The Goldfeldquandt Test

To illustrate the Goldfeld-Quandt test, we present in Table 11.3 data on consumption expenditure in relation to income for a cross section of 30 families. Suppose we postulate that consumption expenditure is linearly related to income but that heteroscedasticity is present in the data. We further postulate that the nature of heteroscedasticity is as given in (11.5.10). The necessary reordering of the data for the application of the test is also presented in Table 11.3.

Dropping the middle 4 observations, the OLS regressions based on the first 13 and the last 13 observations and their associated residual sums of squares are as shown next (standard errors in the parentheses).

Regression based on the first 13 observations:

(8.7049) (0.0744) r2 = 0.8887 RSS-i = 377.17 df = 11 Regression based on the last 13 observations: Yi = - 28.0272 + 0.7941X,

(30.6421) (0.1319) r2 = 0.7681 RSS2 = 1536.8 df = 11

(Continued)

19Technically, the power of the test depends on how c is chosen. In statistics, the power of a test is measured by the probability of rejecting the null hypothesis when it is false [i.e., by 1 — Prob (type II error)]. Here the null hypothesis is that the variances of the two groups are the same, i.e., homoscedasticity. For further discussion, see M. M. Ali and C. Giaccotto, "A Study of Several New and Existing Tests for Heteroscedasticity in the General Linear Model,'' Journal of Econometrics, vol. 26, 1984, pp. 355-373.

20George G. Judge, R. Carter Hill, William E. Griffiths, Helmut Lütkepohl, and Tsoung-Chao Lee, Introduction to the Theory and Practice of Econometrics, John Wiley & Sons, New York, 1982, p. 422.

410 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

EXAMPLE 11.4 {Continued) From these results we obtain

_ RSS2/df _ 1536.8/11 l = RSS1/df = 377.17/11 X = 4.07

The critical F value for 11 numerator and 11 denominator df at the 5 percent level is 2.82. Since the estimated F( = I) value exceeds the critical value, we may conclude that there is heteroscedasticity in the error variance. However, if the level of significance is fixed at 1 percent, we may not reject the assumption of homoscedasticity. {Why?) Note that the p value of the observed I is 0.014.

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