Parameter estimation in linear regression models with stationary ARMA(p,q)-errors using Automatic Differentiation

The use of Automatic Differentiation for Time Series Analysis is considered. Especially the authors discuss the exact ML-estimation for linear regression models with stationary ARMA(p,q) residuals. The gradient and the Hessian matrix of the likelihood function, which has to be minimized, can be computed at fixed but arbitrary chosen points by Automatic Differentiation. The stationarity region for the ARMA(p,q) residuals is represented as a system of nonlinear inequalities. The special behavior of the likelihood function allows the use of well-known methods for solving unconstrained nonlinear programming problems.