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

This preview shows pages 1–8. Sign up to view the full content.

1 Lecture 5 Stock and Watson, Chapter 7 Multiple regression: Inference and Model specification

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

View Full Document
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).
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

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

View Full Document
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
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%)!

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

View Full Document
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
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 .

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 26

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

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online