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This paper deals with the estimation of the risk-return trade-off. We use a MIDAS model for the conditional variance and allow for possible switches in the risk-return relation through a Markov-switching specification. We find strong evidence for regime changes in the risk-return relation. This finding is robust to a large range of specifications. In the first regime characterized by low ex-post returns and high volatility, the risk-return relation is reversed, whereas the intuitive positive risk-return trade-off holds in the second regime. The first regime is interpreted as a "flight-to-quality" regime.

The Intertemporal Capital Asset Pricing Model (ICAPM) of Merton (1973) states that the expected excess return on the stock market is positively related to its conditional variance: equation(1) Et(Rt+1)=μ+γVt(Rt+1),EtRt+1=μ+γVtRt+1, Turn MathJax on formalizing the intuition that a riskier investment should demand a higher expected return (relative to the risk-free return). However, in the empirical literature, there is mixed evidence on whether the parameter γ is indeed positive and statistically significant. Examples include Ghysels et al. (2005), Guo and Whitelaw (2006), and Ludvigson and Ng (2007), who all find a positive risk-return trade-off. 1 In contrast, Glosten et al. (1993), using different GARCH specifications, find a negative relation between risk and return. Similarly, Brandt and Kang (2004) model both the expected returns and conditional variance as latent variables in a multivariate framework and find a negative trade-off. Alternatively, Yu and Yuan (2011) use data on investor sentiment to study the risk-return trade-off. They find that expected returns and conditional variance are positively related in low-sentiment periods, but unrelated during high-sentiment periods. Omitted variables could play a role in explaining these conflicting results. For example, Scruggs (1998) and Guo and Whitelaw (2006) emphasize the need to include additional variables in the risk-return relation to capture shifts in investment opportunities. Lettau and Ludvigson (2001) suggest using the consumption-wealth ratio in the risk-return relation. (Ludvigson and Ng, 2007) instead include factors summarizing the information from a large set of predictors, and Lettau and Ludvigson (2010) find that a positive risk-return relation is uncovered using lagged mean and lagged volatility as additional predictors. Another reason for the conflicting results reported in the literature is in the way that the conditional variance is modelled. Indeed, if one wants to estimate the risk-return trade-off over a long period of time, the conditional variance is not directly observable and must be filtered out from past returns. An attractive approach is the one developed by Ghysels et al. (2005). They introduce a new estimator for the conditional variance - the MIDAS (MIxed DAta Sampling) estimator - where the conditional variance depends on the lagged daily returns aggregated through a parametric weight function. The crucial difference with rolling-window estimators of the conditional variance is that the weights on lagged returns are determined endogenously and in a parsimonious way with the MIDAS approach. In this paper, we follow the approach of Ghysels et al. (2005) and use a MIDAS estimator of the conditional variance, since it is likely that this estimator can more fully describe the dynamics of marketrisk.2 Itis also a convenient approach, since it permits us to easily model the dynamics of the risk-return trade-off at different frequencies. In this paper, we also consider regime changes in the parameter γ entering before the conditional variance to reflect the possibility of a changing relationship between risk and return. 3 The relation between risk and return should not necessarily be linear. For example, Backus and Gregory (1993) and citetWhitelaw2000 show that non-linear models are consistent with a general equilibrium approach. Campbell and Cochrane (1999) underline the time-varying nature of risk premiums. In particular, Whitelaw (2000) estimates a two-regime Markov-switching model with time-varying transition probabilities that include aggregate consumption as a driving variable for the transition probabilities to account for the changes in investment opportunities. He then finds a non-linear and time-varying relation between expected returns and volatility. Alternatively, Tauchen (2000) criticizes the reduced-form nature of the models that estimate the risk-return trade-off. He develops a general equilibrium model where volatility is driven by a two-factor structure, with a risk premium that is decomposed between risk premiums on consumption risk and volatility risk. More recently, Rossi and Timmermann (2010) proposed new evidence on the risk-return relationship by claiming that the assumption of a linear coefficient entering before the conditional variance is likely to be too restrictive. They use an approach based on boosted regression trees and find evidence for a reversed risk-return relation in periods of high volatility, whereas the relation is positive in periods of low volatility.4 They also propose to model risk with a new measure, the realized covariance calculated as the product between the changes in the Aruoba et al. (2009) index of business conditions and the stock returns. We follow their approach and include this new measure of risk as a conditioning variable for estimating the risk-return trade-off. We estimate regime-switching risk-return relations using 1-week, 2-week, monthly and quarterly returns, ranging from February 1929 to December 2010. Our empirical results can be summarized as follows: • There is strong evidence for regime changes in the risk-return relation, as supported by the test for Markov-switching parameters recently introduced by Carrasco et al. (2013). • In the first regime characterized by low ex post returns and high volatility, the risk-return relation is negative, whereas the risk-return relation is positive in the second regime. This is consistent across all the frequencies that we consider and a wide range of specifications (the inclusion of additional predictors, the use of time-varying transition probabilities, the use of Student-t rather than normal innovations and the use of an asymmetric MIDAS estimator of the conditional variance). • The first regime can be interpreted as a "flight-to-quality" regime. This evidence corroborates the findings in (Ghysels et al., 2013), who document that the Merton model holds over samples that exclude financial crises, in particular, the Great Depression and/or the subprime mortgage financial crisis and the resulting Great Recession. They also report that a simple flight-to-quality indicator, based on the ex post extreme tail events, separates the traditional risk-return relation from financial crises, which amount to fundamental changes in that relation. In this paper, we show that a Markov switching regime model is indeed recovering a similar pattern. The paper is structured as follows. Section 2 presents the model we use for estimating the risk-return relation. Section 3 details the main results of the paper and provides a comparison of the estimated conditional variances with GARCH specifications. Section 4 provides a sensitivity analysis across a wide range of models as well as an out-of-sample forecasting exercise. Section 5 concludes.

This paper provides evidence for time instability in the risk-return relation. We allow for regime changes in the risk-return relation through regime switching in the parameter entering before the conditional variance as well as in the intercept of the risk-return relation. The conditional variance is modelled with a MIDAS estimator, which is less prone to misspecifications than GARCH models. We consider as a dependent variable the U.S. excess stock returns ranging from the weekly to the quarterly frequency and use two different estimation samples: (i) from February 1929 to December 2010 and (ii) from February 1964 to December 2010. We find strong statistical evidence for regime changes in the risk-return relation using the test recently introduced by Carrasco et al. (2013) for Markov-switching parameters. In the first regime, we find that the risk-return relation is reversed. Conversely, in the second regime, we uncover the traditional positive risk-return relation. The regime probabilities for the first regime are associated with a decline in stock returns, an increase in volatility and a flattening of the yield curve, which is concomitant with flight-to-quality episodes. Our findings help to explain why the literature has reported conflicting results and are qualitatively close to the recent contribution of Rossi and Timmermann (2010). Our results are also robust to a wide range of modifications: (i) the inclusion of additional predictors, (ii) the use of Student-t rather than Normal innovations, (iii) the use of time-varying rather than constant transition probabilities, and (iv) an asymmetric MIDAS estimator of the conditional variance.