Goldfeld Quandt Test: Comparison of the Variances of Error Terms

The Goldfeld Quandt test is one of two tests proposed in a 1965 paper by Stephen Goldfeld and Richard Quandt. Both parametric and nonparametric tests are described in the paper, but the term “Goldfeld–Quandt test” is usually associated only with the parametric test.
Goldfeld-Quandt test is frequently used as it is easy to apply when one of the regressors (or another r.v.) is considered the proportionality factor of heteroscedasticity. Goldfeld-Quandt test is applicable for large samples. The observations must be at least twice as many as the parameters to be estimated. The test assumes normality and serially independent error terms μ_i.

The Goldfeld–Quandt test compares the variance of error terms across discrete subgroups. So data is divided into h subgroups. Usually, the data set is divided into two parts or groups, and hence the test is sometimes called a two-group test.

Goldfeld-Quandt Test Procedure:

The procedure of conducting the Goldfeld-Quandt Test is;

1. Order the observations according to the magnitude of X (the independent variable which is the proportionality factor).
2. Select arbitrarily a certain number (c) of central observations which we omit from the analysis. (for n=30, 8 central observations are omitted i.e. 1/3 of the observations are removed). The remaining n – c observations are divided into two sub-groups of equal size i.e. (n – c)/2, one sub-group includes small values of X and other sub-group include the large values of X, a data set is arranged according to the magnitude of X.
3. Now Fit the separate regression to each of the sub-group, and obtain the sum of squared residuals form each of them.
   So $\sum c_1^2$ show sum of squares of Residuals from sub-sample of low values of X with $(n – c)/2 – K$ df, where K is the total number of parameters.$\sum c_2^2$ show sum of squares of Residuals from sub-sample of large values of X with $(n – c)/2 – K$ df, where K is the total number of parameters.
4. Compute the Relation $F^* = \frac{RSS_2/df}{RSS_2/df}=\frac{\sum c_2^2/ ((n-c)/2-k)}{\sum c_1^2/((n-c)/2-k) }$

If variances differ, F^* will have a large value. The higher the observed value of the F^*-ratio the stronger the heteroscedasticity of the $u_i$.

References

• Goldfeld, Stephen M.; Quandt, R. E. (June 1965). “Some Tests for Homoscedasticity”. Journal of the American Statistical Association 60 (310): 539–547
• Kennedy, Peter (2008). A Guide to Econometrics (6th ed.). Blackwell. p. 116

Numerical Example of the Goldfeld-Quandt Test.