# Kernel Conditional Moment Test via Maximum Moment Restriction

arXiv: https://arxiv.org/abs/2002.09225 · HTML: https://mlsanity.com/paper/2002.09225/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Feb 2020.

## Built on by (most cited first, 18 of 18)

- [Proximal Causal Learning with Kernels: Two-Stage Estimation and Moment Restriction](https://mlsanity.com/paper/2105.04544/) (2021)
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- [Demystifying Causal Features on Adversarial Examples and Causal Inoculation for Robust Network by Adversarial Instrumental Variable Regression](https://mlsanity.com/paper/2303.01052/) (2023)
- [Deep Learning Methods for Proximal Inference via Maximum Moment Restriction](https://mlsanity.com/paper/2205.09824/) (2022)
- [Spectral Representation Learning for Conditional Moment Models](https://mlsanity.com/paper/2210.16525/) (2022)
- [Falsification of Internal and External Validity in Observational Studies via Conditional Moment Restrictions](https://mlsanity.com/paper/2301.13133/) (2023)
- [Functional Generalized Empirical Likelihood Estimation for Conditional Moment Restrictions](https://mlsanity.com/paper/2207.04771/) (2022)
- [Estimation Beyond Data Reweighting: Kernel Method of Moments](https://mlsanity.com/paper/2305.10898/) (2023)
- [Doubly Robust Proximal Causal Learning for Continuous Treatments](https://mlsanity.com/paper/2309.12819/) (2023)
- [Benchmarking Observational Studies with Experimental Data under Right-Censoring](https://mlsanity.com/paper/2402.15137/) (2024)
- [The Hardness of Validating Observational Studies with Experimental Data](https://mlsanity.com/paper/2503.14795/) (2025)
- [Instrument Space Selection for Kernel Maximum Moment Restriction](https://mlsanity.com/paper/2106.03340/) (2021)
- [Correcting Confounding via Random Selection of Background Variables](https://mlsanity.com/paper/2202.02150/) (2022)
- [Kernel Conditional Moment Constraints for Confounding Robust Inference](https://mlsanity.com/paper/2302.13348/) (2023)
- [Density Ratio-Free Doubly Robust Proxy Causal Learning](https://mlsanity.com/paper/2505.19807/) (2025)
- [A Convex Framework for Confounding Robust Inference](https://mlsanity.com/paper/2309.12450/) (2023)
- [Learning Treatment Representations for Downstream Instrumental Variable Regression](https://mlsanity.com/paper/2506.02200/) (2025)
- [Same Flow, Different Paths: Variance Reduction in Flow Matching](https://mlsanity.com/paper/2609.17287/) (2026)

## Abstract (arXiv)

We propose a new family of specification tests called kernel conditional moment (KCM) tests. Our tests are built on a novel representation of conditional moment restrictions in a reproducing kernel Hilbert space (RKHS) called conditional moment embedding (CMME). After transforming the conditional moment restrictions into a continuum of unconditional counterparts, the test statistic is defined as the maximum moment restriction (MMR) within the unit ball of the RKHS. We show that the MMR not only fully characterizes the original conditional moment restrictions, leading to consistency in both hypothesis testing and parameter estimation, but also has an analytic expression that is easy to compute as well as closed-form asymptotic distributions. Our empirical studies show that the KCM test has a promising finite-sample performance compared to existing tests.