Econometrica: May, 1996, Volume 64, Issue 3

Asymptotically Optimal Smoothing with Arch Models

https://doi.org/0012-9682(199605)64:3<561:AOSWAM>2.0.CO;2-6
p. 561-573

Daniel B. Nelson

Suppose an observed time series is generated by a stochastic volatility 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 be able to consistently (as a continuous time limit is approached) estimate the unobserved volatility process, using information in the lagged 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 Nelson and Foster (1994) and Nelson (1994) to smoothing and to filtering with a random initial condition.