Computing and estimating information matrices of weak ARMA models

Numerous time series admit weak autoregressive-moving average (ARMA) representations, in which the errors are uncorrelated but not necessarily independent nor martingale differences. The statistical inference of this general class of models requires the estimation of generalized Fisher information matrices. Analytic expressions are given for these information matrices, and consistent estimators, at any point of the parameter space, are proposed. The theoretical results are illustrated by means of Monte Carlo experiments and by analyzing the dynamics of daily returns and squared daily returns of financial series.