Principal Component Analysis for High-Dimensional Approximate Factor Models in Time Series: Assumptions, Asymptotic Theory, and Identification

Abstract

We consider estimation of large approximate factor models in high-dimensional panels of stationary time series using Principal Component Analysis (PCA). We review the key results establishing the necessary and sufficient conditions for consistency and asymptotic normality of the estimators which hold when both the cross-sectional dimension $n$ and the sample size $T$ tend to infinity. Special emphasis is placed on identification. First, we show that the common and idiosyncratic components are identified only in the limit $n\to\infty$. Second, we discuss the restrictions required to uniquely determine factors and loadings and examine their consequences for statistical inference.