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 τ-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 parameters 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.
M. L. Diop was supported by the MME-DII center of excellence (ANR-11-LABEX-0023-01). W. Kengne was supported by the ANR BREAKRISK: ANR-17-CE26-0001-01 and the CY Initiative of Excellence (grant “Investissements d’Avenir” ANR-16-IDEX-0008), Project “EcoDep” PSI-AAP2020-0000000013.
The authors are very grateful to the Editor, the Associate Editor and the anonymous Referee for many relevant suggestions and comments which helped to improve the contents of this article.
"Inference and model selection in general causal time series with exogenous covariates." Electron. J. Statist. 16 (1) 116 - 157, 2022. https://doi.org/10.1214/21-EJS1950