Accounting for Spatial Variation of Land Prices in Hedonic Imputation House Price Indexes

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1 Accounng for Spaal Varaon of Land Prces n Hedonc Impuaon House Prce Indexes Jan de Haan a and Yunlong Gong b 9 November 24 Absrac: Locaon s capalzed no he prce of he land he srucure of a propery s bul on, and land prces can be expeced o vary sgnfcanly across space. We accoun for spaal varaon of land prces n hedonc house prce models usng geospaal daa and a nonparamerc mehod known as geographcally weghed regresson. To llusrae he mpac on aggregae prce change, qualy-adjused house prce ndexes and he land and srucures componens are consruced for a cy n he Neherlands and compared o ndexes based on more resrcve models. Keywords: geocoded daa, hedonc modelng, land and srucure prces, non-paramerc esmaon, resdenal propery. JEL Classfcaon: C4, C33, C43, E3, R3. a Correspondng auhor; Dvson of Process Developmen, IT and Mehodology, Sascs Neherlands, and OTB, Faculy of Archecure and he Bul Envronmen, Delf Unversy of Technology; emal: j.dehaan@cbs.nl. b OTB, Faculy of Archecure and he Bul Envronmen, Delf Unversy of Technology; emal: y.gong-@udelf.nl. The vews expressed n hs paper are hose of he auhors and do no necessarly reflec he vews of Sascs Neherlands.

2 . Inroducon Housng markes have wo dsnc feaures: every house s unque and houses are sold nfrequenly. Ths s problemac for he consrucon of house prce ndexes because he usual mached-model mehod, where he prces of goods are racked over me, breaks down. Hedonc regresson mehods and repea sales mehods deal wh hese problems. The unqueness of properes s manly due o locaon. Whn a sngle neghborhood, he value of wo properes wh smlar srucures can dffer sgnfcanly, dependng on he exac localy. Repea sales ndexes fully conrol for locaon snce hey rack he prces of he same properes over me (n a regresson framework). The problem wh repea sales mehods s hreefold. Frs, because hey only use mached pars of houses durng he sample perod, hese mehods hrow away nformaon on sngle sales and are herefore neffcen. Second, sandard repea sales mehods do no adjus for qualy changes of he ndvdual houses. Thrd, hese mehods canno provde nformaon on he shadow prces of he varous propery characerscs and hus do no allow he esmaon of, for example, prce ndexes of he land he srucure ss on. Gven he problems wh repea sales mehods, we focus on hedonc ndexes. Tradonal hedonc prce ndexes also have a number of dsadvanages. Frs, daa on housng characerscs mus be avalable. Second, locaon s ypcally ncluded n hedonc models a some aggregae level, such as poscode areas, raher han a he ndvdual propery level, poenally leadng o locaon bas. Thrd, land s usually no ncluded as an ndependen varable, agan poenally gvng rse o bas and makng mpossble o esmae prce ndexes for land. Geospaal daa,.e. nformaon on he exac locaon n erms of geographc coordnaes such as longude and laude, can help aenuae he laer dsadvanages. Our am s o show how hs can be done and how hedonc house prce ndexes can be consruced accordngly. A general problem wh he esmaon of hedonc models for housng s omed varables bas. No properly accounng for locaon can be a major cause of bas and ofen leads o spaal auocorrelaon of he error erms. As menoned above, he eases way o deal wh he problem s o nclude dummy varables for poscode areas. Anoher sraghforward approach, whch has also been frequenly nvesgaed emprcally, s o nclude explanaory varables for all knds of amenes. Whle beng of neres snce

3 provdes nformaon on he effec of hose amenes on he prces or prce changes of properes, hs approach s very daa nensve. Imporanly, boh mehods canno fully adjus for locaon, and so some omed varables bas and spaal auocorrelaon wll mos lkely reman. In recen years, more sophscaed mehods have been pu forward o handle he problem of spaal auocorrelaon. Spaal error models aemp o explcly model he spaal auocorrelaon whle spaal lag models nclude he value of neghbor properes n he model. Boh mehods can be used n a me dummy hedonc framework, where he model s esmaed on pooled daa for he whole sample perod and prce ndexes are compued from he me dummy coeffcens (Dorsey e al., 2; Hll e al., 29). I s also possble o apply hese mehods n a hedonc mpuaon framework (Rambald and Rao, 2; 23). Anoher mehod uses a spao-emporal fler whch elmnaes spaal auocorrelaon n order o esmae an ndex for a dwellng wh specfc characerscs (Pace e al., 998; Tu e al., 24; Sun e al., 25). A dsadvanage of he above paramerc mehods s ha a spaal wegh marx has o be specfed a pror bu ha s precse srucure s unknown. Nonparamerc or sem-paramerc mehods are more suable o accoun for spaal dependence. Semparamerc mehods have become ncreasngly popular. The effec of varables relang o locaon, for example, can be esmaed nonparamercally n characerscs space whereas he effec of varables relang o he srucure of he propery can be esmaed paramercally, as n radonal hedonc models. In hs paper, we assume ha locaon affecs he prce of land bu no he prce of srucures. Tha s, we posulae ha land prces vary across space whereas he prce of srucures s fxed. We deal wh hs ype of spaal nonsaonary usng a semparamerc approach known as Mxed Geographcally Weghed Regresson (MGWR) n whch he land prces are esmaed by Geographcally Weghed Regresson (GWR), a nonparamerc mehod proposed by Brunsdon e al. (996) and Foherngham e al. (998a). An addonal advanage s ha we wll be able o plo a dealed map of land prces. Apar from he fac ha deals wh spaal nonsaonary n a sraghforward way, GWR enables us o model he local form of auocorrelaon. Moreover, allows land prces o vary no only across space bu also across me by esmang he model for each perod separaely. The laer s a prerequse for he consrucon of hedonc 2

4 mpuaon prce ndexes. In concluson, GWR s a raher flexble mehod, whch can be seen as a generalzaon of radonal hedonc mehods. We are specfcally argeng sascal agences engaged n he complaon of house prce ndexes. Ths has several consequences. The agences should have access o geocoded daa, bu hs s hardly a problem hese days. The mehods appled should be relavely easy o explan. Mos mporanly, he prce ndexes should be non-revsable. Ths means ha he use of he me dummy mehod, where prevously publshed ndex numbers change when he sample perod s exended and new daa s added, s ruled ou. Ths srenghens he case for consrucng hedonc mpuaon ndexes. Furhermore, our paper res o fll a gap n he recen Handbook on Resdenal Propery Prce Indces (Eurosa e al., 23) n whch he use of geospaal daa n he esmaon of hedonc house prce models s no very well covered. The Handbook uses daa for deached dwellngs sold n he Duch cy of A from he frs quarer of 25 o he second quarer of 28 o llusrae he varous mehods. We explo sales daa for he cy of A also bu exend he daa se n hree dmensons. We have daa from he frs quarer of 998 o he second quarer of 28, so our daa se covers a perod of more han years. Noe hough ha we wll use annual raher han quarerly daa n our emprcal work. The range of srucural characerscs s much broader han ha n he Handbook. Fnally, we nclude ypes of houses oher han deached dwellngs. The paper proceeds as follows. Secon 2 oulnes some basc deas. Our hedonc model s lnear, wh non-ransformed propery prce as he dependen varable and sze of land and sze of srucures as explanaory varables. A normalzed verson, wh prce per square meer of lvng space as he dependen varable, s dscussed as well. We also address he ncluson of addonal characerscs o descrbe he qualy of srucures, ncludng age of he srucure o adjus for deprecaon. Secon 3 descrbes how we rea locaon. As menoned before, locaon s capalzed no he prce of land, and we would expec land prces o dffer a he propery level. The GWR and MGWR models and he way n whch hey are esmaed are explaned n deal. Secon 4 shows how we calculae hedonc mpuaon ndexes. Secon 5 presens emprcal evdence for he Duch cy of A. Secon 6 dscusses he resuls, denfes poenal mprovemens and concludes. For an excellen nroducon, see Hll (23). 3

5 2. A smplfcaon of he bulder s model 2. Some basc deas Our sarng pon s he bulder s model proposed by Dewer, de Haan and Hendrks (2) (25). I s assumed ha he value of a propery n perod, no he value L S v L of he land he srucure ss on and he value p, can be spl v S of he srucure: p = v + v. () The value of land for propery s equal o he plo sze n square meers, z L, mes he prce of land per square meer, α, and he value of he srucure equals he sze of he srucure n square meers of lvng space, meer, β. 2 Afer addng an error erm p = zl + β zs + z S, mes he prce of srucures per square u wh zero mean, model () becomes α u. (2) The (shadow) prces of boh land and srucures n (2) are he same for all properes, rrespecve of her locaon. In secon 3 we relax hs assumpon and allow for spaal varaon of, n parcular, he prce of land. The bulder s model akes deprecaon of he srucures no accoun, a opc we address n secon 2.2. Equaon (2) can be esmaed on daa of a sample S of properes sold n perod. Ths approach, however, suffers from a leas hree problems. Frs, he model has no nercep erm, whch hampers he nerpreaon of 2 R and he use of sandard ess n Ordnary Leas Squares (OLS) regresson. Second, a hgh degree of collneary beween land sze and srucure sze can be expeced, so ha α and β wll be esmaed wh low precson. Fnally, heeroskedascy s lkely o occur snce he absolue value of he errors ends o grow wh ncreasng propery prces. Our nex sep s o dvde he lef hand sde and rgh hand sde of equaon (2) by srucure sze z S, gvng p * = α r + β + ε, (3) where p * = p / z s he normalzed propery prce,.e. he value of he propery per S square meer of lvng space, r = z / z denoes he rao of plo sze and srucure L S 2 We follow Dewer, de Haan and Hendrks (25) who used lvng space (usable floor space) n square meers as a measure of sze of he srucures. Alernave measures are also possble, for nsance he volume of he srucure n cubc meers. 4

6 sze, and ε = u / z S. Ths resolves he frs wo problems as he model now has an nercep erm and a sngle explanaory varable. However, he normalzaon s unlkely o resolve he ssue of unsable parameer esmaes. Dvdng by z S s a means of adjusng for heeroskedascy when he error varance n (2) s proporonal o he square of srucure sze; esmang equaon (3) by OLS s equvalen o esmang (2) by Weghed Leas Squares (WLS) usng weghs equal o /( z S ) 2. Ths knd of error varance seems que exreme, and so hs weghng sysem may no be helpful o reduce he heeroskedascy problem. Also, he raos (and he normalzed values * p ) wll exhb relavely lle dsperson. Some sascal agences measure and publsh changes n normalzed propery prces, ofen he prce per square meer of srucures n order o adjus for composonal change of he properes sold. We do no recommend hs approach because s changes n unadjused propery prces and prce changes mos people wll be neresed n. Ye, gven ha (3) s a sraghforward regresson model, ncludng an nercep erm, we favor specfcaon (3) over (2). r 2.2 Addng srucures characerscs A poenal weakness of hedonc modelng for housng s omed varables, leadng o based (OLS) parameer esmaes and predced prces. Omed varables n he models (2) and (3) can relae o land or srucures. Omed facors relang o land are addressed n secon 3. Here we descrbe our approach o ncludng addonal characerscs for srucures. There are wo ssues: deprecaon and renovaon of he srucures has no been aken no accoun so far, and he use of sze as he only measure of qualy of he srucures seems oo smplsc. Followng Dewer, de Haan and Hendrks (25), we nally assume a sragh- lne deprecaon model. The adjused value of he srucure s β ( δ a )z, where δ s he deprecaon rae and he level of ndvdual dwellngs s unavalable so ha a s age of he srucure. Informaon on renovaons a δ a measures he effec of ne deprecaon,.e. he combned effec of rue deprecaon and renovaon. Wren n lnear form, he adjused srucures value s β z β δ a z. Addng he second erm o he rgh-hand sde of equaon (2) yelds p = zl + β zs β δ a zs + α u. (4) S S S 5

7 We do no know he exac age of he srucures, bu we do know he buldng perod n decades, from whch we can calculae approxmae age n decades. Thus, age n our daa se s a caegorcal varable. The ne deprecaon rae s of course caegorcal as well. 3 Usng mulplcave dummy varables D a ha ake on he value f n perod propery belongs o age caegory a ( a =,..., A) and he value oherwse, and afer reparameerzng such ha β z s no longer a separae erm, model (4) s equvalen o A p = α zl + D a = γ a zs + modfy hs model as follows: u S. To be able o use sandard esmaon echnques, we A = zl + a= p α γ D z + u. (5) a a S No resrcons are placed on he parameers γ a, and he new funconal form s neher connuous nor smooh. Ths s somewha problemac from a heorecal pon of vew, because s a odds wh he nal sragh-lne deprecaon model. On he oher hand, our approach nroduces some flexbly. Age of he srucures s no only mporan for modelng deprecaon, can also be seen as an arbue of he dwellng self n ha houses bul n a parcular decade are more n demand han oher houses, perhaps for her archecural syle or for oher reasons. Dewer, de Haan and Hendrks (25) also show how o ncorporae he number of rooms. The new value of he srucures becomes β ( δ a )( + µ z ) z s he parameer for he number of rooms β z β µ z z β δ a z β δ µ a S + R S S R S. Usng dummes z R S, where µ z R. 4 The lnear form for hs expresson s z D r for he number of rooms wh he value f n perod he propery belongs o caegory r ( r =,..., R) and he value oherwse, and reparameerzng agan, he exenson of (5) becomes A R A R = zl + γ ada zs + λr Dr zs + a= r= a= r= p α η D D z + u. (6) ar Nex, n order o save degrees of freedom, we gnore he second-order effecs due o he neracon erms D D a r, yeldng a r S 3 Dewer, de Haan and Hendrks (25) reaed approxmae age as a connuous varable, despe he fac ha s n fac caegorcal. They found ha he esmaed ne deprecaon rae was que volale, whch was no conssen wh her a pror expecaon of a sable deprecaon rae, and subsequenly esmaed models where he deprecaon rae was kep consan over me. However, we are no neresed n he deprecaon rae self and accep any volaly. 4 Noe ha Dewer, de Haan and Hendrks (25) dd no allow he parameer o change over me. 6

8 A R A R p = α zl + γ ada zs + λr Dr zs + u = α zl + γ ada + λr Dr zs + u (7) a= r= a= r= The second expresson shows ha he prce of srucures,.e. he prce per square meer of lvng space, equals γ + λ for properes n age class a ( a =,..., A) and caegory r a r ( r =,..., R) for number of rooms. A hgh degree of mulcollneary can occur among he varous srucures componens, bu we do no worry abou hs because we are only neresed n he combned effec. Mulcollneary beween hese componens and plo sze mgh sll be a problem hough. Dvdng he frs expresson n (7) by z S gves p * A R = + α r + γ ada + a= r= θ λ D + ε. (8) r r We ncluded an nercep erm θ and excluded dummy varables for age class A and caegory R for he number of rooms o denfy he model. Model (8) s a sraghforward esmang equaon for he overall propery prce per square meer of lvng space. Addonal caegorcal varables for srucures can be ncluded n a smlar way as was done for he number of rooms. As a maer of fac, n our emprcal work we wll use ype of house nsead of he number of rooms. 3. Land and spaal nonsaonary 3. Locaon and he prce of land Locaon s he mos mporan omed varable n he hedonc models presened so far. In many emprcal sudes, locaon s reaed as a separae characersc by ncludng addve locaonal dummy varables n models for he overall propery prce. Ths s no he soluon we prefer. Locaon s defnely capalzed no propery prces. However, he prce of srucures s mos lkely o be approxmaely consan across space, a leas whn relavely small regons or ces. I s he prce of he land he srucure s bul on ha can vary sgnfcanly across dfferen locaons, even whn a sngle neghborhood. The queson hen arses as o how hs spaal varaon, or spaal nonsaonary as s somemes referred o, n he prce of land should be modeled. We could make he smplfyng assumpon ha he prce of land vares across poscode areas bu s he same whn each poscode area k ( k =,..., K) and denoed by 7

9 α k. Usng mulplcave poscode dummy varables D k, whch ake on he value of f propery belongs o k and he value oherwse, an mproved verson of model (7) for he unadjused propery prce s K A R k Dk zl + γ a Da zs + k= a= r= p α λ D z + u, (9) = r and an mproved verson of model (8) for he normalzed propery prce s r S p * K A R = + α k ( K ) Dk r + γ ada + k= a= r= θ λ D + ε. () r r The assumpon of equal land prces whn poscode areas could be oo crude, dependng of course on he level of deal of he poscode sysem. Generalzed versons of he models (9) and () are found by assumng ha he prce of land can n prncple dffer a he ndvdual propery level,.e. a he mcro locaon. We denoe he propery- specfc land prce by α, yeldng A R = zl + γ ada zs + a= r= p α λ D z + u () and r r S p * A R = + α r + γ ada + a= r= θ λ D + ε. (2) r r Models () and (2) obvously canno be esmaed by sandard regresson echnques. In secon 3.2 we wll dscuss a sem-paramerc approach ha does allow us o esmae hese models. Because he mehod ulzes daa on he prces of neghborng properes (n addon o he prce of propery self) o esmae α, s no necessarly rue ha he use of models () or (2) wll lead o aggregae prce ndexes ha are very dfferen from hose obaned by usng models (9) or (). 3.2 Accounng for spaal varaon of land prces One mehod ha deals wh spaal nonsaonary of propery prces s he expanson mehod (Case, 972; Jones and Case, 992). The propery prce, or n our case he prce of land, can be seen as an unknown funcon of he propery s locaon n erms of laude x and longude y or a smlar geographc coordnae sysem. Ths funcon can be approxmaed usng a Taylor-seres expanson of some order; ypcally, second- 8

10 order approxmaons are appled. The expanson mehod makes use of geospaal daa bu s bascally paramerc as calbraes a prespecfed paramerc model for he rend of land prces across space (Foherngham e al., 998b). The mehod we wll apply, referred o as Geographcally Weghed Regresson (GWR), deals wh spaal nonsaonary n a ruly nonparamerc fashon (Brunsdon e al., 996; Foherngham e al., 998a). 5 Le us remove he srucural characerscs from model () for a momen and hus consder land as he only ndependen varable. Usng α = α x, y ), he model becomes ( p = α ( x, y ) z + u. (3) L Noe ha we have dropped he superscrp for convenence, bu should be clear ha we esmae all models for each me perod separaely. Noe also ha he prces of land can be esmaed for all pons n space, no jus for he sample observaons, enablng us o depc a surface of land prces for he enre sudy area. Model (3) can be esmaed usng a movng kernel wndow approach, whch s essenally a form of WLS regresson. In order o oban an esmae for he prce of land α x, y ) for propery, a weghed regresson s run where each relaed observaon j ( (.e., each neghborng propery) s gven a wegh w j ( j). The wegh w j should be a monoonc decreasng funcon of dsance d j beween x, y ) and x, y ). There s ( ( j j a range of possble funconal forms. In hs paper we have chosen he frequenly-used b-square funcon gven by: w j 2 2 ( ) 2 j d h f dj < h =, (4) oherwse where h denoes he bandwdh defnng he rae of decrease n erms of dsance. The choce of bandwdh nvolves a rade-off beween bas and varance. A larger bandwdh generaes an esmae wh larger bas bu smaller varance whereas a smaller bandwdh produces an esmae wh smaller bas bu larger varance. Ths bas-varance rade-off moved us o choose he bandwdh by mnmzng he cross-valdaon (CV) sasc CV = n [ y y ( h) ] = 2 ˆ, (5) 5 For a comparson of geographcally weghed regresson and he spaal expanson mehod, see Ber e al. (27). 9

11 where yˆ ( h) s he fed value of y wh he observaons for pon omed from he calbraon process. The nonparamerc GWR approach o dealng wh spaal nonsaonary of he prce of land has o be adjused for he fac ha models () and (2) nclude srucural characerscs wh spaally fxed parameers. Ths leads o a specfc nsance of he sem-paramerc Mxed GWR (MGWR) approach dscussed by Brunsdon e al. (999) n whch some parameers are spaally fxed and he remanng parameers are allowed o vary across space. To descrbe he esmaon procedure, s useful o change over o marx noaon. Denong he number of observaons by n, model () can be wren n marx form as P = Z L α + Z Sβ + u (6) where ( ( x, y ), ( x, y ),..., (, )) T α = α α 2 2 α s a vecor of land prces o be esmaed, x n y n s an operaor ha mulples each elemen of α by he correspondng elemen of and Z S s he marx of srucural characerscs ncluded n model (), gven by Z L, Z S D zs D2 zs L D j zs D2z2 S D22 z2s D2 j z L 2S =, M M O M Dn zns Dn 2znS L Dnj zns T and β = ( β,,..., ) β 2 β n s he vecor of parameers relang o Z S. We follow Foherngham e al. (22), who proposed an esmaon mehod ha s less compuaonally nensve han he mehod descrbed by Brunsdon e al. (999). 6 To economze on noaon, we wre he GWR ha marx as T T z L L ( x, y ) L L ( x, y ) Z W Z Z W T T z 2L ZLW( x2, y2 ) Z L ZLW( x2, y2 ) S =, M T T z nl L ( xn, yn ) L L ( xn, yn ) Z W Z Z W where W x, y ) = dag[ w ( x, y ), w ( x, y ),..., w ( x, y )]. The calbraon of he model ( 2 n consss of four seps: 6 We wll broadly descrbe he acual esmaon procedure and presen he esmaors for he parameers, bu we do no provde he exac MGWR algorhm. For deals, see Foherngham e al. (22), Me e al. (26) and Genaux and Napoléone (28).

12 () regressng each column of Z S agans compung he resduals Q = ( I S) Z ; S Z L usng he GWR calbraon mehod and (2) regressng he dependen varable P agans Z L usng he GWR approach and hen compung he resduals R = ( I S) P ; (3) regressng he resduals R agans he resduals Q usng OLS n order o oban he esmaes βˆ T T = ( Q Q) Q R ; (4) subracng Z β ˆ S from P and regressng hs par agans Z L usng GWR o oban T T esmaes ˆ( α x, y ) = [ Z W( x, y ) Z ] Z W( x, y )( P Z βˆ ). L The predced values for he propery prces can be expressed as L P ˆ = S( P Z βˆ) + Z βˆ S S = LP, (7) S S S S. T T T T wh L = S + ( I S) Z [ Z ( I S) ( I S) Z ] Z ( I S) ( I S) L The parameer esmaes and he predced propery prces depend on he choce of weghs, hence on he choce of bandwdh h. The opmal value for h s deermned by mnmzng he CV score, as menoned above. S 4. Hedonc mpuaon prce ndexes Ths secon addresses he ssue of esmang qualy-adjused propery prce ndexes. 7 Suppose ha sample daa s avalable for perods =,..., T, where s he base perod (he sarng perod of he me seres we wan o consruc), and suppose model () has been esmaed separaely for each perod. The predced propery prces, obaned usng MGWR, are gven by = + ˆ A R ˆ ˆ [ + ˆ + ˆ p α zl θ γ = ] a ada λ r= r Dr zs. For shor, we wre A R he predced prce of srucures, ˆ + D + ˆ θ ˆ γ a= λ a a r= r D r, as βˆ and he predced overall propery prce as pˆ = ˆ α z + ˆ β z ( =,..., T ). L We denoe he sample of properes sold n he base perod by S S. The hedonc mpuaon Laspeyres propery prce ndex gong from perod o perod s defned by pˆ () S P Laspeyres = pˆ, (8) S 7 In hs paper we only dscuss sales-based propery prce ndexes. For an explanaon of he dfference beween sales-based and sock-based ndexes, see Eurosa e al. (23).

13 Equaon (8) may need some explanaon. All quanes are se equal o because each propery s unque. Because he ndex s based on a sngle sample, wll no be affeced by composonal change. Mos, f no all, of he properes raded n perod are no resold n perod, and he mssng prces herefore need o be mpued by also replaced he observable base perod prces known as double mpuaon. 8 The p by predced prces p ˆ (). We have p ˆ, a mehod p ˆ () are esmaed perod consan-qualy propery prces,.e. esmaes of he prces ha would preval n perod for properes sold n perod f he properes prce-deermnng characerscs were equal o hose of he base perod, whch serves o adjus for qualy changes of he ndvdual properes. These consan-qualy prces are () esmaed by = + ˆ A R ˆ ˆ [ + ˆ + ˆ p α zl θ γ = ] a a Da λ r= r Dr zs. For brevy, we use ˆ () β for he esmaed consan-qualy prce of srucures, ˆ A R + ˆ + ˆ θ γ D λ D. P Laspeyres where Subsuon of = S S [ ˆ α z L [ ˆ α z L pˆ ˆ ˆ α zl + β zs = and + ˆ () β zs ] = sˆ + ˆ β z ] S S S L ˆ α z ˆ α z L L + sˆ pˆ a= a ( ) ˆ () ˆ α zl + β S S S a S r= = z no (8) yelds ˆ β () z ˆ β z S S, (9) ˆ ˆ α / S zl α S zl s a prce ndex of land and ˆ ( ) β / ˆ S zs β S s a prce ndex of srucures. Equaon (9) decomposes he overall house prce ndex no srucures and land componens; he weghs ˆ = ˆ / ˆ [ + ˆ sl α ] S zl α S zl β zs and ˆ ˆ / [ ˆ + ˆ s β z α z β z are esmaed shares of land and srucures S = ] S S S L S n he oal value of propery sales n perod. The double mpuaon mehod ensures ha he weghs sum o uny. The prce ndexes of land and srucures n (9) are Laspeyres-ype ndexes and can be wren as weghed averages of prce relaves for he ndvdual properes. For S L L L / ˆ L α L example, he Laspeyres prce ndex of land can be wren as sˆ ( ˆ α / ˆ α ), where he weghs sˆ = ˆ ˆ L zl / α S α z for he prce relaves ˆ α reflec he shares of L he properes n he esmaed value of land (mplcly) sold n perod. Properes wh relavely large value shares, lke properes n wealhy and sough-afer neghborhoods wh large plo szes and hgh land prces, herefore have a bg nfluence on he ndex. r r z S 8 Hll and Melser (28) dscuss dfferen ypes of hedonc mpuaon ndexes n he conex of housng. For a general dscusson of he dfference beween hedonc mpuaon ndexes and me dummy ndexes, see Dewer e al. (29) and de Haan (2). 2

14 An alernave o he Laspeyres prce ndex gven by (9) s he hedonc double mpuaon Paasche prce ndex, defned on he sample ( =,..., T ) : S of properes sold n perod pˆ S P Paasche = ( ) pˆ S. (2) The mpued consan-qualy prces p ˆ are esmaes of he prces ha would preval ( ) n perod f he propery characerscs were hose of perod, whch are esmaed as ( ) ( ) pˆ = ˆ α z ˆ L + β zs, where ˆ ( ) A = ˆ R + ˆ + = ˆ β θ γ a a Da λ r= r Dr denoes he perod consan-qualy prce of srucures. By subsung he consan-qualy prces and he predced prces pˆ = ˆ α z + ˆ β z no equaon (2), he mpuaon Paasche ndex can be wren as P Paasche = S S L L [ ˆ α z + ˆ L β zs ] = sˆ ( ) [ ˆ α z + ˆ β z ] S S () L S S ˆ α z ˆ α z L L + sˆ () S S S ˆ α S zl / ˆ α S zl and ˆ S zs / S () ˆL ˆ β z ˆ β ( ) ˆ ( ) S z S, (2) where β β zs are Paasche prce ndexes of land and srucures, whch are weghed by s = + S zl S z ˆ ( ) ˆ α / [ ˆ α L β zs ] () and s ˆ ˆ z / [ ˆ z + ˆ ( ) β α β z ]. The weghs are now of a hybrd naure S = S S S L S and reflec he shares of land and srucures n he esmaed oal value of propery sales n perod, evaluaed a base perod prces. A drawback of he above ndexes s ha hey are based on he sample of eher he base perod or he comparson perod, bu no on boh samples. When consrucng an ndex gong from o, he sales n boh perods should deally be aken no accoun n a symmerc fashon. The double mpuaon Fsher prce ndex [ P P ] 2 P = (22) Fsher Laspeyres Paasche does so by akng he geomerc mean of he Laspeyres and Paasche prce ndexes. In he emprcal secon of he paper, we wll esmae all hree ypes of ndexes. An exac decomposon of he Fsher ndex no srucures and land componens s no possble. Due o he fxed weghs, he Laspeyres ndex and s decomposon are relavely easy o explan. So, even hough we prefer he Fsher ndex, we are nclned o mplemen he Laspeyres ndex n sascal pracce when he numercal dfferences are small. 3

15 5. Emprcal evdence 5. The daa se The daa se we wll use was provded by he Duch assocaon of real esae agens. I conans resdenal propery sales for a small cy (populaon s around 6,) n he norheasern par of he Neherlands, he cy of A, and covers he frs quarer of 998 o he second quarer of 28. Sascs Neherlands has geocoded he daa. We decded o exclude sales on condomnums and aparmens snce he reamen of land deserves specal aenon n hs case. The resulng oal number of sales n our daa se durng he en-year perod s 6,397, represenng approxmaely 75% of all resdenal propery ransacons n A. The daa se conans nformaon on he me of sale, ransacon prce, a range of characerscs for he srucure, and characerscs for land. We ncluded only hree srucural characerscs n our models,.e., usable floor space, buldng perod and ype of house. For land, we used plo sze and poscode or laude/longude. Afer removng 44 observaons wh mssng values, ransacon prces below,, more han rooms, or raos of plo sze o srucure sze (usable floor space) larger han, we were lef wh 6,353 observaons durng he sample perod. Table A n he Appendx repors summary sascs by year for he numercal varables. The average ransacon prce sgnfcanly ncreased from 998 o 27 and hen slghly decreased durng he frs half of 28 (when he fnancal crss sared). The urban area of he cy of A seems o have expanded along he eas-wes axs; he sandard devaon of he x coordnae n laer years s generally much larger han ha n earler years. 5.2 Esmaon resuls for hedonc models Gven he small sze of he cy of A and he relavely low number of observaons, we decded o use annual daa; n fuure work we wll probably be usng b-annual daa. Three normalzed hedonc equaons were esmaed: model (8), whch has no locaon characerscs a all (denoed as OLS n he ables and fgures below), model () wh 8 poscode dummy varables (OLSD), and model (2) wh propery-specfc land prces 4

16 (MGWR). The las model was esmaed by mxed geographcally weghed regresson usng he sofware package GWR4.. 9 Consderng ha he propery ransacons are no evenly dsrbued across space, we used he adapve b-square funcon o consruc he weghng scheme. In hs case, he bandwdh s generally referred o as he wndow sze, and s selecon procedure s equvalen o he choce of he number of neares neghbors. We derved he opmal bandwdh usng he Golden Secon Search approach based on mnmzng CV scores n a wndow-sze range of % o 9%. There s a unque opmal wndow sze for each annual sample n erms of predcon power; he CV scores ndcaed ha was around % for mos of he years, excep for 998 (5%), 2 (36%), and 23 (29%). Ye, for he consrucon of prce ndexes, we would prefer a fxed wndow sze for all years, especally snce he number of sales s almos evenly spread across he whole perod. So we have chosen a wndow sze of % for every year, leadng o 6 neares neghbors ha were used n he esmaon of he MGWR models. To compare he performance of he hree propery prce models, wo sascs were calculaed, he Correced Akake Informaon Creron (AICc) and he Roo Mean Square Error (RMSE). The AICc akes no accoun he rade-off beween goodness-off and degrees of freedom and s defned for MGWR models by n + r( S) AICc = 2nln( ˆ) σ + nln(2π ) + n n 2 r( S) where σˆ s he esmaed sandard devaon of he error erm and r (S) he race of he ha marx descrbed n secon 3.2. The RMSE measures he varably of he absolue predcon errors of he models and s gven by RMSE = n ( y yˆ ) 2. The AICc and RMSE for each ype of model are shown n Table. Accordng o a rule of humb menoned by Foherngham e al. (22), f he dfference n he AICc for wo models s larger han 3, a sgnfcan dfference exss n erms of performance. I can be seen ha he OLSD model performs much beer han he OLS model n all of he perods, whch s no so surprsng, and n urn ha he MGWR model ouperforms 9 The sofware can be downloaded free of charge from hps://geodacener.asu.edu. The AICc expressons for he OLS and OLSD models may be found n e.g. Hurvch and Tsa (989). 5

17 he OLSD model. The same rankng s found f he RMSE s used o assess he models. These resuls sugges ha land prces ndeed vary across space and ha MGWR does a good job n esmang such nonsaonary. Table : Model esmaon and comparson OLS OLSD MGWR AICc RMSE AICc daic RMSE drmse AICc daic 2 RMSE drmse Noe: daic ndcaes he dfference of AICc beween OLS and OLSD, whle daic 2 ndcaes he dfference of AICc beween OLSD model and MGWR; drmse and drmse 2 have smlar meanngs. Table 2: Summary sascs for esmaed land prces from he MGWR model Mn Max Medan Mean Sd devaon

18 Table 2 conans summary sascs for he prce per square meer of land for he ransaced properes, esmaed usng MGWR. The average esmaed land prce s que volale; he change over me dffers grealy from ha of he average ransacon prce of he properes (see Table A. n he Appendx). Followng a sharp ncrease n 999, he esmaed average land prce peaked n 22, experenced a dramac drop n 23, and hen ncreased agan. The value n he sarng year 998 of approxmaely 45 euros per square meer of land s exremely low. Ths has a bg mpac on he correspondng land prce ndexes, as we wll see n secon 5.3. As an llusraon of he esmaed hedonc models, he 27 parameer esmaes for he srucure characerscs are gven n Table 3. Noe ha almos all esmaes dffer sgnfcanly from zero a he % level. Dummy varables for dwellngs bul afer 2 and for deached houses were no ncluded, and so he nercep erm measures he prce of srucures per square meer of lvng space (n euros) for deached houses bul afer 2. The esmaed nercep for MGWR s raher hgh n comparson wh OLSD. For each model, here s a clear endency for srucures o become less expensve as hey are geng older. Also, deached dwellngs are more expensve han oher ypes of houses, whch accords wh a pror expecaons. Table 3: Parameer esmaes for srucural characerscs, 27 OLS OLSD MGWR Inercep 56.** (46.93) 472.4** (55.59) 633.7** (75.35) Buldng perod: ** (26.85) -3.9** (36.97) -4.55** (45.2) Buldng perod: ** (24.9) ** (35.68) ** (44.86) Buldng perod: ** (24.2) ** (34.25) ** (45.46) Buldng perod: * (22.4) * (27.87) -24.4** (38.28) Terrace ** (35.8) ** (36.78) -39.5** (42.4) Corner ** (32.67) ** (32.67) ** (35.4) Semdeached ** (49.37) ** (49.85) ** (52.39) Duplex -7.43** (3.49) -49.** (3.63) -7.9** (33.94) Noe: Sandard errors n brackes;** and * denoe sgnfcance a he % and 5% level, respecvely. 7

19 5.3 A comparson of dfferen hedonc prce ndexes Changes n average propery prces and her land and srucure componens are affeced by composonal change and qualy change of he raded properes. The hedonc house prce ndexes and he land and srucures componens ha we esmaed conrol for hese effecs, and wll be neresng o see how hey are affeced by he choce of hedonc model (OLS, OLSD, or MGWR). We have esmaed chaned raher han drec ndexes because mpung he mssng prces over a long perod of me may no be useful and because he land and srucures wll be updaed annually. A dsadvanage of channg s ha he resulng ndexes canno be exacly decomposed snce hey are no conssen n aggregaon. In Fgures -3, he esmaed double mpuaon hedonc Laspeyres, Paasche and Fsher prce ndexes for he propery as a whole are ploed, based on he hree models. For each model, he (chaned) Laspeyres ndex ss above he Paasche, as expeced. The ndexes based on OLSD and MGWR are almos he same; he dfferences can hardly be noced n he graphs. So, for he house prce ndex, he ncluson of a lmed number of locaon dummy varables produces sasfacory resuls, despe he fac ha he OLSD model performs no as good as MGWR. No usng locaon nformaon a all makes a dfference hough: he Laspeyres and Paasche ndexes from he OLS model seem o be based downwards and upwards, respecvely. The bases almos cancel ou n he Fsher ndex, whch s very smlar o he Fsher ndexes produced wh he oher wo models. Fgure : Chaned hedonc mpuaon Laspeyres house prce ndex 8

20 Fgure 2: Chaned hedonc mpuaon Paasche house prce ndex Fgure 3: Chaned hedonc mpuaon Fsher house prce ndex and offcal SPAR ndex We enavely conclude ha he double mpuaon Fsher house prce ndex s nsensve o he reamen of locaon n he hedonc model. The offcal house prce ndex for he Neherlands s also ploed n Fgure 3. Our hedonc ndexes show a more modes prce ncrease. There may be a leas wo reasons for hs: house prces n he The offcal ndex s based on he Sale Prce Apprasal Rao (SPAR) mehod. For more nformaon on hs mehod, see de Haan e al. (29) and de Vres e al. (29). 9

21 cy of A apprecaed less compared o he res of he counry, or our ndexes beer adjus for qualy changes. We hnk ha he second reason s more mporan. The pcure changes when we look a he Fsher ndexes for he prce of land n Fgure 4. The OLS- and OLSD-based ndexes are smlar, bu he MWGR-based ndex behaves dfferenly. For example, beween 998 and 999 he MWGR-based ndex rses much faser han he oher wo ndexes, and beween 25 and 26 he MWGR-based ndex rses whereas he oher wo ndexes fall. These resuls are surprsng; for land n parcular, we would expec he OLSD-based ndex o be smlar o he MWGR-based ndex snce boh ndexes explcly accoun for locaon. Fgure 4: Chaned hedonc mpuaon Fsher prce ndexes for land Fgure 5 shows he hedonc mpuaon Fsher prce ndexes for srucures based on he hree models. Whle he dfferences canno be gnored, hey are less pronounced han he dfferences obaned for land. Ths s n accordance wh a pror expecaons: locaon should affec he prce of land, and s modeled as such, bu should leave he prce of srucures unaffeced. Fgures 4 and 5 rase a number of ssues. The frs ssue s wheher he rends of he (Fsher) ndexes for land and srucures are plausble. For land, hs wll be dffcul o check because nformaon on he prce change of land s currenly unavalable for he Neherlands. For srucures we use he naonwde offcal consrucon cos ndex (CCI) for dwellngs, publshed by Sascs Neherlands, as a benchmark. Ths ndex, rebased 2

22 o 998=, s also ploed n Fgure 5. Durng he frs half of he sample perod, our prce ndexes for srucures exhb roughly he same rend as he consrucon cos ndex. Durng he second half of he sample perod, he consrucon cos ndex flaens, bu he srucures prce ndexes keep rsng. A consrucon cos ndex does no necessarly have o be dencal o an mplcly derved prce ndex for srucures, and may suffer from some measuremen problems, 2 bu hs dvergence s neverheless puzzlng. Fgure 5: Chaned hedonc mpuaon Fsher prce ndexes for srucures and offcal consrucon cos ndex Imporanly, he overall propery prce ndexes are affeced mos by he changes n srucures prces; he average esmaed value share for srucures across he sample perod s.73 for he OLS and OLSD models, and.74 for MWGR. Fgure 6 shows he OLSD-based esmaes of he value shares for land and srucures. The volaly of he shares n Fgure 6, and also he volaly of he prce ndexes for land and srucures n Fgures 4 and 5, s srkng. We would no expec he rue shares and prce ndexes o be very volale. The volaly can be caused by problems such as he small number of observaons, mulcollneary, heeroskedascy, or oulers n he daa. Of course, he small-number problem can only be crcumvened by usng daa for a bgger cy, whch could also enable us o esmae b-annual nsead of annual ndexes. 2 The flaenng of he consrucon cos ndex pror beween 23 and 27 has been subjec of debae n he Neherlands. The dscusson arose because he consrucon cos ndex ncreased by only 4.9%, whch was even lower han he ncrease n he CPI of 5.8%, whle house prces were sll rapdly rsng. 2

23 Fgure 6: Esmaes of value shares of land and srucures, OLSD-based Fgure 7: Chaned Fsher prce ndexes for land and srucures, OLSD-based Mulcollneary was a bg problem faced by Dewer e al. (25) n esmang he bulder s model. I resuled n prce changes for land and srucures ha conssenly had oppose sgns. In Fgure 7, he OLSD-based Fsher ndexes for land and srucures from Fgures 4 and 5 are coped. In some years, lke n 22, he prce changes for land and srucures have oppose sgns, bu n oher years he prce changes are n he same drecon. We herefore suspec ha mulcollneary s no he man ssue nvolved. The 22

24 varance nflaon facor (VIF) for he esmaed parameers for he rao of plo sze and srucure sze dd no pon o sgnfcan mulcollneary eher. The use of he propery prce per square meer of lvng space as he dependen varable n he models (.e. he normalzaon) lkely reduced mulcollneary, bu can have led o nsably of he parameer esmaes for land and srucures f resuled n classcal heeroskedascy where he regresson resduals grow wh ncreasng raos of plo sze o srucure sze. For he OLS and OLSD models, he Breusch-Pagan es dd ndeed pon o heeroskedascy. 3 A relaed problem s he relavely small varaon n he plo sze o srucure sze raos. Scaerplos of he normalzed prces agans he plo sze o srucure sze raos showed some exreme oulers; mos of hem are n he hgher ranges of he normalzed prces and raos. To check f deleng oulers would sablze he ndexes, we removed all observaons wh raos of plo sze o srucure sze larger han 5 (nsead of ), reran OLSD regressons and calculaed chaned double mpuaon prce ndexes agan. The new OLSD-based Fsher ndexes for land and srucures are depced by he dashed lnes n Fgure 7. Compared wh he nal ndexes he volaly s slghly reduced, bu he rends have changed dramacally: he new srucure prce ndex ss above he old ndex and he new land prce ndex ss far below he old one. Ths roublng resul s ouched upon n secon 6 below. 6. Dscusson and conclusons Land s ypcally no explcly ncluded n hedonc models for house prces, whch can bas he resuls. Ignorng spaal nonsaonary of land prces can also generae bas. As far as we know, he presen paper s he frs aemp o accoun for nonsaonary of land prces n he consrucon of hedonc mpuaon house prce ndexes usng spaal economercs. We lnearzed he bulder s model proposed by Dewer, de Haan and Hendrks (25), allowed he prce of land o vary a he ndvdual propery level, and esmaed he model for he normalzed propery prce (.e., he prce of he propery per square meer of lvng space) by MGWR, a sem-paramerc mehod, on annual daa for 3 Acually, we should have used a heeroskedascy-conssen esmaor for he sandard errors n he OLS and OLSD models. Noe ha here s no formal heeroskedascy es for he MWGR model. 23

25 he Duch cy of A. We hen consruced chaned mpuaon Laspeyres, Paasche and Fsher ndexes and compared hem wh prce ndexes based on more resrcve models: a model wh no varaon n land prces and a model where land prces can vary across poscode areas, boh esmaed by OLS. The Fsher house prce ndexes were que nsensve o he choce of model, bu he Laspeyres and Paasche ndexes for he fxed land prce model dffered from hose for he models where locaon was explcly ncluded. The use of poscode area dummy varables produced prce ndexes very smlar o ndexes obaned by MGWR. Hll and Scholz (24) also concluded ha he use of geocoded daa and spaal economercs dd no mprove hedonc mpuaon house prce ndexes over models wh poscode dummy varables. 4 Ths resul s reassurng for sascal agences ha do no have he experse or resources o use more sophscaed mehods. For some purposes, separae prce ndexes for land and srucures are needed. As was demonsraed by Dewer, de Haan and Hendrks (25), hs can be a dffcul ask. A poenal problem s mulcollneary, whch arses because (n he bulder s model ) he value of he propery s spl no he value of land and he value of srucures: f he esmaed prce of land s oo hgh, hen he esmaed prce of srucures wll be oo low, gven plo and srucure sze. Probably due o he normalzaon of he propery prce, our esmaes dd no appear o suffer from severe mulcollneary. Ye, our esmaed prce ndexes for land and srucures were very volale. We can hnk of a leas wo reasons. Frs, he normalzaon of he propery prce resuled n heeroskedasc errors (and relavely lle varaon n he plo sze o srucure sze raos), leadng o unsable coeffcens and volale ndexes. Thus, alhough we reduced mulcollneary, a he same me we nroduced heeroskedascy. The second reason for he volaly of he esmaed land and srucures ndexes mgh be he lnear relaon posulaed n our models beween normalzed propery prce and plo sze o srucure sze rao. Mos lkely, he rue relaonshp s nonlnear, and he lnear specfcaon produced oulers n he hgher ranges. The msspecfcaon was confrmed when we deleed all observaons wh plo sze o srucures sze raos larger han 5; he volaly of he land and srucure prce ndexes from he OLSD model (wh poscode dummy varables) was reduced somewha bu he rends changed sgnfcanly. 4 Hll and Scholz (24) reaed locaon as a separae characersc n her hedonc models n ha hey esmaed propery-specfc shf erms for he overall propery prce raher han he prce of land. 24

26 The probable cause s ha he prce of land s dependen on he sze of he land plo: he prce per square meer of land ends o fall wh ncreasng plo sze. Dewer, de Haan and Hendrks (25) adjused for hs ype of nonlneary usng lnear splnes o model he prce of land. In fuure work we wan o modfy our models n he same spr, eher by usng splnes as well or by explcly specfyng some nonlnear funcon. Wha worres us mos s he exreme volaly of he MWGR-based ndexes for land and srucures. The MWGR mehod makes use of prces of neghborng properes, and snce neghborng properes may be expeced o have smlar plo szes, our resuls are unexpeced and counernuve. We lack an explanaon of hs fndng, bu does sugges ha he sem-paramerc MGWR approach produces nherenly unsable resuls. Thus, whle he MWGR model ouperforms he oher wo models n erms of sascal crera (AICc and RMSE) and produces a house prce ndex ha s very smlar o he OLSD model, aggravaes nsably and does no seem approprae for esmang he land and srucures componens. References Ber, C., G.F. Mullgan and S. Dall erba (27), Incorporang Spaal Varaon n Housng Arbue Prces: A Comparson of Geographcally Weghed Regresson and he Spaal Expanson Mehod, Journal of Geographcal Sysems 9, Brunsdon, C., A.S. Foherngham, and M.E. Charlon (996), Geographcally Weghed Regresson: A Mehod for Explorng Spaal Nonsaonary, Geographcal Analyss 28, Brunsdon, C., A.S. Foherngham, and M.E. Charlon (999), Some Noes on Paramerc Sgnfcance Tess for Geographcally Weghed Regresson, Journal of Regonal Scence 39, Case, E. (972), Generang Models by he Expanson Mehod : Applcaons o Geographc Research, Geographc Analyss 4, 8-9. Dewer, W.E., S. Herav and M. Slver (29), Hedonc Impuaon versus Tme Dummy Hedonc Indexes, pp n W.E. Dewer, J. Greenlees and C. Hulen (eds.), Prce Index Conceps and Measuremen, Sudes n Income and Wealh, Vol. 7. Chcago: Unversy of Chcago Press. Dewer, W.E., J. de Haan and R. Hendrks (2), The Decomposon of a House Prce Index no Land and Srucures Componens: A Hedonc Regresson Approach, The Valuaon Journal 6,

27 Dewer, W.E., J. de Haan and R. Hendrks (25), Hedonc Regressons and he Decomposon of a House Prce ndex no Land and Srucure Componens, Economerc Revews 34, DOI:.8/ Dorsey, R.E., H. Hu, W.J. Mayer, and H.C. Wang (2), Hedonc versus Repea- Sales Housng Prce Indexes for Measurng he Recen Boom-Bus Cycle, Journal of Housng Economcs 9, Eurosa, ILO, IMF, OECD, UNECE and World Bank (23), Handbook on Resdenal Propery Prce Indces. Luxemburg: Publcaons Offce of he European Unon. Foherngham, A.S., C. Brunsdon, and M.E. Charlon (998a), Geographcally Weghed Regresson: A Naural Evoluon of he Expanson Mehod for Spaal Daa Analyss, Envronmen and Plannng A 3, Foherngham, A.S., C. Brunsdon, and M.E. Charlon (998b), Scale Issues and Geographcally Weghed Regresson, n N. Tae (ed.), Scale Issues and GIS. Chcheser: Wley. Foherngham, A.S., C. Brunsdon, and M.E. Charlon (22), Geographcally Weghed Regresson: he Analyss of Spaally Varyng Relaonshps. Chcheser: John Wley & Sons. de Haan, J. (2), Hedonc Prce Indexes: A Comparson of Impuaon, Tme Dummy and Re-prcng Mehods, Jahrbücher fur Naonalökonome und Sask 23, de Haan, J., P. de Vres and E. van der Wal (29), The Measuremen of House Prces: A Revew of he Sale Prce Apprasal Rao Mehod, Journal of Economc and Socal Measuremen 34, Genaux, G. and C. Napoléone (28), Sem-paramerc Tools for Spaal Hedonc Models: An Inroducon o Mxed Geographcally Weghed Regresson and Geoaddve Models, pp. -27 n A. Baranzn jr., C. Schaerer and P. Thalmann (eds.), Hedonc Mehods n Housng Markes Prcng Envronmenal Amenes and Segregaon. New York: Sprnger. Hll, R.J. (23), Hedonc Prce Indexes for Resdenal Housng: A Survey, Evaluaon and Taxonomy, Journal of Economc Surveys 27, Hll, R.J. and D. Melser (28), Hedonc Impuaon and he Prce Index Problem: An Applcaon o Housng, Economc Inqury 46, Hll, R.J., D. Melser, and I. Syed (29), Measurng a Boom and Bus: The Sydney Housng Marke 2-26, Journal of Housng Economcs 8, Hll, R.J and M. Scholz (24), Incorporang Geospaal Daa no House Prce Indexes: A Hedonc Impuaon Approach wh Splnes, Graz Economcs Paper 24-5, Deparmen of Economcs, Unversy of Graz. 26

28 Hurvch, C.M. and C.L. Tsa (989), Regresson and Tme Seres Model Selecon n Small samples, Bomerka 76, Jones, J.P. and E. Case (992), Applcaons of he Expanson Mehod. London: Rouledge. Me, C.L., N. Wang and W. X. Zhang (26), Tesng he Imporance of he Explanaory Varables n a Mxed Geographcally Weghed Regresson Model, Envronmen and Plannng A 38, Pace, R.K., R. Barry, J.M. Clapp, and M. Rodrquez (998), Spaoemporal Auoregressve Models of Neghborhood Effecs, Journal of Real Esae Fnance and Economcs 7, Rambald, A.N. and D.S.P. Rao (2), Hedonc Predced House Prce Indces Usng Tme-Varyng Hedonc Models wh Spaal Auocorrelaon, Dscusson paper 432, School of Economcs, Unversy of Queensland. Rambald, A.N. and D.S.P. Rao (23), Economerc Modelng and Esmaon of Theorecally Conssen Housng Prce Indexes, Workng paper WP4/23, Cenre for Effcency and Producvy Analyss, School of Economcs, Unversy of Queensland. Sun, H., Y. Tu, and S. Yu (25), A Spao-Temporal Auoregressve Model for Mul- Un Resdenal Marke Analyss, Journal of Real Esae Fnance and Economcs 3, Tu, Y., S. Yu, and H. Sun (24), Transacon-Based Offce Prce Indexes: A Spaoemporal Modellng Approach, Real Esae Economcs 32, de Vres, P., J. de Haan, E. van der Wal and G. Marën (29), A House Prce Index Based on he SPAR Mehod, Journal of Housng Economcs 8,

29 Appendx Table A : Summary sascs by year Toal Obs Transacon prce (Euro) Mean S.D Un prce (Euro/m 2 ) Mean S.D Parcel sze (m 2 ) Mean S.D Floor space (m 2 ) Mean S.D Rao of parcel o floor space Mean S.D XCoordnae Mean S.D YCoordnae Mean S.D