Asymptotically Optimal Smoothing with Arch Models

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 (19941, 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.
KEYWORDSA:RCH, nonlinear filtering, smoothing, stochastic volatility

THISPAPER MAKES TWO EXTENSIONS to the ARCH asymptotic filtering theory of
Nelson and Foster (1994) and Nelson (1994) (henceforth NF and N respectively).
First, we allow a random initial condition for the filtering error. Once this
extension is made, we are able to consider using both leads and lags of observed
state variables to estimate unobserved state variables-i.e., smoothing. In
econometric practice, the conditional variances generated by an ARCH model
are usually treated as "true7' apart from parameter estimation error (see, for
example, the survey papers of Bollerslev, Chou, and Kroner (1992) or of Engle,
Bollerslev, and Nelson (1994)). NF and N treat them simply as estimates of
unobservable state variables. Clearly, if the ARCH variances are true, then
there is no error in the estimate (conditional on the system parameters) and no
room for smoothing-i.e., all information is contained in the lagged residuals
and none in the led residuals. To the extent that ARCH conditional variances
are noisy estimates, however, there is a role for smoothing, and two-sided
ARCH models can be employed to improve estimates of historical volatilities.