Likelihood Based Inference for ARMA Models

Abstract: Autoregressive moving average (ARMA) models are widely used for analyzing time series data. However, standard likelihood-based inference methodology for ARMA models has avoidable limitations. We show that common ARMA likelihood maximization strategies often lead to sub-optimal parameter estimates. While this possibility has been previously identified, no routinely applicable algorithm has been developed to resolve the issue. We introduce a novel random initialization algorithm, designed to take advantage of the structure of the ARMA likelihood function, which overcomes these optimization problems. Additionally, we show that profile confidence intervals provide superior confidence intervals to those based on the Fisher information matrix. The efficacy of the proposed methodology is demonstrated through a data analysis example and a series of simulation studies. This work makes a significant contribution to statistical practice by identifying and resolving under-recognized shortcomings of existing procedures that frequently arise in scientific and industrial applications.