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雷达卡
The Stata Journal (2003)
3, Number 1, pp. 1–31
Instrumental variables and GMM: Estimation and testing
Christopher F. Baum
Mark E. Schaffer
Steven Stillman
New Zealand Department of Labour
Abstract. We discuss instrumental variables (IV) estimation in the broader context of the generalized method of moments (GMM), and describe an extended IV estimation routine that provides GMM estimates as well as additional diagnostic tests. Stand-alone test procedures for heteroskedasticity,
overidentification, and endogeneity in the IV context are also described.
Keywords: st0030, instrumental variables, generalized method of moments, endogeneity,
heteroskedasticity, overidentifying restrictions, clustering, intra-group correlation
1 Introduction
The application of the instrumental variables (IV) estimator in the context of the classical linear regression model, from a textbook context, is quite straightforward: if the error distribution cannot be considered independent of the regressors’ distribution, IV is called for, using an appropriate set of instruments. But applied researchers often must confront several hard choices.An omnipresent problem in empirical work is heteroskedasticity. Although the consistencyof the IV coefficient estimates is not affected by the presence of heteroskedasticity,the standard IV estimates of the standard errors are inconsistent, preventing valid inference. The usual forms of the diagnostic tests for endogeneity and overidentifying restrictions will also be invalid if heteroskedasticity is present. These problems can be partially addressed through the use of heteroskedasticity-consistent or “robust” standard errors and statistics. The conventional IV estimator (though consistent) is,however, inefficient in the presence of heteroskedasticity. The usual approach today when facing heteroskedasticity of unknown form is to use the generalized method of moments (GMM), introduced by Hansen (1982). GMM makes use of the rthogonality
conditions to allow for efficient estimation in the presence of heteroskedasticity of unknown form.In the twenty years since it was first introduced, GMM has become a very popular tool among empirical researchers. It is also a very useful heuristic tool. Many standard estimators, including IV and OLS, can be seen as special cases of GMM estimators, and are often presented as such in first-year graduate conometrics texts. Most of the diagnostic tests we discuss in this paper can also be cast in a GMM framework. We begin, therefore,with a short presentation of IV and GMM estimation in Section 2. We include here a discussion of intra-group correlation or “clustering”. If the error terms in the regression are correlated within groups, but not correlated across groups, then the consequences for IV estimation are similar to those of heteroskedasticity: the IV coefficient estimates are consistent, but their standard errors and the usual forms of the diagnostic tests are not. We discuss how clustering can be interpreted in the GMM context and how it can be dealt with in Stata to make efficient estimation, valid inference, and diagnostic testing possible.。。。。。。。。。。。。。。。。。
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