Elsevier

Journal of Econometrics

Volume 164, Issue 1, 1 September 2011, Pages 60-78
Journal of Econometrics

Predictability of stock returns and asset allocation under structural breaks

https://doi.org/10.1016/j.jeconom.2011.02.019Get rights and content

Abstract

This paper adopts a new approach that accounts for breaks to the parameters of return prediction models both in the historical estimation period and at future points. Empirically, we find evidence of multiple breaks in return prediction models based on the dividend yield or a short interest rate. Our analysis suggests that model instability is a very important source of investment risk for buy-and-hold investors with long horizons and that breaks can lead to a negative slope in the relationship between the investment horizon and the proportion of wealth that investors allocate to stocks. Once past and future breaks are considered, an investor with medium risk aversion reduces the allocation to stocks from close to 100% at short horizons to 10% at the five-year horizon. Welfare losses from ignoring breaks can amount to several hundred basis points per year for investors with long horizons.

Introduction

Stock market investors face a daunting array of risks. First and foremost is the component of stock returns that cannot be predicted by any model for the return generating process. This source of uncertainty is substantial, given the low predictive power of return forecasting models. Second, even conditional on a particular forecasting model, investors face parameter uncertainty, i.e., the effect of not knowing the true model parameters (Kandel and Stambaugh, 1996, Barberis, 2000). Third, investors do not know the state variables or functional form of the true return process and so face model uncertainty (Avramov, 2002, Cremers, 2002). This paper deals with a fourth source of uncertainty of particular importance to long-run investors, namely model instability, i.e., random changes or “breaks” to the parameters of the return generating process.

Conventional practice in economics and finance is to compute forecasts conditional upon a maintained model whose parameters are assumed to be constant both throughout the historical sample and during future periods to which the forecasts apply. This procedure ignores that, over estimation samples that often span several decades, the relation between economic variables is likely to change. Instability in economic models could reflect institutional, legislative and technological change, financial innovation, changes in stock market participation, large macroeconomic (oil price) shocks and changes in monetary targets or tax policy.1 In the context of financial return prediction models, Merton’s intertemporal CAPM suggests that time-variation in aggregate risk aversion can lead to changes in the relationship between expected returns and predictor variable tracking movements in market risk or investment opportunities.2

Instability in the relation between stock returns and predictor variables such as the dividend yield or short-term interest rates has been documented empirically in several studies. Pesaran and Timmermann (1995), Bossaerts and Hillion (1999), Lettau and Ludvigson (2001), Timmermann and Paye (2006), Ang and Bekaert (2007) and Goyal and Welch (2008) find substantial variation across subsamples in the coefficients of return prediction models and in the degree of return predictability.3 Building on this evidence, recent studies such as Dangl and Halling (2008) and Johannes et al. (2009) capture time-variation in return prediction models by assuming that some parameters follow a random walk and thus change every period.

In this paper we focus instead on the effect of rare but large structural breaks as opposed to small parameter changes occurring every period. The distinction between rare, large breaks versus frequent, small breaks can be difficult to make in practice (Elliott and Mueller, 2006). However, our analysis allows us to pinpoint the most important times where the return prediction model undergoes relatively sharp changes, which provides insights into the interpretation of the economic sources of model instability. Sudden, sharp changes in model parameters are consistent with empirical findings by both Dangl and Halling (2008) and Johannes et al. (2009) that the change in the parameters of return predictability models at times can be large. By considering few, large breaks, our approach is close in spirit to Pastor and Stambaugh (2001) who consider breaks in the risk-return trade-off and Lettau and Van Nieuwerburgh (2008) who consider a discrete break to the steady state value of a single predictor variable (the dividend yield).

Our approach builds on Chib (1998), Pastor and Stambaugh (2001) and Pesaran et al. (2006) in adopting a change point model driven by an unobserved discrete state variable. Specifically, we generalize the univariate model in Pesaran et al. (2006) to a multivariate setting so instability can arise either in the conditional model used to forecast returns, in the marginal process generating the predictor variable(s) or in the correlation between innovations to the two equations. Forecasting returns in this model require accounting for the probability and magnitude of future breaks. To this end, we introduce a meta distribution that characterizes how the parameters vary across different break segments. The resulting hierarchical model nests as special cases both a pooled scenario where the similarity between the parameters in the different regimes is very strong (corresponding to a narrow dispersion in the distribution of parameters across regimes) as well as a more idiosyncratic scenario where these parameters have little in common and can be very different (corresponding to a wide dispersion). Which of these cases is most in line with the data is reflected in the posterior meta distribution.

The proposed model is very general and allows for uncertainty about the timing (dates) of historical breaks as well as uncertainty about the number of breaks and their magnitude. We also extend our setup to allow for uncertainty about the identity of the predictor variables (model uncertainty) using Bayesian model averaging techniques. Hence, investors are not assumed to know the true model or its parameter values, nor are they assumed to know the number, timing and magnitude of past or future breaks. Instead, they come with prior beliefs about the meta distribution from which current and future values of the parameters of the return model are drawn and update these beliefs efficiently as new data are observed.

Our empirical analysis investigates predictability of US stock returns in the context of two popular predictor variables, namely the dividend yield and the short interest rate. We find evidence of multiple breaks in return models based on either of these predictor variables in data covering the period 1926–2005. Many of the break dates coincide with major events such as changes in the Fed’s operating procedures (1979, 1982), the Great Depression, the Treasury-Fed Accord (1951), and the growth slowdown following the oil price shocks in the early 1970s. Variation in model parameters across these regimes is found to be extensive. For example, the predictive coefficient of the dividend yield varies between zero and 2.6, while the coefficient of the T-bill rate varies even more, between −9.4 and 3.3, across break segments.

Instability in model parameters is particularly important to investors’ long-run asset allocation decisions which crucially rely on forecasts of future returns. Long investment horizons make it more likely that breaks to model parameters will occur and some of these breaks could adversely affect the investment opportunity set, thereby significantly increasing investment risks. Asset allocation exercises mostly assume that although the parameters of the return prediction model or the identity of the “true” model need not be known to investors, the parameters of the data generating process remained constant through time (e.g., Barberis (2000) and Pastor and Stambaugh (2009)). Studies that have allowed for time-varying model parameters such as Dangl and Halling (2008) and Johannes et al. (2009) only consider mean-variance investors with single-period investment horizons. Our focus is instead on the effect of model instability on the risks faced by investors with a long investment horizon.

Structural breaks are found to have a large effect on investors’ optimal asset allocations. Moreover, our analysis suggests that model instability is a more important source of investment risk than parameter estimation uncertainty for investors with long horizons and that breaks can lead to a steep negative slope in the relationship between the investment horizon and the proportion of wealth that a buy-and-hold investor allocates to stocks.4 For example, in the model with predictability from the dividend yield but no breaks, the allocation to stocks rises from 40% at short horizons to 60% at the five-year horizon. Allowing for past and future breaks, the allocation to stocks instead declines from close to 100% at short horizons to 10% at the five-year horizon.

Ignoring model instability can also lead to large welfare losses, particularly for low-to-medium risk averse investors who use the dividend yield prediction model. At very short investment horizons where breaks are unlikely to occur, welfare losses are quite modest. However, as the investment horizon grows and the risk of breaks increases, losses from ignoring model instability can rise to several hundred basis points per annum in certainty equivalent returns. Welfare losses from ignoring breaks are more modest for more highly risk averse investors or investors who use the short interest rate to predict returns.

Our portfolio allocation results lend further credence to the finding in Pastor and Stambaugh (2009) that the long-run risks of stocks can be very high. In a model that allows for imperfect predictors and unknown but stable parameters of the data generating process, Pastor and Stambaugh find that the true per-period predictive variance of stock returns can be increasing in the investment horizon due to the combined effect of uncertainties about current and future expected returns (and their relationship to observed predictor variables) and estimation risk. While this finding is similar to ours, the mechanism is very different: Pastor and Stambaugh (2009) derive their results from investors’ imperfect knowledge of current and future expected returns and model parameters, whereas model instability is the key driver behind our results.

The paper is organized as follows. Section 2 introduces the break point methodology and Section 3 presents empirical estimates for return prediction models based on the dividend yield or the short interest rate. Section 4 shows how investors’ optimal asset allocation can be computed while accounting for past and future breaks. Section 5 considers asset allocations empirically for a buy-and-hold investor. Section 6 proposes various extensions to our approach and Section 7 concludes. Technical details are provided in appendices at the end of the paper.

Section snippets

Methodology

Studies of asset allocation under return predictability (e.g., Barberis (2000), Campbell and Viceira (2001), Campbell et al. (2003) and Kandel and Stambaugh (1996)) have mostly used vector autoregressions (VARs) to capture the relation between asset returns and predictor variables. We follow this literature and focus on a simple model with a single risky asset and a single predictor variable. This gives rise to a bivariate model relating returns (or excess returns) on the risky asset to a

Breaks in return forecasting models: empirical results

Using the approach from Section 2, we next report empirical results for two commonly used return prediction models based on the dividend yield or the short interest rate.

Asset allocation under structural breaks

Investors are concerned with instability in the return model because this affects future asset payoffs and therefore could alter their optimal asset allocation. To gauge the economic importance of structural breaks in the return model, we next study the optimal asset allocation under a range of alternative modeling assumptions. Consider a buy-and-hold investor with a horizon of h periods who at time T has power utility over terminal wealth, WT+h, and coefficient of relative risk aversion, γ: u(W

Empirical asset allocation results

We next use the methods from Section 4 to assess empirically the effect of structural breaks on a buy-and-hold investor’s optimal asset allocation. We use the Gibbs sampler to evaluate the predictive distribution of returns under breaks. Details of the numerical procedure used to compute the distributions are provided in the appendices.

Before moving to the results, it is worth recalling two important effects for asset allocation under return predictability from variables such as the dividend

Sensitivity analysis and extensions

This section first explores the robustness of our empirical results with respect to the assumed priors. We then show how the expected volatility of returns change across regimes. Finally, we compute welfare costs for an investor who assumes that the model for returns does not change over time although the true data generating process is subject to breaks.

Conclusion

This paper provides an analysis of the stability of return prediction models and the asset allocation implication of breaks to model parameters. Our analysis accounts for several sources of uncertainty, namely (i) parameter uncertainty; (ii) model uncertainty; (iii) uncertainty about the number, location and size of historical breaks to model parameters; (iv) uncertainty about future (out-of-sample) breaks.

Our empirical results suggest that the parameters of standard return forecasting models

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    Three anonymous referees made constructive and helpful suggestions on an earlier version of the paper. We thank Jun Liu, Ross Valkanov, Jessica Wachter and Mark Watson as well as seminar participants at the Rio Forecasting conference, University of Aarhus (CAF), University of Arizona, New York University (Stern), Erasmus University at Rotterdam, Princeton, UC Riverside, UCSD, Tilburg, and Studiencenter Gerzensee for helpful comments on the paper. Alberto Rossi provided excellent research assistance. Timmermann acknowledges support from CREATES, funded by the Danish National Research Foundation.

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