The Fluctuation of the Quasi-Gaussian Likelihood

Abstract

It has already been noted that use of the quasi-Gaussian likelihood in the case of a causal and invertible ARMA process leads to consistent and asymptotically normal estimates of the unknown parameters of the model. However, in the non-Gaussian context, even though and invertible (that is, minimum phase), the estimates are not efficient. In the nonminimum phase non-Gaussian case the estimates are not even consistent. However, because most estimation procedures use the quasi-Gaussian likelihood and maximize it in the minimum phase case to get estimates, it seems relevant to look at the likelihood as a surface in the parameters. There are good reasons to look at the likelihood surface rather than directly analyze the maximization. The approximation of the likelihood surface globally may yield an effective moderate sample representation that gives better insight than a direct large sample analysis of the estimate. The random fluctuation of the likelihood may lead to several local maxima that could lead a numerical optimization procedure away from the global maximum. In such a case, the quality of the estimate might depend to a great extent on the starting value obtained by an initial estimation procedure. This is especially the case if the local maxima due to random fluctuation occur in the case of likelihood functions that are relatively flat in a neighborhood of the true parameter values.