Lecture 11 Prof. Arkonac's Slides (Ch 7.3 - 7.6) for ECO 4000

Lecture 11 Prof. - Multiple Regression(cont Confidence Intervals in Multiple Regression for ECO 4000 Statistical Analysis for Economics and Finance

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Multiple Regression (cont’) Confidence Intervals in Multiple Regression for ECO 4000, Statistical Analysis for Economics and Finance Fall 2010 Lecture 11 Prof: Seyhan Arkonac, PhD 1
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2 The F -statistic testing 1 and 2 : F = 12 22 1 2 , 1 2 2 , ˆ 2 1 ˆ 21 tt t t t t     The F -statistic is large when t 1 and/or t 2 is large The F -statistic corrects (in just the right way) for the correlation between t 1 and t 2 . The formula for more than two ’s is nasty unless you use matrix algebra. This gives the F -statistic a nice large-sample approximate distribution, which is…
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3 Computing the p-value using the F-statistic : p -value = tail probability of the 2 q / q distribution beyond the F -statistic actually computed. Implementation in STATA Use the “test” command after the regression Example: Test the joint hypothesis that the population coefficients on STR and expenditures per pupil ( expn_stu ) are both zero, against the alternative that at least one of the population coefficients is nonzero.
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4 F-test example, California class size data: reg testscr str expn_stu pctel, r; Regression with robust standard errors Number of obs = 420 F( 3, 416) = 147.20 Prob > F = 0.0000 R-squared = 0.4366 Root MSE = 14.353 ------------------------------------------------------------------------------ | Robust testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- str | -.2863992 .4820728 -0.59 0.553 -1.234001 .661203 expn_stu | .0038679 .0015807 2.45 0.015 .0007607 .0069751 pctel | -.6560227 .0317844 -20.64 0.000 -.7185008 -.5935446 _cons | 649.5779 15.45834 42.02 0.000 619.1917 679.9641 ------------------------------------------------------------------------------ NOTE test str expn_stu; The test command follows the regression ( 1) str = 0.0 There are q=2 restrictions being tested ( 2) expn_stu = 0.0 F( 2, 416) = 5.43 The 5% critical value for q=2 is 3.00 Prob > F = 0.0047 Stata computes the p-value for you
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5 More on F -statistics: a simple F-statistic formula that is easy to understand (it is only valid if the errors are homoskedastic, but it might help intuition). The homoskedasticity-only F -statistic When the errors are homoskedastic, there is a simple formula for computing the “homoskedasticity-only” F -statistic: Run two regressions, one under the null hypothesis (the “restricted” regression) and one under the alternative hypothesis (the “unrestricted” regression). Compare the fits of the regressions – the R 2 ’s – if the “unrestricted” model fits sufficiently better, reject the null
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6 The “restricted” and “unrestricted” regressions Example : are the coefficients on STR and Expn zero? Unrestricted population regression (under H 1 ): TestScore i = 0 + 1 STR i + 2 Expn i + 3 PctEL i + u i Restricted population regression (that is, under H 0 ): TestScore i = 0 + 3 PctEL i + u i ( why ?) The number of restrictions under H 0 is q = 2 ( why ?). The fit will be better ( R 2 will be higher) in the unrestricted regression ( why ?) By how much must the R 2 increase for the coefficients on Expn and STR to be judged statistically significant?
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7 Simple formula for the homoskedasticity-only F-statistic: F = 22 2 ( )/ (1 )/( 1) unrestricted restricted unrestricted unrestricted R R q R n k where: 2 restricted R = the R 2 for the restricted regression 2 unrestricted R = the R 2 for the unrestricted regression q = the number of restrictions under the null k unrestricted
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This note was uploaded on 05/05/2011 for the course ECON 4000 taught by Professor Arkonac during the Spring '11 term at CUNY Baruch.

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Lecture 11 Prof. - Multiple Regression(cont Confidence Intervals in Multiple Regression for ECO 4000 Statistical Analysis for Economics and Finance

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