Identification issues in forward-looking models estimated by GMM, with an application to the Phillips curve

Limited-information methods are commonly used to estimate forward-looking models with
rational expectations, such as the "New Keynesian Phillips Curve" of Galí and Gertler (1999). In this paper, we address issues of identification that have been overlooked due to the incompleteness of the single-equation formulation. We show that problems of weak instruments may arise, depending on the properties of the 'exogenous' variables, and that they are empirically relevant. We also uncover a link between identification and dynamic mis-specification, and examine the (lack of) power of Hansen's (1982) J test to detect invalid over-identifying restrictions. With regards to the New Phillips curve, we find that problems of identification cannot be ruled out, and they deserve further attention.