Second order pseudo-maximum likelihood estimation and conditional variance misspecification

In this paper, we study the behavior of second order pseudo-maximum likelihood estimators under conditional variance misspecification. We determine sufficient and essentially necessary conditions for such a estimator to be, regardless of the conditional variance (mis)specification, consistent for the mean parameters when the conditional mean is correctly specified. These conditions implie that, even if mean and variance parameters vary independently, standard PML2 estimators are generally not robust to conditional variance misspecification. Further, we outline sufficient and essentially necessary conditions for a second order pseudo-maximum likelihood estimator to be consistent for both mean and variance parameters when the conditional mean and the conditional variance are jointly correctly specified, and to remain consistent for the mean parameters when the conditional mean is correctly specified but the conditional variance is not jointly correctly specified. Finally, we provide limiting distribution results for this latter robust to conditional variance misspecification class of estimators under different assumptions regarding the degree of misspecification present in the model, show that its asymptotic covariance matrix is bounded and compare its similarities with QGPML1.