  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 obtained
from the parametric fit and the predictor using the Hilbert--Schmidt
independence criterion (Gretton et al. (2008)). 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 distribution theory for 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 scheme. A simulation study shows that our method has better power
than its main competitors. Two real datasets are analyzed to demonstrate the
scope and usefulness of our method.
