House Age, Price and Rent: Implications from Land-Structure Decomposition

Transcription

1 House Age, Price and Ren: Implicaions from Land-Srucure Decomposiion Yangfei Xu Hang Lung Cener for Real Esae and Deparmen of Consrucion Managemen, Tsinghua Universiy, Qinghua Zhang Guanghua School of Managemen, Peking Universiy, Siqi Zheng (corresponding auhor) Deparmen of Urban Sudies and Planning, Massachuses Insiue of Technology; and Hang Lung Cener for Real Esae and Deparmen of Consrucion Managemen, Tsinghua Universiy, Guozhong Zhu Albera School of Business, Universiy of Albera, Absrac Big ciies ofen winess land price ougrowing srucure price. For such ciies his paper derives wo predicions regarding he dynamics beween house prices, ren and srucure age. Firs, older houses have a higher price growh rae han younger ones, even afer conrolling for locaion and oher aribues; second, he age depreciaion of house price, defined as he decline of house price wih respec o house age, is slower han he similarly-defined age depreciaion of ren. These hypoheses are suppored by he micro-daa on housing marke in Beijing. These wo inferences have implicaions for boh real esae valuaion and house price index consrucion. Key words Land price, srucure price, house prices, ren, depreciaion Acknowledgmens Guozhong Zhu and Qinghua zhang acknowledge suppors from Naional Science Foundaion (# and # ). Yangfei Xu and Siqi Zheng acknowledge suppors from Naional Naural Science Foundaion of China (# , # , # , # ), he 1

2 naional key research & developmen (R&D) plan of China (2016YFC ). 1. Inroducion Researchers have become increasingly aware ha house price should be decomposed ino land price and srucure price. Glaeser, Gyourko and Saks (2005) sugges ha house price appreciaion since he 1970s has been largely driven by increasing land coss in he U.S. Based on he land-srucure decomposiion, Davis and Heahcoe (2007) infers he land price from daa on house price and srucure cos for US ciies, which helps explain rends, flucuaions and regional variaions in he price of housing. Similarly, Bosic, Longhofer and Redfeaarn (2007), and Bourassa, Hoesli, Scognamiglio, e al. (2011) focus on he land leverage of houses (he raio of land o oal value); boh find ha land value accouns for a large share of oal home value, and ha his decomposiion can help explain price flucuaions, regional differences and oher imporan issues in he real esae marke. In his paper we explore his land-srucure decomposiion from anoher perspecive ha complemens he exising lieraure. I has been widely observed ha in many big ciies land price ends o increase in value faser han srucure price due o land scarciy. We begin wih he assumpion ha land price ougrows srucure price, offer wo heoreical predicions based on his assumpion and empirically es hem using daa from Beijing ciy: Firs, older houses (houses wih an older srucure) have a higher price growh rae han younger ones, even afer conrolling for locaion and oher imporan aribues. The inuiion is ha, as a house ages, he share of land value as a percenage of oal value increases. Thus he price of an old house ypically increases quickly provided ha land price ougrows srucure price. Second, he age depreciaion of house price, defined as he decline of house price wih respec o house age, is slower han he similarly-defined age depreciaion of ren. Inuiively, house price more clearly reflecs he invesmen value of a house, while ren represens only is consumpion value. For an old house, he consumpion value declines due o he depreciaion of is srucure, bu a he same ime, invesmen value depreciaes o a lesser exen or even increases because of growing land price. Our heory also shows ha if land price grows less quickly han srucure price, hen boh of hese predicions will be reversed. 2

3 For Chinese ciies we are able o direcly observe he sales price of new land parcels aucioned on he residenial land marke. Alhough land prices are available only for new land parcels, hey are a reasonably good proxy of he price for adjacen residenial land ha has already been developed. Based on he qualiy-conrolled land price (ha is, based on he aucion prices of new land parcels) and house price indices released by Tsinghua Universiy s Hang Lung Cener for Real Esae, in Beijing from , he average annual growh rae of land price was 25.2%, which is abou 25% higher han he annual growh rae of house price (20.3%). This implies ha land price has grown much faser han ha of srucure price, as is consisen wih resuls seen in many oher big ciies wih limied land supply (see Glaeser and Gyourko 2006, Davis and Heahcoe 2007, Davis and Palumbo 2008 for ciies in he US. For evidence in China, see Deng e al ). From his premise, we empirically es he above wo heoreical predicions using a large micro-daa se represenaive of Beijing's housing marke. The daase conains 55,706 second-hand housing sales and 210,600 renal ransacions for he period of Our ransacion-level daa on house price and ren are suiable for he es. In paricular, for each residenial complex 1, we have individual observaions for boh renal and re-sale ransacions, faciliaing a locaion-conrolled comparison of he age depreciaion of house price and renal rae. Since we focus our sudy on house age, one empirical challenge is o effecively separae age, cohor and ime effecs (Coulson and McMillen 2008). Because we have a sufficienly large daase, we adop a mehodology ha conrols for age, cohor and ime effecs in differen funcional forms, so as o avoid he muli-collineariy problem (McKenzie 2006) and obain credible esimaes of he age effecs for boh second-hand house sales and renal samples. Our empirical resuls are consisen wih he wo heoreical predicions. Older houses have a higher price growh rae. This is rue boh a he level of individual houses and a he level of residenial complexes. As for he second hypohesis, we esimae he depreciaion rae wih respec o age for boh house price and house ren, and find ha he depreciaion rae of house ren is 25%-60% higher han ha of house price (afer conrolling for boh cohor effec and ime rend), and his difference 1 Housing developmen in many high-densiy ciies in Mainland China occurs a a uniquely large scale and wih a high degree of homogeneiy in he unis buil wihin he ypical residenial complex. In each complex, a number of buildings are consruced conaining alogeher hundreds or even housands of unis, all wih essenially equivalen locaion, archiecural design, srucure, appliances and finishes. 3

4 is saisically significan. Our heoreical framework and empirical findings shed ligh on he mechanism behind he disinc house price growh raes in wo housing sub-markes markes of older and younger housing unis whose land leverage raios differ. This paper also furhers our undersanding of he disinc age depreciaion raes in he housing sale and renal markes. These wo inferences are useful in real esae valuaion, for boh growh rends and age depreciaion raes are crucial facors in he commonly used valuaion mehodologies such as he hedonic pricing model. Anoher implicaion lies in he repeaed-sales approach which is widely used o consruc house price indices. Since age is no conrolled for in he convenional repeaed-sales regression, he esimaed house price indices will depend on he age disribuion in he repeaed sales sample everyhing else being equal, a larger share of older homes in he sample is likely o generae higher esimaes for house price indices. Housing is boh consumpion good and invesmen good (Henderson and Ioannides 1983). In he lieraure, a sandard way o measure housing invesmen demand is hrough addiionally owned houses oher han primary living residence (Rosenhal and Ioniades 1994). However, due o financial fricion which is prevalen in housing markes, households may have o diver heir invesmen need ino primary living residence when hey are consrained from buying muliple houses. I is challenging o measure he invesmen need for owned-occupied houses. Dusanski and Koc (2007) and Cao, Chen and Zhang (2016) show ha expeced capial gains increase he invesmen need of owned-occupied houses. Our paper complemens heir work by offering anoher way o idenify housing invesmen demand. We explore he decomposiion of housing value ino srucure value and land value. Our heory suggess ha because of he appreciaion of land value, he expeced capial gain of old houses is higher han ha of new houses. Thus we expec a slower age depreciaion rae for house prices (which may carry invesmen purposes) han ha for renal prices. Our empirical analysis finds srong evidence supporing hose predicions. The remainder of his paper is organized as follows: Secion 2 presens he heory and derives he wo hypoheses. Secion 3 inroduces he daa. Secion 4 discusses he empirical sraegies and repors he empirical resuls regarding he wo hypoheses. The final secion concludes. 4

5 2. Theory and Hypohesis Le P denoe he price of a housing uni a ime whose srucure has an age of a years, i can be decomposed ino P L S = q p q p l l s s a (1) i.e., house price is he sum of land value L and srucure value S. boh land value and srucure value are expressed as he produc of quaniy and price. Land price l p and srucure price s p have subscrip, because prices change over ime. The quaniy of srucure s q a is subscriped by age a, indicaing ha srucure depreciaes wih age. L Furher, le / S G L 1 L and G S / 1 S be he growh facors of land value and srucure value. We have G q p p (2) l l l L 1 1 l l l q p p And G q p p (3) s s s S a1 1 1 s s s qa p p s Where / s q 1 1 q deermines he depreciaion of srucure wih age. For simpliciy we a assume i is ime- and age-invarian. a Using equaion (1), he growh facor of house price is P L S G G G P 1 L S P L S L S (4) 5

6 2.1 House Age and House Price Growh Rae We are ineresed in how he growh rae of house price changes wih age, i.e., wha he sigh of P dg da is. Before we proceed, we shall discuss wha exacly P dg da means. Theoreically, price of a house is deermined by a se of facors, including house age, local ameniies, land supply in he surrounding area, employmen opporuniy in he ciy and macroeconomic policies and ohers. A simple view of he complex dynamics of house price is o summarize he ime-varying facors ino he ime effec and age effec. The growh rae of house price, P G, is essenially he derivaive of house price wih respec o ime. Therefore, P dg da is essenially he cross derivaive of house price wih respec o ime and age. The effec of age on house price reflecs he depreciaion of srucure. I is disinc in heory from he effec of fundamenal changes in he economy and he change of people s expecaion over ime. However, empirically i is no easy o disenangle he wo effecs because ime and age have perfec collineariy for a given house. We will reurn o his poin in he empirical par of he paper. I is reasonable o assume he growh facors of land price and srucure price over ime, as given in equaion (2) - (3), are independen of house age. Raher, hey are deermined by fundamenals of he economy such as income growh, migraion, labor cos and he cos of consrucion maerials. Thus he derivaives of P G and P G wih respec o age are boh zero. From equaion (4), we derive he following dg ( G G ) dl ds [ ( )] da L S da da P L S S 2 L ( ) ( G G ) d( q p ) d( q p ) = [ ( )] ( L S ) da da L S l l s s a S 2 L (5) The erm in brackes in he above equaion is generally posiive. Firs, land quaniy and price should no change wih srucure age, so l l d( q p ) =0. Second, while srucure quaniy depreciaes over da 6

7 ime ( s dq a da <0), i is reasonable o assume srucure price is independen of srucure age, herefore s s s d( qa p ) s dqa p <0. 2 Hence da da dl S da ds L( ) 0 (6) da Therefore, from equaion (5) we reach he hypohesis (1). L S Hypohesis (1): if land price grows faser han srucure price, i.e., G G 0, hen he prices of older houses have higher growh raes, i.e., P dg da 0 L S Conversely, if G G 0, hen prices of older houses will have lower growh raes. Inuiively, as a housing uni ages, is price is composed more of land price han of srucure price, and hence is growh rae increases and becomes closer and closer o he growh rae of land price. 2.2 Age Depreciaion of House Price and Ren Given a housing uni, le R be he renal rae a period. Sandard asse pricing heory gives rise o he following equaion 2 A more general condiion for he brackeed erm o be negaive is ha srucure value is more elasic wih respec o age han land value, i.e., dl / da ds / da. This can be easily derived from equaion (5). L / a S / a 7

8 P R E[ D P ] (7) 1 1 Where D 1 is he sochasic discoun facor beween period and +1 and E is he expecaion operaor. 3 Here i is assumed ha ren is colleced in he beginning of he period. Noice ha P +1 conains all he relevan informaion abou fuure rens and fuure discoun facors, herefore equaion (7) is essenially a Gordon growh model wih ime-varying discoun raes. In he appendix we show in deail how o ge from a version of Gordon growh model o equaion (7). P P G We apply 1 P o equaion (7) o ge P R + P E[ D G ] (8) P 1 This equaion makes explici ha house price depends on ren and he fuure growh of house price. Recall ha p G is he growh facor of house price beween period and +1, and hence is sochasic a ime. Clearly P and R are funcions of house age. On he oher hand, he sochasic discoun facor does no depend on house age, bu insead depends on he marginal uiliy of consumpion. Therefore, aking he logarihm of boh sides of equaion (8), and hen aking derivaives wih respec o age, we have p d log R d log P p dg R [ 1 ] [ 1 ] log PE D G PE D d P da da da da R + P E[ D G ] P 1 (9) Where he las expression in he numeraor comes from he Leibniz Rule; ha is, 3 In a consumpion-based asse pricing model, D 1 is he marginal rae of subsiuion beween consumpion bundles in period +1 and period. All he derivaion holds rue if he discoun facor D 1 is non-sochasic. 8

9 de[ D G ] da dg [ ] da p p 1 ED 1 P Afer muliplying boh sides of equaion (9) wih R PE[ D 1G ] and rearranging erms, we have d log P d log R P dg [ ] (10) da da R da P ED 1 I is naural o assume ha he aging of a house negaively affecs is price and ren, i.e., dlog P da dlog R 0 and da 0, hence equaion (10) leads o equaion (11). P d log R d log P P dg E D 1 da da R da (11) where. denoes he absolue value operaor. Therefore, he gap beween age depreciaion of ren P and ha of price depends on dg / da. If he prices of older houses reveal a higher growh rae, hen we expec ren o decline more quickly wih age han house price does. Using equaion (4), equaion (11) becomes L S d log R d log P P ( G G ) dl 1 2 [ ds ( E D S L )] da da R ( L S ) da da (12) Based on (12), we have he following hypohesis. L S Hypohesis (2): If G G 0, hen renal rae depreciaes more quickly wih age han house price, i.e., 9

10 d log R d log P da da (13) L S Conversely, if G G 0, hen renal rae depreciaes more slowly wih age han house price. Inuiively, house price depends on he renal rae and he expeced growh in house price in he fuure. As a house ages, i depreciaes because he renal rae decreases due o he reduced uiliy flow from an older srucure. However, he house price growh rae is higher for older houses given ha land price grows faser han srucure price, as saed in Hypohesis (1). This miigaes he effec of depreciaion of renal rae caused by age, leading o house price depreciaing more slowly. 3. Daa Our micro daa on second-hand housing sales and renal ransacions are from WoAiWoJia, a major real esae broker in Beijing, wih a local marke share of abou 10%. The daase conains 55,706 housing re-sales and 210,600 renal ransacions for he period of The housing uni ransacions come from over 2,500 residenial complexes disribued hroughou Beijing s landscape and hence are represenaive of Beijing's housing marke. I is worh menioning ha he large sample size of our daase provides us a remarkable advanage in he empirical sudy. Because he sample size wihin each complex is large enough o run a complex-specific Hedonic regression, and also because here are a large number of complexes, we can firs esimae he growh rae of house price for each complex, and hen sudy how he growh rae of house price varies wih house age a he complex level. Moreover, since he unobserved variables of housing unis may be correlaed wihin each complex, we cluser he sandard errors by complex in our regressions a he housing uni level. For each ransacion, we have informaion abou ransacion price (price or ren), address, and physical aribues of housing such as uni size (size), level of decoraion (decoraion) and wheher i is on he op floor (op). By geo-coding all sales and renal ransacions on Beijing s GIS map, we consruc several locaion aribue measures for each residenial complex, including disance from 10

11 each residenial complex o he ciy cener (d_cener), wheher he complex is wihin he 2-km reach of a key primary school (school), of a Grade-A hospial (hospial) and wihin he 1-km reach of a subway sop (subway). Summary saisics are provided in Table 1. ** Figure 1 here ** ** Table 1 here ** Figure 1 shows he geographical disribuions for boh he sales sample and renal sample in our daa. A comparison of he lef and righ panels reveals ha he wo samples differ in heir spaial disribuions, alhough boh samples conain housing unis all over he ciy. Therefore, i is necessary o conrol for locaion aribues in our empirical analysis. Table 1 shows oher differences beween sales and renal unis. Sales unis have an average size of 77.6 m 2 and an average price of 18,250 RMB/m 2. On average hey are abou 11.6 km away from he ciy cener. Among hese resale unis, 52% are wihin 2km of he closes key primary school, 45% are wihin 2 km of he closes high-qualiy hospial, and 38% are wihin 1 km of he closes subway saion. In conras, renal unis have an average size of 64.3 m 2 and an average ren of 47.5 RMB/m 2 per monh. Addiionally, renal unis are in slighly beer locaions han he sales unis. On average hey are 11.0 km away from he ciy cener. The percenage of unis wihin 2 km from he key primary school, he Grade-A hospial and wihin 1 km of he subway saion are 64%, 57% and 42% respecively, all of which are higher han he corresponding percenages of sales unis. In our empirical analysis, we conrol for boh physical aribues and locaion aribues. House age a he ime of ransacion (age) is a key variable in our sudy. Alhough he daa do no have direc informaion abou house ages\ for each ransacion, we locaed he building year for each complex based on is address and complex name on he broker s websie, and his enables us o calculae house age. Mos sales unis were buil during he period of While mos renal unis were also buil in abou he same period, he variaion of he building year for he renal sample 11

12 is larger han ha for he sales sample 4. The disribuion of building year for boh he sales sample and renal sample are repored in Figure 2. The average house age is 11.2 for he sales sample and 13.0 years for he renal sample. ** Figure 2 here ** As menioned previously, he disenanglemen of age effec, year effec and cohor effec is a crucial issue in our empirical sudy. To conrol for he cohor effec, we divide all housing ransacions ino 5-year cohor groups according o heir consrucion year. Thus in oal we have 7 cohor groups (ha is, 7 cohor dummies). Table 2 shows he disribuion of ransacions among he 7 groups and he range of house age for each group for boh he sales and renal samples. Thanks o our large daa se, we observe ha he variable age sill has sufficien variaion wihin each cohor. Thus we can conrol for boh he cohor dummy and year effec, and sill have a valid esimaion of age depreciaion in he empirical sudy. ** Table 2 here ** 4. Empirical Analysis In his secion we es he wo hypoheses idenified in secion 2 for he Beijing s housing marke, L S on he premise ha G G 0. The house price and land price indices recenly released by Tsinghua Universiy, as shown in Figure 3, indicae ha land price has grown more quickly han house price from in Beijing; his implies ha he land growh rae is indeed larger han he srucure growh rae. ** Figure 3 here ** 4.1. Tesing Hypohesis (1) 4 In order o have a srong comparison beween he growh rae and depreciaion rae of house price and ren, we only include houses buil afer 1980 in implemening our empirical equaions. 12

13 Because only a very limied number of housing unis have repeaed sales informaion in our sample period, esimaing he price growh rae for individual housing unis is no feasible. To es Hypohesis (1), we esimae he average annual price growh rae for each residenial complex and hen sudy how he growh rae varies by he house age of each residenial complex. We choose residenial complexes wih enough ransacions; ha is, wih more han 50 or 20 ransacions from 2005 o Specifically, for he j- h residenial complex, we esimae Equaion (15) wih a linear ime rend (in years). log( price ) φ X (15) i j j i j i where he subscrips i, denoe he ransacion of housing uni i in year (he earlies year akes he value of 1), price i denoes he uni price of per square meer for he housing uni i (in complex j) in year, and X i includes he physical feaures of he housing uni excluding house age since i is perfecly correlaed wih wihin a complex. The esimae of coefficien j proxies he average annual price growh rae of complex j in our sudy period. Nex, we examine how he esimaed varies by he house age of residenial complex j: j λ Y age u (16) j j j j Where age j is he house age of residenial complex j a he beginning of our sample period. Y j is a se of locaion aribues a he complex level, including d_cener, subway, hospial, and school. The esimaion resuls of equaion (16) are repored in Table 3. The resuls are repored in Columns (1) and (2) respecively. The coefficiens of age in he wo columns are boh significanly posiive, indicaing ha price growh rae does increase wih age. ** Table 3 here ** As an alernaive es of Hypohesis (1), we also pool all sales ransacions ogeher and regress he 13

14 logarihm of house price on ransacion year, age, ineracion of ransacion year and age, and oher physical and locaion aribues a he housing uni level. The coefficien of he ineracion erm indicaes wheher he growh rae of house price over ime varies by house age. If he coefficien is posiive, his means older houses have higher house price growh rae. The regression specificaion is as in Equaion (17). log( price ) δ W S age age * cohor v (17) ij ij ij ij ij ij where W ij is a vecor of conrol variables, including physical aribues of he house such as size, decoraion, floor, op, and house locaion aribues such as log(d_cener), school, hospial, subway, S denoes seasonal dummies 5. Noe ha we analyze he age effec of house price based on repeaed cross secional daa. A classical issue in his kind of regression is he idenificaion of age effec, year effec and cohor effec. To exrac he real age effec, i is necessary o filer ou year and cohor effecs. This is done by including ime variable and cohor vecor cohor j ogeher in he regression as explanaory variables. To address he problem of muli-collineariy among age, cohor and year, we le be he monh in which he ransacion happens (wih 1 denoing he earlies monh), while age j is sill he house age in year. We also le cohor j be a series of dummy variables indicaing he 5-year group during which he ransaced house was buil 6. Sandard errors are clusered by complex. The empirical resuls are repored in Table 4. The coefficiens of physical and locaion aribues are all consisen wih our expecaions. Smaller houses wih beer decoraion and on higher floors (bu no he op floor) have higher prices per square meer, and houses in prime locaions (near Cenral Business Disric, key primary schools and high-qualiy hospials) also have higher prices. The coefficien of is significanly posiive, showing he growing rend for house price from The coefficien of age is significanly negaive, showing ha house price decreases by 0.01% for every 1 year increase in housing age. The coefficien of he ineracion erm age* is significanly posiive, indicaing ha older houses have higher growh raes of heir prices, as is consisen wih Hypohesis (1). In column (2) we replace he variable age wih is logarihm log(age) as a robusness 5 Here we ake monh January o March as he defaul, so we have hree dummies: monh April o June, monh July o Sepember, monh Ocober o December. 6 While we admi ha he cohor seing may be arbirary, we make a robusness check wihou he cohor dummies in he appendix. 14

15 check. Alhough he ineracion erm also has a posiive sign, i is no saisically significan. ** Table 4 here ** 4.2. Tesing of Hypohesis (2) We es Hypoheses (2) by running he following wo regressions of house price and ren separaely log( price ) δ W S age T cohor v (18) ij p p ij p ij p p ij p, ij log( ren ) δ W S age T cohor v (19) ij r r ij r ij r r ij r, ij Where price ij and ren ij denoe he uni sale price and renal price per square meer for he housing uni i in year, respecively.β p and β r denoe he depreciaion rae of house price and house ren for every 1 year increase in housing age, respecively. Again, sandard errors are clusered by complex in boh regressions. We also conrol for ime and cohor effecs (T and cohor). As discussed earlier, here is a mulicollineariy issue among age, cohor and ime since age j + cohor j=. To deal wih his issue, here we le age j be a coninuous variable of housing age in years, and cohor j be a series of dummy variables indicaing he 5-year group during which he ransaced house was buil (we also use a 1- year group and 10-year group as robusness checks). We replace he coninuous variable wih T, which is a polynomial vecor of ime; i.e., T=(, 2, 3, ) where is a coninuous variable denoing he monh in which he ransacion happens wih he earlies monh being 1 7. According o Hypoheses (2), we should have p (20) r The empirical resuls are repored in Table 5. Physical aribues are included in he regressions bu 7 The same as Table 4, we include a robusness check wihou cohor effec in he appendix. The resuls show ha we would underesimae he age depreciaion rae wihou conrolling for cohor effec. 15

16 are no repored. Comparing he coefficiens in column (1) wih hose in column (2), we can see ha p r, which is consisen wih Hypohesis (2). The Wald es 8 esing if hese wo coefficiens are saisically differen also confirms he significan difference beween he depreciaion raes of house price and house ren. As robusness checks, we also re-define our cohor dummies wih 1- year and 10-year cohor groups. Columns (3) and (4) show house price and ren regressions wih 1-year cohor dummies, respecively. Columns (5) and (6) presen resuls wih 10-year cohor dummies. The Wald es confirms he significan difference beween he depreciaion raes in he 10-year cohor regressions, alhough his gap is no ha significan in he 1-year cohor regressions (perhaps due o he limied variaion wihin each 1-year cohor). ** Table 5 here ** 4.3. Robusness Check in erms of he land leverage hypohesis In order o ge a consisen conclusion in erms of he land leverage hypohesis, we also implemen similar ess o oher wo srucural variables op and floor as a robusness check. The regression resuls are repored as Table A3 and Table A4 in he Appendix. In Table A3, we include he floor level of house unis in he Hedonic regression for boh house price and ren. The resuls show ha house unis on higher floors have higher prices bu lower growh rae of he price; and he posiive effec of floor level on he ren is larger han ha on he price. On he conrary, house unis on op of a building have lower prices (due o he less comfor on he op) bu higher price growh rae; and he negaive effec on he ren is larger han ha on he price. In Table A4, we firsly run he Hedonic regressions on he complex level and find ha he variables op and floor have he same impac paern on house price and ren as in Table A3. Then we repea wha we have done in Table 3. We firsly esimae he growh rae of housing price based on he uni-level Hedonic regression of each complex (γ, no repored as a able), and hen use op and floor o explain he difference in γ across complexes. The resuls are consisen wih hose of he ess of housing age, which confirms he 8 As poined ou by he anonymous referee, he wo esimaed depreciaion raes of house price and house ren are correlaed wih each oher, since hey are derived from he regressions for wo highly relaed sub-markes. In his case we employ he Wald es, and use he command sues (designed for seemingly unrelaed esimaion) in STATA o es wheher he wo coefficiens are saisically differen, aking he covariance of he wo coefficiens ino consideraion. 16

17 hypoheses in erms of he land leverage heory. 5. Conclusion By decomposing house price ino separae prices for land and srucure, his paper has sudied an imporan facor in real esae valuaion -- house age. The effec of house age on price, ren, and he growh rae of house price depends on he relaive growh speed of land price and srucure price. For ciies where land price grows more quickly relaive o srucure price, we predic heoreically ha older houses have higher growh raes in heir house price, and ha house ren has a larger depreciaion rae wih respec o age han ha of house price. Boh heoreical predicions are reversed if land price grows less quickly han srucure price. Using our unique micro daa for he Beijing marke, we find ha boh predicions are empirically suppored. Our analysis sheds ligh on he disinc house price growh rends in markes wih differen land leverage raios. I also explains he difference beween age depreciaion raes in he sales and renal markes. These wo inferences have implicaions for boh real esae valuaion and house price index compilaion. While he growh rae of land price is observable for he Beijing marke due o he unique land aucion policy in China, i is usually no direcly observable for markes in many advanced economies. For hese markes, our heoreical predicions provide a novel way of esing he dynamic relaionship beween land price and srucure price. -- researchers can infer wheher land price ougrows srucure price based on he age depreciaion paerns of house price and ren which is more easily observable. 17

18 References [1] Bosic, R., Longhofer, S., Redfearn C. (2007). Land leverage: decomposing home price dynamics. Real Esae Economics, 35(2), [2] Bourassa, S., Hoesli, M, Scognamiglio, D, e al. (2011). Land leverage and house prices. Regional Science and Urban Economics, 41(2), [3] Cao, Y., Cheng, J. and Zhang, Q. (2016). Housing Invesmen in Urban China: Evidence from Chinese Household Survey. Working paper. [4] Coulson, N., McMillen, D. (2008). Esimaing ime, age and vinage effecs in housing prices. Journal of Housing Economics, 17(2), [5] Davis, M., Heahcoe, J. (2007). The price and quaniy of residenial land in he Unied Saes. Journal of Moneary Economics, 54(8), [6] Davis, M., Palumbo, M. (2008). The price of residenial land in large US ciies. Journal of Urban Economics, 63(1), [7] Deng, Y., Gyourko, J., Wu, J. (2012). Land and house price measuremen in China. Naional Bureau of Economic Research. [8] Dusansky, R., Koç, Ç. (2007). The capial gains effec in he demand for housing. Journal of Urban Economics, 61(2), [9] Glaeser, E., Gyourko, J. (2005). Housing dynamics. Naional Bureau of Economic Research. [10] Glaeser, E., Gyourko, J., Saks R. (2005). Why have housing prices gone up? American Economic Review, 95(2), [11] Henderson, J., Yannis, M. (1983). A model of housing enure choice. The American Economic Review, 73(1), [12] Hornsein, A., Greene, W. (2012). Usage of an esimaed coefficien as a dependen variable. Economics Leers, 116(3), [13] Ioannides, Y., Rosenhal, S. (1994). Esimaing he consumpion and invesmen demands for housing and heir effec on housing enure saus. The Review of Economics and Saisics, [14] McKenzie, D. (2006). Disenangling age, cohor and ime effecs in he addiive model. Oxford Bullein of Economics and Saisics, 68, [15] Saxonhouse, G. (1976). Esimaed parameers as dependen variables. The American Economic Review, 66(1),

19 Fig.1 Spaial disribuion of ransacion samples Spaial disribuion of sales unis Spaial disribuion of renal unis Fig.2 Disribuion of building years Sales unis Renal unis 19

20 Fig.3 Hedonic land and house (land + srucure) price indices in Beijing land price index house price index Source: Tsinghua Hang Lung Cener for Real Esae 20

21 Table 1 Summary saisics Sales observaions Renal observaions Variable Definiion Obs. Mean Sd. Obs. Mean Sd. price/ren Transacion price/ren (RMB yuan per square meer for price and RMB yuan per square meer per monh) size Housing uni size (square meer) age House age (years) decoraion Decoraion saus (4=bes; 1=wors) floor Floor number op Wheher he uni is on he op floor (1=yes, 0=no) d_cener Disance o he ciy cener (km) school Wheher he observaion is wihin 2km of key primary school hospial Wheher he observaion is wihin 2km of hospial subway Wheher he observaion is wihin 1km of subway sop

22 Table 2 Sample disribuion by cohor groups and wihin-cohor age range Building year Sales Renals Sample size: 3379 Age range: Sample size: 5539 Age range: Sample size: 7203 Age range: Sample size: Age range: 5-16 Sample size: Age range: 0-11 Sample size: 5626 Age range: 0-6 Sample size: 160 Age range: 0-1 Sample size: Age range: Sample size: Age range: Sample size: Age range: Sample size: Age range: 5-16 Sample size: Age range: 0-11 Sample size: Age range: 0-6 Sample size: 937 Age range:

23 Table 3 House age and house price growh: complex-level regressions >20 ransacions >50 ransacions (1) (2) (3) (4) age *** *** ** *** ( ) ( ) ( ) ( ) log(d_cener) ** ( ) ( ) ( ) ( ) school ** ** ** ( ) ( ) ( ) ( ) hospial e ( ) ( ) ( ) ( ) subway ( ) ( ) ( ) ( ) Consan 0.170*** 0.173*** 0.171*** 0.176*** ( ) ( ) (0.0140) (0.0105) Heeroscedasiciy approach OLS, Robus WLS OLS, Robus WLS Observaions R-squared Noe: (1) Firs, we regress house price wih ime rend (in monh) using all he ransacion daa o ge price growh rae, and hen we regress growh rae wih housing age. Here we choose hose complexes wih enough ransacions o insure he regression. Column (1) and (2) show he resuls of regressions based on complexes wih more han 20 ransacions, column (3) and (4) show he resuls of regressions based on complexes wih more han 50 ransacions. (2) According o Saxonhouse (1976), Hornsein, Greene (2012), e al., we use wo differen mehods o miigae he heeroskedasiciy problem. In column (1) and (3) we use formula allowing for he presence of he heeroscedasiciy in OLS regression and adjus he saisic. In column (2) and (4) we use he weighed leas square mehod, by he inverse of he esimaed sandard error of he prediced in he firs sage. (3) The coefficien of age in his able indicaes ha older houses have, in price per square meer, a higher growh rae. (4) Robus sandard errors in parenheses; *** p<0.01, ** p<0.05, * p<0.1 23

24 Table 4 House age and house price growh: uni-level regressions (1) (2) log(price) Coefficien Sd. Err Coefficien Sd. Err log(size) *** (0.0118) *** (0.0117) decoraion *** ( ) *** ( ) op *** ( ) *** ( ) floor *** ( ) *** ( ) *** ( ) *** ( ) age *** ( ) age* *** ( ) log(age) *** (0.0186) log(age)* ( ) log(d_cener) *** ( ) *** ( ) school *** (0.0112) *** (0.0112) hospial *** (0.0114) *** (0.0113) subway *** ( ) *** ( ) Consan 9.266*** (0.0893) 9.307*** (0.0774) Cohor seings Every 5 years Every 5 years Observaions 55,427 55,427 R-squared Noe: (1) Here we use he radiional Hedonic funcion of house price o es Hypohesis (1). We add in an ineracion erm age* o he Hedonic funcion, and is coefficien is significanly posiive, indicaing ha older houses have a higher growh rae of heir prices. (2) We include cohor dummies in his regression, which are lumped ino groups according o 5 year inervals, based on year of consrucion. (3) Robus sandard errors in parenheses; *** p<0.01, ** p<0.05, * p<0.1 24

25 Table 5 Age depreciaion of house price and ren (1) (2) Wald es (3) (4) Wald es (5) (6) Wald es VARIABLES log(price) log(ren) log(price) log(ren) log(price) log(ren) p r p r cohor seing 5years 5years 1year 1year 10years 10years p r β p β r Chi2=18.25 β p β r Chi2=0.20 β p β r Chi2=7.56 age ** *** P= *** P= *** *** P= ( ) ( ) (0.0223) (0.0120) ( ) ( ) *** *** *** *** *** *** ( ) ( ) ( ) ( ) ( ) ( ) e-05*** 1.66e-05*** -5.57e-05*** 1.70e-05*** -5.61e-05*** 1.72e-05*** (5.33e-06) (2.72e-06) (5.14e-06) (2.70e-06) (5.14e-06) (2.78e-06) Consan 10.26*** 6.194*** 10.84*** 6.879*** 10.26*** 6.112*** (0.116) (0.104) (0.570) (0.307) (0.108) (0.0850) Cohor seings Every 5 years Every 1 year Every 10 years Physical aribues YES YES YES YES YES YES Locaional aribues YES YES YES YES YES YES seasonaliy YES YES YES YES YES YES clusered by complex complex complex complex complex complex Observaions 55, ,995 55, ,995 55, ,995 R-squared Noe: (1) We es Hypohesis (2) wih his regression. Here we include cohor dummies based on year of consrucion using 3 differen seings: every 1 year, every 5 years and every 10 years. 25

26 (2) To miigae he muli-collineariy problem, we use he following funcion form: age is a coninuous variable denoing housing age, cohor is a series of dummy variables indicaing he 5-year group during which he ransaced house is buil, in addiion we use he polynomial funcion of ime rend (, 2 ) where is a coninuous variable denoing he monh in which he ransacion occurs. (3) The coefficien of age indicaes he age depreciaion rae of house price and ren respecively. We find ha house price depreciaes less han house ren, and he Wald es confirms he difference beween hem. The resuls in he 5-year and 10-year cohor seing boh hold, showing he resuls are robus. (4) Robus sandard errors in parenheses; *** p<0.01, ** p<0.05, * p<0.1 26

27 Appendix: Robusness check in erms of he cohor effec Table A1: Robusness check in erms of he cohor effec for Table 4 (1) (2) (3) (4) VARIABLES log(price) log(price) log(price) log(price) log(size) *** *** *** *** (0.0121) (0.0116) (0.0120) (0.0114) decoraion *** *** ** *** ( ) ( ) ( ) ( ) op *** *** *** *** ( ) ( ) ( ) ( ) floor *** *** ** *** ( ) ( ) ( ) ( ) log(d_cener) *** *** *** *** ( ) ( ) ( ) ( ) school *** *** *** *** (0.0111) (0.0110) (0.0110) (0.0110) hospial *** *** *** *** (0.0116) (0.0116) (0.0116) (0.0116) subway *** *** *** *** ( ) ( ) ( ) ( ) *** *** ( ) ( ) *** *** ( ) ( ) age *** *** ( ) ( ) age* *** *** ( ) ( ) log(age) *** *** (0.0127) (0.0122) log(age)* *** *** ( ) ( ) Consan 8.960*** 8.917*** 9.084*** 9.030*** (0.0604) (0.0782) (0.0651) (0.0784) Sale monh dummies NO YES NO YES Seasonaliy YES NO YES NO Observaions 55,427 55,427 55,427 55,427 R-squared Noe: (1) Here we use he radiional Hedonic funcion of house price o es Hypohesis (1). We add in an ineracion erm age* o he Hedonic funcion, and is coefficien is significanly posiive, indicaing ha older houses have a 27

28 higher growh rae of heir prices. We also ake he logarihm funcion of age in Column (3) and (4) as a robusness check. (2) This able acs as a robusness check for Table 4 by excluding he cohor dummies in Column (1) and (3). In Column (2) and (4), we replace he quadraic polynomial funcion of ime rend by he sale monh dummies. (3) Robus sandard errors in parenheses; *** p<0.01, ** p<0.05, * p<0.1 Table A2: Robusness check in erms of he cohor effec for Table 5 (1) (2) Wald es (3) (4) Wald es VARIABLES log(price) log(ren) log(price) log(ren) p r p r age *** *** Chi2= ** *** Chi2= ( ) ( ) P= ( ) ( ) P= *** *** ( ) ( ) *** *** ( ) ( ) Consan 8.899*** 4.732*** 8.863*** 4.747*** (0.0589) (0.0363) (0.0786) (0.0405) Physical aribues YES YES YES YES Locaional aribues YES YES YES YES Sale monh dummies NO NO YES YES Seasonaliy YES YES NO NO Cluser by complex complex complex complex Observaions 55, ,995 55, ,995 R-squared Noe: (1) We es Hypohesis (2) wih his regression. This able acs as a robusness check for Table 5 by excluding cohor dummies. In Column (3) and (4) we furher replace he ime rend funcion wih sale monh dummies. The regression resuls show consisence of he relaionship beween age depreciaion of house price and ren (he age depreciaion rae of house ren is larger han ha of house price). While he coefficiens show ha he cohor effec lead o a under-esimaion of age depreciaion. (2) Robus sandard errors in parenheses; *** p<0.01, ** p<0.05, * p<0.1 28

29 Appendix: Robusness check in erms of he land leverage hypohesis Table A3: Robusness check in erms of he land leverage hypohesis uni level (1) (2) (3) (4) VARIABLES log(price) log(price) log(price) log(ren) log(size) *** *** *** *** ( ) ( ) ( ) ( ) log(age) *** *** *** *** ( ) ( ) ( ) ( ) decoraion *** *** *** *** ( ) ( ) ( ) ( ) op *** *** *** *** ( ) (0.0102) ( ) ( ) floor *** *** *** *** ( ) ( ) ( ) ( ) rend *** *** (5.94e-05) (8.52e-05) rend*op *** ( ) rend*floor -4.23e-05*** (8.61e-06) log(d_cener) *** *** ( ) ( ) school *** *** *** *** ( ) ( ) ( ) ( ) hospial *** *** *** *** ( ) ( ) ( ) ( ) subway *** *** *** *** ( ) ( ) ( ) ( ) Consan 8.938*** 9.089*** 9.065*** 4.853*** (0.0720) (0.0166) (0.0169) (0.0154) Transacion monhly YES NO NO YES dummies Seasonaliy NO YES YES NO Observaions 55,706 55,706 55, ,286 R-squared Noe: (1) Column (1) is he Hedonic funcion for housing price, we conrol for ransacion monhly dummies in he regression. Column (4) is he similar regression for housing ren, compare he coefficiens of op and floor, being on he op of a building has a negaive impac on housing price, bu is smaller han he negaive impac on housing 29

30 ren. While being on higher floors would have a posiive impac on housing price, and also smaller han he posiive impac on housing ren. (2) In column (2) and (3) we employ he ineracion erm rend*op and rend*floor o es he growh rae of housing price wih differen srucural feaures. Trend refers o a coninuous variable showing he ransacion monh. Their coefficiens show ha unis on higher floors have lower growh rae in heir prices, bu hose on he op have higher growh rae. (3) Sandard errors in parenheses; *** p<0.01, ** p<0.05, * p<0.1 Table A4: Robusness check in erms of he land leverage hypohesis complex level (1) (2) (3) (4) VARIABLES log(price) log(ren) log(size) *** *** ( ) ( ) log(age) *** *** ( ) ( ) decoraion *** *** ( ) ( ) op *** *** *** ( ) ( ) (0.0233) floor_mode * *** ** ( ) ( ) ( ) log(d_cener) *** *** ( ) ( ) ( ) ( ) school *** *** *** * ( ) ( ) ( ) ( ) hospial *** *** e-05 ( ) ( ) ( ) ( ) subway *** *** ( ) ( ) ( ) ( ) Consan 8.828*** 4.655*** 0.179*** 0.197*** (0.0725) (0.0216) (0.0103) (0.0126) Transacion monhly YES YES NO NO dummies Heeroscedasiciy WLS WLS approach Observaions 25,215 74, R-squared Noe: (1) In column (1) (2), we simply run a hedonic funcion for housing price and ren separaely on he complex level, where he variable floor_mode here refers o he mode of floor in each complex, while oher srucural 30

31 feaure variables are he mean value for unis in he same complex. The coefficiens of op and floor show he same paern as in column (1) and (4) in Table A3. (2) In column (3) (4), we repea he ess as in Table 3. is he coefficien of ime rend in he Hedonic regression of uni-level for each complex (we also choose hose complexes wih more han 50 ransacions). And hen we use he srucural and locaional feaures o explain he difference in, which is he growh rae of housing price. The coefficiens of op and floor_mode show he same paern as in Column (2) and (3) in Tale A3. (3) Robus sandard errors in parenheses; *** p<0.01, ** p<0.05, * p<0.1 31

32 Appendix: From he Gordon growh model o equaion (7) Le R and D +1 denoe he renal rae a period and he discoun facor beween period and +1, hen house price a period can be wrien as he discouned presen value of rens from he curren and fuure periods, as in he following equaion. j +k 1 P = R + E [ ( D +k j=1 k=1 ) R +j ] A(1) where E is he expecaion operaor ha ake all he informaion available a ime ino accoun. This equaion is he Gordon growh model wih ime-varying sochasic discoun rae. In period +1, we have he following j +k P +1 = R +1 + E +1 [ ( D +k+1 ) R +j+1 ] j=1 k=1 j +k 1 = R +1 + E +1 [ ( D +k j=2 k=2 ) R +j ] A(2) We muliply boh sides of he above equaion wih D +1, and hen ake expecaion using E. This leads o E [D +1 P +1 ] = E [D +1 R +1 ] + E {D +1 E +1 [ ( D +k j=2 j k=2 +k 1 ) R +j ]} +k 1 = E [D +1 R +1 ] + E {[ ( D +k j=2 j k=1 ) R +j ]} A(3) j +k 1 = E [ ( D +k j=1 k=1 ) R +j ] In equaion A(3) we used he law of ieraed expecaions, i.e., E [E +1 (. )] = E (. ). Plugging A(3) ino A(1), we have P = R + E [D +1 P +1 ] A(4) 32

33 Equaion A(4) is exacly equaion (7) in he paper, excep ha we omied he superscrip in he discoun facor. 33