We consider the long-run variance, Ω = avar ( n -1/2 ∑ t=1 n X t ) , where X t is multivariate time series given from a homogeneous finite Markov chain. We show that the maximum likelihood estimator of Ω is consistent and has a Gaussian limit distribution. The estimator can be motivated by a filtering approach, which sheds additional light on the components of the asymptotic variance. We derive a consistent estimator of the asymptotic variance that facilitates inference about Ω , and we compare the finite sample properties of three methods for inference, including a bootstrap implementation. Joint with Guillaume Horel.