Econometric take home APPS_Part_17

Econometric take home APPS_Part_17 - Chapter 10 Systems of...

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67 Chapter 10 ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ Systems of Regression Equations 1. The model can be written as y y i i 1 2 1 2 = + μ ε ε . Therefore, the OLS estimator is m = ( i i + i i ) -1 ( i y 1 + i y 2 ) = ( n y 1 + n y 2 ) / ( n + n ) = ( y 1 + y 2 )/2 = 1.5. The sampling variance would be Var[ m ] = (1/2) 2 {Var[ y 1 ] + Var[ y 2 ] + 2Cov[( y 1 1 , y 2 )]}. We would estimate the parts with Est.Var[ y 1 ] = s 11 / n = ((150 - 100(1) 2 )/99)/100 = .0051 E s t . V a r [ y 2 ] = s 22 / n = ((550 - 100(2) 2 )/99)/100 = .0152 E s t . C o v [ y 1 , y 2 ] = s 12 / n = ((260 - 100(1)(2))/99)/100 = .0061 Combining terms, Est.Var[ m ] = .0079. The GLS estimator would be [( σ 11 + σ 12 ) i y 1 + ( σ 22 + σ 12 ) i y 2 ]/[( σ 11 + σ 12 ) i i + ( σ 22 + σ 12 ) i i ] = w y 1 + (1- w ) y 2 where w = ( σ 11 + σ 12 ) / ( σ 11 + σ 22 + 2 σ 12 ). Denoting Σ = σσ 11 12 12 22 , Σ -1 = 1 11 22 12 2 22 12 12 11 σ . The weight simplifies a bit as the determinant appears in both the denominator and the numerator. Thus, w = ( σ 22 - σ 12 ) / ( σ 11 + σ 22 - 2 σ 12 ). For our sample data, the two step estimator would be based on the variances computed above and s 11 = .5051, s 22 = 1.5152, s 12 = .6061. Then, w = 1.1250. The FGLS estimate is 1.125(1) + (1 - 1.125)(2) = .875. The sampling variance of this estimator is w 2 Var[ y 1 ] + (1 - w ) 2 Var[ y 2 ] + 2 w (1 - w )Cov[ y 1 , y 2 ] = .0050 as compared to .0079 for the OLS estimator. 2. The model is y = y y 1 2 = X β + ε = i0 0x ⎟ + β β 1 2 1 2 ε ε , σ 2 Ω = 11 12 12 22 II . The generalized least squares estimator is βΩ Ω −− = [' ] ' XX Xy 11 1 = 11 12 12 22 1 11 1 12 2 12 1 22 2 ii i 'x i'x x'x iy + iy xy + xy '' ' = n x xs n yy ss xx x x 11 12 12 22 1 11 1 12 2 12 1 22 2 + + where s xx = x x / n , s x1 = x y 1 / n , s x2 = x y 2 / n and σ ij = the ij th element of the 2 × 2 Σ -1 . To obtain the explicit form, note, first, that all terms σ ij are of the form σ ji /( σ 11 σ 22 - σ 2 12 ) But, the denominator in these ratios will be cancelled as it appears in both the inverse matrix and in the vector. Therefore, in terms of the original parameters, (after cancelling n ), we obtain β = 22 12 12 11 1 22 12 1 2 12 12 −+ x xx xx = 1 11 22 12 2 11 12 12 22 22 1 12 2 12 1 11 2 σ sx x xx xx () .
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This note was uploaded on 11/13/2011 for the course ECE 4105 taught by Professor Dr.fang during the Spring '10 term at University of Florida.

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Econometric take home APPS_Part_17 - Chapter 10 Systems of...

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