Instrumental Variables and GMM: Estimation and Testing

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. Instrumental variables and GMM: Estimation and testingdiscussion 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.Efficient GMM brings with it the advantage of consistency in the presence of arbitrary heteroskedasticity, but at a cost of possibly poor finite sample performance. If heteroskedasticity is in fact not present, then standard IV may be preferable. The usual Breusch-Pagan/Godfrey/Cook-Weisberg and White/Koenker tests for the presence of heteroskedasticity in a regression equation can be applied to an IV regression only under restrictive assumptions. In Section 3, we discuss the test of Pagan and Hall (1983) designed specifically for detecting the presence of heteroskedasticity in IV estimation, and its relationship to these other heteroskedasticity tests.Even when IV or GMM is judged to be the appropriate estimation technique, we may still question its validity in a given application: are our instruments "good instruments"? This is the question we address in Section 4. "Good instruments" should be both relevant and valid: correlated with the endogenous regressors and at the same time orthogonal to the errors. Correlation with the endogenous regressors can be assessed by an examination of the significance of the excluded instruments in the first-stage IV regressions. We may cast some light on whether the instruments satisfy the orthogonality conditions in the context of an overidentified model: that is, one in which a surfeit of instruments are available. In that context, we may test the overidentifying restrictions in order to provide some evidence of the instruments' validity. We present the variants of this test due to Sargan (1958), Basmann (1960, and, in the GMM context, Hansen (1982), and show how the generalization of this test, the C or "difference-in-Sargan" test, can be used to test the validity of subsets of the instruments.Although there may well be reason to suspect nonorthogonality between regressors and errors, the use of IV estimation to address this problem must be balanced against the inevitable loss of efficiency vis-à-vis OLS. It is therefore very useful to have a test of whether or not OLS is inconsistent and IV or GMM is required. This is the Durbin-Wu-Hau...

Food away from home expenditures and obesity among older Europeans: are there gender differences?

“…Under i.i.d. assumptions, this endogeneity test statistic is numerically equal to a Hausman test statistic (Baum et al 2003). The reported test statistic in Table A2 is robust to violations of homoskedasticity.…”

“…However, the conventional IV estimator is inefficient in the presence of heteroskedasticity and the usual approach when facing heteroskedasticity of an unknown form is to use the GMM estimator. If heteroskedasticity is indeed present, the GMM estimator is more efficient than the simple IV estimator, whereas if heteroskedasticity is not present the GMM estimator is no worse asymptotically than the IV estimator (Baum et al 2003). However, the GMM estimator may have poor small sample properties and if in fact the errors are homoskedastic, IV would be preferable.…”

“…However, the GMM estimator may have poor small sample properties and if in fact the errors are homoskedastic, IV would be preferable. We conducted tests for the presence of heteroskedasticity, as proposed by Pagan and Hall (1983) 5 (Schaffer 2002;Baum et al 2003). The null of homoskedastic disturbances is rejected for both sub-samples (see Table A1 in the appendix).…”

“…The statistic can be compared to the 7 The partial R 2 is the R 2 of the first stage regression with the included instruments "partialled out," which is equivalent to an F test of the joint significance of the excluded instruments (see Baum et al 2003 for more details). 8 The Shea's partial R 2 and the standard partial R 2 are equivalent when the model contains only one endogenous regressor.…”

Does Social Performance Really Lead to Financial Performance? Accounting for Endogeneity

“…According to Baum et al (2007), if the errors in two-step GMM estimation are heteroskedastic or serially correlated the Cragg-Donald statistic is not valid. In those cases, the Kleibergen and Paap statistic will be a more robust test (Baum et al, 2007).…”