  Inferring linear dependence between time series is central to our
understanding of natural and artificial systems. Unfortunately, the hypothesis
tests that are used to determine statistically significant directed or
multivariate relationships from time-series data often yield spurious
associations (Type I errors) or omit causal relationships (Type II errors).
This is due to the autocorrelation 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 or
prewhitening approaches), and instead that the degrees of freedom in
statistical tests should be altered to account for the effective sample size
induced by cross-correlations in the observations. This insight enabled us to
derive modified hypothesis tests for any multivariate correlation-based
measures of linear dependence between covariance-stationary time series,
including Granger causality and mutual information with Gaussian marginals. We
use both numerical simulations (generated by autoregressive models and digital
filtering) as well as recorded fMRI-neuroimaging data to show that our tests
are unbiased for a variety of stationary time series. Our experiments
demonstrate that the commonly used $F$- and $\chi^2$-tests can induce
significant false-positive rates of up to $100\%$ for both measures, with and
without prewhitening of the signals. These findings suggest that many
dependencies reported in the scientific literature may have been, and may
continue to be, spuriously reported or missed if modified hypothesis tests are
not used when analysing time series.
