On Testing for Uniformity of Fit in Regression: An Econometric Case Study

I T is frequently necessary to test regression results for uniformity of fit—i.e. to test the randomness of the distribution of the error terms. Such a test is necessary because an indication of a systematic trend in the pattern of residuals would suggest that the regression equation fitted to the data did not properly reflect the curvature of, the true relationship between the variables regressed. If, for instance, the re­ siduals after regression fell into three groups of positive residuals, followed by a group of negative residuals, followed again by positive residuals, then the implica­ tion would clearly be that the estimated regression function misrepresented the' true relationship. Indeed, econometricians frequently regard such bunching of residuals as prima facie evidence of serial correlation—see, for example, Leser [9, p. 17]. The problem is, perhaps, most familiar in the analysis of serial cor-, relation errors in time-series analysis. The problem also arises in regressions based on cross-section data, provided that the regression estimates of the residuals after regression are first ordered in ascending order of the independent variable. In a two-variable case, the analogy with time-series studies is then complete, because time is there the independent variable and the residuals are automatically ordered in ascending order of the time variable. In cross-section studies, the independent variable (if there is but one) is the logical, and indeed only, possible choice with which to order the regression residuals. Where there is more than one independent variable, the more important of the two appears the logical choice, though the possibility of ordering the residuals by any other independent variable does exist; for example, one might use the principal component of the independent variables. (Again, the analogy with time-series exists in the case of multiple regression studies: time is generally assumed to be the relevant independent variable when the Durbin-Watson J-test is applied [3, 4]). When applying such tests to crosssection data, the term "uniformity of fit" is used instead of "serial correlation of errors. V ' It is the purpose of this study to describe and compare a number of possible tests for uniformity of fit. It wi l l be obvious from the foregoing that the best known test is a modified version of the Durbin-Watson (/-test (op. cit), where the