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Inferring linear dependence between time series is central to ourunderstanding of natural and artificial systems. Unfortunately, the hypothesistests that are used to determine statistically significant directed ormultivariate relationships from time-series data often yield spuriousassociations (Type I errors) or omit causal relationships (Type II errors).This is due to the autocorrelation present in the analysed time series -- aproperty that is ubiquitous across diverse applications, from brain dynamics toclimate change. Here we show that, for limited data, this issue cannot bemediated by fitting a time-series model alone (e.g., in Granger causality orprewhitening approaches), and instead that the degrees of freedom instatistical tests should be altered to account for the effective sample sizeinduced by cross-correlations in the observations. This insight enabled us toderive modified hypothesis tests for any multivariate correlation-basedmeasures of linear dependence between covariance-stationary time series,including Granger causality and mutual information with Gaussian marginals. Weuse both numerical simulations (generated by autoregressive models and digitalfiltering) as well as recorded fMRI-neuroimaging data to show that our testsare unbiased for a variety of stationary time series. Our experimentsdemonstrate that the commonly used $F$- and $\chi^2$-tests can inducesignificant false-positive rates of up to $100\%$ for both measures, with andwithout prewhitening of the signals. These findings suggest that manydependencies reported in the scientific literature may have been, and maycontinue to be, spuriously reported or missed if modified hypothesis tests arenot used when analysing time series.

Assessing the Significance of Directed and Multivariate Measures of Linear Dependence Between Time Series