Weak instrumental variables due to nonlinearities in panel data: A Super Learner Control Function estimator

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 endogeneous 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.