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Principal component analysis (PCA) is a most frequently used statistical toolin almost all branches of data science. However, like many other statisticaltools, there is sometimes the risk of misuse or even abuse. In this paper, wehighlight possible pitfalls in using the theoretical results of PCA based onthe assumption of independent data when the data are time series. For thelatter, we state with proof a central limit theorem of the eigenvalues andeigenvectors (loadings), give direct and bootstrap estimation of theirasymptotic covariances, and assess their efficacy via simulation. Specifically,we pay attention to the proportion of variation, which decides the number ofprincipal components (PCs), and the loadings, which help interpret the meaningof PCs. Our findings are that while the proportion of variation is quite robustto different dependence assumptions, the inference of PC loadings requirescareful attention. We initiate and conclude our investigation with an empiricalexample on portfolio management, in which the PC loadings play a prominentrole. It is given as a paradigm of correct usage of PCA for time series data.

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