FILTERING AND FORECASTING WITH MISSPECIFIED ARCH MODELS .1. GETTING THE RIGHT VARIANCE WITH THE WRONG MODEL

This paper investigates the properties of the conditional covariance estimates generated by a misspecified ARCH model. For example, suppose that we observe a diffusion process at discrete time intervals of length h. For each h, we use a GARCH(1,1) model to estimate the instantaneous conditional covariance matrix of the diffusion. Under mild regularity conditions, the difference between these conditional covariance estimates and the true conditional covariance converges to zero in probability as h down 0. Many other ARCH models (for example, Exponential ARCH) have similar consistency properties. This may well account for the success of ARCH models in short-term forecasting using high-frequency data, since even misspecified ARCH models can produce 'good' estimates of volatility.