Revisiting Inference for ARMA Models: Improved Fits and Superior Confidence Intervals

TL;DR

This paper introduces a new random initialization algorithm for ARMA models that improves parameter estimation and confidence intervals, addressing limitations of existing likelihood maximization methods.

Contribution

It presents a novel algorithm for ARMA likelihood optimization and demonstrates that profile likelihoods yield better confidence intervals than Fisher-based methods.

Findings

The new algorithm overcomes local optima in ARMA likelihood estimation.

Profile likelihoods provide more accurate confidence intervals.

Method improves ARMA model fitting in practical applications.

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 currently accepted standards for ARMA likelihood maximization frequently lead to sub-optimal parameter estimates. Existing algorithms have theoretical support, but can result in parameter estimates that correspond to a local optimum. While this possibility has been previously identified, it remains unknown to most users, and 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 likelihoods 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.