1. GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL The output above shows the result of regressing EARNINGS, hourly earnings in dollars, on S, years of schooling, and EXP, years of work experience. 1. reg EARNINGS S EXP Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 2, 537) = 67.54 Model | 22513.6473 2 11256.8237 Prob > F = 0.0000 Residual | 89496.5838 537 166.660305 R-squared = 0.2010 -------------+------------------------------ Adj R-squared = 0.1980 Total | 112010.231 539 207.811189 Root MSE = 12.91 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.678125.2336497 11.46 0.000 2.219146 3.137105 EXP |.5624326.1285136 4.38 0.000.3099816.8148837 _cons | -26.48501 4.27251 -6.20 0.000 -34.87789 -18.09213 ------------------------------------------------------------------------------

2. 2 Suppose that you were particularly interested in the relationship between EARNINGS and S and wished to represent it graphically, using the sample data. GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL

3. 3 A simple plot would be misleading. GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL

4. 4 Schooling is negatively correlated with work experience. The plot fails to take account of this, and as a consequence the regression line underestimates the impact of schooling on earnings. GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL. cor S EXP (obs=540) | S ASVABC --------+------------------ S| 1.0000 EXP| -0.2179 1.0000

5. 5 We will investigate the distortion mathematically when we come to omitted variable bias. GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL. cor S EXP (obs=540) | S ASVABC --------+------------------ S| 1.0000 EXP| -0.2179 1.0000

6. 6 To eliminate the distortion, you purge both EARNINGS and S of their components related to EXP and then draw a scatter diagram using the purged variables. GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL. cor S EXP (obs=540) | S ASVABC --------+------------------ S| 1.0000 EXP| -0.2179 1.0000

7. . reg EARNINGS EXP Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 2.98 Model | 617.717488 1 617.717488 Prob > F = 0.0847 Residual | 111392.514 538 207.049282 R-squared = 0.0055 -------------+------------------------------ Adj R-squared = 0.0037 Total | 112010.231 539 207.811189 Root MSE = 14.389 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- EXP |.2414715.1398002 1.73 0.085 -.0331497.5160927 _cons | 15.55527 2.442468 6.37 0.000 10.75732 20.35321 ------------------------------------------------------------------------------. predict EEARN, resid 7 We start by regressing EARNINGS on EXP, as shown above. The residuals are the part of EARNINGS which is not related to EXP. The ‘predict’ command is the Stata command for saving the residuals from the most recent regression. We name them EEARN. GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL

8. . reg S EXP Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 26.82 Model | 152.160205 1 152.160205 Prob > F = 0.0000 Residual | 3052.82313 538 5.67439243 R-squared = 0.0475 -------------+------------------------------ Adj R-squared = 0.0457 Total | 3204.98333 539 5.94616574 Root MSE = 2.3821 ------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- EXP | -.1198454.0231436 -5.18 0.000 -.1653083 -.0743826 _cons | 15.69765.4043447 38.82 0.000 14.90337 16.49194 ------------------------------------------------------------------------------. predict ES, resid 8 We do the same with S. We regress it on EXP and save the residuals as ES. GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL

9. 9 Now we plot EEARN on ES and the scatter is a faithful representation of the relationship, both in terms of the slope of the trend line (the red line) and in terms of the variation about that line. GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL

10. 10 As you would expect, the trend line is steeper that in scatter diagram which did not control for EXP (reproduced here as the black dashed line). GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL

11. 11 Here is the regression of EEARN on ES. GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL. reg EEARN ES Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 131.63 Model | 21895.9298 1 21895.9298 Prob > F = 0.0000 Residual | 89496.5833 538 166.350527 R-squared = 0.1966 -------------+------------------------------ Adj R-squared = 0.1951 Total | 111392.513 539 206.665145 Root MSE = 12.898 ------------------------------------------------------------------------------ EEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ES | 2.678125.2334325 11.47 0.000 2.219574 3.136676 _cons | 8.10e-09.5550284 0.00 1.000 -1.090288 1.090288 ------------------------------------------------------------------------------

12. 12 A mathematical proof that the technique works requires matrix algebra. We will content ourselves by verifying that the estimate of the slope coefficient is the same as in the multiple regression. GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL. reg EEARN ES Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 131.63 Model | 21895.9298 1 21895.9298 Prob > F = 0.0000 Residual | 89496.5833 538 166.350527 R-squared = 0.1966 -------------+------------------------------ Adj R-squared = 0.1951 Total | 111392.513 539 206.665145 Root MSE = 12.898 ------------------------------------------------------------------------------ EEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ES | 2.678125.2334325 11.47 0.000 2.219574 3.136676 _cons | 8.10e-09.5550284 0.00 1.000 -1.090288 1.090288 ------------------------------------------------------------------------------. reg EARNINGS S EXP ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.678125.2336497 11.46 0.000 2.219146 3.137105 EXP |.5624326.1285136 4.38 0.000.3099816.8148837 _cons | -26.48501 4.27251 -6.20 0.000 -34.87789 -18.09213 ------------------------------------------------------------------------------ From multiple regression:. reg EEARN ES Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 131.63 Model | 21895.9298 1 21895.9298 Prob > F = 0.0000 Residual | 89496.5833 538 166.350527 R-squared = 0.1966 -------------+------------------------------ Adj R-squared = 0.1951 Total | 111392.513 539 206.665145 Root MSE = 12.898 ------------------------------------------------------------------------------ EEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ES | 2.678125.2334325 11.47 0.000 2.219574 3.136676 _cons | 8.10e-09.5550284 0.00 1.000 -1.090288 1.090288 ------------------------------------------------------------------------------

13. 13 Finally, a small and not very important technical point. You may have noticed that the standard error and t statistic do not quite match. The reason for this is that the number of degrees of freedom is overstated by 1 in the residuals regression. GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL. reg EEARN ES Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 131.63 Model | 21895.9298 1 21895.9298 Prob > F = 0.0000 Residual | 89496.5833 538 166.350527 R-squared = 0.1966 -------------+------------------------------ Adj R-squared = 0.1951 Total | 111392.513 539 206.665145 Root MSE = 12.898 ------------------------------------------------------------------------------ EEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ES | 2.678125.2334325 11.47 0.000 2.219574 3.136676 _cons | 8.10e-09.5550284 0.00 1.000 -1.090288 1.090288 ------------------------------------------------------------------------------. reg EARNINGS S EXP ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.678125.2336497 11.46 0.000 2.219146 3.137105 EXP |.5624326.1285136 4.38 0.000.3099816.8148837 _cons | -26.48501 4.27251 -6.20 0.000 -34.87789 -18.09213 ------------------------------------------------------------------------------ From multiple regression:. reg EEARN ES Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 131.63 Model | 21895.9298 1 21895.9298 Prob > F = 0.0000 Residual | 89496.5833 538 166.350527 R-squared = 0.1966 -------------+------------------------------ Adj R-squared = 0.1951 Total | 111392.513 539 206.665145 Root MSE = 12.898 ------------------------------------------------------------------------------ EEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ES | 2.678125.2334325 11.47 0.000 2.219574 3.136676 _cons | 8.10e-09.5550284 0.00 1.000 -1.090288 1.090288 ------------------------------------------------------------------------------

14. 14 That regression has not made allowance for the fact that we have already used up 1 degree of freedom in removing EXP from the model. GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL. reg EEARN ES Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 131.63 Model | 21895.9298 1 21895.9298 Prob > F = 0.0000 Residual | 89496.5833 538 166.350527 R-squared = 0.1966 -------------+------------------------------ Adj R-squared = 0.1951 Total | 111392.513 539 206.665145 Root MSE = 12.898 ------------------------------------------------------------------------------ EEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ES | 2.678125.2334325 11.47 0.000 2.219574 3.136676 _cons | 8.10e-09.5550284 0.00 1.000 -1.090288 1.090288 ------------------------------------------------------------------------------. reg EARNINGS S EXP ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.678125.2336497 11.46 0.000 2.219146 3.137105 EXP |.5624326.1285136 4.38 0.000.3099816.8148837 _cons | -26.48501 4.27251 -6.20 0.000 -34.87789 -18.09213 ------------------------------------------------------------------------------ From multiple regression:. reg EEARN ES Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 131.63 Model | 21895.9298 1 21895.9298 Prob > F = 0.0000 Residual | 89496.5833 538 166.350527 R-squared = 0.1966 -------------+------------------------------ Adj R-squared = 0.1951 Total | 111392.513 539 206.665145 Root MSE = 12.898 ------------------------------------------------------------------------------ EEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ES | 2.678125.2334325 11.47 0.000 2.219574 3.136676 _cons | 8.10e-09.5550284 0.00 1.000 -1.090288 1.090288 ------------------------------------------------------------------------------

15. Copyright Christopher Dougherty 2012. 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 3.2 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 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 EC2020 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2012.10.28