  This paper deals with the factor modeling for high-dimensional time series
based on a dimension-reduction viewpoint. Under stationary settings, the
inference is simple in the sense that both the number of factors and the factor
loadings are estimated in terms of an eigenanalysis for a nonnegative definite
matrix, and is therefore applicable when the dimension of time series is on the
order of a few thousands. Asymptotic properties of the proposed method are
investigated under two settings: (i) the sample size goes to infinity while the
dimension of time series is fixed; and (ii) both the sample size and the
dimension of time series go to infinity together. In particular, our estimators
for zero-eigenvalues enjoy faster convergence (or slower divergence) rates,
hence making the estimation for the number of factors easier. In particular,
when the sample size and the dimension of time series go to infinity together,
the estimators for the eigenvalues are no longer consistent. However, our
estimator for the number of the factors, which is based on the ratios of the
estimated eigenvalues, still works fine. Furthermore, this estimation shows the
so-called "blessing of dimensionality" property in the sense that the
performance of the estimation may improve when the dimension of time series
increases. A two-step procedure is investigated when the factors are of
different degrees of strength. Numerical illustration with both simulated and
real data is also reported.
