Lecture 3

Lecture 3 - Econometrics Lecture 3 Properties of the OLS...

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Econometrics 1 Lecture 3: Properties of the OLS Estimator For the OLS estimator y X X X b ' ) ' ( 1 = define: Vector of OLS residuals = = = = K k k nk n K k k k b x y b x y Xb y e 1 1 1 1 #
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Econometrics 2 Vector of OLS fitted or predicted values = = = = K k k nk K k k k b x b x Xb y 1 1 1 ˆ #
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Econometrics 3 Residuals and fitted values have properties a. 0 ' = e X b. If the regression has an intercept b x y ' = with = = = = n i iK n i i x n x n x 1 1 1 1 ) 1 ( 1 # c. If the regression has an intercept y y ˆ =
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Econometrics 4 d. If the regression has an intercept ∑∑ = = = + = n i i n i n i i i e y y y y 1 2 2 11 2 ) ˆ ( ) (
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Econometrics 5 Proofs and remarks: Ad a. 0 ' = e X Proof: 0 ' ' ) ( ' ' = = = Xb X y X Xb y X e X ± This implies that each column of X is orthogonal to vector e , i.e. for K k , , 1 = = = n i i ik e x 1 0 If regression has intercept (and hence X has column equal to ι , a vector of 1-s), then = = = = n i i e e e 1 0 '
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Econometrics 6 The fundamental assumption in CLR model 0 ) | ( = X E ε implies that 0 ) ' ( = X E Hence OLS makes sample analog of ) ' ( X E equal to 0.
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Econometrics 7 Note My y X X X X I Xb y e = = = ) ' ) ' ( ( 1 with ' ) ' ( 1 X X X X I M = The matrix M has a number of properties 0 = MX M M = ' M is symmetric M M = 2 M is idempotent
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Econometrics 8 Also Py y X X X X Xb y = = = ' ) ' ( ˆ 1 with ' ) ' ( 1 X X X X P = Note P P = ' P is symmetric P P = 2 P is idempotent Because Xb y = ˆ the matrix P projects y on the space spanned by the columns of X . For this reason P
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This note was uploaded on 07/22/2008 for the course ECON 513 taught by Professor Rashidian during the Fall '07 term at USC.

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Lecture 3 - Econometrics Lecture 3 Properties of the OLS...

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