The single index model is a generalization of the linear regression model with E(y|x) = g, whereg is an unknown function. The model provides a flexible alternative to the linear regression modelwhile providing more structure than a fully nonparametric approach. Although the fitting of single indexmodels does not require distributional assumptions on the error term, the properties of the estimatesdepend on such assumptions, as does practical application of the model. In this article score testsare derived for three potential misspecifications of the single index model: heteroscedasticity in theerrors, autocorrelation in the errors, and the omission of an important variable in the linear index.These tests have a similar structure to corresponding tests for nonlinear regression models. Monte Carlosimulations demonstrate that the first two tests hold their nominal size well and have good powerproperties in identifying model violations, often outperforming other tests. Testing for the need foradditional covariates can be effective, but is more difficult. The score tests are applied to three realdatasets, demonstrating that the tests can identify important model violations that affect inference, andthat approaches that do not take model misspecifications into account can lead to very different results
