  This paper considers a structural-factor approach to modeling
high-dimensional time series and space-time data by decomposing individual
series into trend, seasonal, and irregular components. For ease in analyzing
many time series, we employ a time polynomial for the trend, a linear
combination of trigonometric series for the seasonal component, and a new
factor model for the irregular components. The new factor model can simplify
the modeling process and achieve parsimony in parameterization. We propose a
Bayesian Information Criterion (BIC) to consistently determine the order of the
polynomial trend and the number of trigonometric functions. A test statistic is
used to determine the number of common factors. The convergence rates for the
estimators of the trend and seasonal components and the limiting distribution
of the test statistic are established under the setting that the number of time
series tends to infinity with the sample size, but at a slower rate. We use
simulation to study the performance of the proposed analysis in finite samples
and apply the proposed approach to two real examples. The first example
considers modeling weekly PM$_{2.5}$ data of 15 monitoring stations in the
southern region of Taiwan and the second example consists of monthly
value-weighted returns of 12 industrial portfolios.
