Mark J. Jensen and John M. Maheu
Working Paper 2012-6
April 2012
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In this paper, we extend the parametric, asymmetric, stochastic volatility model (ASV), where returns are correlated with volatility, by flexibly modeling the bivariate distribution of the return and volatility innovations nonparametrically. Its novelty is in modeling the joint, conditional, return-volatility distribution with an infinite mixture of bivariate Normal distributions with mean zero vectors, but having unknown mixture weights and covariance matrices. This semiparametric ASV model nests stochastic volatility models whose innovations are distributed as either Normal or Student-t distributions, plus the response in volatility to unexpected return shocks is more general than the fixed asymmetric response with the ASV model. The unknown mixture parameters are modeled with a Dirichlet process prior. This prior ensures a parsimonious, finite, posterior mixture that best represents the distribution of the innovations and a straightforward sampler of the conditional posteriors. We develop a Bayesian Markov chain Monte Carlo sampler to fully characterize the parametric and distributional uncertainty. Nested model comparisons and out-of-sample predictions with the cumulative marginal-likelihoods, and one-day-ahead, predictive log-Bayes factors between the semiparametric and parametric versions of the ASV model shows the semiparametric model projecting more accurate empirical market returns. A major reason is how volatility responds to an unexpected market movement. When the market is tranquil, expected volatility reacts to a negative (positive) price shock by rising (initially declining, but then rising when the positive shock is large). However, when the market is volatile, the degree of asymmetry and the size of the response in expected volatility is muted. In other words, when times are good, no news is good news, but when times are bad, neither good nor bad news matters with regards to volatility.
JEL classification: C11, C14, C53, C58
Key words: Bayesian nonparametrics, cumulative Bayes factor, Dirichlet process mixture, infinite mixture model, leverage effect, marginal likelihood, MCMC, non-normal, stochastic volatility, volatility-return relationship
The authors thank the seminar participants at the 2010 Rimini Conference in Economics and Finance, the 10th Annual All-Georgia Financial Conference, the 2011 Seminar on Bayesian Inference in Econometrics and Statistics at Washington University in Saint Louis, the Department of Economics at Istanbul Bilgi University, the Econometrics and Statistics Colloquium at the Booth School Business at the University of Chicago, and Sim Kee Boon Institute for Financial Economics at Singapore Management University. Special thanks go to Mark Fisher, Hedibert Lopes, and Chayawat Ornthanalai for comments and suggestions. 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.
Please address questions regarding content to Mark J. Jensen, Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street, N.E., Atlanta, GA 30309-4470, 404-498-8019, mark.jensen@atl.frb.org, or John M. Maheu, Department of Economics, University of Toronto & RCEA, 150 St. George Street, Toronto, Canada M5S 3G7, 416-978-1495, jmaheu@chass.utoronto.ca.
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