Example 116


From cross-sectional data on 41 countries, Stephen Lewis estimated the following regression model26:

where Y = ratio of trade taxes {import and export taxes) to total government revenue, X2 = ratio of the sum of exports plus imports to GNP, and X3 = GNP per capita; and ln stands for natural log. His hypotheses were that Yand X2 would be positively related {the higher the trade volume, the higher the trade tax revenue) and that Y and X3 would be negatively related {as income increases, government finds it is easier to collect direct taxes—e.g., income tax— than rely on trade taxes).

The empirical results supported the hypotheses. For our purpose, the important point is whether there is heteroscedasticity in the data. Since the data are cross-sectional involving a heterogeneity of countries, a priori one would expect heteroscedasticity in the error variance. By applying White's heteroscedasticity test to the residuals obtained from regression {11.5.24), the following results were obtained27:

Note: The standard errors are not given, as they are not pertinent for our purpose here.

Now n ■ R2 = 41{0.1148) = 4.7068, which has, asymptotically, a chi-square distribution with 5 df {why?). The 5 percent critical chi-square value for 5 df is 11.0705, the 10 percent critical value is 9.2363, and the 25 percent critical value is 6.62568. For all practical purposes, one can conclude, on the basis of the White test, that there is no heteroscedasticity.

A comment is in order regarding the White test. If a model has several regressors, then introducing all the regressors, their squared (or higher-powered) terms, and their cross products can quickly consume degrees of freedom. Therefore, one must use caution in using the test.28

In cases where the White test statistic given in (11.5.25) is statistically significant, heteroscedasticity may not necessarily be the cause, but specification errors, about which more will be said in Chapter 13 (recall point 5 of Section 11.1). In other words, the White test can be a test of (pure) heteroscedasticity or specification error or both. It has been argued that if no cross-product terms are present in the White test procedure, then it is a test of pure heteroscedasticity. If cross-product terms are present, then it is a test of both heteroscedasticity and specification bias.29

26Stephen R. Lewis, "Government Revenue from Foreign Trade,'' Manchester School of Economics and Social Studies, vol. 31, 1963, pp. 39-47.

27These results, with change in notation, are reproduced from William F. Lott and Subhash C. Ray, Applied Econometrics: Problems with Data Sets, Instructor's Manual, Chap. 22, pp. 137-140.

28Sometimes the test can be modified to conserve degrees of freedom. See exercise 11.18.

29See Richard Harris, Using Cointegration Analysis in Econometrics Modelling, Prentice Hall & Harvester Wheatsheaf, U.K., 1995, p. 68.


Other Tests of Heteroscedasticity. There are several other tests of het-eroscedasticity, each based on certain assumptions. The interested reader may want to consult the references.30 We mention but one of these tests because of its simplicity. This is the Koenker-Bassett (KB) test. Like the Park, Breusch-Pagan-Godfrey, and White's tests of heteroscedasticity, the KB test is based on the squared residuals, u2, but instead of being regressed on one or more regressors, the squared residuals are regressed on the squared estimated values of the regressand. Specifically, if the original model is:

Yi = ß1 + ß2 X2i + ß3 X3i + ••• + ßkXki + ui (11.5.26)

you estimate this model, obtain uui from this model, and then estimate ui = «1 + «2(Y )2 + Vi (11.5.27)

where Y are the estimated values from the model (11.5.26). The null hypothesis is that a2 = 0. If this is not rejected, then one could conclude that there is no heteroscedasticity. The null hypothesis can be tested by the usual t test or the F test. (Note that F1k = tk2.) If the model (11.5.26) is double log, then the squared residuals are regressed on (log Y )2. One other advantage of the KB test is that it is applicable even if the error term in the original model (11.5.26) is not normally distributed. If you apply the KB test to Example 11.1, you will find that the slope coefficient in the regression of the squared residuals obtained from (11.5.3) on the estimated Y2 from (11.5.3) is statistically not different from zero, thus reinforcing the Park test. This result should not be surprising since in the present instance we only have a single regressor. But the KB test is applicable if there is one regressor or many.

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