This article proposes the asymmetric linear double autoregression, which jointly models the conditional mean and conditional heteroscedasticity characterized by asymmetric effects. A sufficient condition is established for the existence of a strictly stationary solution. With a quasi-maximum likelihood estimation (QMLE) procedure introduced, a Bayesian information criterion (BIC) and its modified version are proposed for model selection. To detect asymmetric effects in the volatility, the Wald, Lagrange multiplier and quasi-likelihood ratio test statistics are put forward, and their limiting distributions are established under both null and local alternative hypotheses. Moreover, a mixed portmanteau test is constructed to check the adequacy of the fitted model. All asymptotic properties of inference tools including QMLE, BICs, asymmetric tests and the mixed portmanteau test, are established without any moment condition on the data process, which makes the new model and its inference tools applicable for heavy-tailed data. Simulation studies indicate that the proposed methods perform well in finite samples, and an empirical application to S&P500 index illustrates the usefulness of the new model.

中文翻译：

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非对称线性双自回归

本文提出了非对称线性双自回归，它联合建模了以非对称效应为特征的条件均值和条件异方差。存在一个严格平稳解的充分条件。通过引入准最大似然估计 (QMLE) 程序，提出了贝叶斯信息准则 (BIC) 及其修改版本用于模型选择。为了检测波动中的不对称效应，提出了Wald、Lagrange乘数和准似然比检验统计量，并在零备择假设和局部备择假设下建立了它们的极限分布。此外，构建了一个混合 portmanteau 测试来检查拟合模型的充分性。推理工具的所有渐近特性，包括 QMLE、BIC、非对称检验和混合portmanteau检验是在数据过程中没有任何矩条件的情况下建立的，这使得新模型及其推理工具适用于重尾数据。仿真研究表明，所提出的方法在有限样本中表现良好，对标准普尔 500 指数的实证应用说明了新模型的有用性。