Quasi-Maximum Likelihood Estimation of Stochastic Variance Models

Changes in variance or volatility over time can be modelled using stochastic volatility (SV) models. This approach is based on treating the variance as an unobservable variable, the logarithm of which is modelled as a linear stochastic process, usually an autoregression. Although it is not easy to obtain the exact likelihood for SV models, they tie in closely with finance theory and have certain statistical attractions. This article analyses the asymptotic and finite sample properties of a Quasi Maximum Likelihood (QML) estimator based on the Kalman filter applied to the appropriate transformation of the observations. The relative efficiency of the QML estimator when compared with estimators based on the Generalized Method of Moments is shown to be quite high for the typical parameter values often found in empirical applications. The QML estimator can still be employed when the SV model is generalized to allow for distributions with heavier or thinner tails than the normal. If, for example, the errors are assumed to be a Student-t variable, we find the somewhat counter-intuitive result that there is a slight loss in efficiency if the number of degrees of freedom is assumed to be known rather than being estimated. The model is finally fitted to daily observations on the Yen/Dollar exchange rate.