Testing the Normality Assumption in Limited Dependent Variable Models

Limited dependent variable (LDV) models arise when the dependent variable is restricted in some way. The examples are numerous and contain situations (i) where the dependent variable is restricted and takes a limiting value with a positive probability and (ii) where it can take only one of a finite set of values (e.g., when modeling qualitative responses). For the analysis of LDV models, several estimation and inferential results are presently available (e.g., see Tobin 1958, Amemiya 1973, Fair 1977, Olsen 1978, and Pratt 1981) and these are being increasingly used in economic modeling. These results are applicable under a set of assumptions which typically include disturbance normality. The consequences of violation of the normality assumption in LDV situations can be quite severe since, unlike the usual regression model, the maximum likelihood estimators (MLE) can be inconsistent under non-normality. Robinson 1982 has shown this by taking the disturbances as uniform variates. Through a simulation study, Arabmazar and Schmidt 1982 demonstrated that the asymptotic bias (inconsistency) can be quite substantial if normality is wrongly assumed (see also Goldberger 1980, and White 1979). In addition to the logistic and student t distributions, which have closely similar shapes with the normal distribution, and Laplace distribution considered in Goldberger 1980, using the equation (1.4) of Robinson 1982, p. 28, it can be shown that under most of the commonly used non-normal distributions in the statistical literature, the MLE will be inconsistent if normality is a misspecification. For merely general theoretical interest, it is an open question whether normality assumption is necessary for consistency. (While there may be some pathological situations, in particular, with all explanatory variables being discrete in which it may be possible to construct some non-normal distributions which will not lead to inconsistency, however the possibility of inconsistency in the event of non-normality is sufficient reason for testing normality and, for the truncated sample cases, the truncated normal distributions.) Because of sample truncation or censoring, the normality tests developed for