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  • Yukitoshi Matsushita (a1) and Taisuke Otsu (a2)


Hahn and Ridder (2013, Econometrica 81, 315–340) formulated influence functions of semiparametric three-step estimators where generated regressors are computed in the first step. This class of estimators covers several important examples for empirical analysis, such as production function estimators by Olley and Pakes (1996, Econometrica 64, 1263–1297) and propensity score matching estimators for treatment effects by Heckman, Ichimura, and Todd (1998, Review of Economic Studies 65, 261–294). The present article studies a nonparametric likelihood-based inference method for the parameters in such three-step estimation problems. In particular, we apply the general empirical likelihood theory of Bravo, Escanciano, and van Keilegom (2018, Annals of Statistics, forthcoming) to modify semiparametric moment functions to account for influences from plug-in estimates into the above important setup, and show that the resulting likelihood ratio statistic becomes asymptotically pivotal without undersmoothing in the first and second step nonparametric estimates.


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*Address correspondence to Taisuke Otsu, Department of Economics, London School of Economics, Houghton Street, London, WC2A 2AE, UK; e-mail:


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The authors would like to thank the Editor, Co-Editor, and anonymous referees for helpful comments. Financial support from the JSPS KAKENHI (26780133, 18K01541) (Matsushita) and the ERC Consolidator Grant (SNP 615882) are gratefully acknowledged (Otsu).



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  • Yukitoshi Matsushita (a1) and Taisuke Otsu (a2)


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