Chapter 6 THE PARIS OECD-IMF WORKSHOP ON REAL ESTATE PRICE INDEXES: CONCLUSIONS AND FUTURE DIRECTIONS

Transcription

1 Chaper 6 THE PARIS OECD-IMF WORKSHOP ON REAL ESTATE PRICE INDEXES: CONCLUSIONS AND FUTURE DIRECTIONS W. Erwin Diewer 1 1. Inroducion This paper highlighs some of he hemes ha emerged from he OECD-IMF Workshop on Real Esae Price Indexes which was held in Paris, November 6-7, 26. The paper discusses possible uses and arge indexes for real esae price indexes and noes ha a major problem is ha i is no possible o exacly mach he qualiy of dwelling unis over ime due o he fac ha he housing sock changes in qualiy due o renovaions and depreciaion. Four alernaive mehods for consrucing real esae price indexes are discussed: he repea sales model; he use of assessmen informaion along wih propery sale informaion; sraificaion mehods and hedonic mehods. The paper noes ha he ypical hedonic regression mehod may suffer from specificaion bias and suggess a way forward. Problems wih he user cos mehod for pricing he services of owner occupied housing are also discussed. The paper is organized as follows. Secion 2 discusses he quesion: wha are appropriae arge indexes for Real Esae Prices? This secion argues ha he presen Sysem of Naional Accouns is a good saring poin for a sysemaic framework for Real Esae Price indexes bu he presen SNA has o be augmened somewha o mee he needs of economiss who are ineresed in measuring consumpion on a more comprehensive service flow basis and who are ineresed in measuring he produciviy of an economy. Secion 3 noes he fundamenal problem ha makes he consrucion of consan qualiy real esae price indexes very difficul: namely depreciaion and renovaions o srucures make he usual mached model mehodology for consrucing price indexes inapplicable. Secion 4 discusses four classes of mehods ha were suggesed a he workshop o deal wih he above problem and secion 5 discusses some addiional echnical difficulies. 1 This paper is an exended wrien version of my Discussion a he Concluding Overview session of he OECD-IMF Workshop on Real Esae Price Indexes held in Paris, November 6-7, 26. The financial assisance of he OECD and he Ausralian Research Council is graefully acknowledged, as is he hospialiy of he Cenre for Applied Economic Research a he Universiy of New Souh Wales. The auhor hanks Paul Armknech, Sephan Arhur, David Fenwick, Jan de Haan, Johannes Hoffmann, Anne Laferrère, Alice Nakamura, Marc Prud homme, David Robers, Mick Silver, Paul Schreyer and Kam Yu for helpful commens. None of he above individuals or organizaions is responsible for any opinions expressed in his paper. Diewer, W.E. (29), The Paris OECD-IMF Workshop on Real Esae Price Indexes: Conclusions and Fuure Direcions, chaper 6, pp in W.E. Diewer, B.M. Balk, D. Fixler, K.J. Fox and A.O. Nakamura (29), PRICE AND PRODUCTIVITY MEASUREMENT: Volume 1 -- Housing, Trafford Press. Also available as a free e-publicaion a and Alice Nakamura, 29. Permission o link o, or copy or reprin, hese maerials is graned wihou resricion, including for use in commercial exbooks, wih due credi o he auhors and ediors.

2 Secion 6 discusses he problems raised by Verbrugge s (26) conribuion o he Workshop; i.e., why do user coss diverge so much from rens? Finally, secion 7 summarizes suggesions for moving he agenda forward, including a proposal for a new approach o accouning for real esae in measures of inflaion. 2. Wha Are Appropriae Targe Indexes? There are many possible arge real esae price indexes ha could be consruced. Thus i is useful o consider alernaive uses for real esae price indexes ha were suggesed a he workshop since hese uses will largely deermine wha ype of indexes should be consruced. Fenwick (26; 6) suggesed he following lis of possible uses for house price indexes: As a general macroeconomic indicaor (of inflaion); As an inpu ino he measuremen of consumer price inflaion; As an elemen in he calculaion of household (real) wealh, and As a direc inpu ino an analysis of morgage lender s exposure o risk of defaul. Arhur (26) also suggesed some (relaed) uses for real esae price indexes: Real esae price bubbles (and he subsequen collapses) have repeaedly been relaed o financial crises and hus i is imporan o measure hese price bubbles accuraely and in a way ha is comparable across counries, and Real esae price indexes are required for he proper conduc of moneary policy. Fenwick also argued ha various real esae price indexes are required for deflaion purposes in he Sysem of Naional Accouns (SNA): The primary focus of a naional accounan seeking an appropriae deflaor for naional accouns will be differen. Real esae appears in he Naional Accouns in several ways; he impued renal value received by owner occupiers for buildings, as opposed o land, is par of household final consumpion, he capial formaion in buildings, again as opposed o land, is par of gross fixed capial formaion, depreciaion, and he measuremen of he sock of fixed capial, and land values are an imporan par of he Naional sock of wealh. David Fenwick (26; 7-8) 88

3 Fenwick (26; 6) also argued ha i would be useful o develop a coheren concepual framework for an appropriae family of real esae price indexes 2 and he provides such a framework owards he end of his paper. 3 Diewer, in his oral presenaion o he Workshop, followed Fenwick and argued ha in he firs insance, real esae price saisics should serve he needs of he SNA. The reason for his is ha (wih one excepion o be discussed laer) he SNA provides a quaniaive framework where value flows and socks are sysemaically decomposed in an economically meaningful way ino price and quaniy (or volume) componens. The resuling p s and q s are he basic building blocks which are used in virually all macroeconomic models. Hence i seems imporan for price saisicians o do heir bes o mee he deflaion needs of he SNA. Before he main problem area wih he presen SNA reamen of real esae is discussed, i is useful o review a bi of basic economics. There are wo main paradigms in economics: Consumers or households maximizing uiliy subjec o heir budge consrains, and Producers maximizing profis subjec o heir producion funcion (or more generally, heir echnology) consrains. There are one period saic and many period ineremporal versions of he wo paradigms. However, for our purposes, i suffices o say ha he SNA provides he necessary daa o implemen boh models excep for he fac ha he SNA does no deal adequaely wih he consumpion of consumer durables for he needs of eiher consumer or producer modeling. The problem is ha when a consumer or producer purchases a good ha provides services over a number of years, i is no appropriae o charge he enire purchase cos o he quarer or monh when he durable is purchased; he purchase cos should be spread ou over he useful life of he durable. However, wih he imporan excepion of residenial housing, he SNA simply charges he enire cos of he durable o he period of purchase. 4 This is no an appropriae reamen of durables for many economic purposes. Thus, for he SNA household accouns, in addiion o he usual acquisiions approach reamen of consumer durables (which simply charges he enire purchase cos o he period of purchase), i would be useful o have alernaive measures of he service flows generaed by household holdings of consumer durables. There are wo alernaive approaches o consrucing such flow measures: An impued ren approach which impues marke renal prices for he same ype of service (if such prices are available), and A user cos approach which forms an esimae of wha he cos would be of buying he durable a he beginning of he period, using he services of he durable during he period and hen selling i a he end of he period. This esimaed cos also includes he ineres cos ha is 2 I can be seen ha user needs will vary and ha in some insances, more han one measure of house price or real esae inflaion may be required. I can also be seen ha coherence beween differen measures and wih oher economic saisics is imporan and ha achieving his will be especially difficul as saisicians are unlikely o have an ideal se of price indicaors available o hem. David Fenwick (26, p. 8). 3 See Fenwick (26, pp. 8-11). 4 More specifically, for owner occupied residenial housing, he SNA incorporaes esimaes of he period by period flow of housing services. One reason his is done is o improve he comparabiliy of he SNA beween naions where he percenage of households living in owner occupied versus renal housing is very differen. 89

4 associaed wih value of he capial ha is ied up in he purchase of he durable. 5 However, if mos owners of some sor of durable, in fac, coninue o hold i for muliple periods, hen he buy-use-sell sequence migh be priced ou as a per-year average over he expeced holding period, using he available informaion on he beginning and end of period prices and he ransacion coss ha would be involved in buying and hen selling once over he expeced holding period. We discuss he relaive meris of he above wo service flow mehods for valuing housing services in secion 6 below. For addiional maerial on he various economic approaches o he reamen of durables and housing in paricular, see Diewer (22; ), (23), Verbrugge (26), and Chaper 23, Durables and User Coss, in he Inernaional Labour Organizaion (ILO) Consumer Price Index Manual (24). On he producer side of he SNA, he service flows generaed by durable inpus ha are used o produce goods and services are buried in Gross Operaing Surplus. Jorgenson and Griliches (1967) (1972) showed how gross operaing surplus could be decomposed ino price and quaniy componens using he user cos idea and heir work led direcly o he firs naional saisical agency produciviy program; see he Bureau of Labor Saisics (1983). 6 Schreyer, Diewer and Harrison (25) argued ha his produciviy oriened approach o he Sysem of Naional Accouns could be regarded as a naural exension of he presen SNA where he exended version provides a decomposiion of a value flow (Gross Operaing Surplus) ino price and quaniy (or volume) componens. We will argue below ha if he SNA is expanded o exhibi he service flows ha are associaed wih he household and producion secors purchases of durable goods, hen he resuling Durables Augmened Sysem of Naional Accouns (DASNA) 7 provides a naural framework for a family of real esae price indexes. In his augmened sysem of naional accouns, household wealh and consumpion will be measured in real and nominal erms. This will enail measures of he household secor s sock of residenial wealh and i will be of ineres o decompose his value measure ino price and quaniy (or volume) componens. I will also be useful o decompose he residenial housing sock aggregae ino various subcomponens such as: by ype of housing, by locaion or region, by he proporion of land and srucures in he aggregae value, by age (in paricular, new housing should be disinguished), and by wheher he residence is rened or owned. 5 The user cos idea can be raced back o Walras in 1874; see Walras (1954). 6 The lis of counries ha now have official produciviy programs includes he Unied Saes, Canada, he Unied Kingdom, Ausralia, New Zealand and Swizerland. The EU KLEMS projec (for he EU KLEMS daabase and relaed informaion, see hp:// is developing produciviy accouns for many European counries using he Jorgenson and Griliches mehodology, which is described in more deail in Schreyer (21). For recen exensions and modificaions, see Schreyer (26). 7 Such an accouning sysem is laid ou and implemened for he Unied Saes by Jorgenson and Landefeld (26). 9

5 Each of hese subaggregaes should be decomposed ino price and volume componens if possible. The DASNA will also require a measure of he flow of services from households consumpion of he services of heir long lived consumer durables such as moor vehicles and owner occupied housing. 8 Thus i will be necessary o eiher implemen he renal equivalence approach (as is currenly recommended in he SNA) or he user cos approach (or some alernaive) for valuing he services of Owner Occupied Housing in his exended sysem of accouns. 9 Turning now o he producer side of he DASNA, for produciviy measuremen purposes, we will wan user coss for owned commercial, indusrial and agriculural properies. In order o form wealh esimaes, we will require esimaes for he value of commercial, indusrial and agriculural properies and decomposiions of he values ino price and volume componens. The price componens can be used as basic building blocks o form user coss for he various ypes of propery. I will also be useful o decompose he business propery sock aggregaes ino various subcomponens such as: by ype of srucure, by locaion or region, by he proporion of land and srucures in he aggregae value, by age (in paricular, new srucures should be disinguished), and by wheher he srucure is rened or owned. If we hink back o he lis of uses for real esae price indexes suggesed by Fenwick and Arhur earlier in his secion, i can be seen ha if we had all of he price indexes for implemening he DASNA as suggesed above, hen virually all of he user needs could be me by his family of naional accouns ype real esae price indexes. The Durables Augmened SNA is a naural framework for he developmen of real esae price indexes ha would mee comprehensive user needs. We urn now o a discussion of he many echnical issues ha arise when rying o consruc a propery price index. 3. Failure of he Tradiional Mached Model Mehodology in he Real Esae Conex Consider he problems involved in consrucing a consan qualiy price index for a class of residenial dwelling unis or business srucures. The saring poin for consrucing any price 8 For shor lived household durables, i is no worh he boher of capializing hese socks; he usual acquisiions approach will suffice for hese asses. 9 We will reurn o his opic in secion 6 below. 91

6 index beween wo ime periods is o collec prices on exacly he same produc or iem for he wo ime periods under consideraion; his is he sandard mached model mehodology. 1 The fundamenal problem ha price saisicians face when rying o consruc a real esae price index is ha exac maching of properies over ime is ofen no possible for wo reasons: The propery depreciaes over ime (he depreciaion problem), and The propery may have had major repairs, addiions or remodeling done o i beween he wo ime periods under consideraion (he renovaions problem). Because of he above wo problems, some form of impuaion or indirec esimaion will be required. A hird problem ha faces many European counries is he problem of low urnover of properies; i.e., if he sales of properies are very infrequen, hen even if he depreciaion and renovaions problems could be solved, here would sill be a problem in consrucing a saisfacory propery price index because of he low incidence of resales. 11 A fourh problem should be menioned a his poin. For some purposes, i is desirable o decompose he real esae price index ino wo separae consan qualiy componens: A componen ha measures he change in he price of he srucure, and A componen ha measures he change in he price of he underlying land. In he following secion, we will look a some of he mehods ha were suggesed by conference paricipans o consruc consan qualiy real esae price indexes for he land and srucures aken ogeher. The problem of decomposing a real esae price index ino is srucure and land componens is deferred unil secion 5 below. 4. Suggesed Mehods for Consrucing Consan Qualiy Real Esae Price Indexes 4.1 The Repea Sales Mehod The repea sales approach is due o Bailey, Muh and Nourse (1963), who saw heir procedure as a generalizaion of he chained mached model mehodology ha was used by he early pioneers in he consrucion of real esae price indexes like Wyngarden (1927) and Wenzlick (1952). We will no describe he echnical deails of he mehod; we simply noe ha he mehod uses informaion on real esae properies which rade on he marke more han once over he sample period. 12 By uilizing informaion on properies ha are legally he same ha 1 For a deailed descripion of how his mehodology works, see Chaper 2, Elemenary Indices, in he ILO (24). 11 Relaed problems are ha he mix of ransacions can change over ime and in fac enirely new ypes of housing can ener he marke. 12 See Case and Shiller (1989) and Diewer (23, pp ) for deailed echnical descripions of he mehod. Diewer showed how he repea sales mehod is relaed o Summers (1973) counry produc dummy model used in inernaional price comparisons and he produc dummy variable hedonic regression model proposed by Aizcorbe, Corrado and Doms (21). 92

7 rade more han one period, he repea sales mehod aemps o hold he qualiy of he properies consan over ime. We now discuss some of he advanages and disadvanages of he repea sales mehod. 13 The main advanages of he repea sales model are: The availabiliy of source daa from adminisraive and real esae indusry records on propery sales, so ha no impuaions are involved (if no adjusmens are made for renovaions), and Reproducibiliy of he resuls; i.e., differen saisicians given he same daa on he sales of real esae properies will come up wih he same esimae of qualiy adjused price change. 14 The main disadvanages of he repea sales model are: I does no use all of he available informaion on propery sales; i uses only informaion on properies ha have sold more han once during he sample period. 15 I canno deal adequaely wih depreciaion of he dwelling uni or srucure. I canno deal adequaely wih unis ha have undergone major repairs or renovaions. 16 In conras, a general hedonic regression model for housing or srucures can adjus for he 13 Throughou his secion, we will discuss he relaive meris of he differen mehods ha have been suggesed for consrucing propery price indexes. For a similar (and perhaps more comprehensive) discussion, see Hoffmann and Lorenz (26, pp. 2-6). 14 Hedonic regression models suffer from a reproducibiliy problem; i.e., differen saisicians will use differen characerisics variables, differen funcional forms and differen sochasic specificaions, possibly leading o quie differen resuls. However, in acual applied use, he repea sales model is no as reproducible in pracice as indicaed in he main ex because, in some varians of he mehod, houses ha are flipped (sold very rapidly) and houses ha have no sold for long periods are excluded from he regressions or regression-based adjusmens are made o ry o allow for changes in properies in he inervals beween resale. The exac mehods for making hese sors of adjusmens vary among analyss and over ime o ime, leading o a lack of reproducibiliy. 15 Some of he papers presened a he workshop suggesed ha he repea sales mehod migh lead o esimaes of price change ha were biased upwards, since ofen sellers of properies underake major renovaions and repairs jus before puing heir properies on he marke, leading o a lack of comparabiliy of he uni from is previous sale if he pure repea sales approach is used. For example, Erna van der Wal, Dick er Seege and Ber Kroese (26, p. 3) wrie ha: The repea sales mehod does no enirely adjus for changes in qualiy of he dwellings. If a dwelling undergoes a major renovaion or even an exension beween wo ransacion momens, he repea sales mehod will no accoun for his. The las ransacion price may in ha case be oo high, which resuls in an overesimaion of he index. Andrew Levenis (26, p. 9) wries ha: Research has suggesed ha appreciaion raes for houses ha sell may no be he same as appreciaion raes for he res of he housing sock. Levenis goes on o cie maerial by Sephens, Li, Lekkas, Abraham, Calhoun and Kimner (25) on his poin. Finally, Gudnason and Jonsdoir (26) observe ha: The problem wih his mehod is he risk for bias; e.g., when major renovaion and oher changes have been made on he house which increases he qualiy or if he wear of he house has been high, causing a decrease in he qualiy. Such changes are no capured by his mehod. Furhermore, in Iceland, his mehod canno be used because he numbers of housing ransacions are oo few and hus here are no enough repeaed sales o enable calculaion of he repeaed sales index. 16 Case and Shiller (1989) used a varian of he repea sales mehod wih U.S. daa on house sales in four major ciies over he years They aemped o deal wih he depreciaion and renovaion problems as follows: The apes conain acual sales prices and oher informaion abou he homes. We exraced from he apes for each ciy a file of daa on houses sold wice for which here was no apparen qualiy change and for which convenional 93

8 effecs of renovaions and exensions if (real) expendiures on renovaions and exensions are known and can be emporally mached wih he daa on propery ransacions. 17 The mehod canno be used if indexes are required for very fine classificaions of he ype of propery, due o insufficien observaions. For example, if monhly propery price indexes are required, he mehod may fail due o a lack of marke sales for smaller caegories of propery. In principle, esimaes for pas price change obained by he repea sales mehod should be updaed as new ransacion informaion becomes available. 18 Thus he Repea Sales propery price index is subjec o never ending revision. We urn now o anoher class of mehods suggesed by workshop paricipans for forming consan qualiy propery price indexes. 4.2 The Use of Assessmen Informaion Mos counries ax real esae propery. Hence, mos counries have some sor of official valuaion office ha provides periodic appraisals of all axable real esae propery. The paper by van der Wal, er Seege and Kroese (26) presened a he Workshop describes how Saisics Neherlands uses appraisal informaion in order o consruc a propery price index. In paricular, he SPAR (Sales Price Appraisal Raio) Mehod is described as follows: 19 This mehod has been used in New Zealand since he early 196s. I also uses mached pairs, bu unlike he Repea Sales mehod, he SPAR mehod relies on nearly all ransacions ha have occurred in a given housing marke, and hence should be less prone o sample selecion bias. The firs measure in each pair is he official governmen appraisal of he propery, while he second measure is he maching ransacion price. The raio of he sale price and he appraisal of all sold dwellings in he base period, =, serves as he denominaor. The numeraor is he raio of he selling price in he reference period, =, and he appraisal price in he base period for all dwellings ha were sold in he reference period, van der Wal, er Seege and Kroese (26, p. 3). We will follow he example of van der Wal, er Seege and Kroese and describe he SPAR mehod algebraically. Denoe he number of sales of a cerain ype of real esae in he morgages applied (Karl E. Case and Rober J. Shiller, 1989, pp ). I is someimes argued ha renovaions are approximaely equal o depreciaion. While his may be rue in he aggregae, i cerainly is no rue for individual dwelling unis because, over ime, many unis are demolished. 17 However, usually informaion on mainenance and renovaion expendiures is no available in he conex of esimaing a hedonic regression model for housing. Malpezzi, Ozanne and Thibodeau (1987, pp ) commen on his problem as follows: If all unis are idenically consruced, inflaion is absen, and he rae of mainenance and repair expendiures is he same for all unis, hen precise measuremen of he rae of depreciaion is possible by observing he value or ren of wo or more unis of differen ages. To accuraely esimae he effecs of aging on values and rens, i is necessary o conrol for inflaion, qualiy differences in housing unis, and locaion. The hedonic echnique conrols for differences in dwelling qualiy and inflaion raes bu canno conrol for mos differences in mainenance (excep o he exen ha hey are correlaed wih locaion). 18 Anoher drawback on he RS mehod is he fac ha previously published index numbers will be revised when new daa are added o he sample, Erna van der Wal, Dick er Seege and Ber Kroese (26, p. 3). 19 van der Wal, er Seege and Kroese (26, p. 3) noed ha his mehod is described in more deail in Bourassa, Hoesli and Sun (26). The conference presenaion by Saisics Denmark indicaed ha a varian of his mehod is also used in Denmark. Jan de Haan brough o my aenion ha a more comprehensive analysis of he SPAR mehod (similar in some respecs o he analysis in his secion) may be found in de Haan, van der Wal, er Seege and de Vries (26). 94

9 base period by N(), le he sales prices be denoed as [ S,S, K,S ] corresponding official appraisal prices as and denoe he. Similarly, denoe he number of sales of he same ype of propery in he curren period by N(), le he sales prices be denoed as [ S1,S2, K,SN() ] S 1 2 N() S [ A1,A 2, K, A N() ] A and denoe he corresponding official appraisal A prices in he base period as [ A1,A 2, K, A N () ] A. The reason for he double superscrip on he appraisals is ha we are assuming here ha he appraisals are only made periodically; i.e., in period bu no period. Thus he firs superscrip indicaes ha he appraisal was made in period and he second superscrip, or, indicaes ha he propery was sold eiher in period or. The value weighed SPAR index defined by van der Wal, er Seege and Kroese (26; 4) in our noaion is defined as follows: N() N() N() N() i = 1 i i = 1 i n = 1 n n = 1 n (1) P (S,S,A,A ) [ S / A ]/[ S / A ]. DSPAR We have labeled he index defined by (1) as P DSPAR where he D sands for Duo, since he index formula on he righ hand side of (1) is closely relaed o he Duo formula ha occurs in he heory for he elemenary price index componens, which are he lowes level of aggregaion for he componens used in compiling a price index. 2 Wha is he inuiive jusificaion for formula (1)? One way o jusify (1) is o suppose ha he value for each propery ransacion in period is equal o a period common price S n level for he ype of propery under consideraion, say, so ha: n Q n (2) S = P, n = 1,2, K, N(). P say, imes a qualiy adjusmen facor, Nex, we assume ha he period assessed value for ransaced propery n, n Q, is equal o he common price level P imes he qualiy adjusmen facor imes an error erm, which we wrie as 1+ ε n Thus we have n n A n, and which is assumed o be independenly disribued wih zero mean. n (3) A = P Q (1 + ε ), n = 1,2, K, N(), wih (4) [ ε ], n = 1,2, K, N(). E n = where E is he expecaion operaor. Q n 21 N() 1 N() k 1 2 If he erm n = S n / = A n on he righ hand side of (1) is equal o 1, hen he index reduces o a Duo index. For he properies of Duo indexes, see Chaper 2, Elemenary Indices, in ILO (24) or IMF (24). 21 This sochasic specificaion reflecs he fac ha he errors are more likely o be muliplicaive han addiive. 95

10 value Turning now o a model for he period propery price ransacions, we suppose ha he S n for each propery ransacion in period is equal o a period common price level for he given propery ype, P say, imes a qualiy adjusmen facor, say, so ha: i i (5) S = P Q, i = 1,2, K, N(). A i Nex, we assume ha he period assessed value for propery i ransaced in period,, is equal o he period price level P imes he qualiy adjusmen facor imes an independenly disribued error erm, which we wrie as 1+ ε i. 22 Thus we have: i i i (6) A = P Q (1 + ε ) i = 1,2, K, N(). Our goal is o obain an esimaor for he level of propery prices in period relaive o period, which is P / P. Define he share of ransaced propery n in period o he oal value of properies ransaced in period, N() n n S k 1 s n, as follows: (7) s S / = k n = 1,2, K, N(). Similarly, define he share of ransaced propery i in period o he oal value of properies ransaced in period,, as: N() 1 s i (8) s i Si / k= Sk i = 1,2, K, N(). Subsiuing (2)-(6) ino definiion (1), and using definiions (7) and (8), and we obain he following expression for he Duo ype SPAR price index: (9) PDSPAR (S,S,A,A ) N() N() N() N() = [ i = 1 P Qi / i = 1 P Qi (1 + εi )]/[ n = 1 P Q n / n = 1 P Q n (1 + ε n )] =[ P / P ][ 1 + N() i 1 si ε N() = i ]/[1 + n= 1 s n ε n ]. Thus he Duo ype SPAR index will be unbiased for he rue propery price index, P / P, provided ha he share weighed average of he period and qualiy adjusmen errors are equal o zero; i.e., here will be no bias if N() n= 1 n n = (1) s ε and N() n= 1 n n = (11) s ε. I is likely ha he weighed sum of errors in period is equal o zero (a leas approximaely) because i is likely ha he official assessed values for period are Q i Q i 22 I is no longer likely ha he expeced value of he error erm ε i is equal o since he base period assessmens canno pick up any depreciaion and renovaion biases ha migh have occurred beween periods and. 96

11 approximaely equal o he marke ransacion values in he same period; i.e., i is likely ha (1) is a leas approximaely saisfied. However, i is no so likely ha (11) would be saisfied since he period assessed values will no reflec depreciaion and renovaions done beween periods and. If he economy is growing srongly, hen i is likely ha he value of renovaions will exceed he value of depreciaion beween periods and and hence he error erms will end o be less han and PDSPAR (S,S,A,A ) will be biased upwards. On he oher hand, if here is lile growh (or a declining populaion), hen i is likely ha he value of renovaions will be less han he value of depreciaion beween periods and and hence he error erms ε i will end o be greaer han and PDSPAR (S,S,A,A ) will be biased downwards. Varians of he Duo ype SPAR index can be defined; i.e., he equal weighed SPAR index defined by van der Wal, er Seege and Kroese (26; 4) in our noaion is defined as follows: () () (12) PCSPAR (S,S,A,A ) [ (S / A ) / N()]/[ (S n / A n ) / N()] N N i = 1 i i n = 1 N() N() = [ i = 1 {P Qi / P Qi (1 + εi )}/ N()]/[ n = 1 {P Q n / P Qn (1 + ε n )}/ N()] using (2)-(6) N() 1 N() 1 = [P / P ][ i = 1 (1 + εi ) / N()]/[ n = 1 (1 + ε n ) / N()]. We have labeled he index as P CSPAR since looking a he firs line of (12), i can be seen ha he index is a raio of wo equally weighed indexes of price relaives; i.e., he index is a raio of wo Carli indexes. 23 By looking a (12), i can be seen ha if all of he error erms ε i and are equal o zero, hen CSPAR ( S ε i P,S,A,A ) will be equal o he arge index, P / P. Of course, i is much more likely ha he period error erms, ε i, are close o zero han he period erms, ε i. If, in fac, all of he period error erms are equal o, hen i can be N() seen ha S n = A n for all n and P C reduces o he ordinary Carli index, i= 1 (Si / Ai ) / N(), 24 which is known o be biased upwards. The las equaion in (12) gives us an expression ha could be helpful in deermining he bias in his Carli ype SPAR index in he general case of errors in boh periods. However, i 1 proves o be useful o approximae he reciprocal funcion, f ( ε) (1 + ε), by he following second order Taylor series approximaion around ε = : 1 (13) f ( ε) (1 + ε) 1 ε + ε. 2 Subsiuing (13) ino he las line of (12), we find ha he Carli ype SPAR index is approximaely equal o: ε i 23 For he properies of Carli indexes, see Chaper 2, Elemenary Indices, in ILO (24). 24 See Chaper 2, Elemenary Indices, in ILO (24). 97

12 (14) P (S,S,A,A ) CSPAR N() 2 N() 2 [P / P ][ i = 1 (1 εi + [ εi ] ) / N()]/[ n = 1 (1 ε + [ ε n ] ) / N()] N() 2 N() 2 = [P / P ][1 + i = 1 ( εi + [ εi ] ) / N()]/[1 + n = 1 ( ε + [ ε n ] ) / N()] N() 2 N() 2 [P / P ][1 + i = 1 ( εi + [ εi ] ) / N()]/[1 + n = 1 [ ε n ] ) / N()] where he las approximaion follows from he (likely) assumpion ha N() n= 1 (15) ε n = ; i.e., ha he sum of he assessmen measuremen errors in period is zero. Now we can use he las line in (14) in order o assess he likely size of he bias in P CSPAR. If he economy is growing srongly, hen i is likely ha he value of renovaions will exceed he value of depreciaion beween periods and and hence he error erms ε i will end N() N() 2 o be less han so ha n= 1 εi will be posiive. The erms i= 1 [ ε i ] / N() and N() 2 n= 1 [ ε i ] / N() will boh be posiive bu he period squared errors mos likely will be much larger han he period squared errors so, overall, P CSPAR (S,S,A,A ) is likely o have a srong upward bias. If here is lile growh (or a declining populaion), hen he upward bias is likely o be N() 2 smaller. However, an upward bias is sill likely because he erm i= 1 [ ε i ] / N() is likely o N() N( be very much larger han he erms and ) / N() [ ] 2 / N(). = ε n n 1 n = 1 Wha abou he relaive sizes of he bias in he Duo SPAR formula defined by he las line in (9) versus he Carli SPAR formula defined by he las line in (14)? Assuming ha (1) N() 1 holds and using a second order approximaion analogous o (13) for [1 + n= 1 s ε n n ], we obain he following approximaion for he Duo ype SPAR formula: (16) PDSPAR (S,S,A,A ) N() [P / P ]/[1 + n= 1 s n ε n ] N() N() 2 [P / P ]/{1 n = 1 s n ε n + [ k = 1 s n ε n ] }. Comparing (14) wih (16), i can be seen ha he upward bias in he Carli ype index will generally be much greaer han he corresponding bias in he Duo ype index, since he sum of N() 2 he individual period errors divided by he number of observaions, i= 1 [ ε i ] / N(), will usually be very much greaer han he square of he period weighed sum of errors, N() 2 [ n= 1 s n ε n ]. ε n 98

13 I is eviden ha insead of using arihmeic averages of price relaives as in he Carli ype formula (12), geomeric averages could be used, leading o he following Jevons 25 ype SPAR index: N() 1/ N() N() 1/ N() 1 1 n N() 1/ N() N() 1/ N() = [ i = 1 {P Qi / P Qi (1 + εi )}] /[ n = 1 {P Q n / P Qn (1 + ε n )}] (17) PJSPAR (S,S,A,A ) = [ i = (Si / Ai )] /[ n = (S / A n )] [P / P using (2)-(6) () (1 + )] N() 1/ N() N 1/ N() ][ n = 1 (1 + ε n )] /[ i = 1 εi Under he assumpion ha here are no sysemaic appraisal errors in period so ha (4) N() is saisfied, we can assume ha n= 1 (1 + ε n ) is close. In conras, if he value of renovaions N() beween periods and exceeds he value of depreciaion, i is likely ha i= 1 (1 + ε i ) will be 26 less han one and hence P JSPAR (S,S,A,A ) will have an upward bias. I is eviden ha i is no really necessary o have he denominaor erms in he righ hand sides of definiions (1), (12) and (17) above, provided ha he assessmens are reasonably close o marke values in he base period. Thus, we can define he (regular) Duo, Carli and Jevons Marke Value o Appraisal indexes as follows: N() N() i = 1 i i = 1 (18) P (S,A ) [ S / A ]; DSPAR N() CSPAR i= 1 (19) P (S,A ) [ (S / A ) / N()] ; N() 1 i i 1/ N() i (2) PJ (S,A ) [ i= (Si / Ai )]. Using he maerial in Chaper 2 of he ILO CPI Manual (24), i can be shown ha he Jevons index PJ (S,A ) is always sricly less han he corresponding Carli index PC (S,A ), unless all of he raios S i / A i are equal o he same number, in which case he indexes are equal o each oher. I is also shown in he ILO Manual ha he Duo index will normally be fairly 27 close o he corresponding Jevons index. No one of he six index number formulae discussed above is compleely saisfacory because none of hese can deal wih he depreciaion and renovaions problem. However, if exogenous adjusmens can be made o he indexes ha consiue some sor of average adjusmen o he index for renovaions and depreciaion, hen appraisal mehods become quie aracive. If appraisals in he base period are known o be reasonably accurae, hen I would voe for he ordinary Jevons index, P (S,A ), defined by (2). If he appraisals in he base period J. 25 For he properies of Jevons indexes, see Chaper 2, Elemenary Indices, in he ILO (24) Manual. 26 Using second order Taylor series approximaion echniques, i can be shown ha he upward bias in he Jevons ype SPAR index will be less han in he corresponding Carli ype SPAR index. 27 The Manual does no recommend he use of he Carli formula since i fails he ime reversal es wih an upward bias. 99

14 are known o have a sysemaic bias, hen he Jevons ype SPAR index defined by (17), P JSPAR (S,S,A,A ), seems o be he mos aracive index. I is useful o discuss he meris of he above appraisal mehods compared o oher mehods for consrucing real esae price indexes. The main advanages of mehods ha rely on assessmen informaion in he base period and sales informaion in he curren period are: The source daa on assessmen and sales are usually available from adminisraive records. These mehods are reproducible condiional on he assessmen informaion; i.e., differen saisicians given he same daa on he sales of housing unis and he same base period assessmen informaion will come up wih he same esimae of qualiy adjused price change. The assessmen mehods use much more informaion han he repea sales mehod and hence here are fewer problems due o sparse daa. Informaion on housing or srucure characerisics is no required in order o implemen his mehod. The main disadvanages of he assessmen mehods discussed above are: They canno deal adequaely wih depreciaion of he dwelling unis or srucures. They canno deal adequaely wih unis ha have undergone major repairs or renovaions. These mehods are enirely dependen on he qualiy of he base period assessmen informaion. How exacly were he base period assessmens deermined? Were hedonic regression mehods used? Were comparable propery mehods used? 29 How can we be cerain ha he qualiy of hese base period assessmens is saisfacory? 3 The mehods discussed above do no deal wih weighing problems These indexes should be furher adjused o ake ino accoun depreciaion and renovaions bias. 29 Levenis (26) discussed some of he problems wih U.S. privae secor assessmen echniques when he discussed he work of Chinloy, Cho and Megbolugbe (1997) as follows: Using a sample of 1993 purchase price daa for which hey also had he appraisal informaion, hey compared purchase prices agains appraisals o deermine wheher here were sysemaic differences. They esimaed an upward bias of wo percen and found ha appraisals exceeded purchase price in approximaely 6 percen of he cases.... Tha appraisers exrapolae valuaions from recen resuls and have a vesed ineres in ensuring ha heir valuaions appear reasonable (and perhaps consisen) o he originaors sugges ha he volailiy of appraised values may be lower. A he same ime, he auhors believe ha he appraisals reliance on a small number of comparables almos surely leads o more volailiy han marke-wide prices, Levenis (26, pp. 5-6). 3 If he assessmens are used for axaion purposes and hey are supposed o be based on marke valuaions, hen he assessed values canno be oo far off he mark since he governmen has an incenive o make he assessmens as large as possible (o maximize ax revenue) and axpayers have he opposie incenive o have he assessmens as small as possible. 31 This is no really a major problem since he base period assessmen informaion can be used o obain saisfacory weighs. When a new official assessmen akes place, superlaive indexes can be formed beween any wo consecuive assessmen periods and inerpolaion echniques can be used o form approximae weighs for all 1

15 If informaion on housing characerisics is no available, hen he mehod can be used o form only a single index. However, in mos counries, he rae of change in real esae prices is no consan across locaions 32 and differen ypes of housing and so i is useful o be able o calculae more han one real esae price index. These assessmen based mehods canno decompose a propery price index ino srucure and land componens. 33 My overall evaluaion of hese assessmen based mehods is ha hey are quie saisfacory (and superior o repea sales mehods) if: The assessed values are used for axaion purposes; 34 The index is adjused using oher informaion for depreciaion and renovaions bias, and Only a single index is required and a decomposiion of he index ino srucure and land componens is no required. We urn now o anoher class of mehods for consrucing propery price indexes. 4.3 Sraificaion Mehods Possibly he simples approach o he consrucion of a real esae price index is o sraify or decompose he marke ino separae ypes of propery, calculae he mean (or more commonly, he median) price for all properies ransaced in ha cell for he curren period and he base period, and hen use he raio of he means as a real esae price index. The problem wih his mehod can be explained as follows: if here are oo many cells in he sraificaion, hen here may no be a sufficien number of ransacions in any given period in order o form an accurae cell average price. On he oher hand, if here are oo few cells in he sraificaion, hen he resuling cell averages will suffer from uni value bias; i.e., he mix of properies sold in each period wihin each cell may change dramaically from period o period, and hus he resuling sraified indexes do no hold qualiy consan. The sraificaion mehod can work well; for example, see Rosmundur and Jonsdoir (26; 3-5) where hey noe ha hey work wih some 8,-1, real esae ransacions per inervening periods. For descripions of superlaive indexes and heir properies, see Diewer (1976) (1978) or Chapers 15-2 of ILO (24). 32 The paper presened by Girouard, Kennedy, van den Noord and André (26, p. 26) showed ha here are regional differences in he rae of housing price change. This paper also showed ha real esae bubbles were quie common in many OECD counries. In many counries, bubbles lead o differenial raes of housing price increase; i.e., in he upward phase of he bubble, expensive properies end o increase in price more rapidly han cheaper ones and hen in he downward phase, he prices of more expensive properies end o fall more rapidly. A single index will no be able o capure hese differenial raes of price change. 33 We show laer in secion 5.1 ha he hedonic mehod can deal wih his problem. 34 A bi of cauion is called for here: someimes official assessmens are no very accurae for various reasons. 11

16 year in Iceland, which is a sufficien number of observaions o be able o produce 3 monhly subindexes. 35 Wihin each cell, geomeric raher han arihmeic averaging of prices is used: The geomeric mean replaces he arihmeic mean when averaging house prices wihin each sraum a he elemenary level. This is in line wih he calculaion mehod used a he elemenary level in he Icelandic CPI. The geomeric mean is also used in hedonic calculaions and he geomeric mean is a ypical mached model esimaor (Diewer (23b) (23c), de Haan (23)). Rosmundur Gudnason and Guorun Jonsdoir (26; 5). Even hough geomeric averaging is difficul o explain o some users, i has much o recommend i since i is more likely ha random errors in a paricular sraum of real esae are muliplicaive in naure raher han being addiive; see also Chapers 16 and 2 of ILO (24). The Ausralian Bureau of Saisics (ABS) is also experimening wih sraificaion echniques in order o produce consan qualiy housing price indexes: The approach uses locaion (suburb) o define sraa ha group ogeher (or cluser ) houses ha are similar in erms of heir price deermining characerisics. Ideally, each suburb would form is own cluser as his would maximise he homogeneiy of he cluser. However, here are insufficien numbers of observaions from quarer o quarer o suppor his mehodology. The ABS has grouped similar suburbs o form clusers wih sufficien ongoing observaions o deermine a reliable median price. ABS research showed HPI (Housing Price Index) sraa (or clusers of suburbs) were mos effecively deermined using an indicaor of socio-economic characerisics: he median price, he percenage of hree bedroom houses and he geographical locaion of he suburbs Merry Branson (26; 5). The ABS clusering procedures are ineresing and novel bu cauion seems meried in inerpreing he resuling price changes since any individual suburb migh conain a mixure of properies and hus he resuling indexes may be subjec o a cerain amoun of uni value bias. 36 We close his secion wih a discussion of he advanages and disadvanages of he sraificaion approach o he consrucion of real esae price indexes. I is useful o discuss he meris of he above appraisal mehods compared o oher mehods for consrucing real esae price indexes. The main advanages of he sraificaion mehod are: The mehod is concepually accepable, hough i depends crucially on he choice of sraificaion variables. The mehod is reproducible, condiional on an agreed on lis of sraificaion variables. Housing price indexes can be consruced for differen ypes and locaions of housing. The mehod is relaively easy o explain o users. The main disadvanages of he sraificaion mehod are: The mehod canno deal adequaely wih depreciaion of he dwelling unis or srucures. 35 However, he monhly index is produced as a moving average: The calculaion of price changes for real esae is a hree monh moving average, wih a one monh delay. Rosmundur Gudnason and Guorun Jonsdoir (26, p. 4). Gudnason and Jonsdoir (26, p. 3) also noe ha each year abou 8-1 percen of all he housing in he counry is bough and sold. 36 However, Prasad and Richards (26) show ha he sraificaion mehod applied o Ausralian house price daa gave virually he same resuls as a hedonic model ha had locaional explanaory variables. 12

17 The mehod canno deal adequaely wih unis ha have undergone major repairs or renovaions. The mehod requires some informaion on housing characerisics so ha sales ransacions can be allocaed o he correc cells in he classificaion scheme. 37 If he classificaion scheme is very coarse, hen here may be some uni value bias in he indexes. If he classificaion scheme is very fine, he deailed cell indexes may be subjec o a considerable amoun of sampling variabiliy due o small sample sizes. The mehod canno decompose a propery price index ino srucure and land componens. My overall evaluaion of he sraificaion mehod is ha i can be quie saisfacory (and superior o he repea sales and assessmen mehods 38 ) if: An appropriae level of deail is chosen for he number of cells; The index is adjused using oher informaion for depreciaion and renovaions bias, and A decomposiion of he index ino srucure and land componens is no required. I is well known ha sraificaion mehods can be regarded as special cases of general hedonic regressions 39 and so we now urn o his more general echnique. 4.4 Hedonic Mehods Very deailed exposiions of hedonic regression echniques applied o he propery marke can be found in some of he papers presened a his workshop; see, for example, Gouriéroux and Laferrère (26) and Li, Prud homme and Yu (26). Alhough here are several varians of he echnique, he basic model regresses he logarihm of he sale price of he propery on he price deermining characerisics of he propery and a ime dummy variable is added for each period in he regression (excep he base period). Once he esimaion has been compleed, hese ime dummy coefficiens can be exponeniaed and urned ino an index If no informaion on housing characerisics is used, hen he mehod is poenially subjec o remendous uni value bias. 38 The sandard assessmen mehod leads o only a single price index whereas he sraificaion mehod leads o a family of subindexes. However, if sraificaion variables are available, he assessmen mehod can also be used o produce a family of indexes. 39 See Diewer (23b) who showed ha sraificaion echniques or he use of dummy variables can be viewed as a nonparameric regression echnique. In he saisics lieraure, hese pariioning or sraificaion echniques are known as analysis of variance models; see Scheffé (1959). 4 An alernaive approach o he hedonic mehod is o esimae separae hedonic regressions for boh of he periods compared; i.e., for he base and curren period. Prediced prices can hen be generaed in each period using he esimaed hedonic regressions based on a consan characerisics se, say he characerisics of he base period. A raio of he geomeric means of he esimaed prices in each period would yield a pure price comparison based on a 13

18 Since he mehod assumes ha informaion on he characerisics of he properies sold is available, he daa can be sraified and a separae regression can be run for each imporan class of propery. Thus he hedonic regression mehod can be used o produce a family of indexes. 41 The issues associaed wih running weighed hedonic regressions are raher suble and he recen lieraure on his opic will no be reviewed here. 42 Here, we simply noe some of he advanages and disadvanages of he hedonic approach. A he ouside, i should be noed ha he usual hedonic regression model is no able o separae ou he land and srucures componens of he propery class under consideraion. However, in secion 5.1 below, we will explain how he usual mehod can be modified o give us his decomposiion. I is useful o discuss he meris of he hedonic regression mehod compared o oher mehods for consrucing real esae price indexes. The main advanages of he hedonic regression mehod are: Propery price indexes can be consruced for differen ypes and locaions of he propery under consideraion. The mehod is probably he mos efficien one for making use of he available daa. The mehod can be modified o give a decomposiion of propery prices ino land and srucures componens (see secion 5.1 below); none of he oher mehods described so far can handle his decomposiion. If he lis of propery characerisics is sufficienly deailed so ha, for example, i can be deermined wheher major mainenance projecs have been underaken (such as a new roof) and when hey were done, hen i may be possible o deal more generally wih depreciaion and renovaions problems. The main disadvanages of he hedonic mehod are: The mehod is daa inensive (i.e., i requires informaion on propery characerisics) and hus i is relaively expensive o implemen. The mehod is no enirely reproducible; i.e., differen saisicians will ener differen propery characerisics ino he regression, 43 assume differen funcional forms for he consan base period se of characerisics. A hedonic index based on a consan curren period characerisic could also be compiled, as could such indexes based on a symmeric use of base and curren period informaion. Heravi and Silver (27) ouline alernaive formulaions, and Silver and Heravi (27) provide a formal analysis of he difference beween his approach and ha of he ime dummy mehod. The French mehod also does no use he ime dummy mehod bu is oo complex o explain here. 41 This propery of he hedonic regression mehod also applies o he sraificaion mehod. The main difference beween he wo mehods is ha coninuous variables can appear in hedonic regressions (like he area of he srucure and he area of he lo size) whereas he sraificaion mehod can only work wih discree ranges for he independen variables in he regression. 42 Basically, his recen lieraure makes connecions beween weighed hedonic regressions and radiional index number formula ha use weighs; see Diewer (23c) (24) (25a) (25b); de Haan (23) (24); Silver (23), and Silver and Heravi (25). I is worh noing ha a perceived advanage of he sraificaion mehod is ha median price changes can be measured as opposed o he arihmeic mean ones ha are implici in a, say, an ordinary leas squares esimaor. However, regression esimaes can also be derived from robus esimaors from which he parameer esimaes for he price change will be similar o a median. 43 Noe ha he same criicism can be applied o sraificaion mehods; i.e., differen analyss will come up wih differen sraificaions. 14

19 regression equaion, make differen sochasic specificaions and perhaps choose differen ransformaions of he dependen variable 44, all of which can lead o perhaps differen esimaes of he amoun of overall price change. The mehod is no easy o explain o users. My overall evaluaion of he hedonic regression mehod is ha i may be probably he bes mehod ha could be used in order o consruc consan qualiy price indexes for various ypes of propery, provided ha adequae daa are available. 45 Noe ha he paper by Gouriéroux and Laferrère (26) demonsraes ha i is possible o consruc a credible, official, naionwide hedonic regression model for real esae properies. In he following wo secions, we will discuss some addiional echnical issues ha emerged from he workshop. In paricular, in secion 5.1 below, we will show how he hedonic regression echnique can be modified o provide a srucures and land price decomposiion of propery price movemens. 5. Oher Technical Issues 5.1 The Decomposiion of Real Esae Values ino Land and Srucure Componens 46 If we momenarily hink like a propery developer who is planning o build a srucure on a paricular propery, he oal cos of he propery afer he srucure is compleed will be equal o he floor space area of he srucure, say A square meers, imes he building cos per square meer, α say, plus he cos of he land, which will be equal o he cos per square meer, β say, imes he area of he land sie, B. Now hink of a sample of properies of he same general ype, which have prices p n in period and srucure areas A n and land areas B n for n = 1, K, N(), wih hese prices equal o coss of he above ype of srucure imes error erms η n which we assume have mean 1. This leads o he following hedonic regression model for period where α and β are he parameers o be esimaed in he regression: 47 n n n ] n (21) p = [ αa + βb η, n = 1, K, N(). Taking logarihms of boh sides of (21) leads o he following radiional addiive errors regression model: 48 n n n ] (22) ln p = ln[ αa + βb + ε, n = 1, K, N(), n 44 For example, he dependen variable could be he sales price of he propery or is logarihm or he sales price divided by he area of he srucure and so on. 45 This evaluaion agrees wih ha of Hoffmann and Lorenz: As far as qualiy adjusmen is concerned, he fuure will cerainly belong o hedonic mehods. Johannes Hoffman and Andreas Lorenz (26, p. 15). 46 Discussions wih Anne Laferrère helped improve on he iniial oral presenaion of he model presened in his secion. 47 Muliplicaive errors wih consan variances are more plausible han addiive errors wih consan variances; i.e., i is more likely ha expensive properies have relaively large absolue errors compared o very inexpensive properies. The muliplicaive specificaion for he errors will be consisen wih his phenomenon. 48 However, noe ha his model is no linear in he unknown parameers o be esimaed. 15

20 where he new error erms are defined as ε n lnη n for n = 1, K, N() and are assumed o have means and consan variances. Now consider he siuaion in a subsequen period. The price per square meer of his ype of srucure will have changed from α o αγ and he land cos per square meer will have changed from β o βδ where we inerpre γ as he period o price index for he ype of srucure and we inerpre δ as he period o price index for he land ha is associaed wih his ype of srucure. The period counerpars o (21) and (22) are: n n n ] n (23) p = [ αγ A + βδ B η, n = 1, K, N() ; n n n ] (24) ln p = ln[ αγ A + βδ B + ε, n = 1, K, N(), n n n where ε ln η for n = 1, K, N(), he period propery prices are, and he corresponding A n srucure and land areas are and B for n = 1, K, N(). n Equaions (22) and (24) can be run as a sysem of nonlinear hedonic regressions and esimaes can be obained for he four parameers, α, β, γ and δ. The main parameers of ineres are, of course, γ and δ, which can be inerpreed as price indexes for he price of a square meer of his ype of srucure and for he price per meer squared of he underlying land, respecively. The above very basic nonlinear hedonic regression framework can be generalized o encompass he radiional array of characerisics ha are used in real esae hedonic regressions. Thus suppose ha we can associae wih each propery n ha is ransaced in each period a lis of K characerisics X,X, K, X ha are price deermining characerisics for he srucure n1 n2 nk and a similar lis of M characerisics Y n,y n, K, YnM ha are price deermining characerisics for he ype of land beneah he srucure. The equaions which generalize (22) and (24) o he presen seup are he following ones: K k 1 nk n (25) ln p n = ln{[ α + = X α k ] A + [ β + = Ynmβm ]B } + ε n, n = 1, K, N(), K k 1 nk (26) ln p n = ln{ γ [ α + = X αk ]An + δ [ β + = Ynmβm ]Bn} + ε n, n = 1, K, N(), where he parameers o be esimaed are now he K + 1 qualiy of srucure parameers, α, α1, K, αk, he M + 1 qualiy of land parameers, β, β1, K, βm, he period price index for srucures parameer, and he period price index for he land underlying he srucures γ K parameer δ. Noe ha [ α + k= 1 Xnkαk ] in (25) and (26) replaces he single srucures M 1 Y qualiy parameer α in (22) and (24) and [ β + nm β m = m ] in (25) and (26) replaces he single land qualiy parameer β in (22) and (24). 1 2 M m 1 M m 1 n p n 16