  We study a large-dimensional Dynamic Factor Model where: (i)~the vector of
factors $\mathbf F_t$ is $I(1)$ and driven by a number of shocks that is
smaller than the dimension of $\mathbf F_t$; and, (ii)~the idiosyncratic
components are either $I(1)$ or $I(0)$. Under~(i), the factors $\mathbf F_t$
are cointegrated and can be modeled as a Vector Error Correction Model (VECM).
Under (i) and (ii), we provide consistent estimators, as both the
cross-sectional size $n$ and the time dimension $T$ go to infinity, for the
factors, the loadings, the shocks, the coefficients of the VECM and therefore
the Impulse-Response Functions (IRF) of the observed variables to the
shocks.~Furthermore: possible deterministic linear trends are fully accounted
for, and the case of an unrestricted VAR in the levels $\mathbf F_t$, instead
of a VECM, is also studied. The finite-sample properties the proposed
estimators are explored by means of a MonteCarlo exercise. Finally, we revisit
two distinct and widely studied empirical applications. By correctly modeling
the long-run dynamics of the factors, our results partly overturn those
obtained by recent literature. Specifically, we find that: (i) oil price shocks
have just a temporary effect on US real activity; and, (ii) in response to a
positive news shock, the economy first experiences a significant boom, and then
a milder recession.
