Granger causality is a method for identifying directed functional connectivity based on time series analysis of precedence and predictability. The method has been applied widely in neuroscience, however its application to functional MRI data has been particularly controversial, largely because of the suspicion that Granger causal inferences might be easily confounded by inter-regional differences in the hemodynamic response function. Here, we show both theoretically and in a range of simulations, that Granger causal inferences are in fact robust to a wide variety of changes in hemodynamic response properties, including notably their time-to-peak. However, when these changes are accompanied by severe downsampling, and/or excessive measurement noise, as is typical for current fMRI data, incorrect inferences can still be drawn. Our results have important implications for the ongoing debate about lag-based analyses of functional connectivity. Our methods, which include detailed spiking neuronal models coupled to biophysically realistic hemodynamic observation models, provide an important ‘analysis-agnostic’ platform for evaluating functional and effective connectivity methods. ► GC is invariant to confounding times-to-peak in hemodynamic responses applied to fMRI. ► We integrate theoretical analysis, simple simulations, and detailed spiking models. ► Our spiking/balloon model provides a ‘analysis-agnostic’ simulation test-bed. ► GC can't be applied naively to fMRI since downsampling and noise affect inference. ► We establish constraints & principles for functional connectivity analysis of fMRI.