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We study sparse principal component analysis for high dimensional vectorautoregressive time series under a doubly asymptotic framework, which allowsthe dimension $d$ to scale with the series length $T$. We treat the transitionmatrix of time series as a nuisance parameter and directly apply sparseprincipal component analysis on multivariate time series as if the data areindependent. We provide explicit non-asymptotic rates of convergence forleading eigenvector estimation and extend this result to principal subspaceestimation. Our analysis illustrates that the spectral norm of the transitionmatrix plays an essential role in determining the final rates. We alsocharacterize sufficient conditions under which sparse principal componentanalysis attains the optimal parametric rate. Our theoretical results arebacked up by thorough numerical studies.

Sparse Principal Component Analysis for High Dimensional Vector Autoregressive Models