Estimation and Inference by the Method of Projection Minimum Distance: An Application to the New Keynesian Hybrid Phillips Curve

This 2011 International Economic Review paper by Óscar Jordà and Sharon Kozicki proposes “projection minimum distance” (PMD), a two-step semiparametric estimator for macroeconomic models whose likelihood score function is nonlinear in the structural parameters but whose mapping from reduced-form Wold impulse-response coefficients to those parameters is linear – a class that includes forward-looking Euler equations and ARMA-type representations. In the first step, PMD estimates the Wold coefficients b_h semi-parametrically using Jordà’s (2005, 2009) local-projection regressions of y_{t+h} on lagged variables, run separately for each horizon; in the second step it exploits the model-implied linear restrictions on those coefficients via minimum distance (following Ferguson 1958) to estimate the structural parameters, with an information criterion following Hall, Inoue, Nason, and Rossi (2007) selecting the truncation horizon H by trading off identification gain from extra horizons against a many-weak-instruments cost. The paper establishes formally (Propositions 1-2 for the first-stage local-projection estimator; Lemmas 3-4 for the second-stage minimum-distance estimator) that both stages are consistent and asymptotically normal under stated rate conditions, and shows that in the simple case of i.i.d. shocks PMD coincides exactly with a GMM estimator that uses the lagged dependent variable as instrument – but the two estimators diverge once shocks are serially correlated, because the lagged dependent variable is then no longer a valid GMM instrument while the different (longer-lagged) instrument set implicitly used by local projections remains valid. Monte Carlo experiments (Section 4) show that in a univariate ARMA(1,1) design PMD converges to the true parameters even at T=50 and matches MLE’s standard errors closely by T=100-400, while MLE suffers numerical convergence difficulties in the pure-AR(1) and pure-MA(1) special cases that leave PMD unaffected; in a three-equation New Keynesian (Phillips-IS-Taylor) design calibrated from Lindé (2005) across 36 DGP variants, PMD has smaller bias than GMM in the majority of cases – both perform well when shocks are i.i.d. (though GMM’s output-gap estimates are somewhat downward biased), and both deteriorate once serial correlation and additional lagged terms are introduced, with GMM deteriorating more. Applying PMD to re-estimate Fuhrer and Olivei’s (2005) hybrid New Keynesian output and inflation Euler equations on 1966:Q1-2001:Q4 U.S. quarterly data, the paper finds PMD point estimates broadly similar to GMM, MLE, and OI-GMM but generally with similar or smaller standard errors, and uses the instability of the estimated coefficients across different choices of H together with frequent rejections of the overidentifying-restrictions test to argue that the hybrid Phillips curve is dynamically misspecified in ways that the other estimators’ point estimates alone do not reveal.

International Economic Review | Macro Paper Warehouse

Estimation and Inference by the Method of Projection Minimum Distance: An Application to the New Keynesian Hybrid Phillips Curve