The martingale Z-test

TL;DR

The paper introduces a martingale Z-test for assessing the association between two autocorrelated time series, enabling causal inference when one series is randomly generated and potentially history-dependent.

Contribution

It presents a novel martingale-based statistical test for independence in autocorrelated time series with a known randomized component, allowing causal inference.

Findings

The test is asymptotically valid under the null hypothesis.

It can detect immediate causal effects between variables.

The method extends existing tests for autocorrelated data.

Abstract

We describe a statistical test for association of two autocorrelated time series, one of which generated randomly at each time point from a known but possibly history-dependent distribution. The null hypothesis is that at each time point, the two variables are independent, conditional on history until that time point. We define a test statistic that is a martingale under the null hypothesis and describe an asymptotic test for it based on the martingale central limit theorem. If we reject this null hypothesis, we may infer an immediate causal effect of the randomized variable on the measured variable.