Asymptotic Theory for Rotated Multivariate GARCH Models

Abstract

In this paper, we derive the statistical properties of a two step approach to estimating multivariate GARCH rotated BEKK (RBEKK) models. By the definition of rotated BEKK, we estimate the unconditional covariance matrix in the first step in order to rotate observed variables to have the identity matrix for its sample covariance matrix. In the second step, we estimate the remaining parameters via maximizing the quasi-likelihood function. For this two step quasi-maximum likelihood (2sQML) estimator, we show consistency and asymptotic normality under weak conditions. While second-order moments are needed for consistency of the estimated unconditional covariance matrix, the existence of finite sixth-order moments are required for convergence of the second-order derivatives of the quasi-log-likelihood function. We also show the relationship of the asymptotic distributions of the 2sQML estimator for the RBEKK model and the variance targeting (VT) QML estimator for the VT-BEKK model. Monte Carlo experiments show that the bias of the 2sQML estimator is negligible, and that the appropriateness of the diagonal specification depends on the closeness to either of the Diagonal BEKK and the Diagonal RBEKK models.