Inference for Vector AR Models with Heavy-tailed Vector GARCH Noises

This paper first studies the tail-index of a vector GARCH process (VGARCH): εt=Dtηt, Ht=W+r∑i=1Ai→−i+s∑i=1BiHt−i, where Dt=diag (h1/21t, ⋯, h1/2mt), Ht= (h1t, ⋯, hmt) ′, →εt= (ε21t, ⋯, ε2mt) ′ and ηt is a sequence of i. i. d. random vectors. The regular variation condition is established with the tail index α>0. In particular, we show that the tail index of εt is α=2 when −∑rr=1Aizi−∑si=1Bizi\=0 has some roots on the unit circle and other roots lie outside the unit circle. Furthermore, we study the vector heavy-tailed AR-VGARCH model. The limiting distributions of the autocovariance matrices are shown to be the functional of a sequence of vector stable processes. Based on these results, we show that the LSE of parameters of the AR part is not an consistent if the tail-index α of GARCH is in (0, 2), is logn-consistent if α=2, is n1−2/α-consistent if α∈ (2, 4), and n1/2/logn-consistent if α=4, and its limiting distribution is a functional of vector stable processes when α∈[2, 4) and is asymptotical normal when α≥4. The results would offer some helpful insights for the practitioners in economic and financial study.