Conditional Independence Testing in Time Series
Abstract
We consider the problem of testing Granger causality in time series, specifically, whether the future outcome Yt+1 and the exposure history X̅t are conditionally independent given the history of Y̅t and a set of confounding variables Z̅t up to time t. This testing procedure distinguishes true causal effects from associations driven by common external processes, supporting reliable decision-makings in applications such as finance, neuroscience, and climate science. While traditional approaches assume a linear vector autoregressive (VAR) model and are vulnerable to misspecification, we instead address a model-free version of the problem. We propose nonlinearly regressing both the outcome and exposure on the joint history of Y and Z, and calculating a test statistic based on the sample covariance of residuals, we call the Generalised Temporal Covariance Measure (GTCM). To account for heteroscedasticity and improve power against local alternatives, we incorporate variance weights and employ a data-adaptive test based on polynomial lag expansions.The type I error control of the test relies on the relatively weak assumption that user-chosen regression procedures estimate conditional means at a sufficiently fast rate that is slow enough to accommodate nonparametric settings. By further assuming stability of the regression procedures and weak dependence in the time series, we can utilise the entire dataset to estimate the conditional means without splitting the time series into subsets.
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BibTeX
@misc{shi2026conditional,
title = {{Conditional Independence Testing in Time Series}},
author = {Jieru Shi and Rajen D. Shah},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20772},
eprint = {2609.20772},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20772},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Conditional Independence Testing in Time Series
AU - Jieru Shi
AU - Rajen D. Shah
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20772
UR - https://arxiv.org/abs/2609.20772
AB - We consider the problem of testing Granger causality in time series, specifically, whether the future outcome Y_t+1 and the exposure history X̅_t are conditionally independent given the history of Y̅_t and a set of confounding variables Z̅_t up to time t. This testing procedure distinguishes true causal effects from associations driven by common external processes, supporting reliable decision-makings in applications such as finance, neuroscience, and climate science. While traditional approaches assume a linear vector autoregressive (VAR) model and are vulnerable to misspecification, we instead address a model-free version of the problem. We propose nonlinearly regressing both the outcome and exposure on the joint history of Y and Z, and calculating a test statistic based on the sample covariance of residuals, we call the Generalised Temporal Covariance Measure (GTCM). To account for heteroscedasticity and improve power against local alternatives, we incorporate variance weights and employ a data-adaptive test based on polynomial lag expansions.The type I error control of the test relies on the relatively weak assumption that user-chosen regression procedures estimate conditional means at a sufficiently fast rate that is slow enough to accommodate nonparametric settings. By further assuming stability of the regression procedures and weak dependence in the time series, we can utilise the entire dataset to estimate the conditional means without splitting the time series into subsets.
ER -