Asymptotics for Garch Squared Residual Correlations

We develop an asymptotic theory for quadratic forms of the autocorrelations of squared residuals from a GARCH(p,q) model. Denoting by , k ≥ 1, these autocorrelations computed from a realization of length n, we show that the statistic is a matrix computed from the data, converges to the chi-square distribution with K degrees of freedom for any 1 ≤ i1 < ··· < iK. Our results are valid under weak assumptions on the innovations and model coefficients that admit that arbitrary low-order moments of the observations can be infinite. The matrix and its asymptotic limit D depend on the distribution of the innovations. A small simulation study illustrates the theory and shows, in particular, that using the matrix D computed under the assumption of normal innovations may lead to incorrect conclusions if the innovations have a different distribution.We thank the two referees for their comments and Professor Bruce E. Hansen, the co-editor in charge, for his sound advice on how to improve the paper. The work of István Berkes was supported by the Hungarian National Foundation for Scientific Research, grant T 29621. The work of Lajos Horváth and Piotr Kokoszka was supported by NATO grant PST.CLG.977607.