  Identifying the number of factors in a high-dimensional factor model has
attracted much attention in recent years and a general solution to the problem
is still lacking. A promising ratio estimator based on the singular values of
the lagged autocovariance matrix has been recently proposed in the literature
and is shown to have a good performance under some specific assumption on the
strength of the factors. Inspired by this ratio estimator and as a first main
contribution, this paper proposes a complete theory of such sample singular
values for both the factor part and the noise part under the large-dimensional
scheme where the dimension and the sample size proportionally grow to infinity.
In particular, we provide the exact description of the phase transition
phenomenon that determines whether a factor is strong enough to be detected
with the observed sample singular values. Based on these findings and as a
second main contribution of the paper, we propose a new estimator of the number
of factors which is strongly consistent for the detection of all significant
factors (which are the only theoretically detectable ones). In particular,
factors are assumed to have the minimum strength above the phase transition
boundary which is of the order of a constant; they are thus not required to
grow to infinity together with the dimension (as assumed in most of the
existing papers on high-dimensional factor models). Empirical Monte-Carlo study
as well as the analysis of stock returns data attest a very good performance of
the proposed estimator. In all the tested cases, the new estimator largely
outperforms the existing estimator using the same ratios of singular values.
