Testing independence and goodness-of-fit in linear models

We consider a linear regression model and propose an omnibus test to simultaneously check the assumption of independence between the error and predictor variables and the goodness-of-fit of the parametric model. Our approach is based on testing for independence between the predictor and the residual obtained from the parametric fit by using the Hilbert-Schmidt independence criterion (Gretton et al., 2008). The proposed method requires no user-defined regularization, is simple to compute based on only pairwise distances between points in the sample, and is consistent against all alternatives. We develop distribution theory for the proposed test statistic, under both the null and the alternative hypotheses, and devise a bootstrap scheme to approximate its null distribution. We prove the consistency of the bootstrap scheme. A simulation study shows that our method has better power than its main competitors. Two real datasets are analysed to demonstrate the scope and usefulness of our method.