In many fields of science, there is the need of assessing the causal influences among time series. Especially in neuroscience, understanding the causal interactions between brain regions is of primary importance. A family of measures have been developed from the parametric implementation of the Granger criteria of causality based on the linear autoregressive modelling of the signals. We propose a new Bayesian method for linear model identification with a structured prior (GMEP) aiming to apply it as linear regression method in the context of the parametric Granger causal inference. GMEP assumes a Gaussian scale mixture distribution for the group sparsity prior and it enables flexible definition of the coefficient groups. Approximate posterior inference is achieved using Expectation Propagation for both the linear coefficients and the hyperparameters. GMEP is investigated both on simulated data and on empirical fMRI data in which we show how adding information on the sparsity structure of the coefficients positively improves the inference process. In the same simulation framework, GMEP is compared with others standard linear regression methods. Moreover, the causal inferences derived from GMEP estimates and from a standard Granger method are compared across simulated datasets of different dimensionality, density connection and level of noise. GMEP allows a better model identification and consequent causal inference when prior knowledge on the sparsity structure are integrated in the structured prior.