Score Tests for the Single Index Model

The single index model is a generalization of the linear regression model with E yjx g x where g is an unknown function The model provides a exible alternative to the linear regression model while providing more structure than a fully nonparametric approach Although the tting 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 paper score tests are derived for three potential misspeci cations 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 rst 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 e ective but is more di cult The score tests are applied to three real data sets demonstrating that the tests can identify important model violations that a ect inference and that approaches that do not take model misspeci cations into account can lead to very di erent results