Nested Nonparametric Instrumental Variable Regression
This work addresses a methodological bottleneck in econometrics and causal inference for researchers dealing with complex causal parameters in short panel data, offering incremental improvements through new techniques.
The paper tackles the problem of unknown convergence rates for nested nonparametric instrumental variable regression, which prevents inference on causal parameters like mediated and long-term treatment effects, and provides explicit mean square rates and efficient inference using techniques to limit compounding ill-posedness.
Several causal parameters in short panel data models are functionals of a nested nonparametric instrumental variable regression (nested NPIV). Recent examples include mediated, time varying, and long term treatment effects identified using proxy variables. In econometrics, examples arise in triangular simultaneous equations and hedonic price systems. However, it appears that explicit mean square convergence rates for nested NPIV are unknown, preventing inference on some of these parameters with generic machine learning. A major challenge is compounding ill posedness due to the nested inverse problems. To limit how ill posedness compounds, we introduce two techniques: relative well posedness, and multiple robustness to ill posedness. With these techniques, we provide explicit mean square rates for nested NPIV and efficient inference for recently identified causal parameters. Our nonasymptotic analysis accommodates neural networks, random forests, and reproducing kernel Hilbert spaces. It extends to causal functions, e.g. heterogeneous long term treatment effects.