Initial-Condition-Robust Inference in Autoregressive Models

TL;DR

This paper develops an initial-condition-robust (ICR) confidence interval for the AR(1) parameter $\rho$ when $\rho$ may be near or equal to one and initial conditions are arbitrary or explosive. The ICR CI is constructed by augmenting the least-squares regression with an extra regressor to eliminate initial-condition effects, yielding coverage that is invariant to the initial state and robust to conditional heteroskedasticity. The authors prove uniform asymptotic size and asymptotic similarity of the ICR CI over a broad parameter space, and introduce an asymptotically median-unbiased interval estimator (MUE) for $\rho$. Monte Carlo simulations show that the ICR CI maintains near-nominal coverage across diverse initial conditions and error structures, with only a modest length increase relative to existing methods in favorable cases. The work provides practical, tuning-parameter-free inference for near-unit-root AR processes and extends robustness to non-stationary initial states and heteroskedastic errors.

Abstract

This paper considers confidence intervals (CIs) for the autoregressive (AR) parameter in an AR model with an AR parameter that may be close or equal to one. Existing CIs rely on the assumption of a stationary or fixed initial condition to obtain correct asymptotic coverage and good finite sample coverage. When this assumption fails, their coverage can be quite poor. In this paper, we introduce a new CI for the AR parameter whose coverage probability is completely robust to the initial condition, both asymptotically and in finite samples. This CI pays only a small price in terms of its length when the initial condition is stationary or fixed. The new CI also is robust to conditional heteroskedasticity of the errors.

Theorems & Definitions (14)

• Theorem 1
• Corollary 1
• Theorem 2
• Remark 1
• Lemma 1
• Remark 2
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5