  In this paper, we study a general class of causal processes with exogenous
covariates, including many classical processes such as the ARMA-GARCH, APARCH,
ARMAX, GARCH-X and APARCH-X processes.
  Under some Lipschitz-type conditions, the existence of a $\tau$-weakly
dependent strictly stationary and ergodic solution is established.
  We provide conditions for the strong consistency and derive the asymptotic
distribution of the quasi-maximum likelihood estimator (QMLE), both when the
true parameter is an interior point of the parameter's space and when it
belongs to the boundary.
  A significance Wald-type test of parameter is developed. This test is quite
extensive and includes the test of nullity of the parameter's components, which
in particular, allows us to assess the relevance of the exogenous covariates.
  Relying on the QMLE of the model, we also propose a penalized criterion to
address the problem of the model selection for this class. The weak and the
strong consistency of the procedure are established.
  Finally, Monte Carlo simulations are conducted to numerically illustrate the
main results.
