  In this work 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 (RP) 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 RP 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 nonlinearity.
