A Note on Algebraic Equivalence of White’s Test and a Variation of the Godfrey/breusch-pagan Test for Heteroscedasticity

Recently, Godfrey (1978), and Breusch and Pagan (1979) have independently proposed testing for heteroscedasticity based on squared least squares residuals. The squared residuals divided by the mean-squared residual are regressed on a set of regressors chosen by the investigator, and the test statistic (GBP hereafter) is one-half the explained sum of squares from this regression. The test statistic has asymptotically the chi-squared distribution. In another paper, White (1980) develops a test also based on squared least squares residuals. Here, the vector of squared residuals is again the regressand, while the regressors are the squares and crossproducts of the regressors in the initial regression. The test statistic is the sample size times the squared multiple correlation coefficient from this regression. This test statistic also has asymptotically the chi-squared distribution. In each case, the number of degrees of freedom is equal to the number of regressors in the artificial regression minus one (a constant must be included). Both test statistics have unit power asymptotically but are different in finite samples and sometimes give very different results in simulation