Abstract: Estimation based on data with nonignorable nonresponse is considered when the joint distribution of the study variable y and covariate χ is nonparametric and the nonresponse probability conditional on y and χ has a parametric form. The likelihood based on observed data may not be identifiable even when the joint distribution of y and χ is parametric. We show that this difficulty can be overcome by utilizing a nonresponse instrument, an auxiliary variable related to y but not related to the nonresponse probability conditional on y and x. Under some conditions we can apply the generalized method of moments (GMM) to obtain estimators of the parameters in the nonresponse probability and the nonparametric joint distribution of y and x. Consistency and asymptotic normality of GMM estimators are established. Simulation results and an application to a data set from the Korean Labor and Income Panel Survey are also presented.