**Mark J Jensen and John M Maheu, "Estimating a Semiparametric Asymmetric Stochastic Volatility Model with a Dirichlet Process Mixture"**, 2012-04-20

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*Abstract:* 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 a 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 bests 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 forecasting 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.

*Keywords:* Bayesian nonparametrics, cumulative Bayes factor, Dirichlet process mixture, infinite mixture model, leverage eﬀect, marginal likelihood, MCMC, non-normal, stochastic volatility, volatility-return relationship

*JEL Classification:* C11; C14; C53; C58