A condition for the regression predictor to be consistent in a single common factor model

This paper investigates the prediction of a common factor in a single common factor model with infinite items when the structural parameter vector (factor loadings and unique variances) is unknown. A condition in terms of the sample size n and the number of items p is established under which the regression predictor for a unique common factor, in which the parameter vector is replaced by the least squares estimator, converges to it in quadratic mean. The condition is that p → ∞ and p 2 / n → 0 under some mild assumptions.