Identification and Estimation of a Factor Model

Abstract

In this paper, we study identification and estimation of a factor model in classic factor analysis. It is well-known that the factor loading matrix of a classic factor model can only be determined up to a multiplication of an orthogonal matrix on the right if one only considers the first and second moments. To our best knowledge, there are no researches on the analysis of higher order moments in the literature. We take the first step to leverage the information of the fourth-order moments of observable variables and show that, under certain conditions, the factor loading matrix can be identified if the fourth moment of the normalized factor is different from that of a standard normal distribution. The exclusion of normal distributed factor model is necessary since such factor model is deemed to be indeterminate. Although our proof depends on some extra technical assumption, we believe, confirmed by some numerical results, that the factor loading matrix can be identified in general as long as above assumption regarding fourth moment of the normalized factor holds. We also provide an effective algorithm to fit a factor model with a large number of observable variables.