Estimating Forward-Looking Euler Equations with GMM Estimators: An Optimal Instruments Approach Estimating Forward-Looking Euler Equations with GMM Estimators: An Optimal Instruments Approach

November 1, 2010

Motivation for the Research
The basic framework for macroeconomic analysis has the structure of a simple model consisting of a demand or “IS” equation, an inflation or “AS” equation, and a monetary policy reaction function. Over time, this model has evolved from the static Keynesian model into a micro-founded, rational expectations model — often labeled the “New Keynesian” model — in which expectations play a dominant role in the structural equations. Expectations of current and future interest rates affect current aggregate demand, and expectations of current and future aggregate demand affect current inflation.

Different empirical studies have reached different conclusions concerning the importance of expectations regarding future interest rates and future demand in determining the dynamics of current output and inflation in applying the “New Keynesian” model to real-world analysis.

This paper aims to resolve the differences by providing an explanation for the disparate nature of the empirical results on forward-looking demand and inflation relations.

Research Approach
The authors compare different methods for estimating forward-looking output and inflation Euler equations and show that weak identification can be an issue in conventional Generalized Method of Moments (GMM) estimation. Weak instruments lead to GMM point estimates, hypothesis tests, and confidence intervals that are unreliable.

The authors then propose a GMM procedure that uses projections that impose the dynamic constraints implied by the forward-looking relation instead of instrumenting by means of simple linear projections on the instruments set. They label this procedure an “optimal” instruments approach.

Finally, the authors use Monte Carlo simulations to test the performance of the optimal
instruments approach against conventional GMM estimation.

Key Findings

  • The authors find weak identification in the GMM estimation of these macroeconomic relations (as in previous research), and they demonstrate that in a weak instruments context conventional GMM estimates may be biased.
  • In contrast to conventional GMM estimation, GMM estimation with optimal instruments produces estimates that are properly centered around the true values.
  • Estimates obtained by GMM with optimal instruments are comparable to the estimates obtained via maximum likelihood (ML), and, in contrast to ML estimation, GMM estimation does not require assumptions about the type of distribution for the structural shocks.

The authors argue that the disparate nature of the extant empirical findings is largely dependent on the estimation methodology. Since weak identification can be an issue in conventional GMM estimation of output and inflation forward-looking relations, it is important to employ methods that are more reliable than GMM when instruments are weak. Overall, the findings support the use of optimal instruments techniques when estimating output or inflation Euler relations. Optimal instruments methods also provide a tighter test of the Euler relation because they impose a constrained reduced form that is the rational-expectations solution to the relation at hand. In so doing, optimal instruments methods exploit the most distinguishing feature of dynamic rational-expectations models.

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