Before revision:  The test relies on the estimation of the separable IV model and the verification of the independence restriction between the residual and the instrumental variable imposed by the non-separable model. To obtain a large-sample approximation to the test statistics , we develop a novel uniform asymptotic expansion of the empirical distribution function of residuals that lead to a Donsker-type central limit theorem .
After revision:  The large-sample approximation to the test statistics , we develop a novel uniform asymptotic expansion of the empirical distribution function of residuals that lead to a Donsker-type central limit theorem .