Perform a Goldfeld-Quandt test for heteroskedasticity under the assumption the variance of the error increases with population density. Attach your results, and write out the form of the test as well as the value of the statistic and its interpretation.
Answer: This test is frequentely used because it is easy to apply when one of the regressors (or another r.v.)
is considered the proportionality factor of heteroscedasticity.
The test has two limits: its difficulty to reject the null hypothesis of omoscedasticity and the fact that it do not allow to verify other forms of heteroscedasticity. This test is based on the hypothesis that the error variance is related to a regressor X. The test procedure is the following:
1 - the observations on Y and X are sorted following the ascending order of the regressor X which is the proportionality factor;
2 - we divide the sample observations in three subsamples omitting the central one;
3 - we estimate throught OLS the regression models on the rest and third subsample (then on 2.observations each; the number of observations considered has to be sufficiently large);
4 - we calculate the relative RSS, denoted as RSS1 and RSS2;
5 - we derive the Goldfeld-Quandt test: GQ = R =RSS2RSS1;
6 - the test R under the null hypothesis has F distribution with degrees of freedom nc2k 2 both for numerator and denominator.
If the sample value of the test F is greater (in a.v.) than the critical value, at the chosen signicance level, we reject the null hypothesis of omoscedasticity.