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.

中文翻译：

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具有生成回归器的半参数模型的似然推断

哈恩和里德 (2013,计量经济学81, 315–340）制定了半参数三步估计量的影响函数，其中生成的回归量在第一步中计算。这类估计器涵盖了经验分析的几个重要示例，例如 Olley 和 Pakes (1996,计量经济学64, 1263–1297) 和 Heckman、Ichimura 和 Todd (1998,经济研究评论65, 261–294)。本文研究了一种基于非参数似然的推理方法，用于此类三步估计问题中的参数。特别是，我们应用了 Bravo、Escanciano 和 van Keilegom（2018，统计年鉴, 即将推出) 修改半参数矩函数以考虑插件估计对上述重要设置的影响，并表明在第一步和第二步非参数估计中，所得似然比统计量变得渐近关键，而不会出现不平滑。