Title: Sparse Identification and Estimation of High-Dimensional Vector AutoRegressive Moving Averages,
Abstract: The Vector AutoRegressive Moving Average (VARMA) model is fundamental to the
study of multivariate time series. However, estimation becomes challenging in
even relatively low-dimensional VARMA models. With growing interest in the
simultaneous modeling of large numbers of marginal time series, many authors
have abandoned the VARMA model in favor of the Vector AutoRegressive (VAR)
model, which is seen as a simpler alternative, both in theory and practice, in
this high-dimensional context. However, even very simple VARMA models can be
very complicated to represent using only VAR modeling. In this paper, we
develop a new approach to VARMA identification and propose a two-phase method
for estimation. Our identification and estimation strategies are linked in
their use of sparsity-inducing convex regularizers, which favor VARMA models
that have only a small number of nonzero parameters. We establish sufficient
conditions for consistency of sparse infinite-order VAR estimates in high
dimensions, a key ingredient for our two-phase sparse VARMA estimation
strategy. The proposed framework has good estimation and forecast accuracy
under numerous simulation settings. We illustrate the forecast performance of
the sparse VARMA models for several application domains, including
macro-economic forecasting, demand forecasting, and volatility forecasting. The
proposed sparse VARMA estimator gives parsimonious forecast models that lead to
important gains in relative forecast accuracy.