Second Derivative of the Likelihood of an Exact ARMA Model

摘要

The likelihood of an exact Autoregressive Moving Average (ARMA) model under theassumption of normality is investigated. Using a closed form expression of the covariance matrix the elements of the matrix of second derivatives of the concentrated likelihood function are derived. These elements consist in general of four or five terms. Two terms come from the determinant, one of them belonging to the information matrix. These terms are sums of the elements of the covariance matrix or its inverse. The next two terms are quadratic forms of the error vector. The last term is based on the matrix of independent variables, and thus only present in a regression model and not in the pure time series model. The general form of second derivative does not permit conclusions about the existence of global maximum of the likelihood function.