Asymptotic Filtering Theory for Multivariate ARCH Models

ARCH models are widely used to estimate conditional variances and covariances in financial time series models. How successfully can ARCH models carry out this estimation when they are misspecified? How can ARCH models be optimally constructed? Nelson and Foster (1994) employed continuous record asymptotics to answer these questions in the univariate case. This paper considers the general multivariate case. Our results allow us, for example, to construct an asymptotically optimal ARCH model for estimating the conditional variance or conditional beta of a stock return given lagged returns on the stock, volume, market returns, implicit volatility from options contracts, and other relevant data. We also allow for time-varying shapes of conditional densities (e.g., `heteroskewticity` and `heterokurticity'). Examples are provided.