Perturbative methods for non-parametric instrumental variable
This work addresses the curse of dimensionality in nonparametric instrumental variable estimation, a key problem in econometrics and causal inference, with significant improvements in high-dimensional settings.
The paper introduces a perturbative method for nonparametric instrumental variable estimation that extends kernel ridge regression with higher-order corrections, achieving up to 99% reduction in prediction error in high-dimensional ill-defined cases compared to standard approaches.
We introduce a perturbative approach for nonparametric instrumental variable (NPIV) estimation. By drawing inspiration from perturbation theory in physics, we extend standard kernel ridge methods with systematic higher perturbation order corrections that significantly improve estimation accuracy. Spectrally, the perturbation introduces mixing between different eigenmodes of the expectation integral operator, which becomes especially useful when the integral equation is ill-defined. One source for such ill-definedness can be the curse of dimensionality. Our method performs across various dimensionality regimes, particularly when the dimensionality parameter $β$ which is defined through the number of samples $n$ and dimension $d$ as $n^β= d$, becomes large. Experimental results show that our first-order perturbative corrections can reduce prediction error by up to 99\% in high-dimensional ill-defined cases ($β> 0.7$) compared to standard ridge regression approaches. The performance improvement is maintained across a wide range of dimensions, with the advantage becoming more pronounced as dimensionality increases.