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

TL;DR

This paper develops two-stage curvature identification (TSCI) to enable causal inference with potentially invalid IVs by combining a nonlinear treatment model learned via machine learning in the first stage with a bias-corrected second stage that adjusts for violation forms. It introduces a generalized IV strength measure, a data-driven procedure to select violation forms, and a theoretical guarantee of asymptotic normality for the treatment effect estimator under sufficient strength. Through simulations, TSCI shows robustness to various invalidity forms and nonlinearities, outperforming standard TSLS and DML in regimes with invalid IVs and nonlinear treatment models. A real-data application on education and earnings demonstrates practical utility, with TSCI providing selective, data-driven Vi choices and tighter, valid confidence intervals compared to traditional approaches.

Abstract

We discuss causal inference for observational studies with possibly invalid instrumental variables. We propose a novel methodology called two-stage curvature identification (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 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 TSCI to study the effect of education on earnings.

Figures (9)

• Figure 1: Density plot of TSCI and TSCI-Init estimators (with 500 simulations), which are after and before bias correction, respectively. The three panels from left to right correspond to settings with increasing IV strengths; see Section \ref{['sec: thol 40']} in the supplement for details. The black dashed line represents the true value $\beta=0.5$. The green and brown solid lines indicate the means of TSCI and TSCI-Init estimators.
• Figure 2: Comparison of DML and TSCI in terms of RMSE, CI coverage, and length under Settings S1, S2, and S3. The larger value of the constant $a$ in Settings S1 and S2 stands for a higher nonlinearity level in the treatment model, and the larger value of $b$ in Setting S3 for a higher linearity level. In addition, a larger value of $a$ and $b$ indicate larger generalized IV strength.
• Figure 3: Comparison of TSCI, DML and TSLS in terms of RMSE, CI coverage, and length in Settings C1, C2 and C3, where $a$ controls the nonlinearity level in the treatment model. The stacked bar charts show the basis selection of TSCI, where $q = 1$ corresponds to the correct selection.
• Figure 4: The leftmost panel reports the histograms of the TSCI (random forests) estimates with the comparison method, where the estimates differ due to the randomness of different 500 realized sample splittings; the solid red line corresponds to the median of the TSCI estimates, while the solid and dashed black lines correspond to the TSLS and OLS estimates, respectively. The middle panel displays the histogram of the generalized IV strength (after adjusting for $\mathcal{V}_{\widehat{q}_c}$) over the different 500 realized sample splittings; the solid red line denotes the median of all IV strength for TSCI while the solid black line denotes the IV strength of TSLS. The rightmost panel compares different confidence intervals (CIs) produced by OLS, TSLS, DML, and our proposed TSCI with $\mathcal{V}_{\widehat{q}_c}$, and TSCI assuming valid IVs. The CIs of TSCI are adjusted by the multi-splitting method due to finite samples; see Appendix \ref{['sec:finite-sample adjmt']}.
• Figure S1: RMSE, bias and CI coverage of TSCI with different IV strengths. The red dashed line indicates the IV strength as 40. The black dashed line indicates the 95% coverage level.

Theorems & Definitions (16)

• Remark 1
• Remark 2
• Proposition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 2
• Lemma 1
• Theorem 4
• Lemma 2