Quasi-maximum Likelihood Inference for Linear Double Autoregressive Models

TL;DR

This paper develops quasi-maximum likelihood inference methods for linear double autoregressive models, including estimators, model selection criteria, and diagnostic tests, with theoretical guarantees under fractional moments and demonstrated through simulations and real data.

Contribution

It introduces Gaussian and exponential quasi-maximum likelihood estimators for DAR models, establishing their asymptotic properties and proposing model selection and diagnostic tools.

Findings

Proposed G-QMLE and E-QMLE are consistent and asymptotically normal.

New model selection criteria outperform existing methods.

Simulation and real data show the effectiveness of the inference tools.

Abstract

This paper investigates the quasi-maximum likelihood inference including estimation, model selection and diagnostic checking for linear double autoregressive (DAR) models, where all asymptotic properties are established under only fractional moment of the observed process. We propose a Gaussian quasi-maximum likelihood estimator (G-QMLE) and an exponential quasi-maximum likelihood estimator (E-QMLE) for the linear DAR model, and establish the consistency and asymptotic normality for both estimators. Based on the G-QMLE and E-QMLE, two Bayesian information criteria are proposed for model selection, and two mixed portmanteau tests are constructed to check the adequacy of fitted models. Moreover, we compare the proposed G-QMLE and E-QMLE with the existing doubly weighted quantile regression estimator in terms of the asymptotic efficiency and numerical performance. Simulation studies illustrate the finite-sample performance of the proposed inference tools, and a real example on the Bitcoin return series shows the usefulness of the proposed inference tools.