  We reconcile the two worlds of dense and sparse modeling by exploiting the
positive aspects of both. We employ a factor model and assume {the dynamic of
the factors is non-pervasive while} the idiosyncratic term follows a sparse
vector autoregressive model (VAR) {which allows} for cross-sectional and time
dependence. The estimation is articulated in two steps: first, the factors and
their loadings are estimated via principal component analysis and second, the
sparse VAR is estimated by regularized regression on the estimated
idiosyncratic components. We prove the consistency of the proposed estimation
approach as the time and cross-sectional dimension diverge. In the second step,
the estimation error of the first step needs to be accounted for. Here, we do
not follow the naive approach of simply plugging in the standard rates derived
for the factor estimation. Instead, we derive a more refined expression of the
error. This enables us to derive tighter rates. We discuss the implications of
our model for forecasting, factor augmented regression, bootstrap of factor
models, and time series dependence networks via semi-parametric estimation of
the inverse of the spectral density matrix.
