Likelihood Geometry of Moving Average and Autoregressive Processes

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.