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# normal distribution with mean  and variance 

î has exact size . Since in general we do not want to impose homoskedasticity, these results suggest that the most appropriate test is the Wald statistic constructed with the robust heteroskedastic covariance matrix. This can be computed in Stata using the command estat endogenous after ivregress when the latter uses a robust covariance option. Stata reports the Wald st CHAPTER 12. INSTRUMENTAL VARIABLES 446 thus uses the F distribution to calculate the p-value) as ìRobust regression Fî. Using the F rather than the  2 distribution is not formally justiÖed but is a reasonable Önite sample adjustment. If the command estat endogenous is applied after ivregress without a robust covariance option, Stata reports the F statistic as ìWu-Hausman Fî. There is an alternative (and traditional) way to derive a test for endogeneity. Under H0, both OLS and 2SLS are consistent estimators. But under H1, they converge to di§erent values. Thus the di§erence between the OLS and 2SLS estimators is a valid test statistic for endogeneity. It also measures what we often care most about ñthe impact of endogeneity on the parameter estimates. This literature was developed under the assumption of conditional homoskedasticity (and it is important for these results) so we assume this condition for the development of the statistics. Let b =  b 1 ; b 2  be the OLS estimator and let e =  e 1 ; e 2  be the 2SLS estimator. Under H0 (and homoskedasticity) the OLS estimator is Gauss-Markov e¢ cient, so by the Hausman equality var  b 2 e 2  = var  e 2  var  b 2  =  X0 2 (P Z P 1) X2 1 X0 2M1X2 1   2 where P Z = Z (Z 0Z) 1 Z 0 , P 1 = X1 (X0 1X1) 1 X0 1 , and M1 = In P 1. Thus a valid test statistic for H0 is T =  b 2 e 2 0  (X0 2 (P Z P 1) X2) 1 (X0 2M1X2) 1 1 