Smooth tests of goodness of fit for the distributional assumption of regression models

We focus on regression models that consist of (i) a model for the conditional mean of the outcome and (ii) a distributional assumption about the distribution of the outcome, both conditional on the regressors. Generalised linear models form a well-known example. The choice of the outcome distribution is often motivated by prior or background knowledge of the researcher, or it is simply chosen for convenience. We propose smooth goodness of fit tests for testing the distributional assumption in regression models. The tests arise from embedding the regression model in a smooth family of alternatives, and constructing appropriate score tests that correctly account for nuisance parameter estimation. The tests are customised, focussed and comprehensive. We present several examples to illustrate the wide applicability of our method. A small simulation study demonstrates that our tests have power to detect important deviations from the hypothesised model.