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Lecture 14 Hypothesis testing in multiple regression model, Part 2

# Lecture 14 Hypothesis testing in multiple regression model, Part 2

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Economics 326 Methods of Empirical Research in Economics Lecture 14: Hypothesis testing in the multiple regression model, Part 2 Vadim Marmer University of British Columbia May 5, 2010

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Multiple restrictions I Consider the model: ln ( Wage i ) = β 0 + β 1 Experience i + β 2 Experience 2 i + + β 3 PrevExperience i + β 4 PrevExperience 2 i + β 5 Education i + U i , where Experience is the experience at current job, and PrevExperience is the previous experience. I Suppose that we want to test the null hypothesis that, after controlling for the experience at current job and education, the previous experience has no e/ect on wage: H 0 : β 3 = 0 , β 4 = 0 . I We have two restrictions on the model parameters. I The alternative hypothesis is that at least one of the coe¢ cients, β 3 or β 4 , is di/erent from zero: H 1 : β 3 6 = 0 or β 4 6 = 0 . 1/23
t -statistics and multiple restrictions I Let T 3 and T 4 be the t -statistics associated with the coe¢ cients of PrevExperience and PrevExperience 2 : T 3 = ˆ β 3 SE ° ˆ β 3 ± and T 4 = ˆ β 4 SE ° ˆ β 4 ± . I We can use T 3 and T 4 to test signi°cance of β 3 and β 4 separately by using two separate size α tests: I Reject H 0 , 3 : β 3 = 0 in favor of H 1 , 3 : β 3 6 = 0 when j T 3 j > t n ° k ° 1 , 1 ° α / 2 . I Reject H 0 , 4 : β 4 = 0 in favor of H 1 , 4 : β 4 6 = 0 when j T 4 j > t n ° k ° 1 , 1 ° α / 2 . 2/23

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t -statistics and multiple restrictions I Rejecting H 0 : β 3 = 0 , β 4 = 0 in favor of H 1 : β 3 6 = 0 or β 4 6 = 0 when at least one of the two coe¢ cients is signi°cant at level α , i.e. when j T 3 j > t n ° k ° 1 , 1 ° α / 2 or j T 4 j > t n ° k ° 1 , 1 ° α / 2 , is not a size α test! I Recall that if A and B are two sets then ( A \ B ) ± A and therefore P ( A \ B ) ² P ( A ) . I When β 3 = β 4 = 0 : P ( Reject H 0 , 3 or H 0 , 4 ) = = P ( j T 3 j > t n ° k ° 1 , 1 ° α / 2 or j T 4 j > t n ° k ° 1 , 1 ° α / 2 ) = P ( j T 3 j > t n ° k ° 1 , 1 ° α / 2 ) + P ( j T 4 j > t n ° k ° 1 , 1 ° α / 2 ) ° P ( j T 3 j > t n ° k ° 1 , 1 ° α / 2 and j T 4 j > t n ° k ° 1 , 1 ° α / 2 ) = α + α ° P ( j T 3 j > t n ° k ° 1 , 1 ° α / 2 and j T 4 j > t n ° k ° 1 , 1 ° α / 2 ) ³ α . 3/23
Testing multiple exclusion restrictions I Consider the model Y i = β 0 + β 1 X 1 , i + . . . + β q X q , i + β q + 1 X q + 1 , i + . . . + β k X k , i + U i . Suppose that we want to test that the °rst q regressors have no e/ect on Y (after controlling for other regressors).

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