  This paper 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.
