Before revision:  Unfortunately, even testing the significance of simple dependence measures, such as Pearson correlation, is complicated by altered sampling properties when autocorrelation is present in the individual time series . Moreover, it has been recently established that commonly used multivariate dependence measures---such as Granger causality---can produce substantially inaccurate results when applying classical hypothesis-testing procedures to digitally-filtered time series. Here , we suggest thatthe digital filtering-induced bias in Granger causality is an effect of autocorrelation, and we present a principled statistical framework for the hypothesis testing of a large family of linear-dependence measures between multiple autocorrelated time series.
After revision:  Unfortunately, even testing the significance of simple dependence measures, such as Pearson correlation, is complicated by altered sampling properties when autocorrelation is present in the analysed time series -- a property that is ubiquitous across diverse applications, from brain dynamics to climate change. Here we show that, for limited data, this issue cannot be mediated by fitting a time-series model alone (e.g., in Granger causality is an effect of autocorrelation, and we present a principled statistical framework for the hypothesis testing of a large family of linear-dependence measures between multiple autocorrelated time series.