ML vs GMM Estimates of Hybrid Macroeconomic Models (With an Application to the 'New Phillips Curve')

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Abstract
Many macroeconomic models involve hybrid equations, in which some variables are a function of both their lags and their expected future value. The hybrid "New Keynesian" Phillips Curve is a prominent expample. Estimates of such hybrid models have produced conflicting empirical results: Studies which use ML estimation tend to find the forward-looking component to be small, while those using GMM have reported the inflation dynamics to be predominantly forward-looking. This paper provides a rationalization for this empirical conflict. Allowing for two alternative and straightforward mis-specifications (measurement error and omitted dynamics) in a hybrid model, we show that the ML estimator tends to undervalue the weight of the forward-looking component, while the GMM estimator tends to overstate it. This result is shown to hold analytically in a simple DGP. Monte-Carlo experiments indicate that it remains valid in a wide range of more plausible DGPs. Simulations also suggest that the gap obtained between the two estimators in the context of the new Phillips curve can more readily be accounted for by mis-specification, than by the finite-sample biases.