Independent Stepwise Residuals for Testing Homoscedasticity
Abstract Regression models which specify independent, homoscedastic and normally distributed errors may be analyzed in a stepwise manner to produce calculated residuals having this same property. If the nth residual is calculated as the deviation of the nth observation from its predicted value based on a least squares fit to only the first n observations then the resulting sequence of residuals, appropriately normalized, are not only mutually independent and homoscedastic but are also independent of all of the calculated regression functions. If error variance is a monotonic function of the mean then, under certain regularity conditions, the calculated stepwise residuals are likewise monotonically heteroscedastic. Simple linear regression with equally spaced values of the independent variable constitutes one such regular case, and a Monte Carlo study of the “peak-test” of homoscedasticity in this instance shows that for small samples the stepwise residuals are substantially more sensitive to monotonic het...