Asymmetric linear double autoregression

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