Consider a linear model y = X + u with u = (u_1,...,u_T ) and ut a serially correlated linear process given by ut =  ∑^∞_{j=-h}  c_j*e_t-j for a sequence of innovations e_t . Given a set of instruments Z, the optimal GMM estimator based on the moment condition E(Zu) = 0 is by far the most commonly used method to estimate such models. It can, however, be inconsistent unless the instruments are exogenous with respect to past innovations e_t-j for j > 0, when c_j ≠ 0 for j > 0. 

We propose a GLS-IV estimator valid in the general case with instruments exogenous or not, as long as they are predetermined. It is shown to be much more efficient than GMM whether the moment condition E(Zu) = 0 is satisfied or not. We discuss issues of consistency by casting the estimators in a GMM framework with di§erent moment conditions and instruments. To analyze the relative merits of the estimators when all are consistent, we cast them as some GLS estimator using different instruments and first-stage regression. It then becomes clear that GLS-IV involves "stronger instruments", while GMM is more likely to be affected by issues of weak instruments. Other motivating elements and extensive simulations are presented to argue that our proposed GLS-IV estimator has better properties. 

As an empirical application, we revisit the extensive study of Mavroeidis et al. (2014) about the empirical relevance of the forward looking New Keynesian Phillips curve. Using the GLS-IV procedure on the same dataset, our estimates are all in the right quadrant, consistent with theoretical expectations.