Inference and model selection in general causal time series with exogenous covariates

TL;DR

This paper develops inference methods and model selection criteria for a broad class of causal time series models with exogenous covariates, including consistency, asymptotic distributions, and hypothesis testing.

Contribution

It introduces a unified framework for inference and model selection in causal time series with exogenous variables, covering many classical models and establishing new theoretical properties.

Findings

Proved existence of stationary and ergodic solutions under Lipschitz conditions.

Derived the asymptotic distribution of the QMLE for interior and boundary parameters.

Developed a Wald-type test for parameter relevance and covariate significance.

Abstract

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.