Analysis of Financial Time Series, Third Edition

9.5 Statistical Factor Analysis

We now turn to statistical factor analysis. One of the main difficulties in multivariate statistical analysis is the “curse of dimensionality.” For serially correlated data, the number of parameters of a parametric model often increases dramatically when the order of the model or the dimension of the time series is increased. Simplifying methods are often sought to overcome the curse of dimensionality. From an empirical viewpoint, multivariate data often exhibit similar patterns indicating the existence of common structure hidden in the data. Statistical factor analysis is one of those simplifying methods available in the literature. The aim of statistical factor analysis is to identify, from the observed data, a few factors that can account for most of the variations in the covariance or correlation matrix of the data.

Traditional statistical factor analysis assumes that the data have no serial correlations. This assumption is often violated by financial data taken with frequency less than or equal to a week. However, the assumption appears to be reasonable for asset returns with lower frequencies (e.g., monthly returns of stocks or market indexes). If the assumption is violated, then one can use the parametric models discussed in this book to remove the linear dynamic dependence of the data and apply factor analysis to the residual series.

In what follows, we discuss statistical factor analysis based on the orthogonal factor model. Consider the ...