Asymptotic Filtering Theory for Univariate Arch Models

Many researchers have employed ARCH models to estimate conditional variances and
covariances. How successfully can ARCH models carry out this estimation when they are
misspecified? This paper employs continuous record asymptotics to approximate the
distribution of the measurement error. This allows us to (a) compare the efficiency of
various ARCH models, (b) characterize the impact of different kinds of misspecification
(e.g., "fat-tailed" errors, misspecified conditional means) on efficiency, and (c) characterize
asymptotically optimal ARCH conditional variance estimates. We apply our results to
derive optimal ARCH filters for three diffusion models, and to examine in detail the
filtering properties of GARCH(l,1), AR(1) EGARCH, and the model of Taylor (1986)
and Schwert (1989).
KEYWORDSA:RCH, filtering, stochastic volatility

1. INTRODUCTION
MOST ASSET PRICING THEORIES relate expected returns on assets to their
conditional variances and covariances. An enormous literature in empirical
finance has documented that these conditional moments change over time.
Practical experience (as in the 1929 and 1987 stock market crashes) reinforces
this conclusion. Unfortunately, conditional variances and covariances are not
directly observable, and researchers and market participants must use estimates
of conditional second moments. To create these estimates, they rely on models
which are, no doubt, misspecified. How accurate are these estimated variances
and covariances? How can researchers estimate them more accurately?