Asymptotic bias for quasi-maximum-likelihood estimators in conditional heteroskedasticity models

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.