EMPIRICAL TIME SERIES ANALYSIS and MAXIMUM LIKELIHOOD ESTIMATION

Maximum likelihood (ML) estimation maximizes the likelihood function and is a celebrated principle in linear regression analysis. Asymptotically, the Cramér-Rao lower bound for the covariance matrix of unbiased estimated parameters is reached by the maximum likelihood estimator. With asymptotic arguments, it has been proved that this principle can also be applied to autoregression and to the more general autoregressive moving average (ARMA) models in time series analysis. It is at least suggested in textbooks that a closer approximation of the exact likelihood in the maximization will produce a better estimate for time series models. In contrast, the finite sample practice often shows differently. Some finite sample facts and their estimation implications are discussed.