St. Gallen, Switzerland, August 22-28, 2010

Transcription

1 Session Number: Parallel Session 2A Time: Monday, Augus 23, PM Paper Prepared for he 31s General Conference of The Inernaional Associaion for Research in Income and Wealh S. Gallen, Swizerland, Augus 22-28, 2010 The Decomposiion of a House Price index ino Land and Srucures Componens: A Hedonic Regression Approach W. Erwin Diewer, Jan de Haan and Rens Hendriks For addiional informaion please conac: Name: W. Erwin Diewer Affiliaion: Universiy of Briish Columbia Address: diewer@econ.ubc.ca This paper is posed on he following websie: hp://

2 1 The Decomposiion of a House Price index ino Land and Srucures Componens: A Hedonic Regression Approach W. Erwin Diewer, Jan de Haan and Rens Hendriks, 1 Revised May 26, 2010 Discussion Paper 10-01, Deparmen of Economics, The Universiy of Briish Columbia, Vancouver, Canada, V6T 1Z1. diewer@econ.ubc.ca Absrac The paper uses hedonic regression echniques in order o decompose he price of a house ino land and srucure componens using readily available real esae sales daa for a Duch ciy. In order o ge sensible resuls, i proved necessary o use a nonlinear regression model using daa ha covered muliple ime periods. I also proved o be necessary o impose some monooniciy resricions on he price of land and srucures. The resuls of he addiive model were compared wih he resuls of a radiional logarihmic hedonic regression model. Key Words Propery price indexes, hedonic regressions, repea sales mehod, rolling year indexes, Fisher ideal indexes. Journal of Economic Lieraure Classificaion Numbers C2, C23, C43, D12, E31, R21. 1 A preliminary version of his paper was presened a he Economic Measuremen Group Workshop, 2009, December 9-11, Crowne Plaza Hoel, Coogee Beach, Sydney, Ausralia. Revised January 28, W. Erwin Diewer: Deparmen of Economics, Universiy of Briish Columbia, Vancouver B.C., Canada, V6T 1Z1 ( diewer@econ.ubc.ca); Jan de Haan, Saisics Neherlands ( Jhhn@cbs.nl) and Rens Hendriks, Saisics Neherlands ( r.hendriks@cbs.nl ). The auhors hank Chrisopher O Donnell, Alice Nakamura and Keih Woolford for helpful commens. The auhors graefully acknowledge he financial suppor from he Cenre for Applied Economic Research a he Universiy of New Souh Wales, he Ausralian Research Council (LP and LP ) and he Social Science and Humaniies Research Council of Canada. None of he above individuals and insiuions are responsible for he conen of his paper.

3 2 1. Inroducion Our goal in his paper is o use readily available muliple lising daa on sales of residenial properies and o somehow decompose he sales price of each propery ino a land componen and a srucures componen. We will use he daa peraining o he sales of deached houses in a small Duch ciy for 10 quarers, saring in January In secion 2, we will consider a very simple hedonic regression model where we use informaion on only hree characerisics of he propery: he lo size, he size of he srucure and he (approximae) age of he srucure. We run a separae hedonic regression for each quarer which lead o esimaed prices for land and srucures for each quarer. These esimaed characerisics prices can hen be ino land and srucures prices covering he 10 quarers of daa in our sample. We posulae ha he value of a residenial propery is he sum of wo componens: he value of he land which he srucure sis on plus he value of he residenial srucure. Thus our approach o he valuaion of a residenial propery is essenially a crude cos of producion approach. Noe ha he overall value of he propery is assumed o be he sum of hese wo componens. In secion 3, we generalize he model explained in secion 2 o allow for he observed fac ha he per uni area price of a propery ends o decline as he size of he lo increases (a leas for large los). We use a simple linear spline model wih 2 break poins. Again, a separae hedonic regression is run for each period and he resuls of hese separae regressions were linked ogeher o provide separae land and srucures price indexes (along wih an overall price index ha combined hese wo componens). The models described in secions 2 and 3 were no very successful. The problem is he variabiliy in he daa and his volailiy leads o a endency for he regression models o fi he ouliers, leading o volaile esimaes for he price of land and srucures. Thus in secion 4, we noe ha since he median price of he houses sold in each quarer never declined, i is likely ha he underlying separae land and srucures prices also did no decline over our sample period. Thus we imposed his monooniciy resricion on our nonlinear regression model by using squared coefficiens and nonlinear regression echniques in one big regression using all 10 quarers of daa. We obained reasonable esimaes for he land and srucures componens using his echnique.

4 3 Buoyed by he success of our quarerly model, we implemened he model using monhly daa insead of quarerly daa in secion 5. This is more challenging since we had only 30 o 60 observaions for each monh. However, he monhly model also worked reasonably well and when we aggregaed he monhly resuls ino quarerly resuls, we obained quarerly resuls which were similar o he resuls obained in secion 4. In secion 6, we decided o compare our quarerly resuls wih a more radiional hedonic regression model for residenial properies. In his more radiional approach, he log of he propery price is regressed on eiher he logs of he main characerisics of he propery (he land area and he floor space area) or on he levels of he main characerisics, wih dummy variables o represen quarer o quarer price change. We found ha he log-log regression fi he daa much beer han he log-levels regression and he overall index of prices generaed by he log-log regression was quie close o our overall index of prices generaed by he cos of producion model explained in secion 4. However, when we used he log-log model o generae separae price index series for land and for srucures, he resuls did no seem o be credible. Secion 7 concludes wih an agenda for furher research on his opic. 2. Model 1: A Very Simple Model Hedonic regression models are frequenly used o obain consan qualiy price indexes for owner occupied housing. 2 Alhough here are many 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. 3 Since hedonic regression mehods assume 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 hedonic regression mehods can be used o produce a family of consan qualiy price indexes for various ypes of propery. 4 A real esae propery has wo imporan price deermining characerisics: 5 2 See for example, Crone, Nakamura and Voih (2000) (2009) Diewer, Nakamura and Nakamura (2009), Gouriéroux and Laferrère (2009), Hill, Melser and Syed (2009) and Li, Prud homme and Yu (2006). 3 An alernaive approach o he ime dummy hedonic mehod is o esimae separae hedonic regressions for boh of he periods compared; i.e., for he base and curren period. See Haan (2008) (2009) and Diewer, Heravi and Silver (2009) for discussions and comparisons beween hese alernaive approaches. 4 This propery of he hedonic regression mehod also applies o sraificaion mehods. 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 sraificaion mehods can only work wih discree ranges for he independen variables in he regression. Typically, hedonic regressions are more parsimonious; i.e., hey require fewer parameers o explain he daa as opposed o sraificaion mehods. 5 A hird imporan characerisic is he locaion of he propery; i.e., how far is he propery from shopping ceners, places of employmen, hospials and good schools; does he propery have a view; is he propery subjec o noise or pariculae polluion and so on. The presence or lack of hese ameniies will affec he

5 4 The land area of he propery and The livable floor space area of he srucure. For some purposes, i would be very useful o decompose he overall price of a propery ino addiive componens ha refleced he value of he land ha he srucure sis on and he value of he srucure. The purpose of he presen paper is o deermine wheher a hedonic regression echnique could provide such a decomposiion. Diewer (2007) suggesed some possible hedonic regression models ha migh lead o addiive decomposiions of an overall propery price ino land and srucures componens. 6 We will now ouline his suggesed model (wih a few modificaions). 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 S square meers, imes he building cos per square meer, say, plus he cos of he land, which will be equal o he cos per square meer, say, imes he area of he land sie, L. Now hink of a sample of properies of he same general ype, which have prices v n in period 7 and srucure areas S n and land areas L n for n = 1,...,N(), and hese prices are equal o coss of he above ype plus error erms n which we assume have means 0. This leads o he following hedonic regression model for period where and are he parameers o be esimaed in he regression: 8 (1) v n = L n + S n + n ; n = 1,...,N(); = 1,...,T. Noe ha he wo characerisics in our simple model are he quaniies of land L n and he quaniies of srucure S n associaed wih he sale of propery n in period and he wo consan qualiy prices in period are he price of a square meer of land and he price of a square meer of srucure floor space. Finally, noe ha separae linear regressions can be run of he form (1) for each period in our sample. price of land in he neighbourhood and hus i is imporan o sraify he sample in order o conrol for hese neighbourhood effecs. In our example, he Duch own of A is small enough and homogeneous enough so ha hese neighbourhood effecs can be negleced. 6 Two oher recen sudies ha followed up on Diewer s suggesed approach are by Koev and Sanos Silva (2008) and Saisics Porugal (2009). 7 Noe ha we have labeled hese propery prices as v 0 n o emphasize ha hese are values of he propery and we need o decompose hese values ino wo price and wo quaniy componens, where he componens are land and srucures. 8 In order o obain homoskedasic errors, i would be preferable o assume muliplicaive errors in equaion (1) since i is more likely ha expensive properies have relaively large absolue errors compared o very inexpensive properies. However, following Koev and Sanos Silva (2008), we hink ha i is preferable o work wih he addiive specificaion (1) since we are aemping o decompose he aggregae value of housing (in he sample of properies ha sold during he period) ino addiive srucures and land componens and he addiive error specificaion will faciliae his decomposiion.

6 5 The hedonic regression model defined by (1) is he simples possible one bu i is a bi oo simple since i neglecs he fac ha older srucures will be worh less han newer srucures due o he depreciaion of he srucure. Thus suppose in addiion o informaion on he selling price of propery n a ime period, v n, he land area of he propery L n and he srucure area S n, we also have informaion on he age of he srucure a ime, say A n. Then if we assume a sraigh line depreciaion model, a more realisic hedonic regression model han ha defined by (1) above is he following model: (2) v n = L n + (1 A n )S n + n ; n = 1,...,N(); = 1,...,T where he parameer reflecs he depreciaion rae as he srucure ages one addiional period. Thus if he age of he srucure is measured in years, we would expec o be beween 1 and 2%. 9 Noe ha (2) is now a nonlinear regression model whereas (1) was a simple linear regression model. Boh models (1) and (2) can be run period by period; i is no necessary o run one big regression covering all ime periods in he daa sample. The period price of land will he esimaed coefficien for he parameer and he price of a uni of a newly buil srucure for period will be he esimae for. The period quaniy of land for propery n is L n and he period quaniy of srucure for propery n, expressed in equivalen unis of a new srucure, is (1 A n )S n where S n is he floor space area of propery n in period. We implemened he above model (2) using real esae sales daa on he sales of deached houses for a small ciy (populaion is around 60,000) in he Neherlands, Ciy A, for 10 quarers, saring in January 1998 (so our T = 10). The daa ha we used can be described as follows: v n is he selling price of propery n in quarer in unis of 10,000 Euros where = 1,...,10; L n is he area of he plo for he sale of propery n in quarer in unis of 100 meers squared; 10 S n is he living space area of he srucure for he sale of propery n in quarer in unis of 100 meers squared; A n is he (approximae) age (in decades) of he srucure on propery n in period. 11 There were 1404 observaions in our 10 quarers of daa on sales of deached houses in Ciy A. The sample means for he daa were as follows: v = , L = , S = and A = Thus he sample of houses sold a he average price of 111,980 Euros, he average plo size was meers squared, he average living space in he 9 This esimae of depreciaion will be an underesimae of rue srucure depreciaion because i will no accoun for major renovaions or addiions o he srucure. 10 We chose unis of measuremen in order o scale he daa o be small in magniude in order o faciliae he nonlinear regression package used, which was Shazam. 11 The original daa were coded as follows: if he srucure was buil , he observaion was assigned he dummy variable BP = 5; , BP=6; , BP=7; , BP=8. Our Age variable A was se equal o 8 BP. Thus for a recenly buil srucure n in quarer, A n = 0.

7 6 srucure was meers squared and he average age was approximaely 12.6 years. The sample median price was 95,918 Euros. The resuls of our 10 nonlinear regressions of he ype defined by (2) above are summarized in Table 1 below. The Adjused Srucures Quaniies in quarer, AS, is equal o he sum over he properies sold n in ha quarer adjused ino new srucure unis, n (1 A n )S n. Table 1: Esimaed Land Prices, Srucure Prices, Decade Depreciaion Raes, Land Quaniies L and Adjused Srucures Quaniies AS Quarer L AS I can be seen ha he decade depreciaion raes are in he 10 o 18% range which is no unreasonable bu he volailiy in hese raes is no consisen wih our a priori expecaion of a sable rae. Unforunaely, our esimaed land and srucures prices are no a all reasonable: he price of land sinks o a very low level in quarer 6 while he price of srucures peaks in his quarer. Thus i appears ha eiher or model is incorrec or ha our sample is oo small and we are fiing he errors o some exen. I is of some ineres o compare he above land and srucures prices wih he mean and median prices for houses in he sample for each quarer. These prices were normalized o equal 1 in quarer 1 and are lised as P Mean and P Median in Table 2 below. The land and srucures prices in Table 1, and, were also normalized o equal 1 in quarer 1 and are lised as P L and P S in Table 2. Finally, we used he price daa in Table 1, and, along wih he corresponding quaniy daa, L and AS, in Table 1 in order o calculae a consan qualiy chained Fisher house price index, which is lised as P F in Table 2. Table 2: Quarerly Mean, Median and Prediced Fisher Housing Prices and he Price of Land and Srucures Quarer P Mean P Median P F P L P S

8 Noe ha he median price increases in each quarer while he mean price drops (slighly) in quarers 3 and 9. I can be seen ha he overall Fisher housing price index P F is roughly equal o he mean and median price indexes bu again, he separae price series for housing land P L and for housing srucures P S are no realisic. The series in Table 2 are graphed in Char 1 below. Char 1: Quarerly Mean, Median and Prediced Fisher Housing Prices and he Price of Land and Srucures Using Model P Mean P Median P Fisher P Land P Srucures I can be seen ha while he overall prediced Fisher house price index is no oo far removed from he median and mean house price indexes, he separae land and srucures componens of he overall index are no a all sensible. One possible problem wih our highly simplified house price model is ha our model makes no allowance for he fac ha larger sized plos end o sell for an average price ha is below he price for medium and smaller sized plos. Thus in he following secion, we will generalize he model (2) o ake ino accoun his empirical regulariy. 3. Model 2: The Use of Linear Splines on Lo Size

9 8 We broke up our 1404 observaions ino 3 groups of propery sales: Sales involving lo sizes less han or equal o 200 meers squared (Group S); Sales involving lo sizes beween 200 and 400 meers squared (Group M) and Sales involving lo sizes greaer han 400 meers squared (Group L). For an observaion n in period ha was associaed wih a small lo size, our regression model was essenially he same as in (2) above; i.e., he following esimaing equaion was used: (3) v n = S L n + (1 A n )S n + n ; = 1,...,T; n belongs o Group S where he unknown parameers o be esimaed are, and. For an observaion n in period ha was associaed wih a medium lo size, he following esimaing equaion was used: 12 (4) v n = S (2) + M (L n 2) + (1 A n )S n + n ; = 1,...,T; n belongs o Group M where we have now added a fourh parameer o be esimaed, M. Finally, for an observaion n in period ha was associaed wih a large lo size, he following esimaing equaion was used: (5) v n = S (2) + M (4 2) + L (L n 4) + (1 A n )S n + n ; = 1,...,T; n belongs o Group L where we have now added a fifh parameer o be esimaed, L. Thus for small los, he value of an exra marginal addiion of land in quarer is S, for medium los, he value of an exra marginal addiion of land in quarer is M and for large los, he value of an exra marginal addiion of land in quarer is L. These pricing schedules are joined ogeher so ha he cos of an exra uni of land increases wih he size of he lo in a coninuous fashion. 13 The above model can readily be pu ino a nonlinear regression forma for each period using dummy variables o indicae wheher an observaion is in Group S, M or L. The resuls of our 10 nonlinear regressions of he ype defined by (3)-(5) above are summarized in Table 3 below. 12 Recall ha we are measuring land in 100 s of square meers insead of in squared meers. 13 Thus if we graphed he oal cos C of a lo as a funcion of he plo size L in period, he resuling cos curve would be made up of hree linear segmens whose endpoins are joined. The firs line segmen sars a he origin and has he slope S, he second segmen sars a L = 2 and runs o L = 4 and has he slope M and he final segmen sars a L = 4 and has he slope L.

10 9 Table 3: Marginal Land Prices for Small, Medium and Large Los, he Price of Srucures and Decade Depreciaion Raes Quarer S L Obviously, he esimaed prices are no sensible; in paricular, i is no likely ha he cos of an exra uni of land for a large plo could be negaive in quarer 6! Looking a he median price of a house over he 10 quarers in our sample, i was noed earlier ha he median price never fell over he sample period. This fac suggess ha we should impose his condiion on all of our prices; i.e., we should se up a nonlinear regression where he marginal prices of land never fall from quarer o quarer and where he price of a square meer of a new srucure also never falls. We will do his in he following secion and we will also impose a single depreciaion rae over our sample period, raher han allowing he depreciaion rae o flucuae from quarer o quarer. 4. Model 3: The Use of Monooniciy Resricions on he Price of Land and Srucures For he model o be described in his secion, he daa for all 10 quarers were run in one big nonlinear regression. The equaions ha describe he model in quarer 1 are he same as equaions (3), (4) and (5) in he previous secion excep ha he quarer one depreciaion rae parameer, 1, is replaced by he parameer, which will be used in all subsequen quarers. For he remaining quarers, equaions (3), (4) and (5) can sill be used excep ha he parameers S, M, L and are se equal o heir quarer 1 counerpars plus a sum of squared parameers where one squared parameer is added each period; i.e., S, M, L and are reparameerized as follows: (6) S = 1 S + ( S2 ) ( S ) 2 ; = 2,3,...,T; (7) M = 1 M + ( M2 ) ( M ) 2 ; = 2,3,...,T; (8) L = 1 L + ( L2 ) ( L ) 2 ; = 2,3,...,T; (9) = 1 + ( 2 ) ( ) 2 ; = 2,3,...,T; (10) = ; = 2,3,...T.

11 10 Thus our new parameers S2,..., S ; M2,..., M ; L2,..., L and 2,..., and heir squares ener equaions (6)-(9). I can be seen ha his reparameerizaion will preven he marginal price of each ype of land from falling and i will also impose monooniciy on he price of srucures. The resuls of he above reparameerized model were as follows: he quarer 1 esimaed parameers were 1 S = ( ), 1 M = ( ), 1 L = ( ), 1 = ( ) and = ( ), (sandard errors in brackes) wih an R 2 of Thus he overall decade depreciaion rae was a very reasonable 11.5% and he oher parameers seemed o be reasonable in magniude as well. The only mild surprise was he fac ha, a he beginning of he sample period, he marginal valuaion of land for small plos was while he marginal valuaion for medium plos was which was over 6 imes as big. Thus small plos of land suffered a discoun in price per meer squared as compared o medium plos of land, a leas a he beginning of he sample period. 14 Of he 36 squared parameers ha perain o quarers 2 o 10, 23 were se equal o 0 by he nonlinear regression and only 13 were nonzero wih only 8 of hese nonzero parameers having saisics greaer han 2. The quarer by quarer values of he parameers S M L and defined by (6)-(9) are repored in Table 4 below. Table 4: Marginal Prices of Land for Small, Medium and Large Plos and New Consrucion Prices by Quarer Quarer S L The above resuls look reasonable. The impued price of new consrucion,, was approximaely equal o 6.3 o 6.8 over he sample period (his ranslaes ino a price of 630 o 680 Euros per meer squared of srucure floor space). 15 The impued value of land 14 This may no be a genuine effec; i is likely ha he qualiy of consrucion is lower on small plos as compared o he qualiy of medium and larger plos and since we are no aking his possibiliy ino accoun in our model, he lower average qualiy of srucures on small plos may show up as a lower price of land for small plos. We noe also ha by he end of he sample period, he difference in price was grealy reduced. 15 Thus he impued srucures value of a new house wih a floor space area of 125 meers squared would be approximaely 78,000 o 85,000 Euros.

12 11 for a small lo grew from 56 Euros per meer squared in he firs quarer of 1998 o 180 Euros per meer squared in he second quarer of The impued marginal value of land 16 for a lo size in he range of 200 o 400 meers squared grew very slowly from 347 Euros per meer squared o 362 Euros per meer squared over he same period. Finally, he impued marginal value of land 17 for a lo size greaer han 400 meers squared grew very rapidly from 34 Euros per meer squared o 185 Euros per meer squared over he sample period. I is possible o work ou he oal impued value of srucures ransaced in each quarer, V S, and divide his quarerly value by he oal quaniy of srucures (convered ino equivalen new srucure unis), Q S, in order o obain an average price of srucures, P S. Similarly, we can add up all of he impued values for small, medium and large plo sizes for each quarer, say V LS, V LM and V LL, and hen add up he oal quaniy of land ransaced in each of he hree classes of propery, say Q LS, Q LM and Q LL. Finally, we can form quarerly uni value prices for each of he hree classes of propery, P LS, P LM and P LL, by dividing each value series by he corresponding quaniy series. The resuling price and quaniy series are lised in Table 5 below. Table 5: Average Prices for New Srucures, Small, Medium and Large Plos and Toal Quaniies Transaced per Quarer of Srucures and he Three Types of Plo Size Quarer P S P LS P LM P LL Q S Q LS Q LM Noe ha he price of new srucures series, P S, and he price of land for small plos, P LS, in Table 5 coincides wih he series of values for and S lised in Table 4. However, he average prices for land in medium size plos, P LM, and for large size plos, P LL, lised in Table 5 no longer coincide wih he corresponding marginal prices M and L lised in Table 4. This is undersandable since we have used splines o model how he price of a meer squared of land varies as he lo size varies. Noe ha P LM shows a much greaer rae of price increase over he sample period han he corresponding marginal price series M, which hardly changed over he sample period. This is due o he fac ha our model 16 This is our esimae of he value of an exra square meer of land above he hreshold of 200 meers squared (and below he hreshold of 400 meers squared). 17 This is our esimae of he value of an exra square meer of land above he hreshold of 400 meers squared. Q LL

13 12 prices he firs 200 meers squared of a medium sized lo a he average price of a small lo and he price of small los increased quie rapidly over he sample period. Anoher sriking feaure of Table 5 is he endency for he prices of land for small, medium and large los o equalize over ime; i.e., a he beginning of he sample period, he price per meer squared of a small lo was 56 Euros, for a medium lo, 131 Euros and for a large lo, 117 Euros bu by he end of he sample period, he prices were 180 Euros, 232 Euros and 238 Euros, which was a considerable relaive compression in he dispersion of hese prices. A final feaure of Table 5 ha should be menioned is he remendous volailiy in he quaniies ransaced in each quarer. The four price series, P S, P LS, P LM and P LL, were all normalized o equal uniy in quarer 1 and hey are ploed in Char 2 below. Char 2: Prices For Srucures P S and for Three Sizes of Plo P LS, P LM and P LL P S P LS P LM P LL The daa lised in Table 5 were furher aggregaed. We consruced a chained Fisher aggregae for he hree land series and he resuling aggregae land price and quaniy series, P L and Q L, are lised in Table 6 below along wih he srucures price and quaniy series (normalized so ha he price equals 1 in quarer 1), P S and Q S. Finally, a chained Fisher aggregae for srucures and he hree land series was consruced and he resuling aggregae price and quaniy series, P and Q, are also lised in Table 6. Table 6: Aggregae Quarerly Price and Quaniy Series for Housing Quarer P P L P S Q Q L Q S

14 Finally, Char 3 below plos he aggregae house price series P, he land price series P L and he srucures price series P S from Table 6 above along wih he quarerly mean price series P Mean and median series P Median. Char 3: Quarerly Mean Price P Mean, Median Price P Median, Consan Qualiy Housing Price P, Land Price P L and New Srucures Price P S P Mean P Median P P L P S From Char 3, i is eviden ha our esimaed consan qualiy price of housing for Ciy A grew more slowly han he corresponding mean and median series. The major explanaory facor for his difference is probably due o he fac ha he average age of he srucure in he quarerly sample ended o fall as ime marched on. 18 We have used only 3 characerisics of he propery sales: he age of he srucure, he area of he land and he floor space area of he house. Real esae daa bases generally have informaion on many oher characerisics of he house and hese characerisics could be inegraed ino he above hedonic framework. 18 The ime series of average age by quarer in our sample was as follows: 1.38, 1.30, 1.24, 1.06, 1.19, 1.21, 1.16, 1.10, and The average amoun of land ended o increase a bi over ime; he quarerly averages were as follows: 2.30, 2.60, 2.35, 2.48, 2.69, 2.80, 2.75, 2.78, 2.57 and The average srucure size ransaced by quarer was fairly seady: 1.26, 1.28, 1.26, 1.24, 1.28, 1.27, 1.20, 1.26, 1.24 and 1.29.

15 14 In he following secion, we will aemp o implemen he model explained in his secion using monhly daa in place of quarerly daa. 5. A Monhly Model Using Monooniciy Resricions Before we repea he Tables ha were lised in he previous secion using monhly daa insead of quarerly daa, i is useful o lis he descripive saisics ha describe he monhly daa. Thus in Table 7 below, we lis various averages for he 30 monhs of daa in our sample as well as N, he number of observaions in each monh, which range from a low of 26 in monh 9 o a high of 63 in monh 3. Table 7: Descripive Saisics for he Monhly Daa Monh N Mean Median L S A f S f M f L I can be seen ha he monhly means and medians no longer seadily rend upwards; here are now many ups and downs in hese series. The L and S series are he monhly average amouns of land and srucures (in 100s of square meers) sold in each monh.

16 15 There are large flucuaions in some of hese averages: L ranges from a low of 2.10 o a high of 3.05 while S ranges from 1.18 o The average age in decades, A, ranges from a low of 0.81 o The fracion of small los ransaced in a given monh, f S, ranges from a low of o a high of 0.654; he fracion of medium sized los ransaced in a given monh, f M, ranges from a low of o a high of and he fracion of large los ransaced in a given monh, f L, ranges from a low of o a high of Given he magniude of hese flucuaions, i can be seen ha i is unreasonable o expec he mean and median series o give a good approximaion o pure price change because he underlying monhly characerisics are changing so dramaically from monh o monh (and so he mean and median series embody quaniy effecs as well as price effecs). The model described in he previous secion was rerun using he monhly daa so ha we now have 30 monhly ime periods in place of he old 10 quarerly ime periods. The number of parameers o be esimaed has skyrockeed o 121 from he old 41 parameers. The resuls for he monhly model were as follows: he monh 1 esimaed parameers were 1 S = ( ), 1 M = ( ), 1 L = ( ), 1 = ( ) and = ( ) (sandard errors in brackes) wih an R 2 of Recall ha he corresponding quarerly model parameers for quarer 1 were: 1 S = ( ), 1 M = ( ), 1 L = ( ), 1 = ( ) and = ( ), (sandard errors in brackes) wih an R 2 of Thus he monhly model has generaed parameer esimaes ha are quie similar o he quarerly model. Of he 116 squared parameers ha perain o monhs 2 o 30, 97 were se equal o 0 by he nonlinear regression and only 19 were nonzero wih 7 of hese nonzero parameers having saisics greaer han 2. The monh by monh values of he parameers S M L and defined by (6)-(9) are repored in Table 8 below. Table 8: Marginal Prices of Land for Small, Medium and Large Plos and New Consrucion Prices by Monh Monh S L

17 Comparing he enries in Table 8 wih he corresponding quarerly enries in Table 4, i can be seen ha he monhly resuls agree fairly well wih he quarerly resuls wih he excepion of he sudden surge in he marginal price for large los in monh 30 of Table 8 from 1.83 in monh 29 o 3.60 in monh 30. This discrepancy could be due o he fac ha he fracion of large los sold is raher small and so he esimae of he marginal price of large los is paricularly uncerain. Anoher possible explanaion for he large surge in he marginal price for large los in boh he quarerly and monhly models is he fac ha nonparameric ime series models end o be unreliable a he endpoins of he sample period because here is a endency for he model o fi he errors a he endpoins. Our model is very close o being a nonparameric ime series model since i has many free parameers for each ime period and hus, i may be subjec o his ype of bias. 19 As in he previous secion, i is possible o work ou he oal impued value of srucures ransaced in each monh, V S, and divide his monhly value by he oal quaniy of srucures (convered ino equivalen new srucure unis), Q S, in order o obain an average price of srucures, P S. Similarly, we can add up all of he impued values for small, medium and large plo sizes for each monh, say V LS, V LM and V LL, and hen add up he oal quaniy of land ransaced in each of he hree classes of propery, say Q LS, Q LM and Q LL. Finally, we can form monhly uni value prices for each of he hree classes of propery, P LS, P LM and P LL, by dividing each value series by he corresponding quaniy series. The resuling (average) price and quaniy series are lised in Table 9 below. 19 This hypohesis could be checked by adding some addiional monhs of daa o he original sample.

18 17 Table 9: Average Prices for New Srucures, Small, Medium and Large Plos and Toal Quaniies Transaced per Monh of Srucures and he Three Types of Plo Size Monh P S P LS P LM P LL Q S Q LS Q LM Comparing he monhly prices in Table 9 wih heir quarerly counerpars in Table 5, i can be seen ha he prices of srucures and he (average) prices of small los are very similar in he wo ables. However, here are some subsanial differences beween he quarerly and monhly average prices of medium and large los. Moreover, in boh ables, i can be seen ha here are some flucuaions in he average prices of medium and large los, wih he flucuaions being quie subsanial in he case of monhly prices. These flucuaions are due o he smaller sample sizes in he monhly model compared o he quarerly model and o he naure of our spline model for he cos of land. The marginal price of land for an exra uni of land for a medium lo is greaer han he marginal price for an exra uni of land for a small lo. Thus if he average size of a medium lo increases Q LL

19 18 going from one period o he nex, hen he average price for medium los will increase. Similarly, he marginal price of land for large los is less han he marginal price for medium los. Thus if he average size of a large lo increases going from one period o he nex, hen he average price for large los will decrease. Since monhly sample sizes can be small for medium and large los, subsanial flucuaions in he average size of los sold in each monh wihin hese wo caegories of lo size will lead o subsanial flucuaions in he average prices for hese wo ypes of lo. 20 This ype of flucuaion can be conrolled by making he lo size ranges smaller so ha divergences beween marginal and average prices wihin each lo size caegory would be reduced. 21 Anoher mehod for conrolling hese spline model induced flucuaions would be o drop he spline model for he price of land and simply have one price of land for all lo sizes. However, we are relucan o do his since our resuls for he Duch ciy A indicae ha he price levels and rends for he differen sized los differed subsanially. A final mehod for conrolling spline model induced flucuaions in he price of land would be o value he enire sock of deached houses in he ciy using our model. Since he sock of houses changes very lile from monh o monh, his would eliminae large flucuaions in he average amoun of land for medium and large los. 22 We noe ha our model could serve many purposes. As indicaed in he above paragraph, he model could be used o provide up o dae valuaions for he enire sock of deached houses in he ciy, provided ha we had informaion on he age, land area and floor space area for each house in he ciy. The model could also be used o value new addiions o he ciy s housing sock provided ha informaion on he age, land area and floor space area for each newly consruced house in he ciy was available. 23 The daa lised in Table 9 were furher aggregaed. We consruced a chained Fisher aggregae for he hree land series and he resuling aggregae land price and quaniy series, P L and Q L, are lised in Table 10 below along wih he srucures price and quaniy series (normalized so ha he price equals 1 in quarer 1), P S and Q S. Finally, a 20 Analogous flucuaions for small los (and for srucures) canno occur because for hese commodiies, average and marginal prices coincide. 21 A possible problem wih his sraegy is ha he sample sizes wihin each caegory of lo would decline and become zero in some cases. However, his is no necessarily a problem since our spline model does no really require ha he sample size wihin each lo size caegory be nonzero; i.e., our spline model shifs he enire schedule of lo size coss up (or down if we enered he squared erms in equaions (6)-(8) ino he model wih negaive signs insead of posiive signs) and we do no require acual ransacions in a given period for all possible lo sizes. Thus he main cos of increasing he number of spline segmens appears o be he fac ha a large number of addiional parameers would have o be esimaed. 22 This is our preferred mehod for conrolling price flucuaions due o he changing composiion of he houses sold from period o period. However, his mehod requires informaion on he oal sock of housing for he neighbourhood under consideraion. Alernaively, one could simply use he characerisics of a represenaive dwelling uni for he neighbourhood. 23 If he counry uses he acquisiions approach o he reamen of housing in a Consumer Price Index where only he price of he new srucure is o ener he index, hen i can be seen ha our suggesed model could be very useful in his conex. For a review of alernaive ways of reaing housing in a CPI, see Diewer (2002; ) (2007).

20 19 chained Fisher aggregae for srucures and he hree land series were consruced and he resuling aggregae price and quaniy series, P and Q, are also lised in Table 10. Table 10: Aggregae Monhly Price and Quaniy Series for Housing Monh P P L P S Q Q L Q S Comparing he monhly price series in Table 10 wih he corresponding quarerly price series in Table 6, i can be seen ha hey are reasonably close excep ha monhly price of land averaged over he las 3 monhs is 2.23 which is somewha above he corresponding quarerly land index for he las quarer which was The monhly price of srucures for he las 3 monhs was seady a 1.09 which corresponds closely o he quarerly price of srucures for he las quarer, which was As menioned above, we believe he quarerly resuls are likely o be more reliable.

21 20 Char 4 below plos he monhly aggregae house price series P, he land price series P L and he srucures price series P S from Table 10 above along wih he monhly mean price series P Mean and median series P Median. Char 4: Monhly Mean Price P Mean, Median Price P Median, Consan Qualiy Housing Price P, Land Price P L and New Srucures Price P S P Mean P Median P P L P S From Char 4, i is eviden ha our esimaed consan qualiy price of housing for Ciy A grew more slowly han he corresponding mean and median series. As was he case wih he quarerly Char 3, he major explanaory facor for his difference is due o he fac ha he average age of he srucure in he sample ended o fall as ime marched on. I is of ineres o ake he monhly daa from Table 9 and aggregae hese daa ino quarerly uni value prices and he corresponding quarerly quaniies. This was done, generaing hree aggregaed quarerly land price and quaniy series and he aggregaed quarerly srucures price series, AP S. These hree aggregaed land price series were hen aggregaed ino an overall aggregaed quarerly price series AP L using chained Fisher aggregaion. Finally, he hree aggregaed land price series and he aggregaed consan qualiy srucures series AP S were aggregaed ino an overall aggregaed quarerly housing price index, AP, which is lised in Table 11 below along wih AP L and AP S. For comparison purposes, he corresponding quarerly price series for housing, land and srucures, P, P L and P S, from Table 6 in secion 4 (i.e., he esimaes from he original quarerly regression model) are also lised in Table 11. Table 11: Quarerly Price Series for Housing P, Land P L and for Srucures P S and Aggregaed Quarerly Price Series for Housing AP, Land AP L and for Srucures AP S from he Monhly Model

22 21 Quarer AP AP L AP S P P L P S The above series are graphed in Char 5 below. Char 5: Quarerly Consan Qualiy Housing Price P, Land Price P L and New Srucures Price P S and he Corresponding Quarerly Aggregaes Generaed by he Monhly Model, AP, AP L and AP S AP AP L AP S P P L P S I can be seen ha he original quarerly overall house price index series, P, coincides so closely wih he corresponding aggregaed series from he monhly model, AP, ha he wo series can barely be disinguished from each oher in Char 5. Similarly he original quarerly consan qualiy srucures price index, P S, can barely be disinguished from is aggregaed counerpar from he monhly model, AP S. Finally, he original quarerly