Filtering and forecasting with misspecified ARCH models II: Making the right forecast with the wrong model

A companion paper (Nelson, 1992) showed that in data observed at high frequencies, an ARCH model may perform well in estimating the conditional variance of a process, even when the ARCH model is severely misspecified. While such models may perform reasonably well at filtering (i.e., at estimating unobserved instantaneous conditional variances), they may perform disastrously at medium- and long-term forecasting of the process and its volatility. In this paper, we develop conditions under which a misspecified ARCH model successfully performs both tasks, filtering and forecasting. The key requirement (in addition to the conditions for consistent filtering) is that the ARCH model correctly specifies the functional form of the first two conditional moments of all state variables. We apply these results to a diffusion model employed in the options pricing literature, the stochastic volatility model of Hull and White (1987), Scott (1987), and Wiggins (1987).