Lecture 13 Hypothesis testing in multiple regression model, Part 1

Lecture 13 Hypothesis testing in multiple regression model, Part 1

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Unformatted text preview: Economics 326 Methods of Empirical Research in Economics Lecture 13: Hypothesis testing in the multiple regression model, Part 1 Vadim Marmer University of British Columbia May 4, 2009 The model I We consider the classical normal linear regression model: 1. Y i = + 1 X 1 , i + . . . + k X k , i + U i . 2. Conditional on X &s, E ( U i ) = 0 for all i &s. 3. Conditional on X &s, E & U 2 i = 2 for all i &s. 4. Conditional on X &s, E & U i U j = 0 for all i 6 = j . 5. Conditional on X &s, U i &s are jointly normally distributed. I We also continue to assume no perfect multicolinearity: The k regressors and constant do not form a perfect linear combination, i.e. we cannot nd constants c 1 , . . . , c k , c k + 1 (not all equal to zero) such that for all i &s: c 1 X 1 , i + . . . + c k X k , i + c k + 1 = . 1/15 Testing a hypothesis about a single coe cient I Take the j-th coe cient j , j 2 f , 1 , . . . , k g . I Under our assumptions, its OLS estimator j satis&es that conditional on X s: j & N & j , Var & j , where Var & j = 2 / n i = 1 X 2 j , i (see Lecture 12). I Therefore, & j j / r Var & j & N ( , 1 ) . I The conditional variance Var & j is unknown because 2 is unknown.The estimator for Var & j is d Var & j = s 2 n i = 1 X 2 j , i , where s 2 = n i = 1 U 2 i / ( n k 1 ) (see Lecture 11). 2/15 Testing a hypothesis about a single coe cient I We have that conditional on X &s, j & j r d Var & j t n & k & 1 . I Standard error: SE & j = r d Var & j = q s 2 / n i = 1 X 2 j , i . 3/15 Testing a hypothesis about a single coe cient: Two-sided alternatives I Consider testing H : j = j , against H 1 : j 6 = j , ....
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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 13 Hypothesis testing in multiple regression model, Part 1

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