Lecture 12 Properties of OLS in the multiple regression model

Lecture 12 Properties of OLS in the multiple regression model

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Economics 326 Methods of Empirical Research in Economics Lecture 12: Properties of OLS in the multiple regression model Vadim Marmer University of British Columbia March 3, 2009

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Multiple regression and OLS I Consider the multiple regression model with k regressors: Y i = β 0 + β 1 X 1 , i + β 2 X 2 , i + . . . + β k X k , i + U i . I Let ˆ β 0 , ˆ β 1 , . . . , ˆ β k be the OLS estimators: if ˆ U i = Y i ˆ β 0 ˆ β 1 X 1 , i ˆ β 2 X 2 , i . . . ˆ β k X k , i , then n i = 1 ˆ U i = n i = 1 X 1 , i ˆ U i = . . . = n i = 1 X k , i ˆ U i = 0 . 1/16
Multiple regression and OLS I As in Lecture 10, we can write ˆ β 1 as ˆ β 1 = n i = 1 ˜ X 1 , i Y i n i = 1 ˜ X 2 1 , i , where I ˜ X 1 , i ˜ X 1 , i = X 1 , i ˆ γ 0 ˆ γ 2 X 2 , i . . . ˆ γ k X k , i . I ˆ γ 0 , ˆ γ 2 , . . . , ˆ γ k are the OLS coe¢ cients: n i = 1 ˜ X 1 , i = n i = 1 ˜ X 1 , i X 2 , i = . . . = n i = 1 ˜ X 1 , i X k , i = 0 . I Similarly, we can write ˆ β 2 as ˆ β 2 = n i = 1 ˜ X 2 , i Y i n i = 1 ˜ X 2 2 , i , where I ˜ X 2 , i ˜ X 2 , i = X 2 , i ˆ δ 0 ˆ δ 1 X 1 , i ˆ δ 3 X 3 , i . . . ˆ δ k X k , i . I ˆ δ 0 , ˆ δ 1 , ˆ δ 3 , . . . , ˆ δ k are the OLS coe¢ cients: n i = 1 ˜ X 2 , i = n i = 1 ˜ X 2 , i X 1 , i = n i = 1 ˜ X 2 , i X 3 , i = . . . = n i = 1 ˜ X 2 , i X k , i = 0. 2/16

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The OLS estimators are linear I Consider ˆ β 1 : ˆ β 1 = n i = 1 ˜ X 1 , i Y i n i = 1 ˜ X 2 1 , i = n i = 1 ˜ X 1 , i n l = 1 ˜ X 2 1 , l Y i = n i = 1 w 1 , i Y i , where w 1 , i = ˜ X 1 , i n l = 1 ˜ X 2 1 , l . I Recall that ˜ X 1 are the residuals from a regression of X 1 against X 2 , . . . , X k and a constant, and therefore w 1 , i depends only on X 3/16
Unbiasedness I Suppose that 1. Y i = β 0 + β 1 X 1 , i + β 2 X 2 , i + . . . + β k X k , i + U i . 2. Conditional on

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This note was uploaded on 03/24/2012 for the course ECON 326 taught by Professor Whisler during the Spring '10 term at The University of British Columbia.

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Lecture 12 Properties of OLS in the multiple regression model

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