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 bootstrap specification test that detects nonparametric deviations of the link function. The bootstrap is used with the aim to find a more accurate distribution under the null than the normal approximation. We prove that the statistic and its bootstrapped version have the same asymptotic distribution. In a simulation study we show that the bootstrap is able to capture the negative bias and the skewness of the test statistic. It yields better approximations to the true critical values and consequently it has a more accurate level than the normal approximation.