  The Vector AutoRegressive (VAR) model is fundamental to the study of
multivariate time series. Although VAR models are intensively investigated by
many researchers, practitioners often show more interest in analyzing VARX
models that incorporate the impact of unmodeled exogenous variables (X) into
the VAR. However, since the parameter space grows quadratically with the number
of time series, estimation quickly becomes challenging. While several proposals
have been made to sparsely estimate large VAR models, the estimation of large
VARX models is under-explored. Moreover, typically these sparse proposals
involve a lasso-type penalty and do not incorporate lag selection into the
estimation procedure. As a consequence, the resulting models may be difficult
to interpret. In this paper, we propose a lag-based hierarchically sparse
estimator, called "HVARX", for large VARX models. We illustrate the usefulness
of HVARX on a cross-category management marketing application. Our results show
how it provides a highly interpretable model, and improves out-of-sample
forecast accuracy compared to a lasso-type approach.
