Likelihood Geometry of Moving Average and Autoregressive Processes

2026-07-04

AI Summary

This study investigates the solution structure and multiplicity of maximum likelihood estimators in moving average (MA) and autoregressive (AR) time series models, with a focus on classifying critical points that yield non-invertible models. For the first time, algebraic statistics and numerical algebraic geometry are systematically employed to analyze the likelihood equations, leading to closed-form algebraic solutions for parameters in certain cases. The paper also introduces composite likelihood as an alternative estimation strategy. Combining theoretical analysis with simulation experiments, the authors demonstrate that the proposed algebraic methods achieve superior accuracy and numerical stability compared to conventional optimization algorithms, thereby offering a novel theoretical framework and computational toolkit for parameter estimation in time series analysis.

Abstract

We study the problem of maximum likelihood estimation for moving average (MA) time series models from the perspective of algebraic statistics, with a focus on the structure and number of solutions of their likelihood equations. Of particular interest is to classify the critical points that lead to non-invertible models. We consider the composite likelihood as an alternative estimation method and analyze its critical points. We extend our algebraic analysis to autoregressive processes (AR). We provide algebraic closed form formulas for the parameters when possible. We also explore in simulations how methods from numerical algebraic geometry perform against traditional optimization for these models.

Problem

• maximum likelihood estimation
• moving average processes
• autoregressive processes
• likelihood equations
• non-invertible models

Innovation

• algebraic statistics
• maximum likelihood estimation
• moving average processes
• autoregressive processes
• numerical algebraic geometry