House Price Dynamics with Dispersed Information

Transcription

1 House Pice Dynamics with Dispesed Infomation Giovanni Favaa y and Zheng Song z Fist Vesion August 007 This vesion Octobe 009 Abstact We use a use-cost model to study how dispesed infomation among housing maket paticipants a ects the equilibium house pice. In the model, agents consume housing sevices, speculate on pice changes and ae dispaately infomed about local economic conditions. Infomation dispesion leads agent to fom heteogeneous expectations about housing demand and pices. Optimists, who expect high house pice gowth, buy in anticipation of capital gains, while pessimists pefe to ent housing units to avoid capital losses. The upshot is that pessimistic expectations ae not incopoated in the pice of owned houses and, thus, the equilibium pice is highe elative to the benchmak case of common infomation. We test the pedictions of the model on US cities, using the dispesion of city income as a poxy fo the dispesion of infomation of local economic condition. The empiical evidence suppots the pediction that house pices ae highe and moe volatile in cities whee infomation is moe dispesed. JEL Classi cation Numbes: R, R3, G0 Keywods: Housing pices; Infomation dispesion; Income dispesion We ae gateful to Alessando Bebe, Daell Du e, Benad Dumas, Simon Gilchist, John Hassle, Ethan Kaplan, Pe Kusell, Rafael Lalive, Tosten Pesson, Andea Pat, David Rome, Matin Schneide, 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), Shanghai JiaoTong Univesity, the Sveige Riksbank, Bocconi Univesity, Luiss, the Einaudi Institute fo Economics and Finance, IMF, ECB, the 009 Society fo Economic Dynamics Meeting, the 009 LSE-FMG confeence on Housing and the Macoeconomy, the 008 Noth Ameican Summe Meeting of the Econometic Society, the 008 Royal Economic Society Meeting, fo helpful discussions and comments. Financial suppot by the National Cente of Competence in Reseach "Financial Valuation and Risk Management" (NCCR FINRISK, IPA) is gatefully acknowledged. y HEC Univesity of Lausanne, Swiss Finance Institute and IMF. gfavaa@imf.og z School of Economics, Fudan Univesity. zsong@fudan.edu.cn

2 Intoduction The US housing maket has expeienced substantial pice uctuations in the last two decades. Figue gives an example of such uctuations fo the aggegate US economy and a epesentative sample of US cities. As shown, housing pices not only have di eent tends in di eent cities, but display also heteogeneous shot-un dynamics. In the opinion of many housingmaket obseves (see e.g. Glaese and Gyouko, 007) these dynamics ae di cult to explain though the lens of a use cost model in which house pices ae detemined by an indi eence condition between owning and enting. The eason is that in such a model (Poteba, 984, Hendeson and Ioannides, 98), the cost of owning depends on vaiables that eithe do not vay much ove time (e.g., popety taxes) o a constant acoss makets (e.g., inteest ates). The goal of this pape is to popose a vaiant of the use cost model to helps us ationalize the heteogeneous behavio of housing pices in the US. Ou extension of the standad model is that infomation about local economic conditions is dispesed among housing maket paticipants. Agents eceive idiosyncatic noisy signals and thus fom heteogeneous views about the economy and house pices gowth. But since the expected esale value of a house a ects the cost of owning, only optimists pefe to buy houses. As a esult, the pice of housing will be highe and moe volatile the lage the dispesion of infomation about local economic conditions. To the extent house pice expectations depend on local economic conditions and economic conditions vay acoss makets and time, ou model povides a novel intepetation behind the pice uctuations obseved in the US housing maket. Ou analysis is based on thee building blocks, motivated by seveal aspects of the US housing maket. Fist, the available evidence suggests income is the main deteminant of housing demand, eithe because iche agents may a od to spend moe on homes (Poteba, 99, Englund and Ioannides, 997) o because highe income elaxes cedit constaints (Otalo-Magné and Rady, 006, Almeida et al., 006). Second, suveys of housing maket paticipants (Case and Shille, 988, 003, Piazzesi and Schneide, 009) eveal agents desie to buy is stongly in uenced by thei expectations of eselling houses at highe pices. These suveys document also that home buyes disagee about the causes of house pice movements and expectations ae lagely in uenced by past and cuent economic conditions (see, e.g., Case, Quigley and Shille, 003). Thid, housing supply is inelastic and adjust slowly to local demand shocks because of egulations, zoning laws o technological constaints (see, e.g., Glaese and Gyouko (003), Glaese, Gyouko and Saks (005, 006)). Taken togethe, these ingedients suggest a speci c mechanism though which changes in income may geneate lage swings in house pices: if income not only in uences housing demand but also shapes expectations of futue house pices, an income shock may initiate a In some cities, such as Los Angeles, the housing pice has moved in tandem with the oveall national index, though it has moved much moe. 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 expeiencing an exponential incease beginning in 000; in San Antonio, it has declined since the 980s; in Rocheste, it has displayed an invese U shaped histoy; in Memphis the same index has gone though peiodic cycles. While thee is consensus that di eences in state level popety taxes cannot explain the house pice behavio acoss makets, the debate concening the elationship between inteest ates and house pices is less conclusive. McCathy and Peach (004) and Himmelbeg, Maye and Sinai (005) ague that the ecent house pice boom in the US was lagely detemined by low inteest ates. In contast, Shille (005, 006) documents a non-signi cant elationship between house pices and inteest ates ove a longe peiod of time.

3 self-einfocing pocess that, though heteogeneous expectations and the xed housing supply, uns fom expected pices to house demand and back to house pices. To fomalize this mechanism we conside a closed-city model with a xed supply of housing and two goups of agents with no housing endowment. Agents consume housing sevices by eithe buying o enting, and speculate on futue pice changes. The demand fo houses uctuates stochastically because infomation about local economic conditions is dispesed and agents use thei own income, togethe with othe available signals, in estimating the unknown state of the economy. As a esult, idiosyncatic income shocks tanslate into heteogeneous expectations about aggegate housing demand, and given the xed supply into heteogeneous expectations about house pices. As in a standad use-cost model, the equilibium pice is pinned down by an indi eence condition between owning and enting. Howeve, since expectations ae heteogeneous the equilibium pice no longe e ects the indi eence condition of the aveage esident. Instead, it is pinned down by the expectations of the moe optimistic agents in the maket. This is because pessimists, who expect futue capital losses, peceive the use cost to be highe than the cost of enting. Consequently, they pefe to consume housing sevices though the ental maket, whee ental units ae supplied by the optimists who, fo speculative easons, buy units in excess of thei demand fo housing sevices. The diect implication of this model is that the pice of owned houses, e ecting only the opinion of the optimists, is highe and moe volatile elative to a benchmak scenaio whee infomation is not dispesed. In addition, infomation shocks have an asymmetic e ect on pices depending on whethe they convey positive o negative signals: while positive shocks move the equilibium pice upwads negative shocks do not a ect the equilibium pice. This last pediction aises because agents have the option to consume housing sevices though the ental maket. Wee the ental maket absent, pessimists would be foced to consume housing sevices by buying housing units. Hence, pivate signals would cancel out when the demand is aggegated acoss agents and the equilibium pice would depend only on the aveage expectation. It tuns out ou esults suvive even if agents use the equilibium pice to update thei infeence about the state of the economy, povided the house pice is not fully evealing. To test the pedictions of the model we un egessions on a panel of appoximately 350 US cities and use the dispesion of city-industy income shocks as a poxy fo the dispesion of infomation about local income. This poxy is suggested by the logic of the model, which can be intepeted as descibing the dynamics of local housing pices when the demand fo housing depends on expected local economic conditions. Moe pecisely, if city esidents ae employed in di eent industies and ae impefectly infomed about the city income, shocks to industy s income can become a souce of infomation about cuent local economic conditions. In line with ou model pedictions, we nd house pices ae highe and moe volatile in cities whee ou poxy of infomation dispesion is highe. We also nd an asymmetic esponse of housing pices to infomation 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 we elate ou model to the elevant liteatue. In Section 3 we intoduce the baseline model and discuss the deteminants of the equilibium house pice. In Section 4 we study a benchmak scenaio whee agents hold impefect but common infomation about local economic conditions. In Section 5 we deive the main model s pedictions when agents infomation is not only impefect but also dispesed. In this section we also allow agents to update thei infeence though leaning fom 3

4 the equilibium house pice. In Section 6 we discuss ou poxy of infomation dispesion, and pesent ou empiical analysis and esults. Ou conclusions ae in section 7 and all poofs in the Appendix. Related Liteatue Methodologically, ou pape follows the use-cost appoach of Poteba (984) and Hendeson and Ioannides (98) 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 implications fo house pices of changes in taxes (Poteba, 99) and inteest ates (Himmelbeg et al., 006, McCathy and Peach, 004), the ole played by heteogeneity in the expected ate of pice changes has emained so fa unexploed. This is so because di eences in expectations cannot aise in a standad use cost model with homogeneous infomation. We complement this liteatue by showing that infomation dispesion acoss makets, and within makets ove time, help to ationalize pat of the house pice changes documented in Figue, moe than changes in popety taxes which ae faily constant ove time o inteest ates which ae constant acoss makets. The idea of ou pape that changes in income may have moe than popotional e ects on house pices is simila in spiit to the wok of Stein (995) and Otalo-Magné and Rady (006). In these papes agents buy houses by boowing and the ability to boow is diectly tied to the value of houses. Theefoe, a positive income shock that inceases the housing demand and pice elaxes the boowing constaint which inceases futhe the demand fo houses. Ou pape di es fom Stein, and Otalo-Magne and Rady, in thee impotant ways. Fist, in ou stoy thee ae no boowing constaints, and the ampli cation mechanism uns only fom changes in expected pices, via household income, to cuent pices, via changes in the speculative demand. Second, in ou model agents do not need to own houses but can also use the ental maket to consume housing sevices. Thid, it is not only the level but also the dispesion of income that a ects house pices. Fo this eason ou pape is also elated to Gyouko, Maye and Sinai (006) and Van Nieuwebugh and Weil (008). Gyouko et al. ague that the inteaction between an inelastic supply of houses and the skewing of the income distibution geneates signi cant pice appeciations in supesta cities (i.e., cities with unique chaacteistics pefeed by the majoity of the population) because wealthy agents ae willing to pay a nancial pemium to live in these aeas, bidding up pices in the face of a elatively inelastic supply of houses. Van Nieuwebugh and Weil use a simila mechanism to explain the coelation between the dispesion of US house pices and the coss-sectional dispesion of US wages, though in thei model agents move acoss cities fo poductive athe than pefeence easons. Ou pape di es fom these contibutions because it highlights a di eent channel though which income dispesion a ects house pices. In ou famewok income shocks a ect agents s peception of local economic conditions, leading to heteogeneous expectations about cuent and futue economic fundamentals. As a consequence, di eence in expectations ae moe ponounced when, ceteis paibus, income is moe dispesed. Moeove, ou model featues an asymmetic e ect on pices of positive and negative income shocks, a pediction absent in Gyouko et al., and van Nieuwebugh and Weil. In ou model the speculative motive fo housing units is enhanced when agents peceive bette 4

5 economic conditions; convesely, the speculative motive does not aise afte negative income shocks, as in this case agents consume housing sevices though the ental maket. Anothe impotant di eence is methodological. In ou model pices ae detemined by a no-abitage condition between buying and enting, while in Gyouko et al., and van Nieuwebugh and Weil 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 is the main focus of ou analysis. Ou pape 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 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 beliefs and shot-sale constaints (e.g., Mille, 977, Haison and Keps, 979, Hong, Scheinkman, and Xiong, 004 and Sheinkman and Xiong, 003). 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 most optimistic investos. We adapt the same idea to the housing maket. In ou model pessimistic agents would like to shot thei houses if they had the option to do so. Thus, to consume housing sevices pessimists move out of the maket of home fo sale and the pice of owned houses ends up e ecting only the moe optimistic opinion in the maket, athe than the aveage opinion. 3 The Model 3. Infomation The economy is populated by an in nite sequence of ovelapping geneations of agents with constant population. Each geneation has unit mass and lives fo two peiods. In the st peiod, agents supply labo and make saving and housing decisions; in the second peiod, they consume the etun on savings and housing. The wage W j t ; at which labo is supplied inelastically, is equal to W j t = exp t + " j t ; () whee t is the economy income and " j t an individual-speci c wage shock. The individualspeci c shocks, " j t, which ae the only souce of income heteogeneity, ae seially independent and have nomal distibution with zeo mean and vaiance ": We make the assumption that t follows an AR() pocess, t = t + t ; with (0; ] () whee t is independently and nomally distibuted with zeo mean and vaiance. When agents cannot obseve the ealization of t ; " j t becomes a souce of infomation heteogeneity. In othe wods, individual wage W j t is the agent j 0 s pivate signal about the unobsevable aggegate shock, t : 5

6 To make the analysis simple, we conside only two goups of agents, j = 0;, each with equal mass. We also make the standad assumption that idiosyncatic shocks cancel out in aggegate, o equivalently, the aveage pivate signal is an unbiased estimate of t : Assumption : P j "j t = Pefeences Agents have logaithmic pefeences ove housing sevices, V j t ; and second peiod consumption, C j t+ ;3 U j t = Aj t log V j t + E j t log Cj t+ ; (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 a t + j t ; 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 a and. 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 Budget constaint In the st peiod, afte the ealization of the idiosyncatic income, agents decide how many housing units to buy, H j t 0; at the unit pice, P t. They also choose the quantity of housing sevices to consume, V j t ; and the units to ent out, Hj t V j t ; at the ental pice Q t: The stock of houses owned by the old is sold to the young at the beginning of each peiod. At the end of the peiod, the esidual income of the young is saved at the goss inteest ate, R. Fo type-j agents, the esouce constaint is thus: C j t+ = R W j t P t H j t + Q t H j t V j t + P t+ H j t ; (4) with H j t 0: (5) 3.4 Optimal house demand Agents intetempoal 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 V j t and H j t satisfy the following st-ode conditions: " # A j t V j = E j RQ t t ; (6) t 3 Undelying this utility function is the assumption that the demand fo second-peiod housing sevices is constant which, fo simplicity, we nomalize to one. C j t+ 6

7 E j t " # R (U t Q t ) 0; (7) C j t+ whee P t+ U t P t R ; (8) denotes the (pe unit) use cost of housing, which deceases with next peiod house pice, P t+ =R: 4 Accoding to equation (6) agents consume housing sevices until the maginal bene t (the LHS) equal the maginal cost, de ned in tems of next peiod consumption (the RHS). The optimal demand of housing units is implicit in equation (7), which elates the cost of owning, U t ; to the cost of enting housing sevices, Q t. 3.5 The lineaized optimality conditions To delive explicit solutions, we lineaize 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 of (7) leads to E j t u t q t ; (9) whee and R u t = ( + )p t p t+ ; (0) > 0. Futhe, a linea appoximation of (6) leads to v j t = wj t + aj t q t ; () indicating that the demand fo housing sevices inceases with income, is shifted by pefeences shocks, and is negatively elated to the ental pice. Fom now on, we adopt the convention that agents in goup j = ae elatively moe optimistic about the next-peiod house pice; i.e., E t p t+ > E 0 t p t+ : Using (0), equation (9) can be witten as: E 0 t u t > q t and h 0 t = 0 () E t u t = q t and h t > 0: (3) 4 Ou speci cation of the use cost is delibeately simple. We could have assumed that fo each unit owned, agents incu also a cost equal to a faction M t of the nominal value of housing, P th 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 U t = P t( + M t) P 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 moe cumbesome. 7

8 Thus, pessimists choose to own no housing units, h 0 t = 0; as they peceive the cost of owneship to be highe than the cost of enting. Optimists, instead, who expect highe pices in the futue, become indi eent between owning and enting. The upshot is that optimists consume housing sevices, vt ; out of the units of houses owned, h t ; and ent out the di eence, h t vt ; to the pessimists: h t vt = vt 0 : (4) 3.6 The equilibium ental and house pice Assuming a xed housing supply, s, the ental pice is pinned down by the maket cleaing condition fo housing sevices, which, togethe with (), yields whee s = v t + v 0 t ; q t = t + a t s; (5) t = w t + wt 0 and a t = a t + a t ; denote the aveage income and the aveage pefeence shock fo housing sevices. The equilibium house pice is detemined by the indi eence condition (3), which can be ewitten as: p t = + q t + + E t p t+ : (6) Using (5) to substitute out q t, we obtain the following picing equation: whee p t = summaizes aveage fundamental vaiables, and + f t + + E tp t+ + E + e t p t+ ; (7) E t p t+ E t p t+ + E 0 t p t+ f t t + a t s; (8) ; Et ~ p t+ E t p t+ Et 0 p t+ ; denotes, espectively, the aveage expectation and the di eence in expectations about tomoow s pice. In equation (7), as in a standad house picing equation, p t depends on fundamentals, f t ; and the aveage expectation on the futue house pice. The exta tem, e E t p t+ ; is non-standad and aises because agents may hold heteogenous expectations. In the next two sections, we make di eent assumptions about agents infomation sets in ode to evaluate how E t p t+ and ee t p t+ in uence the detemination of the equilibium house pice. 8

9 4 Homogenous Infomation We stat with the benchmak case in which agents ae homogeneously infomed about the state of the economy, t : Speci cally, we assume that agents only ely on the public infomation. t, to infe t. In othe wods, agents shae a common infomation set so that individual expectations coincide with the aveage expectation; i.e., E j t p t+ = E t p t+. In this case, the di eence in expectations is zeo: Et ~ p t+ = 0: Iteating equation (7) fowad and imposing a stationay condition on pices, Appendix II shows that the aveage expectation of tomoow s pice can be witten as E t p t+ = t s; (9) with + : The expectation depends on t since t (which is not obsevable) follows an AR() pocess so that agents estimation of t depends on its past ealization. The pefeence shock, a t, does not in uence the expected pice because it has, by assumption, zeo mean. Inseting (9) into (7), and ecalling that E ~ t p t+ = 0; the equilibium pice unde homogenous infomation, p ; can be witten as p t = f t + t ; (0) whee f t is given in (8) and t t t a t ; + is an expectation eo. We intepet p t as the fundamental pice of owned houses, because it e ects the aveage opinion in the maket which is an unbiased estimate of the unknown fundamental. 5 Heteogeneous Infomation We now conside a setting whee agents use the cuent ealization of thei income, w j t ; as well as the public signal, t, to make an optimal infeence about t. Agent j s infomation set at t is theefoe 5 o j t nw = j t ; t j = 0; : It is impotant to notice that the equilibium house pice is not included in j t : This assumption is made only to simplify the chaacteization of the channels though which infomation dispesion a ects the equilibium pice. As we will discuss in the following section, this assumption is not essential fo ou esults. 6 5 It is supe uous to know the entie histoy of aggegate shocks since t follows an AR() pocess. Similaly, knowing the past ealization of agents pivate signals is ielevant, given the iid assumption fo " j t: 6 A way to think about this assumption is to conside the special case whee the vaiance of the aggegate unobsevable pefeence shock, a; is abitaily lage. In such a case the house pice (7) becomes uninfomative about t and house maket paticipants do not lean much upon obseving p t. In excluding p t fom agents infomation set, we make ou analysis akin to models whee agents do not condition on the equilibium pice because they do not know how to use pices coectly (e.g. they display bounded ationality, as in Hong and Stein, 999) o because they exhibit behavioal biases (e.g., they ae ovecon dent, as in Scheinkman and Xiong, 003). 9

10 With signals w j t and t, the ability of agent j to estimate t depends on the elative magnitude of " and : Because of ou assumption of independently and nomally distibuted eos, the pojection theoem implies E j t t = ( ) t + w j t ; () whee the weight = + " e ects the elative pecision of the two signals. With > 0; expectations among agents ae heteogeneous and both aveage expectations and expectation di eences become impotant deteminants of the equilibium house pice. Moeove, since expectations depend on w j t, the optimistic (pessimistic) ae those with highe (lowe) ealization of the idiosyncatic shock. Iteating equations (7) and () fowad and excluding explosive pice paths, Appendix III shows that di eence in expectations, and the aveage expectation of the futue pice ae, espectively, whee ~E t p t+ = i t ; () E t p t+ = ( t s) + I + ( t t ) ; (3) i t " t " 0 t ; denotes the dispesion of infomation between the two goups of agents and I Z 0 xd (x) ; measues the aveage degee of infomation heteogeneity in the economy, with denoting the distibution of i t. Equation (), stems fom the fact that agents ae dispaately infomed and they assign a positive weight to thei pivate signal in estimating t : Di eences in expectations ae theefoe popotional to the dispesion in pivate signals. Equation (3) is the equivalent of equation (9). It di es, howeve, fom (9) because dispesed infomation intoduces two additional tems, each popotional to the weight that agents assign to thei pivate signals. The st tem, I= aises because pices ae fowad looking: it is not only the cuent dispesion of infomation that in uences the pice of housing, but also the dispesion of futue infomation. The second tem, ( t t ) ; captuing the aveage mispeception in the economy, aises because agents use only pat of the infomation contained in the public signal t ; to make optimal infeence about t. The slow eaction to changes in fundamentals has the e ect of intoducing inetia in the way aveage expectations ae fomed, which accods well with the idea that housing maket expectations tend to be extapolative (see Case and Shille, 988, 003). Plugging these expessions into (7), the equilibium pice can be witten as whee, p t ; is the fundamental pice given in (0), and p t = p t + t ; (4) t ( t t ) + I + ( + ) + i t + (5) 0

11 summaizes the ole of infomation dispesion among agents. With heteogeneous infomation (i.e., > 0), p t is highe than p t fo two easons. The st eason is the unconditional mean of t is positive, implying infomation dispesion leads to a highe equilibium house pice. This is quite intuitive. Optimists expect highe house pices since they estimate a highe t (see equation ()) and, thus, a highe futue pices (see equation (3)). Convesely, pessimists expect lowe futue pices and capital losses. As discussed in Section 3, pessimists pefe to dop out of the maket of homes fo sale and consume housing sevices though the ental maket. Theefoe, the equilibium pice is biased upwad because it e ects only the opinion of the optimists. Th secon eason is the pice misalignment becomes moe ponounced the lage the infomation dispesion, i t. When " t inceases, optimistic agents demand moe houses fo speculative easons, while pessimists continue to demand no housing units. These esults lead to two testable implications. Fist, housing pices incease with infomation dispesion. Second, positive infomation shocks incease housing pices, while negative shocks have no such e ects. A thid testable pediction aises in compaing (4) and (0). It is staightfowad to see that elative to the benchmak case of homogeneous infomation, the volatility of house pices is highe the lage the aveage mispeception in the economy, and the lage the vaiance of infomation dispesion, i : V (p t ) V (p t ) = + i > 0: (6) + This exta souce of pice volatility aises because the equilibium pice with dispesed infomation is not only in uenced by fundamental shocks but also by noise shocks. 5. Leaning fom the equilibium pice In this section we elax the assumption that agents do not use the equilibium pice to infe the unknown state of the economy. This extension is desiable because house pices, like any othe nancial pices, contain useful infomation about the dispesed infomation in the economy. In extending ou analysis to a set-up whee households lean fom the equilibium pice we un, howeve, into a non-tivial poblem. As discussed in the pevious section, if households eceive symmetically dispesed infomation and have the option to consume housing sevices by eithe buying o enting, the housing maket becomes segmented and the equilibium pice depends on the di eence in expectations between optimists and pessimists, i.e., i t = " i t " j t. Howeve, because i t is not nomally distibuted, p t has a non Gaussian distibution and standad linea lteing methods cannot be applied. 7 To cicumvent this poblem we intoduce the assumption that a t the aggegate pefeence shock is an independent and identically distibuted andom vaiable, dawn fom a distibution M, with zeo mean and vaiance a: Moeove, M is such that a t +i t t N ({; ) whee { denotes the unconditional mean of i t and the vaiance of a t + i t : Although ad-hoc, this assumption enables us to use standad methods to chaacteize the lteing poblem, since it ensues that the equilibium pice is Gaussian. In addition, as in a typical noisy ational expectation model à la Gossman and Stiglitz (976) and Hellwig (980), 7 See Appendix IV fo a deivation of the exact distibution of i t.

12 this assumption guaantees that the equilibium pice is not fully evealing about the state of the economy. Speci cally, households cannot tell whethe pices ae high because aggegate economic conditions impove o because unobsevable taste shocks dive housing demand. Using a standad linea solution method, Appendix IV shows that the equilibium pice with leaning can be witten as, p t = p t + t + 3 t ; (7) whee > 0 and 3 > 0 ae the weights on the pivate and the endogenous public signal (the pice), espectively, and t + t + ( + ) ( + ) a t + ( + ) ( + ) i t is a tem that summaizes the degee of magni cation of shocks induced by the pocess of leaning fom the pice. Intuitively, in the pesence of unobsevable shocks, households who obseve a change in house pices do not undestand whethe this change is diven by a change in aggegate income ( t ), pefeences (a t ), o pivate signals (i t ). Thus, with 3 > 0; each of these shocks will have an ampli ed e ect on equilibium pices, since households espond to whateve is the souce of movement in the house pices. A key obsevation to make in compaing equation (7) with (4) is that i t ou measue of infomation dispesion continues to shift the equilibium pice away fom its fundamental value, p t. Moe speci cally, i t exets a diect e ect, via t ; fo the same easons discussed in the pevious section, and an indiect one, via t ; because of the magni cation of shocks induced by the pocess of leaning. The elative impotance of t and t depends, howeve, on and 3. As shown in Appendix IV,! and 3! 0 as a!, while < and 3 > 0 with a nite a. In wods, as the noise in the pefeence fo housing sevices is su ciently lage, the equilibium pice (7) becomes non-infomative, and theefoe, identical to the one pevailing in absence of leaning (4). 8 6 Testing the implications of the model Ou model delives thee main pedictions: ) the deviation of house pices fom the fundamental value inceases with the dispesion of infomation; ) the volatility of house pices is highe the lage the volatility of infomation dispesion; and 3) positive infomation shocks move the house pice upwad, but negative shocks have no e ects. The most di cult pat in testing these pedictions is to obtain data on infomation dispesion. 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 the dispesion of infomation about local housing maket conditions. While debatable, this poxy is motivated by the logic of ou model, which can be seen as descibing the deteminants of house pices in a given city, whee the speculative demand 8 Note that the volatility of the equilibium house pice continues to be highe than in the benchmak scenaio of impefect but homogenous infomation. By compaing (7) with (0), it is immediate to see that (6) holds tue.

13 fo housing depends on local economic conditions. Speci cally, if esidents in each city ae employed in di eent industies, and they ae impefectly infomed about the city income, then industy speci c income shocks can 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 () and () in the model, can be ewitten and eintepeted as follows; w j k;t = k;t + " j k;t and k;t = k;t + k;t (8) whee w j k;t is the time t income of a esident of city k employed in industy j; k;t the aveage city income at time t; and " j k;t the time-t industy-j speci c shock in city k. A poxy fo the dispesion of infomation about k;t can then be computed using a measue of the dispesion of income shocks acoss the j industies Data desciption and summay statistics We collect annual data fo a sample of appoximately 350 US metopolitan aeas (MSA) duing the peiod 980 to 000. To infe the time seies popeties of local income shocks we use (pe employed) eanings data fo 0 one-digit industies, based on the SIC classi cation code. 0 With these data, the dispesion of eanings shocks acoss industies is computed in two steps. Fist, based on equation (8), we un 0 egessions, one fo each industy, in which we pool the gowth ate of industy eanings fo the full sample of MSAs, w j k;t = 0 + k;t + k;t + t + " j k;t fo j = ; ; :::0: (9) Hee is the st di eence opeato, and t is a time xed e ect. In this speci cation, the esiduals " j k;t ecod shocks to eanings gowth in city-k industy-j, contolling fo cityspeci c income dynamics, k;t and k;t ; and nationwide e ects, t. Second, we measue the dispesion of eanings shocks acoss j industies and within each MSA as the weighted aveage of the absolute value of industy-city shocks, i k;t = 0X! j k;t " j k;t j= ; (30) 9 Thee ae also empiical motivations behind ou poxy of infomation dispesion. Thee is widespead consensus that high fequency vaiations in house pices ae mostly local, not national (Glaese and Gyouko, 006) and evidence that the bulk of shot-un movements in house pices is due to changes in demand, diven by local economic conditions, as opposed to changes in pefeences fo local amenities. Endogenous supply-side changes may also a ect movements in house pices (Glaese, Gyoko and Saiz, 008). Howeve, in the shot un, due to egulations and technological constaints, supply changes tend to espond slowly to shifts in demand. 0 Speci cally, we use eanings data fo the following industies: ) Fam, ) 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 0) Govenment and govenment entepises. These data ae available at Ou sample peiod ends in 000 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 000. Available data based on the NICS system cove only the peiod 00 to 006. We use data based on the SIC classi cation codes fo the peiod , to be able to exploit a longe time seies vaiation in the data 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. 3

14 whee the weights! j k;t measue the shae of MSA wokes employed in industy j; to contol fo the size of each industy. 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 conveted in eal dollas using the national CPI index fom the Bueau of Labo Statistics. We use annual obsevations because income data is only available annually. Table epots basic summay statistics. The data display consideable vaiation acoss MSA. Ove the full peiod , ou poxy of infomation dispesion is less than.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 0 yea peiod studied, while Houston, Oklahoma City, and San Antonio expeienced negative house pice changes of appoximately.5%. 6. The baseline egession We stat the analysis by examining the empiical elevance of the pice equation pevailing unde common infomation. To empiically use equation (0), we take st di eences of each vaiable and estimate the following egession: 3 p k;t = 0 + k;t + k;t + t + k + k;t : (3) 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 a standad eo tem. In this egession, and those that follow, we include yea and MSA dummies, t and k ; to account fo unobsevable aggegate and city-speci c deteminants of house pices. Table 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, and ae expected positive and as shown in the st column of Table these pedictions ae stongly suppoted by the data: highe cuent and lagged changes in income ae signi cantly associated with highe housing pices changes. The ole of infomation dispesion is examined in column whee we epot estimates of the empiical countepat of equation (4). Moe speci cally, we add to the baseline egession (3) ou poxy of di eence in expectations, i k;t : In line with the pediction of the model, the esults show a statistically signi cant elationship between ou poxy of infomation dispesion and house pices. The estimated e ects is also sizeable: a % incease in i k;t esults in a 0.% incease in the gowth ate of house pices. To bette gauge the economic e ect of this esult, let us conside an exogenous incease in i k;t ; fom the 0th pecentile value (which is None of the esults pesented below (in tems of economic and statistical signi cance) change if we use squaed deviations athe than absolute deviations. We pefe to use absolute deviations to be able to maintain the same unit as the change in industy eanings, so that the coe cients in the house pice egessions epoted below ae easily intepeted. 3 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. 4

15 appoximately.%) to the 90th pecentile value (which is appoximately 4%). This incease would lead to an acceleation in the gowth ate of house pice by 0.6% pe yea, which is lage consideing that the aveage annual gowth ate of eal house pices is 0.4% ove the peiod. 6.3 Altenative empiical speci cations The esults in Table, although based on the pice equation 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 (989), it is well known 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 of Table 3 indicate house pices indeed exhibits positive coelation at shot lags and negative coelation at longe lags. Howeve, as epoted in column, ou poxy of infomation dispesion continues to play a lage and signi cant ole in explaining house pice changes. Columns 3 to 4 exploe the obustness of ou ndings to an altenative empiical speci cation, suggested by the wok of Lamont and Stein (999). In thei study of the house pice dynamics in US cities, Lamont and Stein nd house pices (a) exhibit shot un movements, (b) espond to contempoaneous income shocks, and (c) display a long un tendency to fundamental evesion. They thus popose to estimate the following egession, p k;t = 0 + p k;t + k;t + 3 (p=) k;t + k + t + k;t (3) whee (p k = k ) t is the lagged atio of house pices to pe-capita income. As shown in column 3, these vaiables have all the expected sign and explain a lage faction of house pice vaiations. To this thee-vaiable speci cation we add ou vaiable of inteest, i k;t, in column 4. In line with the esults in Table, we nd ou poxy of infomation dispesion continues to be elated signi cantly to house pice changes: the gowth ate of house pice is highe in cities whee local income shocks ae moe dispesed. A common objection to the coelations epoted so fa is that we do not contol fo demogaphic factos. Changes in the demand fo housing may be diven not only by economic conditions but also by population changes a shifte of housing demand that has been omitted in ou theoetical analysis. In the attempt to contol fo this e ect, columns 5 and 6 use population gowth as an additional egesso. Population gowth is expected to ente the egession with a positive sign since new potential buyes tend to move housing demand and pices up. The esults show population gowth has indeed a positive and signi cant e ect on house pices. Ou coe ndings, howeve, do not seem to depend on the inclusion of this additional contol. A futhe obustness check of ou main esults is in Table 4. The main pediction of ou model that house pices ae highe the lage the dispesion of infomation holds unde the assumption of xed housing supply. Howeve, if the supply of housing is elastic, changes in demand would have a muted e ect on pices. Table 4 exploes this possibility using Saiz (008) index of housing supply elasticity. The notewothy featue of this index is that it does not depend on local maket conditions but only on geogaphical and topogaphical constaints on house constuctions. Using the median value of this index as a cut-o, we un the same egessions as in Table 3 fo cities with high and low supply elasticity. Although ou sample 5

16 of cities is substantially educed, the esults validate ou pio that the speculative motive fo housing demand has a moe ponounced e ect on pices in cities with tighte supply estictions. The estimated coe cient fo ou poxy of dispesion in infomation is lage and statistically signi cant in cities with low-supply-elasticity and is essentially zeo and neve statistically signi cant in cities with an elastic housing supply. 6.4 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 infomation. 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 then by taking 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. Next, with one obsevation fo MSA, we exploit the coss sectional vaiation of house pice volatility and egess ou measue of house pice volatility on the standad deviation of infomation dispesion in each MSA. The OLS estimates ae in Table 5 and illustated in Figue, which gaphs the volatility of house pice against the tted values fom the egession. As can be seen, MSAs with lage dispesion of infomation have also moe volatile house pices. Inteestingly, this esult holds even if we contol fo the standad deviation of aggegate MSA income, as shown in the second column of Table Positive and negative shocks to industy income gowth The nal implication of ou model is that house pices espond moe to positive than negative infomation shocks. In the model, agents buy housing units fo speculative easons if they eceive signals that convey infomation of highe futue house pices, and ent if they expect house pices to depeciate. This asymmety implies that while negative infomation shocks do not move house pices downwad, positive shocks impat an upwad shift in house pices. It is impotant to check the empiical elevance of this pediction since its validity distinguishes the theoetical implications of ou model fom a setup whee income dispesion a ects house pices because agents moves acoss cities fo poductive o pefeence easons, as fo example in Gyouko, Maye and Sinai (006) o Van Nieuwebugh and Weil (007). Speci cally, we de ne positive and negative infomation shocks as follows: P OS k;t = NEG k;t = X0 j= X0 j=! j k;t "j k;t if " j k;t > 0! j k;t "j k;t if " j k;t < 0: Hee " j k;t ae the esiduals in equation (9), that is peiod t eaning shocks in industy j in city k; and! j k;t ae weights that measue the faction of MSA population employed in industy j. P OS and NEG ae thus the empiical countepats of " t and " 0 t in the model. We ente these vaiables in piecewise linea fom into the empiical speci cation (3) and (3) to allow fo 6

17 di eential e ects between positive and negative shocks to industy eanings. The esults ae in Table 6, using the same speci cations as in Table 3. As shown, the estimates confom with ou model s pedictions that positive and negative shocks have asymmetic e ects on housing pices. In fact positive shocks have a lage and signi cant impact on house pices, while the impact of negative shocks is negligible and not statistically signi cant. 7 Conclusion In this pape we have used a use-cost model to study how infomation dispesion about local economic conditions a ect the equilibium pice of housing. In ou model in which agents consume housing sevices and speculate on futue pice changes the equilibium housing pice is highe, the lage agents di eence in expectations about futue house pices. The intuition is that all agents face de facto a shot-sale constaint in housing. Theefoe, those who hold pessimistic expectations about futue pices decide to ent to avoid capital losses, while those who have optimistic expectations decide to buy to speculate on futue pice inceases. The upshot is that the equilibium pice of owne occupied houses incopoates only the expectations of the optimists and is thus highe and moe volatile elative to an envionment of homogenous infomation. We con m the theoetical pedictions of ou model in a panel data of US cities, using dispesion in industy income shocks as a poxy fo dispesion in infomation about local economic conditions. This poxy is motivated by ou model s assumption that di eent ealizations of individual income lead agents to fom di eent views of the economy. In keeping ou model simple, we have abstacted fom a numbe of issues that might play an impotant ole in the development of a moe complete model. Fo example, we have abstacted fom the geneal equilibium e ects of the inteest ate. Changes in R; howeve, may a ect ou analysis since the etun on the safe asset in uences agents choice of enting and owning, fo a given level of house pice expectations. We have also pevented agents fom e-tading. An extension of the model that allows fo e-tading, as in Stein (995) o Otalo-Magné and Rady (006), may shed new light on whethe infomation dispesion induce a positive coelation between house pices and housing tansactions. These extensions ae left fo futue eseach. Refeences Almeida, H., M. Campello and C. Liu (006), The Financial Acceleato: Evidence Fom Intenational Housing Makets, Review of Finance, 0, -3. Case, K. E., and R. J. Shille (988), The Behavio of Home Buyes in Boom and Post-Boom Makets, New England Economic Review (Novembe/Decembe) Case, K. E., and R. J. Shille (989), The E ciency of the Maket fo Single Family Homes, Ameican Economic Review, 79(), Case, K. E., and R. J. Shille (003) Is Thee a Bubble in the Housing Maket, Booking Papes on Economic Activity,,

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