Before revision:  Specifically, we derive the sampling distributions and subsequent hypothesis tests for any measure that can be decomposed into terms that involve independent partial correlations, which we show includes Granger causality and mutual information under a multivariate linear-Gaussian model. In doing so, we provide the first exact tests for inferring linear dependence between vector autoregressive processes with limited data. Using numerical simulations and brain-imaging datasets, we demonstrate that our newly developed tests maintain the expected false-positive rate (FPR) with minimally-sufficient samples, while the classical log-likelihood ratio tests can yield an unbounded FPR depending on the parameters chosen.
After revision:  Specifically, we derive the sampling distributions and subsequent hypothesis tests for any measure that can be decomposed into terms that involve independent partial correlations, which we show includes 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 rate (FPR) with minimally-sufficient samples, while the classical log-likelihood ratio tests can yield an unbounded FPR depending on the parameters chosen.