Moment–Based Estimation of Stochastic Volatility Models

This chapter reviews the possible uses of the Generalized Method of Moments (GMM) to estimate Stochastic Volatility (SV) models. A primary attraction of the method of moments technique is that it is well suited for identifying and estimating volatility models without a complete parametric specification of the probability distributions. Moreover, simulation-based methods of moments are able to exploit a variety of moments, while avoiding limitations due to a lack of closed form expressions. The chapter first highlights the suitability of GMM for popular regression models of volatility forecasting. Then, it reviews the implications of the SV model specification in terms of higher order moments: skewness, kurtosis, variance of the variance, leverage and feedback effects. The chapter examines the ability of a continuous time version of SV models to accommodate data from other sources like option prices or high frequency data on returns and transactions dates. Simulation-based methods are particularly useful for studying continuous time models due to the frequent lack of closed form expressions for their discrete time dynamics. These simulation-based methods of moments are presented within the unifying framework of indirect inference with a special emphasis on misspecification. Likely misspecification of the parametric model used for simulation requires a parsimonious and well-focused choice of the moments to match.