Notes on the convergence of the estimated risk factor matrix in linear regression models

A two-step iterative estimation of a risk model, alternating between a cross-sectional and time-series regression, aims to achieve an in-sample consistent representation of risk factors, such that the security exposure matrix input of the cross-sectional step is equal to the output exposure matrix estimated in the subsequent time-series step. The sequence of estimated exposure matrices is proven to converge to a fixed point. The condition for a fixed point is identified and proven necessary and sufficient. The presented mathematical proof of viability of the two-step iterative estimation is complementary to earlier research in this area.