Abstract: This study 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 a fractional moment of the observed process. We propose an exponential quasi-maximum likelihood estimator (E-QMLE) for the linear DAR model, and establish its consistency and asymptotic normality. Based on the E-QMLE, we propose a Bayesian information criterion for model selection, and construct a mixed portmanteau test to check the adequacy of the fitted models. Inference tools based on the Gaussian quasi-maximum likelihood estimator (G-QMLE) are also discussed, for comparison. Moreover, we compare the proposed E-QMLE with the G-QMLE and the existing doubly weighted quantile regression estimator in terms of their asymptotic efficiency and numerical performance. Simulation studies illustrate the finite-sample performance of the proposed inference tools, and a real example using a Bitcoin return series shows their usefulness.

Key words and phrases: Double autoregressive models, model selection, portmanteau test, quasi-maximum likelihood estimation.