We develop an asymptotic theory for quadratic forms of the autocorrelations of squared residuals from a GARCH (p,q) model. Denoting by r_n(k), k >=1, these autocorrelations computed from a realization of length n, we show that the statistic n(r_n(i_1),....,r_n(i_K))D_n~(-1) (r_n(i_1),....,r_n(i_k))~T, where D_n is a matrix compound from the data, converges to the chi-square distribution with K degrees of freedom for any 1 <= i_n < … < 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_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.
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