Stochastic Volatility: Univariate and Multivariate Extensions
Discrete time stochastic volatility models (hereafter SVOL) are noticeably harder to estimate than the successful ARCH family of models. Recent advances in the literature now make it possible to produce efficient estimation and prediction for a basic univariate SVOL model. However, the basic model may be insufficient for numerous economics and finance applications. In this paper, we develop methods for finite sample inference, smoothing, and prediction for a number of univariate and multivariate SVOL models. Specifically, we model fat-tailed and skewed conditional distributions, correlated errors distributions (leverage effect), and two multivariate models, a stochastic factor structure model and a stochastic discount dynamic model. We apply some of these extensions to financial series. We find (1) strong evidence of non-normal conditional distributions for stock returns and exchange rates, and (2) evidence of correlated errors for stock returns. These departures from the basic model affect the measured persistence and hence the prediction of volatility. This result as a policy implication on decisions and models such as asset allocation and option pricing which inputs are prediction of volatility. We specify the models as a hierarchy of conditional distributions: p(data | volatilities), p(volatilities | parameters) and p(parameters). Given a model and the data, inference and prediction are based on the joint posterior distribution of the volatilities and the parameters which we simulate via Markov chain Monte Carlo (hereafter MCMC) methods. This approach also provides a sensitivity analysis for parameter inference and an outlier diagnostic. The hierarchical framework is a natural environment for the construction of SVOL models departing from the standard distributional assumptions.
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