Can VAR models capture regime shifts in asset returns? A long-horizon strategic asset allocation perspective

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Abstract

It is often suggested that through a judicious choice of predictors that track business cycles and market sentiment, simple vector autoregressive (VAR) models could produce optimal strategic portfolio allocations that hedge against the bull and bear dynamics typical of financial markets. However, a distinct literature exists that shows that nonlinear econometric frameworks, such as Markov switching (MS), are also natural tools to compute optimal portfolios in the presence of stochastic good and bad market states. In this paper we examine whether simple VARs can produce portfolio rules similar to those obtained under MS, by studying the effects of expanding both the order of the VAR and the number/selection of predictor variables included. In a typical stock–bond strategic asset allocation problem, we compute the out-of-sample certainty equivalent returns for a wide range of VARs and compare these measures of performance with those typical of nonlinear models for a long-horizon investor with constant relative risk aversion. We conclude that most VARs cannot produce portfolio rules, hedging demands, or (net of transaction costs) out-of-sample performances that approximate those obtained from equally simple nonlinear frameworks. We also compute the improvement in realized performance that may be achieved adopting more complex MS models and report this may be substantial in the case of regime switching ARCH.

Introduction

Since the seminal contributions by Brennan et al., 1997, Kandel and Stambaugh, 1996, the empirical finance literature on normative long-run dynamic asset allocation under predictable returns (i.e., how much should a risk-averse investor weight each available asset) has almost exclusively devoted its attention to linear predictability models. In a linear predictability model, asset returns are simply forecast by past values of predictor variables (such as the dividend yield and the term spread) within a vector autoregressive (VAR) framework. The linearity consists of the fact that a movement today in one or more of the predictors commands a proportional response in the expected (predicted) value of future asset returns. However, another strand of the empirical finance literature has in the meantime stressed that returns on most asset classes do contain predictability patterns that are not simply linear, as they involve nonlinear dynamics (such as regimes, thresholds, self-exciting mean reversion, and conditional heteroskedasticity) that often make not only expected asset returns but also higher-order moments predictable.1

Although linear models are key benchmarks in empirical finance and their simplicity makes them obvious choices in many applications, their use in asset allocation exercises has relied on two often-implicit premises. First, that although most normative papers have to be taken only as examples of how practical portfolio choice ought to proceed, when the scope of the investigation is extended beyond the class of small-scale (i.e., with 3–4 predictors at most) VAR (1) models typical in the literature (see e.g., Barberis, 2000, Lynch, 2001), some more complicated VAR must exist that is of practical use in terms of consistently improving realized performances. Hence some VARs can be found that can efficiently summarize the overall balance of predictability in asset returns and making the modeling of any residual nonlinear effects of second-order importance, at least in terms of impact on portfolio weights and performance. Second, that although more complicated, large-scale VAR (p) models may yield complex portfolio strategies, simple, small-scale VAR (1) models must be illustrative already of the first-order effects of linear predictability on dynamic portfolio selection. Our paper tackles both these conjectures at their roots and provides a systematic examination of whether, when, and how small- and medium-scale VAR (p) models may deliver dynamic portfolio choices that are: (i) able to approximate the portfolio choices of an investor that exploits both linear and nonlinear predictability patterns in the data and (ii) competitive in terms of realized portfolio performance.

As econometricians would expect on theoretical grounds, our relatively large set of small- and medium-scale (up to seven predictors are included) VAR (p) models (with p = 1, 2, 4, and 12) fails to imply portfolio choices that approximate those from a rather simple (one may say, “naive”) nonlinear benchmark, represented by a plain vanilla 3-state Markov switching (MS) model. This is of course only an ex-ante perspective on the problem: “different” does not imply “worse” in the view of an applied portfolio manager and what could be misspecified and practically not useful is not the VAR family, but the proposed nonlinear benchmark. More importantly, VARs systematically fail to perform better than nonlinear models in recursive (pseudo) out-of-sample tests, in the sense that VARs generally produce lower realized certainty equivalent returns (i.e., risk-adjusted performances that take into account the curvature of the utility function under which the portfolio choice program has been solved) than multi-state models. This means that VARs provide no approximation tool for more complicated, nonlinear dynamics either ex-ante or ex-post.

We stress that this result that simple VARs cannot capture all predictability patterns typical of well-known, standard financial US data even when all possible combinations of predictors and lag choices are allowed, is obtained with reference to a very simple MS model. Such a choice is motivated by the search for a “lower bound”: if, in the presence of nonlinearities in commonly used data, VARs are not even capable to achieve a “tie” (say, in terms of realized out-of-sample performance) with a “naive” MS model that may itself not perfectly capture the dynamic features of the data, then it may be safe to conclude that the role of nonlinear models (not only MS, of course) ought to be considerably larger than the one they have played so far in the empirical finance literature. To illustrate this fact, we try to tease out from our empirical exercise a measure of the realized utility (certainty equivalent return, CER) gains that a risk-averse investor may derive from adopting realistic MS models that may able to provide a good fit to the dynamics of real asset returns. Here we consider a variety of MS models that include MS VARs, MSVAR ARCH models, and MS models with time-varying transition probabilities. Our key result is that the risk-adjusted performance gains may somewhat exceed the plain vanilla MS findings commented early on. In particular, MS ARCH models may yield substantial CER improvements.

These results are obtained with reference to a strategic asset allocation (SAA) application that appears to have played a key role in the literature (see e.g., Brennan et al., 1997): a risk-averse (constant relative risk aversion) investor wants to allocate at time t her wealth across three macro-asset classes, i.e., stocks (as represented by a standard value-weighted index), long-term default risk-free government bonds, and 1-month Treasury bills. We use monthly US data for the long period 1953–2009 which also includes the recent financial crisis.2 We focus on long-horizon portfolio choices (up to a 5-year horizon) of an investor that recursively solves a portfolio problem in which utility derives from real consumption (i.e., cash flows obtained from dividend and coupon payments and from selling securities in the portfolio) and rebalancing is admitted at the same frequency as the data. This means that even when the problem solved is characterized by a 60-period ahead horizon, the investor decides at time t knowing that at times t + 1, t + 2, …, up to t + 59 she will be allowed to change the structure of her portfolio weights to reflect the fact that, at least in principle, new information will become available at all these future points. Finally, our investor selects optimal portfolio weights taking into account the presence of both fixed and variable transaction costs. This means that – because a given vector of optimal weights at time t may implicitly imply a need to trade in all assets between time t and t + 1 – our investor will also take into account the trading needs of her portfolio choices and especially the impact of the transaction costs incurred on expected utility.

Such a portfolio problem seems to be the most appropriate one, not only for its past role in the development of the literature but also for the specific features of our research design. First, a long-horizon is key when discussing the economic value of predictability or – as in our case – the relative economic value of different types of models. Second, our attention to a problem with continuous/frequent rebalancing of portfolio weights and in which investors care for real consumption streams and real portfolio returns is consistent with the way predictability is exploited in practice, i.e., with full awareness of the fact that its existence not only affects today’s choice but will keep affecting choice in all subsequent periods. We are not aware of previous papers that have jointly solved consumption and portfolio choice problems under MS dynamics. Third, as previously stressed by Balduzzi and Lynch, 1999, Lynch and Balduzzi, 2000, all SAA problems under predictability and active portfolio management ought to carefully consider whether the forecastable variation in investment opportunity sets offers enough welfare gains to exceed the often large trading costs.

The thrust of our exercise does not consist of investigating the different portfolio implications recursive portfolio weights implications of linear vs. nonlinear models, as this operation has already appeared in the literature for specific linear and nonlinear frameworks (see e.g., Ang and Bekaert, 2004, Detemple et al., 2003, Guidolin and Timmermann, 2007).3 These papers measure the economic loss from model misspecification in (density) forecasting applications by resorting to a portfolio choice metric. On the contrary, our aim is to oppose a large set of VAR models, potentially spanning a large portion of the models that have appeared in the literature, to one single, and also relatively simple, nonlinear framework selected to be of a Markov switching type since this class of model has proven relatively popular and intuitive in the recent finance literature (see e.g., Perez-Quiros and Timmermann, 2000). We investigate the implied dynamic recursive portfolio choices and the resulting recursive out-of-sample performance of all VARs one can form using seven predictors besides lagged values of asset returns themselves (in principle this is a total of 1024 different VARs), and experimenting with four alternative lag orders throughout, p = 1, 2, 4, and 12 (with restrictions). The seven predictors are typical in the finance literature and consist of widely employed macro-finance variables, i.e., the dividend yield, the riskless term spread, the default spread between Baa and Aaa corporate bonds, the CPI inflation rate, the nominal riskless 3-month T-bill rate, the rate of growth of industrial production, and the unemployment rate.4

The rest of the paper is structured as follows. Section 2 describes the research design. Section 3 describes the data, the 3-state Markov switching benchmark, and some features of linear predictability. Section 4 computes and presents optimal portfolio weights and hedging demands under the two classes of models. Section 5 computes realized, recursive out-of-sample portfolio performances. Section 6 performs a few robustness checks. Section 7 extends the set of MS models to encompass MS VAR models, MSVAR ARCH models (with leverage), and MS models with time-varying transition probabilities. Section 8 concludes.

Section snippets

Econometric models

We perform recursive estimation, portfolio weight calculation and performance evaluation for three groups of models. First and foremost, we entertain a large class of VAR (p) models. These VARs consist of a linear relationship linking rt+1, a N × 1 vector of risky real asset returns at time t + 1, and yt+1, a M × 1 vector of predictor variables at time t + 1, to lags of both rt+1 and yt+1. For instance, in the case of a VAR (1), we havert+1yt+1=μ+Artyt+εt+1εt+1N(0,Ω),where μ is a (N + M) × 1 vector of

Data and preliminary evidence

Our early tests are based on monthly US data on real asset returns and a standard set of predictive variables sampled over the period 1953:01–2009:12. The data are obtained from CRSP and FRED® at the Federal Reserve Bank of St. Louis. The real asset return data are the CRSP value weighted equity return, the CRSP/Ibbotson 10-year bond return and the 30-day Fama-Bliss Treasury bill return, all deflated by the CPI inflation rate. The predictive variables are the dividend yield on equities

Recursive portfolio weights

Fig. 2 compares recursive optimal portfolio weights (for T = 1 month and 5 years) for two models, MSH and an expanding window VAR (1) in which all predictors are included. The right-most column of plots should be taken as an example of the type of qualitative dynamics commonly observed in VAR optimal weights. The left-hand plots also report optimal weights under the Gaussian IID benchmark. The recursive exercise is performed over the period 1973:01–2009:12 therefore including the deep financial

Realized recursive portfolio performance

Our finding that VAR models produce dynamic (short- and long-run) SAA weights and hedging demands that depart from the implications of a model that accounts for nonlinear patterns is suggestive that naive linear frameworks may be too simple to pick up predictability patterns that are in the data and that may be important in applications. However, these results are suggestive at best. Since a model that fits the data better in-sample than another model does not have to out-perform the latter in

Risk aversion

One potential concern may be that our results are driven by a special (even though, rather typical) assumption on the coefficient of risk aversion. To address this concern we expand the range of portfolio performance results to the cases of γ = 2 and 10. First, inspection of dynamic portfolio weights and hedging demands reveal qualitatively similar structure to Fig. 2 that expand the range of portfolio performance results to the cases of γ = 2 and 10. Interestingly, the general path followed by

How well can Markov switching models fare?

So far our research design has been exclusively targeted to establish a simple result: that no VAR model—even when the search is extended to encompass a variety of predictors, lag structures, and estimation strategies—may produce either in-sample portfolio weights or realized OOS performances that come close to those of a very simple, and yet powerful three-state MSH. On the one hand, in the light of some claims implicit in portions of the empirical finance literature, this is an important

Conclusion

This paper asks whether it is possible for a large class of VAR models that forecast real asset returns on stocks, bonds, and T-bills to imply dynamic strategic asset allocation choices and realized, net of transaction costs performances similar (or superior) to portfolio choices and realized performances typical of slightly more complicated nonlinear frameworks in which the existence of regimes is accounted for. After identifying the nonlinear framework with a simple three-state Markov

Acknowledgments

We would like to thank an anonymous referee, Ike Mathur (the editor), “Paul” Moon Sub Choi (a discussant), Paolo Colla, Carlo Favero, René Garcia, Abraham Lioui, Patrick Minford, Francesco Saita, Andrea Sironi, Nick Taylor, and participants at the INFINITI Conference on International Finance, Dublin 2010, the 3rd International Conference on Computational and Financial Econometrics, Limmasol 2009 and seminar participants at Bocconi University Milan (Finance dept.), Cardiff Business School, EDHEC

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