Can Linear Predictability Models Time Bull and Bear Real Estate Markets? Out-of-Sample Evidence from REIT Portfolios

Abstract

A recent literature has shown that REIT returns contain strong evidence of bull and bear dynamic regimes that may be best captured using nonlinear econometric models of the Markov switching type. In fact, REIT returns would display regime shifts that are more abrupt and persistent than in the case of other asset classes. In this paper we ask whether and how simple linear predictability models of the vector autoregressive (VAR) type may be extended to capture the bull and bear patterns typical of many asset classes, including REITs. We find that nonlinearities are so deep that it is impossibile for a large family of VAR models to either produce similar portfolio weights or to yield realized, ex-post out-of-sample long-horizon portfolio performances that may compete with those typical of bull and bear models. A typical investor with intermediate risk aversion and a 5-year horizon ought to be ready to pay an annual fee of up to 5.7 % to have access to forecasts of REIT returns that take their bull and bear dynamics into account instead of simpler, linear forecast.

Appendix: Solution of Dynamic Asset Allocation Problems by Monte Carlo Methods

Markov Switching Model

Given the optimization problem is solved backwards at each time t (since the portfolio can be rebalanced every month), such that $a\left ( \mathbf {\pi }_{t+1}^{i},t+1\right )$ is known for all values of $i=1,2,\ldots ,Q$ on a discretization grid. Here $a(.)$ is a function of the regime probabilities $\pi _{t+1}$. Computing a Monte Carlo approximation of

$$E_{t}\left[ \left \{ \mathbf{\omega }_{t}\mathbf{R}_{t+1,g}\right \} ^{1-\gamma }a\left( \mathbf{\pi }_{t+1}^{i},t+1\right) \right] $$

requires drawing G random samples of asset returns $\left \{R_{t+1,g}\left (\pi _{t+1}^{i}\right )\right \}_{g=1}^{G}$ from the $t+1$ one-step joint density conditional on the period-t parameter estimates $\hat {\mathbf {\theta }}_{t}=\left (\left \{ \hat {\mathbf {\mu }}_{S},\hat {\Omega }_{S}\right \}_{S=1}^{k},\hat {\mathbf {P}}\right )$ assuming that, at each point $\pi _{t}^{i}$ is updated to $\pi _{t+1}\left (\pi _{t}^{i}\right )$. The algorithm consists of the following steps:

• 1. For each possible value of the current regime $S_{t}$ simulate G returns $\left \{ \mathbf {R}_{t+1,g}(S_{t+1}) \right \} _{g=1}^{G}$ in calendar time from the regime switching model:

  $$\mathbf{R}_{t+1,g}\left( S_{t}\right) =\mathbf{\mu }_{S_{t+1}}+\mathbf{\varepsilon }_{t+1,g}\qquad \mathbf{\varepsilon }_{t+1,g}\sim N\left(\mathbf{0},\Omega_{S_{t+1}}\right). $$

  The simulation enables switching as governed by the transition probability matrix $\hat {\mathbf {P}}_{t}$. For example, starting in state 1, the probability of switching to state 2 between t and $t+1$ is $\hat {p}_{12}\equiv e_{1}^{\prime }\hat {\mathbf {P}}_{t}e_{2}$, while the probability of remaining in state 1 is $\hat {p}_{11}\equiv e_{1}^{\prime }\hat {\mathbf {P}}_{t}e_{1}$. Hence, at each point in time, $\hat {\mathbf {P}}_{t}$ governs possible state transitions.

  Combine the simulated returns $\left \{ \mathbf {R}_{t+1,g}\right \} _{g=1}^{G}$ into a sample of size G, using the probability weights $\pi _{t}^{j}$:

  $$\mathbf{R}_{t+1,g}\left(\mathbf{\pi }_{t}^{i}\right)=\sum \nolimits_{j=1}^{k}\left(\mathbf{\pi}_{t}^{i}\mathbf{e}_{j}\right)\mathbf{R}_{t+1,g}\left( S_{t}=j\right) $$

• 2. Update the future regime probabilities perceived by the investor using the Hamilton-Kim filtering formula

  $$\mathbf{\pi }_{t+1}\left( \mathbf{\pi }_{t}^{i}\right) =\frac{\left( \mathbf{\pi }_{t}^{i}\right) ^{\prime }\hat{\mathbf{P}}\odot \mathbf{\eta }\left(\mathbf{R}_{t+1,g}(\mathbf{\pi }_{t}^{i});\hat{\mathbf{\theta }}_{t}\right) }{\left( \left( \mathbf{\pi }_{t}^{i}\right) ^{\prime }\hat{\mathbf{P}}\odot \mathbf{\eta }\left( \mathbf{R}_{t+1,g}(\mathbf{\pi }_{t}^{i});\hat{\mathbf{\theta }}_{t}\right) \right) \mathbf{\iota}_{k}}. $$

  This gives an $G\times k$ matrix $\left \{\mathbf {\pi }_{t+1}\left ( \mathbf {\pi }_{t}^{i}\right ) \right \} _{g=1}^{G}$, whose rows correspond to simulated vectors of perceived regime probabilities at time $t+1$.

• 3. For all $g=1,2,\ldots ,G$ calculate the value $\tilde {\mathbf {\pi }}_{t+1,g}^{i}$ on the discretization grid $(i=1,2,\ldots ,Q)$ closest to $\pi _{t+1,g}\left (\pi _{t}^{i}\right )$ using the distance measure $\sum _{j=1}^{k-1}\left \vert \mathbf {\pi }_{t+1}^{i}\mathbf {e}_{j}-\mathbf {\pi }_{t+1,g}\mathbf {e}_{j}\right \vert$, i.e.

  $$\tilde{\mathbf{\pi}}_{t+1,g}^{i}\left(\mathbf{\pi }_{t}^{i}\right)\equiv \arg \min \sum \limits_{j=1}^{k-1}\left \vert \mathbf{x}\mathbf{e}_{j}-\mathbf{\pi}_{t+1,g}\mathbf{e}_{j}\right \vert . $$

  Knowledge of thevector $\left \{\tilde {\mathbf {\pi }}_{t+1,g}^{i}\left (\mathbf {\pi }_{t}^{i}\right ) \right \} _{g=1}^{G}$allows us to build $\left \{a\left (\mathbf {\pi }_{t\hspace *{-1.5pt}+\hspace *{-1.5pt}1}^{(i,g)},t\hspace *{-1.5pt}+\hspace *{-1.5pt}1 \right )\right \} _{g=1}^{G}$, where $\pi _{t+1}^{(i,g)}\equiv \tilde {\mathbf {\pi }} _{t+1,g}^{i}\left (\pi _{t}^{i}\right )$ is a function of the assumed, initial vector of regime probabilities $\pi _{t}^{i}$.

• 4. Solve the program

  $$\max \limits_{\mathbf{\omega }_{t}\left(\mathbf{\pi }_{t}^{i}\right)}G^{-1}\sum \limits_{g=1}^{G}\left \{\left[\mathbf{\omega }_{t}\mathbf{R}_{t+1,g}\right]^{1-\gamma }a\left( \mathbf{\pi }_{t+1}^{(i,g)},t+1\right) \right \} $$

  For large values of G this provides a precise Monte Carlo approximation to $E\left [\left \{ \mathbf {\omega }_{t}\mathbf {R}_{t+1,g}\right \} ^{1-\gamma } a\left ( \mathbf {\pi } _{t+1}^{i},t+1\right )\right ]$. The value function evaluated at the optimal weights $\hat {\mathbf {\omega }}_{t}\left (\pi _{t}^{i}\right )$ gives $a\left (\pi _{t}^{i},t\right )$ for the ith point on the initial grid. We also check whether $\omega _{t}R_{t+1,g}$ is negative and reject all corresponding sample paths.

The algorithm is applied to all possible values $\pi _{t}^{i}$ on the discretization grid until all values of $a\left (\pi _{t}^{i},t\right )$ are obtained for $i=1,2,\ldots ,Q$. It is then iterated backwards. We take $a\left (\pi _{t+1}^{i},t+1\right )$ as given and use the actual vector of smoothed probabilities $\pi _{t}$. The resultant vector $\hat {\mathbf {\omega }}_{t}$ gives the optimal portfolio allocation at time t, while $a(\pi _{t},t)$ is the optimal value function. In our application, Q is selected as 5$^{2}=25$ which fits the standard formula $5^{k-1}$ as in Guidolin and Timmermann (2008) and the number of Monte Carlo simulations is 30,000.

VAR Model

Again the optimization problem is solved by backward iteration for each point t so that $a\left ( \mathbf {Z}_{t+1},t+1\right )$. A Monte Carlo approximation of the expectation

$$E_{t}\left[ \left \{ \mathbf{\omega }_{t}\mathbf{R}_{t+1,g}\right \} ^{1-\gamma }a\left( \mathbf{Z}_{t+1}^{i},t+1\right) \right] $$

now requires drawing G random samples of the state variables $\left \{ \mathbf {Z}_{t+1,g}\right \} _{g=1}^{G}$ from the $t+1$ one-step joint density conditional on the period-t parameter estimates $\hat {\mathbf {\theta }}_{t}=(\hat {\mathbf {\mu }},\hat {\mathbf {A}},\hat {\Omega })$. The algorithm is similar but much simpler than for the MRSM. The G returns $\left \{\mathbf {R}_{t+1}(\mathbf {Z}_{t}^{i})\right \} _{g=1}^{G}$ need to be simulated from the VAR model. In this case $Q=20$ delivers quite accurate results (because of the linearity of the prediction framework) and we set again $G=30{,}000$.