POIGNARD Benjamin

This paper deals with the high dimensionality problem in multivariate GARCH processes. The author proposes a new vine-GARCH dynamics for correlation processes parameterized by an undirected graph called "vine". This approach directly generates definite-positive matrices and encourages parsimony. After establishing existence and uniqueness results for stationary solutions of the vine-GARCH model, the author analyzes the asymptotic properties of the model. He then proposes a general framework of penalized M-estimators for dependent processes and focuses on the asymptotic properties of the adaptive Sparse Group Lasso estimator. The high dimension is treated by considering the case where the number of parameters diverges with the sample size. The asymptotic results are illustrated by simulated experiments. Finally in this framework the author proposes to generate the sparsity for dynamics of variance-covariance matrices. To do so, the class of multivariate ARCH models is used and the corresponding processes are estimated by penalized ordinary least squares.

This document contributes to high-dimensional statistics for multivariate GARCH processes. First, the author proposes a new dynamic called vine-GARCH for correlation processes parameterized by an undirected graph called vine. The proposed approach directly specifies positive definite matrices and fosters parsimony. The author provides results for the existence and uniqueness of stationary solution of the vine-GARCH model and studies its asymptotic properties. He then proposes a general framework for penalized M-estimators with dependent processes and focuses on the asymptotic properties of the adaptive Sparse Group Lasso regularizer. The high-dimensionality setting is studied when considering a diverging number of parameters with the sample size. The asymptotic properties are illustrated through simulation experiments. Finally, the author proposes to foster sparsity for multivariate variance covariance matrix processes within the latter framework. To do so, the multivariate ARCH family is considered and the corresponding parameterizations are estimated thanks to penalized ordinary least square procedures.

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