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, expost outofsample longhorizon portfolio performances that may compete with those typical of bull and bear models. A typical investor with intermediate risk aversion and a 5year 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.
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Notes
 1.
Two concurrent explanations exist for the existence of predictability in asset returns. First, numerous asset pricing papers show that predictability results from business cycle movements and changes in investors’ perceptions of risk that are reflected in timevarying risk premia. Second, other research shows that predictability may reflect an inefficient market populated with overreacting and irrational investors. Ling et al. (2000) contain references to this debate. In our paper, we take the existence of both linear and nonlinear predictability patterns as an empirical fact and investigate whether linearizing such patterns may produce the same improvement in portfolio performance (if any) as nonlinear ones would.
 2.
However Campbell and Thompson (2008) defend the finding of predictability in stock returns and show that it is possible to beat the historical average by imposing restrictions on the signs of regression coefficients and return forecasts.
 3.
In the United States, the REIT market offers investors a way to invest in real estate without problems of illiquidity, intense management, and large lot size/high unit cost (see Ciochetti et al. 2002).
 4.
As typical of the literature since Nelling and Gyourko (1998), our VARs also allow the possibility that lagged asset returns forecast future asset returns, also reflecting crossasset patterns (i.e., it is allowed that past stock or bond returns may forecast subsequent REIT returns, etc.).
 5.
However, in the shortrun investment horizon case, the a large set of VAR models provide a superior realized, outofsample CER than a threestate MRSM does.
 6.
Fugazza et al. (2009) report that for longhorizon investors that ignore predictability, ignoring real estate assets results in an approximate CER loss of 200 basis point, which climbs up to 400 basis point under linear predictability. See also welfare loss estimates of omitting real estate from the asset menu in MacKinnon and Al Zaman (2009).
 7.
Other recent papers that have featured either nonlinearities or breaks in patterns of REIT return predictability are Guirguis et al. (2005) and Serrano and Hoesli (2007). In the finance literature, Park (2010) argues that in a subsample of US data that includes the 1990s, the predictive power of the dividend yield disappears. Henkel et al. (2011) find that stock return predictability occurs only during economic contractions but disappears during expansions.
 8.
The 3 derives from the fact that we study the case of \(p=1,\) 2, and 4; the addition of one to \(2^{M}\) depends on the fact that our models always include as a default p lags of past asset returns.
 9.
Notably, (3) can be interpreted as a MRSM under a specific restriction on the transition probabilities, i.e., when an absorbing state in the transition matrix is imposed, such that \(k=1\) obtains.
 10.
Notice that the exact value to which \(\delta \) is calibrated will only affect the intertemporal tradeoff between current and future consumption and not the resulting optimal portfolio weights, which are instead our main focus.
 11.
Detailed evidence and crosscorrelogram plots are available upon request.
 12.
Davies (1977) has derived an upper bound for the significance level of the standard likelihood ratio test under nuisance parameters:
\(\Pr \left( LRT>x\right) \leq \Pr \left( \chi _{1}^{2}>x\right) +\sqrt{2x} \exp \left( \frac{x}{2}\right) \left[ \Gamma \left( \frac{1}{2}\right) \right] ^{1}. \)
This bound holds if the loglikelihood has a single peak.
 13.
For instance, a threestate MSI leads to a maximized loglikelihood of 5026.22 to be contrasted with a singlestate maximized loglikelihood of 4725.92. The increment of 300.30 loglikelihood “points” originates an LRT statistic of 600.6 that however corresponds to a loss of 16 degree of freedom because a MSIH(3) implies the estimation of 28 parameters against the 14 parameters required by a simple Gaussian IID model. Moreover, one has to take into account the nuisance parameter problems related to the fact that when the Markov chain becomes degenerate, a few parametes under the MSIH framework cannot be identified and the loglikelihood is unbounded (see Hamilton 1994). Even taking these issues into account, the pvalue associated with the Davies’ bound is 3.4e06.
 14.
The estimated transition matrix reveals that if in month t the markets are in a bear state, in month \(t+1\) there is an almost 96 % probability of still being a bear regime.
 15.
The reason is that under a Gaussian IID model, the dynamics of portfolio weights does not come from movements in the value of current and past predictors, but simply from the fact that the vector of expected returns and the covariance matrix need to be recursively estimated over time.
 16.
The 90 % empirical range equals the difference between the two values from the time series of weights that leave the 5 % of the recursive holdings in each of the two tails. This measure is computed and reported as a proxy of dispersion that is especially useful when the weights have nonnormal distributions.
 17.
Of course, such lack of important differences between average weights over our OOS, 1995–2009 period does not imply that there cannot be any differences in the timevarying path of the weights, as commented with reference to Figs. 3, 4 and 5. Some minor differences in the average weights assigned to longterm bonds can be found between the Gaussian IID model and the VAR frameworks, with the latter implying higher average weights than the former does.
 18.
We have also tabulated summary statistics similar to those in Table 4 for the cases of \(\gamma =2\) and 10, reaching qualitatively similar conclusions. Detailed results are available upon request.
 19.
Here there are also some important differences within the VAR class, as for instance a full VAR(1) generates larger and more unstable demands than the best performing VAR does. We conjecture that this latent instability in the relationship between long and shortterm demands is responsible for the disappointing OOS performance of the full VAR when compared to a more parsimonious VAR.
 20.
 21.
These 95 % confidence intervals have been computed by applying a block bootstrap to the time series of OOS recursive, realized performance statistics, such as realized utilities, and returns.
 22.
In fact, realized power utility depends on the entire realized density of portfolio returns and as such on all realized moments, not only mean, standard deviation, skewness, and kurtosis (see Campbell and Viceira 2002).
 23.
In more detail, it is easy to show that
\(\kappa _{0,T}(\gamma )\equiv v_{T}^{1\gamma }\left[ (1\gamma )^{1}1\frac{1}{2}\gamma \frac{1}{6}\gamma (\gamma +1)\frac{1}{24}\gamma (\gamma+1)(\gamma +2)\right]\)
\(\begin{array}{rll}\kappa _{1,T}(\gamma ) &\equiv& \frac{1}{6}v_{T}^{\gamma }\left[ 6+6\gamma +3\gamma (\gamma +1)+\gamma (\gamma +1)(\gamma +2)\right] >0\quad \kappa _{2,T}(\gamma )\equiv \frac{1}{4}\gamma v_{T}^{(1+\gamma )} \\&& \qquad\qquad\qquad\quad \left[2+2(\gamma +1)+(\gamma +1)(\gamma +2)\right] <0 \\ \kappa _{3,T}(\gamma ) &\equiv &\frac{1}{6}\gamma (\gamma +1)(\gamma+3)v_{T}^{(2+\gamma )}>0\qquad \kappa _{4,T}(\gamma )\equiv \frac{1}{24} \gamma (\gamma +1)(\gamma +2)v_{T}^{(3+\gamma )}<0.\end{array}\)
This approach is only heuristic because our utility function in (4) links welfare to consumption flows and not directly to realized portfolio returns.
 24.
This is consistent with the fact that most of the predictors often included in the best performing VARs express forecasting power for REIT returns, such as the REIT cap rate, the rate of growth in housing starts, or mortgage rates.
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Appendix: Solution of Dynamic Asset Allocation Problems by Monte Carlo Methods
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
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 \(t+1\) onestep joint density conditional on the periodt 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 HamiltonKim 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 \(\sum _{j=1}^{k1}\left \vert \mathbf {\pi }_{t+1}^{i}\mathbf {e}_{j}\mathbf {\pi }_{t+1,g}\mathbf {e}_{j}\right \vert \), i.e.
$$\tilde{\mathbf{\pi}}_{t+1,g}^{i}\left(\mathbf{\pi }_{t}^{i}\right)\equiv \arg \min \sum \limits_{j=1}^{k1}\left \vert \mathbf{x}\mathbf{e}_{j}\mathbf{\pi}_{t+1,g}\mathbf{e}_{j}\right \vert . $$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 \(5^{k1}\) 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
now requires drawing G random samples of the state variables \(\left \{ \mathbf {Z}_{t+1,g}\right \} _{g=1}^{G}\) from the \(t+1\) onestep joint density conditional on the periodt 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 \(G=30{,}000\).
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Bianchi, D., Guidolin, M. Can Linear Predictability Models Time Bull and Bear Real Estate Markets? OutofSample Evidence from REIT Portfolios. J Real Estate Finan Econ 49, 116–164 (2014). https://doi.org/10.1007/s1114601394116
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Keywords
 REIT returns
 Predictability
 Strategic asset allocation
 Markov switching
 Vector autoregressive models
 Outofsample performance
JEL Classifications
 G11
 C53