Asymptotic theory for multivariate GARCH processes

Multivariate Variance Targeting in the BEKK-GARCH Model

“…We prove this lemma arguing in line with the proof of Theorem 3.2 in Francq and Zakoïan (2010), see also p.77-78 in Comte and Lieberman (2003). By de…nition…”

“…Whereas classical QMLE of the BEKK model has been considered by Comte and Lieberman (2003) and Hafner and Preminger (2009b) (as a special case of the VEC GARCH model), we consider the estimation method of variance targeting. VT estimation is a twostep estimation method where is estimated by a the sample unconditional covariance matrix of X t , and next is estimated by QMLE by optimizing the VT log-likelihood with respect to .…”

“…The proofs are stated in Appendix A. As in Comte and Lieberman (2003), Hafner and Preminger (2009b), and Francq, Horváth, and Zakoïan (2011), we assume thatfX t g t=0;:::;T is strictly stationary and ergodic:…”

“…Remark 4.1 Assumptions 4.2, and 4.3 are in line with Comte and Lieberman (2003) and Hafner and Preminger (2009b).…”

“…E jl t ( 0 ; 0 )j < 1 follows from Lemma B.2. Following the steps from Section 3 in Comte and Lieberman (2003), suppose 6 = 0 and let fe it : i = 1; ::; dg be the (positive) eigenvalues of…”