ASYMPTOTICALLY OPTIMAL SMOOTHING WITH ARCH MODELS

Suppose an observed time series is generated by a stochastic volatilit y model-i.e., there is an unobservable state variable controlling the volatility of the innovations in the series. As shown by Nelson (1992) , and Nelson and Foster (1994), a misspecified ARCH model will often b e able to consistently (as a continuous time limit is approached) esti mate the unobserved volatility process, using information in the lagge d residuals. This paper shows how to more efficiently estimate such a volatility process using information in both lagged and led residuals. In particular, this paper expands the optimal filtering results of Ne lson and Foster (1994) and Nelson (1994) to smoothing and to filtering with a random initial condition.