Lecture_6 - Lecture 5 Stock and Watson Chapter 7 Multiple...

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1 Lecture 5 Stock and Watson, Chapter 7 Multiple regression: Inference and Model specification
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2 Hypothesis Tests and Confidence Intervals for a Single Coefficient in Multiple Regression 11 1 ˆˆ () ˆ var( ) E is approximately distributed N (0,1) (CLT). Thus hypotheses on 1 can be tested using the usual t -statistic, and confidence intervals are constructed as { 1 ˆ 1.96 SE ( 1 ˆ )}. So too for 2 ,…, k . 1 ˆ and 2 ˆ are generally not independently distributed – so neither are their t -statistics (more on this later).
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3 Example : The California class size data (1) TestScore = 698.9 – 2.28 STR (10.4) (0.52) (2) TestScore = 686.0 – 1.10 STR – 0.650 PctEL (8.7) (0.43) (0.031) The coefficient on STR in (2) is the effect on TestScores of a unit change in STR , holding constant the percentage of English Learners in the district The coefficient on STR falls by one-half The 95% confidence interval for coefficient on STR in (2) is {– 1.10 1.96 0.43} = (–1.95, –0.26) The t -statistic testing STR = 0 is t = –1.10/0.43 = –2.54, so we reject the hypothesis at the 5% significance level
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4 Tests of Joint Hypotheses (SW Section 7.2) Let Expn = expenditures per pupil and consider the population regression model: TestScore i = 0 + 1 STR i + 2 Expn i + 3 PctEL i + u i The null hypothesis that “school resources don’t matter,” and the alternative that they do, corresponds to: H 0 : 1 = 0 and 2 = 0 vs. H 1 : either 1 0 or 2 0 or both TestScore i = 0 + 1 STR i + 2 Expn i + 3 PctEL i + u i
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5 Tests of joint hypotheses, ctd. H 0 : 1 = 0 and 2 = 0 vs. H 1 : either 1 0 or 2 0 or both A joint hypothesis specifies a value for two or more coefficients, that is, it imposes a restriction on two or more coefficients. In general, a joint hypothesis will involve q restrictions. In the example above, q = 2, and the two restrictions are 1 = 0 and 2 = 0. A “common sense” idea is to reject if either of the individual t -statistics exceeds 1.96 in absolute value. But this “one at a time” test isn’t valid: the resulting test rejects too often under the null hypothesis (more than 5%)!
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6 The size of a test is the actual rejection rate under the null hypothesis. The size of the “common sense” test isn’t 5%! In fact, its size depends on the correlation between t 1 and t 2 (and thus on the correlation between 1 ˆ and 2 ˆ ). Solution: F - statistic
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7 The F -statistic The F -statistic tests all parts of a joint hypothesis at once. Formula for the special case of the joint hypothesis 1 = 1,0 and 2 = 2,0 in a regression with two regressors: F = 12 22 1 2 , 1 2 2 , ˆ 2 1 ˆ 21 tt t t t t where , ˆ estimates the correlation between t 1 and t 2 .
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Lecture_6 - Lecture 5 Stock and Watson Chapter 7 Multiple...

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