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1 = = = = = = = Working Paper A Regime Shif Model of he Recen Housing Bubble in he Unied Saes Rober Van Order Sephen M. Ross School of Business a he Universiy of Michigan Rose Neng Lai Universiy of Macau Ross School of Business Working Paper Series Working Paper No November 2006 This paper can be downloaded wihou charge from he Social Sciences Research Nework Elecronic Paper Collecion: hp://ssrn.com/absrac= UNIVERSITY OF MICHIGAN

2 A REGIME SHIFT MODEL OF THE RECENT HOUSING BUBBLE IN THE UNITED STATES by Rose Neng LAI and Rober VAN ORDER * November, 2006 Finance Deparmen Key words: Bubbles, House Prices, Regime Shif *Rose Neng Lai is Associae Professor, Faculy of Business Adminisraion, Universiy of Macau, Taipa, Macao, China. RoseLai@umac.mo. Rober Van Order is Professor, Universiy of Aberdeen and Universiy of Michigan, finance Deparmen; rvo@bus.umich.edu.

3 Absrac of A REGIME SHIFT MODEL OF THE RECENT HOUSING BUBBLE IN THE UNITED STATES I has been widely assumed ha here was a bubble in he U.S. housing marke afer1999. This paper analyzes he exen o which ha was rue. We define a bubble as: (1) a regime shif ha is characerized by a change in he properies of deviaions from he fundamenals of house price growh, and (2) where a shock o he fundamenal equaion is more self susaining and volaile han in oher periods. We model he fundamenals of price growh as a lagged adusmen of prices o he expeced presen value of fuure ren. We hen sudy he auoregressive behavior of he residuals hus generaed. We look a changes in momenum (he exen o which a shock o house price growh leads o furher increases in house price growh) of he residuals. Our resuls from 44 Meropolian Saisical Areas for he period of (quarerly daa) are mixed. There is evidence of momenum in house price growh hroughou he period, and momenum did increase afer 1999, indicaing a regime shif; bu by a modes amoun, and while momenum was someimes srong i was no explosive. The regime shif was less apparen in he likely bubble candidae ciies along he coass, which had shown high growh in he pas. The evidence on volailiy is srong. In general, volailiy did no increase in he nonbubble MSAs, and i decreased in he faser-growing bubble MSAs. 1

4 I. Inroducion The recen propery marke in he Unied Saes has been widely perceived as having a bubble. Figure one shows he rae of growh of house prices across nine census regions (no labeled in he figure) from 1975 hrough The figure clearly shows ha house prices end o move ogeher, alhough here are periods of large dispersion across regions. The proposed bubble period is pos Wha makes his period differen is he sharp acceleraion in prices, especially in some regions and especially relaive o inflaion (no shown in he figure), during a period of relaively sable growh and lile change in economic condiions. In earlier periods, he rapid house price change could plausibly be explained by changes in ineres raes (for example, he early 1980s) or on regional recessions or expansions (such as he ups and downs of oil prices). These do no appear o be especially srong candidaes for explanaion in he lae 1990s and afer. A few facors migh show explanaory power on he propery price movemen. Figure wo shows naional raes of growh of house prices, he en-year Treasuries, an index of impued homeowner rens, and he Consumer Price Index (all of which are we use o explain housing fundamenals in he laer par of he paper). In general, propery prices move in sep wih he oher series in Figure wo. However, he acceleraion in house prices afer 1999 does no appear o be consisen wih he oher daa. Alhough ineres rae declines could be a facor (Long-erm Treasuries dropped by 300 basis poins, and real raes fell by close o he same amoun), hey canno explain he regional variaion in Figure One. Hence, a firs glance a he daa suggess ha somehing unusual occurred in U.S. markes afer 1999, and especially afer 2003 This paper analyzes he pos 1999 behavior of house price growh in he U.S. In paricular, we look a he exen o which we can characerize he period as having a 2

5 bubble relaive o he fundamenals of price growh. The definiion of a bubble is ofen vague and no widely agreed on. Our noion of a bubble is ha i is: (1) a regime shif ha is characerized by a change in he properies of deviaions from he fundamenals of house price growh, and (2) where a shock o he fundamenal equaion is more self susaining (increased momenum) and volaile han in oher periods. The fundamenals of price growh come from lagged responses o he presen value of expeced fuure ren. Various mehods have been proposed for esing bubbles in financial markes. Early work relied on economeric models such as variance-bound ess. However, hese mehods, which compare he acual daa wih fundamenals, have been criicized because of he specificaion errors of he fundamenals. Since hen, ess for saionariy and coinegraion as ess for absence of speculaive bubbles have been proposed (see, for example, Diba and Grossman (1988) and Hamilon and Whieman (1985)). Evans (1991), however, shows ha hese mehods end o reec he presence of he bubbles oo ofen even if hey are arificially induced in he Mone Carlo simulaions. The lieraure of esing bubbles hen moved on o he inroducion of he more effecive regime swiching models firs presened by Blanchard and Wason (1982). These models look a bubbles as changes in regime, and hen analyze properies of price processes in ou of he bubble regimes. 1 Our model is a varian of regime shif models. Apar from Roche (2001), who sudies he Dublin marke from 1976 o 1999, regime swiching models have no been widely applied in real esae research in explaining house 1 In a recen sudy, Baddeley (2005) incorporaes desabilizing effecs from bubbles, herding, and frenzies in he sudy of regime shifs condiional on insiuional and poliical changes. She argues ha in a less informed marke such as real esae, hence where herding can be serious, and where financing and uncerainy are crucial facors in deermining he ime o inves, marke booms and buss end o be more pronounced. 3

6 price bubbles. We posulae wo ypes of regimes: he firs is pre bubble, which we assume akes up mos of our sample period ( ) and which is characerized by a process for house prices ha we describe as coming from he fundamenals of he marke in a manner loosely consisen wih price being deermined by expeced presen value of rens, and he second is he bubble candidae period of The srucure of our model is similar o papers on housing bubbles by Black e al. (2006), Chan e al. (2001) and Hwang e al. (2006). We firs develop a model of he fundamenals of house price growh from panel daa on rens (renal equivalen for owner-occupied housing), ineres raes and house prices across 44 Meropolian Saisical Areas (MSAs) in he U.S. We hen use he model, and variaions, as a benchmark from which o generae residuals. Assuming he residuals follow an auoregressive process, we es wheher a bubble exiss, and if so, is magniude, by sudying how he residual processes change afer In paricular, we look a he exen o which momenum in he process (measured by he sum of he coefficiens of he process) increases afer 1999, and wheher he volailiy of he error erms in he residual process increases. We do his for wo ses of panel daa: one for slow growh or nonbubble MSAs, largely ciies in he cener of he counry, which are defined as MSAs whose house prices grew on average less han 2% per year faser han ren; and he oher for a se of bubble candidae MSAs, largely coasal ciies, whose prices grew more han 2% faser han ren (see Appendix 1). 2 We find evidence of momenum hroughou he period and some evidence ha momenum increased afer 1999, bu no by a lo. We find no evidence of an increase in volailiy. We also do no find evidence of explosive momenum (sum of coefficiens 2 The long run rend in our model is for prices o grow a abou a 2% per year faser rae han rens. 4

7 greaer han one) afer 1999, nor do we find much difference in price growh behavior beween he bubble and non bubble candidae ciies. We do find ha momenum operaes wih a long lag. There were always bubbles, bu no a large regime shif, a leas no in our sample period. The paper is organized as follows. The nex secion provides a discussion on bubbles and regime swiching models ha have been widely applied in financial markes. Secion III discusses how he housing price growh can be modeled. In paricular, we sugges he fundamenal equaion from which bubbles in he marke can be esed. Secion IV describes he daa employed, while Secion V presens he resuls. Secion VI discusses he robusness of our ess, and Secion VII concludes he sudy. II. Bubbles and Regime Swiching There has been considerable research on modeling he price movemens of sock markes in he desire of capuring he deviaions from he fundamenal values. 3 Two versions of hese models are he fads model proposed by Summers (1986) and he sochasic bubbles model suggesed by Blanchard and Wason (1982). The laer ype was subsequenly exended by Van Norden and Schaller (1993, 1996), and Van Norden, (1996), who use swiching regressions o describe he ime-varying relaionship beween reurns and deviaions from he fundamenals. The Fads model Borrowing from Fama and French (1988) and Culer, Poerba and Summers (1991), we can describe fads models as follows. The logarihm of marke price of an asse is assumed 3 Oher proposed sources of bubbles are, for example, overconfidence of speculaors coming from wo differen groups such ha he deviaions in price expecaions creae rading (Scheinkman and Xiong (2003)), and money illusion as a resul of reducion in inflaion, and hence nominal morgage coss (Brunnermeier and Julliard (2006)). 5

8 o be divided ino (1) a non-saionary par ha describes he fundamenal price and (2) a saionary componen ha implies he reurns are predicable (from previous reurns). Boh componens are auoregressive and subec o differen whie noises wih heir own disribuions. Given a proxy of he fundamenal price because of measuremen error, all hese imply p p = β + p p + e (1) ( x ) 1 β where p x 2 is he available proxy of he fundamenal price, and iid ( 0, σ ) e ~ ω. This regression equaion gives he excess reurns as a funcion of differences beween he log of he proxy for he fundamenal and he log he observed price. In financial markes, one commonly used proxy is he dividend, and he explanaory variable in he equaion is he lagged log dividend/price raio. Hence, price growh is a funcion of curren price and lagged fundamenals. Furhermore, because curren price (via equaion (1)) depends on he dividend/price raio lagged again, ieraing equaion (1) implies ha price appreciaion depends on a long lagged funcion of he proxy for fundamenals. Applying his model o house prices requires some modificaion. Firs, he assumpion ha he fundamenals follow a random walk and ha he fads par is saionary is no likely o hold. We expec ha o be he case because of obvious inefficiencies in real esae markes: (1) ransacion coss in real esae are high, (2) owner-occupiers are only in he marke occasionally, and (3) he ax benefis accrued o homeowners reduce heir coss bu no coss for speculaors, hus making arbirage difficul. As a resul, here is likely o be momenum even of fundamenal prices all he ime. Beyond ha, we wan o pose expecaions as abou changes in prices raher han levels, and model fundamenals applied o growh raes o see if residuals from his have differen properies in he pos 1999 period. In oher words, we do no impose he assumpion ha residuals from 6

9 equaions like (1) are independenly and idenically disribued (iid). The Regime Swiching Model When he regression error erm, e, is heeroscedasic, he fads model can lead o regime-swiching for sochasic bubbles (which are sochasic because hey eiher survive or collapse, subec o some probabiliies). The exisence of wo possible oucomes of he bubbles means ha here are wo regimes generaing marke reurns, he bubble being he more volaile of he wo. Tess are conduced on wheher he wo volailiies are significanly differen. We exend he regime swiching model by relaxing he assumpion ha he error erm in he auoregressive fundamenal price process is whie noise. We can hen arrive a an equaion similar o equaion (1), wih longer lags, and we assume (and es) ha he e follows an auoregressive process of he form T (2) e = ω e + υ. A regime shif o a bubble regime is characerized by an increase in he volailiy of υ and in he size of he coefficiens of he process for e, which is measured by ω. T III. Modeling House Price Growh In his secion, we develop a model similar o ha in equaion (1) for he housing marke, and adding differen sochasic properies. The basis of he model is he ineremporal behavior of households ha choose beween housing, h, whose purchase price is a represenaive consumer good, c whose price is c P. P and The Basic Model 7

10 Consider a household ha, given informaion se, Ω, abou he uncerain fuure house prices and ineres raes maximizes an ineremporal uiliy funcion of he form (3) E(Σ T U ( c, h ) β over some ime horizon T, subec o he consrain ha he presen value of expendiure (cash flows) equal he presen value of income plus iniial wealh. I is sraighforward o show ha (see Doughery and Van Order (1982) for a derivaion of a nonsochasic version) a firs order condiion can be expressed as (4) U U h c E = r ( P Ω ) +1 P P P P c. where r is approximaely given by ( 1θ ) i π + α, wih θ being he ax rae, i he risk-free ineres rae, π renal growh rae, and α is a consan erm. We can hink of r as he cap rae for our represenaive propery. I capures oher coss like depreciaion and propery axes which migh be assumed proporional o propery value and, if we allow risk, a risk premium. A broader specificaion would ake accoun of possible cash flow effecs. Tha is, high nominal raes can have a cash-flow effec beyond he real rae effec in (4) because of limiaions on he abiliy of borrowers agains fuure income, especially during periods of inflaion. In ha case he implied coefficien on i would be greaer han 1-θ. Equaion (4) says ha he marginal rae of subsiuion beween housing and he oher E P good equals he raio of he implici ren on housing, r ( Ω ) +1 P P P, divided by he price of he oher good. ( U / U ) P can be defined as he household s impued ren. h c c Equaion (4) along wih he oher consrains and parameers can be used o generae he demand for housing. This can hen be aached o a model of housing producion o generae a model of house prices. Building in allowances for ransacions and moving coss in r would imply adusmen o he marginal condiion in equaion (4) wih a lag, and he model would be very complicaed and probably sensiive o paricular specificaions.. 8

11 An alernaive o complicaed model building is o ake advanage of he fac ha housing is rened as well as owned, and ha we can assume ha he owner acs as a landlord rening o him or her self. This implies a useful separaion for modeling. In effec, rens summarize all of he local marke condiions ha are deermined by income and wealh and supply elasiciies, and we can ake ren as given and express price as he presen value of ren. This allows us o employ models like he price-dividend models used o analyze sock prices (he same raionale, and hence modeling approach, is also adoped by Brunnermeier and Julliard (2006)). Consider a household ha is idenical o he one ha we have analyzed excep ha i rens a price R raher han owns. Because prices in his period are known, is firs order condiion corresponding o equaion (4) is nonsochasic and is given by (5) U / U = R / P h c c For a household ha is us indifferen beween owning and rening 4 we can equae expressions (4) and (5) o obain a soluion ha (6) P = R + E ( P Ω ) D +1 where D + = 1 r. The variable r incorporaes a premium for risk for invesing in real esae. Equaion (6) says ha price equals he curren ren plus he sales price in he nex period. This is a raional expecaions (perfec foresigh) model, and like mos such models i is indeerminae. Tha is, here are many curren levels of price ha are consisen wih equaion (6). For example, consider he simples case where rens and r are consan and is equal o R and r. Then a soluion o equaion (6) is R (7) P =. r However, an infiniy of iniial levels of P can be chosen for which (solving equaion (6) for P + 1 ) he pricing equaion (8) P = P D R +1 4 Households ha are approximaely indifferen beween owning and rening are likely o be in he lower ax brackes. Moreover, Cauley and Pavlov (2002) menion ha models using renal coss provide a lower bound of he price because pride of ownership has no been priced. 9

12 sill holds. While equaion (7) reflecs he Gordon model and provides a sable equilibrium price, equaion (8) leads o explosive price moves because D is greaer han one. In oher words, equaion (7) corresponds o wha we hink of as a fundamenal soluion, while equaion (8) corresponds o an explosive bubble. More generally, equaion (8) can be solved recursively o obain i i (9) P = E( R + i / D + i Ω ) + lim E( P + 1+ i / D + 1+ i Ω ). i= 0 The ransversaliy condiion is ha he second erm approaches zero, so ha he fundamenal equaion becomes i= 0 i (10) = E( R / D ) P. + i + i Ω However, as before, his is no he only soluion. A bubble process ha saisfies (11) B +1 = B D + e will also saisfy equaion (6), and i will end o be explosive. Special Cases Equaion (15) is quie complicaed because of covariances, such as hose coming from sock-flow adusmens of rens and prices over ime, among he variables in i. For insance, we should expec ineres raes and fuure rens o be correlaed on he grounds ha a rise in ineres raes will, given rens, lower propery values, bu on he oher hand induce less producion in he fuure, and hus higher rens. Indeed, if supply is perfecly elasic in he long run, a rise in ineres rae will produce a gradual decline in rens wih no long run price change. We consider firs a very simple model wih consan ineres raes and a seady growh rae of expeced rens. Then we can adap he Gordon model (12) P = R /( φ i π * + α). 10

13 Tha is, r is exended o ( φ i π * + α ), where ( ) φ is a coefficien ha incorporaes boh he ax and cash flow effecs on he effec of i, he ineres rae, and π * is he expeced raes of growh of ren. The model does no allow us o predic wheher changes in i or π * will have a larger effec on price. Noice ha he model will always work beer by including more exogenous variables such as level of supply or average personal income. However, if our model is correc, ren should be a summary saisic ha has already accouned for supply and demand. Taking firs differences and logarihms of expression (12), we have (13) GP = π Δ ln( γi π *) where GP is he growh rae of house prices and, π is he curren rae growh of rens. Equaion (13) can be approximaed by (14) ρ = GP π = α β Δi + β π * i π Δ where ρ is he rae of growh of house prices minus he rae of growh of ren and he βs are posiive. This can for insance be esimaed by assuming ha Δ π * is a funcion of pas levels of Δ π. However, preliminary esimaes of (14) do no work well; longer lags are necessary for he model o fi well and/or make sense. So we exend he lag srucure. Adusmen o Equilibrium Equaion (14) only holds in he simple Gordon Model, which involves wo key assumpions: (1) he curren price is he equilibrium price, and (2) ren growh is expeced o be consan. The high ransacion coss in housing markes make he firs assumpion difficul o ake seriously. Moreover, ownership of single family housing in he U.S. is driven in many 11

14 ways by ax advanages 5 ha are received by propery owners only on heir firs or second house, which precludes serious arbirage. Home buyers end o ener he marke and obain informaion abou propery only a imes when hey are seriously ineresed in buying. Hence, he informaion needed for equaion (9) o hold is dispersed only gradually among differen households. For hese reasons, we expec prices o adus wih a lag o he equilibrium price. Furhermore, he second assumpion abou seady price growh is no likely o hold in he shor run. Expeced fuure ren growh is probably no consan and is probably correlaed wih ineres raes and pas growh in rens and prices. Theory does no ell us much abou how o model hese. To solve he problems discussed above, we impose he following srucures on he fundamenals model. Firs, we model he formulaion of expecaions as composed of wo pars: a shor run par ha reflecs curren informaion for he nex few years, and a second longer run, sabilized, 6 par ha looks like he Gordon Model, and o which expecaions adus afer a period of ime. Tha is, we assume ha afer some period he bes ha raders can do is o proec seady growh. Before ha, we allow rens o vary from he rend. Second, we assume ha he presen value formulaion provides he equilibrium price, bu ha he marke price only aduss gradually o i by following a generalized geomerically disribued lag. The combinaion of hese wo srucures implies ha prices or he growh in prices adus gradually o a long run Gordon model. Deermining he Equilibrium Price 5 In paricular, homeowners ge o deduc much of he cos (e.g., morgage ineres and opporuniy cos on equiy) of operaing he house (rening o hemselves) and pay virually no capial gains axes wihou paying ax on he impued ren (see Gyourko and Sinai, 2003, for he sudy of he impac of ax subsidies on home owners in various MSAs). 6 This corresponds o he noion of a sabilized Cap Rae. 12

15 13 Firs we wrie he equilibrium price as he presen value of an irregular ren sream for τ periods, followed by seady sream, or (15) ( ) τ τ π α γ α γ + + = = e i r R i R P ) (1 1 *. Alernaively, adding and subracing ( ) + = + + = τ π α γ i R G * 1, and dividing by R, we have (16) r G r i R R R P e 1 1 ) (1 + = = + = τ α γ δ. We assume ha acual price aduss o he difference beween curren and equilibrium price, in logarihms. Le ) / log( R P p = and ) / log( * * R P p =. We exend he adusmen model in equaion (1) o include possibly longer lags, ha is, (17) = + = T e p p p p 0 1 ) ( λ. Taking firs differences and rearranging, we have (18) ) 1/ 1/ ( ) ( = = = = T T r r G G p p p p θ β ρ. A linear approximaion o his can be wrien as (19) = + Δ + Δ + Δ = T G i G i 0 * * ) ( ρ β β π β β α ρ ρ π Boh Δπ* and ΔG are expecaions variables. We assume ha hey are aken from he curren informaion se, which conains recen and pas levels of ineres raes, rens and prices. We can hen rewrie equaion (19) as (20) = + Δ + Δ = T i i 0 ) ( ρ γ π γ γ α ρ ρ π, which says ha he curren rae of growh of house prices relaive o rens is a linear funcion of pas change in ineres raes, ren growh, and lagged changes in price growh ne of ren growh.

16 A less srucured version of his is T + T ' i 0 i π (21) ρ = α ( γ Δ + γ Δπ ) in which lagged ρ is dropped, while he lag is lenghened by T. Boh versions impose he consrain ha in he long run an increase in ren of 1% will increase house price growh by 1%. Hence, afer T or T periods he model revers o he Gordon model if α is zero. The presence of α allows rens and prices o have differen rends. Reasons why his migh be he case, primarily measuremen error, are discussed below. Esimaes of equaions (20) and (21) will generae residuals, e. And insead of imposing e as iid, we assume ha i follows he auoregressive process T (22) e = ω e + υ where = υ is iid. The process for e is a varian of he B process in expression (11). For a raional bubble, he sum of coefficiens mus be greaer han one. Our ess are (1) of he amoun of, and changes in, momenum as measured by wheher ω is greaer han T one and/or increased during he pos 1999 period, and (2) wheher he variance of υ was higher during he pos 1999 period. IV. Daa and Esimaion Our measure of house price is he quarerly house price index released by he Office of Federal Housing Enerprise Oversigh (OFHEO), which provides he widely quoed residenial (single-family) house price index for over 100 individual Meropolian 14

17 Saisical Areas (MSAs) since The ren series is he owner s equivalen ren of primary residence obained from he Bureau of Labor Saisics, from which we also acquire he local Consumer Price Indices. Afer maching hese hree series, daa for a oal of 44 MSAs can be used. 7 We use he 10-year Treasury as a measure of nominal risk-free rae. 8 Three daa concerns are in order. Firs, he price index may no hold qualiy consan. The OFHEO index looks a he same house wice bu does no adus for home improvemen beween observaions, so i may over esimae growh in house prices. Second, measured ren may grow oo slowly because of he agency cos of rening and measuremen errors in he renal index. Tha is, even if we have mached prices and rens for owners indifferen beween owning and rening, here is reason o believe ha reners ake less good care of propery han do owners. Crone e al (2006) and Gordon and van Goehem, (2004) boh discuss he exen o which he CPI renal index has underesimaed ren growh over ime (especially before 1985). If any of he above is he case, hen here will be a endency for our measure of P o grow faser han our measure of R (ha is, for α o be posiive), which is indeed wha we find. Third, he price and ren series do no necessarily mach up in he sense of he price series represening price growh for a household ha is indifferen beween owning and rening, probably a household in a relaively low ax bracke. We noe here ha he OFHEO index only covers prices of houses whose morgages can be purchases by Fannie Mae or Freddie Mac. This imposes a limi, which is indexed o house prices over ime and excludes approximaely he op 10% of he marke (by number of loans). Hence, he price daa do a leas exclude hose 7 In order o maximize he lengh of he ime-series, we eliminae hose MSAs ha have shor ren indices. 8 We have ried o proxy he real ineres rae by he en year Treasury Inflaion-Proeced Securiies (TIPS). However, since he earlies available TIPs begins lising in 1997, we are no able o obain a reliable real ineres rae series. 15

18 owners who, say, for ax reasons, are he furhes from being indifferen beween owning and rening. Wih hese daa, we firs esimae varians of he fundamenals of price growh from he specificaions of ρ in equaions (20) or (21). We esimae fundamenals over he enire period. We vary hese models by changing lag lengh. From hese are generaed residuals, which we model as given by equaion (22) for various lag lenghs. For each fundamenal equaion, we esimae four residual equaions for a given lag lengh. These groups of regressions come from dividing he MSA sample ino wo groups: fas growing (bubble candidae) MSAs (hose ha are widely perceived as overheaed markes, and mosly whose house prices over he period grew on average a a 2% per year faser rae han rens grew), and he res as non bubble saes. These bubble saes are depiced wih aserisks in Appendix 1. This grouping is mean o capure he possibiliy ha he bubble candidaes are more suscepible o bubbles. We also divide he sample ino (1) he 1999 and earlier, pre bubble period, and (2) he pos 1999 regime shif candidae period. Our ess are of he exen o which here was a regime shif. If here was a regime shif, we should expec he sum of he coefficiens in esimaes of equaion (22) o be larger in he pos 1999 period, perhaps larger in he bubble MSAs, and he variance of he residuals in he error regression o be higher pos 1999, and perhaps higher in he bubble MSAs. V. Resuls Esimaes of he Fundamenals Tables 1 and 2 summarize he esimaes of our wo fundamenal equaions (20) and (21) respecively using he enire panel of daa across MSAs for he enire sample period. The 16

19 variables in general have he righ signs. The signs wihin groups are also consisen, generally negaive for ineres rae and posiive for pas ren growh and lagged ρ. Consider he 8-lag model in Table 1 (he hird column). Long run effecs are given by he sum of coefficiens for he hree variables. The sum of he ineres rae coefficiens is around -2.5, ha of he ren growh is 2.3, while he sum of lagged ρ coefficiens is 0.6 (see Table 3). Tha he sum of coefficiens of ineres rae changes is close o, bu slighly bigger han, he sum of he ren growh change coefficiens is consisen wih he noion ha price is driven by real ineres raes wih he ax effec being more han offse by he cash flow effec. However, ha resul does hold for our oher specificaion wihou lagged ρ. 9 All versions of our models have consan erms of around , reflecing a quarerly difference in growh raes of abou 0.25%, or 1% per year. In he models wihou lagged ρ he consan erm was around.5. The resuls sugges ha in he long run prices grow a close o a 2% faser annual rae han rens. The long run effec of a change has o ake ino accoun feedback hrough he gradual adusmen of ρ. Long run equilibrium in growh raes akes place when pas and curren levels of ρ are he same. The cumulaive effec of a one-ime shock o i on ρ afer T periods is given (rounded) by T i ρ (23) ρ = ( γ ) Δi + ( γ ) ρ ) = -2.5Δi ρ. 0 T 0 The daa used in he model are all a quarerly raes, including he 10-year Treasuries. Hence a 100 basis poins increase in 10-year Treasury raes is a 25 basis poins increase in he quarerly rae. As a resul, he long run effec of he 10-year Treasuries mus be divided by four. Thus, a 100 basis poins increase in ineres raes will have a long run 9 For insance we migh expec he sum of coefficiens of lagged ren growh changes o be low because as proxies for fuure expecaions hey are subec o measuremen error. 17

20 impac of / (1-0.6), or abou -1.6% on price relaive o ren. The long run impac of a change in raes in he model is like duraion. One migh expec a somewha bigger number for a long erm asse. However, as was discussed above, i is likely ha when ineres raes change, expecaions abou fuure rens also change; in he usual sock flow model of housing adusmen, a decrease in ineres raes will cause consrucion o increase, which will decrease fuure rens, lowering he numeraor in he presen value formula. The model wih 4 lags does no fi well or make much sense. Wih 12 lags, he resuls are similar, alhough he fi is somewha beer. The sum of ineres rae coefficiens is abou -1.7, he sum of ren growh 1.6, and he long run effec of an ineres rae change is -1.3 (see Table 3). In he 16 lag case, he model reecs he fixed effecs. Neverheless, we sill obain similar resuls. This appears o be a respecable model of fundamenals in he sense of having sensible coefficiens. I also suggess ha, because of he significanly posiive coefficiens for lagged ρ, here is momenum in house price growh over he enire period. A oneime shock o ρ feeds back ino he model gradually and fades gradually. The srengh of he momenum will also depend on he auoregressive properies of he errors in equaion (22), which are analyzed below. The model implies a significan lag in he effec of an ineres rae change on house prices. Table 2 has esimaes of he fundamenal equaion wihou lagged ρ for 8, 12, and 16 quarer lags. The lag lenghs are longer han hose in Table 1 because, by no including he lagged ρ, he effecs capured by oher explanaory variables should end o be longer. I is obvious ha no including he lagged dependen variable significanly reduces he 18

21 explanaory power of he fundamenal equaion. Saed differenly, he memory effec previously shown by he lagged dependen variables o capure he momenum has o be shifed o oher exogenous variables, hus requiring even longer hisory from hese variables. Resuls are however in some ways similar. The sum of coefficiens of ineres rae change is -6.2, which gives abou he same long run effec of ineres raes on price. On he oher hand, he effec of ren inflaion is much less a 3.5. Shorer lag specificaions produce worse fis, and he coefficiens make less sense. Table 3 presens a summary of he resuls of he coefficiens for he various specificaions of he fundamenals, as well he effecs of a one ime increase in he 10 Treasury rae, and adused R-squared. An obvious resul is ha longer lag specificaions fi beer, and heir coefficiens make more economic sense. 10 Error Equaions We use he fundamenal equaion(s) o generae errors equaions. In paricular, we employ he 8-lag and 12-lag fundamenal regression equaions as depiced in Table 1 o generae residuals, which are hen used o esimae varians of he auoregressive model as in equaion (22). As described above, we divide he available daa ino bubble and nonbubble MSAs, and we produce separae esimaes of he error model by hese MSA divisions in he pre- and pos-1999 period. The resuls using residuals from he fundamenal equaion wih 8 lags and 12 lags are depiced respecively in Appendices 2 and 3. Resuls for he same error equaions for residuals from he 8-lag fundamenal equaion wihou lagged ρ (ha is, regression resuls from Table 2) are shown in Appendix 4, while Appendix 5 depics he corresponding findings for he 12-lag model. 10 We iniially ried o esablish he fundamenal model wih local CPI o capure he MSA specific inflaion, and in a way, deduce he real ineres rae. However, adding he variable does no increase he explanaory power and inuiion of he model significanly. We herefore mainain he curren model for parsimony. 19

22 Our concern is wih he sum of he coefficiens and he volailiy of he disurbance. Table 4 presens summary resuls for he sum of he coefficiens by model and lag srucure. Consider Panel B, which presens resuls for he model wihou lagged dependen variable, ρ. In all of he specificaions, he sum of coefficiens is posiive before 1999; and in all cases he sum increased afer On average, he increase in sums was around 0.2 or 0.3. While he bubble MSAs had higher sums, he increase in sum was, if anyhing, lower in he bubble MSAs. Running across he able, i is easy o see ha while he sums of he coefficiens for he non-bubble MSAs show mixed paerns in boh ypes of fundamenal equaions, hose for he bubble MSAs almos ubiquiously (excep for 12-lagged error erms in he 12-lagged fundamenal) decrease. A closer look a Appendices 2 o 5 reveals ha he lagged errors of he bubble MSAs are significan mosly for only he firs few lags; and his is especially rue when more lags are included. The MSAs had faser, more fron-loaded, adusmens in he pos-bubble period. Somehing did happen pos There was a regime shif, bu no a large one. In fac, he regime shif ends o be smaller when longer lags are considered in he case of bubble MSAs. Furhermore, in none of he cases was he sum of coefficiens close o one; ess on his were reeced in all cases. Momenum increased, bu i was no explosive. Panel A produces resuls for cases wih lagged ρ. I is more complicaed because he presence of lagged ρ adds momenum o he sysem along wih momenum added by he errors. In he pre-1999 period, he coefficien sums were generally negaive, hus offseing some of he posiive momenum from he posiive coefficiens of lagged ρ. The resuls for before and afer 1999 were similar; he sums ended o increase by around 0.3, hough wih more variabiliy. 20

23 Summing up he basic resuls for he coefficiens sum, we are able o observe momenum hroughou he period, and he adusmen lags were long. There is also some evidence of a regime shif in he pos-1999 period. However, unlike financial markes, here is no evidence of an explosive bubble associaed wih he regime shif. Finally, here is no evidence ha he bubble MSAs were worse. Indeed, our model is able o explain he price movemens in he bubble MSAs relaively beer han he non-bubble ones. Volailiy Table 5 presens resuls for esing he changes in he volailiy of he errors in equaion (22). Panel A corresponds o resuls generaed from he fundamenal wih lagged regressand, ρ in Appendices 2 and 3; while Panel B presens resuls wihou lagged ρ from Appendices 4 and 5. We apply he Goldfield-Quand es for he differences in variances. The resuls of he ess can be read from he Pre/Pos-1999 Tes rows. Bold face numbers show cases where he hypohesis ha he variances are differen is acceped. In he nonbubble MSAs he hypohesis is always reeced. In he bubble MSAs he hypohesis is almos always acceped. However, in all hose case he variance fell afer Hence, once again, here is some evidence of a regime shif in he bubble MSAs. However, he shif is oward a more sable regime afer Reviewing he acual marke movemen may promp a query on he choice of he cuoff period. Tha is, i is possible ha he bubble became bigger in he laer par of he pos-bubble period. We herefore run error equaions from he fundamenals in Appendix 2 wih 8 lags (resuls omied here). In boh periods of 2002 hrough 2005 and We have also esed he variances of he 44 individual MSAs. There is only an average of one or wo MSAs ha have saisically significan change in variance beween o pre- and pos-bubble periods in any of he cases. We herefore omi he resuls here for purpose of simpliciy. 21

24 hrough 2005, he coefficien sums are eiher he same as hose in he pos-1999 period as a whole or, in he case of he bubble MSAs, lower. Table 6 depics he variances in he various sub-periods. The rows noed as GQ Tes: 99 vs 02 and GQ Tes: 99 vs 03 exhibi ess for increases in variance. Variances did go up in boh he pos-1999 versus and pos-1999 versus he periods, bu differences are no saisically significan. We can hus conclude ha our resuls are no sensiive o he cuoff poin a Fundamenals wihou Lagged Regressands As menioned earlier, he fundamenal equaion wihou lagged dependen variable, ρ, requires longer lags because he effecs capured by oher explanaory variables should end o be longer. We run he ess again wih 20 lags and 24 lags on he explanaory variables. We hen adop he 12-lag error equaion model. Resuls are shown in Appendix 6. Panel A of he Table presens he panel regression resuls, while Panels B and C depic he regression resuls of he 12-lag error equaion and he es of difference in variances beween he pre and pos bubble periods respecively. As expeced, he explanaory power increases wih increase in he number of lags included, albei marginal. The error equaion resuls also show ha he MSAs price versus ren growh raes adus relaively faser in he pos-bubble period. As more lags are included in he fundamenal, he errors end o carry more momenum (sum of coefficiens is bigger). Neverheless, hey are sill non-explosive. Finally, only he bubble MSA group shows a change in he volailiy from he pre versus pos bubble period in he 24-lag fundamenal equaion case; and is only barely saisically differen. Once again, volailiy in general decreases in he pos-bubble period. 22

25 VI. Robusness of he Fundamenals A maor complicaion in our sudy is ha he resuls of he error equaions migh be sensiive o he fundamenal equaions employed. We herefore esimae some variaions of he fundamenals o see if he error equaions sill lead o findings ha are similar o he ones we obained in he previous secion. We firs separae he daa se for he fundamenals hose for bubble MSAs and non-bubble MSAs and esimae separae panel regressions (regression resuls depiced in Appendix 7). The raionale is ha, assuming bubble and non-bubble markes are separae groups, inra-group markes migh share idenical effecs from he facors in he fundamenal equaion, bu no iner-group markes. As expeced, he error equaions (wih 8, 12, and 16 lags) shown in Appendix 8 are differen beween he wo groups of MSAs, as well as differen from he previous es wihou he separaion, because he panel regressions should be able o beer capure he common characerisics of he wo MSA groups. The sensiiviy o a change in he regression does no however aler our previous conclusion. Firs, he sums of he coefficiens of he lagged errors are far less han one. Second, he volailiies in he pre- and pos-bubble period are very similar (see Table 7). Even if he hypohesis ha variances in he wo periods are saisically he same is occasionally reeced, he difference is minimal. This is similar o he findings in he previous secion. Our second variaion is o include he inflaion rae ino he fundamenal equaion. This allows he discoun rae o be hough of composed of a real rae plus real ren growh, and hese migh no have he same coefficiens (e.g., because of differen measuremen errors). Furhermore, local inflaion may conain informaion abou ren, or is deerminans ha is no found in he renal equivalen index (e.g., he ren numbers migh be oo smooh or grow oo slowly relaive o he rue numbers. We herefore obained 23

26 changes in inflaion raes for each individual MSA from he local CPI series available from he Bureau of Labor Saisics, and included he series in he fundamenal equaion, again wih 8 lags, from which we obain he variance ess from he error equaions. 12 The error equaion resuls are abulaed in Appendix 9, while he comparison of variances in he pre- and pos-1999 period is exhibied Table 8. I is clear from he ables ha, again, albei he high sensiiviy of he resuls o he fundamenal equaion, he basic resul is sill ha here are only small races of bubbles/regime shifs in he propery marke in he U.S. in he period of sudy. Anoher robusness check is o es if he behavior of he models differs when he pre-bubble period is separaed from he pos-bubble period in esimaing he fundamenal equaion. However, he very shor pos-bubble period daa does no have enough degrees of freedom for esing on he wo periods separaely. VII. Conclusions Perhaps he bes way o characerize housing markes during our sample period is ha here were always small bubbles, bu no a large regime shif. There does appear o have been a small regime shif afer 1999, bu i was weaker in he likely bubble candidae ciies along he coass, ciies which had shown high growh hroughou he period. There is evidence of momenum in house price growh hroughou he period, and he momenum did increase afer 1999, bu no by a lo. These resuls appear o hold if we consider pos-2002 as he bubble period. The evidence for volailiy is srong. In general, volailiy no only did no increase in he nonbubble MSAs, bu acually decreased in he faser-growing bubble MSAs. Hence, evidence for a bubble across regions is modes, and 12 We do no presen he panel regression resuls wih local inflaion because our focus is on he behavior of he residuals from he error equaions hus generaed. 24

27 somewha miigaed by he long lags suggesed by he model. Resuls are no very sensiive o he variaions in lag lengh and lag srucure ha we ried. We have no esed for local resuls, so we canno exclude srong local bubbles. We have also no ruled ou ha income (see Black e al.) migh be a beer proxy for ren han our curren ren series exraced from he CPI. Asian immigrans mosly o coasal ciies are anoher possible explanaion for overheaing he real esae marke, bu are unlikely o be a source of bubbles. I is for fuure exension and more complicaed modeling o es effecs of such immigrans, or oher demographic paerns, as an explanaion for why growh raes in bubble ciies end o be more susainable. References Abraham, Jesse and Paric Hendersho, (1996) Bubbles in Meropolian Housing Markes, Journal of Housing Research, 9(2) pp Baddeley, Michelle (2005) Housing bubbles, herds and frenzies: Evidence from Briish Housing Markes, CCEPP Policy Brief No , Cambridge Cenre for Economic and Public Policy, Cambridge. Black, Angela, Paricia Fraser and Marin Hoesli, (2006) House Prices, Fundamenal and Bubbles, Journal of Business Finance and Accouning, forhcoming Blanchard, Olivier J., and Mark Wason (1982). Bubbles, Raional Expecaions and Financial Markes, In P. Wachel (ed.), Crisis in he Economic and Financial Srucure, Lexingon MA: Lexingon Books. Brunnermeier, Markus K. and Chrisian Julliard (2006). Money Illusion and Housing Frenzies, manuscrip, Princeon Universiy Cauley, Sephen Day, and Andrey D. Pavlov (2002). Raional Delays: he Case of Real Esae, Journal of Real Esae Finance and Economics, 24(1/2), Chan, In Cahn, Shy Kam Lee and Kai Yin Woo (2001) Deecing Raional bubbles in he Residenial Housing Markes in Hong Kong, Economic Modelling,18, Crone, Theodore M., Leonard Nakamura and Richard Voih, (2006) The CPI for Rens: A Case of Undersaed Inflaion. Working Paper Nol Federal Reserve Bank of Philadelphia. 25

28 Culer, David M., James M. Poerba, and Lawrence H. Summers (1991). Speculaive Dynamics, Review of Economic Sudies, 58(3), Doughery, Ann J., and Rober Van Order Inflaion, Housing Coss and he Consumer Price Index, American Economic Review, 72, Diba, Behzad T. and Herschel I. Grossman (1988). Explosive Raional Bubbles in Sock Prices? American Economic Review, 78(3), Fama, Eugene F. and Kenneh R. French (1988). Dividend yields and Expeced Sock Reurns, Journal of Financial Economics, 22(1), Funke, Michael, Sephen Hall, and Marin Sola (1994) Raional Bubbles during Poland s Hyperinflaion Implicaions and Empirical Evidence, European Economic Review, 38, Glaseser, Edward, Joseph Gyourko and Raven Saks (2004). Why Have Housing Prices Gone Up? Gordon, Rober., and Todd vangoehem, (2004) Downward Bias In The Mos Imporan CPI Componen: The Case Of Renal Sheler, In Erns Bernd and Charles Hulen, eds, Hard o Measure Goods and Services, Essays in Memory of vi Griliches, Univ. of Chicago Press, Chicago, forhcoming. Gyourko, Joseph and Todd Sinai (2003). The Spaial Disribuion of Housing-Relaed Ordinary Income Tax Benefis, Real Esae Economics, 31(4), Hamilon, James B. (1989). A New Approach o he Economic Analysis of Nonsaionary Time Series and he Business Cycle, Economerics, 57(2), Heibling, Thomas,(2003) Housing Price Bubbles- A Tale Based on Housing Price Booms and Buss, on Bis Papers, Number 21g Himmelberg, Charles, Chrisopher Mayer, and Todd Sinai. (2005). Assessing High House Prices: Bubbles, Fundamenals and Mispercepions, Journal of Economic Perspecives, 19(4), Hwang, Min, John Quigley and Jae-young Son, (2006) The Divedend Pricing Model: New Evidence from he Korean Housing Marke, Journal of Real Esae Finance and Economics, 32, Poerba, James M Tax Subsidies o Owner-Occupied Housing: An Asse-Marke Approach, Quarerly Journal of Economics, 99(4), Roche, Maurice J. (2001) The rise in House Prices in Dublin: Bubble, Fad or Jus Fundamenals, Economic Modelling, 18, Schaller, Hunley and Simon Van Norden (1997). Feds or Bubbles, Bank of Canada Working Paper No Scheinkman, José A. and Wei Xiong (2003). Overconfidence and Speculaive Bubbles, Journal of Poliical Economy, 111(6), Van Norden, Simon (1996). Regime Swiching as a Tes for Exchange Rae Bubbles, Journal of Applied Economerics, 11,

29 Van Norden, Simon and Rober Vigfusson (1996). Regime-Swiching Models: A Guide o he Bank of Canada Gauss Procedures, Bank of Canada Working paper Figure one Annual Growh Raes of House Prices by 9 Census Regions Figure Couresy of Freddie Mac 27

30 Figure Two Growh Raes of House Price Index, Ten-year Treasury Bonds, Consumer Price Index, and Ren Index, a he Naional Level percenage CPI Ren index Price index 10-Yr Treasury 28

31 Table 1 Basic Regression Resuls for he Fundamenal Equaion wih Lagged Regressands (Various Lags) Equaion being Difference beween Growh Raes of House Prices versus Ren (Regressand) on Lagged Changes in 10-year Treasury, Local Ren Growh, and Regressands (MSA Fixed Effecs Omied) Variables 4 Lags 8 Lags 12 Lags Nominal Ineres Lag ** ** * *** ** *** *** Ren Growh Lag *** *** ** *** *** *** *** *** *** *** *** *** *** *** *** *** ** ** *** *** ** Regressand Lag *** * *** *** *** *** *** *** *** *** *** *** *** ** *** *** *** *** ** *** Adused R-square

32 ***, **, * represen significance a 1%, 5% and 10% respecively. Table 2 Basic Regression Resuls for he Fundamenal Equaion Wihou Lagged Regressands (Various Lags) Equaion being Difference beween Growh Raes of House Prices versus Ren (Regressand) on Lagged Changes in 10-year Treasury and Local Ren Growh (MSA Fixed Effecs Omied) Variables 8 Lags 12 Lags 16 Lags Nominal Ineres Lag *** * *** *** ** *** *** *** *** *** ** *** Ren Growh Lag *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** ** *** *** *** *** *** *** Adused R-square ***, **, * represen significance a 1%, 5% and 10% respecively. 30

33 Table 3 Comparison of Sum of Coefficiens From Fundamenal Equaions, Equaions (40) and (41). Coefficiens are from Tables 1 and 2. Wih Regressand on RHS Wihou Regressand on RHS Variables Lag = 4 Lag = 8 Lag = 12 Lag = 8 Lag = 12 Lag = 16 Change in Ineres Rae Change in Ren Growh Change in lagged LHS LR effec of Change in Ineres rae 100bp Adused R-squared N/A N/A N/A