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sjpdf - The Stata Journal(2003 3 Number 1 pp 131...

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The Stata Journal (2003) 3 , Number 1, pp. 1–31 Instrumental variables and GMM: Estimation and testing Christopher F. Baum Boston College Mark E. Schaffer Heriot–Watt University 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, en- dogeneity, heteroskedasticity, overidentifying restrictions, clustering, intra-group correlation 1 Introduction The application of the instrumental variables ( IV ) estimator in the context of the clas- sical 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 con- sistency of the IV coefficient estimates is not affected by the presence of heteroskedas- ticity, the standard IV estimates of the standard errors are inconsistent, preventing valid inference. The usual forms of the diagnostic tests for endogeneity and overiden- tifying 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 orthogonal- ity 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 econometrics 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 c circlecopyrt 2003 Stata Corporation st0030
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2 Instrumental variables and GMM: Estimation and testing 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
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