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Estimation based on data with nonignorable nonresponse is considered when the joint distribution of the study variable y and covariate is nonparametric and the nonresponse probability conditional on y and has a parametric form. The likelihood based on observed data may not be identifiable even when the joint distribution of y and 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. 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. 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.