TITLE:
Constructing Confidence Regions for Autoregressive-Model Parameters
AUTHORS:
Jan Vrbik
KEYWORDS:
Markov, Yule and Autoregressive Models, Maximum Likelihood Function, Asymptotic Variance-Covariance Matrix, Confidence Intervals, Nuisance Parameters
JOURNAL NAME:
Applied Mathematics,
Vol.14 No.10,
October
31,
2023
ABSTRACT: We discuss formulas and techniques for finding maximum-likelihood estimators of parameters of autoregressive (with particular emphasis on Markov and Yule) models, computing their asymptotic variance-covariance matrix and displaying the resulting confidence regions; Monte Carlo simulation is then used to establish the accuracy of the corresponding level of confidence. The results indicate that a direct application of the Central Limit Theorem yields errors too large to be acceptable; instead, we recommend using a technique based directly on the natural logarithm of the likelihood function, verifying its substantially higher accuracy. Our study is then extended to the case of estimating only a subset of a model’s parameters, when the remaining ones (called nuisance) are of no interest to us.