A diagnostic for heteroscedasticity based on the spearman rank correlation

Consider a regression situation in which one wants to understand whether variability of the response is related to a scalar predictor z; the latter could be the predicted value. A diagnostic for heteroscedasticity is the score test (Cook and Weisberg, 1983), which is equivalent to computing the Pearson correlation between the squared residuals from a preliminary fit to the data and the predictor z. As such, the score test is highly nonrobust, both to outlying residuals, which are squared, and leverage points in z. Carroll and Ruppert (1988, pp. 98-99) propose that instead of using the Pearson correlation, one could use the Spearman correlation because it is no more difficult to compute in practice and is intuitively robust. We study this test theoretically, obtaining its limit distribution and influence function.