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 (r) over cap (n)(k), k greater than or equal to 1, these autocorrelations computed from a realization of length n, we show that the statistic n((r) over cap (n)(i(1)),..., (r) over cap (n)(i(K))) (D) over cap (-1)(n)((r) over cap (n)(i(1)),...,(r) over cap (n)(i(K)))(T), where (D) over cap (n) is a matrix computed from the data, converges to the chi-square distribution with K degrees of freedom for any 1 less than or equal to i(1) < (...) < i(K). 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 (D) over cap (n) 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.