Factor models of security returns decompose the random return on each of a cross section of assets into factor-related and asset-specific returns. Letting r denote the vector of random returns on n assets, and assuming k factors, a factor decomposition has the form
where B is an n × k-matrix of factor betas, f is a random k-vector of factor returns, and ε is an n-vector of asset-specific returns. The n-vector of coefficients a is set so that E[ε] = 0. By defining B as the least squares projection , it follows that cov(f, ε) = 0k×n.
The factor decomposition (1) puts no empirical restrictions on returns beyond requiring that the means and variances of r and f exist. So in this sense it is empty of empirical content. To add empirical structure, it is commonly assumed that the asset-specific returns ε are cross-sectionally uncorrelated, E[εε′] = D where D is a diagonal matrix. This implies that the covariance matrix of returns can be written as the sum of a matrix of rank k and a diagonal matrix:
This is called a strict factor model. Without loss of generality, one can assume ...