An extension of a standard test for heteroskedasticity to a systems framework

Abstract

The Glejser (1969) test for heteroskedasticity concerning the disturbance terms of a regression model is widely referenced - see, e.g., Goldfeld and Quandt (1972), Johnston (1972), Theil (1971) and Maddala (1978) among others. As originally proposed, Glejser suggested estimating the model’s disturbance terms via least squares, and then regressing their absolute values on certain known functions of the regressors; the suggested test for heteroskedasticity then relates to the significance of the ‘slope’ coefficients. Obvious modifications of this procedure, such as using squared estimated disturbance terms, have been suggested - see Goldfeld and Quandt (1972), and Kelejian and Oates (1974); still others have suggested modifications of the procedure which involve iterations - see Maddala (1978). Glejser noted that there may be shortcomings in the procedure which arise due to the use of the estimated disturbances in the second-stage regression. Goldfeld and Quandt (1972), among others, also noted shortcomings in the procedure but recognized that conditions may exist under which these shortcomings are of no consequence in the relevant asymptotic distribution; they went on to note, however, that at the time, a demonstration of such conditions was not available. In recent papers Amemiya (1977) and White (1980) gave results which justify the large sample version of the Glejser test based on squared estimated disturbances. However, their results assumed the absence of lagged dependent variables, and were given in the single-equation context. The purpose of this paper is to extend the Amemiya (1977) and White (1980) results to the case of a simultaneous equation framework, which may or may not contain lagged endogenous variables. We consider two cases. The first is the one in which the researcher suspects that heteroskedasticity may exist in only one of the system’s equations. This extension is not trivial due to, among other things, feedbacks involving the endogenous regressors. Nevertheless, it turns out that if the least-squares procedure in the first stage is replaced by virtually any consistent procedure, such as two-stage least- squares - henceforth 2SLS, no additional complexities arise. The importance of this result is that under typical modelling specifications, a computationally simple large sample test for heteroskedasticity, which is associated with one equation of a system, can be carried out in the context of that system. This test should be especially useful in those cases in which the exact specification of the ‘suspected’ heteroskedasticity is not known1 The second case we consider is the one in which the researcher suspects that heteroskedasticity may be associated with more than one equation of the system. As expected, the resulting test is computationally more ‘demanding’. The model is specified in section 2, and the basic results are given in section 3. Suggestions for further work are given in section 4; technical details are relegated to the appendix.