Econometric take home APPS_Part_11

Econometric take home APPS_Part_11 - +-+-+-+-+-+-+...

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43 +--------+--------------+----------------+--------+--------+----------+ |Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X| +--------+--------------+----------------+--------+--------+----------+ Constant| -856.107861 221.141722 -3.871 .0002 YT | 1.21490273 .32340906 3.757 .0003 4987.32410 CT1 | .98759074 .04395654 22.467 .0000 4465.65542 CY | -1.13474451 .31933175 -3.553 .0006 4503.23012 ? ? The results are essentially the same. This suggests ? that neither model is right. The regressions are based on real consumption and real disposable income. Results for 1950 to 2000 are given in the text. Repeating the exercise for 1980 to 2000 produces: for the first regression, the estimate of α is 1.03 with a t ratio of 23.27 and for the second, the estimate is -1.24 with a t ratio of -3.062. Thus, as before, both models are rejected. This is qualitatively the same results obtained with the full 51 year data set.
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44 Chapter 8 The Generalized Regression Model and Heteroscedasticity Exercises 1. Write the two estimators as ˆ β = β + ( X ′Ω -1 X ) -1 X ′Ω -1 ε and b = β + ( X X ) -1 X ′ε . Then, ( ˆ β - b ) = [( X ′Ω -1 X ) -1 X ′Ω -1 - ( X X ) -1 X ] ε has E [ ˆ β - b ] = 0 since both estimators are unbiased. Therefore, Cov[ ˆ β , ˆ β - b ] = E [( ˆ β - β )( ˆ β - b ) ]. Then, E {( X ′Ω -1 X ) -1 X ′Ω -1 εε′ [( X ′Ω -1 X ) -1 X ′Ω - 1 - ( X X ) -1 X ] } = ( X ′Ω -1 X ) -1 X ′Ω -1 ( σ 2 Ω )[ Ω -1 X ( X ′Ω - 1 X ) -1 - X ( X X ) -1 ] = σ 2 ( X ′Ω -1 X ) -1 X ′Ω -1 ΩΩ -1 X ( X ′Ω -1 X ) -1 - ( X ′Ω -1 X ) -1 X ′Ω -1 Ω X ( X X ) -1 = ( X ′Ω -1 X ) -1 ( X ′Ω -1 X )( X ′Ω -1 X ) -1 - ( X ′Ω -1 X ) -1 ( X X )( X X ) -1 = 0 once the inverse matrices are multiplied. 2 First, ( R ˆ β - q ) = R [ β + ( X ′Ω -1 X ) -1 X ′Ω -1 ε )] - q = R ( X ′Ω -1 X ) -1 X ′Ω -1 ε if R β - q = 0 . Now, use the inverse square root matrix of Ω , P = Ω -1/2 to obtain the transformed data, X * = PX = Ω -1/2 X , y * = Py = Ω -1/2 y , and ε * = P ε = Ω -1/2 ε . Then, E [ ε * ε * ] = E [ Ω -1/2 εε′Ω -2 ] = Ω -1/2 ( σ 2 Ω ) Ω -1/2 = σ 2 I , and, ˆ β = ( X ′Ω -1 X ) -1 X ′Ω -1 y = ( X * X * ) -1 X * y * = the OLS estimator in the regression of y * on X * . Then, R ˆ β - q = R ( X * X * ) -1 X * ′ε * and the numerator is ε * X * ( X * X * ) -1 R [ R ( X * X * ) -1 R ] -1 R ( X * X * ) -1 X * ′ε * / J . By multiplying it out, we find that the matrix of the quadratic form above is idempotent. Therefore, this is an idempotent quadratic form in a normally distributed random vector. Thus, its distribution is that of σ 2 times a chi-squared variable with degrees of freedom equal to the rank of the matrix. To find the rank of the matrix of the quadratic form, we can find its trace. That is tr{ X * ( X * X * ) -1 R [ R ( X * X * ) -1 R ] -1 R ( X * X * ) -1 X * } = t r { ( X * X * ) -1 R [ R ( X * X * ) -1 R ] -1 R ( X * X * ) -1 X * X * } = t r { ( X * X * ) -1 R [ R ( X * X * ) -1 R ] -1 R } = t r { [ R ( X * X * ) -1 R ][ R ( X * X * ) -1 R ] -1 } = tr{ I J } = J , which might have been expected. Before proceeding, we should note, we could have deduced this outcome from the form of the matrix. The matrix of the quadratic form is of the form Q = X * ABA X * where B is the nonsingular matrix in the square brackets and A = ( X * X * ) -1 R , which is a K × J matrix which cannot have rank higher than J .
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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_11 - +-+-+-+-+-+-+...

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