  We propose a new estimator of high-dimensional spectral density matrices,
called UNshrunk ALgebraic Spectral Estimator (UNALSE), under the assumption of
an underlying low rank plus sparse structure, as typically assumed in dynamic
factor models. The UNALSE is computed by minimizing a quadratic loss under a
nuclear norm plus $l_1$ norm constraint to control the latent rank and the
residual sparsity pattern. The loss function requires as input the classical
smoothed periodogram estimator and two threshold parameters, the choice of
which is thoroughly discussed. We prove consistency of UNALSE as both the
dimension $p$ and the sample size $T$ diverge to infinity, as well as algebraic
consistency, i.e., the recovery of latent rank and residual sparsity pattern
with probability one. The finite sample properties of UNALSE are studied by
means of an extended simulation exercise as well as an empirical analysis of US
macroeconomic data.
