Estimating Bidirectional Causal Effects with Large Scale Online Kernel Learning

TL;DR

This paper introduces a scalable online kernel learning framework for accurately estimating bidirectional causal effects in complex systems with mutual dependence, heteroskedasticity, and high-dimensional data.

Contribution

It combines heteroskedasticity-based identification with large scale online kernel learning, enabling flexible nonlinear modeling and efficient computation for bidirectional causal inference.

Findings

Outperforms baseline methods in accuracy and stability

Achieves lower bias and root mean squared error

Demonstrates near-linear computational scaling

Abstract

In this study, a scalable online kernel learning framework is proposed for estimating bidirectional causal effects in systems characterized by mutual dependence and heteroskedasticity. Traditional causal inference often focuses on unidirectional effects, overlooking the common bidirectional relationships in real-world phenomena. Building on heteroskedasticity-based identification, the proposed method integrates a quasi-maximum likelihood estimator for simultaneous equation models with large scale online kernel learning. It employs random Fourier feature approximations to flexibly model nonlinear conditional means and variances, while an adaptive online gradient descent algorithm ensures computational efficiency for streaming and high-dimensional data. Results from extensive simulations demonstrate that the proposed method achieves superior accuracy and stability than single equation and polynomial approximation baselines, exhibiting lower bias and root mean squared error across various data-generating processes. These results confirm that the proposed approach effectively captures complex bidirectional causal effects with near-linear computational scaling. By combining econometric identification with modern machine learning techniques, the proposed framework offers a practical, scalable, and theoretically grounded solution for large scale causal inference in natural/social science, policy making, business, and industrial applications.