Goldfeld–Quandt test

There is an additional assumption here, that the design matrices for the two subsets of data are both of full rank.

The test statistic used is the ratio of the mean square residual errors for the regressions on the two subsets.

This test statistic corresponds to an F-test of equality of variances, and a one- or two-sided test may be appropriate depending on whether or not the direction of the supposed relation of the error variance to the explanatory variable is known.

[3][4] The second test proposed in the paper is a nonparametric one and hence does not rely on the assumption that the errors have a normal distribution.

For this test, a single regression model is fitted to the complete dataset.

The parametric Goldfeld–Quandt test offers a simple and intuitive diagnostic for heteroskedastic errors in a univariate or multivariate regression model.

[6] Primarily, the Goldfeld–Quandt test requires that data be ordered along a known explanatory variable.

A parametric test for equal variance can be visualized by indexing the data by some variable, removing data points in the center and comparing the mean deviations of the left and right side.
The nonparametric test can be visualized by comparing the number of 'peaks' in the residuals from a regression ordered against a pre-identified variable with how many peaks would arise randomly. The lower figure is provided only for comparison, no part of the test involves visual comparison with a hypothetical homoskedastic error structure.