Novel approaches to multivariate GARCH models in high dimension.

Summary 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.