RKHS-Based Inference for Nonlinear Granger Causality via Conditional Centering
Abstract
Granger causality is commonly formulated through linear prediction in vector autoregressive models, limiting its ability to detect nonlinear predictive relationships. We propose a reproducing kernel Hilbert space (RKHS)-based test for nonlinear Granger non-causality in conditional mean for nonlinear autoregressive processes. The key idea is a conditional-centering decomposition of the target regression function into an own-history component and an orthogonal component capturing the additional predictive contribution of the potential source history. Non-causality is characterized by the vanishing of the latter component. The own-history component is estimated by kernel ridge regression, and the residuals are embedded in a second RKHS using a conditionally centered kernel. This yields an RKHS-valued residual moment whose squared norm forms the test statistic. We establish a weighted chi-square null limit and consistency against fixed alternatives for the population-centered statistic. An empirically centered version is shown to retain the null limit and enables spectral calibration of critical values and p-values without resampling. Simulation studies demonstrate accurate size control and power against nonlinear alternatives, and real-data applications illustrate the usefulness of the method for detecting nonlinear predictive relationships in time series.
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BibTeX
@misc{tian2026rkhsbased,
title = {{RKHS-Based Inference for Nonlinear Granger Causality via Conditional Centering}},
author = {Yuhan Tian and Adam Waterbury and Marie-Christine Düker},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22007},
eprint = {2609.22007},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22007},
}
RIS
TY - UNPB
TI - RKHS-Based Inference for Nonlinear Granger Causality via Conditional Centering
AU - Yuhan Tian
AU - Adam Waterbury
AU - Marie-Christine Düker
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22007
UR - https://arxiv.org/abs/2609.22007
AB - Granger causality is commonly formulated through linear prediction in vector autoregressive models, limiting its ability to detect nonlinear predictive relationships. We propose a reproducing kernel Hilbert space (RKHS)-based test for nonlinear Granger non-causality in conditional mean for nonlinear autoregressive processes. The key idea is a conditional-centering decomposition of the target regression function into an own-history component and an orthogonal component capturing the additional predictive contribution of the potential source history. Non-causality is characterized by the vanishing of the latter component. The own-history component is estimated by kernel ridge regression, and the residuals are embedded in a second RKHS using a conditionally centered kernel. This yields an RKHS-valued residual moment whose squared norm forms the test statistic. We establish a weighted chi-square null limit and consistency against fixed alternatives for the population-centered statistic. An empirically centered version is shown to retain the null limit and enables spectral calibration of critical values and p-values without resampling. Simulation studies demonstrate accurate size control and power against nonlinear alternatives, and real-data applications illustrate the usefulness of the method for detecting nonlinear predictive relationships in time series.
ER -