A triangular structural panel data model with additive separable individual-specific effects is used to model the causal effect of a covariate on an outcome variable when there are unobservable confounders with some of them time-invariant. In this setup, a linear reduced-form equation might be problematic when the conditional mean of the endogenous covariate and the instrumental variables is nonlinear. The reason is that ignoring the nonlinearity could lead to weak instruments (instruments are weakly correlated with the endogenous covariate). As a solution, we propose a triangular simultaneous equation model for panel data with additive separable individual-specific fixed effects composed of a linear structural equation with a nonlinear reduced form equation. The parameter of interest is the structural parameter of the endogenous variable. The identification of this parameter is obtained under the assumption of available exclusion restrictions and using a control function approach. Estimating the parameter of interest is done using an estimator that we call Super Learner Control Function (SLCF) estimator. The estimation procedure is composed of two main steps and sample splitting. First, we estimate the control function using a super learner . In the following step, we use the estimated control function to control for endogeneity in the structural equation. Sample splitting is done across the individual dimension. The estimator is consistent and asymptotically normal achieving a parametric rate of convergence. We perform a Monte Carlo simulation to test the performance of the estimators proposed. We conclude that the Super Learner Control Function Estimators significantly outperform Within 2SLS estimators. Finally, we show that the SLCF estimator differs from both the plug-in IV estimator and a naive plug-in 2SLS estimator.

翻译：本文采用具有可加可分离个体特定效应的三角结构面板数据模型，用于在存在不可观测混杂变量（其中部分为时不变变量）的情况下，研究协变量对结果变量的因果效应。在此设定下，当内生协变量与工具变量的条件均值存在非线性关系时，线性简化型方程可能存在问题。原因在于忽略非线性可能导致弱工具变量（工具变量与内生协变量的相关性较弱）。作为解决方案，我们提出了一种具有可加可分离个体特定固定效应的面板数据三角联立方程模型，该模型由线性结构方程和非线性简化型方程构成。感兴趣参数是内生变量的结构参数。该参数的识别基于可用排除性约束假设，并采用控制函数方法实现。我们使用一种称为超级学习器控制函数估计量的方法对感兴趣参数进行估计。该估计程序包含两个主要步骤和样本分割。首先，我们使用超级学习器估计控制函数。随后，利用估计的控制函数校正结构方程中的内生性问题。样本分割沿个体维度进行。该估计量具有一致性且渐近正态，能达到参数收敛速率。我们通过蒙特卡洛模拟检验所提估计量的性能。结果表明，超级学习器控制函数估计量显著优于组内2SLS估计量。最后，我们证明SLCF估计量既不同于插件IV估计量，也不同于朴素插件2SLS估计量。