A Bootstrap Test for Single Index Models

Single index models are frequently used in econometrics and biometrics. Logit and probit models are special cases with fixed link functions. In this paper we consider a specification test that detects nonparametric deviations of the link function, e. g., testing against a semiparametric alternative. Simulations with single index models have shown that the empirical power of this test may be affected in small samples. In this work the bootstrap is used with the aim to find a more accurate distribution under the null than the standard normal. We prove that the statistic and its bootstrapped version have the same asymptotic distribution. A simulation study is performed to investigate the empirical behaviour of this approach. The bootstrapped critical values yield better approximations to the true values.