A state-space formulation is introduced for estimating multivariate autoregressive (MVAR) models of cortical connectivity from noisy, scalp-recorded EEG. A state equation represents the MVAR model of cortical dynamics, while an observation equation describes the physics relating the cortical signals to the measured EEG and the presence of spatially correlated noise. We assume that the cortical signals originate from known regions of cortex, but the spatial distribution of activity within each region is unknown. An expectation-maximization algorithm is developed to directly estimate the MVAR model parameters, the spatial activity distribution components, and the spatial covariance matrix of the noise from the measured EEG. Simulation and analysis demonstrate that this integrated approach is less sensitive to noise than two-stage approaches in which the cortical signals are first estimated from EEG measurements, and next, an MVAR model is fit to the estimated cortical signals. The method is further demonstrated by estimating conditional Granger causality using EEG data collected while subjects passively watch a movie.