Risk, Return and Volatility Feedback: A Bayesian Nonparametric Analysis

In this paper, we let the data speak for itself about the existence of volatility feedback and the often debated risk–return relationship. We do this by modeling the contemporaneous relationship between market excess returns and log-realized variances with a nonparametric, infinitely-ordered, mixture representation of the observables’ joint distribution. Our nonparametric estimator allows for deviation from conditional Gaussianity through non-zero, higher ordered, moments, like asymmetric, fat-tailed behavior, along with smooth, nonlinear, risk–return relationships. We use the parsimonious and relatively uninformative Bayesian Dirichlet process prior to overcoming the problem of having too many unknowns and not enough observations. Applying our Bayesian nonparametric model to more than a century’s worth of monthly US stock market returns and realized variances, we find strong, robust evidence of volatility feedback. Once volatility feedback is accounted for, we find an unambiguous positive, nonlinear, relationship between expected excess returns and expected log-realized variance. In addition to the conditional mean, volatility feedback impacts the entire joint distribution.