If we have a parametric model for the invariant distribution of a Markov chain but cannot or do not want to use any information about the transition distribution (except, perhaps, that the chain is reversible) --- what, then, is the best use we can make of the observations? It is not optimal to proceed as if the observations were i.i.d. We determine a lower bound for the asymptotic variance of estimators and construct efficient estimators. The results apply in particular to discretely observed diffusions. 1 Introduction Suppose we have a parametric diffusion model and observe one path at a large number of discrete time points. Then the observations form a Markov chain. The transition distribution is often difficult to calculate, but the invariant distribution is usually tractable. Recent references on estimation in such models are Bibby and Sørensen (1995), Pedersen (1995), Kessler and Sørensen (1995), Ait-Sahalia (1996a, 1996b, 1997) and Elerian, Chib and Shephard (1998). If we canno..