Learning about Volatility: The Local Scale Model with Homoskedastic Innovations, with Application to Stock Returns

The Local Scale Model (LSM) of Shephard (1994) is a state-space model of volatility clustering similar in effect to IGARCH, but with an unobserved volatility that realistically evolves independently of the observed errors, instead of being mechanically determined by them. It has one fewer parameter to estimate than IGARCH, and a closed form likelihood, despite the unobservability of the volatility. Although the errors are assumed to be Gaussian conditional on the unobserved stochastic variance, they are Student t when conditioned on experience, with degrees of freedom that grow to a finite bound. The present paper improves on the Shephard (1994) model by assigning equal variance to the innovations to the volatility. The implied volatility gain at first declines sharply as in the Local Level Model, rather than being constant throughout as in traditional IGARCH. The improved model is fit to monthly stock returns by Maximum Likelihood. The parameter estimates imply 7.76 steady-state degrees of freedom. A short-lived “Great Moderation” is evident during the mid-1990’s, but expires by 1998. Otherwise the period since 1970 was generally more volatile than the 1950s and 60s, though less so than the 1930s and 40s. The LSM volatility responds more nimbly to the data than does an IGARCH model. Although the Student t densities generated by the Gaussian-based LSM account for much of the conditional leptokurtosis in the data, further refinements will be required to adequately model the pronounced negative skewness and/or residual leptokurtosis in stock returns.