Econometrics-I-7

# 2435 part 7 estimating the variance of b 2535 part 7

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Part 7: Estimating the Variance of  b ™    25/35
Part 7: Estimating the Variance of  b The NIST Filipelli Problem ™    26/35

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Part 7: Estimating the Variance of  b Certified Filipelli Results ™    27/35
Part 7: Estimating the Variance of  b Stata Filipelli Results ™    28/35

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Part 7: Estimating the Variance of  b Regression of x2 on all other variables ™    29/35
Part 7: Estimating the Variance of  b Using QR Decomposition ™    30/35

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Part 7: Estimating the Variance of  b Multicollinearity There is no “cure” for collinearity. Estimating something else is not helpful (principal components, for example). There are “measures” of multicollinearity, such as the condition number of X and the variance inflation factor. Best approach: Be cognizant of it. Understand its implications for estimation. What is better: Include a variable that causes collinearity, or drop the variable and suffer from a biased estimator? Mean squared error would be the basis for comparison. Some generalities. 1 ™    31/35
Part 7: Estimating the Variance of  b Specification and Functional Form: Nonlinearity 2 2 1 2 3 4 1 2 3 4 2 3 2 3 2 2 Population Estimators ˆ [ | , ] ˆ 2 2 ˆ Estimator of the variance of ˆ . [ ] [ ] 4 x x x x y x x z y b b x b x b z E y x z x b b x x EstVar Var b x Va = β +β + ε = + + + δ = = β + β δ = + δ δ = + 3 2 3 [ ] 4 [ , ] r b xCov b b + ™    32/35

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Part 7: Estimating the Variance of  b Log Income Equation ---------------------------------------------------------------------- Ordinary least squares regression ............ LHS=LOGY Mean = -1.15746 Estimated Cov[b1,b2] Standard deviation = .49149 Number of observs. = 27322 Model size Parameters = 7 Degrees of freedom = 27315 Residuals Sum of squares = 5462.03686 Standard error of e = .44717 Fit R-squared = .17237 --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- AGE| .06225*** .00213 29.189 .0000 43.5272 AGESQ| -.00074*** .242482D-04 -30.576 .0000 2022.99 Constant| -3.19130*** .04567 -69.884 .0000 MARRIED| .32153*** .00703 45.767 .0000 .75869 HHKIDS| -.11134*** .00655 -17.002 .0000 .40272 FEMALE| -.00491 .00552 -.889 .3739 .47881 EDUC| .05542*** .00120 46.050 .0000 11.3202 --------+------------------------------------------------------------- Average Age = 43.5272. Estimated Partial effect = .066225 – 2(.00074)43.5272 = .00018. Estimated Variance 4.54799e-6 + 4(43.5272)2(5.87973e-10) + 4(43.5272)(-5.1285e-8) = 7.4755086e-08. Estimated standard error = .00027341. ™    33/35
Part 7: Estimating the Variance of  b Specification and Functional Form: Interaction Effect 1 2 3 4 1 2 3 4 2 4 2 4 2 2 Population Estimators ˆ [ | , ] ˆ z ˆ Estimator of the variance of ˆ . [ ] [ ] x x x x y x z xz y b b x b z b xz E y x z b b z x EstVar Var b z Va = β +β + ε = + + + δ = = β +β δ = + δ δ = + 4 2 4 [ ] 2 [ , ] r b zCov b b + ™    34/35

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Part 7: Estimating the Variance of  b Interaction Effect ---------------------------------------------------------------------- Ordinary least squares regression ............ LHS=LOGY Mean = -1.15746 Standard deviation = .49149 Number of observs. = 27322 Model size Parameters = 4 Degrees of freedom = 27318 Residuals Sum of squares = 6540.45988 Standard error of e = .48931 Fit R-squared = .00896 Adjusted R-squared = .00885 Model test F[ 3, 27318] (prob) = 82.4(.0000) --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- Constant| -1.22592*** .01605 -76.376 .0000 AGE| .00227*** .00036 6.240 .0000 43.5272 FEMALE| .21239*** .02363 8.987 .0000 .47881 AGE_FEM| -.00620*** .00052 -11.819 .0000 21.2960 --------+------------------------------------------------------------- Do women earn more than men (in this sample?) The +.21239 coefficient on FEMALE would suggest so. But, the female “difference” is +.21239 - .00620*Age. At average Age, the effect is .21239 - .00620(43.5272) = -.05748.
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