Asymptotic Theory of Principal Component Analysis for Time Series Data with Cautionary Comments

Abstract Principal component analysis (PCA) is a most frequently used statistical tool in almost all branches of data science. However, like many other tools, there sometimes the risk misuse or even abuse. In this paper, we highlight possible pitfalls using theoretical results PCA based on assumption independent when are time series. For latter, state with proof central limit theorem eigenvalues and eigenvectors (loadings), give direct bootstrap estimation their asymptotic covariances, assess efficacy via simulation. Specifically, pay attention to proportion variation, which decides number principal components (PCs), loadings, help interpret meaning PCs. Our findings that while variation quite robust different dependence assumptions, inference PC loadings requires careful attention. We initiate conclude our investigation an empirical example portfolio management, play prominent role. It given as paradigm correct usage for series data.