Analysis of Financial Time Series, Third Edition

9.1 A Factor Model

Suppose that there are k assets and T time periods. Let rit be the return of asset i in the time period t. A general form for the factor model is

9.1

where αi is a constant representing the intercept, {fjt|j = 1, … , m} are m common factors, βij is the factor loading for asset i on the jth factor, and ϵit is the specific factor of asset i.

For asset returns, the factor is assumed to be an m-dimensional stationary process such that

and the asset specific factor ϵit is a white noise series and uncorrelated with the common factors fjt and other specific factors. Specifically, we assume that

Thus, the common factors are uncorrelated with the specific factors, and the specific factors are uncorrelated among each other. The common factors, however, need not be uncorrelated with each other in some factor models.

In some applications, the number of assets k may be larger than the number of time periods T. We discuss an approach to analyze such data in Section 9.6. It is also common to assume that the factors, hence , are serially uncorrelated in factor analysis. In applications, ...