Orthogonal Machine Learning: Power and Limitations
This work addresses the robustness limitations in causal inference and econometrics for researchers using machine learning with high-dimensional nuisance parameters, offering incremental improvements to existing methods.
The paper tackles the problem of improving the robustness of double machine learning by introducing higher-order orthogonal moment equations, which relax the nuisance parameter estimation rate from n^{-1/4} to n^{-1/(2k+2)}. It demonstrates this in partially linear regression, showing that second-order orthogonal moments are constructible if and only if the treatment residual is non-normal, with proof based on Stein's lemma.
Double machine learning provides $\sqrt{n}$-consistent estimates of parameters of interest even when high-dimensional or nonparametric nuisance parameters are estimated at an $n^{-1/4}$ rate. The key is to employ Neyman-orthogonal moment equations which are first-order insensitive to perturbations in the nuisance parameters. We show that the $n^{-1/4}$ requirement can be improved to $n^{-1/(2k+2)}$ by employing a $k$-th order notion of orthogonality that grants robustness to more complex or higher-dimensional nuisance parameters. In the partially linear regression setting popular in causal inference, we show that we can construct second-order orthogonal moments if and only if the treatment residual is not normally distributed. Our proof relies on Stein's lemma and may be of independent interest. We conclude by demonstrating the robustness benefits of an explicit doubly-orthogonal estimation procedure for treatment effect.