Lecture 12 Properties of OLS in the multiple regression model

Lecture 12 Properties of OLS in the multiple regression model

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 17

Lecture 12 Properties of OLS in the multiple regression model

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online