Factor
and sparse models are two widely used methods to
impose a low-dimensional struc- ture in
high-dimension. They are seemingly mutually
exclusive. In this paper, we propose a simple
lifting method that combines the merits of these
two models in a supervised learning methodology
that allows to efficiently explore all the
information in high-dimensional datasets. The
method is based on a flexible model for panel
data, called factor-augmented regression model
with both observable, latent common factors, as
well as idiosyncratic components as
high-dimensional covariate variables. This model
not only includes both factor regression and
sparse regression as specific models but also
significantly weakens the cross-sectional depen-
dence and hence facilitates model selection and
interpretability. The methodology consists of
three steps. At each step, the remaining
cross-section dependence can be inferred by a
novel test for covariance structure in
high-dimensions. We developed asymptotic theory
for the factor-augmented sparse regression model
and demonstrated the validity of the multiplier
bootstrap for testing high-dimensional
covariance structure. This is further extended
to testing high-dimensional partial covariance
structures. The theory and methods are further
supported by an extensive simulation study and
applications to the construction of a partial
covariance network of the financial returns and
a prediction exercise for a large panel of
macroeconomic time series from FRED-MD database.