Asymptotic inference of least absolute deviation estimation for AR(1) processes

Abstract

In this article, we consider a first-order autoregressive process $y_{\mathrm{t}}={\rho }_{\mathrm{n}}{y}_{\mathrm{t}-1}+u_{\mathrm{t}}$ with $n|1-{\rho }_{\mathrm{n}}|\to \infty$ as $n\to \infty$. The Gaussian limit theory and the Cauchy limit theory of the least absolute deviation estimator for the near-stationary process (${\rho }_{\mathrm{n}}\in [0,1)$) and the mildly explosive process (${\rho }_{\mathrm{n}}>1$) are derived, respectively. The results are complementary to the uniform limit theory of least squares estimators for stationary autoregressions in Giraitis and Phillips (2006). Some simulations are carried out to assess the performance of our procedure.

Keywords:

• Uniform limit
• nearly stationary autoregression
• mildly explosive autoregression
• least absolute deviation estimation

MR(2010) Subject Classification:

• 60G10
• 62M10
• 62F12

Acknowledgment

The authors thank the Editor in Chief Prof. N. Balakrishnan, an associate editor and anonymous referees for their helpful comments and valuable suggestions that greatly improved the article.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (11701005, 11871072, 11671012), the Natural Science Foundation of Anhui Province (1608085QA02, 1508085J06) and Introduction Projects of Anhui University Academic and Technology Leaders.