Bootstrap Con\u85dence Sets under Semiparametric Conditional Moment Restrictions with Single-Index Components

This paper investigates models of semiparametric conditional moment restric-tions where the restrictions contain a nonparametric function of a single-index as a nuisance parameter. It is assumed that this nonparametric function and the single-index are identi\u85ed and estimated as a \u85rst step prior to the estimation of the parameter of interest under conditional moment restrictions. This paper nds that the estimated parameter of interest is robust to the quality of the esti-mated single-index component. More speci\u85cally, based on symmetrized nearest neighborhood estimation of this nonparametric function, this paper shows that the inuence of the estimated single-index is asymptotically negligible even when the estimated single-index follows cube-root asymptotics. Using this \u85nding, this paper proposes a method to construct bootstrap con\u85dence sets that have three characteristics. First, the con\u85dence sets are asymptotically valid in the presence of n1=3-converging single-index components. Second, the con\u85dence sets accom-modate conditional heteroskedasticity. Third, the method is computationally easy as it does not require re-estimation of the single-index for each bootstrap sample. Some results from Monte Carlo simulations are presented and discussed