Econometric take home APPS_Part_5

Econometric take home APPS_Part_5 - Chapter 5 Inference and...

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Chapter 5 Inference and Prediction Exercises 1. The estimated covariance matrix for the least squares estimator is s 2 ( X X ) -1 = 20 3900 3900 29 0 0 08 0 1 01 0 8 / 0 0 = where s 2 = 520/(29-3) = 20. Then, the test may be based on t = (.4 + .9 - 1)/[.410 + .256 - 2(.051)] 1/2 = .399. This is smaller than the critical value of 2.056, so we would not reject the hypothesis. . .. 69 0 0 0 40 051 00 5 1 2 5 6 2. In order to compute the regression, we must recover the original sums of squares and cross products for y. These are X y = X Xb = [116, 29, 76] . The total sum of squares is found using R 2 = 1 - e e / y M 0 y , so y M 0 y = 520 / (52/60) = 600. The means are x 1 = 0, x 2 = 0, y = 4, so, y y = 600 + 29(4 2 ) = 1064. The slope in the regression of y on x 2 alone is b 2 = 76/80, so the regression sum of squares is b 2 2 + 1 2 0 0 β β ( (80) = 72.2, and the residual sum of squares is 600 - 72.2 = 527.8. The test based on the residual sum of squares is F [(527.8 - 520)/1]/[520/26] = .390. In the regression of the previous problem, the t -ratio for testing the same hypothesis would be t = .4/(.410) 1/2 = .624 which is the square root of .39. 3. For the current problem, R = [ 0 , I ] where I is the last K 2 columns. Therefore, R ( X X ) -1 R N is the lower right K 2 × K 2 block of ( X X ) -1 . As we have seen before, this is ( X 2 M 1 X 2) -1 . Also, ( X X ) -1 R is the last K 2 columns of ( X X ) -1 . These are ( X X ) -1 R = Finally, since q = 0 , Rb - q = ( 0b 1 + Ib 2 ) - 0 = b 2 . Therefore, the constrained estimator is -X X X X X MX XMX 11 12 2 12 21 2 (') ' (' ) (' ) −− 1 b * = ( X 2 M 1 X 2 ) b 2 , where b 1 and b 2 are the multiple regression coefficients in the regression of y on both X 1 and X 2 . Collecting terms, this produces b * = . But, we have from Section 6.3.4 that b 1 ( X 1 X 1 ) -1 X 1 y - ( X 1 X 1 ) - 1 X 1 X 2 b 2 so the preceding reduces to b * = which was to be shown. b b 1 2 - 2 ) ) 1 b b 1 2 -XX XXb b 12 2 2 (') ' 1 XX Xy 0 1 1 If, instead, the restriction is β 2 = β 2 0 then the preceding is changed by replacing R β - q = 0 with R β - β 2 0 = 0 . Thus, Rb - q = b 2 - β 2 0 . Then, the constrained estimator is b * = ( X 2 M 1 X 2 )( b 2 - β 2 0 ) b b 1 2 2 ) ) 1 or b * = b b 1 2 2 2 Using the result of the previous paragraph, we can rewrite the first part as (') ' ( ) ) XX XXb - b 122 b = ( X 1 X 1 ) -1 X 1 y - ( X
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Econometric take home APPS_Part_5 - Chapter 5 Inference and...

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