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The vector autoregressive (VAR) model is a powerful tool in modeling complextime series and has been exploited in many fields. However, fitting highdimensional VAR model poses some unique challenges: On one hand, thedimensionality, caused by modeling a large number of time series and higherorder autoregressive processes, is usually much higher than the time serieslength; On the other hand, the temporal dependence structure in the VAR modelgives rise to extra theoretical challenges. In high dimensions, one popularapproach is to assume the transition matrix is sparse and fit the VAR modelusing the "least squares" method with a lasso-type penalty. In this manuscript,we propose an alternative way in estimating the VAR model. The main idea is,via exploiting the temporal dependence structure, to formulate the estimatingproblem into a linear program. There is instant advantage for the proposedapproach over the lasso-type estimators: The estimation equation can bedecomposed into multiple sub-equations and accordingly can be efficientlysolved in a parallel fashion. In addition, our method brings new theoreticalinsights into the VAR model analysis. So far the theoretical results developedin high dimensions (e.g., Song and Bickel (2011) and Kock and Callot (2012))mainly pose assumptions on the design matrix of the formulated regressionproblems. Such conditions are indirect about the transition matrices and nottransparent. In contrast, our results show that the operator norm of thetransition matrices plays an important role in estimation accuracy. We provideexplicit rates of convergence for both estimation and prediction. In addition,we provide thorough experiments on both synthetic and real-world equity data toshow that there are empirical advantages of our method over the lasso-typeestimators in both parameter estimation and forecasting.

A Direct Estimation of High Dimensional Stationary Vector Autoregressions