Using Invalid Instruments on Purpose: Focused Moment Selection and Averaging for GMM

TL;DR

A novel moment selection procedure for GMM is proposed–the Focused Moment Selection Criterion (FMSC)–in which moment conditions are chosen not based on their validity but on the MSE of their associated estimator of a user-specified target parameter.

Abstract

In finite samples, the use of a slightly endogenous but highly relevant instrument can reduce mean-squared error (MSE). Building on this observation, I propose a novel moment selection procedure for GMM -- the Focused Moment Selection Criterion (FMSC) -- in which moment conditions are chosen not based on their validity but on the MSE of their associated estimator of a user-specified target parameter. The FMSC mimics the situation faced by an applied researcher who begins with a set of relatively mild "baseline" assumptions and must decide whether to impose any of a collection of stronger but more controversial "suspect" assumptions. When the (correctly specified) baseline moment conditions identify the model, the FMSC provides an asymptotically unbiased estimator of asymptotic MSE, allowing us to select over the suspect moment conditions. I go on to show how the framework used to derive the FMSC can address the problem of inference post-moment selection. Treating post-selection estimators as a special case of moment-averaging, in which estimators based on different moment sets are given data-dependent weights, I propose simulation-based procedures for inference that can be applied to a variety of formal and informal moment-selection and averaging procedures. Both the FMSC and confidence interval procedures perform well in simulations. I conclude with an empirical example examining the effect of instrument selection on the estimated relationship between malaria and income per capita.

Figures (9)

• Figure 1: RMSE values for the two-stage least squares (TSLS) estimator, the ordinary least squares (OLS) estimator, and the post-Focused Moment Selection Criterion (FMSC) estimator based on 10,000 simulation draws from the DGP given in Equations \ref{['eq:OLSvsIVDGP1']}--\ref{['eq:OLSvsIVDGP3']} using the formulas described in Section \ref{['sec:OLSvsIVExample']}.
• Figure 2: RMSE values for the post-Focused Moment Selection Criterion (FMSC) estimator, Durbin-Hausman-Wu pre-test estimators with $\alpha = 0.1$ (DWH90) and $\alpha = 0.05$ (DHW95), and the minmum-AMSE averaging estimator, based on 10,000 simulation draws from the DGP given in Equations \ref{['eq:OLSvsIVDGP1']}--\ref{['eq:OLSvsIVDGP3']} using the formulas described in Sections \ref{['sec:OLSvsIVExample']} and \ref{['sec:momentavgexample']}.
• Figure 3: RMSE values for the valid estimator, including only $(z_1, z_2, z_3)$, the full estimator, including $(z_1, z_2, z_3, w)$, and the post-Focused Moment Selection Criterion (FMSC) estimator based on 20,000 simulation draws from the DGP given in Equations \ref{['eq:chooseIVDGP1']}--\ref{['eq:chooseIVDGP3']} using the formulas described in Section \ref{['sec:chooseIVexample']}.
• Figure 4: RMSE values for the post-Focused Moment Selection Criterion (FMSC) estimator and the GMM-BIC, HQ, and AIC estimators based on 20,000 simulation draws from the DGP given in Equations \ref{['eq:chooseIVDGP1']}--\ref{['eq:chooseIVDGP3']} using the formulas described in Section \ref{['sec:chooseIVexample']}.
• Figure G.1: RMSE values for the post-Focused Moment Selection Criterion (FMSC) estimator and the downward $J$-test estimator with $\alpha = 0.1$ (J90) and $\alpha = 0.05$ (J95) based on 20,000 simulation draws from the DGP given in Equations \ref{['eq:chooseIVDGP1']}--\ref{['eq:chooseIVDGP3']} using the formulas from Sections \ref{['sec:chooseIVexample']}.

Theorems & Definitions (17)

• Theorem 2.1: Consistency
• Theorem 2.2: Asymptotic Normality
• Corollary 2.1: Limit Distribution of Valid Estimator
• Corollary 3.1: AMSE of Target Parameter
• Theorem 3.1: Asymptotic Distribution of $\widehat{\tau}$
• Corollary 3.2: Asymptotically Unbiased Estimator of $\tau \tau'$
• Theorem 3.2: OLS and TSLS Limit Distributions
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5: Choosing IVs Limit Distribution