221 Week 3 A /For WIZARD KIM

2017 V45 2: pp. 259–300

DOI: 10.1111/1540-6229.12127

REAL ESTATE ECONOMICS

U.S. House Prices over the Last 30 Years: Bubbles, Regime Shifts and Market (In)Efficiency Rose Neng Lai* and Robert Van Order**

This paper studies U.S. house prices across 45 metropolitan areas from 1980 to 2012. It applies a version of the Gordon dividend discount model for long-run “fundamentals” and uses Mean Group and Pooled Mean Group estimation to estimate long-run and short-run determinants of house prices. We find great similarity across cities in that the long-run house prices are largely explained by the same fundamentals; the long-run rent to price ratio is approximately 5% plus 0.75 times the real interest rate (which is on the order of 2%). However, ad- justments to deviations from the fundamentals are slow, in the long-run, closing the gap at a rate of around 10% per year. We find sharp differences in short- run adjustments (momentum) away from the fundamentals across cities, and the differences are correlated with local supply elasticities (more momentum with lower elasticity). Analysis of residuals suggests strong cyclical deviations, which are mean-reverting.

Introduction

The U.S. real estate market underwent a boom (or “bubble”) period after 2000 until about 2006, when property prices started to fall and mortgage default started to rise, leading to the collapse of the securitized mortgage market and then the collapse of many financial institutions. Since then house prices have largely recovered. The fluctuations varied widely across cities. The purpose of this paper is to use a simple asset pricing model as the basis for explaining house price fluctuations across cities and time in the United States. The model allows easy separation of adjustments into long-run fundamentals and short-run and long-run dynamics.

Studies of the housing bubble and house price dynamics have become abun- dant by now. Examples are Capozza, Hendershott and Mack (2004), Chan, Lee

*Faculty of Business Administration, University of Macau, Taipa, Macau SAR, China or [email protected].

**George Washington University, Washington D.C. or [email protected].

C© 2016 American Real Estate and Urban Economics Association

260 Lai and Van Order

Figure 1 � Ratio of U.S. National Rent Index to National House Price Index.

and Woo (2001), Chang, Cutts and Green (2005), Black, Fraser and Hoesli (2006), Coleman, LaCour-Little and Vandell (2008), Hwang, Quigley and Son (2006), Lai and Van Order (2010), Taipalus (2006), Wheaton and Nechayef (2008), and Nneji, Brooks and Ward (2013). Case, Cotter and Gabriel (2011) explain the speculative forces with housing asset pricing models, while Ling, Ooi and Le (2015) use nonfundamentals-based sentiments of home buyers, builders, and lenders in explaining how the feedback effects result in housing boom and bust periods.

We analyze price fluctuations over the past few decades by exploiting U.S. data on equivalent rents for owner-occupied housing; we use it in the same way as dividends in pricing shares. This allows us to cut out the use of variables like local income, employment, housing supply and other factors that represent local market conditions. We can then focus on an asset pricing approach. Many papers (see e.g., Clark 1995, Ayuso and Restoy 2006, Lai and Van Order 2010, Sommer et al. 2011) have studied the determinants of rent to price ratios as a way of estimating determinants of house prices. It is clear from Figure 1, which depicts rent to price ratio in the aggregate, over time, that the ratio was relatively stable before the “bubble years,” decreased sharply and then increased thereafter, almost returning to prebust levels, suggesting momentum away from fundamentals but also mean reversion. What follows

U.S. House Prices over the Last 30 Years 261

assesses the extent to which Figure 1 can be explained over time and across cities.

Our contribution is the use of more recent data and, more importantly, in the structure of our model, which provides a clean separation among: long-run effects, which are given by the fundamentals (through a dividend discount model); long-run adjustment to the fundamentals (given by a speed of ad- justment coefficient); and short-run momentum away from the fundamentals (given by the sums of coefficients of lagged rent to price ratios). We use a version of the Gordon dividend discount model to model long-run fundamen- tals, and analyze data on house prices across 45 metropolitan areas (MSAs) from 1980 to 2012. The Mean and Pooled Mean Group estimations allow us to constrain the long-run to look like the Gordon model, and tease out long-run adjustment speeds, while allowing a looser specification of short- run variations, including momentum. To the best of our knowledge, this is the first paper to exploit the data, the Gordon model and the estimation technique in this manner.

We find an intuitively plausible result, that in the long-run the ratio of rents to prices (aka cap rate) is around 5% plus 0.75 times the real interest rate, and that this result is very similar across specifications and cities. However, there is a long lag in adjustment—the gap between current and long-run rent to price ratios closes at a rate of about 10% per year, and is similar across cities. We find considerable momentum which, although not explosive, varies considerably across cities. The variation is related to a well-known measure (Saiz 2010) of housing supply elasticity. Hence, we find that some rather simple rules of thumb are broadly consistent with house price behavior in the United States over the past few decades.

However, while house prices are somewhat predictable, because of the het- erogeneity of momentum government policies applied throughout the country (e.g., monetary policy) in an attempt to curb bubbles might not be effective. We also find that while the boom from 2000 to 2006 was longer than usual, examination of the residuals reveals that the boom was on the verge of cool- ing off around 2002 or 2003, but started up again in a way that is consistent with stimulus from the newly emerging subprime securitization business.

Fundamental Models and Models for Estimation

Modeling House Price Growth

The equilibrium condition for holding property is that the current dividend, or rent, from the property equal the appropriate (risk-adjusted) interest rate

262 Lai and Van Order

plus expected capital gains over the period. Then, given an information set, �t , the equilibrium condition for holding property at time t is given by

1

Rt /Pt = it + α − E (( Pt +1/Pt ) − 1|�t ) ≡ it + α − πt , (1) where Pt is the price of a constant quality house, Rt is corresponding net rental income, which in our case is the imputed net rent for owner-occupied housing, it is the risk-adjusted hurdle rate, α is (constant) depreciation, and πt is expected house price growth. Equation (1) applies to a particular location. We add location notation when we perform estimation for individual MSAs later.

Equation (1) can be used to determine house prices given expected future prices. Because future prices depend on future rents, current price depends on future rents via the expected present value relationship:

Pt = ∞∑

i =0 E ( Rt +i /It +i |�t ) + lim E (1/It +i |�t ) , (2)

where the discount factor is given by It = 1+ it, and therefore It+i is the discount rate for an i-period loan at time t. Assuming that the second term approaches zero, and dividing through by Rt, gives the usual expected present value formulation:

Pt /Rt = ∞∑

i =0 E

( 1/Dt +i |�t

) , (2’)

where Dt +i = (1 + it +i )/(1 + πt +i ∗), and π t+i* is the expected rate of growth of rent from period t to period i. If D and the rate of growth of rents are constant in the long-run, then the reciprocal of (2’) will converge to (1), which gives the long-run fundamentals. We can interpret bubbles as situations where the second term in (2) does not converge.

We take the imputed rent from owner-occupied property to be the market rent of comparable properties, which works if Equation (2’) is applied to an owner who is indifferent between owning and renting. Rent is observable at any time t, and it is assumed to be forecastable thereafter. The advantage of this approach is that it does not require development of a model of housing demand and supply. For instance, Glaeser, Gyourko and Saks (2005) emphasizes the role of inelastic supply in house price growth, especially due to local policy variation. Our rent variable captures this effect without having to estimate supply (or demand) elasticities across cities and time. This, together with

1See Lai and Van Order (2010).

U.S. House Prices over the Last 30 Years 263

allowing momentum to vary across cities, facilitates capturing heterogeneity across cities.

The model is not operational in its current form because it requires a model of how expectations are formed. More broadly, it needs to acknowledge transaction costs that can make adjustment to (2’) gradual. It has been well known at least since Case and Shiller (1989) that house prices adjust slowly to shocks, making them more predictable than is consistent with standard notions of market efficiency. We take Glaeser and Nathanson (2015) as our point of departure. They develop a pricing model for house prices where traders are “almost” rational. The “almost” is because rational expectation models are subject to big errors for small mistakes; as a result their optimal forecasting procedure uses past prices to forecast housing in a way that allows short-run momentum (positive feedback), long-run mean reversion, and excess volatility. Our estimation allows for all of these properties. We add the restriction that the long-run mean is given by the Gordon dividend model.

Long-Run Specification

Theory suggests that we should expect prices and rents to move together in the long-run, in a way that depends on real interest rates. In the long-run, the Gordon model implies

Rt Pt

= it − πt + α ≡ rt + α, (1’)

where rt is the real rate. For housing, we should allow for possible tax and other effects. For instance, if the focus is on the tax break for not paying tax on imputed rent for owner-occupied housing and not taxing capital gains on housing, then

Rt Pt

= (1 − θ )it − πt + α ≡ −θ i + r + α, (3)

where θ is marginal tax rate for the marginal homeowner (marginal in the sense of being indifferent between owning and renting). However, it may be the case that in an inflationary world high nominal interest rates provide a cash-flow problem for home buyers (even if real rates are constant), who can- not draw down savings or borrow against human capital. Then the coefficient of i is ambiguous.

264 Lai and Van Order

We formulate the long-run as:

Rt Pt

= ct , (4)

where ct = αi it − απ πt + α ≡ γi it + γr rt + α.

Then c is the “cap rate” for housing. Our tests are of whether property values converge to rent divided by cap rate, how fast they converge, the nature of short-run deviations, and whether coefficients make sense. We use long-run risk-free rates for i, so that estimates of α contain risk adjustments as well as depreciation and long-run expected future rent growth, that vary across cities but do not change over time. We also use a direct measure of real risk-free rates.

Instead of defining short-run fundamentals, we analyze how short-run devi- ations move over time. In general, we expect γr to be close to 1, α to be a number of around 4% or 5% and undetermined about γi which would be expected to be zero absent tax and liquidity effects. In our data set we have R and P only in the form of indices. Hence, testing for the magnitude of coefficients (e.g., whether γr is close to 1) requires calibration assumptions, which are made below.

Dynamic Heterogeneous Panel Estimation

Following a variation on Glaeser and Nathanson (2015), we assume that R/P depends on a lagged function of past levels of R, P and i. We decompose the relationship into long-run and short-run effects using the Pooled Mean Group (PMG) and Mean Group (MG) estimation models developed in Pesaran, Shin and Smith (1997, 1999).2 Our hypothesis is that the information set �t in (1), (2) and (2’) contains only past rents, prices and interest rates, and that prices ultimately adjust to fundamentals.

The MG and PMG models are restricted maximum likelihood estimations, based on an autoregressive distributed lag (ARDL) model (see Pesaran and Shin 1997). Traditionally economic analysis has focused on long-run rela- tionships among the dependent variables and the regressors. PMG estimation facilitates identifying common long-run relationships (expression (1’)) and individual short-run dynamics separately. The intercepts that reflect the fixed effect, the short-run coefficients and the error variances are allowed to differ across cities, but the long-run coefficients are constrained to be identical. MG

2Ott (2014) uses PMG to study the house price dynamics in the Euro area.

U.S. House Prices over the Last 30 Years 265

estimation is different in that the long-run coefficients are also allowed to vary across cities.

Our model can be represented by:

� Rc,t Pc,t

= l∑

j =1 λc, j,�

Rc,t − j Pc,t − j

+ q∑

j =0

n∑ k=1

δ k

c, j x k

c,t − j + δc + εc ,t , (5)

where Rc,t Pc,t

is property rent to price ratio in city c, at time t

δc captures city specific fixed effects xkc,t-j is the kth of n regressors for city c δkc, j is the coefficient of the kth regressor for city c λc,j are scalars εc,t are the city specific errors c represents panels or cities, i = 1,2, . . . ,N t represents time in quarters, t = 1,2, . . . ,T j is an indicator of lags j = 0,1,2, . . . ,l for lagged dependent variable j = 0,1,2, . . . ,q lags for regressors

Letting ρ = R P

, (3) can be written as:

�ρct = λcρc,t −1 + q∑

j =0

n∑ k=1

δ k

c, j �x k

c,t − j + δc + εc,t (6)

which, when written in error correction form, yields:

�ρct = ϕc {

ρc,t −1 − n∑

k=1 βc

k x kc,t

} −

q∑ j =0

n∑ k=1

δ k c, j �x

k c,t − j + δc + εc,t , (7)

where

ϕc = −(1 − λc ), βkc = δk c,0

(1 − λc ) .

Expression (7) is used for the MG estimation model. It allows us to restrict some of the parameters inside the brackets to be zero so we can get to a long-run specification that looks like the Gordon model, as given in (1’), but with fewer restrictions on short-run adjustment parameters across cities. Among the items inside the bracket in (7) are long-run fixed effects, αc, and αc = δc/ϕc . The coefficients (one for each city) before the brackets, ϕc , denote the speed of reversion to the long-run, after short-run deviations. The

266 Lai and Van Order

adjustment outside the brackets is momentum, which will disappear if the model is not explosive.

For PMG we assume homogeneous long-run relations; i.e., βc k = βk for all

cities. Then:

�ρct = ϕc {

ρc,t −1 − n∑

k=1 β

k x kc,t

} −

q∑ j =0

n∑ k=1

δ k c, j �x

k c,t − j + δc + εct . (8)

The double summation term in (7) and (8) can include lagged values of changes in the dependent variable, which is our measure of momentum. We measure the level of momentum by the sum of these coefficients. We expect the error correction coefficients, ϕc , to be negative and the sums of the coefficients of lagged changes in R/P (momentum) to be positive but less than 1 (in order that the model converge). Hence, the model can have the properties of short-run momentum and long-run mean reversion in Glaeser and Nathanson (2015), to which we add the effect of forcing the reversion to look like the Gordon model and testing to see if all of it holds together.

Note that the model requires rents and prices to grow at a constant rate within each city in the long-run,3 but the presence of δc allows the growth rates to vary across cities in the long-run, which in turn causes the long-run level of R/P to differ across cities. Long-run equilibrium is given by:

ρc = n∑

k=1 β

k x kc − δc/ϕc. (9)

Recall that the last term in (9), which is the negative of the ratio of the constant term in (8) (short-run constant term) divided by the correction speed (which is negative), is the long-run constant term, αc. This allows for differences in risk premia and growth rates across cities that are not time-varying.

Preliminary Tests

Before testing for the existence of a long-run relationship, however, we check if the series are stationary. If some or all the rental income relative to house prices and interest rates are nonstationary, and are integrated of the same order, we can check for their long-run relationship with cointegration tests. Hence, the first step is to test if these series are unit roots.

3We also tried to relax this condition by adding a linear time trend, common to all cities inside the brackets in (6). Results are similar, and therefore are omitted here.

U.S. House Prices over the Last 30 Years 267

We perform cointegration analysis tests developed by Westerlund (2007) to confirm the existence of long-run relationships among the series. Specifically, Westerlund (2007) relies on the error correction based cointegration. That is, as in expression (7), when ϕi , the error correction parameter, is significantly different from zero, then there is a long-run relationship (i.e., cointegration). Formally, H0: ϕi = 0 and H1: ϕi < 0. Westerlund (2007) proposes four tests. The first two are “group mean statistics” which state that rejecting the null of no cointegration means that at least one or more cities are cointegrated. The test statistics are

Gτ = 1

N

N∑ i =1

ϕ̂i

SE(ϕ̂i ) and Gα =

1

N

N∑ i =1

T ϕ̂i ϕ̂i (1)

, (10)

where N is the number of cities, SE(ϕ̂i ) is the usual standard error of ϕ̂i , and ϕ̂i (1) is the kernel estimator of ϕi (1) = 1 −

∑l j =1 ϕi j . The first expression is

the t-ratio while the latter is the coefficient statistics (analogous to the rho- statistics of Phillips and Perron (1988)).The other two are “panel statistics,” where a rejection of the null of no cointegration means rejection for the panel as a whole. Formally, they are

Pτ = ϕ̂

SE(ϕ̂) and Pα = T ϕ̂. (11)

Again, the first expression is the t-ratio while the latter is the coefficient statistic. Westerlund (2007) shows that these statistics are more accurate than the widely used cointegration test due to Pedroni (2004) when the residuals in expression (3), εi,t, are moving average series.

Given that long-run cointegration exists, we next find the long-run and short- run effects among variables using the MG and PMG models. The Hausman test can be used to check if a common long-run coefficient exists. That is, not rejecting the null hypothesis of common coefficients between the MG and PMG means common coefficients should be adopted.

Testing

Our measure of house price is the quarterly house price index released by the Federal Housing Finance Administration (FHFA), which provides a repeat sales house price index for over 100 individual Metropolitan Statistical Areas (MSAs) since 1980. This is primarily an index of sales price of owner- occupied houses. The rent series is the “owner’s equivalent rent of primary residence” obtained from the Bureau of Labor Statistics. It is an estimate of what owner-occupied units would rent for if rented in the market.

268 Lai and Van Order

We use 10-year Treasury bonds as a measure of nominal long term discount rate; we also use the 10-year Treasury Inflation-Protected Securities (TIPS) (bonds issued by the U.S. Treasury that are indexed to inflation) as a direct measure of real interest rates in some variations of the model. This requires assuming expected rent growth to be the same as expected CPI growth; using it allows elimination of expected inflation from our cap rate. TIPS data are available only after 1998. We interpolate the series back to 1979 Q4, as is explained in Section 3, to obtain a TIPs series for the entire period. Since it is also possible that market risk could affect the cap rate, we use the Merrill Lynch 1-year high yield rates minus the 1-year Treasury to generate a yield spread to represent market-wide risk.

There is a total of 45 MSAs that have all data available for the required sample period. Since some cities that are more prone to boom might behave differently from those less prone to boom, for purposes of comparison, we follow Lai and Van Order (2010) in classifying the MSAs into bubble MSAs and nonbubble MSAs, based on house price growth rates in previous periods (see Appendix A for names of the cities). This classification is for compar- ison only; we do not have separate estimates for the two categories, unless specified. Our purpose is to examine whether the classifications of bubble and nonbubble cities in Lai and Van Order (2010) (for which data stopped before the recovery) continue to hold after the bust, in the sense of whether the differences in momentum found in this paper correspond to the bubble city classifications.

Panel Unit Root and Cointegration Tests

If property markets were efficient in the usual sense, house prices relative to rents would resemble random walk series, and therefore be nonstationary. If these series are not integrated of order 1 (i.e., I(1)), cointegration tests fail, and MG and PMG estimations cannot be applied. We perform panel unit root tests for the rent to price ratio for different sample periods. Several panel unit root tests are adopted here, such as Harris and Tzavalis (1999) test, Breitung test due to Breitung (2000) and Breitung and Das (2005), test due to Hadri (2000), and the IPS, and Fisher-type, due to Im, Pesaran and Shin (2003), and Choi (2001) respectively, which are suitable for unbalanced data (not all the MSAs time series have the same length). Except for the test due to Hadri (2000), all tests have the null hypotheses as existence of unit root, and alternative hypotheses as at least one panel stationary. The null hypothesis of Hadri (2000) is that all panels are stationary, while the alternative is to have some panels containing unit root.

U.S. House Prices over the Last 30 Years 269

Table 1 shows that the rent–price ratio is nonstationary in all the tests, while all the differenced series are stationary, whether de-meaned or not. We also test for stationarity for the interest rate series using the Phillips–Perron unit root test and the Augmented Dickey Fuller test, which also show that interest rates are in general nonstationary, or vaguely stationary, while their differenced series are stationary.

We then verify that there is a long-run relationship with the Westerlund (2007) panel cointegration test. We perform cointegration tests separately with nonbubble MSAs and bubble MSAs. Results are shown in Table 2. While the results are not very strong throughout, there is pair-wise cointegration between the rent to price ratio and the various interest rates, particularly with the 10-year Treasury rates, and the 10-year TIPs. Apparently the bubble MSAs have stronger cointegration with interest rates than the nonbubble MSAs. For instance, while the overall and the nonbubble MSAs rent to price ratios do not seem to be cointegrated with the high yield rate, there is strong cointegration in the bubble MSAs. This is a strong hint that bubble MSAs might be riskier than their counterparts, and they might be driven by different forces. Taken as a whole, the tests suggest that the study of long term relationships is feasible.

Model Estimation and Tests on Regime Shift

An important feature of MG and PMG is that the models are able to show regime shifts across time because there are both long-run and short-run com- ponents, the latter of which would reflect intertemporal differences, while avoiding problems of insufficient data series length. Therefore, testing for subperiods to account for possible regime shifts is not essential.4

We run MG and PMG estimation with variations on expressions (5) and (6), taking account of various lags of the short-term variables. We tried 1-, 2- and 4-lag models and found that the 4-lag (four quarters) models in general were more stable across subperiods and tests than the other two. We used variations on lagged R/P and real and nominal interest rates as short-run factors.

PMG is chosen over MG if a small Hausman test statistic is coupled with a correspondingly large p-value; that is, the null hypothesis that there is a com- mon long-run effect is not rejected. Otherwise, MG estimation is better, and

4We have run the tests for various subperiods including 1980–1990, 1991–1998, 1999–2006 and 2007–2013. Tests results are roughly similar although not very stable, probably because of loss of degrees of freedom for such short sample periods. More importantly, while MG estimations are valid, PMG estimations fail in several cases. Even for MG estimations, some variables do not have reasonable coefficients.

270 Lai and Van Order

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U.S. House Prices over the Last 30 Years 271

Table 2 � Cointegration tests of rent to price and various rates based on Westerlund (2007).

Gτ Gα Pτ Pα

Rent to Price Ratio and 10-Year Rate All MSAs –1.413*** –1.650 –8.576*** –1.486 Nonbubble MSAs –1.036 –0.814 –4.539** –0.713 Bubble MSAs –2.110*** –3.193 –7.63*** –3.166***

Rent to Price Ratio and 10-Year TIPs All MSAs –2.133*** –1.91 –13.643*** –2.112**

Nonbubble MSAs –1.958*** –1.561 –10.717*** –1.75 Bubble MSAs –2.456*** –2.555 –8.443*** –2.547**

Rent to Price Ratio and High Yield Rate All MSAs –0.880 –1.709 –6.97*** –1.732*

Nonbubble MSAs –0.266 –0.400 –1.443 –0.329 Bubble MSAs –2.014*** –4.125*** –7.442*** –4.17***

Rent to Price Ratio and High Yield Spread All MSAs –0.566 –0.575 –4.292 –0.674 Nonbubble MSAs –0.323 –0.253 –1.675 –0.259 Bubble MSAs –1.014 –1.17 –3.877** –1.272

Rent to Price Ratio and Interpolated 10-year TIPs All MSAs –1.868*** –2.262 –11.531*** –2.255***

Nonbubble MSAs –1.575*** –1.433 –7.293*** –1.348 Bubble MSAs –2.41*** –3.791 –8.758*** –3.769***

Note: *, ** and *** denote significance at the 10%, 5% and 1% levels, respectively.

the reported long-run coefficients are the means of the long-run coefficients for individual MSAs.

The residuals from the regressions also provide insights into bubble formation of the MSAs. For instance, a sum of coefficients of the autoregressive tests of the residuals equal to or greater than 1 indicates an explosive bubble. Also, a change in the variance of the residuals from these autoregressive tests indicates a shift in risk patterns. We therefore run panel autoregressive tests on the residuals from both the MG and PMG results using 2, 4, and 8 lags. Our models do not need very long lags because long-run phenomena are reflected in the gradual adjustment to long-run effects in the MG and PMG estimations.

We extract the residuals from the panel autoregressive tests to find the variance of the residuals in subperiods of 1990–1999, 2000–2006 and 2007–2013, which roughly represent prebubble, bubble and post-bubble periods. We then use the Goldfeld-Quandt test to check whether the variances across periods are statistically significantly different, and hence existence of regime shifts.

272 Lai and Van Order

For instance, a variance in the bubble period higher than that of the prebubble period indicates increased risk during the bubble period. We also conjecture that volatility of these MSAs after the bubble burst will return to prebubble levels. We test this with all MSAs, bubble MSAs and nonbubble MSAs.

Test Results

We use several combinations of long-run and short-run variables in the MG and PMG tests. The idea is to have different rates to represent the right-hand side of expression (2’) because, as seen from Figure 2(A), while the few selected interest rate series exhibit similar movements, we are not sure which can best represent real rates, and different representations might generate different effects in our model. Reckoning that the best real rates are actually the TIPs, which however start only in 1998, we interpolate the data first by regressing TIPS on 10-year Treasury rates and inflation with the data from 1998 to 2013 and then obtaining the estimated 10-year TIPS series from 1980 to 1997 based on the regression thus generated. The interpolated TIPS and the actual TIPS are shown in Figure 2(B). We use interpolated TIPs before 1998 and actual TIPs thereafter.

For long-run variables, we have included combinations of nominal 10-year Treasuries and/or the real interest rates represented by 10-year Treasury rates minus rent growth, Merrill-Lynch 1-year high-yield spread, and/or TIPS. Short-run variables include the lagged rent to price ratio to capture the mo- mentum, and the lagged long-run variables to capture the short-run effects. In some cases, we also added high yield rates minus rent growth together with the 10-year Treasury rates minus rent growth in the short-run variables in order to capture the effects from risky versus riskless rates.

All the tests generate consistent explanatory powers from the variables used, with signs as expected. For instance, the error correction coefficients are all negative and significant; and short-run momentum is strong. However, for some of the variations, PMG estimation does not outperform MG estima- tion (for example, see Appendix D where the Hausman test statistics are significant).

Calibration

Because rent and price are both indexes, we cannot judge magnitudes of coefficients without calibration assumptions. However, from Figure 1, rent divided by price was close to 1 for a long period before the boom and bust. Note also that our interest rate values are in units like 5.0% rather than 0.05. The S&P 500 price-earnings ratio, which is comparable to the inverse of our rent to price ratio, was around 20 for much of this period. Hence, if we take

U.S. House Prices over the Last 30 Years 273

Figure 2 � Movements of various interest rates used for the tests.

Notes: Panel (A): plots of 10-year Treasuries, Merrill Lynch high-yield rate, and corresponding yield spread. Panel (B): plots of actual TIPS and interpolated TIPS.

274 Lai and Van Order

20 as a comparable, and a 5% as the benchmark rent to price ratio, then multiplying the coefficients of interest rates and other variables other than lagged rent to price on the right-hand side by 5 will approximately calibrate them. Then, if the long-run coefficient of the real rate from our model is, say, 0.15, multiplying it by 5 to get 0.75 is consistent with the impact being close to, but a bit less than,1. Hence, coefficients between 0.10 and 0.20 are consistent with our expectations regarding magnitude.

Model Estimation with Nominal Rates—Model A

As our first set of tests, Model A, we take combinations of 10-year TIPS, 10-year Treasuries minus rent growth, and the 10-year Treasuries as long term variables, and lagged values of changes in the ratio of rent to price index, yield spread, 10-year Treasuries, and 10-year Treasuries minus rent growth as short-term variables. The latter variable choices are mostly to include the same variables as in the long-term variables, plus the lagged values of the dependent variable (the ratio of rent to price), which captures momentum effects.

The MG and PMG results are depicted in Panel 3A of Table 3. Where coefficients are allowed to vary across MSAs (MG models) the table presents averages of the coefficients. In general, the results across the four models are similar. All but one (Model A2) show that PMG performs better than MG (low Hausman tests coupled with the p-values of 10% or higher); that is, the long-run variables share the same parameters across MSAs. Even when the test fails to show that PMG is better than MG in Model A2, the model is successful in showing that there are long-run versus short-run relationships in the variables. We focus on the PMG results in the following analysis.

The 10-year TIPS dominate the long-run effect, with coefficients in the neigh- borhood of 0.15 in models A1 and A2. The 10-year Treasury minus rent growth, taken to be a way to generate real rates, is not significant, nor are 10- year Treasuries, except for Model A3. In general, the magnitudes are roughly consistent with our expectations, given our suggested calibration. The error correction coefficients are almost identical throughout. An error correction co- efficient of about –0.027 from the quarterly data translates into about 10.8% per annum, meaning that the deviation from the long-run is corrected at a rate of about 10% each year. The momentum (from the 4 lagged rent to price ratios) is strong and significant, with an average sum of coefficients of 0.4. The yield spread does not show strong short-term effects.

When both the 10-year Treasuries and the 10-year Treasuries minus rent growth enter the short run separately, neither is especially significant. But

U.S. House Prices over the Last 30 Years 275

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280 Lai and Van Order

when both are added together, as in Models A3 and A4, both become signif- icant for all lagged values, but with opposite signs and approximately equal in magnitude. In these cases the long-run 10-Year TIPS have a smaller ef- fect; however, the strange pattern of short-run signs makes these two columns difficult to interpret.

The constant term, which is about 0.027 across the four models, shows the average of the fixed effects of all the MSAs. The different values of this constant term, divided by the adjustment speeds (see expression (8’)) for different MSAs identify which cities are long-run “growth stocks” relative to others. From (8’) and the average adjustment coefficient we see that the average constant term in the long-run equation for the rent to price ratio is about 1% per year, or 5% after calibration.

Panel 3B summarizes the averages, maxima and minima of the sums of the coefficients of the short-run variables in Model A1 for the bubble and nonbubble MSAs. Sums of coefficients for individual MSAs are listed in Appendix A, Panel A for nonbubble MSAs and Panel B for bubble MSAs. On average, the error correction coefficients of the bubble MSAs (–0.0327) are more negative than those of the nonbubble MSAs (–0.0229). This means that there is stronger correction back to the long-run in bubble MSAs.

The average value of momentum for nonbubble MSAs is 0.3495 while that of bubble MSAs is 0.5967. The bubble MSAs have stronger momentum than the nonbubble MSAs; the differences between the maxima and minima of these sums of coefficients show a wider range for bubble MSAs than non- bubble counterparts. Similarly, the average and the range (maximum minus minimum) of the constant terms for bubble MSAs are bigger than those of the nonbubble MSAs. This shows that the bubble MSAs are a bit more like “growth stocks”. In particular, the average for the nonbubble MSAs (1.0504, or 5.3 after calibration) is slightly higher than that of the bubble MSAs (0.9227, or 4.6 after calibration).

Figure 3(A) shows the residuals and the moving averages (over 4 periods) of the residuals of the PMG results of Model A1. The residuals series oscillates quite a lot, which shows autocorrelation. The moving average shows more clearly a regime shift in the residuals for the period of 2000–2006 (the bubble period) and then another shift thereafter (the crisis and post-crisis period). Figure 3(B) shows the residuals of Model A1 on bubble and nonbubble MSAs separately. The bubbles MSAs generate more volatile residuals. The residuals of the nonbubble MSAs during the bubble period are somewhat smaller than other periods, but fluctuate to a bigger extent in the post-bubble period. We note that for the boom from 2000 to 2006 there is an apparent

U.S. House Prices over the Last 30 Years 281

Figure 3 � Residuals from Model A1: pooled mean group estimation.

Notes: Panel (A): average and moving averages of residuals. (B): residuals of bubble versus nonbubble MSAs.

282 Lai and Van Order

pause around 2003, which suggests two parts to the boom, the second one roughly coincides with the rise of the subprime securitization market.

We ran residual tests on all models. Here we explain the effects of the variables, focusing only on the results from Model A1. Panel 3C depicts the autoregression equations of the residuals from the PMG estimation. We test AR(2), AR(4) and AR(8) equations for three subperiods, 1990–1998, 1999– 2006 and 2007–2013, for all MSAs, and for bubble and nonbubble MSAs separately. In general, the AR(2) equation shows the worst explanatory power. We sum all the coefficients of the lagged error term in panel 3D. Also listed in the same panel are the variances of the residuals from these autoregressive equations.

The results for all MSAs show that the “bubbles” were not explosive; the sums of the coefficients are all less than 1 (see Panel 3D). The sums of the coefficients of the lagged residuals increased in the middle period (1999– 2006), and then decreased after the financial crisis. However, when further testing the equations by separating bubble MSAs from nonbubble MSAs, the residual autoregression in the bubble MSAs dropped significantly after the bubble burst. This decrease is even more severe and becomes negative, a level lower than the precrisis period, for bubble MSAs. Those of the nonbubble MSAs are higher than the prebubble period, and the drop from the crisis period is either minimal (for 2-lag model) or actually increases for the other two models.

The variances of the residuals from the autoregression equations of bubble MSAs are bigger than those of nonbubble MSAs, indicating that the former experienced bigger price fluctuations than the latter. Means of the standard deviations of the residuals across cities reveal that bubble MSAs are mostly more volatile than the nonbubble MSAs, as expected. Also, the variances of the residuals are smaller in the bubble period (1999–2006) than in the prebub- ble period. They then increase after the bubble burst. A possible explanation is that volatility tended to be lower during the bubble period when people over- whelmingly anticipated that house prices would continue to rise, but prices fluctuated more after the crisis when expectations were more chaotic.

The question is how statistically different these variances are. Panel 3E shows the comparison with the Goldfeld-Quandt test. Across the table is the compar- ison with different autoregression equations for the different sets of MSAs. The different AR equations mostly do not generate different variances ex- cept during the bubble period. Of greater importance is the second half of the table, which compares the variances across the different periods. In the “All MSAs” case, the variances in the prebubble and bubble periods are

U.S. House Prices over the Last 30 Years 283

significantly different, and for bubble versus post-bubble period. An excep- tion is the variance of bubble MSAs during and after the bubble periods. Hence, we see some evidence of a temporary regime shift during the bubble period in terms of volatility.

As mentioned earlier, Table 3 shows that the MG results are better than those from PMG in Model A2. We therefore also show the short-run coefficients from the MG estimation with Model A2 by MSAs in Appendix B. Comparing this with the figures in Appendix A, we find that the coefficients for the same set of short-run variables for individual MSAs are quite close. Furthermore, since the long-run coefficients generated from the MG estimation are averages of the coefficients across all MSAs, we also list in Appendix C the long-run coefficients of all MSAs, bubble and nonbubble, for Models A2, and B1 to B3, for comparison. Those coefficients that deviate for more than 1 standard deviation away from the mean are highlighted in bold. Not many MSAs have coefficients too far off the mean. Most of those coefficients are insignificant coefficients from the tests.

Explanatory Power of the Model

In the previous subsection, we analyzed the PMG model, which provides long-run and short-run analysis, and how an additional residual autoregression model can further capture the movement of the rent to price ratio. The next issue is how much explanatory power each of these components can offer. To answer this, we compare the sum of squares of the estimated values from the PMG model to the actual rent to price ratios, which is similar to finding coefficients of determination (R2). Furthermore, to separate the short- run from the long-run effects, we borrow the concept of coefficients of partial determination (Partial r2). Note that the total explanatory power is not the sum of the two coefficients of partial determination (see Borcard (2002)).

Figure 4 shows the fraction of the actual rent to price ratio explained by the PMG model and the residual autoregression model. Figure 5 shows the coefficients of partial determination of long-run and short-run compo- nents. Both figures consist of two panels, Panel A for nonbubble MSAs and Panel B for bubble MSAs. Recall that the tests are not run separately for bubble MSAs versus nonbubble MSAs.

The PMG model performs better in explaining the bubble MSAs; the mean explanatory power shown in Figure 4 is 30.34% for nonbubble MSAs and 41.98% for bubble MSAs. Furthermore, Figure 5 shows that the fundamentals as represented in the long-run component do not explain much of the rent to price ratio, although bubble MSAs are better explained than the nonbubble

284 Lai and Van Order

Figure 4 � Fraction of actual rent to price ratio explained by pooled mean group model and residual autoregression model: Model A1.

Notes: Panel (A): Nonbubble MSAs. Panel (B): Bubble MSAs.

U.S. House Prices over the Last 30 Years 285

Figure 5 � Coefficients of partial determination of long-run and short-run components of model A1.

Notes: Panel (A): Nonbubble MSAs. Panel (B): Bubble MSAs.

286 Lai and Van Order

Table 4 � Explanatory Power of Various Components in Model A1.

Mean Maximum Minimum Max - Min

Nonbubble MSAs Long-Run Partial R2 0.1324 0.4437 0.0299 0.4138 Short-Run Partial R2 0.5145 0.8857 0.3006 0.5851

PMG Model 0.3034 0.7819 0.1301 0.6519 Residual Autregression 0.5035 0.7636 0.2059 0.5577 Total = PMG + Residual AR(4) 0.7345 0.9849 0.4137 0.5712 Bubble MSAs Long-Run Partial R2 0.2313 0.5533 0.0667 0.4866 Short-Run Partial R2 0.6472 0.9150 0.2506 0.6644 PMG Model 0.4198 0.7925 0.0902 0.7023 Residual Autregression 0.3635 0.7956 0.0293 0.7663 Total = PMG + Residual AR(4) 0.6790 0.9612 0.0464 0.9149

MSAs. Instead, it is the short-run momentum from lagged rent to price ratio that does most of the job. The mean long run and short run coefficients of partial determination for Bubble MSAs, shown in Table 4, are 23.13% and 64.72%; while those of the Nonbubble MSAs are 13.24% and 51.45%. The short-run component explains more of the bubble MSAs than the nonbubble counterparts, as seen from higher partial r2 in most of the bubble MSAs. Finally, Figure 6 shows residuals from the residual autoregressions, which suggests something unusual about the 2003–2006 period and perhaps the period after it.

Model Estimation with Other Rates—Models B and C

Table 5 lists the PMG and MG results for two other sets of models for com- parison. Model B uses only the 10-year TIPS, while Model C has similar long-run variables to Model A except replacing the 10-year TIPS with high yield spread. Model B echoes the results of Model A in that the 10-year TIPS exert long-run effects on the housing rent-to-price dependent variable. Other analyses are similar to the four variations of Model A. All the long-run vari- ables in Model C are significant, suggesting that these variables are capturing effects that the 10-year TIPS have been influencing, but are absent there. In sum, the different variations of the different models that we have tested (lengthy results of some of which are not shown here, and are available upon requests) all show strong effects from the long-run interest rate variables, with the TIPS fitting the explanation best and consistently, therefore supporting the fundamental model.

U.S. House Prices over the Last 30 Years 287

Figure 6 � Residuals of the AR(4) residual equation from Model A1.

Notes: Panel (A): all MSAs. Panel (B): Bubble and Nonbubble MSAs.

288 Lai and Van Order

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From the calibration we get an approximation to the magnitude of the effects of real rates (i.e., the long-run 10-yr TIPS in Models A and B) on R/P by mul- tiplying the relevant coefficients inside the brackets by 5. The Tables suggest that the calibrated coefficients of TIPS are less than 1, mostly around 0.75, which is consistent with the theory. Hence, on average the long-run rent to price ratio is around 5% plus around 0.75 times the real rate. This is consistent with a marginal tax rate of 25% and no money illusion or liquidity problem.

Determinants of Momentum

One needs to be careful in using “bubble city” as a classification—especially when based on a small historical sample. Here we look at whether there is persistence in where “bubbles” tend to happen, and why. Recall that we have followed the classifications of bubble and nonbubble MSAs in Lai and Van Order (2010) for comparison sake. Here, we apply those classifications to see if the differences in levels of momentum found in this paper are con- sistent with the classifications. Appendices A and B give results for mo- mentum (sums of own-lag coefficients) by previously categorized bubble and nonbubble MSAs for two versions of the model. Over the entire sample, the previously-classified bubble MSAs have more momentum in both the tables—approximately 0.6 versus 0.3, which suggests some consistency over time.

The model suggests that prices fluctuate considerably more than rents, and that the fluctuations (momentum) vary widely across cites. A possibility is that the differences across MSAs are due to a common factor, like supply elasticity, in the sense that supply-constrained cities might show faster short- run price adjustments (momentum) in the presence of sticky rents, with longer run adjustments back to the fundamentals. Saiz (2010) estimates supply elas- ticities by city and argues that the elasticities are inversely related to housing prices (Wheaton et al. (2014) also draw to the same conclusion). Here we test for the correlations between our momentum measures and those elasticities for the cities for which the two series overlap (34 cities). We expect that cities with low supply elasticity will have stronger momentum.

We ran Ordinary Least Squares regressions to check the relationship between momentum and supply elasticity. Figure 7 shows the plots of the regression results and the actual observations. Supply elasticity is significantly negatively related to our momentum measure in the test for the regression for all MSAs. The figure shows that the bubble MSAs have a strong tendency toward lower elasticity. Hence, we can conclude that the bubble MSAs as classified in Lai and Van Order (2010) indeed exhibit more momentum over the sample and some of this is due to inelastic supply.

U.S. House Prices over the Last 30 Years 291

Figure 7 � Plots of the OLS regression and actual observations of sum of changes in rent-to-price ratio on supply elasticity (*implies significance).

It is also possible that the second last term in expression (8), which is the long-run constant term and measures whether a city is a growth stock, is related to the supply elasticities of the cities. We also ran OLS regressions for the long-run constants on the supply elasticities, but found that there is no significant relationship between the two, and therefore do not report the detailed results here.

Comments, Policy Implications and Conclusions

We use quarterly data, from 1980 to 2013 to estimate long-run and short-run relationships among house prices and rental rates and interest rates. Using Pooled Mean Group estimation and Mean Group estimation, we find that a simple Gordon dividend discount model explains long-run behavior well; the rent to price ratio adjusts back to the long-run fundamentals given by the Gordon model in quite similar ways across cities, albeit slowly. The funda- mentals do not satisfactorily explain quarter-to-quarter changes in housing

292 Lai and Van Order

prices, which are largely governed by momentum, in quite different ways across cities, and the differences across cities, are, in part, explained by dif- ferences in supply elasticities.

Our results suggest that policy makers should be cautious when trying to curb housing booms and busts with aggregate policies—like monetary policy— because short-run price movements are so different across cities. In fact, the model suggests patience in the sense that the long-run prices appear to be priced in about the same way almost everywhere. Government policies to curb the market will cause overshooting in some declining markets as it tries to slow down hot markets.

Our model is also consistent with the long-standing proposition that housing price processes do not appear to come from an efficient market, but are instead forecastable. Long adjustment lags in house prices and the associated inefficient pricing are well known, at least going back to Case and Shiller (1989). Our contribution is to tie the long lags more precisely to fundamentals and to show that the fundamentals look very similar across cities, with sensible coefficient magnitudes. While the data suggest that we can be confident that house prices across cities look similar in the sense of returning to the Gordon model in the long-run, they adjust slowly at about the same speed (the adjustment speeds are consistently around 10% per year across cities), which indicates that it takes around 6 years to get half-way to the long-run.

The model is coherent across time in that the current period is governed by the same fundamentals as that in the 1980s and 1990s, and that the recent cycle, while worse than previous ones, was not qualitatively different. We find regime shifts in a narrow sense—that there have been periods when deviations from our estimated equations were consistently positive or negative for several quarters and where volatility was higher than in other periods. However, we have not found evidence of permanent regime changes. The residual plots from our model (fundamental plus short-term momentum and auto-correlated residuals all added together) show that the 2000–2006 bubble period includes a second surge in 2003, a year that coincides with the surge of “private label” securitization.

We have received helpful comments from Min Hwang, the editor and two referees, and participants at the 2013 AsRES Annual Conference at Kyoto, Japan, and the 2015 AREUEA Annual Conference, Boston, and excellent research assistance from Nicole Rui-hui Xu.

U.S. House Prices over the Last 30 Years 293

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U.S. House Prices over the Last 30 Years 295

Appendix A: Sum of Short-Run Coefficients for Individual MSAs from PMG Estimation of Model A1

Panel A: Nonbubble MSAs

Long-Run Constant= [Constant/

Sum of Sum of absolute Nonbubble Error Sum of (�Yield (�10Y value error MSAs Correction (�R/P) spread) Treasury) Constant correction]

Akron −0.0303 0.2463 0.0077 0.0177 0.0352 1.1617 Ann Arbor −0.0353 0.4618 0.0000 −0.0107 0.0344 0.9745 Atlanta −0.0158 0.4285 0.0038 0.0009 0.0146 0.9241 Atlantic City −0.0201 0.6690 0.0090 0.0178 0.0243 1.2090 Baltimore −0.0252 0.8728 0.0048 −0.0004 0.0111 0.4405 Boulder −0.0252 −0.0868 0.0132 −0.0092 0.0161 0.6389 Bremerton −0.0179 0.5212 0.0078 −0.0158 0.0189 1.0559 Chicago −0.0281 0.7512 0.0059 0.0112 0.0366 1.3025 Cincinnati −0.0130 0.6262 0.0048 0.0150 0.0152 1.1692 Cleveland −0.0296 0.1864 0.0050 0.0188 0.0362 1.2230 Dallas −0.0107 0.2651 0.0041 0.0039 0.0095 0.8879 Denver −0.0182 0.4579 0.0054 0.0034 0.0145 0.7967 Detroit −0.0379 0.5713 0.0030 −0.0031 0.0411 1.0844 Flint −0.0117 −0.3830 0.0156 0.0012 0.0144 1.2308 Fort Worth −0.0120 0.1211 0.0040 0.0033 0.0117 0.9750 Gary −0.0351 0.3488 0.0102 0.0140 0.0515 1.4672 Greeley −0.0268 −0.0648 0.0036 −0.0017 0.0257 0.9590 Houston −0.0105 0.2116 0.0031 −0.0013 0.0069 0.6571 Kansas City −0.0294 0.0580 0.0031 −0.0014 0.0270 0.9184 Lake County −0.0164 0.4308 0.0042 0.0124 0.0237 1.4451 Milwaukee −0.0240 0.4724 0.0055 0.0127 0.0311 1.2958 Minneapolis −0.0255 0.3067 0.0050 −0.0095 0.0204 0.8000 Philadelphia −0.0208 0.6959 0.0031 0.0293 0.0258 1.2404 Pittsburgh −0.0338 −0.2266 −0.0008 0.0124 0.0361 1.0680 Racine −0.0170 0.5750 0.0047 −0.0053 0.0208 1.2235 St.Louis −0.0249 0.4735 0.0014 0.0029 0.0252 1.0120 Wilmington −0.0220 0.4458 0.0116 0.0110 0.0264 1.2000 Average −0.0229 0.3495 0.0055 0.0048 0.0242 1.0504 Maximum −0.0105 0.8728 0.0156 0.0293 0.0515 1.4672 Minimum −0.0379 −0.3830 −0.0008 −0.0158 0.0069 0.4405

296 Lai and Van Order

Appendix A: � Continued.

Panel B: Bubble MSAs

Long-Run Constant= [Constant/

Sum of Sum of absolute Bubble Error Sum of (�Yield (�10Y value error MSAs Correction (�R/P) spread) Treasury) Constant correction]

Anchorage -0.0345 -0.1519 -0.0053 -0.0113 0.0204 0.5913 Boston -0.0332 0.7903 0.0066 0.0059 0.0337 1.0151 Fort Lauderdale -0.0333 0.8314 0.0045 -0.0070 0.0272 0.8168 Los Angeles -0.0264 0.8309 0.0052 -0.0077 0.0239 0.9053 Miami -0.0356 0.3745 0.0090 -0.0103 0.0306 0.8596 New York -0.0251 0.8122 0.0054 0.0093 0.0287 1.1434 Phoenix -0.0380 1.0877 0.0004 -0.0022 0.0083 0.2184 Portland -0.0119 0.4946 0.0054 -0.0031 0.0121 1.0168 Riverside -0.0292 0.8154 0.0083 -0.0129 0.0275 0.9418 Salem -0.0247 0.7440 0.0061 0.0146 0.0334 1.3522 San Diego -0.0351 0.8033 0.0081 -0.0070 0.0379 1.0798 San Francisco -0.0239 0.7903 0.0087 -0.0052 0.0236 0.9874 San Jose -0.0234 0.7344 0.0122 -0.0133 0.0223 0.9530 Seattle -0.0200 0.6085 0.0107 0.0057 0.0200 1.0000 Tacoma -0.0294 0.6889 0.0035 -0.0046 0.0335 1.1395 Tampa -0.0324 0.9046 0.0040 -0.0001 0.0233 0.7191 Honolulu -0.1059 -1.2240 0.0055 0.1443 0.1587 1.4986 Washington -0.0267 0.8056 0.0050 -0.0037 0.0099 0.3708 Average -0.0327 0.5967 0.0057 0.0051 0.0319 0.9227 Maximum -0.0119 1.0877 0.0122 0.1443 0.1587 1.4986 Minimum -0.1059 -1.2240 -0.0053 -0.0133 0.0083 0.2184

Appendix B: Sum of Short-run Coefficients for Individual MSAs from MG Estimation of Model A2

Panel A: Nonbubble MSAs

Sum of Sum of Nonbubble Error Sum of (�Yield (�10Y Treasury MSAs Correction (�R/P) spread) – Rent Growth) Constant

Akron –0.0322 0.2152 0.0064 0.0049 0.0387 Ann Arbor –0.0378 0.4740 0.0000 –0.0262 0.0361 Atlanta –0.0316 0.4564 0.0024 –0.0031 0.0380 Atlantic City –0.0206 0.6690 0.0109 0.0183 0.0252

U.S. House Prices over the Last 30 Years 297

Appendix B: � Continued.

Panel A: Nonbubble MSAs

Sum of Sum of Nonbubble Error Sum of (�Yield (�10Y Treasury MSAs Correction (�R/P) spread) – Rent Growth) Constant

Baltimore –0.0264 0.9038 0.0055 –0.0025 0.0088 Boulder –0.0450 –0.1161 0.0057 –0.0340 0.0235 Bremerton –0.0293 0.5172 0.0156 –0.0082 0.0446 Chicago –0.0317 0.7618 0.0078 0.0142 0.0430 Cincinnati –0.0141 0.5324 0.0054 0.0124 0.0167 Cleveland –0.0350 0.0900 0.0074 0.0131 0.0433 Dallas –0.0110 0.2479 0.0029 –0.0006 0.0097 Denver –0.0231 0.4495 0.0035 0.0017 0.0173 Detroit –0.0393 0.5247 0.0103 –0.0221 0.0378 Flint –0.0277 –0.4631 0.0152 –0.0195 0.0378 Fort Worth –0.0120 0.1042 0.0043 0.0002 0.0114 Gary –0.0593 0.3670 0.0120 0.0208 0.1003 Greeley –0.0278 –0.0570 0.0099 –0.0035 0.0297 Houston –0.0133 0.1927 0.0043 –0.0030 0.0101 Kansas City –0.0204 –0.1626 0.0072 –0.0066 0.0120 Lake County –0.0165 0.3692 0.0087 0.0105 0.0257 Milwaukee –0.0192 0.3822 0.0056 0.0095 0.0168 Minneapolis –0.0243 0.2697 0.0057 –0.0115 0.0170 Philadelphia –0.0229 0.6889 0.0035 0.0223 0.0264 Pittsburgh –0.0753 –0.1545 –0.0014 0.0033 0.0959 Racine –0.0130 0.5510 0.0071 –0.0060 0.0187 St.Louis –0.0171 0.2864 0.0032 0.0054 0.0131 Wilmington –0.0247 0.4518 0.0117 –0.0026 0.0365 Average –0.0278 0.3167 0.0067 –0.0005 0.0309 Maximum –0.0110 0.9038 0.0156 0.0223 0.1003 Minimum –0.0753 –0.4631 –0.0014 –0.0340 0.0088

Panel B: Bubble MSAs

Sum of Sum of Bubble Error Sum of (�Yield (�10Y Treasury MSAs Correction (�R/P) spread) – Rent Growth) Constant

Anchorage –0.0374 –0.1426 –0.0023 –0.018 0.0228 Boston –0.0396 0.8257 0.0066 0.0077 0.041

298 Lai and Van Order

Appendix B: � Continued.

Panel B: Bubble MSAs

Sum of Sum of Bubble Error Sum of (�Yield (�10Y Treasury MSAs Correction (�R/P) spread) – Rent Growth) Constant

Fort Lauderdale –0.0426 0.8612 0.0103 0.0018 0.0435 Los Angeles –0.0338 0.838 0.0057 –0.0062 0.0285 Miami –0.0337 0.359 0.0179 0.004 0.0241 New York –0.0298 0.8328 0.0039 0.0065 0.0412 Phoenix –0.0568 1.0931 0.0014 0.0158 0.0305 Portland –0.0207 0.4741 0.0081 0.0027 0.0161 Riverside –0.0380 0.8306 0.0117 –0.0055 0.0462 Salem –0.0487 0.8205 0.0062 0.0073 0.0433 San Diego –0.0366 0.7826 0.0093 –0.0075 0.0331 San Francisco –0.0470 0.8003 0.0043 –0.0169 0.0373 San Jose –0.0405 0.7424 0.0088 –0.0215 0.0297 Seattle –0.0269 0.6138 0.0146 0.0076 0.0308 Tacoma –0.0363 0.7028 0.0129 0.0002 0.0487 Tampa –0.0292 0.998 0.0087 –0.0085 –0.004 Honolulu –0.2046 –1.0474 –0.0149 0.0006 0.2264 Washington –0.0208 0.8639 0.0051 –0.0117 –0.0081 Average –0.0457 0.6249 0.0066 –0.0023 0.0406 Maximum –0.0207 1.0931 0.0179 0.0158 0.2264 Minimum –0.2046 –1.0474 –0.0149 –0.0215 –0.0081

Appendix C: Long–run Coefficients for Individual MSAs from MG Estimation of Models B1–B3

Panel A: Nonbubble MSAs

Model A2 Model B1 Model B2 Model B3

Nonbubble 10-year 10-yr 10-yr 10-yr 10-yr MSAs Treasury TIPs TIPs TIPs TIPs

Akron –0.0401 0.2329 0.1318 0.1319 0.1064 Ann Arbor 0.0166 0.1284 0.1673 0.1610 0.1239 Atlanta –0.0039 0.0722 0.0622 0.0614 0.0816 Atlantic City 0.0109 0.1222 0.1505 0.1521 0.1620 Baltimore 0.0294 0.1469 0.1376 0.1247 0.0341 Boulder 0.0683 0.0896 0.2658 0.2647 0.2593 Bremerton 0.0622 –0.1296 –0.0017 –0.0038 0.0289 Chicago 0.0202 0.0900 0.1408 0.1390 0.1363 Cincinnati –0.1353 0.4562 0.0996 0.0989 0.1077

U.S. House Prices over the Last 30 Years 299

Appendix C: � Continued.

Cleveland –0.0722 0.3101 0.1322 0.1302 0.1170 Dallas –0.0454 0.2602 0.1242 0.0984 0.0322 Denver 0.0750 0.0071 0.2378 0.2362 0.2125 Detroit –0.0706 0.3557 0.1869 0.1864 0.1839 Flint –0.2238 0.5983 0.0617 0.0556 0.0629 Fort Worth –0.0344 0.2439 0.1417 0.1102 –0.0079 Gary 0.0116 0.0342 0.0608 0.0598 0.0556 Greeley 0.0137 0.0844 0.1079 0.1078 0.0904 Houston 0.0110 0.0921 0.1283 0.1052 0.0018 Kansas City –0.1501 0.6175 0.1619 0.1613 0.1446 Lake County –0.1472 0.4481 0.0990 0.0994 0.1093 Milwaukee –0.0233 0.3593 0.2867 0.2846 0.2709 Minneapolis –0.0107 0.2109 0.1820 0.1786 0.2178 Philadelphia 0.0390 0.0922 0.2084 0.2030 0.2209 Pittsburgh 0.0284 0.0214 0.1042 0.1025 0.1066 Racine –0.1211 0.3384 0.0739 0.0707 0.0625 St.Louis –0.1780 0.6481 0.1273 0.1247 0.0746 Wilmington –0.0516 0.1782 0.0553 0.0509 0.0626 Mean –0.0341 0.2263 0.1346 0.1295 0.1543 Standard Dev. 0.0795 0.1987 0.0658 0.0664 0.1040 Maximum 0.0750 0.6481 0.2867 0.2846 0.3836 Minimum –0.2238 –0.1296 –0.0017 –0.0038 –0.0079

Panel B: Bubble MSAs

Anchorage 0.0221 0.1000 0.1717 0.1681 0.2153 Boston 0.0479 0.0441 0.1505 0.1457 0.2208 Fort Lauderdale 0.0460 –0.0180 0.0961 0.0928 0.1349 Los Angeles 0.0821 –0.0054 0.2262 0.2247 0.2050 Miami 0.0096 0.1827 0.2080 0.2097 0.2188 New York 0.0413 –0.0171 0.0746 0.0736 0.0841 Phoenix –0.0636 0.1095 0.1734 0.0842 0.1439 Portland 0.1313 –0.0331 0.3090 0.3101 0.3076 Riverside 0.0316 –0.0067 0.0714 0.0697 0.0635 Salem 0.1025 0.0861 0.3631 0.3632 0.3613 San Diego 0.0476 0.1078 0.2403 0.2385 0.2265 San Francisco 0.1292 –0.0519 0.2848 0.2819 0.2866 San Jose 0.1100 –0.0021 0.2910 0.2904 0.2957 Seattle 0.0818 –0.0681 0.0804 0.0759 0.1334 Tacoma 0.0481 –0.0197 0.0842 0.0806 0.1284 Tampa 0.1300 0.3073 0.1498 0.1271 0.0206 Honolulu 0.1458 –0.0290 0.3780 0.3762 0.3836 Washington 0.0521 0.4123 0.3866 0.4023 0.3529 Mean 0.0664 0.0610 0.2077 0.2008 0.2102 Standard Dev. 0.0530 0.1293 0.1082 0.1149 0.1064 Maximum 0.1458 0.4123 0.3866 0.4023 0.3836 Minimum –0.0636 –0.0681 0.0714 0.0697 0.0206

Note: Numbers in bold are greater than 1 standard deviation away from the mean.

300 Lai and Van Order

Appendix D: Examples of Other PMG and MG Estimations

Long-Run Is 10-yr Long-Run Is High Treasury – Rent Growth Yield – Rent Growth

PMG MG PMG MG

long-run coefficients Trend 0.0194*** –0.0095 –0.0059*** –0.0052***

10Y- RentG 0.2926*** –0.0523 HY - RentG –0.0366*** –0.0601**

short-run coefficients Error Correction –0.0181*** –0.0289*** –0.0257*** –0.0352***

�R/Pt –1 0.039 0.0269 0.0653 * 0.0635*

�R/Pt –2 0.0860 *** 0.0748*** 0.1060*** 0.1040***

�R/Pt –3 0.1569 *** 0.1536*** 0.1828*** 0.1840***

�R/Pt –4 0.1118 *** 0.1123*** 0.1358*** 0.1374***

�Yield spreadt 0.0013 ** 0.0012** 0.0023*** 0.0025***

�Yield spreadt– 1 0.0015 0.0014 0.0022 * 0.0023

�Yield spreadt– 2 0.0015 0.0015 0.0024 ** 0.0027***

�Yield spreadt– 3 0.0015 ** 0.0016** 0.0025*** 0.0027***

�Yield spreadt– 4 –0.0004 –0.0004 0.0006 0.0008 �10Yt – RentGt –0.0056

** –0.0056** –0.0016 –0.0016 �10Yt –1 – RentGt –1 0.0072

** 0.0070** 0.0104*** 0.0102***

�10Yt –2 – RentGt –2 –0.0030 ** –0.0029* 0.0002 0.0004

�10Yt –3 – RentGt –3 –0.0040 *** –0.0038*** –0.0006 –0.0005

�10Yt –4 – RentGt –4 –0.0018 –0.0017 0.0012 0.0014 Constant –0.0331*** –0.0074 0.0592*** 0.0850***

Log Likelihood 12725 12808 12688 12740 Hausman statistic 7.17 5.85 p-Value 0.0277 0.0535

Note: *,** and ***denote significance at the 10%, 5% and 1% levels respectively.

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