Analysis of Financial Time Series, Third Edition

9.4 Principal Component Analysis

An important topic in multivariate time series analysis is the study of the covariance (or correlation) structure of the series. For example, the covariance structure of a vector return series plays an important role in portfolio selection. In what follows, we discuss some statistical methods useful in studying the covariance structure of a vector time series.

Given a k-dimensional random variable with covariance matrix , a principal component analysis (PCA) is concerned with using a few linear combinations of ri to explain the structure of . If denotes the monthly log returns of k assets, then PCA can be used to study the main source of variations of these k asset returns. Here the keyword is few so that simplification can be achieved in multivariate analysis.

9.4.1 Theory of PCA

Principal component analysis applies to either the covariance matrix or the correlation matrix of . Since the correlation matrix is the covariance matrix of the standardized ...