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12.20 Functions of Parameters Given the distribution theory in Theorems 12.2 and 12.3 it is straightforward to derive the asymptotic distribution of smooth nonlinear functions of the coe¢ cients. SpeciÖcally, given a function r ( ) : R k !  R q we deÖne the parameter  = r ( ): Given b 2sls a natural estimator of  is b 2sls = r  b 2sls . Consistency follows from Theorem 12.1 and the continuous mapping theorem. Theorem 12.4 Under Assumptions 12.1 and 7.3, as n ! 1, b 2sls p ! : If r ( ) is di§erentiable then an estimator of the asymptotic covariance matrix for b is Vb  = Rb 0 Vb Rb Rb = @ @ r( b 2sls) 0 : We similarly deÖne the homoskedastic variance estimator as Vb 0  = Rb 0 Vb 0 Rb : The asymptotic distribution theory follows from Theorems 12.2 and 12.3 and the delta m CHAPTER 12. INSTRUMENTAL VARIABLES 430 Theorem 12.5 Under Assumptions 12.2 and 7.3, as n ! 1, p n  b 2sls   d ! N (0;V ) where V  = R0V R R = @ @ r( ) 0 and Vb  p ! V : When q = 1, a standard error for b 2sls is s(b 2sls) = q n1Vb  . For example, letís take the parameter estimates from the Öfth column of Table 12.1, which are the 2SLS estimates with three endogenous regressors and four excluded instruments. Suppose we are interested in the return to experience, which depends on the level of experience. The estimated return at experience = 10 is 0:047 0:032 2 10=100 = 0:041 and its standard error is 0:003. This implies a 4% increase in wages per year of experience and is precisely estimated. Or suppose we are interested in the level of experience at which the function maximizes. The estimate is 50 0:047=0:032 = 73. This has a standard error of 249. The large standard error implies that the estimate (73 years of experience) is without precision and is thus uninforma