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.
Electron. J. Statist.
16(1):
116-157
(2022).
DOI: 10.1214/21-EJS1950
Bardet, J. M., Kengne, K. and Wintenberger, O. Multiple breaks detection in general causal time series using penalized quasi-likelihood. Electronic Journal of Statistics 6, (2012), 435-477. MR2988415Bardet, J. M., Kengne, K. and Wintenberger, O. Multiple breaks detection in general causal time series using penalized quasi-likelihood. Electronic Journal of Statistics 6, (2012), 435-477. MR2988415
Bardet, J.-M. and Wintenberger, O. Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes. The Annals of Statistics 37, 5B, (2009), 2730-2759. MR2541445Bardet, J.-M. and Wintenberger, O. Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes. The Annals of Statistics 37, 5B, (2009), 2730-2759. MR2541445
Bardet, J.-M., Kamila, K. and Kengne, W. Consistent model selection criteria and goodness-of-fit test for common time series models. Electron. J. Stat. 14, (2020), 2009-2052. MR4091861Bardet, J.-M., Kamila, K. and Kengne, W. Consistent model selection criteria and goodness-of-fit test for common time series models. Electron. J. Stat. 14, (2020), 2009-2052. MR4091861
Bierens, H. J. Topics in advanced econometrics: estimation, testing, and specification of cross-section and time series models. Cambridge University Press, (1996). MR1291390Bierens, H. J. Topics in advanced econometrics: estimation, testing, and specification of cross-section and time series models. Cambridge University Press, (1996). MR1291390
Deistler, M. The properties of the parameterization of ARMAX systems and their relevance for structural estimation and dynamic specification. Econometrica: Journal of the Econometric Society, (1983), 1187-1207. MR0710226Deistler, M. The properties of the parameterization of ARMAX systems and their relevance for structural estimation and dynamic specification. Econometrica: Journal of the Econometric Society, (1983), 1187-1207. MR0710226
Francq, C. and Sucarrat, G. An equation-by-equation estimator of a multivariate log-GARCH-X model of financial returns. Journal of Multivariate Analysis, 153, (2017), 16-32. MR3578836Francq, C. and Sucarrat, G. An equation-by-equation estimator of a multivariate log-GARCH-X model of financial returns. Journal of Multivariate Analysis, 153, (2017), 16-32. MR3578836
Guo, S., Ling S. and Zhu, K. Factor double autoregressive models with application to simultaneous causality testing. Journal of Statistical Planning and Inference 148, (2014), 82-94. MR3174149Guo, S., Ling S. and Zhu, K. Factor double autoregressive models with application to simultaneous causality testing. Journal of Statistical Planning and Inference 148, (2014), 82-94. MR3174149
Han, H. and Kristensen, D. Asymptotic Theory for the QMLE in GARCHX Models With Stationary and Nonstationary Covariates. Journal of Business & Economic Statistics 32, (2014), 416-429. MR3238595Han, H. and Kristensen, D. Asymptotic Theory for the QMLE in GARCHX Models With Stationary and Nonstationary Covariates. Journal of Business & Economic Statistics 32, (2014), 416-429. MR3238595
Hannan, E.J. The identification and parametrization of ARMAX and state space forms. Econometrica: Journal of the Econometric Society, (1976), 713–723. MR0479509Hannan, E.J. The identification and parametrization of ARMAX and state space forms. Econometrica: Journal of the Econometric Society, (1976), 713–723. MR0479509
Ing, C.-K. and Wei, C.-Z. On same-realization prediction in an infinite-order autoregressive process. Journal of Multivariate Analysis, 85(01), (2003), 130-155. MR1978181Ing, C.-K. and Wei, C.-Z. On same-realization prediction in an infinite-order autoregressive process. Journal of Multivariate Analysis, 85(01), (2003), 130-155. MR1978181
Ing, C.-K. and Wei, C.-Z. Order selection for same-realization predictions in autoregressive processes. The Annals of Statistics, 33(05), (2005), 2423-2474. MR2211091Ing, C.-K. and Wei, C.-Z. Order selection for same-realization predictions in autoregressive processes. The Annals of Statistics, 33(05), (2005), 2423-2474. MR2211091
Nana, G.N., Korn, R. and Elwein-Sayer, C. GARCH-extended models: theoretical properties and applications. arXiv: 1307.6685v1, (2013).Nana, G.N., Korn, R. and Elwein-Sayer, C. GARCH-extended models: theoretical properties and applications. arXiv: 1307.6685v1, (2013).
Ng, K. Y. and Awang, N. Multiple linear regression and regression with time series error models in forecasting PM 10 concentrations in Peninsular Malaysia. Environmental monitoring and assessment 190(2), (2018), 1-11.Ng, K. Y. and Awang, N. Multiple linear regression and regression with time series error models in forecasting PM 10 concentrations in Peninsular Malaysia. Environmental monitoring and assessment 190(2), (2018), 1-11.
Souza, J. B., Reisen, V. A., Franco, G. C., Ispány, M., Bondon, P., and Santos, J. M. Generalized additive models with principal component analysis: an application to time series of respiratory disease and air pollution data. Journal of the Royal Statistical Society: Series C (Applied Statistics) 67, 2 (2018), 453–480. MR3758775Souza, J. B., Reisen, V. A., Franco, G. C., Ispány, M., Bondon, P., and Santos, J. M. Generalized additive models with principal component analysis: an application to time series of respiratory disease and air pollution data. Journal of the Royal Statistical Society: Series C (Applied Statistics) 67, 2 (2018), 453–480. MR3758775
Sucarrat, Genaro and GrØnneberg, S. and Escribano, A. Estimation and inference in univariate and multivariate log-GARCH-X models when the conditional density is unknown. Computational Statistics & Data Analysis 100 (2016), 582-594. MR3505819Sucarrat, Genaro and GrØnneberg, S. and Escribano, A. Estimation and inference in univariate and multivariate log-GARCH-X models when the conditional density is unknown. Computational Statistics & Data Analysis 100 (2016), 582-594. MR3505819
Zambom, A. Z. and Gel, Y. R. Testing for local covariate trend effects in volatility models. Electronic Journal of Statistics 14(2) (2020), 2529-2550. MR4121798Zambom, A. Z. and Gel, Y. R. Testing for local covariate trend effects in volatility models. Electronic Journal of Statistics 14(2) (2020), 2529-2550. MR4121798