## The Annals of Statistics

### Statistical inference for autoregressive models under heteroscedasticity of unknown form

Ke Zhu

#### Abstract

This paper provides an entire inference procedure for the autoregressive model under (conditional) heteroscedasticity of unknown form with a finite variance. We first establish the asymptotic normality of the weighted least absolute deviations estimator (LADE) for the model. Second, we develop the random weighting (RW) method to estimate its asymptotic covariance matrix, leading to the implementation of the Wald test. Third, we construct a portmanteau test for model checking, and use the RW method to obtain its critical values. As a special weighted LADE, the feasible adaptive LADE (ALADE) is proposed and proved to have the same efficiency as its infeasible counterpart. The importance of our entire methodology based on the feasible ALADE is illustrated by simulation results and the real data analysis on three U.S. economic data sets.

#### Article information

Source
Ann. Statist., Volume 47, Number 6 (2019), 3185-3215.

Dates
Revised: August 2018
First available in Project Euclid: 31 October 2019

https://projecteuclid.org/euclid.aos/1572487390

Digital Object Identifier
doi:10.1214/18-AOS1775

Mathematical Reviews number (MathSciNet)
MR4025739

#### Citation

Zhu, Ke. Statistical inference for autoregressive models under heteroscedasticity of unknown form. Ann. Statist. 47 (2019), no. 6, 3185--3215. doi:10.1214/18-AOS1775. https://projecteuclid.org/euclid.aos/1572487390

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#### Supplemental materials

• Supplement to “Statistical inference for autoregressive models under heteroscedasticity of unknown form”. The supplement includes additional simulation results, applications, some technical lemmas and the remaining proofs.