Low-Rank and Sparse Modeling of High-dimensional Vector Autoregressions

Network modeling of high-dimensional time series in presence of unobserved latent variables is an important problem in macroeconomics and finance. In macroeconomic policy making and forecasting, it is often impossible to observe and incorporate all the relevant series in the analysis. Failure to include these variables often results in spurious connectivity among the observed time series in structural analyses, which may have serious policy implications. In order to accurately estimate a network of Granger causal interactions after accounting for latent effects, we introduce a novel approach of low-rank and sparse vector autoregression (VAR). We argue that in presence of a few latent pervasive factors, the transition matrix of a misspecified VAR model among the observed series can be approximated as the sum of a low-rank and a sparse component. Exploiting this connection, we consider estimating low-rank plus sparse VAR models using a combination of nuclear norm and lasso penalties. We establish non-asymptotic upper bounds on the estimation error rates of the low-rank and the sparse components and demonstrate the advantage of the proposed methodology over ordinary and sparse VAR estimates via numerical experiments.