The relationship between risk and return is one of the most studied topics in finance. The majority of the literature is based on a linear, parametric relationship between expected returns and conditional volatility. This paper models the contemporaneous relationship between market excess returns and contemporaneous log-realized variances nonparametrically with an infinite mixture representation of their joint distribution. The conditional distribution of excess returns given log-realized variance will also have an infinite mixture representation but with probabilities and arguments depending on the value of realized variance. Our nonparametric approach allows for deviation from Gaussianity by allowing for higher-order nonzero moments and a smooth nonlinear relationship between the conditional mean of excess returns and contemporaneous log-realized variance. We find strong robust evidence of volatility feedback in monthly data. Once volatility feedback is accounted for, there is an unambiguous positive relationship between expected excess returns and expected log-realized variance. This relationship is nonlinear. Volatility feedback impacts the whole distribution and not just the conditional mean.
JEL classification: C11, C14, C32, G12
Key words: Dirichlet process prior, slice sampling, dependent Bayesian nonparametrics
The authors are grateful for helpful comments from Tolga Cenesizoglu, Christian Dorion, and Georgios Skoulakis. They also thank conference participants at the 2012 International Conference on Computational and Financial Econometrics, the National Bureau of Economic Research-National Science Foundation's 2013 Seminar on Bayesian Inference in Econometrics and Statistics, the Rimini Centre for Economic Analysis's 2013 Bayesian workshop, and the 2014 workshop on applied financial time-series at HEC Montreal. They also thank seminar participants at McMaster University and University of Toronto. A previous version of this work was titled "A Bayesian Nonparametric Analysis of the Relationship between Returns and Realized Variance." They are also grateful to Tom McCurdy, who supplied the data. The views expressed here are the authors' and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors' responsibility. Maheu is grateful to the Social Sciences and Humanities Research Council for financial support.
Please address questions regarding content to Mark J. Jensen, Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street NE, Atlanta, GA 30309-4470, 404-498-8019, Mark.Jensen@atl.frb.org; or John M. Maneu, DeGroote School of Business, McMaster University, 1280 Main Street W, Hamilton, Ontario, Canada, L8S4M4, and the University of Toronto, Canada, and RCEA, Italy, email@example.com.
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