Lecture31-2010 - Exceptions to Ordinary Least Squares:...

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Exceptions to Ordinary Least Squares: Lecture XXXI Charles B. Moss November 23, 2010 I. Heteroscedasticity A. Using the derivation of the variance of ordinary least squares es- timator ˆ β = β +( X 0 X ) 1 ( X 0 ± ) V ± ˆ β ² =( X 0 X ) 1 X 0 ±± 0 X ( X 0 X ) 1 V ± ˆ β ² X 0 X ) 1 X 0 SX ( X 0 X ) 1 3 : S =E[ ±± 0 ] (1) under the Gauss-Markov assumptions S ±± 0 ]= σ 2 I T × T . B. However, if we assume that S =E [ ±± 0 ] 6 = σ 2 I T × T the ordinary least squares estimator is still unbiased, but is no longer efficient. In this case, we use the generalized least squares estimator ˜ β X 0 AX ) 1 ( X 0 Ay )( 2 ) 1. The estimator under heteroscedasticity (generalized least squares) implies ˜ β X 0 AX ) 1 ( X 0 AXβ + X 0 ) X 0 AX ) 1 ( X 0 AX ) β X 0 AX ) 1 X 0 = β X 0 AX ) 1 X 0 (3) 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XXXI Fall 2010 The variance of the generalized least squares estimator then becomes V ± ˜ β β ² =( X 0 AX ) 1 X 0 A±± 0 A 0 X ( X 0 AX ) 1 X 0 AX ) 1 X 0 ASA 0 X ( X 0 AX ) 1 (4) Setting A = S 1 V ( ˜ β ) X 0 AX ) 1 X 0 AX ( X 0 AX ) 1 3 : A 0 = A X 0 AX ) 1 (5) C. Seemingly Unrelated Regressions 1. One of the uses of generalized least squares is the estimation of simultaneous systems of equations without endogeneity. a) Derived input demand equations derived from cost mini- mization implies relationship between the parameters x 1 = α 1 + A 11 w 1 + A 12 w 2 11 y + ± 1 x 2 = α 2 + A 21 w 1 + A 22 w 2 21 y + ± 2 (6) where x 1 and x 2 are input levels, w 1 and w 2 are the respec- tive input prices, y is the level of output, and α 1 , α 2 , A 11 , A 12 , A 21 , A 22 11 and Γ 21 are estimated parameters. b) Both relationships can be estimated simultaneously by forming the regression matrices as x 11 x 12 .
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.

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Lecture31-2010 - Exceptions to Ordinary Least Squares:...

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