On identification of multi-factor models with correlated residuals

Abstract

We specify some conditions for the identification of a multi-factor model with correlated residuals, uncorrelated factors and zero restrictions in the factor loadings. These conditions are derived from the results of Stanghellini (1997) and Vicard (2000) which deal with single-factor models with zero restrictions in the concentration matrix. Like these authors, we make use of the complementary graph of residuals and the conditions build on the role of odd cycles in this graph. However, in contrast to these authors, we consider the case where the conditional dependencies of the residuals are expressed in terms of a covariance matrix rather than its inverse, the concentration matrix. We first derive the corresponding condition for identification of single-factor models with structural zeros in the covariance matrix of the residuals. This is extended to the case where some factor loadings are constrained to be zero. We use these conditions to obtain a sufficient and a necessary condition for identification of multi-factor models. © 2004 Biometrika Trust.