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# lecture_6_slides - 7-1 Outline 1. Hypothesis tests and...

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7-1 Hypothesis Tests and Confidence Intervals in Multiple Regression (SW Chapter 7) Outline 1. Hypothesis tests and confidence intervals for a single coefficient 2. Joint hypothesis tests on multiple coefficients 3. Other types of hypotheses involving multiple coefficients 4. How to decide what variables to include in a regression model?

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7-2 Hypothesis Tests and Confidence Intervals for a Single Coefficient in Multiple Regression (SW Section 7.1) 1 ˆ ( ) var( E β - is approximately distributed N (0,1) (CLT). Thus hypotheses on can be tested using the usual t statistic, and confidence intervals are constructed as { ± 1.96 × SE )}. So too for 2 ,…, k . and are generally not independently distributed – so neither are their -statistics (more on this later).
7-3 Example : The California class size data (1) · TestScore = 698.9 – 2.28 × STR (10.4) (0.52) (2) = 686.0 – 1.10 – 0.650 PctEL (8.7) (0.43) (0.031) The coefficient on in (2) is the effect on TestScores of a unit change in , holding constant the percentage of English Learners in the district falls by one-half The 95% confidence interval for coefficient on in (2) is {–1.10 ± 1.96 0.43} = (–1.95, –0.26) The t -statistic testing β = 0 is = –1.10/0.43 = –2.54, so we reject the hypothesis at the 5% significance level

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7-4 Standard errors in multiple regression in STATA reg testscr str pctel , robust; Regression with robust standard errors Number of obs = 420 F( 2, 417) = 223.82 Prob > F = 0.0000 R-squared 0.4264 Root MSE 14.464 ------------------------------------------------------------------------------ | Robust testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- str | -1.101296 .4328472 -2.54 0.011 -1.95213 -.2504616 pctel | -.6497768 .0310318 -20.94 0.000 -.710775 -.5887786 _cons | 686.0322 8.728224 78.60 668.8754 703.189 ------------------------------------------------------------------------------ · TestScore = 686.0 – 1.10 × STR – 0.650 PctEL (8.7) ( 0.43 ) 0.031) We use heteroskedasticity-robust standard errors – for exactly the same reason as in the case of a single regressor.
7-5 Tests of Joint Hypotheses (SW Section 7.2) Let Expn = expenditures per pupil and consider the population regression model: TestScore i = β 0 + 1 STR 2 3 PctEL u The null hypothesis that “school resources don’t matter,” and the alternative that they do, corresponds to: H : = 0 and vs. either 0 or or both

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7-6 Tests of joint hypotheses, ctd. H 0 : β 1 = 0 and 2 vs. either 0 or 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, = 2, and the two restrictions are and = 0. A “common sense” idea is to reject if either of the individual t -statistics exceeds 1.96 in absolute value.
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## This note was uploaded on 05/25/2011 for the course ECON 2007 taught by Professor J during the Spring '11 term at UCL.

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lecture_6_slides - 7-1 Outline 1. Hypothesis tests and...

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