Orthogonal Series Estimation for the Ratio of Conditional Expectation Functions

TL;DR

This paper introduces a flexible orthogonal series estimation framework for the ratio of conditional expectation functions, enabling robust inference in causal analysis with machine learning techniques.

Contribution

It develops a novel series estimator combined with debiased machine learning signals for accurate CEFR estimation and inference, including asymptotic properties and bootstrap validity.

Findings

The proposed estimator performs well in finite samples.

Asymptotic results validate the inference procedures.

Application to 401(k) data demonstrates practical utility.

Abstract

In various fields of data science, researchers are often interested in estimating the ratio of conditional expectation functions (CEFR). Specifically in causal inference problems, it is sometimes natural to consider ratio-based treatment effects, such as odds ratios and hazard ratios, and even difference-based treatment effects are identified as CEFR in some empirically relevant settings. This chapter develops the general framework for estimation and inference on CEFR, which allows the use of flexible machine learning for infinite-dimensional nuisance parameters. In the first stage of the framework, the orthogonal signals are constructed using debiased machine learning techniques to mitigate the negative impacts of the regularization bias in the nuisance estimates on the target estimates. The signals are then combined with a novel series estimator tailored for CEFR. We derive the pointwise and uniform asymptotic results for estimation and inference on CEFR, including the validity of the Gaussian bootstrap, and provide low-level sufficient conditions to apply the proposed framework to some specific examples. We demonstrate the finite-sample performance of the series estimator constructed under the proposed framework by numerical simulations. Finally, we apply the proposed method to estimate the causal effect of the 401(k) program on household assets.