House Pice Dynamics with Heteogeneous Expectations Giovanni Favaa y and Zheng Song z This vesion August 2008 Abstact This pape pesents a dynamic equilibium model of the housing maket in which agents consume housing sevices and speculate on futue pice changes. The model featues a xed supply of housing and a andom vaiation in demand, oiginating fom the fact that agents hold heteogeneous expectations about the futue couse of pices. The impotant featue of the model is that heteogeneous expectations geneate a non linea demand fo housing: agents expecting highe futue pices buy in anticipation of capital gains; agents holding pessimistic expectations pefe to ent to avoid capital losses. Because pessimistic agents ent thei expectations ae not incopoated in the pice fo owned houses. As a consequence, the equilibium pice e ects only the expectations of optimistic agents and is thus biased upwad. We test the pedictions of the model with US city data, using the dispesion in city income as a poxy fo infomation dispesion. The empiical evidence suppots the pediction that house pices ae highe in cities with moe dispesed beliefs about futue economic conditions. We ae gateful to Philippe Bacchetta, Alessando Bebe, Daell Du e, Benad Dumas, Simon Gilchist, John Hassle, Ethan Kaplan, Pe Kusell, Rafael Lalive, Tosten Pesson, Pascal St. Amou, Gianluca Violante, Alexande Ziegle, Fabizio Zilibotti, and semina paticipants at HEC Lausanne (DEEP and IBF), the Univesity of Zuich (IEW and ISB), Jiao Tong Univesity, the Univesity of Malaga, Bocconi Univesity, Luiss, the Einaudi Institute fo Economics and Finance, IMF, ECB, the 2008 Noth Ameican Summe Meeting of the Econometic Society, the 2008 Swiss Economic Society Annual Meeting, the 2008 Royal Economic Society Meeting, the 1 st Nodic Summe Symposium in Macoeconomics, fo helpful discussions and comments. y HEC Univesity of Lausanne. Email: gfavaa@unil.ch z Fudan Univesity. E-mail: zsong@fudan.edu.cn 1
1 Intoduction The US housing maket has expeienced substantial pice vaiations in the last two decades. Figue 1 gives an example of such vaiations fo the aggegate US economy and a epesentative sample of US cities. 1 As shown, in some cities, such as Los Angeles, the housing pice has moved in tandem with the oveall national index, yet in othe cities pices movements have been quite heteogenous. In Miami, fo example, the house pice index has been steady fo almost two decades befoe inceasing exponentially in 2000. In San Antonio, the same index has declined since the 1980s and has not ecoveed since then. In Rocheste, the eal index of house pices has displayed an invese U shaped histoy, while in Memphis it has gone though peiodic cycles. In the opinion of many housing-maket obseves (see e.g. Glaese and Gyouko, 2007), these high fequency pice vaiations ae di cult to explain though the lens of a standad use cost appoach (e.g., Poteba, 1984, 1991) in which house pices ae detemined by an indi eence condition between owning and enting given cuent and expected futue fundamentals, such as the use cost, income, and constuction costs. Fo US cities, fo example, Case and Shille (2003) nd that house pices movements cannot be explained by income vaiations alone; Himmelbeg, Maye and Sinai (2005) ague that the pice-income atio and the pice-ent atio cannot account fo the bulk of house pice changes; Glaese, Gyouko and Saks (2005b) nd a weak elationship between house pices and constuction costs. The pupose of this pape is to pesent a simple vaiant of the use cost appoach to ationalize some of the pice movements displayed in Figue 1. Specifically, we build a model featuing a xed supply of housing and a demand that uctuates stochastically because households hold heteogeneous beliefs about the futue couse of house pices. Heteogeneous beliefs aise because households ae impefectly infomed about the state of the economy and use thei own income, togethe with othe signals, in the estimation of the undelying fundamentals. Thus, idiosyncatic income shocks tanslate into heteogeneous expectations about futue aggegate demand and housing demand, and a fotioi given the xed supply into heteogeneous expectations of house pices. Guided by the logic of this model, we ceate an index of di eence in expectations based on the dispesion of local income shocks and nd that this index in uences signi cantly the dynamics of house pices acoss and within US cities. Ou analyis ests on thee building blocks household income, heteogeneous expectations, and xed housing supply which ae motivated by seveal aspects of the US housing maket. Fist, the available evidence suggests that income emains 1 We use the OFHEO constant quality house pice index fo single-family one-unit popeties nanced with a motgage below the confoming loan limit. 2
the main deteminant of housing demand, eithe because iche households tend to demand moe (Poteba, 1991, Englund and Ioannides, 1997) o because highe income elaxes cedit constaints (Otalo-Magne and Rady, 2006, Almeida et al., 2006, Benito, 2005). Second, suveys of households expectations (Case and Shille, 1988, 2003) eveals a stong investment motive of home-buyes: agents desie to buy is lagely in uenced by thei expectations of eselling houses at highe pices. These suveys document also that home buyes expectations tend to be extapolative and lagely in uenced by past and cuent economic conditions (see Case, Quigley and Shille, 2003). Thid, the supply of houses is inelastic in the shot-un, and adjust only slowly to local demand shocks because of locals egulations, zoning laws o technological constaints (Glaese and Gyouko, 2003, Glaese, Gyouko and Saks, 2005, 2007, Gyouko, Maye and Sinai, 2006). Taken togethe, these obsevations suggest a speci c mechanism though which vaiations in income geneate lage swings in house pices: if household income is an impotant deteminant of housing demand and shapes expectations of futue house pices, a small income shock may initiate a dynamic pocess that, though heteogeneous expectations and the xed supply of housing, uns fom expected pices to house demand and back to house pices. To fomalize this mechanism we conside a model in which two goups of households, with di eent expectations about futue pices, paticipate in the housing maket to consume housing sevices (by eithe owning o enting) and to speculate on futue pice changes. The equilibium pice is pinned down by a non abitage condition equating the cost of enting to the cost of owning. Howeve, since in ou model expectations ae heteogenous and anticipated capital gains educes the cost of owning, this non abitage condition holds only fo optimistic households, who ae willing to buy in anticipation of futue capital gains. Fo pessimistic households, instead, who expects futue capital losses, the use cost is peceived highe then the cost of enting. Consequently, they stictly pefe to move out of the maket of homes fo sale and to use the ental maket to consume housing sevices, whee ental units ae supplied by the optimists who buy units in excess of thei demand fo housing sevices. The nal esult is that the ental pice e ects the aveage opinion in the maket, while the equilibium pice of owned occupied houses is biased upwad since it e ects only the views of the optimists. Ou model delives two main pedictions. Fist, house pices and thei volatility ae highe the lage the di eence in expectations between pessimistic and optimistic households. Second, infomational shocks have an asymmetic e ect on pices: positive shocks bias the equilibium house pice upwads while negative shocks ae moot. Both pedictions stem fom the fact that the maket fo owne occupied houses is segmented, because households use thei pivate infomation to make infeence about the unobsevable aggegate income and housing demand. 3
In contast, if households had homogenous expectations, the demand fo housing would be linea, the individual pivate signals would get washed out in aggegate and the equilibium pice would depend only on aveage income. It tuns out that ou esults suvive even if agents use the equilibium ental and house pices which ae summay statistics of the dispesed infomation in the economy to update thei beliefs about the economy-wide income, povided these pices ae not pefectly evealing due to unobsevable pefeence shocks fo housing sevices. To test the pedictions of ou model we un panel egessions using US city data. Lacking a diect measue of impefect infomation, we use the city-level dispesion of industy income shocks to poxy fo di eence in beliefs. This choice is motivated by the fact that ous can be intepeted as a closed city model whee moving costs pevent households to move in and out of cities, and the speculative demand fo housing depends only on expected local economic conditions. If city esidents ae employed in di eent industies and they ae impefectly infomed about the city income, then industy income shocks become a souce of confusion about the cuent and futue local economic conditions. In line with the model s pedictions, we nd that house pices ae highe and moe volatile in cities with moe heteogenous expectations. We also nd an asymmetic esponse of house pices to positive and negative infomational shocks: positive shocks explain signi cantly house pice inceases, while negative shocks lack statistical pedictive powe. In the est of the pape we poceed as follows. In Section 2 we elate ou model to the elevant liteatue. In Section 3 we intoduce the baseline model and discuss the deteminants of the equilibium ental and house pices. In Section 4 we study the benchmak case in which agents hold impefect but common infomation about local economic conditions, while in Section 5 we deive the main pedictions of the model when the infomation is not only impefect but also dispesed. Section 6 discusses the obustness of ou model s pedictions when the infeence poblem depends also on the equilibium house and ental pices. In Section 7 we intoduce ou poxy of infomation dispetion and pesent ou empiical analysis and esults. We conclude in Section 8. All poofs ae in the Appendix. 2 Related Liteatue Methodologically, ou pape follows the use-cost appoach of Poteba (1984) and Hendeson and Ioannides (1982) in which a pospective buye is indi eent between enting and owning, and the cost of owning depends, among othe vaiables, on popety taxes, the oppotunity cost of capital and the expected capital gains on the housing unit. While some papes have studied the house pice implications of changes in taxes (Poteba, 1991) and inteest ates (Himmelbeg et al., 2006, 4
McCathy and Peach, 2004) the ole played by di eences in the expected ate of pice changes has so fa emained unexploed. In pat this is because buyes expectations ae di cult to measue, but also because di eence in expectations cannot aise in the standad use cost famewok, given the symplifying assumption that the housing maket is populated by a epesentative agent. We complement this liteatue by showing that di eence in expectations can lead house pices to deviate fom thei fundamental values and that a given degee of vaiation in expected pices acoss makets and within makets ove time can account fo some of the house pice changes documented in Figue 1, moe so than changes in popety taxes which ae faily constant ove time o inteest ates which ae constant acoss makets. In the housing liteatue, Stein (1995) and Otalo-Magne and Rady (2006) explain lage swings in house pices by using boowing constaints and household leveage. In thei models buyes nance the puchase of houses by boowing and the ability to boow is diectly tied to the value of the houses they own. Theefoe, a positive income shock that inceases the demand fo houses and hence thei pices elaxes the boowing constaint, futhe inceasing the demand fo houses and so on. Ou pape is elated to both studies because changes in households income may have a moe than popotional e ects on house pices. Thee ae, howeve, thee impotant di eences. Fist, in ou stoy thee ae no boowing constaints. Instead, the ampli cation mechanism opeates fom expected pice, via household income, back to cuent pices, via changes in speculative demand. Second, households do not need to own houses to consume housing sevices; they can also use the ental maket. Finally, in ou set-up not only the level of income but also its dispesion mattes fo explaining the dynamic of house pices. In this egad, ou pape elates to the ecent woks of Gyouko, Maye and Sinai (2006) and Van Nieuwebugh and Weil (2007). The st pape agues that the inteaction between an inelastic supply of houses and the skewing of the income distibution geneates a signi cant pice appeciation in supesta cities cities with unique chaacteistics pefeed by the majoity of the population. Wealthy households ae willing to pay a signi cant nancial pemium to live in these aeas, bidding up pices in the face of a elatively inelastic supply of houses. Van Nieuwebugh and Weil (2006) use a simila mechanism (though in thei model households move acoss cities fo poductive athe than pefeence easons) to explain why the dispesion and the level of house pices inceases with the cosssectional wage dispesion in US cities. Ou pape di es fom these contibutions because it highlights a di eent channel though which income dispesion mattes. In ou setup, income shocks a ect households s peception of local economic conditions, leading to the fomation of heteogeneous beliefs about futue economic fundamentals. As a consequence, heteogenous expectations ae moe ponounced 5
when, ceteis paibus, income is moe dispesed. Moeove, in ou model positive and negative income shocks have an asymmetic e ect on pices, a pediction absent in Gyouko et al., and van Nieuwebugh and Weil. In ou famewok the speculative motive fo buying housing units is enhanced when households expect bette economic conditions to pevail in the futue; on the contay, the speculative motive is moot following negative income shocks, since in this case households use the ental maket. Anothe impotant di eence is methodological. In ou model pices ae detemined by a non abitage condition between buying and enting, while Gyouko et al., and van Nieuwebugh and Weil follow the uban economics tadition in which house pices ae detemined by a spatial no abitage condition, with ownes indi eent between di eent locations, given local wages and amenities. The spatial equilibium appoach is, howeve, moe suitable fo studying the long un distibution of housing pices as opposed to high fequency pice vaiations, which ae the main focus of ou analysis. Ou analysis is also elated to a lage liteatue in macoeconomics and nance that studies the ole of impefect infomation among decision makes. In fact, ou stoy can be seen as an adaptation of the Phelps-Lucas hypothesis to the housing maket, in the sense that impefect infomation about the natue of distubances to the economy makes di eent economic agents eact di eently to changes in maket conditions. Pat of ou wok shaes also many featues with the liteatue on the picing of nancial assets in the pesence of heteogeneous expectations and shotsale constaints (i.e., Mille, 1977, Haison and Keps, 1979, Hong, Scheinkman, and Xiong, 2004 and Sheinkman and Xiong, 2003). In this liteatue, if agents have heteogeneous beliefs about asset fundamentals and face shot sales constaint, the equilibium asset pice e ects the opinion of the moe optimistic investos, and it is thus biased upwad. We adapt the same idea to the housing maket whee the shot sale constaint aises as a natual constaint when households have the option to consume housing sevices by eithe enting o owning. In fact, in ou model enting act as if elatively pessimistic investos wee facing a binding shotsale constaint. Expecting lowe pices in the futue, this goup of households would like to shot thei houses but cannot do so since they have to consume housing sevices. Consequently, they move out of the maket of home fo sale and the pice of owned houses ends up e ecting only the moe optimistic view in the maket, athe than the aveage opinion. 6
3 The Model 3.1 Infomation The economy is populated by an in nite sequence of ovelapping geneations of households with constant population. Each geneation has unit mass and lives fo two peiods. In the st peiod, households supply labo and make saving and housing decisions; in the second peiod, they consume the etun on savings. The wage W j t ; at which labo is supplied inelastically, is equal to W j t = exp t + " j t ; (1) whee t is the economy income and " j t is an individual-speci c wage shock. We make the assumption that t follows an AR(1) pocess, t = t 1 + t ; with 2 (0; 1] (2) with t independent and nomally distibuted innovations with zeo mean and vaiance 2. The individual-speci c shocks, " j t, which ae the only souce of income heteogeneity, ae seially independent and have nomal distibution with zeo mean and vaiance 2 ": When households cannot obseve the ealization of t at time t; " j t is also a souce of infomation heteogeneity. In othe wods, the wage W j t is the household j 0 s pivate signal about the unobsevable aggegate shock, t : As usual in this context, we make the assumption that idiosyncatic shocks cancel in aggegate, o equivalently the aveage pivate signal is an unbiased estimate of the undelying fundamental: Assumption 1: R " j tdj = 0 3.2 Pefeences Households have logaithmic pefeences ove housing sevices, V j t ; and second peiod consumption, C j t+1; 2 U j t = A j t log V j t + E j t log C j t+1; (3) whee E j t denotes the expectation opeato based on household j s infomation set at time t (to be speci ed late) and the paamete A j t is a pefeence shock, A j t = exp 2 a t + j t ; 2 Undelying this utility function is the assumption that the demand fo second-peiod housing sevices is constant which, fo simplicity, we nomalize to zeo. 7
which consists of an aggegate taste shock, a t, and an idiosyncatic noise j t: We assume that a t and j t ae independent and nomally distibuted with zeo mean and vaiance 2 a and 2. We also conside the limiting case whee the vaiance of j t is abitaily lage, so that knowing one s own individual taste povides no infomation about the aggegate taste. 3 3.3 Budget constaint In the st peiod, afte the ealization of the idiosyncatic income, households decide how many housing units to buy, H j t 0; at unit pice, P t. They also choose the quantity of housing sevices to consume, V j t ; and by implication the units of housing to ent out, H j t V j t ; at the ental pice Q t : At the end of the st peiod, the esidual income is saved at the goss inteest ate, R; and at the beginning of the second peiod the stock of owned houses is sold to the young of the new geneation, at the pice P t+1. Fo a the typical household j the ow of funds constaint is thus: C j t+1 = R W j t P t H j t + Q t H j t V j t + Pt+1 H j t ; (4) with H j t 0: (5) 3.4 Optimal house demand Households inte-tempoal decisions consist of choosing H j t and V j t to maximize (3) subject to (4) and (5). It is immediate to establish that the optimal demand fo housing sevice, V j t ; and housing units, H j t ; satisfy the following st-ode conditions, " # A j t = E j RQ t t ; (6) whee E j t V j t C j t+1 " # R (U t Q t ) 0; (7) C j t+1 U t = P t P t+1 R ; (8) 3 While this assumption implies that a t is unobsevable, the law of motion of a t is known by all agents. 8
denotes the (pe unit) use cost of housing, which inceases with the cuent house pice, p t ; and is invesely elated to the next peiod house pice, P t+1 =R: 4 Equation (6) establishes that households consume housing sevices until the cuent peiod maginal utility of housing sevices, A j t=v j t ; equal the expected maginal cost in tems of next peiod consumption utility, E j t RQt =C j t+1. The optimal quantity of housing units to buy is implicit in equation (7), which elates the use cost, U t ; to the cost of enting housing sevices, Q t. Whethe this condition holds with equality depends on households expectations about futue pices. Fo households holding pessimistic expectations the use cost is peceived to be highe than the cost of enting. Thus, constaint (5) binds and (7) is satis ed with inequality. The opposite holds fo households with elatively optimistic expectations. They demand a stictly positive amount of housing units, H j t > 0; and (7) holds with equality. 3.5 The lineaized optimality conditions To delive explicit solutions, we nd it convenient to wok with a linea appoximation of equations (6) and (7) aound the cetainty equilibium, i.e., the equilibium pevailing when both aggegate and idiosyncatic shocks ae zeo. Denoting with lowe case lettes vaiables in pecentage deviations fom the equilibium with cetainty, Appendix I shows that a linea appoximation to (7) leads to whee E j t u t q t ; (9) u t = (1 + )p t p t+1 ; (10) and is the net inteest ate R 1 > 0. Similaly, a linea appoximation of (6) leads to v j t = w j t + a j t q t ; (11) indicating that the consumption of housing sevices is positively elated to individual income and pefeences, and negatively elated to the ental pice. 4 Ou speci cation of the use costs is delibeately simple. Altenatively, we could have assumed that fo each unit owned, households also incu a cost equal to a faction M t of the nominal value of housing, P t H j t : M t can be thought of as including maintenance and depeciation costs, popety taxes, inteest payments on motgages, etc. Unde this altenative speci cation, the use cost of housing would be P t+1 U t = P t (1 + M t ) R : As long as house maket paticipants ae homogeneously infomed about M t ; none of the esults pesented below ae a ected, though the algeba would be much moe cumbesome. 9
Fom now on, in ode to make the analysis tactable, we conside only two goups of households, j = 1 and j = 0; each with equal mass, and adopt the convention that households in the st goup eceive a elatively moe optimistic signal about the cuent fundamental, i.e., " 1 t > " 0 t : As a consequence E 1 t p t+1 > E 0 t p t+1 ; and using (10) E 0 t u t > E 1 t u t : Equation (9) can then be witten as E 0 t u t > q t and h 0 t = 0 (12) E 1 t u t = q t and h 1 t > 0: (13) In othe wods, pessimistic households choose to own no housing units, h 0 t = 0; as they hold lowe expectations about next peiod pices and thus peceive the cost of owneship to be highe than the cost of enting. On the contay, optimistic households peceive the use cost to be equal to the cost of enting and ae thus indi eent between owning and enting. The implication is that optimists choose the units of housing sevices to consume, v 1 t ; out of those owned, h 1 t ; and ent out the di eence, h 1 t v 1 t = v 0 t to the elatively moe pessimists households, at the equilibium ent, q t. 3.6 The equilibium house pice and ental pice Using (10), the indi eence condition (13) can be witten as p t = 1 + q t + 1 1 + E1 t p t+1 ; (14) suggesting that fo a given ental pice, the equilibium house pice e ects only the expectations of optimistic households. With the maintained assumption of xed housing supply, s; the equilibium ental pice, q t ; is pinned down by the maket cleaing condition, which using (11) yields whee s = v1 t + v 0 t 2 ; q t = t + a t s; (15) t = w1 t + wt 0 and a t = a1 t + a 2 t ; 2 2 ae, espectively, the aveage income and the aveage pefeence fo housing sevices. Plugging (15) back into (14), the equilibium pice can be witten as p t = 1 + f t + 1 1 + E tp t+1 + 1 E 1 + e t p t+1 ; (16) 10
whee summaizes fundamental vaiables, and E t p t+1 E1 t p t+1 + E 0 t p t+1 2 f t ( t + a t s) ; (17) ; Et ~ p t+1 E1 t p t+1 Et 0 p t+1 ; 2 denotes, espectively, the aveage expectation and the di eence in expectations about tomoow s pice. In equation (16), as in a standad asset picing equation, the equilibium pice, p t ; depends, on fundamentals, f t ; and on the expected aveage capital gain fom house pice appeciation. The exta tem, e Et p t+1 ; is non-standad and aises because in ou setup households hold heteogenous expectations. In the next two sections, we make di eent assumptions about households infomation sets in ode to evaluate how E t p t+1 and e E t p t+1 in uence the elationship between house pices and fundamental vaiables. 4 Homogenous Infomation In ode to have a benchmak against which to compae the esults, we stat with the case whee households ae impefectly but homogeneously infomed about the state of the economy, t : In othe wods, households eceive identical infomation about the undelying unobsevable aggegate fundamental: " 1 t = " 0 t : In this benchmak case individual expectations coincide with aveage expectations, E j t p t+1 = E t p t+1 ; and di eences in expectation ae zeo, E ~ t p t+1 = 0: As Appendix II shows, iteating equation (16) fowad and imposing a tansvesality condition on housing pices, the aveage expectation of tomoow s pice can be witten as, E t p t+1 = E t f t = t 1 s; (18) whee 1 + : The second equality in the equation above aises because a t ; has mean zeo and t, which is not obsevable, follows an AR(1) pocess. Thus, households foecast of t depends on its past ealization. Inseting (18) into (16), and ecalling that ~E t p t+1 = 0; the equilibium pice unde impefect but homogenous infomation, p ; can be witten as p t = f t + t ; (19) 11
whee f t is given in (17) and t t 1 t a t ; 1 + is an expectation eo. In what follows, we intepet p t as the fundamental house pice, because it e ects the aveage opinion in the maket which, by Assumption 1, is an unbiased estimate of the unknown fundamental. As we will see, when infomation is not only impefect but also heteogeneous among house maket paticipants, the maket becomes segmented, in the sense that not all households paticipates in the maket of homes fo sale. The upshot is that idiosyncatic shocks ae not washed out in aggegate and the equilibium pice ends up e ecting only the most optimistic view in the maket. 5 Heteogeneous Infomation We now conside a setting whee households use the cuent ealization of thei income, w j t ; and the exogenous public signal, t 1, to make an optimal infeence about t. Household j s infomation set at t is theefoe 5 j t = w j t ; t 1 j = 0; 1: Unde the assumption that households do not shae thei pivate infomation with each othe, and because w j t is bu eted by idiosyncatic shocks, households end up holding heteogeneous infomation about t. Befoe poceeding, it is impotant to notice that the equilibium pices (both the housing and ental pices) ae not included in j t: This assumption is made only to simplify the chaacteization of the channels though which infomation dispesion a ect the equilibium pice. As we will discuss in the following section this assumption is inessential fo ou esults. A way to think about this assumption is to conside the special case whee the vaiance of the aggegate unobsevable pefeence shock, 2 a; is abitaily lage. In such a case the house pice (16) and the ental pice (15) become uninfomative about t and house maket paticipants do not lean much upon obseving p t o q t. 6 5 To know the entie histoy of aggegate shocks is supe uous since t follows an AR(1) pocess. Similaly, knowing the past ealization of household pivate signals is ielevant, given the iid assumption fo " j t: 6 In excluding the equilibium pices fom the household infomation set, me make ou analysis akin to a di eence of opinion model, widely used in the nance liteatue. In such a model investos agee to disagee about the distibution of payo and signals and theefoe do not use 12
With signals w j t and t 1, the ability of household j to estimate t fom available data depends on the elative magnitude of 2 " and 2 : Because of ou assumption of independent and nomally distibuted eos, the pojection theoem implies E j t t+1 = E j t t = (1 ) t 1 + w j t ; (20) whee the weight 2 = 2 + 2 " e ects the elative pecision of the two signals. Thus, with > 0; expectations among households ae heteogeneous and both aveage expectations and expectations di eences become impotant deteminants of the equilibium house pices. Iteating equations (16) and (20) fowad and excluding explosive pice paths, Appendix III shows that the di eence in expectations, and the aveage expectation of futue pices ae, espectively, whee ~E t p t+1 = i t ; (21) E t p t+1 = ( t 1 s) + I + ( t t 1 ) ; (22) i t " 1 t " 0 t ; denotes the infomational di eence between the two goups of households and I Z 1 0 xd (x) ; measues the aveage degee of infomation heteogeneity in the economy, with denoting the distibution of i t. Equation (21), stems fom the fact that households ae dispaately infomed and in estimating t they assign a positive weight to thei pivate signal, w j t : Di eences in expectations, E ~ t p t+1 ; ae theefoe popotional to the di eence in pivate signals, i t " 1 t " 0 t. Equation (22), is the equivalent of (18), that is the aveage expectation unde impefect but homogeneous infomation. It di es, howeve, fom (18) because heteogenous infomation intoduces two additional tems, each popotional to the weight that households give to thei pivate infomation. The st tem, I= aises because pices ae fowad looking: it is not only the cuent degee of infomation heteogeneity that mattes but also the aveage di eence in expectation that is expected to pevail in the futue, given that the futue dispesion in infomation will also a ect the couse of house pices. The second tem, the equilibium pices to infe othe investos beliefs. An altenative eason households may not condition on the equilibium pices is because they do not know how to use pices coectly (e.g. they display bounded ationality, as in Hong and Stein, 1999) o because they exhibit behavioal biases (e.g., they ae ovecon dent, as in Scheinkman and Xiong, 2003). 13
( t t 1 ) ; e ects instead the aveage degee of mispeception in the economy and aises because households use only patially the public signal, t 1; to infe the cuent state of the wold. The lage the weight assigned to the pivate signal and the lage the degee of mispeception in the economy, ( t t 1 ) ; the moe the expected pice deviates fom the one pevailing unde homogenous infomation. This patial eaction of households to shifts in the fundamentals has the e ect of intoducing inetia in the way expectations ae fomed, which accods well with the idea that housing maket expectations ae fomed in an extapolative manne (see Case and Shille, 1988, 2003). Plugging these expessions into (16), the equilibium house pice can be witten as p t = p t + t : (23) whee, p t ; is the fundamental pice given in (19), and t ( t t 1 ) 1 + I + (1 + ) + i t 1 + (24) summaizes the ole of infomation heteogeneity among households. Thus, in the pesence of heteogenous infomation, i.e., > 0; p t di es fom p t ; due to shifts in t : In tun, shifts in t ae moe ponounced the lage is the cuent and expected degee of infomation heteogeneity: i t ; and I: The eason is quite intuitive. Households that eceive a positive signal expect highe income in the futue and thus highe housing demand and pices. Convesely, households with a negative signal expect lowe futue pices. Howeve, while households with optimistic expectations demand moe houses fo speculative easons the highe the expected pice, pessimistic households pefe to consume housing sevices though the ental maket. In othe wods, enting act as if elatively pessimistic investos wee facing a binding shot-sale constaint: holding pessimistic expectations about tomoow s pices these households would like to shot thei houses. Since they cannot do so, pessimists dop out of the maket of homes fo sale and thei beliefs ae not fully e ected in the equilibium pice. The nal esult is that the equilibium pice e ects only the expectations of elatively optimistic households, and is thus biased upwad. Equation (23) suggests that the lage the divegence in beliefs among households, the moe ponounced is this e ect. Moeove, in compaing (23) with (19) it is staightfowad to see that heteogenous infomation inceases also the volatility of house pices by a facto popotional to the noise in the di eence in beliefs: 2 V (p t ) V (p t ) = 2 + 2 i : 1 + 14
6 Leaning fom the equilibium pice The esult just deived that di eences in beliefs impat an upwad shift in the level and volatility of house pices hold unde the assumption that the vaiance of the unobsevable aggegate pefeence fo housing sevices, a t ; is abitaily lage, so that to infe t households ely only on the exogenous public and pivate signals ( t 1 and w j t ); but not on the endogenous public signals, i.e. the house pice, p t ; and the ental pice, q t. We now elax this assumption and allow households to condition on pices. This extension is desiable because house and ental pices, like any othe nancial pices, ae a useful summay statistics of the dispesed pivate infomation in the economy. If households use these endogenous public signals to update thei beliefs on the undelying fundamental, dispesion in beliefs may vanish, and in the limit the upwad bias in the equilibium house pice may disappea. As shown in Appendix IV, of the two available endogenous public signals, q t and p t ; households use only the infomation contained in the equilibium ental pice. The eason p t is edundant is because q t conveys the same infomation on t as in p t ; but it is not in uenced by the exta souce of noise, i t ; oiginating fom the fact that house maket paticipants hold di eent expectations about the futue couse of house pices (compae equations (15) and (23). Howeve, since q t depends also on a t ; the undelying fundamental, t ; is not fully evealed. Speci cally, households cannot tell whethe the equilibium ent is high because aggegate economic conditions impove o because unobsevable taste shocks dive the demand fo housing sevices. Fo a given noise in the exogenous public and pivate signals, the infomative content of the ental pice will be deceasing in the pefeence noise, 2 a; as typical in a noisy ational expectation model à la Gossman and Stiglitz (1976) and Hellwig (1980). Using a standad linea solution method, Appendix IV shows that the equilibium pice with leaning can be witten as, p t = p t + 2 t + 3 t ; (25) whee 2 and 3 ae weights on the pivate and the endogenous public signals (the equilibium ent), t is given in (24) and t 1 + (( t t 1 ) + a t ) is a tem that summaizes the degee of magni cation of shocks induced by the pocess of leaning. Intuitively, in the pesence of unobsevable shocks, households who obseve a change in the ental pice do not undestand whethe this change is diven by a change in the aggegate income o by changes in pefeences. Thus, 15
when 3 > 0; each of these shocks will have an ampli ed e ect on the house pice, since households espond to whateve is the souce of movement in the ental pice. A key obsevation to make in compaing equation (25) with (23) is that i t ou measue of dispesion in beliefs continues to shift the equilibium pice away fom its fundamental value, p t, fo the same easons discussed in the pevious section. The elative impotance of t and t depends, howeve, on 2 and 3 which, as shown in Appendix IV, ae elated to as follows, > 2 R 3 ; with the popety that 2! ; and 3! 0; as 2 a! 1: In wods, as the noise in the pefeence fo housing sevices becomes lage that is the ental pice gets less and less evealing the equilibium pice (25) becomes identical to the one pevailing in absence of leaning (23). This is illustated in Figue 2, whee we plot the pecentage deviation of the equilibium pice fom its fundamental level, p t p t, fo di eent values of the pefeence noise, 2 a: 7 As the gue shows, egadless of whethe households ae able to lean, the pice misalignments emain of compaable size, fo lage values of 2 a. When the pefeence shock is smalle, q t is infomative about t and households ely less on thei pivate souce of infomation, i.e. 2 < : Nonetheless, povided pefeence shocks pevent pices fom being fully evealing, i.e., 2 a is not vey small, the pivate signals continue to povide some infomation in estimating the undelying fundamental. This is illustated in Figue 3, which plots the pice di eential, p t p t, fo di eent values of the pivate signal s noise 2 ": As it can be seen, the misalignment of the equilibium pice fom its fundamental level is hump-shaped in the signal noise, egadless of whethe households lean fom the equilibium pice. The bias is inceasing fo intemediate values of the pivate signals noise, and it is falling fo smalle values of 2 "; since in this case households eceive simila infomation. The bias will also be deceasing fo lage values of 2 " because households tend to assign less and less weight to thei pivate signals. Finally, when the equilibium ent is not pefectly evealing, the volatility of the equilibium house pice continues to be highe than in the benchmak scenaio of impefect but homogenous infomation. By compaing (25) with (19), it is easy to see that the di eence of vaiances 2 V (p t ) V (p 2 2 t ) = 2 + 2 3 i + 2 + 2 a 1 + 1 + 7 Figue 2 plots the aveage pice di eential, p t p t ; of 1000 simulated seies using the following paamete values: t 1 = a t 1 = s = 0 and = 0:01: These paamete values ae chosen without loss of geneality given that we conside the deviation of the equilibium pice fom its fundamental value. 16
is popotional to the noise in di eence in beliefs as in the case without leaning and to the noise in pefeence shocks because of the ampli cation of shocks induced by the pocess of leaning fom the equilibium ent. 7 Testing the implications of the model Ou model delives thee main pedictions: 1) the deviation of house pices fom thei fundamental value is highe the lage the dispesion of beliefs; 2) the dispesion of beliefs is positively elated to the volatility of house pices; and 3) optimistic expectations bias the equilibium pice upwad moe than pessimistic expectations. The most di cult pat of testing these pedictions is to have data on heteogeneous infomation among house maket paticipants. These data, in fact, ae conspicuous by thei absence. To deal with this poblem we adopt the following stategy. We take US cities as units of obsevation and use the dispesion (within cities) of shocks to industy eanings as a poxy fo dispesion of beliefs about local housing maket conditions. While debatable, this poxy is motivated by empiical and theoetical consideations. Fom an empiical viewpoint, thee is widespead consensus that high fequency vaiations in house pices ae mostly local, not national (Glaese and Gyouko, 2006). In addition, the evidence suggests that the bulk of shot un movements in local house pices is due to changes in demand, diven by economic conditions within a egion as opposed to changes in pefeences fo local amenities. 8 Fom a theoetical pespective, ou poxy of infomation dispesion is motivated by the logic of ou model, which can be thought as descibing the time seies dynamics of house pices in a typical city, whee the speculative demand fo housing ultimately depends on the ealization of income shocks. If esidents in each cities ae employed in di eent industies, and they ae impefectly infomed about the city income, then industy speci c income shocks become a souce of confusion about the city aveage income, as in the signal extaction poblem discussed in ou theoetical famewok. With this intepetation, equation (1) and (2) in the model, which govens the dynamics of households income, can be eintepeted and ewitten as follows; w l k;t = k;t + " l k;t and k;t = k;t 1 + k;t (26) whee wk;t l is the time t eanings of a epesentative household esident in city k and employed in industy l; k;t is the aveage city income at time t; and " l k;t is 8 Endogenous supply-side changes may also a ect vaiations in house pices (Glaese, Gyoko and Saiz, 2008). Howeve, in the shot un, due to egulations and technological constaints, quantity changes tend to espond slowly to shifts in demand. 17
a time-t industy-l speci c shock. At any point in time, a poxy fo the degee of infomation dispesion in city k can then be computed using a measue of the dispesion of eanings shocks acoss industies, as we now explain. 7.1 Data desciption and summay statistics We collect annual data fo a sample of appoximately 350 metopolitan aeas (MSA) duing the peiod 1980 to 2000. To infe the time seies popeties of local income shocks we use (pe employed) eanings data fo 10 one-digit industies, based on the SIC classi cation code. 9 With these data, the dispesion of eanings shocks acoss industies is computed in two steps. Fist, based on equation (26), we un 10 egessions, one fo each industy, in which we pool the gowth ate of industy eanings fo the full sample of MSAs, w l k;t = 0 + 1 k;t + 2 k;t 1 + t + " l k;t fo l = 1; 2; :::10; (27) whee is the st di eence opeato, and t is a time xed e ect. In this speci cation, the esiduals " l k;t measue shocks to eanings gowth in each cityindusty, contolling fo city-speci c income dynamics and nationwide e ects. 10 Second, we measue the dispesion of eanings shocks acoss industies and within each MSA as the weighted aveage of the absolute value of industy-msa esiduals, X10 i k;t =! l " l k;t ; (28) l=1 with weights! l equal to the shae of MSA wokes employed in industy l; to contol fo the size of each industy. 11 9 Speci cally, we use eanings data fo the following industies: 1) Fam, 2) Mining, 3) Constuction, 4) Manufactuing, 5) Tanspotation and public utilities, 6) Wholesale tade, 7) Retail tade, 8) Finance, insuance, and eal estate, 9) Sevices, and 10) Govenment and govenment entepises. These data ae available at http://www.bea.gov/egional/eis/. Ou sample peiod stops in 2000 because in that yea the Standad Industial Classi cation (SIC) system has been eplaced by the Noth Ameican Industy Classi cation System (NAICS). This di eent system fo classifying economic activity makes it impossible to extend ou data beyond 2000. Available data based on the NICS system cove only the peiod 2001 to 2006. To be able to bette exploit the time seies vaiation in the data we have pefeed to use data based on the SIC classi cation codes fo the peiod 1980-2000. 10 We have also expeimented with speci cations that include lags of w l k;t to contol fo industy-city speci c dynamics. All the esults epoted below ae obust to such changes. 11 None of the esults pesented below (in tems of economic and statistical signi cance) ae a ected by using squaed deviations athe than absolute deviations. We pefe to model absolute deviations to be able to maintain the same unit as the change in industy eanings, so the coe cients in the house pice egessions epoted below ae easily intepeted. 18
We focus ou analysis on MSAs because these ae the smallest units of obsevation fo which income and quality-adjusted housing pice data is available. Fo each MSA we take the nominal house pice index fo single-family houses fom the O ce of Fedeal Housing Entepise Ovesight, and pe capita income data fom the Bueau of Economic Analysis. Nominal vaiables ae then conveted in eal dollas using the national CPI index less sheltes fom the Bueau of Labo Statistics. We use annual obsevations because income data is only available annually. Table 1 epots basic summay statistics. As shown, ou data display consideable vaiation acoss MSA. Ove the full peiod 1980-2000, ou poxy of infomation dispesion is less than 1.5% in Minneapolis, Cleveland, Kansas City and Tampa but geate than 4% in Chicago, Dallas, Los Angeles, and New Yok, among othe cities. Real house pice changes also exhibits consideable vaiation. Fo instance, Boston, San Fancisco, and San Jose all expeienced gowth ates in house pices ove 3% pe annum ove the 20 yea peiod studied, while Houston, Oklahoma City, and San Antonio expeienced house pice changes of less than minus 1.5%. We now exploit the ich coss-city vaiation of these data to test the implications of ou model. 7.2 The baseline egession We stat the analysis by examining the empiical elevance of the pice equation pevailing unde common infomation. To empiically use equation (19), we take st di eences of each vaiable and estimate the following egession 12 : p k;t = 0 + 1 k;t + 2 k;t 1 + t + k + k;t : (29) Hee, p k;t is the log change of the eal house pice index in MSA k in yea t, k;t the log change in eal pe capita income and k;t is a standad eo tem. In this egession, and those that follow, we include also yea and MSA dummies, t and k ; to account fo unobsevable aggegate and city-speci c deteminants of house pice changes. Thus, each vaiable in any yea is measued in deviation fom both national aveage and long un city aveage. Table 2 epots OLS estimates of this baseline egession, with standad eos clusteed at the MSA level to allow fo within-city autocoelation in the eos. Accoding to the model, 1 and 2 ae expected positive and as shown in the st column of Table 2 these pedictions ae stongly suppoted by the data: highe cuent and lagged income changes ae signi cantly associated with highe housing pices changes. 12 We use each vaiable in st di eence because the OFHEO house pice index is not standadized to the same epesentative house acoss makets. Thus, pice levels cannot be compaed acoss cities, but they can be used to calculate gowth ates. 19
The ole of heteogeneous infomation is examined in columns 2 whee we epot estimates of the empiical countepat of equation (23). Moe speci cally, we add to the baseline egession (29) ou poxy of di eence in beliefs, i k;t : In line with the pediction of the model, the esults show a statistically signi cant elationship between di eence in beliefs and house pices. Ou indicato of infomation dispesion entes the egession positively and with a sizeable estimated e ect: a 1% pe cent incease in i k;t esults in a 0.2% incease in the gowth ate of house pices. To bette gauge the economic e ect of these esults conside an exogenous incease of i k;t ; fom the 10th pecentile value (which is appoximately 1.2%) to the 90th pecentile value (which is appoximately 4%). This incease would lead to an acceleation of the gowth ate of house pice by 0.6% pe yea, which is lage consideing that the mean change in eal house pices ove the 1980-2000 peiod is 0.4%. 7.3 Altenative empiical speci cations The esults in Table 2, although based on the equilibium pice equations implied by the theoetical model, do not contol fo some pattens of the house pice dynamics that pio woks have documented to be impotant. Fo example, stating with Case and Shille (1989), it is well known that house pices exhibit momentum and mean evesion ove time. To contol fo these e ects, we add thee lags of the dependent vaiable to ou baseline egession. The esults shown in column 1 of Table 3 indicate that house pices indeed exhibits positive coelation at shot lags and negative coelation at longe lags. Howeve, as epoted in column 2, ou poxy of infomation dispesion continue to play a lage and signi cant ole in explaining house pice changes ove time. Column 3 to 4 exploe the obustness of ou ndings to an altenative empiical speci cation, suggested by the wok of Lamont and Stein (1999). In thei study of the house pice dynamics in US cities, Lamont and Stein show that house pices (a) exhibit shot un movements, (b) espond to contempoaneous income shocks, and (c) exhibit a long un tendency to fundamental evesion. They thus popose to estimate the following egession, p k;t = 0 + 1 p k;t 1 + 2 k;t + 3 (p=) k;t 1 + k + t + k;t (30) whee (p k = k ) t 1 is the lagged atio of house pices to pe-capita income. 13 Column 3 shows that these vaiables have all the expected sign and explain signi cantly a lage faction of house pice vaiations. To this thee-vaiable speci cation we add one. 13 The inclusion of this vaiable implies that the long un elasticity of pices to city income is 20
ou vaiable of inteest, i k;t, in column 4. In line with the esults in Table 2, we nd that ou poxy of infomation dispesion continues to be elated signi cantly to house pice vaiations. House pices ae highe in cities whee infomation about local income shocks is moe dispesed. A common objection to the coelations epoted so fa is that they disappea when we contol fo demogaphic factos. In fact, high fequency changes in the demand fo housing may be diven not only by changing economic conditions within a egion but also by population movements, a shifte of housing demand that has been omitted in ou theoetical analysis. In the attempt to contol fo these e ects, columns 3 and 4 epot estimates with the ate of population gowth as an additional egesso. Population gowth is expected to ente the egession with a positive sign since new buyes willing to move into a city push up housing demand and pices. 14 The esults in columns 3 and 4 show that population gowth has indeed a positive and signi cant coe cient. Ou coe esults, howeve, do not seem to depend on the inclusion of this additional contol. The estimated e ects fo ou poxy of infomation dispesion continue to be lage and signi cant, suppoting the model s pediction that dispesion in infomation exets an upwad impact on house pices. 7.4 The volatility of house pices We now tun to the second pediction of the model that the volatility of house pices inceases with the vaiance in the dispesion of beliefs. To examine the stength of this pediction we compute the volatility of house pices by unning a pooled egession fo the change in house pices, contolling fo yea e ects, and we then take the standad deviation of the esiduals in each MSA. This gives us a measue of the volatility of house pices, within a metopolitan aea, contolling fo aggegate e ects. With one obsevation fo MSA, we exploit the coss sectional vaiation of house pice volatility and egess ou measue of volatility of house pices on the standad deviation of dispesion of beliefs acoss MSA. Figue 2 gaphs the volatility of house pice against the tted values fom this egession, fo a sample of 321 MSAs: volatility = :0127 (11:37) + 1:3034 (6:00) s.d of dispesion of beliefs, whee obust t-statistics ae in paenthesis, and the R 2 = 0:12: The esults highlight the empiical validity of ou model s pediction: MSAs with highe dispesion 14 Given that the population of a city is almost pefectly coelated with the size of its housing stock (Glaese, Gyouko and Saks, 2006) the inclusion of population gowth in ou egessions seves also to contol fo changes in housing supply. In this case, the estimated coe cient is expected positive povided constuction ms ae fowad looking and new homes ae built in esponse to ising pices. 21