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

TL;DR

This paper examines the theoretical underpinnings of PCA when applied to time series data, highlighting potential pitfalls, providing new asymptotic results, and emphasizing careful interpretation of loadings in practical applications.

Contribution

It establishes a central limit theorem for PCA eigenvalues and eigenvectors in time series, and offers methods for accurate inference and interpretation of PCA results under dependence.

Findings

Proportion of variation is robust to dependence assumptions.

Inference of PC loadings requires careful attention.

Empirical example demonstrates correct PCA usage in portfolio management.

Abstract

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