A Test of Normality in Nonlinear Systems of Consumer Demand Equations

Abstract

Largely because of its increasingly heavy use in the consumer demand systems literature, the class of nonlinear systems of simultaneous (or joint) equation systems has become important in applied econometrics.1 The FIML (full information maximum likelihood) estimator is the most commonly used estimator for those models. Yet hypothesis tests based upon FIML estimation of simultaneous equation systems commonly condition upon an untested maintained hypothesis of normality of the error structure. In addition, when the system is nonlinear in its parameters, many of the asymptotic properties of widely used parameter estimators are known only subject to a normality assumption.2 Furthermore, a fully developed theory of quasi-maximum likelihood estimation does not yet exist in the nonlinear case.3 Hence the assumption of normally distributed disturbances inherently plays a central role in statistical inference with nonlinear simultaneous equation systems. Nevertheless, that normality assumption, unlike other assumptions on the error structure, appears never to have been tested in the relevant applied consumer demand systems literature.4 We shall present and apply a test for error structure normality.