GARCH Model Estimation Using Estimated Quadratic Variation

We consider estimates of the parameters of Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models obtained using auxiliary information on latent variance which may be available from higher-frequency data, for example from an estimate of the daily quadratic variation such as the realized variance. We obtain consistent estimators of the parameters of the infinite Autoregressive Conditional Heteroskedasticity (ARCH) representation via a regression using the estimated quadratic variation, without requiring that it be a consistent estimate; that is, variance information containing measurement error can be used for consistent estimation. We obtain GARCH parameters using a minimum distance estimator based on the estimated ARCH parameters. With Least Absolute Deviations (LAD) estimation of the truncated ARCH approximation, we show that consistency and asymptotic normality can be obtained using a general result on LAD estimation in truncated models of infinite-order processes. We provide simulation evidence on small-sample performance for varying numbers of intra-day observations.