🤖 AI Summary
In short panel data, key causal parameters—including mediation effects, time-varying effects, and long-run treatment effects—are functionals of nested nonparametric instrumental variable (NPIV) regression. However, their mean-squared convergence rates have remained unknown, hindering valid statistical inference for general-purpose machine learning estimators. To address this, we introduce two novel concepts: *relative well-posedness* and *robustness to multiple ill-posedness*, which overcome the compounded ill-posedness arising from nested inverse problems. Leveraging these, we derive explicit mean-squared convergence rates for nested NPIV. Building on this theoretical foundation, we propose a unified estimation framework integrating neural networks, random forests, and reproducing kernel Hilbert spaces—enabling non-asymptotic analysis and yielding provably valid, efficient inference for complex heterogeneous causal functions, such as long-run treatment effects. Our approach substantially extends the applicability of high-dimensional, nonlinear causal inference.
📝 Abstract
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.