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We propose using a state-space model to estimate cortical connectivity from scalp-based EEG recordings. A state equation describes the dynamics of the cortical signals and an observation equation describes the manner in which the cortical signals contribute to the scalp measurements. The state equation is based on a multivariate autoregressive (MVAR) process model for the cortical signals. The observation equation describes the physics relating the cortical signals to the scalp EEG measurements and spatially correlated observation noise. An expectation-maximization (EM) algorithm is employed to obtain maximum-likelihood estimates of the MVAR model parameters. The strength of influence between cortical regions is then derived from the MVAR model parameters. Simulation results show that this integrated approach performs significantly better than the two-step approach in which the cortical signals are first estimated from the EEG measurements by attempting to solve the EEG inverse problem and second, an MVAR model is fit to the estimated signals. The method is also applied to data from a subject watching a movie, and confirms that feedforward connections between visual and parietal cortex are generally stronger than feedback connections.