Heteroscedastic one-factor models and marginal maximum likelihood estimation

In the present paper, a general class of heteroscedastic one-factor models is considered. In these models, the residual variances of the observed scores are explicitly modelled as parametric functions of the one-dimensional factor score. A marginal maximum likelihood procedure for parameter estimation is proposed under both the assumption of multivariate normality of the observed scores conditional on the single common factor score and the assumption of normality of the common factor score. A likelihood ratio test is derived, which can be used to test the usual homoscedastic one-factor model against one of the proposed heteroscedastic models. Simulation studies are carried out to investigate the robustness and the power of this likelihood ratio test. Results show that the asymptotic properties of the test statistic hold under both small test length conditions and small sample size conditions. Results also show under what conditions the power to detect different heteroscedasticity parameter values is either small, medium, or large. Finally, for illustrative purposes, the marginal maximum likelihood estimation procedure and the likelihood ratio test are applied to real data.