Econometrica: May 1997, Volume 65, Issue 3

Asymptotic Bias for Quasi-Maximum-Likelihood Estimators in Conditional Heteroskedasticity Models

https://doi.org/0012-9682(199705)65:3<587:ABFQEI>2.0.CO;2-V
p. 587-599

Douglas G. Steigerwald, Whitney K. Newey

Virtually all applications of time-varying conditional variance models use a quasi-maximum-likelihood estimator (QMLE). Consistency of a QMLE requires an identification condition that the quasi-log-likelihood have a unique maximum at the true conditional mean and relative scale parameters. We show that the identification condition holds for a non-Gaussian QMLE if the conditional mean is identically zero or if a symmetry condition is satisfied. Without symmetry, an additional parameter, for the location of the innovation density, must be added for identification. We calculate the efficiency loss from adding such a parameter under symmetry, when the parameter is not needed. We also show that there is no efficiency loss for the conditional variance parameters of a GARCH process.

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