Chapter 9

Principal Component Analysis and Factor Models

Most financial portfolios consist of multiple assets, and their returns depend concurrently and dynamically on many economic and financial variables. Therefore, it is important to use proper multivariate statistical analyses to study the behavior and properties of portfolio returns. However, as demonstrated in the previous chapter, analysis of multiple asset returns often requires high-dimensional statistical models that are complicated and hard to apply. To simplify the task of modeling multiple returns, we discuss in this chapter some dimension reduction methods to search for the underlying structure of the assets. Principal component analysis (PCA) is perhaps the most commonly used statistical method in dimension reduction, and we start our discussion with the method. In practice, observed return series often exhibit similar characteristics leading to the belief that they might be driven by some common sources, often referred to as common factors. To study the common pattern in asset returns and to simplify portfolio analysis, various factor models have been proposed in the literature to analyze multiple asset returns. The second goal of this chapter is to introduce some useful factor models and demonstrate their applications in finance.

Three types of factor models are available for studying asset returns; see Connor (1995) and Campbell, Lo, and MacKinlay (1997). The first type is the macroeconomic factor models that use macroeconomic variables such as growth rate of GDP, interest rates, inflation rate, and unemployment rate to describe the common behavior of asset returns. Here the factors are observable and the model can be estimated via linear regression methods. The second type is the fundamental factor models that use firm or asset specific attributes such as firm size, book and market values, and industrial classification to construct common factors. The third type is the statistical factor models that treat the common factors as unobservable or latent variables to be estimated from the returns series. In this chapter, we discuss all three types of factor models and their applications in finance. Principal component analysis and factor models for asset returns are also discussed in Alexander (2001) and Zivot and Wang (2003).

The chapter is organized as follows. Section 9.1 introduces a general factor model for asset returns, and Section 9.2 discusses macroeconomic factor models with some simple examples. The fundamental factor model and its applications are given in Section 9.3. Section 9.4 introduces principal component analysis that serves as the basic method for statistical factor analysis. The PCA can also be used to reduce the dimension in multivariate analysis. Section 9.5 discusses the orthogonal factor models, including factor rotation and its estimation, and provides several examples. Finally, Section 9.6 introduces asymptotic principal component analysis.