Fan Yang Wallentin: Asymptotic Efficiency of the Pseudo-Maximum Likelihood Estimator in Multi-group Factor Models with Pooled Data

In practice, the presence of different strata is typically unknown; pooling observations from several normal populations is an example. Distribution of pooled data becomes a mixture of normal distributions. In this study, the effect of pooling data is investigated through a two-group factor model. Two independent normal populations are pooled together. A single-group factor model is fitted to the pooled data set using pseduo-maximum likelihood (PML) where the data are treated as normally distributed and the normal theory ML is applied. The asymptotic standard errors of factor loadings for the single-group factor model are computed and compared with the asymptotic standard errors from the multi-group ML approach. Theoretically, the multi-group ML estimators should be asymptotically efficient. However, the results from our numerical study show that the PML is more efficient than the multi-group ML. A mathematical rationale shows that the standard errors from the PML are underestimated. Such underestimation is due to the ignorance of the effects of factor means and covariances in different groups. Therefore, the normal theory ML is not robust for pooled data. Especially, it largely underestimates the variances of factor loadings when error variances are larger and the group size is small.