Robustness Against Weak or Invalid Instruments: Exploring Nonlinear Treatment Models with Machine Learning

We discuss causal inference for observational studies with possibly invalid instrumental variables. We propose a novel methodology called two-stage curvature identification (\texttt{TSCI}) by exploring the nonlinear treatment model with machine learning. The first-stage machine learning enables improving the instrumental variable's strength and adjusting for different forms of violating the instrumental variable assumptions. The success of \texttt{TSCI} requires the instrumental variable's effect on treatment to differ from its violation form. A novel bias correction step is implemented to remove bias resulting from the potentially high complexity of machine learning. Our proposed \texttt{TSCI} estimator is shown to be asymptotically unbiased and Gaussian even if the machine learning algorithm does not consistently estimate the treatment model. Furthermore, we design a data-dependent method to choose the best among several candidate violation forms. We apply \texttt{TSCI} to study the effect of education on earnings.