Generalized C \(\left ( \alpha \right )\) Tests with Nonstandard Convergence Rates

We study hypothesis testing of linear and nonlinear restrictions on a finite-dimensional parameter vector, using estimating functions (or moment equations), when nuisance parameter estimators and the estimating functions converge at nonstandard rates. We focus on generalized \(C(\alpha )\) tests [Dufour et al., Generalized \(C(\alpha )\) tests for estimating functions with serial dependence. In Advances in Time Series Methods and Applications (pp. 151–178). Springer, 2016], which allow one to use a wide class of root-n consistent restricted estimators, under weak assumptions on the asymptotic distribution of the estimators. However, root-n consistency remains notably restrictive, because it precludes estimators which converge at a slow rate, e.g., many estimators based on nonparametric regressions. We establish conditions under which generalized \(C(\alpha )\) -type statistics follow the usual chi-square distribution (under the null hypothesis) when the statistic is based on a restricted estimator which converges at a rate slower than the usual \(n^{1/2}\) rate. We also allow for nonstandard convergence rates on the estimating functions and their derivatives. The conditions given depend on the relation between the different convergence rates. As a special case, when the estimating function converges to its limit at rate \(n^{1/2}\) , we show that the convergence rate of the restricted estimator need only be faster than \(n^{1/4}\) . We apply the proposed procedure to a testing problem on derivatives of the conditional expectation, involving multiple nonstandard rates.