Score Tests for the Single Index Model

Abstract

The single index model is a generalization of the linear regression model with E(y|x) = g, where g is an unknown function. The model provides a flexible alternative to the linear regression model while providing more structure than a fully nonparametric approach. Although the fitting of single index models does not require distributional assumptions on the error term, the properties of the estimates depend on such assumptions, as does practical application of the model. In this article score tests are derived for three potential misspecifications of the single index model: heteroscedasticity in the errors, 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 Carlo simulations demonstrate that the first two tests hold their nominal size well and have good power properties in identifying model violations, often outperforming other tests. Testing for the need for additional covariates can be effective, but is more difficult. The score tests are applied to three real datasets, demonstrating that the tests can identify important model violations that affect inference, and that approaches that do not take model misspecifications into account can lead to very different results.