A Closer Look at the Relation between GARCH and Stochastic Autoregressive Volatility

We show that, for three common SARV models, fitting a minimum mean square linear filter is equivalent to fitting a GARCH model. This suggests that GARCH models may be useful for filtering, forecasting, and parameter estimation in stochastic volatility settings. To investigate, we use simulations to evaluate how the three SARV models and their associated GARCH filters perform under controlled conditions and then we use daily currency and equity index returns to evaluate how the models perform in a risk management application. Although the GARCH models produce less precise forecasts than the SARV models in the simulations, it is not clear that the performance differences are large enough to be economically meaningful. Consistent with this view, we find that the GARCH and SARV models perform comparably in tests of conditional value-at-risk estimates using the actual data. keywords: GARCH, stochastic volatility, volatility forecasting, value-at-risk, particle filter, Markov chain Monte Carlo. There is no shortage of research on estimating volatility. Nonetheless, there are key aspects of the relation between the two main classes of volatility models ---generalized autoregressive conditional heteroscedasticity (GARCH) and stochastic autoregressive volatility (SARV) ---that warrant further investigation. Although GARCH models dominate in terms of popularity, this stems more from computational convenience than anything else. Indeed, theory suggests strong arguments for modeling volatility as stochastic and, if these arguments are valid, then the usual approach to estimating GARCH models is fundamentally misspecified. It is natural to ask, therefore, whether we can reconcile the We thank Barbara Ostdiek for providing many useful comments on an earlier draft as well as the editor (Eric Renault), an associate editor, three anonymous referees, Tim Bollersleve, Neil Shephard, and seminar participants at the Australian Graduate School of Management. Address correspondence to Jeff Fleming, Jones Graduate School of Management, Rice University, P.O. Box 2932, Houston, TX 77252-2932, or e-mail: