Sélection de modèles pour les séries affines causales
Kare Kamila · HAL (Le Centre pour la Communication Scientifique Directe) · 2021
Time series analysis is a very active research topic in Statistics and Data Science. Theabundance of this type of data has created a huge need for efficient and accurate methodologies.Thus, several families of models have emerged. Given this multitude of models,how do you choose one to model a time series? The purpose of this thesis is to proposeand study model selection criteria for a large family of models containing autoregressivetime series such as ARMA and conditionally heteroscedastic time series such as GARCH.We start with a brief presentation of time series, in particular of the family of causalaffine models, while recalling some previous results useful for this thesis. We then describesome classical model selection criteria obtained for time series and end with a brief summaryof our main contributions.The rest of our work presents our four contributions. The first chapter gives sufficientconditions on the penalty depending on the regularity of the dependence of the process onits past in order to obtain a consistent criterion. We also propose a goodness-of-fit test ofthe selected model based on the autocorrelation of the square of the model residuals. Thenumerical simulations have shown satisfactory results.In Chapter 3, we propose a generalization of the Hannan and Quinn criterion to theclass of causal affine series. This generalization induces a certain constant known for classicalmodels (ARMA, GARCH or APARCH type) and can be data-driven estimated forcomplex models like ARMA-GARCH. Here again, some simulation studies have attestedto the quality of the criteria obtained.In the third contribution, we construct asymptotically efficient criteria. We propose ageneralization of Akaike’s AIC criterion based on the so-called ideal penalty. The asymptoticbehavior of this ideal penalty suggested a penalty term which is exactly $2\,D_m$ asin the AIC for simple models, and for complex models, we give a less explicit formula.Following Schwartz, we also derive the BIC criterion based on the maximization of the aposteriori probability of choosing the true model.In Chapter 5, we restricted ourselves to the non-asymptotic study of a particular processof the class of causal affines. A penalized least-squares estimator is built on a datadriven selected model among a collection of linear models. We showed that the final estimatorperforms almost as well as the best over the considered collection, i.e. it achieves,up to a constant, the bias-variance tradeoff. The penalty obtained generalizes Mallows’penalty and depends on a constant that is estimated with data-driven calibration algorithms.Finally, we give some research directions in the General Conclusion of the work.