First Order Conditions for the Maximum Likelihood Estimation of an Exact ARMA Model

摘要

Using the exact covariance matrix of ARMA(p,q) errors, first order conditions for the parameters are derived and solved. This is done for the pure MA case, the pure AR case and the general ARMA model. The approach applies both to maximum likelihood and minimum distance estimation. The exact covariance is written in the form of lag matrices, which can simply be differentiated. The resulting first order conditions have at least one solution. The difference between maximum likelihood and minimum distance estimation amounts to a function of the elements of the covariance matrix. This function is simple in case of the pure MA or AR case, but more complicated in the general ARMA case. Of course, the solution for the AR and MA parameters are in general conditional. Only in the pure MA and AR case of a time series model without explanatory variables direct solutions are found.