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Continuous Record Asymptotics for Rolling Sample Variance Estimators

Dean P. Foster
Dan B. Nelson

Abstract

It is widely known that conditional covariances of asset returns change over time. Researchers doing empirical work have adopted many strategies for accommodating conditional heteroskedasticity. Among the popular strategies are: (a) chopping the available data into short blocks of time and assuming homoskedasticity within the blocks, (b) performing one-sided rolling regressions, in which only data from, say, the preceding five year period is used to estimate the conditional covariance of returns at a given date, and (c) performing two-sided rolling regressions, in which covariances are estimated for each date using, say, five years of lags and five years of leads. Another model--GARCH--amounts to a one-sided weighted rolling regression. We develop continuous record asymptotic approximations for the measurement error in conditional variances and covariances when using these methods. We derive asymptotically optimal window lengths for standard rolling regressions and optimal weights for weighted rolling regressions. As an empirical example, we estimate volatility on the S & P 500 stock index using daily data from 1928 to 1990.