Bias corrections in testing and estimating semiparametric, single index models

Semiparametric methods are widely employed in applied work where the ability to conduct inferences is important. To establish asymptotic normality for making inferences, bias control mechanisms are often used in implementing semiparametric estimators. The first contribution of this paper is to propose a mechanism that enables us to establish asymptotic normality with regular kernels. In so doing, we argue that the resulting estimator performs very well in finite samples. Semiparametric models are commonly estimated under a single index assumption. Because the consistency of the estimator critically depends on this assumption being correct, our second objective is to develop a test for it. To ensure that the test statistic has good size and power properties in finite samples, we employ a bias control mechanism similar to that underlying the estimator. Furthermore, we structure the test so that its form adapts to the model under the alternative hypothesis. Monte Carlo results confirm that the bias control and the adaptive feature significantly improve the performance of the test statistic in finite samples.