The paper describes two regression models—principal components and maximum-likelihood factor analysis—which may be used when the stochastic predictor varibles are highly intereorrelated and/or contain measurement error. The two problems can occur jointly, for example in social-survey data where the true (but unobserved) covariance matrix can be singular. Departure from singularity of the sample dispersion matrix is then due to measurement error. We first consider the more elementary principal components regression model, where it is shown that it can be derived as a special case of (i) canonical correlation, and (ii) restricted least squares. The second part consists of the more general maximum-likelihood factor-analysis regression model, which is derived from the generalized inverse of the product of two singular matrices. Also, it is proved that factor-analysis regression can be considered as an instrumental variables estimator and therefore does not depend on whether factors have been “properly” identified in terms of substantive behaviour. Consequently the additional task of rotating factors to “simple structure” does not arise.