  This paper aims to decompose a large dimensional vector autoregessive (VAR)
model into two components, the first one being generated by a small-scale VAR
and the second one being a white noise sequence. Hence, a reduced number of
common components generates the entire dynamics of the large system through a
VAR structure. This modelling, which we label as the dimension-reducible VAR,
extends the common feature approach to high dimensional systems, and it differs
from the dynamic factor model in which the idiosyncratic component can also
embed a dynamic pattern. We show the conditions under which this decomposition
exists. We provide statistical tools to detect its presence in the data and to
estimate the parameters of the underlying small-scale VAR model. Based on our
methodology, we propose a novel approach to identify the shock that is
responsible for most of the common variability at the business cycle
frequencies. We evaluate the practical value of the proposed methods by
simulations as well as by an empirical application to a large set of US
economic variables.
