The Long-Run Relationship between House Prices and Income: Evidence from Local Housing Markets

The Long-Run Relationship between House Prices and Income: Evidence from Local Housing Markets Joshua Gallin, Federal Reserve Board (cid:3) April, 2003 Abstract Thepropositionthat\housingpricescan’tcontinuetooutpacegrowth inhouseholdincome"(Wall Street Journal;July25,2002) isthereceived wisdom among many housing-market observers. More formally, many in the housing literature argue that house prices and income are cointegrated. In this paper, I show that the data do not support this view. Standardtestsusing27yearsofnational-leveldatadonot(cid:12)ndevidenceof cointegration. However, it is known that tests for cointegration have low power,especiallyinsmallsamples. Iusepanel-datatestsforcointegration thathavebeenshowntobemorepowerfulthantheirstandardtime-series counterparts to test for cointegration in a panel of 95 metro areas over 23 years. Using a bootstrap approach to allow for cross-correlations in city-level house-price shocks, I show that even these more powerful tests donotrejectthehypothesisofnocointegration. Thustheerror-correction speci(cid:12)cationforhousepricesandincomecommonlyfoundintheliterature may beinappropriate. (cid:3)Thanks to Doug Elmendorf, Steve Oliner, Jeremy Rudd, Dan Sichel, Bill Wascher, participants at the 2002 Federal Reserve System Conference on Regional Economics and the 2002RegionalScience AssociationInternational Conference,andseminarparticipants atthe University of Georgia. Special thanks to Norm Morin and Peter Pedroni for their help and comments. Theviews presented aresolelythose of the author and donot representthose of theFederalReserveBoardoritssta(cid:11). 1

1 Introduction Inthesecondhalfofthe1980s,realhousepricesroseabout3percentperyear.1 Thenin1990alone,pricestumbledalmost5percent. Pricescontinuedtofall,on balance, through the end of 1994, reversing more than half of the gains posted inthelate1980s;pricesdidnotreturnto their1989leveluntil almosttenyears later. Many coastal cities experienced even wilder swings: Real house prices rose about 65 percent in Los Angeles and about 45 percent in Boston in the secondhalfofthe1980s,onlytofall30percentinL.A.and20percentinBoston during the next (cid:12)ve years. Many real-estate market observers think that housing prices got too far ahead of fundamentals in the 1980s|especially in many coastal cities|and that the poor performance of house prices during the (cid:12)rst half of the 1990s wasthe inevitable aftermath. During the past(cid:12)ve years,realhouse prices have moved up almost 5 percent per year, out-pacing the gains from a decade ago and sparking fears that the housing market is once again over-valued. Ofparticularconcerntomanyisthefactthathouse-pricegainshavedwarfed per capita income gains in recent years. As can be seen in Chart 1, the ratio of house prices to per capita personal income moved up recently after trending down for most of the previous 20 years.2 More speci(cid:12)cally, from the middle of 1997 to the middle of 2002, real house prices rose about 28 percent while real per capita personal income rose about 15 percent. In contrast, during the previous 20-year period, real house prices rose only 8 percent while real per capita income rose 35 percent. The recent performance of house prices relative to income is taken as evidence by some that house prices are out of line with \fundamentals," and that prices must stagnate or fall to allow income to catch up. Thisideaiscommonlyformalizedinthehousingliteraturebypositingacointegrating relationship between house prices and fundamentals such as income, and then estimating an error-correction speci(cid:12)cation (Abraham and Hendershott, 1996; Malpezzi, 1999; Capozza et al., 2002; Meen, 2002). That is, house prices and income are thought to be linked by a stable long-run relationship; 1Allhousepricemeasuresinthispaper arefromthe O(cid:14)ceofFederal HousingEnterprise Oversight’sweighted repeat-sales priceindex unlessotherwisenoted. Nominal pricesaredeflatedbythepersonalconsumptiondeflatorfromtheNationalIncomeandProductAccounts. 2Inthispaper,Ifocusonpercapitaincomeratherthanhouseholdincomefortworeasons. First, household size|the link between per capita and household income|is endogenous to the household formation problem, and therefore to housing demand. Second, I do not have city-leveldataonthenumberofhouseholds. 2

they may drift apart temporarily, but their tendency is to return to their longrun equilibrium. Thepurposeofthispaperistotestthisviewofthehousingmarket. Ifprices and income are cointegrated, then the gap between the two may be a useful indicator of when house prices are above or below their equilibrium values, and therefore a useful predictor of future house-price changes. Conversely, if prices and income are not cointegrated, then the error-correctionspeci(cid:12)cations commonin the literature areinappropriate,andhouse prices neednotstagnate or fall just because they have grown more quickly than has income of late.3 Why might such a relationship exist? Imagine a simple supply-and-demand framework. Suppose that the supply of housing, including land, slopes up either because there is a limited supply of attractive land or because of zoning restrictions (Glaeser and Gyourko, 2002). Because demand-side shocks such as population and income are not stationary, we should expect house prices to be non-stationary as well. However, if other shocks to the demand curve and all shocks to the supply curve are stationary, then population, income, and price will be cointegrated in a way that depends on the elasticities of supply and demand. Many researchers simply assume that house prices and fundamentals are cointegrated (Abraham and Hendershott, 1996; Capozza et al., 2002). Others implicitly assume that they are not (Poterba, 1991). Meen (2002) is the only paperIamawareofthattestsforcointegrationofpricesandfundamentalsusing national-leveldata. His reported tests do not (cid:12)nd evidence for cointegrationat conventional signi(cid:12)cance levels. However, Meen argued that the test statistics are \near" their critical values, and therefore concluded that prices and fundamentals are cointegrated. One contribution of this paper is to show that using 27 years of national-level data, one does not (cid:12)nd evidence that prices, income, and other fundamentals are cointegrated. Thus my results, if not my interpretation, are in accord with Meen, and suggest that it is inappropriate to model house-price dynamics using an error-correctionspeci(cid:12)cation. However, cointegration tests are known to have low power, particularly in small samples (Banerjee, 1999). A time span of 27 years may be too short to estimate whatmaybe agenuine long-runrelationshipwithslow adjustment. StartingwithQuah(1990)andLevinandLin(1992),researchershavedeveloped panel tests for unit roots and cointegration that are more powerful than their 3The absence of cointegration does not preclude house-price bubbles. For example, one mightnot(cid:12)ndcointegration, whichisalinearrelationship,ifthehousingmarketisbeset by non-linearrationalbubbles. 3

standard time-series counterparts. The second, and main, contribution of this paper is to apply recently developed tests of Pedroni (1999) and Maddala and Wu (1999) to a panel of 95 U.S. cities over 23 years. I show that even these morepowerfultestscannotrejectthehypothesisthatpricesandincomearenot cointegrated. My results contradict those of Malpezzi (1999), who found that onecan rejectthenullofnocointegrationinasimilarpanel. However,Malpezzi used a panel unit root test, which overstates the likelihood of cointegration because it ignoresthe (cid:12)rst-stageestimation in his residuals-basedcointegration test. The rest of this paper is organized as follows. In the following section I briefly describe a simple model of housing supply and demand. The purpose of the model is to motivate why prices, per capita income, and perhaps other variablesmight be cointegrated. In Section 3, I describe the national-leveldata andshowthatthereislittleevidenceforcointegrationusingEngleandGranger’s (1987)AugmentedDickey-Fuller(ADF)test. Section4istheheartofthepaper. In it, I describe several tests for cointegration in panel data. Three are from Pedroni (1999): one is a panel version of Phillips and Ouliaris’ (1990) variance ratio test, one is a panel version of their Z(cid:11) test, and one is a panel version of Engle and Granger’s (1987) ADF test. My fourth test is a Maddala and Wu (1999) version of Engle and Granger’s ADF test. I also describe a bootstrap approach that allows for arbitrary cross correlations of the city-level shocks. I describe the city-level data and show that, using the bootstrapped critical values,none of the tests rejects the nullof no cointegration. A secondaryresult is that the correlationsamong localhousing markets canhave a largee(cid:11)ect the tests. I conclude in Section 5. 2 Housing Prices and Fundamentals House prices andfundamentals like income|orsome transformationofthem| will be cointegrated if they are linked by a long-run equilibrium relationship. The interaction of housing supply and demand o(cid:11)ers the most obvious and simple way to characterize such a relationship. Consider a simple supply and demand model of housing. In it, the demand for owner-occupied housing depends on income, Y; population, N; wealth, W; the user cost of housing, UC; and other demand shifters, (cid:18)d. The supply of housing depends on the price of housing, P; the cost of new construction, C; and other supply shifters, (cid:18)s: Qd = D(Y;N;W;UC;(cid:18)d) 4

Qs = S(P;C;(cid:18)s): The user cost of capital, in turn, depends on the price of housing; mortgage rates, m; income and property taxes, (cid:28)y and (cid:28)p; maintenance and depreciation, (cid:14); and expected capital gains, cg: UC =P[(1−(cid:28)y)(m+(cid:28)p)+(cid:14)−cg]=P (cid:1)A where A represents the term in brackets. Then the price of housing can be written as a function of all the other variables: P =F(Y;N;W;C;A;(cid:18)d;(cid:18)s) Alog-linearizedsolutiontothemodelwouldrelatetheloghousepricetothe logsofallthedrivingvariables. Undertheassumptionthattheunobservedcomponents of the model are stationary and the coe(cid:14)cients of the log-linearization do not change, house prices will be cointegrated with those fundamentals that also have a unit root, and the relationship will depend on the elasticities of supply and demand. The point is not that house pricesmust be cointegratedwith fundamentals. Indeed this simple model illustrates that there are many reasons why such a cointegrating relationship need not exist. For instance, the price elasticity of supplymaynotbestableovertimebecauseofchangesinregulatoryconditions, the price elasticity of demand may not be stable because of changing demographics, or demand shifters such as local taxes may not be stationary. The modeldoesshowwhatkindofassumptionsareneededtogenerateacointegrating relationship. Whether one exists is an empirical question. 3 National-Level Tests for Cointegration In this sectionI presenttests ofcointegrationof national-levelhouse prices and various fundamentals. Suppose that the hypothesized cointegrating regression is given by XM x 0;t =(cid:11)+(cid:14)t+ (cid:12)mxm;t+et; (1) m=1 where m = 1;:::;M indexes I(1) variables and t = 1;:::;T indexes time. If the residual et is stationary, then we say that the x’s are cointegrated. Here, I use the common two-step procedure for testing for cointegration suggested by Engle and Granger (1987), sometimes called an augmented Engle-Granger 5

(AEG) (cid:28)-test. In the (cid:12)rst stage, I estimate Equation (1) by OLS to get e^t. In the second stage, I conduct an augmented Dickey-Fuller (ADF) (cid:28)-test on the residuals. Thecriticalvaluesdi(cid:11)erfromthoseofthestandardADFtestbecause the residuals are estimated in the (cid:12)rst stage. My source for house-price data is the repeat-sales price index for existing homes,whichispublishedbytheO(cid:14)ceofFederalHousingEnterpriseOversight (OFHEO).4 The index is based on price changes for homes that are resold or re(cid:12)nanced,but does not controlfor changes to the house throughimprovement or neglect; that is, while it does hold some characteristics constant, it is not a true quality-adjusted price. In addition, the repeat-sales sample excludes homes with jumbo, FHA, or VA mortgages (Calhoun, 1996). I used the BEA’s measure of total personal income, the Census Bureau’s measure of population, and the BLS’s measure of average hourly wages for construction workers. I used the Standardand Poor’s500 stock index to measure stock-marketwealth. IincludedthelevelofthepersonalconsumptiondeflatorfromtheBEAtocontrol for inflation. All data are quarterly. To calculate the user cost of housing, I used a weighted averageof the rates on(cid:12)xed-rateand1-yearadjustable-ratecontractsfor 30-yearloans;theweights were the origination shares.5 I set expected capital gains to the average percentage increase in the house-price index during the previous three years. The standard unit root tests,which are available upon request, do not reject the hypotheses that house prices, per capita income, population, the stock market, and construction wages all have a unit root, but that the non-price component of the user-cost, A, does not. The results were the same when I assumed rational and myopic, as opposed to backward-looking,expectations. I therefore did not include non-price part of the user cost in the regressions.6 Table 1 displays AEG (cid:28)-tests for cointegration of several sets of variables using quarterly data from 1975:Q1 to 2002:Q2. The (cid:12)rst-stage levels equation yields an estimate of the cointegrating vectors, which are shown in the upper panelofthetable. Thecoe(cid:14)cientestimatesindicatethatpercapitaincomeand the constructionwagehavepositiveandstatisticallysigni(cid:12)cante(cid:11)ectsonhouse prices, but that population has no e(cid:11)ect and the price level and stock market have negative e(cid:11)ects. Of course, if there is no cointegrating relationship, these 4AlternativetestsbasedontheaverageexistinghousepricefromtheNationalAssociation ofRealtorsyieldedsimilarresults. 5My measures of Federal and state and local tax rates are from the FRB/US model See Reifschneider,Tetlow,andWilliams(1999) formoreinformationabouttheFRB/USmodel. 6Theinclusionofthenon-pricepartoftheusercostorthesimplemortgageratesdoesnot a(cid:11)ecttheresults. 6

Table 1 National-Level Tests for Cointegration of House Prices and Fundamentals nullofnocointegration,1975:Q1to2002:Q1 First-Stage Levels Regressions dependent variable is log(price) Independent variables (logvalues) 1 2 3 4 per capita income :70(cid:3)(cid:3) 1:45(cid:3)(cid:3) 1:57(cid:3)(cid:3) 1:71(cid:3)(cid:3) (:05) (:20) (:15) (:17) population | −2:94(cid:3)(cid:3) | −1:57 (:76) (1:26) stock market | | | −:13(cid:3)(cid:3) (:02) construction wage | | | :25(cid:3)(cid:3) (:13) PCE deflator :03 −:38(cid:3)(cid:3) −:76(cid:3)(cid:3) −1:04(cid:3)(cid:3) (:10) (:14) (:15) (:14) trend (cid:2)10 | | −:74(cid:3)(cid:3) −:10 (:12) (:24) Second-Stage Test Results AEG (cid:28)-stat -2.2 -2.0 -1.9 -3.4 critical value (10%) -3.5 -3.8 -3.8 -4.7 Notes: (cid:3)(cid:3) -Signi(cid:12)cantat:05. Standarderrorsareinparentheses. 7

levels regressions are spurious. The lower panel presents the second-stage tests with 10 percent criticalvalues that are based on the dimension of the proposed cointegrating vector and the presence or absence of a time trend (Davidson and MacKinnon, 1993).7 The lower panel of Table 1 shows that in none of the cases can we (cid:12)nd strong evidence for the cointegration of house prices and fundamentals. In other words, the national-level data do not support the view that the log levels of house prices and various fundamentals are linked by a long-run stationary relationship.8 Onecriticismofthesetestsisthattheyareknowntohavelowpoweragainst thealternativeofcointegration,particularlywhen,asisthecasehere,thesample size is small. Thus the evidence in Table 1 may not be convincingto those who havestrongpriorsthathouse pricesandfundamentals, particularlyincome,are cointegrated. 4 City-Level Tests for Cointegration Quah(1990,1994)andLevinandLin(1992)wereamongthe(cid:12)rsttodevisepanel tests for unit roots and to show that they can o(cid:11)er a substantial improvement in power relative to separate tests for each cross-sectional unit of the panel.9 The literature has blossomed since then. Some, like Im, Pesaran, and Shin (1997)havedevisedteststhatimposefewerrestrictionsthandidLevinandLin. Others,likePedroni(1997,1999,2001)developedrelatedtestsforcointegration. Banerjee (1999) provides an overview of the literature. In this section I briefly describe three panel cointegration tests of Pedroni (1999) and of Maddala and Wu (1999). One of the underlying assumptions of all the tests is that shocks are either independent across cross sections or thatthecross-sectionaldependencecanbemodeledasanaggregatetime e(cid:11)ect. For the purposes of this paper, that assumption implies that shocks to housing markets in, say, San Francisco and Seattle have the same correlation as shocks to housing markets in Philadelphia and New York. As this is an unattractive feature, I discuss bootstrapped versions of the tests that relax this assumption byallowingforthe cross-sectionaldependence amongcities evidentinthe data. I then describe the city-level data and present the test results. 7The time trend is not statistically signi(cid:12)cant in Column 4. The stationarity results did notchangewhenIexcludedthetimetrend. 8Meen (2002) conducted similar tests and concluded that prices and fundamentals are cointegrated. However, his reported test statistics were quite far from conventional critical values. 9SeealsoLevin,Lin,andChu(2002). 8

4.1 Some Panel Cointegration Tests Suppose thatthe hypothesizedcointegratingregressionforeachcity is givenby XM x 0;i;t =(cid:30)i+(cid:14)it+ (cid:12)m;ixm;i;t+eit; (2) m=1 where i = 1;:::;N indexes the city, m = 1;:::;M indexes variables, and t = 1;:::;T indexes time. Notice that this speci(cid:12)cation admits city-speci(cid:12)c intercepts and time trends. In this paper, I use Pedroni’s panel-data versions of Phillips and Ouliaris’ (1990)varianceratiotest,P−v;PhillipsandOuliaris’Z(cid:11) test,P−Z(cid:11);andEngle andGranger’s(1987)ADFtest,P−AEG. Theseresidual-basedpaneltestshave the same structure as do their time-series counterparts, but are constructed by poolinginformationfromthecrosssectionalunits.10 SeeAppendixAfordetails on how the pooling is done. Pedroni(1999)showsthatthe appropriatelystandardizedversionofhis test statistics are asymptotically normal.11 As with the standard time-series cases, underthe alternative,theP−v testdivergestopositivein(cid:12)nity, sotherighttail of the normal distribution is used for rejection, and the P−Z(cid:11) and P−AEG tests diverge to negative in(cid:12)nity, so the left tail is used to reject. Maddala and Wu (1999) present an alternative, and very general, test for cointegrationbased on Fisher (1932). Maddala and Wu’s test is based on averaging the p-values for any test from each cross-sectionalunit. Suppose that we implement any cointegration test for each cross-sectional unit. Under the null, the signi(cid:12)cance levels for each test, pi are distributed uniformly over (0;1). This implies that −2logpi (cid:24)(cid:31)2(2): Under the assumption that the tests are independent, the Maddala-Wu (MW) test statistic is XN MW =−2 logpi (cid:24)(cid:31)2(2N); i=1 10Pedroni also presents a panel version of Phillips and Ouliaris’Zt test. The results from the P−Z(cid:11) test and P−Zt were almost identical, so I only report the P−Z(cid:11) test results in this paper. In addition, the tests I use here require that the AR(1) terms in the second stage residual regressions for each city be equal to each other (and less than 1) under the alternative hypotheses. Pedroni also proposes versions of his tests that do not impose this equalityrestriction. Theresultsarenotsensitivetothechoiceoftests. 11Pedronicalculated each teststatistics’ meanandvariancebysimulation. Pedroni generouslygavemetheRATScodetoconstructtheteststatistics. 9

andtherighttailofthedistributionisusedforrejection. Thistestcanbeapplied usinganyunderlyingtest. However,onemustsimulateanapproximationtothe entiredistributionofwhateverteststatistic oneis usingforeachcross-sectional unitinordertocalculatethepi’s. TheMaddala-WutestsIpresentinthispaper are based on AEG (cid:28)-tests, where I simulated the AEG (cid:28)-test distributions for a 23-year time series. That is, I did not use the asymptotic distribution of the (cid:28) statistic. The validity of Pedroni’s tests depends in part on the assumption that any cross-sectionalcorrelationsareadequatelycapturedbyanaggregatetimee(cid:11)ect; Maddala and Wu’s test requires complete independence. The assumption of cross-sectional independence is clearly violated in the data, and local housing market shocks will likely be correlated in ways that are not captured by a simple time e(cid:11)ect. One way to address this problem is to bootstrap empirical distributions of the test statistics under the null to calculate critical values for thetest. Thekeyistomaintainthecross-sectionaldependencewhileresampling. More speci(cid:12)cally, following Maddala and Wu (1999), one can get the bootstrap sample by estimating (cid:1)xm;i;t =(cid:17)m;i(L)(cid:1)xm;i;t−1 +um;i;t t=1;:::;T (3) for each series m in city i, and then calculating the residuals, u^m;i;t. Then, let (t)(cid:24)uniform(1;T) index the random resampling and construct u~m;i;t =u^m;i; (t) 8m;i;t: Note that the same (t) is used for each city i, thereby maintaining the crosscorrelationthat exists in the data. Given each set of resampledu0s, construct XJ v~m;i;0 = (cid:17)^m;i(L)u(cid:20)m;i;−j j=0 v~m;i;t = (cid:17)^m;i(L)v~m;i;t−1 +u~m;i;t x~m;i;0 = 0 x~m;i;t = x~m;i;t−1 +v~m;i;t where the u(cid:20)m;i;−j are drawn as a separate bootstrap sample of size J.12 The null of no cointegrationis true by construction for the resampled data, so one can build up the empirical distribution of the test statistic under the 12Thisisequivalenttosettingu~m;i;0=0andsimulatingJ+T observations,butonlyusing thelastT observations. 10

null by replicating the process of resampling and re-estimation enough times. The bootstrapped test results presented later in this paper are basedon 20,000 replications of 23 \years" of data for each city, where J = 100. These bootstrapped distributions are, by construction, the small-sample distributions and arethereforenotdirectlycomparabletotheasymptoticallynormaldistributions of Pedroni’s tests, but are comparable to the Maddala-Wu test. To help with comparisons, I also constructed the small sample distributions for Pedroni’s test statistics under the assumption of cross-sectional independence. That is, for each city I simulated 20,000 replications of 23 \years" of data for three unit-rootprocessesunder the assumptions ofno cointegrationandno cross-city correlations,and then tabulated the the values of Pedroni’s test statistics. 4.2 The City-Level Data Mydatasourcesatthecityandnationallevelsarethesame. OFHEOpublishes a quarterly repeat-sales price index for over 300 metropolitan areas. However, onlyashorttimesampleexistsformanyofthesecities. Forthepurposesofthis paper,Irestrictedmyattentiontoasampleof95citiesforwhichthepricedata begin in 1978. See Appendix B for a list of the cities. Total personalincome at the city levelis availableonlyatanannualfrequency, andonlythrough2000.13 Charts2through6displayhousepricesandpercapitaincomeonalogscale for 15 cities, all in current dollars.14 I chose these cities because they provide a good picture of the behavior of prices and income in most U.S. cities. The house-price run up of the late 1980’sis apparent in most of the cities, with Los Angelesprovidingthe mostdramaticexample. The averagehouse pricein L.A. rose29percentinthetwoyearsleadinguptothethirdquarterof1990,reaching apeakofabout$255,000. Duringthesameperiod,percapitaincomerosemore than 12 percent, a rapid increase, but well short of house-price gains. During the next four years, nominal prices fell more than 20 percent and per capita income was about flat. But the charts also show that not all cities saw such dramatic swings. Indeed, Midwesterncities saw little, if any,price appreciation during the late 1980s, and essentially no depreciation after. Turning to the more recent period, house prices have risen in all 15 cities 13Shillerand Perron(1985) showedthat the power ofunitroottests depends moreonthe number of years covered by the dataset than onthe number of observations. Thus, the cost intermsofpowerofusingannualdataforthecity-levelanalysisislikelysmall. 14I calculated the current-dollar values by multiplying the median house price in 2001:Q4 published by the National Association of realtors by the OFHEO index, re-indexed to that quarter,foreachcity. 11

shown, and indeed in just about every U.S. city, during the past few years. For example,duringthe twoyearsleading uptothe (cid:12)rstquarterofthis year,house pricesroseabout25percentinBoston,NewYork,andWashington,D.C.;prices rose only a bit less in San Francisco and Los Angeles and even slower-growth cities like Chicago, Cleveland, and Saint Louis saw double-digit price growth. AlthoughIonlyhavepercapita income(cid:12)guresfor cities through2000,it seems likely that income growth did not keep pace in 2001.15 Taking a longer view in these cities, one can see that although both prices and per capita income have trended up over time, their levels can diverge for many years. It is not at all clear from these pictures that house prices and per capita income are cointegrated. 4.3 Results and Interpretation Thecity-levelcointegrationtestsarebasedon(cid:12)rst-stagecointegratingrelationships ofhouseprices onper capita income andpopulation. Iestimatedthe (cid:12)rst stage regressions, which all include a time trend, separately for each city. It is important to note that the bootstrap approach can account for any aggregate e(cid:11)ect due to factors such as monetary or (cid:12)scal policy, nationaleconomic conditions,changestothehousing-(cid:12)nanceindustry,orcostshocks. Thus,Iamreally testing for the cointegration of prices, income, and population conditional on these other, potentially non-stationary,factors. Table 2 displays the main results of the paper. The (cid:12)rst column contains the value of each test statistic, the second column contains their asymptotic p-values, the third column contains their small-sample p-values, and the fourth column contains their bootstrapped p-values. The conclusions one might draw fromtheasymptoticp-valuesareinsevereconflict. TheP−v andP−AEGtests alone (lines 1 and 3) seem to provide strong evidence for cointegration, while the P−Z(cid:11) test (line 2) seems to provide none. Usingthesmall-samplep-values,thecaseforcointegrationlooksmuchweaker. Onecannolongerrejectthe nullusingtheP−AEGtest, andonecannotreject using the MW statistic. However, one can still reject using the P −v statistic. NoticethattheP−AEGtestclearlysu(cid:11)ersfromextremesizedistortioninsmall samples. Pedroni (2001) found a less extreme, but still signi(cid:12)cant distortion in the case where N = 20 and T = 40. My results indicate that the distortions become much worse as T gets even smaller. Column 4 shows the bootstrapped p-values. Note that allowing for arbi- 15Theavailablestate-level (cid:12)guresforpercapitaincomesupportthisview. 12

Table 2 City-Level Test for Cointegration of House Prices, Per Capita Income, and Population nullofnocointegration;N=95;T =23 Test Asymptotic Small-Sample Bootstrapped Statistic p-value p-value p-value 1. P−v 2:7 :00 :01 :15 2. P−Z(cid:11) 4:1 1:0 1:0 :31 3. P−AEG −3:0 :00 :95 :65 4. MW 158:4 | :95 :14 Notes: P−visasymptoticallynormalandtherighttailisusedforrejection. P−Z(cid:11) andP−AEG areasymptoticallynormalandthelefttailisusedforrejection. TheMaddala-Wutestisdistributed chi-squaredandtherighttailisusedforrejection(MaddalaandWu,1999). Thebootstrapped MaddalaWutesthasanon-standarddistributionthatmustbeestimatedbysimulation;theright tailisusedforrejection. ThePedronitestsincludeaggregatetimee(cid:11)ects. trary cross correlations has a signi(cid:12)cant e(cid:11)ect on the simulated distribution of the test statistics. The p-values for the P−Z(cid:11), P−AEG, and MW statistics are all smaller when one uses the bootstrapped distributions instead of the small-sample distributions, but one would still not reject the null of no cointegration using conventional signi(cid:12)cance levels. In contrast, the p-value for the P−v statistic is larger using the bootstrapped distribution, and the di(cid:11)erence is enough to change the results of the test from rejection to no rejection of the null. Thus, using conventional signi(cid:12)cance levels, the tests are all in accordance: House prices, income, and population do not appear to be cointegrated at the national or local levels. These results contradict those of Malpezzi (1999), who foundthatonecan rejectthenullofnocointegrationinasimilarpanel. Malpezzi wasabletorejectthenullofaunitrootintheresidualsofcity-levelcointegrating regressions of prices and income using a Levin-Lin test. His interpretation was thatpricesandincomearethereforecointegrated. However,theLevin-Lintestis inappropriateinthiscontextbecausethe criticalvaluestabulatedbyLevinand Lin(1992)arenotadjustedforthe factthatMalpezzi’sresidualsareestimated. In other words, the Levin-Lin tests is a unit root test, not a residuals-based 13

cointegration test. While the two types of tests are clearly related, the Levin- Lin test is only applicable to the case of cointegration when the researcher imposes the cointegrating relationship. In addition, Malpezzi did not correct for the fact that the Levin-Lin test requires independence across local housing markets. If house prices and fundamentals are not cointegrated, how should we interpret the error-correction type models of Abraham and Hendershott (1996), Malpezzi (1999), and Capozza et al. (2002)? All three (cid:12)nd that house prices increasemore slowly when actualhouse prices are abovea measureof the longrun equilibrium price level, and all three base their equilibrium measure on a (cid:12)rst-stagelevelsregression. Astrictinterpretationofthe resultsinthispaperis thatanerror-correctionmodelis amis-speci(cid:12)cation,andthatresultsfromsuch amodelarespurious. Alooserinterpretationis thatevenifprices,income,and population are cointegrated, we cannot verify this relationship. Our inability to verifythe relationshipimplies aninability to accuratelyestimate it. Inother words,evenifwe think a long-runrelationshipexists,we cannotsaywithmuch certainty when prices are in line with fundamentals and when they are not. Forecasts based on our best guess as to the degree of \disequilibrium," such as those from an error-correctionmodel, are therefore highly suspect. 5 Conclusion Many housing market observers have become concerned that house prices have growntooquicklyoflate,andthatpricesarenowtoohighrelativetopercapita incomes. Prices will likely stagnate or fall, the argument goes, until they are better aligned with income. This idea is often formalized in the housing literature by asserting a long-run equilibrium relationship between house prices and fundamentals such as income, population, and user cost. The validity of this assumption has important implications for how we model house price dynamics. If the assumption is accurate|so that house prices and fundamentals are cointegrated|thenthe error-correctionspeci(cid:12)cations of Abraham and Hendershott (1996), Malpezzi (1999), and Capozza et al. (2002) are appropriate. In this paper I have usedstandardtests to show that there is little evidence forcointegrationofhousepricesandvariousfundamentalsatthe nationallevel. Ihavealsoshownthatbootstrappedversionsofmorepowerfulpanel-datatests, applied to a panel of 95 U.S. metropolitan areas over 23 years, also do not (cid:12)nd evidencefor cointegration. This does notmeanthat fundamentalsdo nota(cid:11)ect house prices,but it does mean thatthe level ofhouse prices does not appearto 14

be tied to the level of fundamentals. Thus, the levels regressions found in the literaturearelikelyspurious,andtheassociatederror-correctionmodelsmaybe inappropriate. 15

A Statistical Appendix This appendix provides de(cid:12)nitions of the panel cointegration tests of Pedroni (1999). Suppose that the hypothesized cointegrating relationship for each city is given by XM x 0;i;t =(cid:11)i+(cid:14)it+ (cid:12)m;ixm;i;t+eit; (4) m=1 Pedroni suggests the following test statistics: ! XN XT −1 P−v = T2N 3 2 L^− 11 2 i e^2 i;t−1 i=1 t=1 ! p XN XT −1XN XT P−Z(cid:11) = T N L^− 11 2 i e^2 i;t−1 L^− 11 2 i (e^i;t−1 (cid:1)e^it −(cid:21)^ i) i=1 t=1 i=1 t=1 ! −1 XN XT 2 XN XT P−AEG = s~ (cid:3)2 L^−2e^ (cid:3)2 L^−2e^ (cid:3) (cid:1)e^ (cid:3) N;T 11i i;t−1 11i i;t it i=1 t=1 i=1 t=1 where (cid:18) (cid:19) 1 Xki j XT (cid:21)^ i = 1− (cid:22)^i;t(cid:22)^i;t−j T ki+1 j=1 t=j+1 XT 1 s^2 = (cid:22)^2 i i;t T t=1 XT 1 s^ (cid:3)2 = (cid:22)^ (cid:3)2 i i;t T t=1 XN 1 s~ (cid:3)2 = s^ (cid:3)2 N;T i N i=1 (cid:18) (cid:19) 1 XT 2 Xki j XT L^− 11 2 i = T (cid:17)^ i 2 ;t + T 1− ki+1 (cid:17)^i;t(cid:17)^i;t−j t=1 j=1 t=j+1 andwherethe residuals(cid:22)^i;t,(cid:22)^(cid:3) i;t ,and(cid:17)^i;t arefromthefollowingregressions: e^i;t = γ^ie^i;t−1 +(cid:22)^i;t XKi (cid:3) e^i;t = γ^ie^i;t−1 + γ^i;k(cid:1)e^i;t−k+u^ i;t k=1 XM (cid:1)yi;t = ^bmi(cid:1)xmi;t+(cid:17)^i;t: m=1 16

B Data Appendix Metropolitan StatisticalAreas 1978to2000 Akron,OH Minneapolis-St. Paul,MN-WI Albuquerque,NM Modesto,CA AnnArbor,MI Monmouth-Ocean,NJ Atlanta,GA Nassau-Su(cid:11)olk,NY Austin-SanMarcos,TX NewOrleans,LA Bakers(cid:12)eld,CA NewYork,NY Baltimore,MD Newark,NJ BatonRouge,LA Norfolk-VirginiaBeach-NewportNews,VA-NC Bergen-Passaic,NJ Oakland,CA Birmingham,AL OklahomaCity,OK Boston,MA-NH Omaha,NE-IA Boulder-Longmont,CO OrangeCounty,CA Bu(cid:11)alo-NiagaraFalls,NY Orlando,FL Canton-Massillon,OH Philadelphia,PA-NJ Charlotte-Gastonia-RockHill,NC-SC Phoenix-Mesa,AZ Chicago,IL Pittsburgh,PA Cincinnati,OH-KY-IN Portland-Vancouver,OR-WA Cleveland-Lorain-Elyria,OH Raleigh-Durham-ChapelHill,NC Columbus,OH Richmond-Petersburg,VA Dallas,TX Riverside-SanBernardino,CA Dayton-Spring(cid:12)eld,OH Rockford,IL Denver,CO Sacramento,CA DesMoines,IA St. Louis,MO-IL Detroit,MI Salinas,CA Eugene-Spring(cid:12)eld,OR SaltLakeCity-Ogden,UT Flint,MI SanDiego,CA FortCollins-Loveland,CO SanFrancisco,CA FortLauderdale,FL SanJose,CA FortWayne,IN SanLuisObispo-Atascadero-PasoRobles,CA FortWorth-Arlington,TX SantaBarbara-SantaMaria-Lompoc,CA Fresno,CA SantaCruz-Watsonville,CA GrandRapids-Muskegon-Holland,MI SantaRosa,CA Greensboro-Winston-Salem-HighPoint,NC Sarasota-Bradenton,FL Hamilton-Middletown,OH Seattle-Bellevue-Everett,WA Honolulu,HI Spokane,WA Houston,TX Stockton-Lodi,CA Indianapolis,IN Syracuse,NY Kalamazoo-BattleCreek,MI Tacoma,WA KansasCity,MO-KS Tampa-St. Petersburg-Clearwater,FL Lansing-EastLansing,MI Tucson,AZ LasVegas,NV-AZ Tulsa,OK LosAngeles-LongBeach,CA Vallejo-Fair(cid:12)eld-Napa,CA Louisville,KY-IN Ventura,CA Madison,WI Visalia-Tulare-Porterville,CA Memphis,TN-AR-MS Washington,DC-MD-VA-WV Miami,FL WestPalmBeach-BocaRaton,FL Middlesex-Somerset-Hunterdon,NJ Wichita,KS Milwaukee-Waukesha,WI 17

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