Myths and Facts about the Alleged Over-Pricing of U.S. Real Estate

Abstract

This paper uses a multi-factor pricing model with time-varying risk exposures and premia to examine whether the 2003–2006 period has been characterized, as often claimed by a number of commentators and policymakers, by a substantial mispricing of publicly traded real estate assets (REITs). The estimation approach relies on Bayesian methods to model the latent process followed by risk exposures and idiosynchratic volatility. Our application to monthly, 1979–2009 U.S. data for stock, bond, and REIT returns shows that both market and real consumption growth risks are priced throughout the sample by the cross-section of asset returns. There is weak evidence at best of structural mispricing of REIT valuations during the 2003–2006 sample.

Appendix

The assumption of $\boldsymbol {\epsilon } _{t}\equiv (\epsilon _{1,t},\epsilon _{2,t},...,\epsilon _{N,t})^{\prime }\sim N(0,\mathbf {I}_{N})$ in Eq. 5 allows to estimate independently parameters for each asset $i=1,...,N$. For the sake of brevity, we rewrite the model in Eq. 5 for each i as

$$\begin{array}{rll} x_{t}&=&\beta _{0,t}+\displaystyle\sum\limits_{j=1}^{K}\beta _{j,t}f_{j,t}+\sigma _{t}\epsilon _{t} \\ \beta _{j,t}&=&\beta _{j,t-1}+k_{j,t}\eta _{j,t}\quad j=0,...,K\qquad \\ \ln (\sigma _{t}^{2})&=&\ln \left(\sigma _{t-1}^{2}\right)+k_{2,t}\upsilon _{t}\quad i=1,...,N \text{,} \end{array} $$

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where the subscript i has been omitted, $\mathbf {\epsilon }_{t}\sim N(0,1), (\boldsymbol {\eta } _{t},\upsilon _{t})^{\prime }\sim N(0,\mathbf {Q})$ with $\mathbf {Q}$ a diagonal matrix characterized by the parameters $q_{0}^{2}, q_{1}^{2},...,q_{K}^{2}, q_{\upsilon }^{2}$, and $\boldsymbol {\kappa }_{t}\equiv (\kappa _{0,t},...,\kappa _{K,t},k_{2,t})^{\prime }$ is a $(K+2)\times 1$ vector of unobserved uncorrelated 0/1 processes with $\Pr [\kappa _{j,t}=1]=\pi _{j}$ for $j=0,...,K$ and $\Pr [k_{2,t}=1]=\pi _{2k}$. The model parameters are the structural break probabilities $\mathbf {\pi }\equiv (\pi _{0},...,\pi _{K},\pi _{2})^{\prime }$ and the vector of variances of the break magnitude $\mathbf {q}^{2}\equiv (q_{0}^{2},q_{1}^{2},...,q_{K}^{2},q_{\upsilon }^{2})$. They are collected in a $(2(K+2)\times 1)\ \text {vector}\ \boldsymbol {\theta } \equiv (\mathbf {\pi }^{\prime },(\mathbf {q}^{2})^{\prime })^{\prime }$.

Independent conjugate priors are used to ease posterior simulation. Priors could differ across assets $i=1,...,N$. For the break probability we assume simple Beta distributions,

$$ \pi _{j}\sim \text{Beta}(a_{j},b_{j})\qquad \pi _{j}^{k_{2}}\sim \text{Beta}\left(a_{j}^{k_{2}},b_{j}^{k_{2}}\right), $$

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where the hyperparameters $a_{j}$ and $b_{j} (j=0,...K)$ reflect prior beliefs about the occurrence of breaks. For the variance parameters the inverted Gamma-2 prior is chosen,

$$ q_{j}^{2}\sim IG(\nu _{j},\delta _{j})\qquad q_{\upsilon }^{2}\sim IG(\nu _{\upsilon },\delta _{\upsilon })\text{,} $$

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where $\nu _{j}$ expresses the strength of the prior mean.

For posterior simulation we run the Gibbs sampler in combination with the data augmentation technique by Tanner and Wong (1987). The latent variables $B=\left \{ \beta _{t}\right \} _{t=1}^{T}$, $R=\left \{ \sigma _{t}^{2}\right \} _{t=1}^{T},$ and $\mathcal {K}=\left \{\kappa _{t}\right \} _{t=1}^{T}$ are simulated alongside the model parameters, $\boldsymbol {\theta }$. Define $x=\left \{ x_{t}\right \} _{t=1}^{T}$ and $f=\left \{ \left \{ f_{j,t}\right \} _{j=1}^{K}\right \} _{t=1}^{T}=\left \{ f_{t}\right \} _{t=1}^{T}$, the complete data likelihood function is given by

$$\begin{array}{rll} p(x,{} B,{} \mathcal{K}, {} R|\boldsymbol{\theta} , {} f) &\,=\,&\prod\limits_{t=1}^{T}p {} \left(x_{t}|f_{t},\beta _{t},\sigma _{t}^{2}\right){} \prod\limits_{j=0}^{m}p {} \left(\beta _{jt}|\beta _{jt-1},\kappa _{jt},q_{j}^{2}\right) \\ &&\times p {} \left(\sigma _{t}^{2}|\sigma _{t-1}^{2},k_{2,t},q_{\upsilon }^{2}\right) {} \prod\limits_{j=0}^{k}\pi _{j}^{k_{jt}}(1\, - \,\pi _{j})^{1-k_{jt}}\pi _{2k}^{k_{2t}}(1\, - \,\pi _{2k})^{1-k_{2t}}.\\ \end{array} $$

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Combining prior and likelihood, we obtain the posterior density $p(\theta ,B,\mathcal {K},R|x,f) \propto p(\theta ) p(x,B,\mathcal {K},R|\theta ,f)$. Defining $\mathcal {K}_{\beta }=\left \{ \kappa _{0,t},...,\kappa _{K,t}\right \} _{t=1}^{T}$ and $\mathcal {K}_{\sigma }=\left \{ k_{2t}\right \} _{t=1}^{T}$, the sampling scheme consists of the iterative steps:

• 1. Draw $\mathcal {K}_{\beta }$ conditional on $R,\mathcal {K}_{\sigma },\boldsymbol {\theta }$, x and f.

• 2. Draw B conditional on $R,\mathcal {K},\boldsymbol {\theta }$, x and f.

• 3. Draw $\mathcal {K}_{\sigma }$ conditional on $B,\mathcal {K}_{\beta },\boldsymbol {\theta }$, x and f.

• 4. Draw R conditional on $B,\mathcal {K},\boldsymbol {\theta }$ , x and f.

• 5. Draw $\boldsymbol {\theta }$ conditional on $B,\mathcal {K}$, x and f.

The first step applies the efficient sampling algorithm of Gerlach et al. (2000), the main advantage being drawing $\kappa _{j,t}$ without conditioning on the states $\beta _{j,t}$, as Carter and Kohn (1994) instead do. The conditional posterior density for $\kappa _{\beta ,t}$, $t=1,..,T$ unconditional on B is:

$$\begin{array}{rll} p(\kappa _{\beta ,t}|\mathcal{K}_{\beta ,-t},\mathcal{K}_{\sigma },R,\boldsymbol{ \theta },r) &\propto& p(r|\mathcal{K}_{\beta },\mathcal{K}_{\sigma },R, \boldsymbol{\theta} )p(\kappa _{t}|\mathcal{K}_{\beta ,-t},\boldsymbol{\theta} ) \\ &\propto& p(x^{t+1,T}|f^{t+1,T},\mathcal{K},R,\boldsymbol{\theta} ) \\ && \times p(x_{t}|x_{1,t-1},f_{t},\kappa _{\beta ,1,t-1},R,\boldsymbol{\theta} ,x) \\ && \times p(\kappa _{\beta ,t}|\mathcal{K}_{\beta ,-t},\boldsymbol{\theta} ). \end{array} $$

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Gerlach et al. (2000) show how to evaluate the first two terms while the last one is obtained from the prior. When $\mathcal {K}_{\beta ,t}$ and $\beta _{j,t}$ are highly dependent, the sampler of Carter and Kohn (1994) breaks down completely: the higher the correlation (dependence), the bigger the efficiency gain. The latent process for the betas is estimated by means of the forward-backward algorithm of Carter and Kohn (1994). $\mathcal {K}_{\sigma }$ and R are drawn in the same way as $\mathcal {K}_{\beta }$ and B. To do so we follow Kim et al. (1998) and approximate the log of a $\chi ^{2}(1)$ distribution by means of a mixture of seven normals. In each iteration of the Gibbs sampler we simulate a component of the mixture distribution in order to get a conditional linear state space model for $\ln (\sigma _{t}^{2})$. Finally, $\boldsymbol {\theta }$ is easily sampled as we use conjugate priors. We use a burn-in period of 1,000 and draw 5,000 observations storing every other of them to simulate the posterior distributions of parameters and latent variables. The resulting autocorrelations of the draws are very low.³²

To estimate the cross section in Eq. 6 at each time t and for each draw of $B_{t|t-1}=(\beta _{1,t|t-1},...,\beta _{N,t|t-1})$ where each $\beta _{j,t|t-1}$ is a $((K+1)\times 1)$ vector and N is the total number of assets, we use natural conjugate priors. In particular, $p(\lambda ,\sigma ^{2})=p(\lambda |\sigma ^{2})\times p(\sigma ^{2})$ where

$$ (\lambda |\sigma ^{2})\sim N(\underline{\lambda },\sigma ^{2}\underline{V}) \text{ and }(\sigma ^{2})\sim IG\left(\frac{\underline{\nu }}{2}\underline{s}^{2}, \frac{1}{2}\underline{\nu }\right) $$

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Combining them with the data likelihood we obtain a joint posterior density with convenient analytical form. The resulting marginal posterior distributions are

$$ (\lambda |x)\sim t\left(\overline{\lambda },\overline{s}^{2}\overline{V}, \overline{\nu }\right)\qquad (\sigma ^{2}|x)\sim IG\left(\frac{\overline{\nu }}{2} \overline{s}^{2},\frac{1}{2}\overline{\nu }\right) $$

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with

$$\begin{array}{rll} E(\lambda |x)&=&\overline{\lambda }\quad Var(\lambda |x)=\frac{\overline{\nu } \overline{s}^{2}}{\overline{\nu }-2}\overline{V}\quad E(\sigma ^{2}|x)=\frac{ \overline{\nu }\overline{s}^{2}}{\overline{\nu }-2} \\ Var(\sigma ^{2}|x)&=&\frac{(\overline{\nu }\overline{s}^{2})^{2}}{(\overline{\nu }-2)^{2}(\frac{\overline{\nu }}{2}-2)} \end{array} $$

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where

$$ \overline{V}=(\underline{V}^{-1}+(B^{\prime -1}))^{-1}\qquad \overline{ \lambda }=(\underline{V}^{-1}+(B^{\prime -1}))^{-1}(\underline{V}^{-1} \underline{\lambda }+(B^{\prime -1}\hat{\lambda})) $$

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where $B=\left \{ B_{t|t-1}\right \} _{t=2}^{T}$, $\overline {\nu }=\underline {\nu }+N,$ and $\hat {\lambda }$ is the OLS estimate. Results were obtained under two different sets of priors. In the former case priors were weak informative ($\underline {\nu }=0$ and $\underline {V}^{-1}=0$) and exploit the well known Jeffreys’ prior while in the latter case we impose some prior information. In more detail, we opted for a small amount of strength ($\underline {\nu }=5$) supporting a prior view for premiums with zero mean and standard deviation equal to a twelfth of the maximum absolute return observed in the sample. When informative priors were used, we recorded a striking reduction in the variability of the estimated posterior distributions for the risk premia relative to the baseline case.³³ Finally, the prior residual variance is centered at about 10, a value that appears in the higher range of the maximum likelihood estimates.