103lect10hand6

103lect10hand6 - Dummy Variables in MR - VIII We also may...

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Dummy Variables in MR - VIII We also may be interested in the effects of multiple sets of categorical variables, e.g. the effects of both the sex of an individual, and their ethnicity. In this case, there needs to be an “excluded” group for each set of categories, e.g. – Ethnicity: let’s choose “Other” as the excluded group • Hispanic i = 1 if observation i is Hispanic, Hispanic i = 0 otherwise •B l a c k i = 1 if observation i is Black, Black i = 0 otherwise – Sex: let’s choose “Male” as the excluded group •F em a l e i = 1 if observation i is Female, Female i = 0 otherwise What do these regression results say about average wages for Hispanic Females, Black Males, Other Males, etc………? n Wage = 17.40 6.08Hispanic 4.59Black 2.90Female (1.02) (1.10) (1.04) (0.97) ii i i −− Hypothesis Testing in MR • There are two types of hypothesis tests we can do in multiple regression. – 1) Hypothesis tests involving a single coefficient, e.g. a test whether β 5 =1, or a test whether β 3 =0. – 2) Hypothesis tests involving multiple coefficients, e.g. a test whether β 5 =2* β 3 , or a test whether β 1 = β 2 = β 4 =0. These are sometimes called joint hypothesis tests, because they test conditions on multiple coefficients jointly, or together. • Tests of the form 1) can be done using t STAT ’s, just like in basic regression analysis. • Tests of the form 2) will be done with a new statistic, the F-statistic, or F STAT . Hypothesis Tests for a Single Coefficient - I • Recall that under A1) – A4), each estimated parameter β j satisfies: • Note: STATA reports SE( β j ) for each coefficient. • Hence, we can test hypotheses regarding a single β j in the same way as before, i.e. • 1) State hypotheses and choose significance level, e.g. significance level = 0.05 H 0 : β j = c vs. H A : β j c () ˆ ˆˆ ,Var and 0,1 ˆ jj jjj j NN SE ββ βββ β ∼∼ Hypothesis Tests for a Single Coefficient - II • 2) Compute t STAT • 3) Compare t STAT to the appropriate critical value. Reject H 0 if the absolute value of the t STAT is greater than the critical value. • Just as before: – 1) We could alternatively do this test using the p-value associated with the t STAT – 2) STATA reports t STAT ’s and associated p-values for the tests that each coefficient equals zero. – 3) Confidence Intervals for β j can be formed as before, e.g. a 99% confidence interval is given by: ˆ ˆ j STAT j c t SE = ( )( ) ( ) 2.58 , 2.58 SE SE −+ Hypothesis Tests for a Single Coefficient - III Last note: When using dummy variables, remember that we had to choose an excluded group. Again, with PS2 data we had: Regression A (with “Other” as the excluded group) Regression B (with “Hispanic” as the excluded group) Again, these regressions are really exactly the same (they both say average wages for Others is 16.08, average wages for Blacks is 11.57, and average wages for Hispanics is 10.19) But because the variables are coded differently, the coefficients measure different aspects of the relationship, so the t STAT ’s on the coefficients test different things.
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103lect10hand6 - Dummy Variables in MR - VIII We also may...

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