Econometrics-I-14

0 part 14 generalized regression groupwise

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Part 14: Generalized Regression Groupwise Heteroscedasticity Regression of log of per capita gasoline use on log of per capita income, gasoline price and number of cars per capita for 18 OECD countries for 19 years. The standard deviation varies by country. The “solution” is “weighted least squares.” Countries are ordered by the standard deviation of their 19 residuals. ™  8/44
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Part 14: Generalized Regression White Estimator +--------+--------------+----------------+--------+--------+----------+ |Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X| +--------+--------------+----------------+--------+--------+----------+ Constant| 2.39132562 .11693429 20.450 .0000 LINCOMEP| .88996166 .03580581 24.855 .0000 -6.13942544 LRPMG | -.89179791 .03031474 -29.418 .0000 -.52310321 LCARPCAP| -.76337275 .01860830 -41.023 .0000 -9.04180473 | White heteroscedasticity robust covariance matrix | +----------------------------------------------------+ Constant| 2.39132562 .11794828 20.274 .0000 LINCOMEP| .88996166 .04429158 20.093 .0000 -6.13942544 LRPMG | -.89179791 .03890922 -22.920 .0000 -.52310321 LCARPCAP| -.76337275 .02152888 -35.458 .0000 -9.04180473 ™  9/44
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Part 14: Generalized Regression Autocorrelated Residuals logG=β1 + β2logPg + β3logY + β4logPnc + β5logPuc + ε ™  10/44
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Part 14: Generalized Regression Newey-West Estimator ™  11/44 n 2 0 i i i i 1 L n 1 l t t l t t l t l t l 1 t l 1 l Heteroscedasticity Component - Diagonal Elements 1 e n Autocorrelation Component - Off Diagonal Elements 1 w e e ( ) n l w 1  =  "Bartlett weight" L 1 1 Est.Var[ ]= n = - - - = = + = = + = - + ∑ ∑ S x x ' S x x x x X b 1 1 0 1 [ ] n n - - + ÷ ÷ 'X X'X S S
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Part 14: Generalized Regression Newey-West Estimate --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- Constant| -21.2111*** .75322 -28.160 .0000 LP| -.02121 .04377 -.485 .6303 3.72930 LY| 1.09587*** .07771 14.102 .0000 9.67215 LPNC| -.37361** .15707 -2.379 .0215 4.38037 LPUC| .02003 .10330 .194 .8471 4.10545 --------+------------------------------------------------------------- --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X Robust VC Newey-West, Periods = 10 --------+------------------------------------------------------------- Constant| -21.2111*** 1.33095 -15.937 .0000 LP| -.02121 .06119 -.347 .7305 3.72930 LY| 1.09587*** .14234 7.699 .0000 9.67215 LPNC| -.37361** .16615 -2.249 .0293 4.38037 LPUC| .02003 .14176 .141 .8882 4.10545 --------+------------------------------------------------------------- ™  12/44
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Part 14: Generalized Regression Generalized Least Squares A transformation of the model: P = -1/2. P’P = -1 Py = PX + P or y * = X * + *. Why? E[ * *’| X *]= P E[ ’|X* ] P’ = P E[ ’|X ] P’ = σ2 PP’ = σ2 -1/2  -1/2 = σ2 0 = σ2 I ™  13/44
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Part 14: Generalized Regression Generalized Least Squares Aitken theorem. The Generalized Least Squares estimator, GLS. Py = PX + P or y * = X * + *. E[ * *’| X *]= σ2 I Use ordinary least squares in the transformed model. Satisfies the Gauss – Markov theorem. b * = ( X*’X* ) -1X*’y* ™  14/44
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Part 14: Generalized Regression Generalized Least Squares Efficient estimation of and, by implication, the inefficiency of least squares b . = ( X*’X* ) -1X*’y* = ( X’P’PX )-1 X’P’Py = ( X’Ω-1X )-1 X’Ω-1y ≠ b . is efficient, so by construction, b is not. ™  15/44 ˆ β ˆ β ˆ β
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Part 14: Generalized Regression Asymptotics for GLS Asymptotic distribution of GLS. (NOTE. We apply the full set of results of the classical model to the transformed model. Unbiasedness Consistency - “well behaved data” Asymptotic distribution Test statistics ™  16/44
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Part 14: Generalized Regression Unbiasedness ™  17/44 1 1 1 ˆ ( ) ( ) ˆ E[ ] ( ) E[ | ]            =     if  E[ | ] - - - = = + + = -1 -1 -1 -1 -1 -1 X' X X' y β Ω Ω     X' X X' β Ω Ω ε | X = X' X X' X β β Ω Ω ε X 0 β ε
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Part 14: Generalized Regression Consistency ™  18/44 1 2 n i i i 1
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