Sample and Implied Volatility in GARCH Models

The unconditional variance of various GARCH-type models is a function h(theta) of the parameter vector theta which is estimated by theta. For most models used in practice, closed-form expressions of h(.) have been found. On the contrary, the unconditional variance can be estimated by the sample variance sigma^2. This article establishes the asymptotic distributions of the differences sigma^2 - h(theta) and &sigma^2 - h(theta) for broad classes of GARCH-type models. Even though both limit distributions are normal, the asymptotic variances are not equal. Potential practical consequences of these results are discussed. Copyright 2006, Oxford University Press.