  This paper considers the estimation and testing of a class of
high-dimensional non-stationary time series factor models with evolutionary
temporal dynamics. In particular, the entries and the dimension of the factor
loading matrix are allowed to vary with time while the factors and the
idiosyncratic noise components are locally stationary. We propose an adaptive
sieve estimator for the span of the varying loading matrix and the locally
stationary factor processes. A uniformly consistent estimator of the effective
number of factors is investigated via eigenanalysis of a non-negative definite
time-varying matrix. A high-dimensional bootstrap-assisted test for the
hypothesis of static factor loadings is proposed by comparing the kernels of
the covariance matrices of the whole time series with their local counterparts.
We examine our estimator and test via simulation studies and a real data
analysis.
