Goodness-of-fit tests for high dimensional linear models

We propose a framework for constructing goodness-of-fit tests in both low and high dimensional linear models. We advocate applying regression methods to the scaled residuals following either an ordinary least squares or lasso fit to the data, and using some proxy for prediction error as the final test statistic. We call this family residual prediction tests. We show that simulation can be used to obtain the critical values for such tests in the low dimensional setting and demonstrate using both theoretical results and extensive numerical studies that some form of the parametric bootstrap can do the same when the high dimensional linear model is under consideration. We show that residual prediction tests can be used to test for significance of groups or individual variables as special cases, and here they compare favourably with state of the art methods, but we also argue that they can be designed to test for as diverse model misspecifications as heteroscedasticity and non-linearity.