A Trend and Variance Decomposition of the Rent-Price Ratio in Housing Markets

A Trend and Variance Decomposition of the Rent-Price Ratio in ∗ Housing Markets Sean D. Campbell, Morris A. Davis, Joshua Gallin, and Robert F. Martin Federal Reserve Board April, 2006 Abstract We use the dynamicGordon-growthmodeltodecomposethe rent-priceratioforowner-occupiedhousing in the U.S., four Census regions,and twenty-three metropolitan areas into three components: The expected presentvalue ofrealrentalgrowth,realinterestrates,andfuture housing premia. We usethese components to decomposethe trendandvariancein rent-priceratiosfor 1975-2005,for anearlysub-sample (1975-1996), and for the recent housing boom (1997-2005). We have three main findings. First, variation in expected future real rents accounts for a small share of variation in our sample rent-price ratios; variation in real interestratesandhousing premiaaccountfor mostofthe variability. Second, expectedfuture realratesand housing premia were so strongly negatively correlated prior to 1997 that changes to real interest rates did not affect the rent-priceratio. After 1997,rates and premia have been positively correlated,and the decline in the rent-price ratio that has occurred in almost every geographic area in our sample since 1997 reflects both declining realrates and declining premia. Third, we show that in the recenthousing boom, 65 percent of the decline in the aggregate rent-price ratio is due to a declining housing premium. Keywords: Rent-price ratio, house price, housing rents, interest rate. ∗We would like to thank Mike Gibson, Michael Palumbo, David Reifschneider, and Tom Tallarini for helpful suggestions. TheviewsexpressedinthispaperarethoseoftheauthorsandshouldnotbeattributedtotheBoardofGovernorsoftheFederal ReserveSystemorothermembersofitsstaff. Email: sean.d.campbell@frb.gov,modavis99@yahoo.com,joshua.h.gallin@frb.gov, robert.f.martin@frb.gov.

A Trend and Variance Decomposition of the Rent-Price Ratio in Housing Markets Abstract We use the dynamicGordon-growthmodeltodecomposethe rent-priceratioforowner-occupiedhousing in the U.S., four Census regions,and twenty-three metropolitan areas into three components: The expected presentvalue ofrealrentalgrowth,realinterestrates,andfuture housing premia. We usethese components to decomposethe trendandvariancein rent-priceratiosfor 1975-2005,for anearlysub-sample (1975-1996), and for the recent housing boom (1997-2005). We have three main findings. First, variation in expected future real rents accounts for a small share of variation in our sample rent-price ratios; variation in real interestratesandhousing premiaaccountfor mostofthe variability. Second, expectedfuture realratesand housing premia were so strongly negatively correlated prior to 1997 that changes to real interest rates did not affect the rent-priceratio. After 1997,rates and premia have been positively correlated,and the decline in the rent-price ratio that has occurred in almost every geographic area in our sample since 1997 reflects both declining realrates and declining premia. Third, we show that in the recenthousing boom, 65 percent of the decline in the aggregate rent-price ratio is due to a declining housing premium.

Decomposition of Rent-Price Ratio 1 1 Introduction AccordingtotheFlowofFundsAccountsoftheUnitedStates,housingwealthaccountedforabout40percent of the overall net worth of households at the end of 2005. Housing’s share of household net worth has increasedmorethan10percentagepointssincethestartof1997,whichwedateasthebeginningofthemost recent housing boom. Unfortunately, there does not seem to be consensus among practitioners and housing economistsabouthowto characterizethe gainsinhouseprices,either nationallyorinanyparticularregion. Forexample,aSeptember2005reportbyeconomistsattheAndersonSchoolatUCLAconcludesthathouse prices in California were overvalued by 45 percent. Gallin (2004) argues that housing may be overvalued because the nationwide rent-price ratio appears significantly lower than one would expect given historical statistical relationships among rents, house prices, and interest rates. Case and Shiller (2003) suggest that fundamentals may explain house prices in certain metro areas, but not in the more expensive cities of the East and West Coasts. In contrast, Himmelberg, Mayer, and Sinai (2005) argue that there is no evidence of a bubble in 2004 in any of the regional markets that they study, and McCarthy and Peach (2004) argue that there is no overvaluationat the national level. In this paper, we use the dynamic Gordon growth framework of Campbell and Shiller (1988, 1989) to study housing valuations. The framework is an accounting identity based on the definition of one-period asset returns. It shows how the ratio of an asset’s flow of fundamental value to its price can be decomposed into the expected present value of future growthrates of fundamentals, the expected presentvalue of future interest rates, and the expected present value of future premia over those interest rates. Campbell and Shiller (1988, 1989), Campbell (1991), Campbell and Ammer (1993), and Shiller and Beltratti (1992) have successfullyusedthisframeworktostudyvaluationsinthestockandbondmarkets. Inthispaper,weextend thiskindofanalysistothehousingmarketbyequatinghousingfundamentalswithhousingrentsandhousing valuations with the (log) rent-price ratio. The dynamic Gordon growth framework provides a quantitative decomposition of housing market valu-

Decomposition of Rent-Price Ratio 2 ations that can be consistently compared across different geographic locations, across time within a single location,andacrossotherassetmarkets. Weestimatetherealinterestrateandrealrentgrowthcomponents usingsimpleforecastingmodels;thehousingpremiumisbydefinitiontheresidualintheaccountingidentity. While we are not the first to appeal to some form of the Gordongrowth model in an analysis of housing markets (Cutts et al. 2005; Himmelberg, Mayer, and Sinai 2005), other studies assume that the housing premium is fixed across time, locations, or both. One of the major contributions of this paper is that we explicitlyallowforhousingpremiatovaryovertimeandbylocation. Specifically,weexaminehowmovements in expected rent growth, interest rates, and housing premia account for both the trend and variability in housing valuations using semi-annual data over the 1975 to 2005 period. We conduct a separate analysis at the national and regional levels, and for twenty-three large metropolitan areas. Further, we examine how theserelationshipshavechangedovertimebycomparingthebehaviorofthecomponentsintherecentboom years,1997to 2005to theirbehaviorpriorto 1997. We willshowthattime-varyinghousingpremia arevital for understanding the behavior of housing valuations. Wefindconsiderablevariationinhousingpremiaacrossbothtimeandlocation. Averagehousingpremia, expressed at an annual rate, range from less than 1 percent per year in Honolulu and New York City to about 3.5 percent per year in Denver and Philadelphia. Premia also exhibit substantial variation across time. At the national level, premia range from roughly 2.5 percent per year to less than 1 percent per year. Time-seriesvariationinhousing premia are evenmoreextreme at the regionaland metropolitanarealevels. Fundamentals – growth in rents – explain little of the trend and variance of rent-price ratios for all locations we study. This implies that the expected future real return to housing is the primary determinant of the trend and variance of rent-price ratios. In this way, the housing market is remarkably similar to the stock and bond markets. Wefindthatrealinterestratesandhousingpremiacontributeaboutequallytothe trendandvarianceof rent-priceratios. However,we alsofind that the relationshipbetween realinterest ratesand housing premia

Decomposition of Rent-Price Ratio 3 haschangedimportantly since 1997. Before1997,the estimatedrealrateandhousingpremium components werestronglynegativelycorrelated: Changesto realrateswerelargelyoffsetby opposite changesto housing premia. The negative correlation of real rates and housing premia resulted in rent-price ratios that were relatively insensitive to movements in real interest rates. Since 1997, however,changes to real interest rates and housing premia have been positively correlated. As a result, housing valuations appear to be more sensitive to changes in real interest rates in recent years. Our results provide a new interpretation of the recent boom in house prices. We find that the recent run-up in house prices owes significantly to a decline in real interest rates and a decline in the housing premium. However, the predominant feature of the recent boom has been the change in the relationship between these two variables. Therefore, in our view, rent-price ratios in 2005 are considerably lower than could have been projected in 1997, even given advance knowledge of the decline in real interest rates that has occurred since 1997. 2 The Campbell-Shiller Decomposition Consider the one-period gross return to housing: (1) R t+1 ≡ P t+1+V t+1 , P t whereRisthegrossreturntoahousinginvestment,P isthepriceofhousingandV istheflowoffundamental value. Inthis paper we equate fundamentals withhousing services,and measurethese serviceswith housing rents. Campbell and Shiller (1988,1989)re-write equation (1) using a log-linearapproximationthat relates the current log rent-price ratio to expected future rates of return and expected future growth in rents, ∞ ∞ (2) v t −p t = k+E t [ ρjr t+1+j − ρj∆v t+1+j ] j=0 j=0 X X ρ = (1+ev−p) −1 k = (1−ρ) −1 [ln(ρ)+(1−ρ)ln(1/ρ−1)],

Decomposition of Rent-Price Ratio 4 where lowercase letters represent logs of their uppercase counterparts, ρ is a discount factor related to the long run rent-price ratio (written as ev−p), and k is a constant of linearization. By defining the return to housing as the sum of an interest rate, i, and the per-period premium over that rate, π = r −i, we can express the log rent-price ratio as the sum of three pieces: future expected real rates, housing premia, and rent growth, ∞ ∞ ∞ (3) v t −p t =k+E t ρji t+1+j +E t ρjπ t+1+j −E t ρj∆v t+1+j ], j=0 j=0 j=0 X X X or (4) v −p =k+i +π −∆v . t t t t t We refer to π as the housing premium and π as the e per-peeriod f housing premium. t t The above representation of the (log) rent-price ratio in terms of future expected real interest rates, e premia, and growth in real rents is a dynamic version of the classic Gordon growth model of asset prices in whichtheconstantrent-priceratioisrelatedtoaconstantrealinterestrate,constantpremiumandconstant growthrate of realrents, V =i+π−∆v. The dynamic Gordongrowthmodel abovewas firstdevelopedby P CampbellandShiller(1988,1989)toanalyzethedeterminantsofdividend-priceratiovariability. Sincethen, it has been applied in the analysis of a variety of other markets. Shiller and Beltratti (1992) and Campbell and Ammer (1993), for example, use the dynamic Gordon growth model to decompose the variability of long-term bond yields. To implement the dynamic Gordon-growthmodel, in this paper we use data on real interestratestoestimatetheinterestratecomponentoftherent-priceratio,i ,anddataonrealrentgrowth t to estimate the rent-growth component, ∆v t . Then, given data on rent-perice ratios, v t −p t , we use the accounting identity above to identify the hfousing premium component, π t ≡v t −p t −k+i t +∆v t . A popular alternative to equation (4) used by some authors to analyeze housing valuatioens efxpresses the level of the rent-price ratio as (5) V t /P t =i t +π−g t+1, where i t is the current real interest rate, π is a constant housing premium and g t+1 represents the expected

Decomposition of Rent-Price Ratio 5 capital gain or loss on housing. Versions of this expression can be found in Gallin (2004), Cutts et al. (2005), Himmelberg, Mayer and Sinai (2005), Verbrugge (2005), and elsewhere. Equations (4) and (5) both recognizethatpricesareforward-looking,sothatvaluationsmightappeartobeatatoddswithcurrentrents and interest rates even though they are in line with the long-term paths of these variables. Equation (5), however,combinesallfutureconsiderationsintoasinglevariable: expectedfuturecapitalgains. Incontrast, ourframework(a)recognizesandaccountsforthedynamicsineachofthecomponentsoftherent-priceratio but also (b) imposes a very strong restriction on long run capital gains that is obscured by equation (5). In particular, the dynamic Gordon growth model imposes that long-run real capital gains are identical to the long run growth rate of real rents. There is no room for expected future price gains in housing that are unrelated to future rental growth. For example, two cities cannot have permanently different growth rates of prices without permanently different growthrates of rent. In our analysis, if households expect relatively robustpricegrowthfarintothefuture inanygivenarea,they mustexpectrobustrentalgrowthinthatarea as well. 3 Data All data in this study are at a semi-annual frequency and all of our calculations are appropriate for semiannual data. In our tables and graphs, and throughout our analysis, however, we have annualized our data and results for expository purposes. 3.1 House Rent and Price Data We assume that real rents represent the fundamental service flow from housing. Implicit rents on housing earnedbyowner-occupiersareunobservedandwethereforeusedataonrentspaidbyrenterstoimputerents accruing to owner-occupiers. Our source for nominal rents is the index for tenants’ rent from the Bureau of LaborStatistics’(BLS)ConsumerPriceIndex(CPI).Weuseinformationfromthe 2000DecennialCensuses

Decomposition of Rent-Price Ratio 6 of Housing (DCH) to convert the CPI rent indexes to nominal dollar rents earned by owner-occupiers.1 To convert nominal rents to real rents, we deflate using the CPI excluding shelter. We use the BLS semiannual rent index for the nation, four Census regions, and twenty-three separate metropolitan areas. The metropolitan areas used in this study are listed in Table 1. Column 1 of Table 1 lists the beginning sample date for each metropolitan area. The data for all metropolitan areas extends through the first half of 2005 (2005:H1).2 The national data span 1975:H1 to 2005:H1.3 In Figure 1, we graph real rental growth at annual rates for the nation and the four Census regions. Over the full sample, except in a few periods, annual real rental growth ranged between -2 percent and 2 percent. At the national level (top panel), real rents increased at a relatively quick pace during the mid- 1980s, stagnated from 1986 to 1996, and increased again from 1996 to 2001. Since 2001, real rents have decelerated. Real rental growth in the four Census regions (bottom panel) looks quite similar to rental growth in the aggregate,and the averagecorrelationbetween rent growth across the four regions is roughly 0.75. We measure house-price changes using the weighted repeat-transactions house-price index published by 1Amoredetaileddataappendixisavailablefromtheauthorsuponrequest. SeeDavis,Lehnert,Martin(2005)foracomplete descriptionofthemethodusedtobenchmarkowner-occupied rentsusingthe2000DecennialCensusofHousing. 2Theregion-levelrentindexesarenotsimpleaveragesofthecomponentcitiesinouranalysis. TheBLScollectsrentdatafor over80metroareasandusesthosedataforregionalandnationalestimates,butonlypublishesmetro-areaindexesforselected cities. 3Apotentialconcernwiththesedataisthatwedonotmeasurerentsnetofdepreciation,taxes,andothercosts. Inparticular, iftaxesanddepreciationreducetheserviceflowofhousingbyτt percentperperiod,thenthedynamicGordongrowthrelation takes theform(Sharpe2002), ∞ vt −pt=k+it+πt −∆vt −(1−ρ)Et ρj(1−τt+j+1) , " # j=0 X andourestimatesofthehousingpremium, e πt,aereco f ntaminatedbyanychangesinexpectedfuturedepreciationratesortaxes. e

Decomposition of Rent-Price Ratio 7 theOfficeofFederalHousingEnterpriseOversight(OFHEO).4 TheOFHEOindexisbasedonpricechanges for owner-occupied homes that are resold or refinanced and therefore partially controls for changes in the composition of homes sold.5 We use the quarterly price index (converted to a semi-annual frequency) for the nation, the four Census regions, and the twenty-three metro areas for which we have rent data. As in the case of the rent data, the region-level price indexes are not weighted averages of the component cities in our analysis. OFHEO collects price data for over 300 metro areas and uses those data for regional and national estimates. Like our rent data, the national price data span 1975:H1 to 2005:H1; the regional and metro-area data begin at different dates but all end in 2005:H1. The ratio of rents to prices (annual rate) for the nation and four regions are graphed in Figure 2. At the national level and across all four Census regions, the rent-price ratio was fairly flat from 1975 to about 1995,andthenfellduringthe lastdecadeofthe sample. Nationally,the rent-priceratiodeclinedfromabout 5.6 percent in 1975:H1 to 3.8 percent in 2005:H1. The South declined the least over the sample period (5.4 percent to 4.2 percent) while the West declined the most (4.8 percent to 2.8 percent).6 Also, while rent-price ratios vary considerably from period to period, they have declined since 2000 in every region and metropolitan area in our study (not graphed). Whentakentogether,figures1and2showthatrent-priceratiosvarymoreacrossregionsthanthegrowth rate of rents. Since real interest rates are roughly constant across geographic areas,and since rental growth rates in each region are very highly correlated, housing premia likely account for a substantial fraction of the variance in rent-price ratios. 4As in the case of the rent indexes, we convert OFHEO index values into dollar values by using information from the 2000DCH.SeeDavis,Lehnert,andMartin(2005)fordetails. 5TheHPIsampleexcludes homes withjumbo,FHA,orVAmortgages. SeeCalhoun(1996)foradescriptionoftherepeattransactions methodologyusedbyOFHEO. 6Unlikethethreeother U.S.regions,rent-priceratiosactuallyincreasedintheSouthpriortothelate1990s.

Decomposition of Rent-Price Ratio 8 3.2 Real Interest Rates Weuseanestimateoftherealexpectedyieldona10-yearU.S.Treasurybondasourrealinterestrateinthe implementation of equation (4). As a result, our estimates of housing premia reflect the premium paid to housing overand above the yield on a 10-yearTreasury. In principle, we could havechosenany realinterest rate. Weusethereal10-yearTreasuryasourrealinterestratesoourresultsaremorecomparablewithother recent studies of housing valuations, such as Himmelberg et al. (2005), Cutts et al. (2005), Gallin (2004), andMeeseandWallace(1994).7 We constructedrealinterestratesusingmedianinflationexpectationsfrom theBlueChipEconomicIndicatorforecastfrom1975:H1through1991:H1,andthemedian10-yearexpected inflation forecasts from the Livingston survey from 1991:H2 to 2005:H1. The nominal 10-year yield, 10-year expected inflation, and the real 10-year yield are plotted at annual rates in Figure 3. The nominal yield, expected inflation, and the real yield all increased prior to 1982, and have been declining since then. The real 10-year Treasury varies considerably in our sample. It increases from 2 percent at the start of our sample to about 8 percent in 1982, and then falls after that. At the end of the sample, the real 10-year Treasury yield is about 1.7 percent at an annual rate. 4 Time Variation in Housing Premia A main goal of this analysis is to document the importance of time variation in the expected present value of housing premia, π . However, all the empirical studies on housing that we know of assume a constant t premium. So, a natural starting point for our analysis is to check that variation in housing valuations can e not be attributed to changes in real interest rates and rental growth rates alone. 7In an earlier draft of this paper, we estimated the expected present value of real interest rates using a forecasting model for the real six-month Treasury. We also experimented estimating the expected present value of real interest rates using the observedterm-structureofTreasuryrates(allowingfortime-varyingtermpremiaonlonger-datedTreasuries). Theresultsfrom thesealternativemethodswerequalitativelysimilartothoseinthispaper.

Decomposition of Rent-Price Ratio 9 Consider the per-period approximate log premium, (6) π t+1 =ρp t+1+(1−ρ)v t+1 −p t −i t+1. If the per-period housing premium is predictably different from a constant, then the expected present value of all future housing premia must vary over time. Following Campbell and Shiller (1988, 1989),we test this hypothesis by regressingthe approximate excess return onto lagged values of v −p and i −v .8 A finding t t t t that v −p and i −v predict excess returns providesevidence againstthe hypothesis of constant expected t t t t premia. We carry out this test on each geographic area used in this study using two lags of the (log) rent price ratio and two lags of i −v . We conduct the test over three different periods: 1975:H1 to 2005:H1, t t 1975:H1 to 1996:H2 (the early period), and 1997:H1 to 2005:H1 (the late period). The results of these tests are listed in table 2. The test statistics and associatedp-values are displayedin columns 1 and 2 for the full period, columns 3 and 4 for the early period, and columns 5 and 6 for the late period. The results show that housing premia are likely time-varying across most geographic locations for both timeperiodsthatweconsider. IneveryCensusregionandatthenationallevel,thenullhypothesisofconstant housingpremiaisrejectedinthefullsampleandineachperiod. Insomemetroareas,thestatisticalevidence suggeststhatpremiamaynotbe time-varyinginsomeperiods. Onthe whole,however,theevidenceagainst constant housing premia is remarkably strong. The results of the forecasting regression (not reported here) indicate that the rejection of constant housing premia arises because lagged house prices changes forecast future house prices changes, a well-known result documented by Case and Shiller (1989). 8CampbellandShiller’s(1988,1989)preferredversionofthistestisanon-linearWaldtestoftherestrictionsimposedbya VAR representation of the dynamic Gordon growth model. As they show, this test is equivalent to the test on excess returns discussedhere.

Decomposition of Rent-Price Ratio 10 5 The Components of the Rent-Price Ratio Inthissectionwediscussourestimatesoftheexpectedpresentvaluesofrealrates,i ,realrentgrowth,∆v , t t andultimatelyhousingpremia,π t . Tobegin,recallthattheexpectedpresentvaluesedependonadiscounfting parameter,ρ=(1+ev−p) −1. We use the firstobservationofthe rent-priceratioinoursampleto compute ρ e for each geographic area; column 3 of table 1 displays their values. We use the initial level of the rent-price ratio to fix ρ (as opposed to the average level over the sample period) to ensure that trends in house prices themselves do not influence our estimates.9 Note that areas with lower values for the rent-price ratio have higher values for ρ. 5.1 Estimating Discounted Expected Future Real Rates, i t e Our decomposition requiresthat for eachperiod t in our sample, we have estimates of the expected value of the entire future sequence of real interest rates from period t forward. Campbell and Shiller (1989, 1989), Shiller and Beltratti (1992), and Campbell and Ammer (1993) use vector auto-regressivemodels to forecast future real rates. In this spirit, we use a simple first-order auto-regressive model to forecast the expected present value of future real 10-yearTreasury yields, (7) i t+1 =i(1−β)+βi t +σε t+1, where i is the average (time-invariant) real interest rate, β captures the persistence in real interest rates, ε is a random shock with zero mean and unit variance, and σ scales the shocks. Over the sample period, our annualized estimates of the model parameters are, i = 3.6 percent, β = 0.87 and σ = 0.7 percent.10 The estimated persistence coefficient, β =0.87 implies that the half life of a shock to the 10-yearTreasury yield 9Ourresultsarenotsensitivetothischoice. 10Estimatesofpersistenceparametersinfirstorderautoregressivemodelsarebiaseddownwardsinsmallsamples. Wecorrect forthissmallsamplebiaswiththebias-correctioninKendall(1954).

Decomposition of Rent-Price Ratio 11 is slightly less than 5 years,indicating that real yields are quite persistent.11 Using these estimates of β and i, we construct the present discounted value of future real interest rates as ∞ i i −i (8) i t = ρji t+1+j = 1−ρ + 1 t −ρβ . j=0 X e Accordingly, the effect of current real rates on the present discounted value of future real rates depends on both the long-run level of the rent-price ratio, the discount factor ρ, and the persistence of shocks to real rates, β. We plot (1−ρ)i and i in figure 4 (for ρ = 0.95). The scaling puts i on an annualized basis. Figure 4 t t t showsthatbecauseeβ <1,the discountedpresentvalue offuture realrateesincompletelyincorporatesshocks to the current level of real rates, and therefore i is considerably smoother than i . Note that at the end t t of our sample, the present discounted value of exepected future real rates is roughly 3 percent while the real yield on the 10-year Treasury bond stands at roughly 1.7 percent. Obviously, if the dynamics of real interest rates are systematically different than those of the first-order auto-regressive model, our analysis will be inappropriate. In particular, our modeling framework assumes that real interest rates have a constant unconditional mean. This assumption is important in light of the rather low value of the real interest rate near the end of the sample period. If there has been a structural break in the level of the real 10-year Treasury, then our forecast model is mis-specified. We address this issue in the sensitivity analysis at the end of the paper. 11Wealsoconsideredmorecomplicatedforecastingmodels(longerlaggedAR,ARMA,andVARspecifications). Ourresults indicatethattheAR(1)specificationisareasonablebenchmark

Decomposition of Rent-Price Ratio 12 5.2 Estimating Discounted Expected Future Real Rent Growth, ∆v t g As with real interest rates, we use a simple first-order auto-regressive model to produce forecasts of the expected present value of future rent growth. Specifically, for each geographic area we estimate (9) ∆v t+1 =∆v(1−γ)+γ∆v t +ση t+1, where ∆v is the average(time-invariant) growthrate of real rents, γ measures the persistence in real rental growth, η is a random shock with zero mean and unit variance, and σ scales the rental growth shocks.12 Withestimatesofγ and∆v inhand,wecalculatethe discountedexpectedfuture valueofrealrentalgrowth as (10) ∆v t = ∞ ρjE t ∆v t+1+j = 1 ∆ − v ρ + ∆v 1 t+ − 1 − ργ ∆v . j=0 (cid:0) (cid:1) X f The estimated parameters of the rental growth model are displayed in table 3. The table displays the averagerateofrealannualgrowth,∆v,foreachgeographicarea(column1),theestimatedannualizeddegree of persistence, γ (column 2), the associated t-statistic (column 3), and the estimated standard deviation of real rental shocks, σ (column 4).13 The average real growth rate of rents is smaller than 1 percent per year in most areas we consider (column 1). In addition, the persistence of rental growth rates is also modest everywhere. Ourdata thereforesuggestthatalthoughshocksto realrentgrowthcanbe quite large(column 4), these shocks are largely transitory, with a half-life of five quarters or less. 12Asinthe caseofthe realrate, wealsoconsideredmorecomplicated models(longer lagged AR,ARMA,andVARspecifications). Morecomplicatedmodelsdidnotproducemateriallydifferentforecasts. 13Asinthecaseoftherealinterestrate,weuseabias-correctedversionoftheOLSestimatorofγ.

Decomposition of Rent-Price Ratio 13 5.3 Computing Discounted Expected Future Housing Premia, π e Giveni and∆v ,wecomputeestimatesofthediscountedexpectedfuturevalueofhousingpremia,π ,using t t t the ideentity, f e (11) π =(v −p )−k−i +∆v , t t t t t e e f for the nation, regions, and metro areas. Table 4 reports the average value of π , the average level of the t rent-price ratio (V/P), the averagereal interest-rate component i , and the averagerent-growthcomponent t e ∆v t over the full period (columns 1-4), the early period (columnes 5-8), and the late period (columns 9-12); agll variables are displayed at an annual rate. Figure 5 graphs time-series of π t . Thetoppanelofthefirstpageoffigure5displays(1−ρ)π theestimatedhousingpremiumatanannual t e rate at the national level.14 Over much of the sample, the annualized premium ranges between 1 percent e and 2 percent with three notable exceptions. Estimated premia are near 2.5 percent at the beginning of the sample, and below 1 percent in the early 1980s and at the end of the sample. Premia at the national level have averaged roughly 1.6 percent over the entire sample (table 4, column 4). To give some perspective, one standard deviation of the national-level housing premium is 0.5 percentage point. All else equal, a one standard deviation increase in the national-level housing premium would imply a 10 percent decline in real house prices. The data in table 4 and the plots of (1−ρ)π for the regions (figure 5, first page, bottom panel) show t that the regional premia exhibit similar time-series patterns as the national premia. In contrast, average e estimated premia vary considerably across metro areas. For example, New York, Los Angeles, and San Francisco have average premia over the full sample below 1 percent, whereas Denver, Dallas, and Houston have average premia (over the full sample) in excess of 2.5 percent. Note that not every city on the East and West coasts has low average housing premia. The average premium in Philadelphia is about 3 percent 14Inthediscussionofthehousingpremiumthatfollows,wedescribetheannualizedpremium,(1−ρ)πt. e

Decomposition of Rent-Price Ratio 14 over the full sample, whereas the averagepremium in Milwaukee is 1.1 percent. Estimatedaveragepremiahavedeclinedsharplyinsomemetroareasbetweenthetwoperiodsweconsider. Forexample,averagepremiafellfrom1.6percentto0.1percentinDetroit,from1.7percentto0.8percentin Boston, and from 2.4 percent to 0.7 percent in Seattle. Importantly, the average housing premium appears to have declined in every metro area in our West Coast sample, as well as in Boston and New York, and in four of our Midwestern metro areas. However, the average premium paid to housing appears to have increased in Atlanta, Dallas, Houston, and Kansas City. 6 Trends and Variability of Rent-Price Ratios 6.1 Decomposing Trends in Rent-Price Ratios Figure5revealsarecentdownwardtrendinthehousingpremiainmanygeographicareas. Indeed,estimated premiaattheendofthesamplearenearalltimelowsformanyareas. Inthissection,weattributeanytrend inthe(log)rent-priceratiotounderlyingtrendsintherealratecomponent,i ,rentgrowthcomponent,∆v , t t and the housing premium component, π t over the full, early, and late periodes. f Consider the sample trend in the log rent-price ratio and its components, e v t −p t = a0+b0t+ε0,t , i t = a1+b1t+ε1,t , ∆ve t = a2+b2t+ε2,t , fπ t = a3+b3t+ε3,t . e The identity relating the components to the rent-price ratio in equation (4) implies that, (12) b0 =b1 −b2+b3, so that the trend in the (log) rent-price ratio over any period in time may be attributed to the trends in

Decomposition of Rent-Price Ratio 15 each of the individual components. Table5presentsestimatesofthistrendrelationshipforthefullsampleincolumns1-4,theearlysamplein columns 5-8,and the late sample in columns 9-12. The shares aredefined as b1/b0, −b2/b0, andb3/b0. Over the full sample, the rent-price ratio has trended down at the national level, in the four regions, and in most of our metro areas (column 1); Dallas, Houston, and Kansas City are the only exceptions. At the national level, roughly 40 percent of the decline in rent-price ratios is attributable to declining real interest rates (column 2). Most of the rest of the trend decline is attributable to the downward trend in housing premia (column4); trendsinrealrentgrowthareofminimalimportance(column3). Byregion,therent-priceratio hasfallenfastestintheNortheastandWest,whilethetrenddeclineintheMidwestandespeciallytheSouth has been less pronounced. In the Midwest and West, nearly all of the decline in the log rent-price ratio is attributabletodecliningrealrates,whileintheNortheastonly67percentofthetrendinhousingvaluations is attributable to declining real rates.15 Columns 5-12 highlight the fact that trends in housing valuations (rent-price ratios) reported over the full sample are largely reflective of changes to valuations since 1997. In the early sample, trend changes to housingvaluationsare,literally,alloverthemap. Forexample,therent-priceratiodeclinedintheNortheast and West, but increased in the South and to a lesser extent in the Midwest. Moreover, only the Northeast shows a significant trend decline in the rent-price ratio over this period. Trends in housing valuations, real rates, and housing premia are much more pronounced in the late period (column 9). Rent-price ratios uniformly trended downward over this period. At the national level, the downwardtrend in realinterest rates accounts for 38 percent of the trend decline in the rent-price ratio andthe downwardtrend inhousing premiaaccounts for65 percentofthe trend. In the Midwest andSouth, 15Note that the regional-level results arenot the weighted average of the metroareas we display. Recall, the BLS does not reportmetro-areadataforallcomponent cities. Also,thestartdates forthevarious areas aredifferent, and estimated trends aresensitivetothechoiceofendpoints.

Decomposition of Rent-Price Ratio 16 the trend in real rates accounts for more than two-thirds of the trend in housing valuations, but in the Northeast and West, the trend in real rates accounts for less than half of the trend decline in rent-price ratios. In metro areas such as Boston, Miami, Los Angeles, and San Francisco, declining premia account for over 70 percent of the trend in rent-price ratios, while in metro areas such as Chicago, Houston, Kansas City, Pittsburgh, and St. Louis, the trend in real rates accounts for over 70 percent of the trend decline in housing valuations. 6.2 Decomposing the Variability in Rent-Price Ratios Figure 6 displays the decomposition of the log rent-price ratio for the nation as a whole. Note that we plot the sum of the log rent-price ratio and the constantof linearizationand the negative of the real rentgrowth component. Thus, the three components sum to the total. In addition, we plot a vertical line in 1997. By plotting all the components on one graph, several features become immediately apparent. First, the real rent growth component is less variable than the log rent-price ratio, implying that variation in fundamentals do not explain much of the variation in housing valuations. Second, the real interest rate and housingpremium componentsareeachmore variablethanthe rent-priceratioitself, but they arenegatively correlated,sothatmovementsinonearelargelyoffsetby the other. Third,the negativecorrelationbetween therealrateandhousingpremiumcomponentsappearstohaveweakenedstartingin1997: Since1997,both real rates and housing premia have trended down. Toaddressthese featuresin a moresystematic way,considerthe followingvariancedecompositionof the rent-price ratio, (13) Var(v −p ) = Var i +Var(π )+Var ∆v t t t t t (cid:16) (cid:17) (cid:16) (cid:17) +2Coev π ,i −e2Cov π ,∆fv −2Cov i ,∆v . t t t t t t (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) e e e f e f We report the results from this variance decomposition over the full period in table 6a; table 6b presents the results of the variance decomposition over the early and late periods. In table 6a, column 1 lists the

Decomposition of Rent-Price Ratio 17 variance of the (log) rent-price ratio. Columns 2-4 report the share of the rent-price ratio’s variance that owes to variance of ∆v , i , and π . Finally, columns 5-7 report the share of the rent-price ratio’s variance t t t that owes to the covafrianece betweeen each of the three components. Considering the four Census regions, housing valuations are most volatile in the Northeast and West, andleastvolatileinthe South(table 6a,column1). This resultis consistentwiththe findingthatrent-price ratios have trended downward fastest in the Northeast and West and the least in the South over the full sample period. As figure 6 suggests, the rent-growth component explains very little of the variability in rent-price ratios (column 2). At the national level and in each of the four regions, variability in the rent growth component typically accounts for less than 10 percent of the variability in rent-price ratios. The finding that rentgrowthonly accountsfor a smallshare ofthe variability in housing valuations is consistent with the findings from other asset markets. In the case of the stock market, for example, Campbell and Ammer (1993)estimate that roughly 10 percent of the volatility of the dividend price ratio is accounted for by the volatility in dividends. In this way, houses behave like other financial assets. Since rent growth volatility only accounts for a small portion of the volatility in rent-price ratios, the remainderofthevolatilityinrent-priceratiosisaccountedforbyvariabilityinrealratesandpremia. Columns 3 and 4 indicate that at the national and regional levels, the real rate and housing premium components account for more than 50 percent of the variation in rent-price ratios. In many cases these components are each more variable than rent-price ratios. But, the interest-rate and premium components are negatively correlated (column 7), so that movements in one component are largely offset by the other over the full sample. The negative correlation of the components of housing returns highlights the fact that over the early sample, rent-price ratios were insensitive to changes in real interest rates. As an example, consider the case of the decline in the real interest rate component in the mid-1980’s. From 1985:1 through 1990:1 the real interest rate component of house prices declined by 0.8 percentage point, from roughly 4.6 percent per year

Decomposition of Rent-Price Ratio 18 to 3.8 percent per year (figure 3). At the national level, the rent-price ratio only declined by 0.4 percentage point over this same period from, 5.2 percent to 4.8 percent (figure 2). Our accounting identity requires an increase in the housing premium of roughly 0.4 percentage point.16 The lack of responsiveness of rentprice ratios to movements in real interest rates over this period is a distinguishing feature of these data. More generally, over the full sample, movements in rent-price ratios typically reflected only a portion of the underlying movements in real interest rates, highlighted by the negative covariance in column 7. The metro area results are similar. First, in nearly every metro area, the volatility of the rent growth componentaccountsforlessthan10percentofthevolatilityoftherent-priceratioandinonlyonecasedoes it account for more than 15 percent of the volatility of the rent-price ratio. Second, both the real rate and housing premium components are highly variable and arenegatively correlatedwith eachother overthe full sample. Accordingly, the lack of sensitivity of rent-price ratios to real rates is a widespread feature of these data that is apparent at the national, regional, and metro-area levels. Intable6bwerepeatthevariancedecompositionfortheearlyandlateperiods;thetablelayoutissimilar to that of table 6a. The results in columns 1 and 8 show that, at the national level, there has been a large increase in the variability of rent-price ratios since 1997. Specifically, rent-price ratios have been more than fourtimesasvolatile,atthenationallevel,duringthelateperiodthanduringtheearlyperiod. Thispattern however, is not shared across all regions. While the South and West have exhibited more volatile housing valuationsoverthelaterperiod,housingvaluationsintheMidwestandNortheastwereaboutequallyvolatile overbothperiods. The resultsatthe metro-arealevelarealsomixed, withsomeareas,suchasMinneapolis, Miami and San Diego, experiencing large increases in rent-price ratio variability while other metro areas including Boston, Detroit, Houston, and Portland experiencing sharp declines in the variation. 16Of course, the small decline in the rent-price ratio could also be explained by an offsetting increase in the rent growth component overthisperiodwithnochange inthehousingpremium. Astable6ashows,however, changes inrentgrowthplay aminorroleindeterminingrent-priceratios.

Decomposition of Rent-Price Ratio 19 As in the case of the full sample, rent growth plays a minor role in accounting for the variability in rent-price ratios (columns 2 and 9). Indeed, the results suggest that the importance of rent growth has declined somewhat over time. Also, as in the case of the full sample, the insignificance of the rental-growth component implies that the realrate and housing premium components accountfor most of the variationof the rent-price ratio. Table 6b highlights what we think is a key change in the relationship over time between the real rate and housing premium components. Over the early period, the covariancebetween the realrate and housing premium component is estimated to be negative at the national level, regional level, and in every metro area(column 7). Moreover,the size of the covarianceis large. In nearly everycase the estimated covariance between the realrate and premium componentis estimated to be largerthan the variance ofthe underlying rent-priceratio,indicatingastrongtendencyformovementsinonecomponenttobeoffsetbytheother. Over the late period, the picture is quite different. At the national, regional and metro-area level the estimated covariance between these two components has increased substantially (column 14). At the national level, in the Northeast and West Census regions, and in roughly half of the metro areas, the covariance between the real rate and housing premium components is positive. Further, in all but one of the metro areas, the covariance increased. Thus,table6bdemonstratesthatrent-priceratioshavebecomemoresensitiveto changesinrealinterest rates since the onset of the most recent housing boom. It is helpful to compare the behavior of rent-price ratios from 2000:H1 through 2005:H1 with their behavior from 1985:H1 through 1990:H1. In both periods, the real interest rate component declined about 3/4 percentage point. From 2000:H1 to 2005:H1, at the national level, the rent-price ratio fell a full percentage point, implying that housing premia also declined about 0.3 percentage points. In contrast, recall from the earlier discussion that from 1985:H1 to 1990:H1, rent-price ratios declined less than did real rates, indicating that housing premia increased. The results in column 7 and 14 of table 6a indicate that the change in the housing market’s responsiveness to changes in

Decomposition of Rent-Price Ratio 20 real rates is widespread, affecting every region and metro area that we study. Since 1997, rent-price ratios have become more sensitive to changes in real rates and the tendency for real rate movements to be offset by movements in housing premia has diminished considerably. In table 7, we further explore how the links between the real interest rate and housing premium components have changed. Columns 1 and 2 display the estimated coefficient and R2 from regressions of π on i t t for the earlyperiod, andcolumns 3 and4display the same informationforthe laterperiod. Finally, e columen 5 displays the t-statistic of the null hypothesis that the regression coefficients are identical across the two periods. The tendency for movements in real rates to be offset by movements in housing premia was strong over the early period (column 1). At the nationallevel and in the South and West, movementsin realrates were met, nearly one for one, with opposing movements in housing premia. In the Midwest and the Northeast, the tendency of real rate movements to be offset by movements in housing premia was somewhat weaker, but still strong. The R2 measures from the early period reveal that the degree of association between the two components was quite high. At the national level, and in all Census regions except the Northeast, at least 68 percent of the variation in housing premia can be explained by variation in real rates in the early period. Qualitatively and quantitatively, the metro-area results are quite similar. Column 3 of table 7 shows that the tendency for movements in real rates to be offset by movements in housingpremiahaslargelybeenreversedintherecentboomperiod. Allareasshowatleastsomeattenuation ofthe negativecorrelationbetweenrealrates andhousing premia. Furthermore,atthe nationallevel,in the Northeast and West, and in many metro areas, housing premia have tended to move in the same direction as real rates beginning in 1997. The sign of the regression coefficient switched from negative to positive between the two periods in thirteen out of twenty-three metro areas, and these changes are statistically significant in almost every geographic area we study (column 5). A comparison of columns 2 and 4 shows that the degree of association between real rate and housing premium movements has also attenuated over

Decomposition of Rent-Price Ratio 21 the two periods. At the national level, for example, the estimated R2 between housing premia and real rate movements has declined from 76 percent over the early period to only 21 percent in the later period. This decline in predictive power has also occurred in each of the regions and in a majority of the metro areas. Thus, when compared to the pre-1997 data, the recent behavior of housing valuations is more consistent with an environment in which housing premia are independent of real interest rates. The results presented in tables 6a, 6b, and 7 provide a new perspective on the recent boom in housing markets. The relationship between real rates and housing premia appears to have undergone an important change. One interpretation is that, prior to 1997, a decline in real rates was typically accompanied by an increase in housing premia, implying only a small increase in housing returns, and a muted impact on rent-price ratios. Since 1997, the tendency of housing premia to offset the effect of real rates has largely disappeared. In this sense, the predominant feature of the most recent housing boom is not necessarily the decline in real rates or changes to housing premia, but rather a change in the interaction between these two variables. Of course this interpretationis basedon evidence from a shortsample. The reactionof rent-price ratios to rising real rates would provide an important out-of-sample test of the hypothesis that housing premia and real rates no longer offset each other. 7 Sensitivity Analysis Inthis section,we examinethe sensitivityofour resultsto changesinassumptionsaboutthe value ofγ (the persistence of real rent growth) and β (the persistence of real interest rates). We focus mainly on the late period and consider two scenarios. In the first scenario, we maintain β at the baseline value used so far in thepaperandweincreasethepersistenceofrealrentalgrowthbysettingγ =0.99(annualizedvalue). Inthe second scenario, we maintain γ at the baseline and increase the persistence of real interest rates by setting β =0.99 (annualized value). Increasing the persistence of both real interest rates and rental growth results in expectations that are heavily influenced by current conditions. Thus, our sensitivity analysis allows for

Decomposition of Rent-Price Ratio 22 the possibility that developments in interest rate and rental markets over the late period represent a break from the past that will be expected to persist for many years into the future. The results of this sensitivity analysis are shown in table 8. Columns 1 through 4 reproduce the trend decomposition for the late period (table 5,columns 9-12). Columns 5through7 reportthe effectofincreasingthe persistence ofrentalgrowth rates and columns 8 through 10 report the effect of increasing the persistence of real interest rates. At the national level, real rental growth declined from roughly 2 percent per year in 1997 to roughly 0percentperyearin2005,exertingupwardpressureontherent-priceratio. Ifrentalgrowthwereexpectedto beneararandomwalk,theupwardpressurewouldbemuchgreaterthanforourbaselinecase,andtherefore observed housing valuations could only be justified with lower housing premia (relative to baseline). Thus, increasingthe persistenceofrentalgrowthratesto γ =0.99forthe nationasawholeincreasesthe extentto whichtherecentdeclineinrent-priceratiosisattributabletodecliningpremia. Thispatternisfoundineach region except for the Northeast, and in most metro areas (columns 4 and 7). Two interesting exceptions are Los Angeles and Miami, where increasing the persistence of rent growth actually reduces the extent to which declining premia accounts for the decline in rent-price ratios because rent growth increased in these areas over the late period. On balance, however, columns 5-7 of table 8 indicate that assuming extremely persistent rental growth over the late period typically results in a steeper decline in housing premia than in our baseline estimates. Increasing the persistence in the real interest rate yields very different results. If market participants behave as if real interest rates are near a random walk, then the recent downward trend in real rates has a much larger effect on the expected present value of all future rates than in the baseline. As a result, more of the decline in rent-price ratios in the late period can be attributed to interest rates. For example, if β = .99 (annual rate), the recent downward trend in the rent-price ratio for the nation is consistent with essentially no trend in the housing premium (columns 8 and 10). Furthermore, when compared to baseline, the trend in the housing premium in every geographic area we study explains a

Decomposition of Rent-Price Ratio 23 smallershareofthe downwardtrendinthe rent-priceratiounder the alternativeassumptionfor β (columns 4 and 10). However,evenunder the alternative,the housing premium is a veryimportant componentof the rent-price ratio in the late period in areas such as Boston, Miami, Denver, San Diego, and San Francisco. 8 Conclusion We use Campbell and Shiller’s (1988,1989)dynamic Gordongrowthmodel to decompose the log rent-price ratio into three components: the expected present value of real rent growth, the expected present value of real interest rates (the real10-yearTreasury),and the expected present value of the housing premium. Our approach explicitly allows for time variation in housing premia. This stands in contrast to previous studies of housing markets in which authors have assumed that returns to housing in excess of interest rates and rentalgrowthare constant. We show that a Wald test soundly rejects the assumption of a constanthousing premium. Thedecompositionyieldsseveralresults. First,housingpremiavarysignificantlyovertime. Forexample, aonestandarddeviationincreaseinthehousingpremiumatthenationallevel(0.5percentagepoint,annual rate) is, all else equal, associated with a 10 percent decrease in real house prices. Second, housing premia varysignificantlyacrosslocations. Forexample,theaveragehousingpremiuminHoustonfrom1975to2005 was about 2.7 percent at an annual rate, while the average housing premium in New York during the same period was about zero. A third result is that “fundamentals”, housing rents in our case, are of little importance for explaining the trend or variance of housing valuations. By definition, then, housing returns explain most of the trend and variation in valuations. In this way, the housing market is remarkably similar to the stock and bond markets. Ourfourthresultisthatrealinterestratesandrealhousingpremiaplayroughlyequalrolesindetermining trends and variancesin housing valuations. For example, changesto realrates accountfor about 40 percent

Decomposition of Rent-Price Ratio 24 of the downward trend in the national log rent-price ratio from 1975 to 2005; housing premia account for about 50 percent. The housing premium plays a relatively more important role from 1997 onward, and can explain about 65 percent of the recent run-up of house prices relative to rents. Finally, the decomposition reveals that the real interest rate and housing premium components display a strong negative correlation from 1975 to 1996 in every location we study. During this period, the ratio of rents to house prices was insensitive to movements in interest rates. Thus, by definition, housing premia movedtooffsettheeffectofinterestrates. Since1997,thisnegativecorrelationappearstohavedisappeared. In particular, real interest rates and housing premia both have fallen from 1997 to 2005 in almost all the locations we study. Our results offer a novel way to characterize the recent housing boom. A substantial portion of the increase in prices relative to rents can be attributed to falling real rates, and an even more substantial portion (65 percent at the national level) can be attributed to a decline in the housing premium. However, wethink thatthe predominantfeature ofthe mostrecenthousingboomis notnecessarilythe decline inreal rates or housing premia, but rather the change in the interaction between these two variables. In our view, rent-price ratios in 2005 are considerably lower than could have been projected in 1997,even given advance knowledge of the decline in real interest rates that has occurred since then. In this paper we have measured housing premia using a well-established accounting framework. Given the history of treating the housing premium as constant, we think it is a useful and necessary first step. However, our work leaves many questions for future research. For example: What determines the housing premium? How much is a true “risk” premium, how much is a liquidity premium, and how much simply reflects transactions costs in the housing market? In other words, in this paper we have described the behavior of housing premia, but we think future work should be directed to understanding its fundamental determinants. Regardless, the housing premium appears to have important implications for valuations in the housing

Decomposition of Rent-Price Ratio 25 market. Prior to 1997, housing premia effectively moved to smooth valuations in the face of interest-rate volatility. If the break we observed in 1997 proves to be permanent, we should expect housing prices to be much more volatile in the future.

Decomposition of Rent-Price Ratio 26 References [1] Calhoun, C. (1996). “OFHEO House Price Indexes: HPI Technical Description.” Office of Federal Housing Enterprise Oversight.http://www.ofheo.gov/house/download.html. [2] Campbell, J. (1991), “A Variance Decomposition for Stock Returns”, The Economic Journal, 101, 157-179. [3] Campbell, J.and J.Ammer (1993),“What Movesthe Stock andBond Markets: A VarianceDecomposition for Long Term Asset Returns”, The Journal of Finance, 48, 3-37. [4] Campbell, J. and R. Shiller (1988), “Stock Prices, Earnings and Expected Dividends”, Journal of Finance, 43, 661-676. [5] Campbell, J. and R. Shiller (1989), “The Dividend Price Ratio and Expectations of Future Dividends and Discount Factors”, Review of Financial Studies, 1, 195-227. [6] Campbell,J.andR.Shiller(2001),“ValuationRatiosandtheLong-RunStockMarketOutlook”,NBER Working Paper 8221. [7] Case, K. and R. Shiller (1989), “The Efficiency of the Market for Single-Family Homes”, American Economic Review, 79, 125-37. [8] Case,K.andR.Shiller(2003),“IsThereaBubbleintheHousingMarket”,BrookingsPanelonEconomic Activity, manuscript. [9] Cutts,A.,GreenR.,andY.Chang(2005),“DidChangingRentsExplainChangingHousePricesDuring the 1990s”,Manuscript. [10] Davis,M., Lehnert, A., and R. Martin(2005),“The Rent-PriceRatio for the Owner-OccupiedStock of Housing”, Manuscript. [11] Gallin, J. (2004), “The Long-Run Relationship between House Prices and Rents”, Finance and Economics Discussion Series 2004-50. [12] Himmelberg,C.,Mayer,C.andT.Sinai.(2005),“AssessingHighHousePrices: Bubbles,Fundamentals, and Misperceptions”, Working paper. [13] Kendall, M. (1954), “Note on the Bias in the Estimation of Autocorrelation”, Biometrika, 41, 403-404. [14] McCarthy, J. and R. Peach (2004), “Are Home Prices the Next ”Bubble”” Federal Reserve Bank of New York Economic Policy Review, 10, 1-17. [15] Meese, R. and N. Wallace (1994), “Testing the Present Value Relation for Housing Prices: Should I Leave My House in San Francisco?”, Journal of Urban Economics, 35, 245-266. [16] Sharpe, S. (2002), “Reexamining Stock Valuationand Inflation: The Implication of Analysts’ Earnings Forecasts”,Review of Economics and Statistics, 84,632-648. [17] Shiller,R.andA.Beltratti(1992),“StockPricesandBondYields: CantheirComovementsbeExplained in Terms of Present Value Models”, Journal of Monetary Economics, 30, 25-46. [18] Verbrugge,R. (2005),“The Puzzling Divergence of Rents and User Costs, 1980-2004”,Working Paper.

Decomposition of Rent-Price Ratio 27 Table1 SampleInformation (1) (2) (3) Area 1st obs. V0/P0 ρ USA 1975:H1 5.56 0.95 Midwest 1978:H1 4.69 0.95 Chicago 1975:H2 5.90 0.94 Cincinnati 1976:H2 5.10 0.95 Cleveland 1976:H1 5.65 0.95 Detroit 1975:H2 6.23 0.94 KansasCity 1976:H2 5.82 0.94 Milwaukee 1977:H2 4.79 0.95 Minneapolis 1976:H2 6.40 0.94 St. Louis 1976:H1 5.88 0.94 Northeast 1978:H1 6.16 0.94 Boston 1976:H2 7.19 0.93 NewYork 1975:H2 4.92 0.95 Philadelphia 1976:H1 7.03 0.93 Pittsburgh 1977:H1 5.20 0.95 South 1978:H1 5.21 0.95 Atlanta 1976:H1 5.86 0.94 Dallas 1976:H1 6.17 0.94 Houston 1976:H2 6.58 0.94 Miami 1978:H1 5.92 0.94 West 1978:H1 4.55 0.96 Honolulu 1977:H2 4.81 0.95 Denver 1976:H2 8.07 0.92 LosAngeles 1975:H1 6.03 0.94 Portland 1976:H2 6.77 0.94 SanDiego 1976:H1 6.03 0.94 SanFrancisco 1975:H2 5.06 0.95 Seattle 1975:H2 8.22 0.92 Notes: (1)1st obs. isthestartingdateofthesample;(2)V0/P0 isthefirstobservedrent-priceratio(annualized percent); and(3)ρisthediscountfactor(annualized) usedintheCampbell-Shilerdecomposition.

Decomposition of Rent-Price Ratio 28 Table2 WaldTestsforTime-VaryingRiskPremia FullSample 1975-1996 1997-2005 (1) (2) (3) (4) (5) (6) City WaldTest p-value WaldTest p-value WaldTest p-value USA 118.01 0.00 130.18 0.00 14.66 0.01 Midwest 140.97 0.00 61.71 0.00 137.11 0.00 Chicago 63.44 0.00 78.38 0.00 36.75 0.00 Cincinnati 52.75 0.00 32.48 0.00 19.57 0.00 Cleveland 22.38 0.00 30.30 0.00 30.55 0.00 Detroit 17.91 0.00 29.93 0.00 172.31 0.00 KansasCity 31.48 0.00 35.09 0.00 8.61 0.07 Milwaukee 17.35 0.00 21.26 0.00 13.36 0.01 Minneapolis 83.33 0.00 40.84 0.00 43.33 0.00 St. Louis 12.55 0.01 5.16 0.27 56.83 0.00 Northeast 41.66 0.00 25.12 0.00 88.13 0.00 Boston 43.61 0.00 51.23 0.00 11.77 0.02 NewYork 48.51 0.00 115.05 0.00 32.66 0.00 Philadelphia 24.64 0.00 15.99 0.00 91.01 0.00 Pittsburgh 5.75 0.22 6.15 0.19 26.78 0.00 South 41.44 0.00 61.49 0.00 58.96 0.00 Atlanta 8.24 0.08 44.41 0.00 30.84 0.00 Dallas 29.24 0.00 14.10 0.01 31.81 0.00 Houston 75.56 0.00 46.32 0.00 1.97 0.74 Miami 82.52 0.00 9.11 0.06 2076.91 0.00 West 58.54 0.00 123.54 0.00 126.32 0.00 Honolulu 4.62 0.33 8.15 0.09 77.07 0.00 Denver 46.24 0.00 63.10 0.00 22.43 0.00 LosAngeles 236.75 0.00 270.62 0.00 73.58 0.00 Portland 14.96 0.00 17.96 0.00 46.20 0.00 SanDiego 95.22 0.00 143.95 0.00 26.88 0.00 SanFrancisco 64.44 0.00 144.42 0.00 10.08 0.04 Seattle 18.53 0.00 13.00 0.01 13.66 0.01 Notes: (1) “Wald Test” is the test statistic under the null hypothesis of constant housing premia; (2) “p-value” is the associated p-value; columns (3)-(4) are the same as (1)-(2) for the 1975-1996 sample; and columns (5)-(6) are the same as (1)-(2)forthe1997-2005sample.

Decomposition of Rent-Price Ratio 29 Table3 EstimatesoftheAR(1)modelforrealrentgrowth(equation11) (1) (2) (3) (4) Area ∆v γ t-stat σ USA 0.35 0.39 6.37 1.07 Midwest -0.04 0.51 6.25 1.02 Chicago 0.48 0.33 4.98 1.45 Cincinnati -0.03 0.12 2.43 1.53 Cleveland -0.03 0.20 3.38 1.61 Detroit -0.17 0.32 4.65 1.47 KansasCity 0.06 0.46 5.87 1.50 Milwaukee -0.07 0.13 2.48 1.81 Minneapolis 0.32 0.19 3.10 1.70 St. Louis -0.13 0.25 3.98 1.40 Northeast 0.74 0.60 7.54 1.24 Boston 1.15 0.56 7.08 1.79 NewYork 0.69 0.54 7.15 1.33 Philadelphia 0.55 0.48 6.35 1.48 Pittsburgh -0.20 0.17 3.05 1.80 South 0.02 0.38 4.98 1.06 Atlanta 0.10 0.48 6.48 1.82 Dallas 0.17 0.48 6.35 1.77 Houston -0.26 0.39 5.34 2.41 Miami 0.16 0.02 0.89 2.18 West 0.87 0.21 3.33 1.46 Honolulu 0.26 0.12 2.38 2.09 Denver 0.46 0.52 6.80 1.88 LosAngeles 1.20 0.41 6.24 1.54 Portland 0.10 0.40 5.37 1.32 SanDiego 1.48 0.60 7.86 1.78 SanFrancisco 1.53 0.30 4.42 2.28 Seattle 0.68 0.33 4.73 1.79 Notes: (1)∆vistheaveragerealgrowthrateofrents(annualizedpercent);(2)γ istheestimateofthe(annualized)AR(1) coefficient for real rental growth; (3) “t-stat” is the t-statistic on the estimate of γ in column (2); and (4) σ is the standard erroroftheregression,inannualizedpercentterms.

Decomposition of Rent-Price Ratio 30 Table4 Averagevaluesoftherent-priceratioandcomponents FullSample 1975-1996 1997-2005 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) Area V/P i ∆v π V/P i ∆v π V/P i ∆v π USA 4.93 3.57 0.35 1.59 5.09 3.69 0.34 1.66 4.49 3.27 0.39 1.42 e f e e f e e f e Midwest 4.76 3.75 -0.04 0.89 4.99 3.95 -0.05 0.92 4.23 3.31 0.00 0.84 Chicago 4.92 3.60 0.48 1.63 5.07 3.78 0.46 1.61 4.53 3.12 0.52 1.69 Cincinnati 5.08 3.64 -0.03 1.32 5.27 3.84 -0.03 1.32 4.62 3.19 -0.03 1.33 Cleveland 5.21 3.62 -0.03 1.46 5.40 3.82 -0.04 1.45 4.74 3.12 -0.01 1.47 Detroit 5.20 3.60 -0.17 1.17 5.71 3.79 -0.18 1.60 3.90 3.10 -0.13 0.06 KansasCity 5.79 3.64 0.06 2.11 5.89 3.85 0.03 1.97 5.55 3.15 0.13 2.44 Milwaukee 4.99 3.71 -0.07 1.10 5.27 3.89 -0.07 1.20 4.34 3.31 -0.06 0.88 Minneapolis 5.75 3.64 0.32 2.25 6.11 3.86 0.31 2.45 4.88 3.12 0.34 1.78 St. Louis 5.56 3.62 -0.13 1.70 5.81 3.82 -0.14 1.76 4.93 3.11 -0.12 1.54 Northeast 4.78 3.75 0.74 1.48 4.97 3.98 0.70 1.45 4.36 3.23 0.83 1.55 Boston 4.70 3.64 1.15 1.44 5.07 3.87 1.09 1.71 3.81 3.09 1.29 0.78 NewYork 3.39 3.60 0.69 0.02 3.61 3.77 0.65 0.14 2.81 3.16 0.76 -0.28 Philadelphia 6.23 3.62 0.55 2.98 6.43 3.84 0.53 2.97 5.73 3.07 0.59 3.01 Pittsburgh 5.31 3.67 -0.20 1.34 5.53 3.87 -0.20 1.35 4.79 3.21 -0.19 1.31 South 5.20 3.75 0.02 1.40 5.30 3.96 0.01 1.28 4.98 3.28 0.05 1.66 Atlanta 5.62 3.62 0.10 2.01 5.73 3.82 0.11 1.93 5.33 3.11 0.09 2.20 Dallas 6.05 3.62 0.17 2.49 5.89 3.83 0.17 2.12 6.44 3.10 0.18 3.40 Houston 6.69 3.64 -0.26 2.65 6.52 3.86 -0.28 2.23 7.11 3.12 -0.20 3.64 Miami 5.55 3.75 0.16 1.83 5.79 3.98 0.16 1.88 5.01 3.24 0.16 1.70 West 4.29 3.75 0.87 1.33 4.50 3.94 0.86 1.36 3.82 3.32 0.88 1.26 Honolulu 3.54 3.71 0.26 -0.23 3.71 3.89 0.27 -0.17 3.14 3.31 0.25 -0.35 Denver 6.29 3.64 0.46 2.73 6.66 3.89 0.42 2.94 5.38 3.06 0.54 2.21 LosAngeles 3.92 3.57 1.20 0.95 4.13 3.69 1.17 1.14 3.37 3.25 1.26 0.46 Portland 5.64 3.64 0.10 1.81 6.15 3.87 0.10 2.23 4.42 3.11 0.09 0.82 SanDiego 4.61 3.62 1.48 2.15 4.88 3.82 1.39 2.24 3.94 3.11 1.69 1.94 SanFrancisco 3.25 3.60 1.53 0.67 3.43 3.77 1.52 0.78 2.80 3.16 1.56 0.41 Seattle 5.58 3.60 0.68 1.94 6.05 3.82 0.68 2.42 4.39 3.03 0.68 0.73 Allresults aredisplayed inannualized percentage terms. Column(1) willnot equal (2) -(3) + (4) because wereportthe averageleveloftherentpriceratioandnottheaveragelog-level,and,weomittheconstantoflinearization. Notes: (1)V/P is the average value of therent-priceratio, fullsample; (2) iis the average value ofthe net present value ofthe expected future sequenceofrealrisk-freeratesfromequation4,fullsample;(3)∆vistheaveragevalueofthenetpresentvalueoftheexpected e future sequence of real rental growth from equation 4, full sample; (4) π is the average value of the net present value of the f expected future sequence of real housing premiafrom equation 4, full sample; columns (5)-(8) are the same as (1)-(4) for the 1975-1996sample;andcolumns(9)-(12)arethesameas(1)-(4)forthe1e997-2005sample.

Decomposition of Rent-Price Ratio 31 Table5 Trendchanges tothelogrent-priceratioanditscomponents FullSample 1975-1996 1997-2005 trend Shares(sumto1.0) trend Shares(sumto1.0) trend Shares(sumto1.0) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) Area v−p i ∆v π v−p i ∆v π v−p i ∆v π USA -0.69 0.41 0.07 0.53 -0.25 -0.62 0.11 1.51 -3.83 0.38 -0.04 0.65 e f e e f e e f e Midwest -0.83 1.09 0.11 -0.20 0.07 -8.36 -1.93 11.29 -2.47 0.87 -0.15 0.28 Chicago -0.66 0.71 0.11 0.18 -0.40 -0.35 0.27 1.08 -2.64 0.81 -0.08 0.26 Cincinnati -0.49 1.30 0.00 -0.30 0.30 -0.22 -0.01 1.22 -1.67 1.36 -0.05 -0.31 Cleveland -0.60 0.93 0.04 0.03 -0.13 -0.45 0.18 1.27 -1.03 2.21 -0.17 -1.04 Detroit -1.90 0.24 0.02 0.74 -0.86 -0.16 0.01 1.14 -2.40 0.88 -0.08 0.20 KansasCity 0.04 -16.68 -1.48 19.15 0.95 -0.06 0.00 1.06 -2.10 1.02 -0.32 0.29 Milwaukee -0.78 0.96 0.02 0.02 0.59 -0.62 -0.06 1.68 -3.49 0.58 -0.00 0.43 Minneapolis -1.02 0.57 0.00 0.43 0.28 -0.21 0.11 1.10 -5.27 0.39 -0.04 0.65 St. Louis -0.74 0.75 0.01 0.25 0.08 0.71 0.39 -0.10 -3.20 0.70 -0.01 0.32 Northeast -1.22 0.67 0.10 0.23 -1.43 0.37 0.05 0.57 -4.82 0.40 0.01 0.59 Boston -2.18 0.25 0.05 0.70 -2.45 0.02 -0.01 0.99 -5.53 0.35 -0.09 0.73 NewYork -2.13 0.24 0.07 0.69 -2.78 -0.05 0.05 1.00 -6.53 0.36 0.01 0.63 Philadelphia -1.00 0.51 0.04 0.45 -1.12 -0.05 0.01 1.04 -4.27 0.48 0.03 0.49 Pittsburgh -0.72 0.99 0.01 0.01 -0.09 2.24 -0.08 -1.16 -2.32 0.95 -0.02 0.06 South -0.22 3.87 0.19 -3.06 0.51 -1.11 -0.07 2.18 -3.00 0.69 -0.04 0.35 Atlanta -0.27 2.02 0.04 -1.05 0.33 0.17 -0.33 1.16 -3.31 0.67 -0.26 0.59 Dallas 0.69 -0.78 0.11 1.67 1.00 0.05 0.15 0.79 -2.13 1.02 -0.50 0.48 Houston 0.92 -0.63 -0.07 1.69 1.64 -0.04 -0.02 1.06 -2.12 0.96 -0.14 0.17 Miami -0.86 0.96 0.01 0.02 0.29 -1.87 -0.02 2.89 -7.20 0.27 0.01 0.72 West -0.98 0.93 0.00 0.06 -0.16 3.74 -0.30 -2.44 -4.83 0.45 -0.02 0.57 Honolulu -1.72 0.43 -0.01 0.58 -2.56 0.14 0.00 0.85 -5.24 0.38 0.03 0.59 Denver -1.04 0.50 0.05 0.45 -0.06 0.85 0.77 -0.62 -3.75 0.49 -0.33 0.84 LosAngeles -1.45 0.19 0.00 0.81 -1.31 -0.11 -0.13 1.24 -6.86 0.21 0.06 0.73 Portland -1.67 0.34 0.00 0.66 -0.54 0.11 0.14 0.75 -2.98 0.67 -0.16 0.49 SanDiego -1.31 0.42 0.06 0.52 -0.57 -0.10 -0.63 1.73 -7.01 0.31 -0.00 0.69 SanFrancisco -1.51 0.33 0.00 0.66 -1.37 -0.11 -0.03 1.14 -6.75 0.34 -0.11 0.77 Seattle -2.03 0.20 -0.01 0.82 -1.78 -0.07 -0.03 1.10 -3.90 0.47 -0.14 0.67 Notes: (1)v−pis100timestheannualizedtrendchangeinthelogrent-priceratio,fullsample;(2)iistheshareoftrend changeinlogrent-priceratioaccounted forbythetrendchangeinnetpresentvalueofrealrisk-freerates,fullsample;(3)∆v e is the share of trend change in log rent-price ratio accounted for by the trend change in net present value of real real rental f growth rates, full sample; (4) π is the share of trend change in log rent-price ratio accounted for by the trend change in net presentvalueofrealhousingpremia,fullsample;columns(5)-(8)arethesameas(1)-(4)forthe1975-1996sample;andcolumns (9)-(12) arethesameas(1)-(4)eforthe1997-2005sample.

Decomposition of Rent-Price Ratio 32 Table6a Time-seriesvariancedecompositionofthelogrent-priceratio,fullsample FullSample Shares(sumto1.0) Variances Covariances (1) (2) (3) (4) (5) (6) (7) Area v−p ∆v i π i,∆v ∆v,π i,π USA 0.73 0.03 1.12 1.32 -0.09 0.01 -1.40 f e e e f f e ee Midwest 0.94 0.05 1.83 0.98 -0.05 -0.13 -1.68 Chicago 0.78 0.03 2.24 1.35 -0.14 0.02 -2.50 Cincinnati 0.64 0.01 3.08 1.79 0.00 -0.05 -3.83 Cleveland 0.71 0.01 2.77 1.54 -0.00 -0.07 -3.25 Detroit 3.75 0.01 0.45 0.76 0.01 -0.02 -0.20 KansasCity 0.61 0.12 2.91 2.86 -0.28 -0.11 -4.49 Milwaukee 1.68 0.00 0.91 0.70 -0.02 -0.02 -0.56 Minneapolis 1.85 0.00 0.88 0.91 -0.02 0.00 -0.78 St. Louis 0.96 0.01 1.97 1.08 -0.05 0.01 -2.03 Northeast 1.99 0.08 0.70 0.80 -0.09 0.09 -0.58 Boston 5.08 0.04 0.29 0.93 -0.07 0.10 -0.29 NewYork 5.73 0.02 0.36 1.11 -0.02 0.12 -0.58 Philadelphia 1.29 0.06 1.24 1.35 -0.19 0.05 -1.51 Pittsburgh 0.93 0.01 2.00 0.78 0.00 -0.04 -1.76 South 0.35 0.05 4.58 4.01 -0.12 -0.05 -7.47 Atlanta 0.37 0.33 5.07 4.42 -0.79 -0.28 -7.75 Dallas 0.77 0.15 2.36 4.83 -0.07 -0.32 -5.95 Houston 1.17 0.09 1.35 3.63 0.09 -0.44 -3.72 Miami 1.66 0.00 0.87 0.98 0.01 0.01 -0.86 West 1.41 0.01 1.25 0.96 -0.05 0.01 -1.17 Honolulu 3.63 0.00 0.42 0.77 -0.01 0.01 -0.19 Denver 1.54 0.11 0.85 1.14 -0.05 -0.07 -0.98 LosAngeles 3.24 0.02 0.24 1.02 -0.02 -0.00 -0.25 Portland 3.22 0.01 0.48 0.77 0.03 -0.08 -0.21 SanDiego 2.82 0.11 0.66 1.10 -0.06 -0.22 -0.59 SanFrancisco 3.17 0.01 0.63 1.27 -0.06 -0.04 -0.82 Seattle 3.62 0.01 0.36 0.93 0.03 -0.07 -0.26 Notes: (1)v−pis100timesthevarianceofthelogrent-priceratio;(2) ∆vistheshareofthevarianceofv−paccounted for by the variance of the net present value of real rental growth; (3) i is the share of the variance of v−p accounted for by thevarianceofthenetpresentvalueofrealrisk-freeinterestrates;(4)π isth f eshareofthevarianceofv−paccounted forby thevarianceofthenetpresentvalueofrealhousingpremia;(5)i,∆vi e stheshareofthevarianceofv−paccounted forbythe covariance ofthe netpresent valueofrealrisk-freeinterestrates andreeal rentalgrowth; (6)∆v,π isthe shareof thevariance of v−p accounted forbythe covariance ofthe net presentvalue e o f f realrental growth andthe real housingpremium;and (7) i,π istheshareofthevarianceof v−paccounted forbythecovariance ofthenetpresent va f lueeofrealrisk-freeinterestrates andtherealhousingpremium. ee

Decomposition of Rent-Price Ratio 33 Table6b Time-seriesvariancedecompositionofthelogrent-priceratio,earlyandlatesamples 1975-1996 1997-2005 Shares(sumto1.0) Shares(sumto1.0) Variances Covariances Variances Covariances (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) v−p ∆v i π i,∆v ∆v,π i,π v−p ∆v i π i,∆v ∆v,π i,π USA 0.19 0.13 4.82 5.82 -0.68 0.17 -9.26 0.93 0.01 0.21 0.58 -0.01 -0.11 0.32 f e e e f f e ee f e e e f f e ee Midwest 0.33 0.15 5.23 3.68 -0.42 -0.39 -7.25 0.38 0.09 1.09 0.71 -0.17 -0.35 -0.37 Chicago 0.55 0.05 3.47 2.46 -0.39 0.12 -4.70 0.48 0.03 0.86 0.62 -0.05 -0.18 -0.28 Cincinnati 0.34 0.01 6.24 4.56 0.00 -0.11 -9.70 0.18 0.01 2.54 1.19 -0.08 -0.06 -2.60 Cleveland 0.50 0.02 4.21 2.86 -0.05 -0.12 -5.92 0.07 0.11 6.55 3.46 -0.52 -0.07 -8.53 Detroit 1.07 0.02 1.70 1.89 0.00 -0.16 -2.46 0.38 0.05 1.04 0.67 -0.06 -0.17 -0.53 KansasCity 0.65 0.11 2.91 3.30 -0.47 0.07 -4.92 0.28 0.19 1.48 1.01 -0.52 -0.54 -0.62 Milwaukee 0.93 0.01 1.66 1.48 -0.07 -0.04 -2.04 0.77 0.00 0.47 0.48 0.02 -0.04 0.08 Minneapolis 0.33 0.02 5.18 4.95 -0.13 0.08 -9.11 1.70 0.01 0.22 0.53 -0.04 -0.07 0.35 St. Louis 0.31 0.04 6.66 4.24 -0.31 0.09 -9.72 0.63 0.01 0.70 0.45 0.00 -0.07 -0.10 Northeast 1.72 0.11 0.81 1.14 -0.27 0.21 -1.00 1.45 0.03 0.23 0.52 0.03 -0.09 0.28 Boston 4.20 0.05 0.37 1.33 -0.15 0.16 -0.76 1.86 0.07 0.18 0.72 -0.05 -0.26 0.34 NewYork 5.38 0.02 0.41 1.51 -0.07 0.16 -1.04 2.63 0.01 0.18 0.50 0.02 -0.02 0.31 Philadelphia 0.90 0.11 1.91 2.44 -0.48 0.13 -3.10 1.26 0.02 0.30 0.47 0.06 -0.06 0.21 Pittsburgh 0.58 0.02 3.38 1.66 -0.03 -0.07 -3.95 0.34 0.02 1.28 0.51 0.07 -0.13 -0.75 South 0.11 0.18 14.65 15.78 -1.10 0.44 -28.96 0.59 0.02 0.65 0.52 -0.01 -0.14 -0.04 Atlanta 0.09 1.44 23.41 23.40 -3.56 -0.00 -43.69 0.69 0.16 0.64 0.77 -0.34 -0.54 0.30 Dallas 0.72 0.15 2.71 5.20 0.02 -0.28 -6.81 0.30 0.46 1.40 1.18 -0.89 -1.06 -0.10 Houston 1.30 0.10 1.30 3.45 0.05 -0.40 -3.51 0.29 0.12 1.29 1.01 -0.05 -0.47 -0.90 Miami 0.12 0.01 12.38 12.50 -0.02 -0.00 -23.86 3.34 0.00 0.10 0.57 0.00 0.01 0.31 West 0.45 0.02 3.91 3.48 -0.30 0.09 -6.20 1.60 0.00 0.26 0.53 -0.01 -0.05 0.26 Honolulu 3.54 0.00 0.44 0.98 -0.01 0.01 -0.42 2.10 0.00 0.17 0.53 0.02 0.02 0.26 Denver 0.45 0.38 3.05 4.14 -0.15 0.04 -6.45 0.86 0.20 0.35 0.93 -0.36 -0.68 0.57 LosAngeles 2.13 0.03 0.40 1.62 -0.08 -0.08 -0.89 3.04 0.01 0.06 0.62 0.03 0.04 0.24 Portland 1.11 0.03 1.48 1.77 0.18 -0.26 -2.20 0.67 0.05 0.54 0.60 -0.14 -0.19 0.15 SanDiego 1.22 0.24 1.63 2.71 -0.51 -0.60 -2.48 3.18 0.05 0.14 0.74 0.03 -0.23 0.27 SanFrancisco 2.04 0.02 1.07 2.27 -0.12 0.00 -2.23 2.90 0.02 0.16 0.72 -0.07 -0.18 0.35 Seattle 1.82 0.02 0.76 1.70 0.11 -0.14 -1.45 1.01 0.04 0.30 0.70 -0.11 -0.23 0.31 Columns (1)-(7) refer to the 1975-1996 sample, columns (8)-(14) refer to the 1997-2005 sample. Notes: (1) v−p is 100 times thevariance ofthelogrent-priceratio;(2) ∆v isthe shareof thevariance ofv−paccounted forbythevariance ofthe netpresentvalueofrealrentalgrowth;(3)iistheshareofthevarianceofv−paccountedforbythevarianceofthenetpresent valueofrealrisk-freeinterestrates;(4)π isthesh f areofthevarianceofv−paccounted forbythevarianceofthenetpresent value ofreal housing premia; (5) i,∆v ist e he shareof the variance ofv−p accounted for bythe covariance of the net present valueofrealrisk-freeinterestratesanderealrentalgrowth;(6) ∆v,π istheshareofthevarianceofv−paccounted forbythe e f covarianceofthenetpresentvalueofrealrentalgrowthandtherealhousingpremium;and(7)i,πistheshareofthevariance ofv−paccounted forbythecovarianceofthenetpresentvalu f eoferealrisk-freeinterestratesandtherealhousingpremium. ee

Decomposition of Rent-Price Ratio 34 Table7 Co-movementofthenetpresentvalueofrealrisk-freeratesandhousingpremia 1975-1996 1997-2005 (1) (2) (3) (4) (5) Area coef π oni R2 coef π oni R2 (1)-(3) t-stat USA -(cid:0)0.96 (cid:1) 0.76 0(cid:0).77 (cid:1) 0.21 3.44 e e e e Midwest -0.69 0.68 -0.17 0.04 1.81 Chicago -0.68 0.65 -0.16 0.04 1.84 Cincinnati -0.78 0.83 -0.51 0.56 1.43 Cleveland -0.70 0.73 -0.65 0.80 0.40 Detroit -0.72 0.47 -0.25 0.10 1.54 KansasCity -0.85 0.63 -0.21 0.06 2.05 Milwaukee -0.61 0.42 0.08 0.01 1.82 Minneapolis -0.88 0.81 0.79 0.26 3.70 St. Louis -0.73 0.84 -0.07 0.01 2.42 Northeast -0.62 0.27 0.62 0.17 2.48 Boston -1.03 0.30 0.94 0.22 2.62 NewYork -1.26 0.43 0.85 0.26 3.47 Philadelphia -0.81 0.52 0.36 0.08 2.74 Pittsburgh -0.59 0.70 -0.29 0.21 1.81 South -0.99 0.91 -0.03 0.00 3.08 Atlanta -0.93 0.87 0.23 0.05 3.22 Dallas -1.26 0.82 -0.03 0.00 4.28 Houston -1.35 0.69 -0.35 0.16 2.99 Miami -0.96 0.92 1.51 0.41 4.28 West -0.79 0.71 0.50 0.12 2.80 Honolulu -0.48 0.10 0.75 0.18 1.89 Denver -1.06 0.83 0.81 0.25 4.59 LosAngeles -1.11 0.30 1.97 0.38 3.24 Portland -0.75 0.46 0.14 0.02 2.37 SanDiego -0.76 0.35 0.99 0.18 2.42 SanFrancisco -1.05 0.52 1.08 0.26 3.38 Seattle -0.95 0.41 0.53 0.12 2.60 Notes: (1)coef π oni isthecoefficientonifromaregressionofπoniforthe1975-1996sample;(2)R2istheR-Squared of the regression of π on i for the 1975-1996 sample; columns (3)-(4) are the same as (1)-(2) for the 1997-2005 sample; and (cid:0) (cid:1) column(5)reportstheeNe e wey-Westt-statisticfo e rthedifferenceincoeefficien e ts incolumns(1)and(3). e e

Decomposition of Rent-Price Ratio 35 Table8 Trendchanges tothelogrent-priceratio,1997-2005,anditscomponents, alternativesimulations β baseline,γ baseline β baseline,γ=0.99 β=0.99,γ baseline trend Shares(sumto1.0) Shares(sumto1.0) Shares(sumto1.0) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Area v−p i ∆v π i ∆v π i ∆v π USA -3.83 0.38 -0.04 0.65 0.38 -0.45 1.06 1.09 -0.04 -0.05 e f e e f e e f e Midwest -2.47 0.87 -0.15 0.28 0.87 -1.69 1.82 1.93 -0.15 -0.78 Chicago -2.64 0.81 -0.08 0.26 0.81 -1.02 1.21 1.50 -0.08 -0.42 Cincinnati -1.67 1.36 -0.05 -0.31 1.36 -1.20 0.84 2.68 -0.05 -1.62 Cleveland -1.03 2.21 -0.17 -1.04 2.21 -2.89 1.68 3.99 -0.17 -2.83 Detroit -2.40 0.88 -0.08 0.20 0.88 -1.00 1.13 1.58 -0.08 -0.50 KansasCity -2.10 1.02 -0.32 0.29 1.02 -3.26 3.24 1.90 -0.32 -0.59 Milwaukee -3.49 0.58 -0.00 0.43 0.58 -0.09 0.51 1.35 -0.00 -0.34 Minneapolis -5.27 0.39 -0.04 0.65 0.39 -0.71 1.32 0.70 -0.04 0.34 St. Louis -3.20 0.70 -0.01 0.32 0.70 -0.22 0.53 1.24 -0.01 -0.23 Northeast -4.82 0.40 0.01 0.59 0.40 0.04 0.55 0.79 0.01 0.20 Boston -5.53 0.35 -0.09 0.73 0.35 -0.61 1.25 0.61 -0.09 0.48 NewYork -6.53 0.36 0.01 0.63 0.36 0.10 0.54 0.70 0.01 0.29 Philadelphia -4.27 0.48 0.03 0.49 0.48 0.26 0.26 0.80 0.03 0.17 Pittsburgh -2.32 0.95 -0.02 0.06 0.95 -0.30 0.35 1.89 -0.02 -0.88 South -3.00 0.69 -0.04 0.35 0.69 -0.59 0.90 1.46 -0.04 -0.42 Atlanta -3.31 0.67 -0.26 0.59 0.67 -2.59 2.92 1.20 -0.26 0.06 Dallas -2.13 1.02 -0.50 0.48 1.02 -4.66 4.63 1.79 -0.50 -0.29 Houston -2.12 0.96 -0.14 0.17 0.96 -1.49 1.53 1.71 -0.14 -0.57 Miami -7.20 0.27 0.01 0.72 0.27 0.16 0.57 0.55 0.01 0.45 West -4.83 0.45 -0.02 0.57 0.45 -0.43 0.98 1.01 -0.02 0.01 Honolulu -5.24 0.38 0.03 0.59 0.38 0.65 -0.03 0.89 0.03 0.08 Denver -3.75 0.49 -0.33 0.84 0.49 -2.31 2.82 0.81 -0.33 0.52 LosAngeles -6.86 0.21 0.06 0.73 0.21 0.62 0.17 0.57 0.06 0.38 Portland -2.98 0.67 -0.16 0.49 0.67 -1.69 2.02 1.18 -0.16 -0.02 SanDiego -7.01 0.31 -0.00 0.69 0.31 -0.03 0.71 0.55 -0.00 0.45 SanFrancisco -6.75 0.34 -0.11 0.77 0.34 -1.75 2.41 0.66 -0.11 0.45 Seattle -3.90 0.47 -0.14 0.67 0.47 -1.47 2.00 0.77 -0.14 0.38 β istheAR(1)parameterforthereal10-yearTreasuryBillandγ istheAR(1)parameterfortherealgrowthrateofrents. ThebaselineβfortheUSAis0.87(annualizedvalue)andthebaselineγvaluesarereportedincolumn(2)oftable3. Notes: (1) v−pis100timestheannualizedtrendchangeinthelogrent-priceratio,itmatchescolumn(9)oftable5;(2)iistheshareof trendchangeinlogrent-priceratioaccountedforbytrendchangeinnetpresentvalueofthereal10-yearTreasuryBill,baseline e β; (3) ∆v is the share of trend change in log rent-price ratio accounted for by trend change in net present value of real real rental growth rates, baseline β; (4) π isthe shareof trendchange inlogrent-priceratioaccounted for bytrend change innet f presentvalueofrealhousingpremia,baselineβ;columns(5)-(8)arethesameas(1)-(4)forβ baselineandγ=0.99(annualized value);andcolumns(9)-(12) aretheesameas(1)-(4)forγ baselineandβ=0.99(annualizedvalue).

Decomposition of Rent-Price Ratio 36 Figure1 GrowthRateofRealRents(AnnualizedPercent) National 8 6 4 2 0 -2 -4 1975 1980 1985 1990 1995 2000 2005 Midwest Northeast 8 8 6 6 4 4 2 2 0 0 -2 -2 -4 -4 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 Midwest National Northeast National South West 8 8 6 6 4 4 2 2 0 0 -2 -2 -4 -4 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 South National West National

Decomposition of Rent-Price Ratio 37 Figure2 Rent-PriceRatio(AnnualizedPercent) National 6 5 4 3 1975 1980 1985 1990 1995 2000 2005 Midwest Northeast 6 6 5 5 4 4 3 3 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 Midwest National Northeast National South West 6 6 5 5 4 4 3 3 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 South National West National

Decomposition of Rent-Price Ratio 38 Figure3 Nominal10-YearTreasuryInflation Expectations andReal10-YearTreasury(AnnualizedPercent) 16 12 8 4 0 1975 1980 1985 1990 1995 2000 2005 Nominal 10-Year Treasury 10-Year Inflation Expectations Real 10-Year Treasury

Decomposition of Rent-Price Ratio 39 Figure4 Real10-YearTreasuryanditsFutureExpected PresentValue(AnnualizedPercent) 8 7 6 5 4 3 2 1 1975 1980 1985 1990 1995 2000 2005 Real 10-Year Treasury Future EPV of Real 10-Year Treasury

Decomposition of Rent-Price Ratio 40 Figure5 HousingPremia(AnnualizedPercent) National 4 3 2 1 0 -1 1975 1980 1985 1990 1995 2000 2005 Midwest Northeast 4 4 3 3 2 2 1 1 0 0 -1 -1 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 Midwest National Northeast National South West 4 4 3 3 2 2 1 1 0 0 -1 -1 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 South National West National

Decomposition of Rent-Price Ratio 41 Figure5(Contd.) HousingPremia(AnnualizedPercent), MidwestCities Chicago Cincinnati 6 6 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 Cleveland Detroit 6 6 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 Kansas City Milwaukee 6 6 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 Minneapolis St. Louis 6 6 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005

Decomposition of Rent-Price Ratio 42 Figure5(Contd.) HousingPremia(AnnualizedPercent), NortheastCities Boston New York 6 6 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 Philadelphia Pittsburgh 6 6 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005

Decomposition of Rent-Price Ratio 43 Figure5(Contd.) HousingPremia(AnnualizedPercent),SouthCities Atlanta Dallas 6 6 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 Houston Miami 6 6 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005

Decomposition of Rent-Price Ratio 44 Figure5(Contd.) HousingPremia(AnnualizedPercent), WestCities Honolulu Denver 6 6 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 Los Angeles Portland 6 6 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 San Diego San Francisco 6 6 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 Seattle 6 5 4 3 2 1 0 -1 -2 1975 1980 1985 1990 1995 2000 2005

Decomposition of Rent-Price Ratio 45 Figure6 NationalLogRent-PriceRatioanditsComponents 6 5 4 3 2 1 0 -1 1975 1980 1985 1990 1995 2000 2005 Annualized Log Rent-Price Ratio plus Constant Annualized NPV of Real Interest Rates Annualized NPV of Real Housing Premia (Negative) Annualized NPV of Real Rent Growth