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.

翻译：本文研究线性双自回归（DAR）模型的拟极大似然推断，包括参数估计、模型选择与诊断检验，所有渐近性质仅在观测过程分数阶矩条件下建立。我们针对线性DAR模型提出了高斯拟极大似然估计量（G-QMLE）与指数拟极大似然估计量（E-QMLE），并证明了这两个估计量的相合性与渐近正态性。基于G-QMLE与E-QMLE，我们提出了两种用于模型选择的贝叶斯信息准则，并构建了两种混合港口曼特奥检验以评估拟合模型的充分性。此外，我们从渐近效率与数值表现两方面，将所提出的G-QMLE和E-QMLE与现有双加权分位回归估计量进行了比较。模拟研究展示了所提推断工具在有限样本下的表现，而基于比特币收益率序列的实际案例则验证了这些推断工具的有效性。