Econometrics-I-14

N the white estimator n

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Unformatted text preview: n The White estimator n Newey-West. ˜™™™™ ™ 6/44 Part 14: Generalized Regression The White Estimator ˜™™™™ ™ 7/44 n 1 2 1 i i i i 1 n 2 i 2 i 1 2 i i i 2 1 2 2 2 Est.Var[ ] ( ) e ( ) e Use ˆ n ne ˆ ˆ = , =diag( ) note tr( )=n ˆ ˆ ˆ ˆ ˆ Est.Var[ ] n n n n ˆ Does ˆ n n-- = =- = σ = ϖ ϖ σ σ = ÷ ÷ ÷ ÷ σ- σ ÷ ÷ ∑ ∑ b X'X x x ' X'X Ω Ω X'X X' X X'X Ω b X' X X' X Ω Ω → ÷ 0? 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 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 Part 14: Generalized Regression Autocorrelated Residuals logG=β1 + β2logPg + β3logY + β4logPnc + β5logPuc + ε ˜˜™™™™ ™ 10/44 Part 14: Generalized Regression Newey-West Estimator ˜˜™™™™ ™ 11/44 n 2 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 1 [ ] n n-- + ÷ ÷ 'X X'X S S 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--------+-------------------------------------------------------------...
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n The White estimator n Newey-West˜™™™™ ™...

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