  This article proposes a new approach to modeling high-dimensional time series
by treating a $p$-dimensional time series as a nonsingular linear
transformation of certain common factors and idiosyncratic components. Unlike
the approximate factor models, we assume that the factors capture all the
non-trivial dynamics of the data, but the cross-sectional dependence may be
explained by both the factors and the idiosyncratic components. Under the
proposed model, (a) the factor process is dynamically dependent and the
idiosyncratic component is a white noise process, and (b) the largest
eigenvalues of the covariance matrix of the idiosyncratic components may
diverge to infinity as the dimension $p$ increases. We propose a white noise
testing procedure for high-dimensional time series to determine the number of
white noise components and, hence, the number of common factors, and introduce
a projected Principal Component Analysis (PCA) to eliminate the diverging
effect of the idiosyncratic noises. Asymptotic properties of the proposed
method are established for both fixed $p$ and diverging $p$ as the sample size
$n$ increases to infinity. We use both simulated data and real examples to
assess the performance of the proposed method. We also compare our method with
two commonly used methods in the literature concerning the forecastability of
the extracted factors and find that the proposed approach not only provides
interpretable results, but also performs well in out-of-sample forecasting.
Supplementary materials of the article are available online.
