Testing for Granger Causality in the Frequency Domain: A Phase Resampling Method

abstract

This article introduces phase resampling, an existing but rarely used surrogate data method for making statistical inferences of Granger causality in frequency domain time series analysis. Granger causality testing is essential for establishing causal relations among variables in multivariate dynamic processes. However, testing for Granger causality in the frequency domain is challenging due to the nonlinear relation between frequency domain measures (e.g., partial directed coherence, generalized partial directed coherence) and time domain data. Through a simulation study, we demonstrate that phase resampling is a general and robust method for making statistical inferences even with short time series. With Gaussian data, phase resampling yields satisfactory type I and type II error rates in all but one condition we examine: when a small effect size is combined with an insufficient number of data points. Violations of normality lead to slightly higher error rates but are mostly within acceptable ranges. We illustrate the utility of phase resampling with two empirical examples involving multivariate electroencephalography (EEG) and skin conductance data.

KEYWORDS:

• Frequency domain
• generalized partial directed coherence
• Granger causality
• partial directed coherence
• vector autoregressive model

Notes

² For simplicity, here we introduce Granger causality testing with bivariate time series. However, the models and techniques discussed can be easily extended to higher-dimensional data.

³ Takahashi, Baccalá, and Sameshima (2007) developed the asymptotic theoretical distributions for the PDC, but theoretical distributions for other measures (e.g., gPDC, DTF) are not yet available.

⁴ The algorithm for implementing phase resampling is available on request from the corresponding author.

⁵ We also examine type I and type II error rates with alpha at the 1% and 10% levels. Considering both types of errors, phase resampling has the best performance with an alpha of 0.05.