Econometrics-I-11

V v v ˆ ˆ ˆ and if f θ θ θ θ θ θ θ θ θ v

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v v ... v ˆ ˆ ˆ and if f( , ,..., ) = θ θ θ θ θ θ θ θ θ V 1 1 K 11 12 1K 21 22 2K 1 2 K K1 K2 KK  a continuous function with continuous derivatives, then ˆ ˆ ˆ the asymptotic variance of f( , ,..., ) is v v ... v v v ... v f(.) f(.) f(.)  =   ... ... ... ... ... v v ... v θ θ θ ∂θ ∂θ ∂θ g'Vg 1 K K 2 kl k 1 l 1 k l K f(.) f(.) f(.) f(.) V ... f(.) = = ∂θ ∂θ = ∂θ ∂θ ∂θ ∑∑

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Part 11: Asymptotic Distribution  Theory 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 --------+------------------------------------------------------------- ™  31/42
Part 11: Asymptotic Distribution  Theory Age-Income Profile: Married=1, Kids=1, Educ=12, Female=1 ™  32/42

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Part 11: Asymptotic Distribution  Theory Application: Maximum of a Function AGE| .06225*** .00213 29.189 .0000 43.5272 AGESQ| -.00074*** .242482D-04 -30.576 .0000 2022.99 ™  33/42 2 1 2 1 1 2 2 1 1 2 1 2 2 2 2 log ... At what age does log income reach its maximum? log .06225 2 0 => Age* = 42.1 2 2( .00074) * 1 1 =g = =675.68 2 2( .00074) * .06225 2 Y Age Age Y Age Age Age Age g = β + - = β + β = = = β - - - = ∂β β - β = = = ∂β β 2 56838.9 2( .00074) = -
Part 11: Asymptotic Distribution  Theory Delta Method Using Visible Digits ™  34/42 2 6 2 10 8 675.68 (4.54799 10 ) 56838.9 (5.8797 10 ) 2(675.68)(56838.9)( 5.1285 10 ) .0366952 standard error = square root = .1915599 - - - × + × + - × =

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Part 11: Asymptotic Distribution  Theory Delta Method Results ----------------------------------------------------------- WALD procedure. --------+-------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] --------+-------------------------------------------------- G1| 674.399*** 22.05686 30.575 .0000 G2| 56623.8*** 1797.294 31.505 .0000 AGESTAR| 41.9809*** .19193 218.727 .0000 --------+-------------------------------------------------- (Computed using all internal digits of regression results) ™  35/42
Part 11: Asymptotic Distribution  Theory Krinsky and Robb Descriptive Statistics ============================================================ Variable Mean Std.Dev. Minimum Maximum --------+--------------------------------------------------- ASTAR | 42.0607 .191563 41.2226 42.7720 ™  36/42

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Part 11: Asymptotic Distribution  Theory More than One Function and More than One Coefficient ™  37/42 1 1 K 11 12 1K 21 22 2K 1 2 K K1 K2 KK 1 1 K ˆ ˆ ˆ If   , ,...,  are K consistent estimators of K parameters  v v ... v v v ... v , ,...,  with asymptotic covariance matrix  = , ... ... ... ... v v ... v ˆ ˆ ˆ and if f1( , ,..., ), θ θ θ θ θ θ θ θ θ V 1 2 K 1 2 1 1 K 1 1 K f1 f1 f1 f 2 f 2 f ˆ ˆ ˆ ˆ ˆ ˆ  f2( , ,..., ), ...,fJ( , ,..., )  =  J continuous ˆ ˆ functions with continuous derivatives, then the asymptotic covariance matrix of f1,...,fJ is ... ...  =   ∂θ ∂θ ∂θ ∂θ ∂θ θ θ θ θ θ θ GVG' 1 1 1 K 2 2 2 1 2 K K K K f1 f 2 fJ 11 12 1K 2 f1 f 2 fJ 21 22 2K fJ fJ fJ f1 f 2 fJ K1 K2 KK ...
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