Simultaneous confidence bands for Yule-Walker estimators and order selection

TL;DR

This paper develops simultaneous confidence bands for Yule-Walker estimators in autoregressive models, enabling improved order selection and proposing new consistent estimators that outperform traditional criteria in simulations.

Contribution

It introduces the limiting distribution of the maximum deviation of Yule-Walker estimators and proposes new order estimators based on confidence bands, enhancing model selection accuracy.

Findings

Limiting distribution of maximum estimator deviation is Gumbel-type.

New estimators outperform traditional criteria like AIC, BIC in simulations.

BIC, HQC, SIC are shown to be consistent for certain model orders.

Abstract

Let $\{X_k,k\in{\mathbb{Z}}\}$ be an autoregressive process of order $q$. Various estimators for the order $q$ and the parameters ${\bolds \Theta}_q=(\theta_1,...,\theta_q)^T$ are known; the order is usually determined with Akaike's criterion or related modifications, whereas Yule-Walker, Burger or maximum likelihood estimators are used for the parameters ${\bolds\Theta}_q$. In this paper, we establish simultaneous confidence bands for the Yule--Walker estimators $\hat{\theta}_i$; more precisely, it is shown that the limiting distribution of ${\max_{1\leq i\leq d_n}}|\hat{\theta}_i-\theta_i|$ is the Gumbel-type distribution $e^{-e^{-z}}$, where $q\in\{0,...,d_n\}$ and $d_n=\mathcal {O}(n^{\delta})$, $\delta >0$. This allows to modify some of the currently used criteria (AIC, BIC, HQC, SIC), but also yields a new class of consistent estimators for the order $q$. These estimators seem to have some potential, since they outperform most of the previously mentioned criteria in a small simulation study. In particular, if some of the parameters $\{\theta_i\}_{1\leq i\leq d_n}$ are zero or close to zero, a significant improvement can be observed. As a byproduct, it is shown that BIC, HQC and SIC are consistent for $q\in\{0,...,d_n\}$ where $d_n=\mathcal {O}(n^{\delta})$.