A THEORETICAL AND PRACTICAL PERSPECTIVE ON THE EQUITY RISK PREMIUM*

Abstract In historical perspective, equity returns have been higher than interest rates but have also varied a good deal more. However, the average excess return has been larger than what could be expected based on classical equilibrium theory: the equity risk premium (ERP) puzzle. This paper has two objectives. First, the paper presents a comprehensive overview of the vast literature developed aimed at adjusting theory and testing the robustness of the puzzle. Here we will show that the failure of theory to link asset prices to economics is mostly quantitative by nature and not qualitative (anymore). Second, beyond providing a survey of theory, we aim for a relevant practical angle as well. Our main contribution is that we spend time on why returns have been higher than investors reasonably could have expected. We present evidence that forecasts of equity returns can be enhanced by valuation models: low valuation levels (low price‐to‐earnings ratios) portend high subsequent returns. While conventional wisdom (several years ago) was to use historical returns to forecast future returns, a growing consensus now recognizes that the predictive power of valuation ratios is preferred. Finally we provide some practical implications based on this predictability. While the ERP is essentially a long‐term issue, the likelihood of a lower risk premium increases risk for many and means that short‐term volatility might not be neglected.     

The relation between the equity risk premium and the bond maturity premium in the UK: 1900–2006

Using a rich data set for the UK for over a century, we find that the relation between the equity risk premium and the government bond maturity premium is nonlinear and subject to stochastic regime switching. We identify a regime in which both premia are jointly characterized by low volatility and another regime in which both premia are characterized by high volatility. The occurrence of the high volatility regime chronologically coincides with major changes in the pound exchange rate. The low volatility regime has a higher probability of turning up over two consecutive years than the high volatility regime, but it is not perceived by investors to be an absorbing regime. The lagged maturity premium is a strong predictor of the equity risk premium only in the regime of low volatility. In addition, the lagged equity premium is a predictor of the maturity premium also in the low volatility regime. This result on regime-dependent bidirectional predictability is robust to alternative definitions of the equity premium, and to the inclusion of real interest rate and real growth effects.

Pricing of the time-change risks

We develop an equilibrium endowment economy with Epstein-Zin recursive utility and a Lévy time-change subordinator, which represents a clock that connects business and calendar time. Our setup provides a tractable equilibrium framework for pricing non-Gaussian jump-like risks induced by the time-change, with closed-form solutions for asset prices. Persistence of the time-change shocks leads to predictability of consumption and dividends and time-variation in asset prices and risk premia in calendar time. In numerical calibrations, we show that the risk compensation for Lévy risks accounts for about one-third of the overall equity premium.     

A Simple Approach for Diagnosing Instabilities in Predictive Regressions

We introduce a method for detecting the presence of structural breaks in the parameters of predictive regressions linking noisy variables such as stock returns to persistent predictors such as valuation ratios. Our approach relies on the least squares‐based squared residuals of the predictive regression and is straightforward to implement. The distributions of the two test statistics we introduce are shown to be free of nuisance parameters, valid under dependent errors, already tabulated in the literature and robust to the degree of persistence of the chosen predictor. Our proposed method is subsequently applied to the predictability of US stock returns.     

Predictive regression under various degrees of persistence and robust long-horizon regression

The paper proposes a novel inference procedure for long-horizon predictive regression with persistent regressors, allowing the autoregressive roots to lie in a wide vicinity of unity. The invalidity of conventional tests when regressors are persistent has led to a large literature dealing with inference in predictive regressions with local to unity regressors. Magdalinos and Phillips (2009b) recently developed a new framework of extended IV procedures (IVX) that enables robust chi-square testing for a wider class of persistent regressors. We extend this robust procedure to an even wider parameter space in the vicinity of unity and apply the methods to long-horizon predictive regression. Existing methods in this model, which rely on simulated critical values by inverting tests under local to unity conditions, cannot be easily extended beyond the scalar regressor case or to wider autoregressive parametrizations. In contrast, the methods developed here lead to standard chi-square tests, allow for multivariate regressors, and include predictive processes whose roots may lie in a wide vicinity of unity. As such they have many potential applications in predictive regression. In addition to asymptotics under the null hypothesis of no predictability, the paper investigates validity under the alternative, showing how balance in the regression may be achieved through the use of localizing coefficients and developing local asymptotic power properties under such alternatives. These results help to explain some of the empirical difficulties that have been encountered in establishing predictability of stock returns.     

Contagion between real estate and financial markets: A Bayesian quantile-on-quantile approach

We study contagion between Real Estate Investment Trusts (REITs) and the equity market in the U.S. over four sub-samples covering January, 2003 to December, 2017, by using Bayesian nonparametric quantile-on-quantile (QQ) regressions with heteroskedasticity. We find that the spillovers from the REITs on to the equity market has varied over time and quantiles defining the states of these two markets across the four sub-samples, thus providing evidence of shift-contagion. Further, contagion from REITs upon the stock market went up during the global financial crisis particularly, and also over the period corresponding to the European sovereign debt crisis, relative to the pre-crisis period. Our main findings are robust to alternative model specifications of the benchmark Bayesian QQ model, especially when we control for omitted variable bias using the heteroskedastic error structure. Our results have important implications for various agents in the economy namely, academics, investors and policymakers.     

This is what the leading indicators lead

We propose an optimal filter to transform the Conference Board Composite Leading Index (CLI) into recession probabilities in the US economy. We also analyse the CLI's accuracy at anticipating US output growth. We compare the predictive performance of linear, VAR extensions of smooth transition regression and switching regimes, probit, non‐parametric models and conclude that a combination of the switching regimes and non‐parametric forecasts is the best strategy at predicting both the NBER business cycle schedule and GDP growth. This confirms the usefulness of CLI, even in a real‐time analysis. Copyright © 2002 John Wiley & Sons, Ltd.     

Robust Non‐nested Testing for Ordinary Least Squares Regression when Some of the Regressors are Lagged Dependent Variables*

The problem of testing non‐nested regression models that include lagged values of the dependent variable as regressors is discussed. It is argued that it is essential to test for error autocorrelation if ordinary least squares and the associated J and F tests are to be used. A heteroskedasticity–robust joint test against a combination of the artificial alternatives used for autocorrelation and non‐nested hypothesis tests is proposed. Monte Carlo results indicate that implementing this joint test using a wild bootstrap method leads to a well‐behaved procedure and gives better control of finite sample significance levels than asymptotic critical values.     

Market efficiency,asset returns,and the size of the risk premium in global equity markets

An important economic insight is that observed equity prices must equal the present value of the cash flows associated with the equity claim. An implication of this insight is that present values of cash flows must also quantitatively justify the observed volatility and cross-correlations of asset returns. In this paper, we show that parametric economic models for present values can indeed account for the observed high ex post return volatility and cross-correlation observed across five major equity markets—the U.S., the U.K., France, Germany, and Japan. We present evidence that cash flow growth rates contain a small predictable long-run component; this feature, in conjunction with time-varying systematic risk, can justify key empirical characteristics of observed equity prices. Our model also has direct implications for the level of equity prices and specific versions of the model can, in many cases, capture observed price levels. Our evidence suggests that the ex ante risk premium on the global market portfolio has dropped considerably—we show that this fall in the risk premium is related to a decline in the conditional variance of global real cash flow growth rates.     

REVERSE REGRESSIONS AND LONG‐HORIZON FORECASTING

Long‐horizon predictive regressions in finance pose formidable econometric problems when estimated using available sample sizes. Hodrick in 1992 proposed a remedy that is based on running a reverse regression of short‐horizon returns on the long‐run mean of the predictor. Unfortunately, this only allows the null of no predictability to be tested, and assumes stationary regressors. In this paper, we revisit long‐horizon forecasting from reverse regressions, and argue that reverse regression methods avoid serious size distortions in long‐horizon predictive regressions, even when there is some predictability and/or near unit roots. Meanwhile, the reverse regression methodology has the practical advantage of being easily applicable when there are many predictors. We apply these methods to forecasting excess bond returns using the term structure of forward rates, and find that there is indeed some return forecastability. However, confidence intervals for the coefficients of the predictive regressions are about twice as wide as those obtained with the conventional approach to inference. We also include an application to forecasting excess stock returns. Copyright © 2011 John Wiley & Sons, Ltd.     

Daily House Price Indices: Construction,Modeling, and Longer‐run Predictions

We construct daily house price indices for 10 major US metropolitan areas. Our calculations are based on a comprehensive database of several million residential property transactions and a standard repeat‐sales method that closely mimics the methodology of the popular monthly Case–Shiller house price indices. Our new daily house price indices exhibit dynamic features similar to those of other daily asset prices, with mild autocorrelation and strong conditional heteroskedasticity of the corresponding daily returns. A relatively simple multivariate time series model for the daily house price index returns, explicitly allowing for commonalities across cities and GARCH effects, produces forecasts of longer‐run monthly house price changes that are superior to various alternative forecast procedures based on lower‐frequency data. Copyright © 2015 John Wiley & Sons, Ltd.     

Improved instrumental variables estimation of simultaneous equations under conditionally heteroskedastic disturbances

In this paper we develop estimation techniques and a specification test for the validity of instrumental variables allowing for conditionally heteroskedastic disturbances. We propose modified two‐stage least squares (2SLS) and modified 3SLS procedures where the conditional heteroskedasticity is taken into account, which are natural extensions of the traditional 2SLS and 3SLS estimators and which achieve a lower variance. We recommend the use of these modified 2SLS and 3SLS procedures in practice instead of alternative estimators like limited‐information maximum likelihood/full‐information maximum likelihood, where the non‐existence of moments leads to extreme values, and also for ease of computation. It is shown theoretically and with simulation that in some cases 2SLS, 3SLS and our modified 2SLS and 3SLS procedures can have very severe biases (including the weak instruments case), and we present bias correction procedures to apply in practice along the lines of Flores‐Lagunes ( 2007 ). Our new estimation procedures can also be used to extend the test for weak instruments of Stock and Yogo ( 2005 ) and to allow for conditional heteroskedasticity. Finally, we show the usefulness of our estimation procedures with an application to the demand and supply of fish. Copyright © 2010 John Wiley & Sons, Ltd.     

Does competition prevent industrial pollution? Evidence from a panel threshold model

The objective of this paper is to assess the impact of competition on industrial toxic pollution by using, for the first time, a panel threshold model which allows evaluations of the main drivers of toxic releases under two different market regimes. The empirical analysis is based on a micro‐level panel dataset over the five‐year period 1987–2012. We show that this relationship is statistically significant and robust above and below the threshold, even after accounting for alternative specifications of market concentration. Specifically, we unmask an inverted V‐shaped relationship between market concentration and industrial pollution. We argue that the increasing non‐parametric regression line up to a certain concentration (threshold) level indicates a negative effect on facilities' emissions levels, whereas a decreasing line indicates a positive effect. This relationship provides new insights into environmental policy design towards abatement of industrial releases and sustainability. Finally, our empirical model remains robust under different specifications properly accounted for possible endogeneity.     

Understanding saving and portfolio choices with predictable changes in assets returns

We consider an expected-utility-maximizing consumer living two periods who can invest in two assets, one of which is risk free. We do not restrict relative risk aversion to be constant. We first examine the effect that a change in the opportunity set in the second period has on the optimal saving in the first period. We show that an increase in the future risk free rate (keeping the equity premium unchanged) reduces savings if relative risk aversion is uniformly larger than unity. An increase in the equity premium or a reduction in the volatility of the risky asset raises savings if the index of cautiousness, i.e., the derivative of absolute risk tolerance, is smaller than unity. In a second stage, we use these results to determine the sign of the hedging demand for the risky asset for the following three types of predictability: predictable changes in the interest rate, mean-reversion in stock returns, and predictable changes in volatility. Depending upon the type of predictability under scrutiny, what matters to sign the hedging demand is whether relative risk aversion or cautiousness is smaller or larger than unity.     

Estimating and predicting multivariate volatility thresholds in global stock markets

We propose a general double tree structured AR‐GARCH model for the analysis of global equity index returns. The model extends previous approaches by incorporating (i) several multivariate thresholds in conditional means and volatilities of index returns and (ii) a richer specification for the impact of lagged foreign (US) index returns in each threshold. We evaluate the out‐of‐sample forecasting power of our model for eight major equity indices in comparison to some existing volatility models in the literature. We find strong evidence for more than one multivariate threshold (more than two regimes) in conditional means and variances of global equity index returns. Such multivariate thresholds are affected by foreign (US) lagged index returns and yield a higher out‐of‐sample predictive power for our tree structured model setting. Copyright © 2006 John Wiley & Sons, Ltd.     

Forecasting Bitcoin risk measures: A robust approach

Over the last few years, Bitcoin and other cryptocurrencies have attracted the interest of many investors, practitioners and researchers. However, little attention has been paid to the predictability of their risk measures. This paper compares the predictability of the one-step-ahead volatility and Value-at-Risk of Bitcoin using several volatility models. We also include procedures that take into account the presence of outliers and estimate the volatility and Value-at-Risk in a robust fashion. Our results show that robust procedures outperform non-robust ones when forecasting the volatility and estimating the Value-at-Risk. These results suggest that the presence of outliers plays an important role in the modelling and forecasting of Bitcoin risk measures.     

On Business Cycle Asymmetries in G7 Countries

We investigate whether business cycle dynamics in seven industrialized countries (the G7) are characterized by asymmetries in conditional mean. We provide evidence on this issue using a variety of time series models. Our approach is fully parametric. Our testing strategy is robust to any conditional heteroskedasticity, outliers, and/or long memory that may be present. Our results indicate fairly strong evidence of nonlinearities in the conditional mean dynamics of the GDP growth rates for Canada, Germany, Italy, Japan, and the US. For France and the UK, the conditional mean dynamics appear to be largely linear. Our study shows that while the existence of conditional heteroskedasticity and long memory does not have much effect on testing for linearity in the conditional mean, accounting for outliers does reduce the evidence against linearity.     

The VIX,the variance premium and stock market volatility

We decompose the squared VIX index, derived from US S&P500 options prices, into the conditional variance of stock returns and the equity variance premium. We evaluate a plethora of state-of-the-art volatility forecasting models to produce an accurate measure of the conditional variance. We then examine the predictive power of the VIX and its two components for stock market returns, economic activity and financial instability. The variance premium predicts stock returns while the conditional stock market variance predicts economic activity and has a relatively higher predictive power for financial instability than does the variance premium.     

Estimating the Kullback–Liebler risk based on multifold cross‐validation

This paper concerns a class of model selection criteria based on cross‐validation techniques and estimative predictive densities. Both the simple or leave‐one‐out and the multifold or leave‐m‐out cross‐validation procedures are considered. These cross‐validation criteria define suitable estimators for the expected Kullback–Liebler risk, which measures the expected discrepancy between the fitted candidate model and the true one. In particular, we shall investigate the potential bias of these estimators, under alternative asymptotic regimes for m. The results are obtained within the general context of independent, but not necessarily identically distributed, observations and by assuming that the candidate model may not contain the true distribution. An application to the class of normal regression models is also presented, and simulation results are obtained in order to gain some further understanding on the behavior of the estimators.     

Data snooping in equity premium prediction

We analyze the performance of a comprehensive set of equity premium forecasting strategies. All strategies were found to outperform the mean in previous academic publications. However, using a multiple testing framework to account for data snooping, our findings support Welch and Goyal (2008) in that almost all equity premium forecasts fail to beat the mean out-of-sample. Only few forecasting strategies that are based on Ferreira and Santa-Clara’s (2011) sum-of-the-parts approach generate robust and statistically significant economic gains relative to the historical mean even after controlling for data snooping and accounting for transaction costs.