CONTINUOUS RECORD ASYMPTOTICS FOR ROLLING SAMPLE VARIANCE ESTIMATORS

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