# A bootstrap test to detect prominent Granger-causalities across frequencies

## TL;DR

This paper introduces a bootstrap test for detecting significant Granger-causalities across frequencies in time series, providing a non-parametric method that is especially effective in non-stationary contexts.

## Contribution

It proposes a novel bootstrap-based approach to identify prominent causality cycles in the frequency domain, accounting for non-stationarity and conditioning variables.

## Key findings

- Money stock M1 significantly impacts GDP at all frequencies.
- The opposite causality from GDP to M1 is only significant at high frequencies.
- The method outperforms parametric tests in non-stationary scenarios.

## Abstract

Granger-causality in the frequency domain is an emerging tool to analyze the causal relationship between two time series. We propose a bootstrap test on unconditional and conditional Granger-causality spectra, as well as on their difference, to catch particularly prominent causality cycles in relative terms. In particular, we consider a stochastic process derived applying independently the stationary bootstrap to the original series. Our null hypothesis is that each causality or causality difference is equal to the median across frequencies computed on that process. In this way, we are able to disambiguate causalities which depart significantly from the median one obtained ignoring the causality structure. Our test shows power one as the process tends to non-stationarity, thus being more conservative than parametric alternatives. As an example, we infer about the relationship between money stock and GDP in the Euro Area via our approach, considering inflation, unemployment and interest rates as conditioning variables. We point out that during the period 1999-2017 the money stock aggregate M1 had a significant impact on economic output at all frequencies, while the opposite relationship is significant only at high frequencies.