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OLS Differentiation

# OLS Differentiation - The population regression is Y =X u...

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The population regression is Y = + u where x i = 2 6 6 4 x i 1 x i 2 ::: x ik 3 7 7 5 k 1 ;X = 2 6 6 4 x 0 1 x 0 2 ::: x 0 n 3 7 7 5 = 2 6 6 4 x 11 x 12 :::x 1 k x 21 x 22 :::x 2 k ::: x n 1 x n 2 ::: x nk 3 7 7 5 n k ;Y = 2 6 6 4 y 1 y 2 ::: y n 3 7 7 5 n 1 ; = 2 6 6 4 1 2 ::: k 3 7 7 5 k 1 ;u = 2 6 6 4 u 1 u 2 ::: u n 3 7 7 5 n 1 ;i = 1 ; 2 :::n i denotes the i th observation and k denotes the k th variable. By choosing to minimize ( Y ) 0 ( Y ) , we get b = ( X 0 X ) ± 1 X 0 Y = ( X 0 X ) ± 1 X 0 ( + u ) = + ( X 0 X ) ± 1 X 0 u = + ( X 0 X n ) ± 1 X 0 u n = + ( 1 n n X i =1 x i x 0 i ) ± 1 ( 1 n n X i =1 x i u i ) = + S ± 1 XX g where S XX = 1 n P n i =1 x i x 0 i and g = 1 n P n i =1 x i u i . So both S XX and g as sample means. When n is large, we can apply the law of large number. That is, 1 n P n i =1 x i x 0 i p E ( x i x 0 i ) ,and g p E ( g ) . Denote XX = E ( x i x 0 i ) . By central limit theorem, g d

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OLS Differentiation - The population regression is Y =X u...

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