Limited Information Estimation and Testing of Discretized Multivariate Normal Structural Models

We consider the estimation of multivariate normal structural models that have been discretized according to a set of thresholds. A popular estimation procedure for this restricted multinomial model consists in the following three stage estimator: First, estimate by maximum likelihood the thresholds for each variable separately from the univariate marginals of the contingency table. Then, estimate by maximum likelihood each of the polychoric correlations separately from the bivariate marginals of the contingency table given the estimated thresholds. Finally, if restrictions are imposed on the thresholds and polychoric correlations, estimate the underlying parameters from the estimated thresholds and polychoric correlations by a weighted least squares procedure. An unresolved issue is how to perform goodness of fit tests in this context.We show that the first, second and third stage estimates can be expressed asymptotically as a linear function of the bivariate marginal proportions. Using this result, we propose limited information tests of discretized multivariate normality, as well as of the overall restrictions imposed by the model