Testing single-index restrictions with a focus on average derivatives

This paper considers a situation where the violation of a single-index restriction is a concern only to the extent that it causes bias to the estimates of the average derivatives. We propose a method to construct tests that concentrate their asymptotic powers upon only such interesting alternatives. The test has a χ12 limiting distribution under the null hypothesis, and even accommodates the case where the parameter estimates have a convergence rate slower than n as in the case of maximum score estimation. The testing procedure can be easily modified when the main interest lies in average increment effects of binary covariates, multivariate average derivatives or linear restrictions other than those of average derivatives. Results from Monte Carlo experiments show that the asymptotic theory is a reasonable approximation of the finite-sample distributions and an application of our methods to female labor market participation illustrates the importance of this non-omnibus approach.