e al. (2) for applcaons o Toyo; and Shller (99, 993, and 24) on repea-sales mehodology., hough see Case, Pollaows, and Wacher (23) for a hybrd repea sales and hedonc model appled o house prce ndexes. 2 Hll (23, 96) concludes hs survey paper: Hedonc ndexes seem o be gradually replacng repea sales as he mehod of choce for consrucng qualy-adjused house prce ndexes. Ths rend can be arbued o he nheren weanesses of he repea sales mehod (especally s deleon of sngle-sales daa and poenal lemons bas) and a combnaon of he ncreasng avalably of dealed daa ses of house prces and characerscs, ncludng geospaal daa, ncreases n compung power, and he developmen of more sophscaed hedonc models ha n parcular ae accoun of spaal dependence n he daa.

6 3 The am of hs paper s o furher develop a bes pracce mehodology grounded n boh he praccal consderaons and mehodologcal rgor requred for such an mporan sasc. The mehodology s conssen wh, bu exends he provsons n, he 23 Handboo on RPPIs (Eurosa e al., 23) ha form he nernaonal sandards n hs area. The hedonc approach denfes properes as ed bundles of characerscs. The characerscs are he prce-deermnng ones, ncludng sze of propery, number of bedrooms, locaon and so forh, and he sense n whch hey are ed s ha he characerscs are no sold separaely here s no prce n he mare for each characersc, only one for he house, srucure and land, as a whole. Were here a prce for a say addonal bahroom, and houses ransaced n he curren perod had more bahrooms, on average, we would have he means by whch consan qualy propery prce changes could be esmaed. A hedonc regresson of propery prces on propery characerscs provdes he means by whch we can unbundle he overall prce and arbue esmaed margnal values o he ndvdual characerscs. Ths paper acles he mporan queson as o how, gven esmaed hedonc regressons, do we bes comple hedonc, consan-qualy, propery prce ndexes? The Handboo on Resdenal Propery Prce Indces (RPPIs) (Eurosa e al., 23) provdes nernaonal gudelnes on RPPI measuremen and chaper 5 conans hree hedonc approaches he hedonc me dummy approach, characerscs approach, and mpuaon approach. Ths follows prevous leraure n hs area ncludng Trple (26), Slver and Herav (27a) and Hll (23). A problem s ha here are many alernave forms for each approach dependng on whch perod esmaed hedonc coeffcens, characersc bases, and weghs are held consan, wheher dual or sngle mpuaon s used for eher prces or weghs, a drec of ndrec formulaon s used, chaned, rollng wndow or fxed bases of characercs, and more. We frs oulne n secon II he alernave approaches o hedonc propery prce ndexes o ground he analyss. Throughou he paper hs s underaen for boh lnear and log-lnear hedonc specfcaon. In secon III we demonsrae, for reasonable specfcaons of hedonc regressons, equvalences beween he approaches and consoldae hem o show ha hedonc

7 4 mpuaon and characerscs approaches yeld he same resul and he me dummy can be formulaed as beng a close approxmaon. The resulng formulas benef from beng jusfed by he dfferen nuons of he approaches. In secon III we devse a weghng sysem for propery prce change a he elemenary level, n hs case for he prce change of each ndvdual propery an ssue hghlghed by Dewer (25a). Ths s underaen for he hedonc mpuaon approach bu, due o he equvalences of he approaches, can also be mrrored n he characerscs approach o gve he same resul. Whle arhmec (lnear) formulaon has a foruous mplc weghng sysem; however he log-lnear (geomerc) prce ndex equally weghs propery prce changes. We develop for he log-lnear (geomerc) case a means by whch explc weghs can be readly appled. Havng done so, a naural nex sep s o defne a superlave hedonc prce ndex ha maes symmerc use of reference perod and curren perod weghs. Ths s underaen n wo seps by defnng hedonc quas-superlave and re-defnng hedonc superlave propery prce ndexes, o advance on exsng formulaons n he leraure of hese arge measures. Praccal problems are consdered arsng ou of a concern wh hn mares sparse ransacon prce daa. Conrollng for he effec of heerogeneous properes requres a concoman generous hedonc specfcaon and care wh esmaon ha s ll-served by frequen reesmaon usng sparse daa. I s parcularly mporan o ground he hedonc prce comparsons n a reference perod ha s relavely exhausve of he propery mx ha arses n subsequen perods. The concern of he proposed mehods are for parsmony of esmaon, ha s o no rely on esmaes n successve perods and ha beer formulaed o deal wh sparse daa. In secon IV a useful praccal measure for counres s developed. The measure () benefs from a focus on he mpuaon approach, whch s conducve o weghng, whch provdes equvalen resul o he characerscs approach; () requres ha a hedonc regresson only be

8 5 run for he reference perod; 3 () beer accommodaes sparse ransacon daa n hn mares; (v) ncorporaes a quas-superlave weghng sysem a he elemenary level; (v) adops an ndrec approach o faclae he use of dual mpuaons bu also ads n nerpreaon; and (v) can be readly exended as a convenonal hedonc superlave ndex for rerospecve sudes. II. MEASURES OF HEDOIC COSTAT-QUALITY PROPERTY PRICE CHAGE A. Hedonc regressons The prce ndex number problem for real esae s ha measures of changes n he average prce of properes reflec n par changes n he qualy-mx of properes ransaced. For example, here may be more 2-bedroom aparmens sold n he curren perod han n some reference perod. One way of aclng hs problem s o deermne he (margnal) value of an addonal un of each prce-deermnng qualy characersc, such as he number of bedrooms, bahrooms, square fooage of propery, floor of aparmen, possesson or oherwse of parng, balcony, poscode, proxmy o a mero, qualy ndcaor of local school, and so forh. Bu such characerscs are no prced on he mare, only he propery as a whole. Esmaed hedonc regresson equaons explan varaon n propery prces, on he lef hand sde (LHS) of he equaon, n erms of explanaory prce-deermnng characerscs on he rgh hand sde (RHS). The coeffcens on each RHS characersc are esmaes of he margnal value of each respecve characersc. 4 By consderng properes as ed bundles of characerscs wh assocaed esmaed margnal values, we are equpped o solve he problem of adjusng changes n average propery prces for changes n he qualy-mx of properes ransaced. 3 Though some re-esmaon, say every year or wo years, much le rebasng a consumer prce ndex, would be advsed, as would separae esmaes for meanngful sraa, say defned by locaon and ype, for example, sngle famly homes n he capal cy. 4 See Rosen (975), Feensra (995), Dewer (23b), Paes (23), and Slver (24) for he heorecal bass of prce ndexes based on hedonc regressons.

9 6 Our sarng pon s an esmaed hedonc regresson for a sraum of properes n a counry, say aparmens n he nner area of a capal cy. The prncples governng he specfcaon and esmaon of hedonc regressons are no he subjec of hs paper. 5 Our concern s how hedonc regressons are used o derve propery prce ndexes. Ye here s one ssue ha has a drec bearng on he dervaon of hedonc prce ndexes and ha s he funconal form of he hedonc regresson. Oulned here are wo funconal forms ha are wdely used, he laer more so: a lnear and log-lnear form. Choce beween hese forms should be based on a pror and emprcal grounds (esng), as oulned n Halvorsen and Pallaows (98), Cassel and Mendelsohn (985), Can (992), and Trple (26). Funconal forms of he hedonc regresson: a lnear form Consder a lnear hedonc funconal form. An esmaed hedonc regresson would have he prces, p of an ndvdual propery on he LHS and her assocaed characerscs, z, on he RHS as explanaory varables. Such hedonc regressons may be esmaed for each defned sraum n a perod reference perod (ndex =.) and each successve perod (=,2,..,T). The lnear funconal form for perod s gven by: (). p = γ + γz, + ε = h ( z) + ε and esmaed as: = (2). p = γ + γz, = h ( z) where p (and = p ) are he predced (acual) prce of propery n perod ;, z are he values of each =,., prce-deermnng characersc for propery n perod ; γ andγ (and, below, β and β below) are he coeffcens from a lnear (and log-lnear) hedonc equaon; ε (and ν )..d errors; and h ( z ) a shorhand for a lnear hedonc funcon esmaed usng perod daa and perod characerscs. 5 Readers are referred o Bernd (99) and Trple (26) for a clear overvew of hedonc regresson mehods, albe no n he conex of house prces, and for real esae: Srmans e al.(26) on explanaory varables for he hedonc regresson, de Haan and Dewer (23), Coulson (28), and Pace and LeSage (24), Hll and Scholz (23) and Slver and Graf (22) for he ncreasng wor on he spaal economerc modelng of house prces.

10 7 Equaon () has prces explaned by a consan, γ, slope coeffcens γ for each prce- deermnng characercs, z,, of whch here are, and an error erm, ε. I s a lnear relaonshp dcaed, n equaon (2), by he esmaed consan and he slope coeffcens, represened as has ^ over he coeffcens; for a sngle characersc: p = γ+ γz,. The acual relaonshp may be non-lnear and here wll be omed varable bas n usng a lnear form o (ms)represen he relaonshp. To couner hs bas one possbly s o nroduce some curvaure va a squared erm, ( ) 2 p = γ + γ z + γ z, and es a null hypohess as o, 2, wheher γ 2 =, ha s, wheher he squared erm has any explanaory power over and above ha due o samplng error, say a a 5 percen level of sgnfcance. Ineracon erms beween more han one explanaory varable may also be nroduced, Maddala and Lahr (29). Funconal forms of he hedonc regresson: a log(arhmc)-lnear form An alernave funconal form s a log(arhmc)-lnear also referred o as a semlogarhmc form of he hedonc regresson. Ths form arses from a hedonc relaonshp beween z p and, gven by: z, z,2 z, (3). = ( ) ( 2 ),...,( ) p β β β β ε The log-lnear form frs allows for curvaure n he relaonshps say beween square fooage and prce, and second, for a mulplcave assocaon beween qualy characerscs,.e. ha possesson of a garage and addonal bahroom may be worh more han he sum of he wo. The esmaon of ordnary leas squares regresson (OLS) equaons requres a lnear form; we ransform he non-lnear funconal relaonshp n equaon (3) no a lnear form by ang logarhms of boh sdes of he equaon and use OLS: (4). ln p = ln β + z, ln β + lnε = h ( z) + ε = where he lde across h ( z ) desgnaes a log-lnear funconal form. An OLS regresson esmaed for he logarhm of prces, ln p, on characerscs, z,, s gven as:

11 8 (5). ln ln p = β + z, ln β = z, ln β = h ( z) = = I s mporan o noe ha he log-lnear regresson oupu from esmang equaon (4), ha s ln p on z,, provdes us wh he logarhms of he coeffcens from he orgnal log-lnear formulaon n equaon (3). Exponens of he esmaed coeffcens from he oupu of he sofware have o be aen f he parameers of he orgnal funcon, ha s equaon (3), are o be recovered, ha s: ( β) exp ln = β. 6 Snce many explanaory varables are dummy varables ang a value of zero or possesson or oherwse of a characersc and snce logarhms canno be aen of zero values, he loglnear form s more convenen han a double-logarhmc ransformaon ha would requre logarhms be aen of he z, on he RHS. I should be noed ha he nerpreaon of coeffcens from a log-lnear form dffers from ha of coeffcens from a lnear form. For a log-lnear form our esmaed coeffcens are he logarhms of β, β 2,andβ : a un change n 3 he say square fooage, z, leads o a β percen change n prce, whle for a dummy, explanaory varable, say possesson of a balcony, z 2, = as opposed o z 2, = oherwse, ( 2 ) leads o an esmaed ( ) deal n he nex secon. exp β percen change n prce, as wll be explaned n more We consder n hs paper ha hedonc regressons ae a generally applcable lnear and lolnear forms gven by equaons (2) and (5) and ha hese have been esmaed. Oulnes of he hree man hedonc approaches o dervng consan qualy prce ndexes from hese esmaed equaons, along her relave mers, are gven below n secons B, C and E. These approaches are he () hedonc me dummy varable, () hedonc characerscs and () hedonc mpuaon approaches. The approaches are oulned and dscussed n he conex of blaeral perod (reference perod =.) and curren perod prce level comparsons where 6 Agan squared erms and cross-produc neracon erms can be added o ncrease he flexbly of he funconal form o beer represen underlyng relaonshps.

12 9 =,2,.,T. Whle our man concern wll be wh quarer-on-quarer nflaon raes, he prncples can be readly exended o quarer-on-same quarer n prevous year, hough see Rambald and Rao (23). The concern of secon F s wh he perodc updang or channg of he reference perod esmaes. The mehod B. The me dummy varable approach. A sngle hedonc regresson equaon may be esmaed from daa across properes over several me perods ncludng he reference perod and successve subsequen perods. Prces of ndvdual properes are regressed on her characerscs, bu also on dummy varables for me, ang he values of 2 δ f he house s sold n perod, and zero oherwse, δ f he house T s sold n perod 2 and zero oherwse,., δ f he house s sold n perod T and zero oherwse. We exclude n hs case a perod dummy me varable and nerpre he δ as he dfference beween he curren perod and reference perod average prces, havng conrolled for qualymx change va he varables n he hedonc regresson on her characerscs. The mehod has been wdely appled ncludng Fsher, Gelner, and Webb (994), Hansen, (29), and Shmzu e al. (2). Consder a lnear form of he hedonc regresson gven by equaon () bu esmaed over say wo adjacen perods, and :,,,, = (6). p = β + δ D + β z + ε The daa for prces and characerscs exend over he wo perods and, ye only a sngle parameer, β, s esmaed for each characersc s slope coeffcen. The resrcon s ha he slopes of he regresson lnes for perod and perod are he same: β = β = β for each of =,., characerscs. For smplcy, consder a sngle explanaory varable, he square fooage of an ndvdual aparmen, z or z n perods and respecvely. Separae regresson equaons can be

13 esmaed for each of perod and perod, bu he slope coeffcen, he esmaed margnal value of an addonal square foo, s resrced o be he same n each perod, namely β : (7a). p = β + β z + ε for perod, and (7b). p = β + β z + ε for perod. The esmaed coeffcens on he nerceps n each perod are respecvely β and β. These are esmaes of he average prce n perods and havng conrolled for varaon n he square fooage of he aparmens he average s an arhmec mean for hs lnear formulaon (and a geomerc mean for a log-lnear formulaon). We can represen equaons (7a and b) n a sngle hedonc regresson: (8). ( ) p = β + δ D + β z + ε = β + β β D + β z + ε,,,,, The dummy varable D n equaon (8) s equal o f he daa are n perod, and zero oherwse and s esmaed coeffcen δ ( ) β β =. Ths represenaon of equaons 7a and 7b can be seen by nserng D = (perod daa) no he RHS erm of equaon (8) o gve equaon (7a) and nserng D = (perod daa) o gve equaon (7b), assumng, ( ε ) ( ε ) ( ε) E = E = E. The esmaed coeffcen on he dummy varable, δ, s he bass for an esmae of a consan qualy propery prce ndex beween perods and. The esmae s of he dfference beween he perod and perod nerceps, 7 ha s he dfference n he average prces of perod and perod ransacons from her regresson lnes for perod and perod havng conrolled for varaon n he qualy characerscs = β z,,, as n equaon (6), whereby each characersc s valued a s assocaed β. 7 I may be hough ha hs nerpreaon s for he nerceps only when he explanaory varables are zero, bu hs s no he case. By resrcng he slope coeffcens o be he same, he regresson lnes for equaons 7(a) and 7(b) run n parallel and he dfference n he nerceps s he same for any value of he z,, characerscs.

14 A log-lnear specfcaon s gven by:,, (9). ln p = β + z ln β + δ D + ε, = = T The δ are esmaes of he proporonae change n prce arsng from a change beween he reference perod = he perod no specfed as a dummy me varable and successve perods =,,T havng conrolled for changes n he qualy characerscs va he erm, β z,. = The consan-qualy prce ndex s gven for each perod =,..,T, wh respec o perod =, whch equals., by exp( δ ). In prncple exp( δ ) requres an adjusmen for o be a conssen (and almos unbased) approxmaon of he proporonae mpac of he me dummy. The adjusmen s gven by: ( δ ) V ( δ ) exp exp( / 2)),where V ( δ ) s he varance (sandard error squared) of δ and s generally very small; he esmae of consan-qualy prce change s gven by: 8 ( exp β + var( β) / 2 + δ + var( δ ) / 2) () exp( ) exp( ) P TD = δ = β β exp( β + var( β) / 2) 8 We follow ennedy (98) and use for hs log-lnear form as he esmae of he proporonae mpac of he perod me dummy, he conssen (and almos unbased) approxmaon: exp( δ ) exp( V ( δ / 2)) where δ s he OLS esmaor of δ n equaon (9) above and V ( δ ) s esmaed varance. The approxmaon s shown by Gles (2) o be exremely accurae, even for que small samples. The δ esmaed mpac of he perod me dummy s proporonae o he esmaed consan, β he base (omed) perod = nercep, acng as a benchmar. The consan-qualy ndex s gven by equaon (). The numeraor s he consan (perod = nercep) plus he nercep shf of he me dummy, and he denomnaor he esmaed perod = nercep. The smplfed rgh-hand-sde of equaon () s derved by assumng he correcon s mnmal, as s usually he case bu should be emprcally checed n any applcaon and, for he ndex measuremen, cancellng ou he β. Ths leaves a readly nerpreable approxmaon of exp( δ ) see also Van Garderen and Shah (22) and he oe a he end of Hll (23).

15 2 The me dummy mehod has many posve feaures. Gven daa have been colleced over me on prce and qualy characerscs, s relavely easy o apply smply requrng he ncluson of me dummy varables no he panel (cross-secon (propery) me seres) daa se a daa se ha requres no machng of properes snce = β z,, conrols for changes n he qualy mx over me. The esmaes are readly derved from he esmaed coeffcens of he me-dummy varables, δ. Feaures of he mehod The mehod mplcly resrcs he coeffcens on he qualy characerscs o be consan over me: for example, for adjacen perod and regressons, β = β = β, as apparen from equaons (6) and (9). Ths regresson lne for perod s parallel o ha of perod. The exen of hs resrcon depends on he lengh of he me perod over whch he regresson s run. 9 If, for example, he regressons are run over quarerly daa for a rollng -year wndow, a propery prce comparson beween say 26Q and 26Q wh valuaons of characerscs held consan may srech credbly, hough hs can be allevaed by shorer wndows and or adjacen perod regressons as oulned below. The me dummy mehod s crczed hroughou he leraure for holdng he esmaed coeffcens consan. However, as wll be oulned below n secon C and D, a consan qualy prce ndex has o hold somehng consan over me o separae ou he prce change from he qualy-mx change. In wha we wll erm he drec mehod, he quanes of prce deermnng (qualy) characerscs are held consan over me, for example for aparmen 9 The resrcon s also for a parcular sraum. If separae me dummy regressons are run by sraa for ypes of house by major ces, he coeffcens on qualy characerscs for such properes have he flexbly o dffer from hose for oher sraa. Indeed null hypoheses of no dfference beween one or more coeffcen beng he same across sraa can be readly esed. These ess may be nesed F-ess or lelhood rao ess on he hedonc regressons ha resrc such esmaed coeffcens o be he same and hen allow hem, say hrough dummy slope varables, o be dfferen (Maddala and Lahr, 29). Ths can help nform praccal consderaons as o he deal a whch sraa can be defned.

16 3 prces, ha he average number of bedrooms s held consan a 3.2, he square fooage a,5, and so forh and re-prced each perod. An advanage of he me dummy approach s ha he esmaes are generaed for a regresson formulaon. Ths faclaes he exploraon of how he addon and deleon of explanaory varables, changes n he funconal form and esmaor have on he resulng prce ndex number esmaes. I also allows for confdence nervals o be drawn up around hese esmaes and, as Hll (23, secon 5) oulnes, geo-spaal daa ad spaal dependence can be readly negraed no he esmang framewor (see also Pace and LeSage, 24). The me dummy approach uses he ndrec mehod and adjuss (dvdes) he change n mean prces by changes n he volume of characerscs over me. However, hs adjusmen requres he esmaed coeffcens (characersc prces) o be consan so ha only changes n he volume of characerscs are measured. I s dffcul o argue ha consranng characersc prces, he margnal value gven o an addonal bedroom and so forh, s less enable han consranng average characerscs, he say average number of bedrooms n houses ransaced n perod compared wh. There are no grounds for dsmssng he me dummy approach on he grounds of consraned coeffcens. Indeed, we show n secon IV, and n Dewer, Herav and Slver (29), an equvalence beween he drec and ndrec mehods. If used for regular ndex number producon, pas values of he ndex wll be revsed each perod as new daa ener he regresson. A problem wh he revson of pas values of he ndex should no be oversaed. The hree man RPPIs long-esablshed and well-publczed n he Uned Saes, he Case-Shller, FHFA, and CoreLogc ndexes, are all repea-sales ndexes whose pas values are revsed each perod whou publc concern. Second, he esmaed coeffcens for he qualy characerscs are deermned usng daa on prce and quany characerscs over he whole perod of he regresson. Thus some elemen of he esmae of Our neres s wh confdence nervals, no sgnfcance ess. The laer ae he form of havng a null hypohess of say he me dummy beng zero. I may be ha acual house prce nflaon s zero, or close o. A sgnfcance es as o wheher he dfference beween he (exponen of he) esmaed coeffcen and zero s over and above ha due samplng errors a a gven level of sgnfcance s of lle meanng.

17 4 propery prce nflaon for he curren perod compared wh he prevous perod s deermned by pas, f no que dsan, daa. Ths lends some sably o he propery prce ndex, bu may also smooh he resuls and rs some credbly when here s apparen volaly n he prces no mrrored n he ndex. The rollng wndow approaches dffer from he me dummy mehod n he mporan respec ha esmaed coeffcens are no resrced o be consan over me: hey are me varyng. A say perod o + rollng adjacen perod ndex s based on daa n hese wo perods of concern, raher han he whole perod. Rambald and Flecher (24) provde an exensve oulne, and an emprcal sudy, of he use of a alman Fler Smooher (S) as agans he rollng adjacen-perod wndow approach. They argue ha he alman Fler Smooher s preferred on he grounds ha opmally weghs pas values of he seres when esmang he regresson raher han jus weghng he observaons n he curren wndow. The parameer esmaes vary over me bu are modeled as sochasc processes and can be appled o he medummy hedonc ndexes (Schwann, 998 and France, 28) and he hedonc mpuaon approach (Rambald and Rao, 2 and 23). There s a rade-off beween he exen o whch an ndex s smoohed and volaly dampened, by drawng on more dsan daa eher hrough a longer rollng wndow or alman Fler Smooher, and s ably o reflec curren prce changes n he mare, albehey subjec o more volaly. Smoohng mehods are parcularly suable when daa are sparse, ha s n hn mares, as dscussed below n secon IV. Channg, rollng wndows and smoohng We can mlae agans he crcsms of undue resrcon of coeffcens, revsably, and sale daa by usng a chaned rollng wndow for llusraon here, 4 quarers. Consder a fxed base ndex of he ype descrbed by equaons () and (2) n whch each perod s ndex, say 25Q4, s derved from he coeffcen of he dummy varable on me for he perod n queson, compared wh he (omed) perod =, say 25Q. The example s hus of he equaon (6), or An alernave smoohng esmaor, as oulned n Rambald and Flecher (24), s a alman Smooher hough hs requres boh pas and fuure observaons, no avalable for he real-me complaon of an ndex.

18 5 n log-lnear form, equaon (9), esmaed on a quarerly bass over say years. The fxed base esmaed ndex from equaons (9) for 25Q4, where 25Q=. s: (). ( 25 4 ) Q RP Q Q TD δ = exp The adjacen perod ndex s derved from successve mulplcaon channg of regresson esmaes based on successve adjacen perods,.e. a regresson s frs run on 25Q and 25Q2 daa wh a me dummy ha s equal o f he ransacon s n 25Q2 and zero oherwse. The esmaed coeffcen on hs me dummy s an esmae of he change n prce beween he wo perods, conrollng for changes n qualy (2). ( 25 2 ) Q RP Q Q AJ δ = exp. The chaned adjacen perod ndex for 25Q o 25Q4 s: 25Q 25Q4 25Q 25Q2 25Q2 25Q3 25Q3 25Q4 (3). RPCAJ RPAJ RPAJ RPAJ =... The leas resrcve formulaon, n erms of assumpon f consan coeffcens, s o use a rollng wndow of adjacen perods only (Dewer (25b). However, he mehod requres an adequae sample sze of ransacons over he wo perods. Gven he same number of ransacons n each quarer, n hs example say, he fxed base equaons (6) and (9) formulaon use 4 = 4, observaons over say years whle he adjacen perod formulaon uses 2 = 2 each quarer. There may well be degrees of freedom problems n esmang he hedonc regresson, especal f here are many locaonal varables such as dummy varables for each poscode. Furher, n usng rollng wndow adjacen perod regressons, complers have o bear n mnd wo hngs: () s desrable o comple RPPIs as weghed sums of consan-qualy prce ndexes across sraa of dfferen ypes of houses, locaons, and oher meanngful and useful facors. Larger samples enable a more dealed srafcaon; and () sample szes of ransacons for some sraa may appear adequae say f he ndex s developed ousde of a recesson, bu may become nadequae as an economy moves no and durng a recesson, when measuremen really maers. 2 2 A less-dealed srafcaon or esmaon over more han wo me perods could of course be used n such an even.

19 6 A more general formulaon s o use a rollng wndow me dummy regresson. For example, for 25Q o 25Q4, where 25Q=., a 4-quarers rollng wndow has he frs regresson esmaed over he frs four quarers, 25Q o 25Q4, he second regresson drops he frs observaon n hs wndow, 25Q, and adds he nex quarer, 26Q, and so forh. For example, where RP 25Q2 RW 25Q Q4 s he ndex for 25Q2, wh 25Q =., from a rollng wndow regresson based on 25Q o 25Q4 daa, RW 25Q 25Q4 : (4). 25 Q 25 Q 4 25 Q 2 25 Q 3 25 Q 4 ( 25 Q 4 26 Q RP RP RP RP O RP TDMW MW 25Q Q 4 MW 25Q Q 4 MW 25Q Q 4 verlap MW 25Q2 26Q) = ( 26 Q 26 Q 2 ) ( 25 Q 3 25 Q O RP,..., O RP 4 verlap MW 25Q3 26Q2 verlap MW 25Q 25Q4 ). Table, Illusrave lnng of resuls from rollng wndow regresson Perod Rollng wndow 25Q Rollng wndow 25Q2 4-quarer rollng o 25Q4 o 26Q wndow ndex me dummy me dummy 25Q=. 25Q=. 25Q2=. 25Q.. 25Q Q Q Q = The overlap erms requres explanaon. Table shows llusrave resuls for he frs four perods of he ndex smply based on he resuls for ( ) exp δ. from a rollng wndow regresson for 25Q o 25Q4. The nex wndow regresson s esmaed from 25Q2 o 26Q daa. Ths wndow exends he resuls no he nex quarer, 26Q (25Q2=.). There s a need o smlarly exend he 25Q=. ndex. An overlap of he wo ndexes for 25Q4 allows us o rescale he 26Q ndex from he 25Q2 = wndow o 25Q =, ha s: =.

20 7 There s a rade-off here. The 4-quarers rollng wndow smoohes and lags he RPPI resuls o her dermen gven he need for a mely ndcaor. However, wh lmed sample szes avalable, can provde more relable resuls hrough more dealed srafcaon and smaller sandard errors and hus confdence nervals. Complers of he ndex would gan from expermenal RPPIs beng esmaed a dfferen frequences of rollng wndows, ncludng where possble, adjacen-perod regressons and, where approprae, provde users wh sudes of/regular daa on smoohed as well as adjacenperod resuls, an o he spr of measures of core nflaon and consumer prce ndexes. C. The characerscs approach The characerscs approach n a Laspeyres-ype form aes as s sarng pon he average characerscs of properes n a reference perod, say perod, and revalues hese characerscs n successve perods. 3 A hedonc regresson s run o deermne he prcedeermnng characerscs of properes n say perod ; he average propery n perod can hen be defned as a ed bundle of he averages of each prce-deermnng characersc, for example, 2.8 bahrooms, 3.3 bedrooms,.8 garages and so forh our sarng pon. 4 The characerscs approach ae he predced prce of hese perod average characerscs from a perod regresson n he numeraor and hen compares wh he predced prce of hese perod average characerscs from a perod regresson n he denomnaor. The resul s a consan (perod ) qualy propery prce ndex. I s a prce ndex of a consan qualy snce he characerscs are held consan n perod and valued (for he denomnaor) 3 A characerscs approach n a Paasche-ype form, for an ndex comparng perod wh perod, would ae he average characerscs of properes n curren perod and revalues hese characerscs n a precedng reference perod. 4 Indeed he resuls from a hedonc regresson can also be used o help defne sraa. Say locaonal dummy varables of major conurbaons are ncluded along wh slope neracon erms for characerscs n hese locaons. For example, he number of bedrooms n aparmens had a dummy varable as o wheher he aparmen was locaed n he nner or ouer area of a capal cy. The -es on he dummy varable s of a null hypohess of no dfference n her respecve margnal values. If he es s rejeced a an accepable level of sgnfcance, here would be a case for havng separae sraa, sample sze permng.

21 8 and revalued (for he numeraor) usng perod and perod hedonc regressons respecvely. The numeraor provdes an answer o a counerfacual queson: wha would be he esmaed ransacon prce of a propery wh perod average characerscs f was on he mare n perod? For llusraon: f only he sze (square fooage) of an aparmen deermned s prce and he esmaed regresson equaon for aparmens n an nner cy area were, for perod, p = Sqf and for perod, p = Sqf. Say he average sze n perod, z =, 23.4 square fee; he consan (perod ) qualy ndex s: p z ,23.4 (5). = p = ,23.4, a percen prce ncrease. z As a noaonal maer, he predced prce s no longer for propery, prevously used as a subscrp, bu for he average of z, now desgnaed as a subscrp n equaon (5). Before connung we need o say somehng abou he concep of he average characerscs values. Wha averages of characersc values o use? Means, medan, and represenave characersc values The average values may be a mean, medan, or pre-defned represenave propery. The means are generally no of acual values for an ndvdual propery. For example, he mean square foo and mean number of bedrooms for aparmens may ncrease from,29.6 o,227. and from.7 o.9 respecvely over perods and. The medan s a beer represenaon of a ypcal aparmen say ncreasng from,5. o,75. square fee and possessng 2-bedrooms n each perod. The medan wll no be affeced by oulers even f hey exend o an abnormal al n up o half of he daa. Represenave aparmens have her characerscs held consan by defnon; say wo bedroom, o,3 square foo aparmens. The assumpon s ha prce changes of all aparmens follow he measured prce changes of he represenave one. 5 5 These model or represenave properes mgh be jusfed on pragmac grounds f, for a sraum of properes, here s a szable cohor of well-defned smlar properes of a specfc ype sold over me, wh ransacons for he remanng properes n he sraum of mxed characerscs wh nadequae daa on he characersc change.

22 9 Where he dsrbuon of characerscs s hghly sewed here s a case for preferrng geomerc means or medans o arhmec means o downplay exreme values on he als of he dsrbuons of characerscs, or for ha maer prces. 6 However, an alernave, and more nformed approach, s o denfy and valdae, or oherwse, oulers pror o runnng he regressons, wh furher valdaon by examnng he resduals of he regresson. The am s no jus o clean he daa, bu o denfy clusers of characerscs responsble for exreme prces and ncorporae hem no he modelng. Indeed, exreme values may also sgnal an nadequae samplng of a cluser of perfecly vald observaons and a need for a sraegy o ncrease he sample sze n hs regard. Hedonc characerscs ndexes: a lnear funconal form Consder frs wo lnear hedonc regresson, as gven by equaon (2), and repeaed below as equaons (6) and (7) bu adopng he smplfcaon ha he consans γ and γ are ncluded n he summaons as = where z, = and z, = n her respecve reference perod and successve perods =,.,T: (6). p = γ + γz, = γz, = h z = = ( ) (7). p = γ + γz, = γz, = h ( z) = = and for smplcy of exposon, hereafer = desgnaes he consan for whch z, =. 6 Whle he choce beween he geomerc mean and mean s argued o be dependen on he funconal form of he hedonc regresson, he dfference beween he averages may no be as grea as frs consdered. I can be demonsraed ha he rao of an ndex based on arhmec means o geomerc means s gven by he dfference beween he changes n half he varance of prces as he varance of prces of properes, her heerogeney, ncreases, so oo wll he arhmec mean ndex exceed a geomerc mean one. However, Slver and Herav (27b) show ha for a consan-qualy prce ndex, he varances wll be reduced along wh he dfferences beween he arhmec and geomerc ndexes demonsraed n Annex. Use of a hedonc regresson ha beer explans propery prce varably and beer removes prce varably, leads o smaller dfferences beween arhmec and geomerc consan-qualy propery prce ndexes, and hus more confdence n her use.

23 2 (8). z = z and z = z,, Consan qualy hedonc propery prce ndexes can be defned n wo mmedaely apparen ways. Boh requre a comparson of he prce change of a consan base of characerscs prced from a hedonc regresson n perod and agan n perod, ye n he frs defnon s a consan perod base and n he second a consan perod base. Consder a consan perod base of characerscs; we ae he averages of each qualy characersc z n perod, and as wha would be he prce of a propery wh hese average characerscs f sold n perod. Ths predced prce s hen compared wh a valuaon of he self-same average characerscs usng he esmaed perod hedonc regresson. We compare esmaed prces of consan perod average characerscs. A consan perod base of characerscs z s smlarly defned. The Duo (rao of arhmec means) hedonc base (reference) perod ndex (DHB) 7 has n he numeraor perod mean characerscs valued a perod characersc-prces and n he denomnaor perod mean characerscs valued a perod characersc-prces: 7 We depar from he namng sandards n he RPPI Handboo (Eurosa (23) and de Haan and Dewer (23) n parcular). We denfy wo levels of weghng and commensurae formulas n hs paper. The frs s based on sample selecon, ha s, for a blaeral prce comparson beween perod and perod, wheher we use he ransacons n perod (also mpued o perod ), or he ransacons n perod (also mpued o perod ). Eurosa (23) refer o hese as hedonc Laspeyres and Paasche ndexes respecvely, even hough hey are unweghed. The second level of weghng s based on he wegh (expendure share) a he elemenary level gven o a prce change for an ndvdual propery. More wegh s gven o he prce change of more expensve properes for a pluocrac ndex. Reasonable weghed formulaons nclude weghs for he reference perod, curren perod, and some average of he wo. We use he erms hedonc base and curren perod Duo (HBD, HCD) and hedonc base and curren perod Jevons (HBJ, HCJ) as arhmec and geomerc forms of hese aggregaors for unweghed ndexes De Haan and Dewer (23) refer o hs nomenclaure n paragraph 5.4 ff.6. In secon IV we refer o hedonc Laspeyres, hedonc Paasche, and hedonc geomerc Laspeyres and hedonc geomerc Paasche and so forh for weghed ndexes ncludng superlave ndexes. A hrd form of weghng s ha gven o characerscs; hese are weghed by her esmaed coeffcens, explcly n he characerscs approach and mplcly n he dervaon of predced mpued values. The use of Jevons or Duo s argued here o arse from he choce beween a lnear (Duo) of log-lnear (Jevons) hedonc funconal form and mpacs on he weghs gven o he characerscs.

24 2 (9). P γ z h = = γ z h = HDB: z = ( z ) ( z ) and a Duo hedonc curren perod qualy ndex s defned as: (2). P γ z h = = γ z h = HDC: z = ( z ) ( z ) If, n a perfec mare, preferences change and he mplc prces of one characersc, say an addonal bedroom, ncrease a an above average rae; oher hngs beng equal, ulymaxmzng buyers would subsue expendure owards oher characerscs, say more overall space. The use of a consan perod characersc base, z would undersae prce ncreases he γ z γ z = expendure weghs n equaon (2) do no reflec he subsuon away from characerscs wh above average prce ncreases and of a consan perod characersc base, z, oversae. Ths s because, as we show n secon VC, he consan qualy prce change of each characersc from equaons (9) and (2) are mplcly weghed by he esmaed relave values of he characersc. For example, usng he noaon n equaon (5) and equaons (6) and (9): p (2). p γ γ γ z z z = = γ = = z γ z γz = =. For he aforemenoned subsuon bas relang o characerscs, a geomerc mean of equaons (9) and (2) a hedonc Fsher-ype prce ndex number s jusfable on grounds of economc heory, axomac properes, and nuon. 8 8 The Consumer Prce Index (CPI) Manual (ILO e al., 24) recommends superlave prce ndexes he Fsher, Törnqvs, and Walsh ndexes as he arge formulas for he hgher-level ndexes. These formulas generally produce smlar resuls, usng symmerc weghs based on quany or expendure nformaon from boh he reference and curren perods. They derve her suppor as superlave ndexes from economc heory. A uly funcon underles he defnon of (consan uly) cos of lvng ndex (COLIs) n economc heory. Dfferen ndex number formulas can be shown o correspond wh dfferen funconal forms of he uly funcon. (connued )

25 22 (22). P = P P DHF DHB DHC z z z z The heory of hedonc regressons can be found n Rosen (974), Trple (987), Feensra (995) and for an applcaon, Slver (999) Dewer (23b), and Slver (24); he heory of Laspeyres and Paasche bounds s n onus (924) and of subsuon effecs warranng a (superlave) geomerc mean of a Laspeyres and Paasche formula, n Dewer (976, 978 and 24). oe ha he denomnaor n equaon (9) s he mpued or predced prce, raher han acual prce, n perod, ( ) h z, and smlarly n he numeraor of equaon (2) we use he mpued or predced prce raher han acual prce n perod, ( ) h z. In calculang equaon (9) we ae he rao of wo mpuaons: he mpued prce of z valued a perod characersc prces n he numeraor and a perod characersc prces n he denomnaor a dual mpuaon. For a lnear form he average predced prce n perod from an Ordnary leas squares regresson s equal o he average acual prce, ( ) h z = p and, hough equaon (9) s hardly complex, can be calculaed wh a sngle mpuaon as he much smpler: h (23). p secon IV. ( z ). We reurn o ssues of dual versus sngle mpuaon laer n hs secon and n Types of hedonc characerscs ndexes: log-lnear funconal form Laspeyres, for example, corresponds o a hghly resrcve Leonef form. The underlyng funconal forms for superlave ndexes, ncludng Fsher and Törnqvs, are flexble: hey are second-order approxmaons o oher (wce-dfferenable) homohec forms around he same pon. I s he generaly of funconal forms ha superlave ndexes represen ha allows hem o accommodae subsuon behavor and be desrable ndexes. The Fsher prce ndex s also recommended on axomac grounds and from a fxed quany base perspecve (ILO e al., 24).

26 23 A consan-qualy characerscs prce ndex for a log-lnear hedonc regresson equaon follows smlar prncples: for properes, n a gven sraum, for he reference perod and successve perods =,.,T he esmaed hedonc regressons are: (24). ln p = z ln β = h ( z ), = (25). ln p = z, ln β = h ( z) = The lde above h denoes a log-lnear funconal form, he consan s ncluded as z, * β for whch =, and smlarly for perod, over all observaons, and perods and average values of each characersc are arhmec means: 9 (26). z = z and z = z,, Consan qualy propery prce ndexes can be defned n wo mmedaely apparen ways. A hedonc geomerc Laspeyres-ype consan perod characerscs ndex aes he means of a se of characersc z for he reference perod =, and values hem n he numeraor n equaon () by her respecve margnal valuaons β from a log-lnear hedonc regresson, esmaed jus from daa on ransaced properes n perod, and compares hs overall valuaon wh he same se of characerscs valued usng perod = esmaed coeffcens, ha s, β, n he denomnaor. The ndex s a rao of geomerc means wh characerscs held consan n he base (reference) perod: ln p = ln β + z ln β + z ln β + z ln β + lnε, ha 9 I s apparen from he log-lnear ransformaon, 2, 2 3, 3 he z, are no n logarhms and arhmec averages of, z are approprae. The average of he characerscs for, boh a lnear and log-lnear formulaons are an arhmec means. There s n any even an mmedae problem wh ang a geomerc mean of dummy varables snce logarhms canno be aen of zero values. However, here s a wor-around. Where s he sample sze, and here are n values of and n of zero, he geomerc mean s ( n Geomean( n ) + n Geomean( n )) / equals( n Geomean( n )) /. Snce he Geomean( n ) = Arhmean( n ) for he n values of uny, s que sraghforward. For example wh =6, of whch 6 are uny, he mean s he smple proporon, 6/6=.267.

27 24 (27). P ( z ) β exp z ln β h, ( z ) h z = = = = = = HGMB: z β ( z ) exp ln h ( z ) p z β = = ( ) Equaon (27) holds he (qualy) characersc se consan n perod, hough a smlar ndex could be equally jusfed by valung n each perod a consan perod average qualy se. A hedonc geomerc Laspeyres-ype consan-perod (arhmec mean) characerscs ndex s gven by: (28). P ( z ) β exp z ln β h z, = = p == = = = HGMC: z β ( z ) exp ln h z h z z β = = ( ) ( ) ( ) Dual mpuaons A naural queson arses as o he phrasng of he second o las erms n equaons (27) and (28) as dual mpuaons, ha s hey use predced (mpued) prces n boh he denomnaor and numeraor Slver (2) and de Haan (24a). As we wll see n secon IV, he use of equaon (28) only requres ha a hedonc regresson be esmaed for he reference perod, ha acual perod prces may be used, and we lose hs feaure f we adop dual mpuaons. Here we explan ha whle here s a well esablshed logc for he use of dual mpuaons, need no hold n hs nsance, hough s mporan n our wor on weghng as explaned n secon IV. Dual mpuaon requres a predced (mpued) prce n boh he denomnaor and numeraor of equaons (27) and (28) as opposed o a sngle mpuaon, he las erm n boh equaons (27) and (28), for whch h ( ) z = p and ( ) h z = p. For example n equaon (27) he sngle mpuaon hedonc approach uses he acual prce n he denomnaor, and predced prce n he numeraor. The logc for he need for dual mpuaons s ha he above equales only hold for perfecly specfed hedonc regressons esmaed whou bas. However, hs would lead o a based prce comparson f here were subsanve omed varables n he hedonc specfcaon. For example, cheaper erraced houses may have no fron yard (garden) openng drecly ono he sree. Ths poorer feaure would be refleced n he acual prce (denomnaor) of a consan perod ndex, bu may be excluded or no properly represened n he hedonc specfcaon

28 25 and hus predced prce (numeraor). The numeraor would be based upwards and ndex downwards. The dual mpuaon hedonc ndex would o some exen offse an upwards bas by usng predced prces n boh he numeraor and denomnaor. Dual mpuaons are generally advsed for hedonc prce ndexes, see Slver (2 and 24), de Haan (24a), Hll and Melser (28), Dewer, Herav and Slver (29), assocaed commens (de Haan 29) and response, Hll (23) and secon IV, where we consder an alernave woraround. Ye a feaure of he OLS esmaor s ha he mean of acual prces s equal o he mean of predced prces; p = z p and p = z p. Thus he las erms n equaons (27) and (28) see also (de Haan and Dewer, 23, paragraph 5.38). A problem arses, however, wh he use of weghs a hs lower level, as explaned n secon IV, for whch we need dual mpuaons. eher a perod consan-characerscs ndex nor a perod consan-characersc quany base can be consdered o be superor, boh acng as bounds for her heorecal counerpars. Some average or compromse soluon s requred. Dewer (976, 978) defned n economc heory a class of ndex number o be superlave. We consder defnons of superlave ndexes n secon III. Ths ncludes he Törnqvs ndex formula gven n hs log-lnear conex by: (29). P τ exp ln z β THB = = z z where z τ = ( z + z) τ β ( z ) h = = = β τ τ h exp z ln ( z ) β = = /2 τ ( z ) τ ( z ) D. The mpuaon approach The mpuaon approach dffers from he characerscs approach. For he characerscs approach he average (arhmec mean) values of characerscs were derved n, for example, perod as 3. bedrooms,.7 possesson of a garage,,25 square fee, and hen revalued usng esmaed hedonc characersc coeffcens esmaed from daa n perod. The characerscs approach answered a counerfacual queson: wha would be he prce change of

29 26 a se of average perod characerscs valued frs, a perod hedonc valuaons, and second, a perod hedonc valuaons? In conras he mpuaon approach wors a he level of ndvdual properes, raher han he average values of her characerscs. I acles a smlar counerfacual queson: wha would a propery wh s gven characerscs n perod be worh f he same such characerscs were revalued usng perod hedonc valuaons? An average of hese s hen aen over he ndvdual properes, and compared wh an average of mached perod valuaons of perod properes. The summaon s over he predced prces of =,., perod properes. The raonal for he mpuaon approach les n he mached model mehod. Consder a se of properes ransaced n perod. We wan o compare her perod prces wh he prces of he same mached properes n perod. In hs way here s no conamnaon of he measure of prce change by changes n he qualy-mx of properes ransaced. However he perod properes were no sold n perod here s no correspondng perod prce. The soluon s o mpue he perod prce of each perod propery. We use a perod regresson o predc prces of properes sold n perod o answer he counerfacual queson: wha would a propery wh perod characerscs have sold a n perod? Equaon (25) provdes he answer. I s a hedonc regresson usng perod daa, o esmae perod characersc prces and hen apply hem o perod characerscs values. The requremens of he mpuaon mehod for a lnear funconal form usng consan perod characerscs are o: () esmae a hedonc regresson for he reference perod and each successve perod ; () denfy he values of he characerscs of each propery sold n perod, say propery had 4 bedrooms, 2 bahrooms and so forh; () usng he hedonc regressons mpue/predc he prce of each ndvdual perod propery would have sold a n perods and perod ; and (v) usng mpued propery prces, deermne he average prce of perod properes n perod and perod and as a rao, he change n he average perod consanqualy prces he dfferen formulaons of hedonc mpuaon ndexes are oulned n Slver and Herav (27a).