Alternative Goodness-of-Fit Tests for Linear Models

Abstract

Fan and Huang (2001) presented a goodness-of-fit test for linear models based on Fourier transformations of the residuals of the fitted model. We present two more theoretically appealing tests in which the Fourier transforms are incorporated into a fitted model. We show that when suitably normalized, the new test statistics have the same asymptotic distribution as Fan and Huang’s test. We propose modifications to the asymptotic normalization constants to improve the small sample sizes of our tests while retaining their asymptotic distributions. Small sample sizes and powers are examined via simulations. An illustration is given.