On 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 the predictor variables, and the goodness-of-fit of the parametric model. Our approach is based on testing for independence between the residual and the predictor using the recently developed Hilbert-Schmidt independence criterion, see [GFT+08]. The proposed method requires no user-defined regularization, is simple to compute, based merely on pairwise distances between points in the sample, and is consistent against all alternatives. We develop the distribution theory of the proposed test-statistic, both under the null and the alternative hypotheses, and devise a bootstrap scheme to approximate its null distribution. We prove the consistency of the bootstrap procedure. The superior finite sample performance of our procedure is illustrated through a simulation study.