The Long-Run Relationship between House Prices and Rents

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. The Long-Run Relationship between House Prices and Rents Joshua Gallin 2004-50 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

The Long-Run Relationship between House Prices and Rents Joshua Gallin, Federal Reserve Board (cid:3) September 2004 Abstract Ishowthatwhenhousepricesarehighrelativetorents(thatis,when the rent-price ratio is low) changes in real rents tend to be larger than usual and changes in real prices tend to besmaller than usual. Standard error-correction models provide inconclusive results about the predictive poweroftherent-priceratioataquarterlyfrequency. Iusealong-horizon regressionapproachtoshowthattherent-priceratiohelpspredictchanges inrealrentsandrealpricesoverthree-yearperiods. Thisresultwithstands theinclusion of ameasure of theuser cost of capital. I show that a longhorizon regression approach can yield biased estimates of the degree of errorcorrectionifpriceshaveaunitrootbutdonotfollowarandomwalk. Iconstructbootstrapdistributionstoconductappropriateinferenceinthe presenceofthisbias. Theresultslendempirical supporttotheviewthat the rent-priceratio is an indicator of valuation in thehousing market. (cid:3)Thanks to Amy Crews-Cutts, Douglas W. Elmendorf, Gregg Forte, Norman Morin, Stephen D. Oliner, Jeremy Rudd, Charles S. Struckmeyer, and William L. Wascher. The views presented are solely those of the author and do not represent those of the Federal Reserve Board or its sta(cid:11). Please do not cite without the author’s permission. 1

1 Introduction Nominalhouse prices in the United States haverisenby about70 percentsince 1994. Over the same period, the indexes for tenants’ and owners’ equivalent rent in the consumer price index have increased less than half as much.1 The resulting high level of house prices relative to rents has raised concerns that housing is overvalued. Such concerns are based on the idea that rents are a fundamental determinant of the value of housing and as such should not move too far out of line withprices. Theanalogytothe stockmarketisstraightforward: Therent-price ratio in the housing market is like the dividend-price ratio in the stock market (Leamer, 2002). Campbell and Shiller (2001) showed that when stock prices have been high relative to dividends, future price growth for stocks has been subdued. One might reasonably expect the analogous statement to be true for the housing market. I examinedthe time-series relationshipbetweenhouse prices andrents from 1970:Q1 to 2003:Q4, using both standard error-correction models and longhorizon regression models to examine how well the rent-price ratio predicts futurechangesinrealrentsandprices;therent-priceratiomusthavepredictive power for house prices for it to be a useful measure of valuation in the housing market. Although the results from a standard error-correction model suggests that rents and prices correct back toward each other, the point estimates are imprecisely estimated, and the results are therefore inconclusive. My long-horizon approach is quite similar to the one Campbell and Shiller (2001) used to study how well the dividend-price ratio helps predict changes in stock prices and dividends and to the one Mark (1995) used to study exchange rates. I show that if prices follow a general unit root as opposed to a strict random walk (with or without drift), long-horizon regression coe(cid:14)cients will yield biased estimates of the degree of error-correction;Mark’s (1995) and Campbell’s and Shiller’s (2001) approaches do not allow for this possibility. I use a bootstrap approach to adjust for the bias induced by serially correlated shocks to price changes. 1Exceptasnoted,IuseFreddieMac’sConventionalMortgageHousePriceindextomeasure houseprices,thetenants’rentseriesfromtheconsumerpriceindextomeasurerents,andthe price deflator for personal consumption expenditures excluding food and energy to deflate nominalvalues. 2

My main(cid:12)ndings are thatperiods inwhich houseprices are highrelativeto rents appear to be followed by periods in which real rent growth is faster than usual, and real house-price growth is slower than usual, and that the response of prices dominates that of rents. I show that it canbe di(cid:14)cult to compare the long-horizon results to those from a standard error-correction model because the signs and magnitudes of the estimated coe(cid:14)cients on the rent-price ratio in a long-horizon regression can yield biased estimates of the degree of errorcorrection. I use a bootstrap approachto correctfor such biases and show that we can reject the null hypothesis that rents \do all the correcting." In other words, the view that a low rent-price ratio indicates that house prices could be too high appears to have some empirical basis. Including a measure of the user cost of housing capital does not alter the result. 2 A brief review of the theory and existing literature Inthestandardtextbookmodelofhousepricesandrentsinafrictionlessmarket, rent should cover the user cost of housing: R t =P t[(i t+(cid:28) t p )(1−(cid:28) t y )+(cid:14) t+(cid:21) t −E t G t+1 ]; (1) where i t is the real interest rate, (cid:28) t p is the property tax rate, (cid:28) t y is the marginal incometaxrate,(cid:14) t isthecombinedmaintenanceanddepreciationrate,(cid:21) t isthe risk premium associated with housing, and E t G t+1 is expected capital gains.2 Equation (1) yields the standard result that in a frictionless market, prices should be high relative to rents when, among other things, interest rates are low and expected capital gains are high. I de(cid:12)ne C t as the direct user cost of housing capital, C t =(i t+(cid:28) t p )(1−(cid:28) t y )+(cid:14) t : (2) That is, C t is the cost of housing excluding the risk premium and expected capital gains. With a log-linearizationsimilar to that used in the dividend ratio model for the stock market (Campbell, Lo, and MacKinlay, 1997),expected capital gains 2I ignore transaction costs; I assume that all local property taxes and interest payments aredeductiblefromfederaltaxesandthathouses arefully(cid:12)nanced. 3

can be approximated by X1 E t G t+1 (cid:25)E t (cid:26)j [(1−(cid:26))(cid:1)R t+1+j −C t+1+j −(cid:21) t+1+j]; (3) j=0 where (cid:26) depends on the levels of rents and prices around which the approximation is taken.3 Thus, prices should be high relative to rents when, among other things, expected future interest rates are low and expected future changes in rents are high. Only a handful of papers deal directly with the question of how much the rent-price ratio helps predict future changes in rents and prices. Capozza and Seguin (1996) used decennial census data to examine how cross-sectional differences in the rent-price ratio among metropolitan areas in the United States are related to ten-year changes in prices in those areas. They tested whether the expected capital gains implicitly needed to support an area’s rent-price ratio were closely related to actual capital gains. For each metropolitan area, CapozzaandSeguintriedtocontrolforthefactthatrentalandowner-occupied housingcandi(cid:11)erinqualitybyusingdataonhousingcharacteristics. Theyalso decomposedtherent-priceratiointoacomponentexplainedbylocalconditions and an unexplained residual. They found that the predictable part of the rentpriceratiowasnegativelyrelatedto subsequentpricechanges. Thatis,cities in which prices were high relative to rents for reasons associated with local conditions typically saw their relatively high prices justi(cid:12)ed by higher capital gains. They also found that the unpredictable part of the rent-price ratio, which they calledthe disequilibrium component, waspositively relatedto subsequentprice changes. That is, cities in which prices were high relative to rents for reasons not associated with local conditions had smaller realized capital gains. Clark(1995)usedanapproachsimilartothatofCapozzaandSeguintotest whether the rent-price ratio helped predict future changes in rents. He found that the rent-price ratio is signi(cid:12)cantly and negatively related to subsequent changesin rents. That is, pricesappear to be higher in areasthat subsequently have larger increases in rents. These cross-sectional studies provide useful insights into the long-run predictive power of the rent-price ratio. However, the approach is less useful for examiningtherelationshipbetweenrentsandhousepricesatahigherfrequency. 3SeeCampbell,Lo,andMacKinlay(1997)forthederivations. 4

MeeseandWallace(1994)usedtime-seriesdataonprices,rents,andthecostof capitalforAlamedaandSanFranciscocountiestoshowthatpricesandrentsare cointegrated. However,they didnotexaminehowpricesandrentsadjustinthe short run to reestablish the long-run equilibrium implied by the cointegrating relationship. In related research, Blackley and Follain (1996) examined the link between rentsandusercost. Theyfoundthatincreasesinusercostarenotfullymatched by increases in rents and that rents adjust very slowly. However, they did not examine the predictive power of the rent-price ratio. Mankiw and Weil (1989) touched on the forecasting value of the rent-price ratio in a time-series setting, but they had a very short time series available to them. They found that the relationship between the rent-price ratio and future price growth was not statistically signi(cid:12)cant. Case and Shiller (1989) used high-frequency price and rent data to construct estimates of the return on housing but did not examine whether the rent-price ratio helps forecast future changes in rents and prices. 3 The data De(cid:12)nitions of the variables and measurement issues Four high-frequency measures of house prices are available for national-level studies of the housing market: (1) the new home price series (Census Bureau), (2) the existing home price series (National Association of Realtors), (3) the quality-adjusted price index for new homes sold (Census Bureau), and (4) the ConventionalMortgage House Price Index (Freddie Mac).4 The series for new and existing home prices are reported each month, but theyarenotadjustedforthetypesofhomessoldandthereforecannotaccurately measure the price of a home separately from its quality. The quality-adjusted price of new homes sold is based on hedonic regressionsthat include characteristics such as region of the country, whether the home is inside a metropolitan area, and the number of bedrooms (Census, 2004). However, the regressions do not include a measure of the location of the home within its fairly broad geographic designation. Because new homes are typically built on relatively cheapland,thepriceindexcannotaccuratelyreflectthelandpricesrelevantfor 4TheO(cid:14)ceofFederalHousingEnterpriseOversightpublishesahouse-priceindexsimilar tothatofFreddieMac. Bothindexesarebasedonaweightedrepeat-salesmethod. 5

the existing stock of housing.5 Davis and Heathcote (2004) showed that land’s share of the house price is quite large. In this paper I use the ConventionalMortgageHouse Price Index (CMHPI) published by Freddie Mac. The index is based on price changes for homes that are resold or re(cid:12)nanced and is therefore not a(cid:11)ected by changes in the composition of homes sold. In addition, the CMHPI sample excludes homes with jumbo, FHA, or VA mortgages. According to several researchers, the repeat-sales methodology used in the CMHPI yields estimates of house-price growth that are upward biased because homesthatchangehandsmorefrequentlytendtohavegreaterpriceappreciation (Gatzla(cid:11) and Haurin, 1997; Case, Pollakowski, and Wachter, 1997). Gatzla(cid:11) and Haurin show that the repeat-sales methodology created an upward bias of 0:33 percentage point per year using micro data for the Miami metropolitan area from 1971 to 1995. In related research, Dreiman and Pennington-Cross (2004) show that the standard methods for constructing a weighted repeat-sales price index are too restrictive. They show that the variance of price changes for individual properties are di(cid:11)erent for those properties that are either above or below the mean rateofappreciationandforpropertiesthatarein di(cid:11)erentprice tiers. Dreiman and Pennington-Cross’s more flexible speci(cid:12)cations yielded price indexes with signi(cid:12)cantly smaller averageannualincreases than those from the standard approach;the biasesrangedfrom0.1percentagepointto 0.6percentagepointper year.6 AlthoughIdonotknowhowmuchthetransactionsbiasesofGatzla(cid:11)and Haurin(1997)andthevariancebiasesofDreimanandPennington-Cross(2004) overlap, the e(cid:11)ects would seem unlikely to cancel each other. Still, I chose to be conservative and reduce the growth rate of the CMHPI only 0:3 percentage points per year.7 5For example, consider a monocentric city with (cid:12)xed agricultural land prices, (cid:12)xed constructioncosts,andagrowingpopulation(asetupalongthelinesofthatinHenderson(1977)). Theaverage priceof allhomes willincreasesteadily withpopulation even though each wave ofnewhomessellsforaconstantpriceequaltothesumofthepricesforagriculturallandand thestructure. 6They calculated alternative indexes for California, Georgia, Florida, Illinois, Kansas, North Carolina, Nevada, New York, Texas, and Washington. Anthony Pennington-Cross generouslyprovidedthedataforthealternativepriceindexes forthesetenstates. 7TheCMHPIignoresthee(cid:11)ectofimprovementstoanddeteriorationofthesampleshouses. DatafromtheBureauofEconomicAnalysis(BEA)onimprovementstoandphysicaldepreci- 6

My source for rent data is the index for tenants’ rent from the Consumer PriceIndex(CPI).Onecouldarguethattheowners’equivalentrentseriesfrom the CPI is preferable because it is a measure of the rent that owners implicitly pay to themselves. As such, it is closer to housing \dividends" for owners. In contrast,tenants’rentmeasuresrentspaidbyrenters. Becauserentalunitsdi(cid:11)er fromowner-occupiedunits,theymaynotaccuratelyreflecttruealternativesfor owners.8 The tenants’ rent series has one crucial advantage: It is available for a much longer time series. Owners’ equivalent rent is available from only 1983; the tenants’ rent series begins well before the 1970:Q1 starting date for the house price data. The additional 13 years of data are vital for estimating the time-series relationship between prices and rents. I adjusted the published rent data in two closely related ways. First, I boostedthegrowthrateoftheindex0.3percentagepointperyearpriorto1988 totrytomatchtheadjustmentthattheBureauofLaborStatistics(BLS)made to the published series beginning in 1988 to better reflect the aging of rental units (Moulton,1997;Crone,Nakamura,and Voith, 2000and 2004;Lebowand Rudd, 2003). Second, I increased the growth rate of the entire series an additional 0.2 percentage point per year during the entire sample period because several researchers have argued that BLS’s age adjustment still does not adequatelyadjustforaginge(cid:11)ects (Crone,Nakamura,andVoith,2004;Lebowand Rudd, 2003). I constructed an estimate of the direct user cost of housing capital on the basis of Equation (2). I used the 30-year (cid:12)xed-rate mortgage rate relative to inflation expectations from the Philadelphia Federal Reserve Bank Survey, the property tax rate and marginal income tax rate (at twice the median income) usedintheFRB/USmodelattheFederalReserveBoard(Reifschneider,Tetlow, andWilliams, 1999),andthe rateofdepreciationonresidentialstructuresfrom the National Income and Product Accounts.9 ationofowner-occupiedhomeswithonetofourunitsindicatethatimprovementso(cid:11)setmost ofthee(cid:11)ectofdepreciation. Inaddition,becausetheBEAdoesnotmeasuretheimplicitlabor costofdo-it-yourselfhomeimprovements,thepublisheddatalikelyunderstatetheactualpace ofimprovements. 8Forexample,the2001AmericanHousingSurveyshowedthattheproportionofdwellings that were detached single-family homes was 82 percent for owner-occupied homes but only 23percentforrentalhomes. 9Thedepreciationrateonstructureslikelyoverstatestherelevantdepreciationratebecause land,whichisincludedinthepriceofahouse,probablydoesnotdepreciate. 7

Rents, prices, and the direct user cost of housing capital From 1976:Q1 to 1979:Q4, real house prices relative to the deflator for personalconsumption expenditures excluding food and energy (core PCE) rose an average of about 4-1/4 percent per year for a total gain of about 17 percent (Figure 1,toppanel). Realhouseprices thenwentthroughasimilar cyclefrom the mid-1980s through the mid-1990s before beginning a spectacular decade: From the end of 1994 to the end of 2003, they increased 45 percent. Over the entire period, realhouse prices increaseda total of63 percent, or 1-1/2 percent per year. Rents relative to the core PCE deflator have been less volatile than house prices(Figure1,bottompanel). Theyfellsharplyin1974beforestartingalong andfairlysteadyrisethatacceleratedinthe mid-1980s;bytheendof1986,real rentswereup12percentfromtheirtrough. Realrentsthenstagnatedforabout 10yearsbefore risingabout18percentfromthe endof1993to the endof2003. The log ratio of rents to prices shows a distinct trough in the late 1970s and a smaller one in the late 1980s (Figure 2, top panel).10 More recently, as house prices reached record highs relative to rents, the rent-price ratio reached a record low. Simple augmented Dickey-Fuller tests (which are available from the author upon request) show that while we cannot reject the null hypothesis that prices and rents each have a unit root, we can reject that null for the rent-price ratio. In other words, prices and rents are cointegrated. The direct cost of housing declined signi(cid:12)cantly from early 2000 through the end of 2003 to a level below the readings of most of the 1980s and 1990s (Figure 2, bottom panel). Even so, the cost was lower still in the late 1970s, when higher inflation and income tax rates helped keep the direct user cost of housing low even though nominal mortgage rates were high. Long-horizon relationships A scatterplot of the data from 1970 through 2003 with a (cid:12)tted regression line shows that when prices have been high relative to rents (the rent-price ratio is low) rent increases during the subsequent three years have tended to be large, 10Because prices and rents areboth measured as indexes, the absolute level of theratio is meaningless. Theaveragelevelofthelogindexis97:8. 8

and that when house prices have been low relative to rents, subsequent rent increaseshavetendedtobesmall(Figure3,toppanel). Thisresultisconsistent both with the theory and with Clark’s (1995) results: Prices at least partially capitalize the present value of future rents, and relatively high prices therefore signal larger increases in rents. A similar scatterplot (Figure 3, bottom panel) indicates that when prices are high relative to rents, subsequent price increases are small. This result appearsto be consistentwith the view takenby Leamer(2001)andothers that a low rent-price ratio is a sign of overvaluation in the housing market that is subsequently,ifslowly,eliminated. However,itdoesnotappeartobeconsistent withthe theorythathighpricesaretypicallysupportedby highcapitalgains| the required capital gains do not seem to materialize. The scatterplots providesuggestiveevidence that when prices are highrelative to rents, the two series move towardeach other. However,the calculations on which this simple relationship is based ignore two potentially important econometric issues. First, the bivariate scatterplots do not include the e(cid:11)ect of changes in the direct user cost of housing; the rent-price ratio may have little predictive power after such costs are included. Second, the quarterly observations of long-horizon di(cid:11)erences are not independent of each other. In the following sections I address these shortcomings by using error-correction and long-horizonmodels to examine the predictive power of the rent-price ratio. 4 An error-correction model of house prices and rents Error-correctionmodels provide a simple way to examine the predictive power of the rent-price ratio. Let y t =(logR t logP t)0. The model is (cid:1)y t =A 0 (L)(cid:1)y t−1 +A 1 y t−1 +A 2 (L)x t−1 +(cid:17) t ; (4) where x t−1 includes other variables that can a(cid:11)ect (cid:1)y t. Because rents and prices are cointegrated, we can include their levels in the regression.11 I restricted x t−1 to include levels and changes of the direct user cost of housing. I included the level of the direct cost because Equation (1) suggests 11Thematrixofcoe(cid:14)cients,A 1,canbethoughtofastheproductofthecointegratingvector andamatrixoferror-correctioncoe(cid:14)cients. 9

that the levels of prices, rents, and the direct user cost should be related. I calculatedordinaryleastsquaresestimates of twoversionsofEquation(4) using quarterly data from 1970:Q1 to 2003:Q4 and from 1970:Q1 to 2001:Q4. The (cid:12)rstversionincluded both the log rent-price ratio and the log of the direct costofcapitalandthe secondversionincluded onlythe logrent-priceratio. All the models include four lags of the changes in rents, prices, and the direct user cost,allinrealterms.12 Forthe estimatesbasedonthe entire availablesample, the signs of all the coe(cid:14)cients are consistent with Figure 3 (Table 1, (cid:12)rst two columns). Inparticular,thenegativecoe(cid:14)cientonthelaggedrent-priceratioin the rentequationandthe positivecoe(cid:14)cientonthatratiointhe price equation imply the convergence of the rent and house price series. Because the absolute value of the coe(cid:14)cient onthe laggedrent-price ratio is much largerin the price equation than it is in the rent equation, prices apparently correct more than rents do. In addition, the results suggest that when the level of the direct user cost is high, subsequent rent growth is large and subsequent price growth is small. Unfortunately,allthepointestimatesareimpreciselyestimatedandnotstatistically signi(cid:12)cant at conventional levels. The data apparently do not permit identi(cid:12)cation of the nature of the error-correction process that maintains the long-run equilibrium implicit in the cointegrating relationship. Ialsoestimatedthemodelwithatimeperiodthatexcludesthelasttwoyears ofthesample(Table1,lasttwocolumns). Pricegainsduringthesetwoomitted yearsfar outstripped rent gains and drovethe rent-price ratio downfrom levels that were already very low by historical standards (Figure 2, top panel). The resultsfromtheshorterperiodtellaqualitativelysimilarstory: Rentsandprices bothappeartodosomecorrecting,butpricesappeartoadjustmorethanrents. Theestimatedcoe(cid:14)cientonthelaggedrent-priceratioislargerwhenestimated on the shorter period, and is statistically signi(cid:12)cant in the price equation. 12IchosefourlagsbasedontheSchwartzcriterion. Theresultsarenotsensitivetomodest changes inthenumberoflags. 10

Table 1 Error-CorrectionModels of Housing Prices 1970:Q1 to 2003:Q4 1970:Q1 to 2001:Q4 rent price rent price model model model model Model 1 lagged rent-price ratio −:028 :093 −:031 :158(cid:3) (:023) (:066) (:026) (:070) lagged direct user cost :005(cid:3) −:005 :004(cid:3) −:005 (:002) (:006) (:002) (:005) Model 2 lagged rent-price ratio −:013 :079 −:015 :142(cid:3) (:025) (:063) (:026) (:067) Notes: Standarderrorsareinparentheses. (cid:3) indicatesasigni(cid:12)cancelevelof:05. Allthemodels includefourlagsofthechangesinrents,prices,andthedirectusercost,allinrealterms. Coe(cid:14)cientsareexpressedatanannualrate. It is not surprising that excluding data from two years in which the rentpriceratiowasverylowandpricegainsfaroutstrippedrentgainsyieldedlarger estimatedcoe(cid:14)cients in the error-correctionmodel. It is more di(cid:14)cult to know what to conclude. By extending the estimation to 2003:Q4, we are perhaps catching the long downturn in the rent-price ratio without catching the inevitable upturn. The smallsample size maytherefore make it impossible to get a precise estimate of the rate of error correction. Of course, the upturn may not occur|after all, the continuing decline in the rent-price ratio casts doubt on the premise that the ratio (as measured) is a useful predictor of prices. Thus, although the error-correction model provides some support for the viewthatalowrent-priceratioisasignofovervaluationinthehousingmarket, theresultsarefarfromconvincing. Inthefullsample,rentsandpricesappearto becointegratedbutthe error-correctiontermsarenotstatisticallysigni(cid:12)cantin either equation;this combinationsuggeststhatusinganerror-correctionmodel at a quarterly frequency may be \asking" too much of the data. 11

5 A long-horizon model of house prices and rents Method My examination of the rent-price ratio as a predictor of changes in rents and prices at horizons longer than one quarter closely follows that of Campbell and Shiller’s (2001) examination of how well the dividend-price ratio predicts stock dividends and prices and Mark’s (1995) examination of how well a nation’s moneystockanddomesticincomepredictexchangerates.13 LikeCampbelland Shiller (2001),and, indeed, like Capozzaand Seguin (1996)and Clark(1995),I donotconductastricttestofthepresent-valuemodelinEquations(1)and(3). Rather, these equations serve as the motivation for the common use of the rent-price ratio as a measure of valuation in the housing market. The following equations form the heart of my long-horizon empirical strategy: r t+12 −r t = a 0 +a 1 (r t −p t)+a 2 c t+u t (5) p t+12 −p t = b 0 +b 1 (r t −p t)+b 2 c t+v t (6) where lower-caseletters denote log values and t indexes quarters,so thatt+12 indicates t plus three years. The coe(cid:14)cients a andb have,at(cid:12)rstglance,simple interpretations. They 1 1 show how changes in rents and prices over a three-year horizon are related to the rent-price ratio at the beginning of the three-year period (after controlling for the e(cid:11)ect of c t). It is tempting to employ the language commonly used to describe error-correctionmodels: one would say that the signs and magnitudes of a and b tell us whether, and by how much, rents \correct" to prices and 1 1 prices\correct"torents. However,trueerror-correctionmodelstypicallyinclude laggedvaluesofthedi(cid:11)erenceinthevariableofinterest. Inthiscase,onewould includemeasuresofr t+11 −r t−1 andp t+11 −p t−1 andtheirlagsontherightsideof theregressions. Along-horizonmodel,byde(cid:12)nition,cannotincludetheseterms becausetodosowoulde(cid:11)ectivelymakeitanerror-correctionmodel. Excluding these variables induces the statistical problems associated with autocorrelated residuals in models with lagged dependent variables. The following example illustrates this point. 13Hodrick(1992) andFamaandFrench(1988)areearlierantecedents. 12

Suppose that r t and p t are generated by the following processes: r t = p t+(cid:15) r;t (7) (cid:1)p t = (cid:11)(cid:1)p t−1 +(cid:15) p;t (8) where ! ! (cid:15) (cid:27)2 (cid:27) r;t =iidN 0; r rp : (9) (cid:15) (cid:27) (cid:27)2 p;t rp p Note that rents and prices are cointegrated by assumption (because (cid:15) r;t is stationary) but that rents do all the correcting. That is, the level of p t does not react to the level of r t, but the level of r t does react to the level of p t. Supposethatinsteadofthethree-yearhorizonusedinEquations(5)and(6), we were interested in a model of horizon s. Then Equations (7) and (8) imply that r t+s −r t = p t+s −p t+(cid:15) r;t+s −(cid:15) r;t (10) Xs X1 p t+s −p t = (cid:11)k(cid:15) p;t+j−k : (11) j=1k=0 Ordinaryleast squares estimationof the long-horizonmodels would asymptotically yield the following estimates: (cid:11)(1−(cid:11)s) a^ 1 = 1−(cid:11) (cid:27) rp −1 (12) (cid:11)(1−(cid:11)s) ^b 1 = 1−(cid:11) (cid:27) rp : (13) Thus,the nullhypothesisthat \rentsdo allthe correcting"does notimply that b 1 = 0. The term (cid:11)( 1 1 − − (cid:11) (cid:11)s)(cid:27) rp captures the fact that shocks to the rent-price ratio can be correlated with shocks to the change in log real prices. Any autocorrelation in the change in prices can therefore induce a correlation between the rent-price ratio and long-horizondi(cid:11)erences in prices. Several additional points merit mention. First, b will equal zero under the 1 null hypothesis if (cid:11) = 0. Thus the approaches of Mark (1995) and Campbell and Shiller (2001), which assume (cid:11) = 0, are valid if that assumption holds. Second,morecomplicatedprocessesforrentsandpricesyieldmorecomplicated bias terms. Third, because cov(p t+1 −p t ;r t −p t jp t −p t−1 )=cov((cid:15) p;t+1 ;(cid:15) r;t)=0; (14) error-correctionmodels do not su(cid:11)er from this problem. 13

I use a bootstrap approach to correct for the biases inherent in the longhorizonapproach. MyapproachfollowscloselythatofCampbellandShiller(2001) and Mark (1995). I built small- and large-samplebootstrap distributions using restrictedautoregressionsthat, by construction,matchthe nullhypothesis that rents do all the correcting. Let z t = r t −p t. The bootstrap distributions are based on X4 (cid:1)z t = γ 0 +γ 1 z t−1 + γ 2;j(cid:1)z t−j +(cid:15) r;t (15) j=1 XJ (cid:1)p t = (cid:11) 0 + (cid:11) 1;j(cid:1)p t−j +(cid:15) p;t : (16) j=0 Equation (15) ensures that rents and prices are cointegrated; Equation (16) ensures that prices have a unit root but that the level of rents does not a(cid:11)ect the level of prices; rents do all the correcting.14 IestimatedEquation(15)usingfourlagsof(cid:1)z t−1−j. Ibasedthe laglength on the Schwartz criterion. I found that γ^ = −:05 with an Augmented Dickey 1 Fuller test statistic of 3:45,largeenoughto reject the hypothesis of a unit root; indeed, this is the equation to which I referred in Section 3 to establish the stationarity of the rent-price ratio. The four versions of Equation (16) that I estimatedweredeterminedbyletting J equal0,1,4,and8quarters(Appendix Table A.1). Let(cid:15)^t =((cid:15)^r;t ;(cid:15)^p;t)0 andγ^ 0 ; γ^ 1 ; γ^ 2;j ; (cid:11)^ 0 ; (cid:11)^ 1;j betheestimatedresidualsand coe(cid:14)cients. I constructed the large-sample bootstrap distribution of a^ and^b 1 1 by running 10;000 replications of the following procedure, indexed by i: 1. Let T =4;000. Draw with replacement T +100 values from (cid:15)^t; call them f(cid:15)igT+100. t t=1 2. With sample means for initial values, generate sequences of arti(cid:12)cial observations using X4 (cid:1)z t i = γ^ 0 +γ^ 1 z t−1 + γ^ 2;j(cid:1)z t−j +(cid:15)i r;t (17) j=1 XJ (cid:1)pi t = (cid:11)^ 0 (cid:11)^ 1;j(cid:1)p t−j +(cid:15)i p;t : (18) j=0 14Asanalternative,onecouldimposearestrictionconsistentwiththenullhypothesisthat prices do all the correcting. However, because shocks to the rent-price ratio are not highly correlatedwithshockstorents,thebiasunderthisnullissmall. 14

3. Construct the levels of pi from Equation (18) using the sample mean as t the initial value, and the levels of ri from zi+pi. t t t 4. Dropthe(cid:12)rst100observations,constructpi −pi,ri −ri,andregress t+12 t t+12 t them on zi. Keep the estimated coe(cid:14)cients a^i and^bi. t 1 1 The mean values of a^i and ^bi provide estimates of the values of a and b 1 1 1 1 under the null hypothesis implied by Equations (17) and (18). Call them a0 1 and b0. 1 I conducted a similar procedure to generate small-sample distributions for a^ and^b inwhichIsetT =124tomatchmyactualsamplesize. Using100;000 1 1 replications of the above procedure, indexed by k, I collected the small-sample t-statistics de(cid:12)ned as t(a^ ) = (a^ k−a0)=(cid:27)^ k (19) 1 1 1 a1 t(^b ) = (^bk−b0)=(cid:27)^ k : (20) 1 1 1 b1 Results The results from regressions of the three-year-ahead changes in real rents and prices on the log rent-price ratio and on the log of the direct user cost of housing capital are qualitatively similar to those from the error-correction models (Table 2). The resultsimplythatforeach10percentagepointdi(cid:11)erencebetweenrents andprices,thechangeinrealrentsis0:61percentagepointlessperyear,andthe changeinrealpricesis onaverage1:69percentagepoints more peryear,during the subsequent three years (Table 2, columns 1 and 3, which correspond to the regression lines in Figure 3). Thus, periods in which prices are high relative to rents are typically followed by periods of relatively larger changes in rents and relatively smaller changes in prices and the e(cid:11)ect on prices is more than twice as large. Inclusion of the direct user cost of housing capital does not signi(cid:12)cantly a(cid:11)ectthe estimatedcoe(cid:14)cientsonthe rent-priceratioineither model(Table1, columns2and4). Thedirectusercostofhousingcapitalispositivelyrelatedto futurerentchangesandnegativelyrelatedtofuturepricechanges. Thusperiods in which the direct user cost of housing capital is higher are typically followed by periods in which the change in rents is larger and the change in prices is 15

Table 2 Long-HorizonModels of the Change in Rents and Prices (three-year-aheadchnages;1970:Q1to2003:Q4) Rent Model Price Model 1 2 3 4 log rent-price ratio −:061 −:083 :169 :174 (:017) (:015) (:038) (:039) direct user cost | :030 | −0:007 (:004) (:011) R-squared :09 :33 :14 :14 Notes: OLSstandarderrorsareinparentheses. Coe(cid:14)cientsareexpressedatanannualrate. smaller. This result should not be surprising to those who think that rents and pricesadjustslowlytoshocks. However,moreimportantforpresentpurposesis the factthatincluding the levelofthe directuser costdoesnotappearto a(cid:11)ect the relationship between the rent-price ratio and subsequent changes in rents and prices. Given the insensitivity of the results to the inclusion of the direct cost of housing capital, I focus the bootstrap exercise on the simpler models shown in columns 1 and 3. Table 3 contains the results of the bootstrap exercise. The (cid:12)rst column of the table simply re-displays the coe(cid:14)cient estimates from columns 1 and 3 of Table 2. The remainingcolumns containthe values ofthe coe(cid:14)cients a andb 1 1 under the null hypotheses that rents and prices are cointegrated, that rents do all the correcting, and that the change in the log of real rents follows either a unit root or an autoregressive process with 1, 4, or 8 lags; the log of real rents is a random walk with drift if there are zero lags. I calculated these null values using the large-sample bootstrap described above. The table also displays the p-values of the tests that the estimated coe(cid:14)cients are di(cid:11)erent from the null produced by the large-sample bootstrap; the p-values are based on the smallsample bootstrap described above. The results of the large-sample bootstrap described above for the null hypothesis for a (the rent model) imply that for each 10 percentage point di(cid:11)er- 1 16

ence between rents and prices, the annualized change in real rents is 2:58 percentage points to 4:55 percentage points smaller during the subsequent three years (Table 3). Although the actual estimated value for a , −:061, indicates 1 that periods of relatively high prices are typically followed by periods in which the change in rents is relatively large, the e(cid:11)ect is quite small compared with what one would expect if rents did all the correcting. Indeed, regardless of the presumed time-series properties of prices, the small-sample bootstrapped p-values for a indicate that the estimated coe(cid:14)cient is in the far right tail of 1 the small-sample bootstrap distributions, suggesting that we should reject the null hypothesis that rents do all the correcting. Inthepricemodel,thenullhypothesisthatrentsdoallthecorrectingimplies a negative estimate of b as long as prices follow a unit rootrather that a strict 1 random walk. Recall from Equation (13) that the bias term depends on the covariance of the shocks to (cid:1)p t and z t and the time-series properties of (cid:1)p t. Theactualestimateforb ,:169,indicatesthatpricegrowthisslowerinyears 1 that follow periods of low rent-price ratios than one would expect if prices did notcorrecttorents. Forexample,underthenullhypothesisthatpricesfollowa randomwalk, one would expect b to equal zero. However,although the actual 1 estimate is above the null’s value, the small-sample p-value of :14 indicates that this event is not rare enough to reject the null hypothesis at conventional levels of signi(cid:12)cance. The assumption that changes in house prices follow an AR(1)processyieldsasimilarresults. However,underthemorereasonablenull hypothesis that changes in prices are persistent (with lags of 4 quarters or 8 quarters), the p-values are low enough to reject the null hypothesus that rents do all the correcting. Thus, the long-horizon regression results and bootstrap Monte Carlossuggestthat periods in which prices are high relativeto rents are typicallyfollowedbyperiodsinwhichchangesinrealrentsarelargerthanusual and changes in real prices are smaller than usual. 17

Table 3 Bootstrapped Signi(cid:12)cance Level Lags=0 Lags=1 Lags=4 Lags=8 coe(cid:11) null p-value null p-value null p-value null p-value Rent model a −:061 −:258 :01 −:261 :01 −:455 :01 −:390 :01 1 Price model b :169 0 :14 −:002 :14 −:196 :03 −:113 :05 1 Notes: Lags=0impliesthatpricesfollowarandomwalkwithdrift. 18

6 Summary TheevidenceIpresentedinthispapersuggeststhatwhenhousepricesarehigh relative to rents, subsequent changes in real rents are larger than usual and subsequent changes in real house prices are smaller than usual. The conclusion is based on the results from three related analyses in which I measured house pricesusingFreddie Mac’sConventionalMortgageHouse PriceIndex andrents using the CPI index for tenants’ rent. In the (cid:12)rst analysis, simple regessions of the data from the house-price and rentseriessuggestedthatrentsandpricestendtocorrectbacktoeachotherover three-year horizons. But the regressions do not include the e(cid:11)ects of changes in the direct user cost of housing nor do they account for the interdependence of quarterly observations over a long period. In the second analysis, I showed thatstandarderror-correctionmodelscorroboratedtheevidencefromtheinitial regressions, but were not de(cid:12)nitive: Although rents and prices appear to be cointegrated,andalthoughthepointestimatesfromtheerror-correctionmodels showthatrentsandpricesbothcorrecttowardeachother,noneofthecoe(cid:14)cient estimatesofthe speedofcorrectionwerestatisticallysigni(cid:12)cantwhenIusedall the available data from 1970:Q1 to 2003:Q4. The third analysis provided the most conclusive evidence that house prices correct back to rents. I used a bootstrap procedure to construct arti(cid:12)cial data that conform to the null hypothesis that rents and prices are cointegrated, but that rents do all the correcting. I then used these arti(cid:12)cial data with a long-horizon regressionmodel to examine how the rent-price ratio is related to changes to real rents and prices over three-year horizons. Under the null hypothesis, we would expect rents to \correct" much faster than they do in the actualdata. Inaddition,wewouldexpectto(cid:12)ndanegativecorrelationbetween the rent-price ratio and the change in real house prices instead of the positive correlation apparent in the data. These results provide evidence against the null that rents do all the correcting and that prices do none. Because a low rent-price ratiohas been a harbingerof sluggishprice growth since 1970, it seems reasonable to treat the rent-price ratio as a measure of valuation in the housing market. Indeed, one might be tempted to cite the currently low level of the rent-price ratio as a sign that we are in a houseprice\bubble." However,severalimportantcaveatsargueagainstsuchastrong 19

conclusion and in favor of further research. First, the data I used in this paper are imperfect. The greatest concern is thatneitherthe rentdatanorthehouse-pricedataaccuratelymeasurerentand price changes. In this paper, I used the latest results from the literature on the measurement of house prices and rent to adjust the published data. However, westillneedbetter measuresofhousepricesandrentstofully understandtheir relationship. Second, the motivating model in this paper, in which rents equal the textbook versionofuser cost,is too simple. For instance, the analysisin this paper essentially ignores potential transactions costs and the risks involvedin renting and owning (Sinai and Souleles, 2003). Third,eveniftherent-priceratiocanbethoughtofasameasureofvaluation in the housing market, we should not expect it to be a precise indicator of if, when, and by how much house prices will change direction. Asset price movementsarenotoriouslyhardto predict;the housingmarketis noexception. 20

Appendix Table A.1 Autoregressions for Changes in Real House Prices (1970:Q1to2000:Q4) Number of Lags 1 4 8 (cid:1)p t−1 0:204 0:167 0:414 (:091) (:081) (:096) (cid:1)p t−2 | 0:126 0:132 (:083) (:101) (cid:1)p t−3 | −0:059 0:101 (:079) (:092) (cid:1)p t−4 | 0:501 0:277 (:078) (:084) (cid:1)p t−5 | | −0:216 (:079) (cid:1)p t−6 | | −0:008 (:082) (cid:1)p t−7 | | −0:190 (:082) (cid:1)p t−8 | | 0:197 (:080) R-squared :156 :415 :505 Notes: OLSstandarderrorsareinparentheses. 21

References [1] American Housing Survery for the United States: 2001 U.S. Department of Commerce, 2001 [2] Blackley, Dixie M. and Follain, James R. (1996) \In Search of Empirical Evidence that Links Rent and User Cost." Regional Science and Urban Economics 26:409-431. [3] Calhoun, Charles, A. (1996). \OFHEO House Price Indexes: HPI Technical Description." O(cid:14)ce of Federal Housing Enterprise Oversight. http://www.ofheo.gov/house/download.html. [4] Campbell,Lo,andMacKinlay.(1997).Theeconometricsof(cid:12)nancialmarkets Princeton, N.J. : Princeton University Press, 1997. [5] Campbell,JohnY.andShiller,RobertJ.(1988).\StockPrices,Earnings, and Expected Dividends", Journal of Finance, 43, 661-676. [6] Campbell, John Y. and Shiller, Robert J. (2001). \Valuation Ratios and theLong-RunStockMarketOutlook: AnUpdate",NBERWorkingPaper W8221. [7] Capozza,Dennis R.andSeguin, PaulJ.(1996).\Expectations,e(cid:14)ciency, and euphoria in the housing market", Regional Science and Urban Economics 26, 369-385. [8] Case,BradfordandHenryO.PollakowskiandSusanM.Wachter.(1997). \Frequency of Transaction and House Price Modeling", Journal of Real Estate Economics, 14, 173-187. [9] Case,KarlE.;Shiller,RobertJ.(1989).\TheE(cid:14)ciencyofthe Marketfor Single-Family Homes", American Economic Review, 79, 125-37. [10] Clark, Todd E. (1995).\Rents and Prices of Housing Across Areas of the UnitedStates: ACross-SectionExaminationofthePresentValueModel," Regional Science and Urban Economics 25:237-247. [11] Crone,Theodore,Nakamura,Leonard,andVoith,Richard.(2000).\Measuring Housing Services Inflation," Journal of Economic and Social Measurement, 26:153-171. 22

[12] Crone, Theodore, Nakamura, Leonard, and Voith, Richard. (2004). \Hedonic Estimates of the Cost of Housing Services: Rental and Owner- Occupied Units," Presented at the International Conference on Index Number Theory and the Measurement of Prices and Productivity, June, 2004, Vancouver, British Columbia. [13] Davis, Morris and Jonathan Heathcote. (2004).\The Price and Quantity of Residential Land in the United States", Federal Reserve Board FEDS paper 2004-37 [14] Dreiman, Michelle H. and Pennington-Cross, Anthony. (2004). \Alternative Methods of Increasing the Precisionof Weighted Repeat Sales House PricesIndices."JournalofRealEstateFinanceandEconomics,28(4):299- 317. [15] Fama, Eugene F. and French, Kenneth R (1988). \Dividend Yields and Expected Stock Returns." Journal of Financial Economic, 22(1):3-25. [16] Gatzla(cid:11), Dean H. and Donald R. Haurin. (1997). \Sample Selection Bias and Repeat-Sales Index Estimates", Journal of Real Estate Economics, 14, 33-50. [17] Henderson,J.V.(1977)Economic Theory and the Cities NewYork: Academic Press, 1977. [18] Hodrick,RobertJ.(1992).\DividendYieldsandExpectedStockReturns: AlternativeProceduresforInferenceandMeasurement."review of Financial Studies, 5(3):357-386. [19] Leamer, Edward E. (2002). \Bubble Trouble? Your Home Has a P/E Ratio Too", UCLA Anderson Forecast, June 2002. [20] Mankiw, N. Gregory and David N. Weil. (1989). \The baby boom, the baby bust, and the housing market", Regional Science and Urban Economics 19, 235-258. [21] Mark, Nelson C. (1995). \Exchange Rates and Fundamentals: Evidence on Long-Horizon Predictability?", American Economic Review 85, 201- 218. 23

[22] Lebow,DavidandJeremyRudd.(2003).\MeasurementErrorintheConsumer Price Index: Where Do We Stand?", Journal of Economic Literature, 41, 159-201. [23] Reifschneider, D., Tetlow, R., Williams, J. (1999) \Aggregate disturbances, monetary policy, and the macroeconomy: The FRB/US perspective" Federal Reserve Bulletin, 85, Issue 1, 1-19. [24] Sinai, Todd and Nicholas S. Souleles. (2003). \Owner-Occupied Housing as a Hedge against Rent Risk", NBER working paper 9462. 24

Figure 1 Real House Prices and Rents (1970:Q1 to 2003:Q4) Real house prices log index, 1996:Q1 = 100 log index, 1996:Q1 = 100 1 40 140 Real house price 130 130 120 120 110 110 100 100 90 90 80 80 70 70 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 Source: House Price Index from Freddie Mac deflated by the core PCE deflator from the BEA. See text for a description of how I adjusted the data. Real rents log index, 1996:Q1 = 100 log index, 1996:Q1 = 100 1 40 140 Real tenants’ rent 130 130 120 120 110 110 100 100 90 90 80 80 70 70 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 Sources: Tenants rent index from the Consumer Price Index deflated by the core PCE deflator from the BEA. See text for a description of how I adjusted the data.

Figure 2 The log rent-price ratio and the direct user cost of housing capital (1970:Q1 to 2003:Q4) Rent-price ratio log index, 1996:Q1 = 100 log index, 1996:Q1 = 100 1 20 120 log rent-price ratio 110 110 100 100 90 90 80 80 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 Sources: Same as Figure 1. See text for a description of how I adjusted the data. User cost of capital Percent Percent 1 0 10 direct cost of housing capital 8 8 6 6 4 4 2 2 0 0 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 Sources: BEA, Freddie Mac, and Census. See text for details.

Figure 3 The Log Rent-Price Ratio and Subsequent Changes in Rents and Prices Three Years Ahead (1970:Q1 to 2003:Q4) Percent Percent 8 8 6 slope: .169 6 standard error: .038 4 4 2 2 0 0 -2 -2 -4 -4 85 90 95 100 105 110 etar launna ,sraey eerht tneuqesbus ,htworg ecirp laeR Percent Percent 8 8 6 slope: -.061 6 standard error: .017 4 4 2 2 0 0 -2 -2 -4 -4 85 90 95 100 105 110 Log Rent-Price Ratio etar launna ,sraey eerht tneuqesbus ,htworg tner laeR Log Rent-Price Ratio