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# Collinearity Errors

Asymptotic Standard Errors As described in Section 4.15, a standard error is an estimator of the standard deviation of the distribution of an estimator. Thus if Vb b is an estimator of the covariance matrix of b, then standard errors are the square roots of the diagonal elements of this matrix. These take the form s( bj ) = q Vb bj = rh Vb b i jj : Standard errors for b are constructed similarly. Supposing that  = h( ) is real-valued then the standard error for b is the square root of (7.30) s(b) = r Rb 0 Vb bRb = q n1Rb 0 Vb Rb : When the justiÖcation is based on asymptotic theory we call s( bj ) or s(b) an asymptotic standard error for bj or b. When reporting your results, it is good practice to report standard errors for each reported estimate, and this includes functions and transformations of your parameter estimates. This helps users of the work (including yourself) assess the estimation precision. We illustrate using the log wage regression log(W age) = 1 education + 2 experience + 3 experience2 =100 + 4 + e: Consider the following three parameters of interest. 1. Percentage return to education: 1 = 100 1 (100 times the partial derivative of the conditional expectation of log wages with respect to education.) 2. Percentage return to experience for individuals with 10 years of experience: 2 = 100 2 + 20 3 (100 times the partial derivative of the conditional expectation of log wages with respect to experience, evaluated at experience = 10.) 3. Experience level which maximizes expected log wages: 3 = 50 2= 3 (The level of experience at which the partial derivative of the conditional expectation of log wages with respect to experience equals 0.) The 4 1 vector R for these three parameters is R = 0 BB@ 100 0 0 0 1 CCA ; 0 BB@ 0 100 20 0 1 CCA ; 0 BB@ 0 50= 3 50 2= 2 3 0 1 CCA ; respective CHAPTER 7. ASYMPTOTIC THEORY FOR LEAST SQUARES 238 We use the subsample of married black women (all experience levels), which has 982 observations. The point estimates and standard errors are log(\W age) = 0:118 (0:008) education + 0:016 (0:006) experience 0:022 (0:012) experience2 =100 + 0:947 (0:157) : (7.31) The standard errors are the square roots of the Horn-Horn-Duncan covariance matrix estimate V b = 0 BB@ 0:632 0:131 0:143 11:1 0:131 0:390 0:731 6:25 0:143 0:731 1:48 9:43 11:1 6:25 9:43 246 1 CCA 104 : (7.32) We calculate that b1 = 100 b1 = 100 0:118 = 11:8 s(b1) = p 1002 0:632 104 = 0:8 b2 = 100 b2 + 20 b3 = 100 0:016 20 0:022 = 1:16 s(b2) = s 100 20   0:390 0:731 0:731 1:48   100 20  104 = 0:55 b3 = 50 b2= b3 = 50 0: