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Christopher Dougherty EC220 - Introduction to econometrics (chapter 6) Slideshow: f test of a linear restriction Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 6). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/132/http://learningresources.lse.ac.uk/132/ Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms. http://creativecommons.org/licenses/by-sa/3.0/ http://creativecommons.org/licenses/by-sa/3.0/ http://learningresources.lse.ac.uk/

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1 F TEST OF A LINEAR RESTRICTION In the last sequence it was argued that educational attainment might be related to cognitive ability and family background, with mother's and father's educational attainment proxying for the latter.

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. reg S ASVABC SM SF Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686 -------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943 ------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ASVABC |.1257087.0098533 12.76 0.000.1063528.1450646 SM |.0492424.0390901 1.26 0.208 -.027546.1260309 SF |.1076825.0309522 3.48 0.001.04688.1684851 _cons | 5.370631.4882155 11.00 0.000 4.41158 6.329681 ------------------------------------------------------------------------------ 2 However, when we run the regression using Data Set 21, we find the coefficient of mother's education is not significant. F TEST OF A LINEAR RESTRICTION

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. reg S ASVABC SM SF Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686 -------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943 ------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ASVABC |.1257087.0098533 12.76 0.000.1063528.1450646 SM |.0492424.0390901 1.26 0.208 -.027546.1260309 SF |.1076825.0309522 3.48 0.001.04688.1684851 _cons | 5.370631.4882155 11.00 0.000 4.41158 6.329681 ------------------------------------------------------------------------------ 3 As was noted in one of the sequences for Chapter 3, this might be due to multicollinearity, because mother's education and father's education are correlated.. cor SM SF (obs=540) | SM SF --------+------------------ SM| 1.0000 SF| 0.6241 1.0000 F TEST OF A LINEAR RESTRICTION

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4 In the discussion of multicollinearity, several measures for alleviating the problem were suggested, among them the use of an appropriate theoretical restriction. F TEST OF A LINEAR RESTRICTION

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5 In particular, in the case of the present model, it was suggested that the impact of parental education might be the same for both parents, that is, that 3 and 4 might be equal. F TEST OF A LINEAR RESTRICTION

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6 If this is the case, the model may be rewritten as shown. We now have a total parental education variable, SP, instead of separate variables for mother’s and father’s education, and the multicollinearity caused by the correlation between the latter has been eliminated. F TEST OF A LINEAR RESTRICTION

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. g SP=SM+SF. reg S ASVABC SP Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 2, 537) = 156.04 Model | 1177.98338 2 588.991689 Prob > F = 0.0000 Residual | 2026.99996 537 3.77467403 R-squared = 0.3675 -------------+------------------------------ Adj R-squared = 0.3652 Total | 3204.98333 539 5.94616574 Root MSE = 1.9429 ------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ASVABC |.1253106.0098434 12.73 0.000.1059743.1446469 SP |.0828368.0164247 5.04 0.000.0505722.1151014 _cons | 5.29617.4817972 10.99 0.000 4.349731 6.242608 ------------------------------------------------------------------------------ 7 Here is the regression with SP replacing SM and SF. F TEST OF A LINEAR RESTRICTION

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. reg S ASVABC SM SF ------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ASVABC |.1257087.0098533 12.76 0.000.1063528.1450646 SM |.0492424.0390901 1.26 0.208 -.027546.1260309 SF |.1076825.0309522 3.48 0.001.04688.1684851 _cons | 5.370631.4882155 11.00 0.000 4.41158 6.329681 ------------------------------------------------------------------------------. reg S ASVABC SP ------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ASVABC |.1253106.0098434 12.73 0.000.1059743.1446469 SP |.0828368.0164247 5.04 0.000.0505722.1151014 _cons | 5.29617.4817972 10.99 0.000 4.349731 6.242608 ------------------------------------------------------------------------------ 8 A comparison of the regressions reveals that the standard error of the coefficient of SP is much smaller than those of SM and SF, and consequently its t statistic is higher. Its coefficient is a compromise between those of SM and SF, as might be expected. F TEST OF A LINEAR RESTRICTION

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. reg S ASVABC SM SF ------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ASVABC |.1257087.0098533 12.76 0.000.1063528.1450646 SM |.0492424.0390901 1.26 0.208 -.027546.1260309 SF |.1076825.0309522 3.48 0.001.04688.1684851 _cons | 5.370631.4882155 11.00 0.000 4.41158 6.329681 ------------------------------------------------------------------------------. reg S ASVABC SP ------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ASVABC |.1253106.0098434 12.73 0.000.1059743.1446469 SP |.0828368.0164247 5.04 0.000.0505722.1151014 _cons | 5.29617.4817972 10.99 0.000 4.349731 6.242608 ------------------------------------------------------------------------------ 9 However, the use of a restriction will lead to a gain in efficiency only if the restriction is valid. If it is not valid, its use will lead to biased coefficients and invalid standard errors and tests. F TEST OF A LINEAR RESTRICTION

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. reg S ASVABC SM SF ------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ASVABC |.1257087.0098533 12.76 0.000.1063528.1450646 SM |.0492424.0390901 1.26 0.208 -.027546.1260309 SF |.1076825.0309522 3.48 0.001.04688.1684851 _cons | 5.370631.4882155 11.00 0.000 4.41158 6.329681 ------------------------------------------------------------------------------. reg S ASVABC SP ------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ASVABC |.1253106.0098434 12.73 0.000.1059743.1446469 SP |.0828368.0164247 5.04 0.000.0505722.1151014 _cons | 5.29617.4817972 10.99 0.000 4.349731 6.242608 ------------------------------------------------------------------------------ 10 Do the coefficients of SM and SF in the unrestricted regression look as if they satisfy the restriction? Not really, in this case. The coefficient of SM is much smaller than that of SF, but then it should be noted that the standard errors are quite large. F TEST OF A LINEAR RESTRICTION

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. reg S ASVABC SM SF Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686 -------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943. reg S ASVABC SP Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 2, 537) = 156.04 Model | 1177.98338 2 588.991689 Prob > F = 0.0000 Residual | 2026.99996 537 3.77467403 R-squared = 0.3675 -------------+------------------------------ Adj R-squared = 0.3652 Total | 3204.98333 539 5.94616574 Root MSE = 1.9429 11 We will now perform a proper test. The imposition of a restriction makes it more difficult for the regression model to fit the data because there is one fewer parameter to adjust. There will therefore be an increase in RSS (and a decrease in R 2 ) when it is imposed. F TEST OF A LINEAR RESTRICTION

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. reg S ASVABC SM SF Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686 -------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943. reg S ASVABC SP Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 2, 537) = 156.04 Model | 1177.98338 2 588.991689 Prob > F = 0.0000 Residual | 2026.99996 537 3.77467403 R-squared = 0.3675 -------------+------------------------------ Adj R-squared = 0.3652 Total | 3204.98333 539 5.94616574 Root MSE = 1.9429 12 If the restriction is valid, the deterioration in the fit should be a small, random amount. However, if the restriction is invalid, the distortion caused by its imposition will lead to a significant deterioration in the fit. F TEST OF A LINEAR RESTRICTION

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. reg S ASVABC SM SF Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686 -------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943. reg S ASVABC SP Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 2, 537) = 156.04 Model | 1177.98338 2 588.991689 Prob > F = 0.0000 Residual | 2026.99996 537 3.77467403 R-squared = 0.3675 -------------+------------------------------ Adj R-squared = 0.3652 Total | 3204.98333 539 5.94616574 Root MSE = 1.9429 13 In the present case, we can see that the increase in RSS is very small, and hence we are unlikely to reject the restriction. F TEST OF A LINEAR RESTRICTION

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14 The null hypothesis is that the restriction is valid, and the alternative one is that it is invalid. F TEST OF A LINEAR RESTRICTION

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15 The test statistic is a member of the family of F tests where the numerator is the improvement in the fit on relaxing the restriction, divided by the cost of relaxing it (one degree of freedom, because one additional parameter has to be estimated). F TEST OF A LINEAR RESTRICTION

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16 The denominator of the test statistic is RSS after making the improvement (that is, RSS for the unrestricted model), divided by n – k, the number of degrees of freedom remaining. k is the number of parameters in the unrestricted model. F TEST OF A LINEAR RESTRICTION

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17 The F statistic is 0.90. An F statistic below 1 is never significant (look at the F table), so we do not reject H 0. The restriction appears to be valid. At least, it is not rejected by the data. F TEST OF A LINEAR RESTRICTION

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Copyright Christopher Dougherty 2011. These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 6.5 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course 20 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 11.07.25

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