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A key challenge of network neuroscience is to understand the role of interactions between brain regions and how they contribute to the encoding and broadcasting of information within cognitive processes. This demands computational tools to infer directional relations between brain regions from neural time series. For fMRI, the most common approaches are based on Granger causality (GC) analysis and effective connectivity (EC) models. Despite their different conceptual framing, GC and EC models for fMRI are based on similar mathematical assumptions, grounded on continuous- and discrete-time linear stochastic models. Based on a mapping between multivariate Ornstein-Uhlenbeck (MOU) and multivariate autoregressive (MVAR) processes, we analytically obtain an approximately quadratic relation between EC and GC, after rescaling the GC to compensate for unequal noise variances of the source and target. Simulations show that these relations can be observed in finite time series only if a large amount of data is available, implying that they may emerge only at a group level in real fMRI experiments. We verified this prediction by systematically comparing EC and GC in a large-scale fMRI data set from the Human Connectome Project. Overall, our findings can provide methodological and interpretational guidance in the usage of GC and EC for brain network reconstruction, by clearly elucidating what is common between the two methods, but also their respective specificities and biases.