Transition Matrix Estimation in High Dimensional Vector Autoregressive Models

We propose a new method for estimating transition matrices of high dimensional stationary vector autoregressive models. Our method explicitly takes into account the temporal dependence structure of time series and can be efficiently solved by a linear program. Our theoretical results allow the dimension $d$ to increase in a faster rate than the sample size $T$ and provide explicit nonasymptotic rates of convergence in estimation accuracy under different matrix norms. We sharply characterize the impact of data dependence of the time series on transition matrix estimation and provide sufficient conditions under which the proposed estimator is consistent even when $d$ is nearly exponentially larger than $T$. Experiments on both synthetic and real equity data show the advantage of our method over the existing ones. The results of this paper have potential applications in different areas, including finance, genomics and brain imaging.