1. Christopher Dougherty EC220 - Introduction to econometrics (chapter 3) Slideshow: precision of the multiple regression coefficients Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 3). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/129/http://learningresources.lse.ac.uk/129/ 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/

2. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS This sequence investigates the variances and standard errors of the slope coefficients in a model with two explanatory variables. 1

3. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 2 The expression for the variance of b 2 is shown above. The expression for the variance of b 3 is the same, with the subscripts 2 and 3 interchanged.

4. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS The first factor in the expression is identical to that for the variance of the slope coefficient in a simple regression model. 3

5. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 4 The variance of b 2 depends on the variance of the disturbance term, the number of observations, and the mean square deviation of X 2 for exactly the same reasons as in a simple regression model.

6. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS The difference is that in multiple regression analysis the expression is multiplied by a factor which depends on the correlation between X 2 and X 3. 5

7. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 6 The higher is the correlation between the explanatory variables, positive or negative, the greater will be the variance.

8. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 7 This is easy to understand intuitively. The greater the correlation, the harder it is to discriminate between the effects of the explanatory variables on Y, and the less accurate will be the regression estimates.

9. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 8 Note that the variance expression above is valid only for a model with two explanatory variables. When there are more than two, the expression becomes much more complex and it is sensible to switch to matrix algebra.

10. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 9 The standard deviation of the distribution of b 2 is of course given by the square root of its variance. standard deviation of

11. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 10 With the exception of the variance of u, we can calculate the components of the standard deviation from the sample data. standard deviation of

12. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 11 The variance of u has to be estimated. The mean square of the residuals provides a consistent estimator, but it is biased downwards by a factor (n – k) / n, where k is the number of parameters, in a finite sample. standard deviation of

13. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS Obviously we can obtain an unbiased estimator by dividing the sum of the squares of the residuals by n – k instead of n. We denote this unbiased estimator s u. 12 2 standard deviation of

14. 13 PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS Thus the estimate of the standard deviation of the probability distribution of b 2, known as the standard error of b 2 for short, is given by the expression above. standard deviation of

15. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS We will use this expression to analyze why the standard error of S is larger for the union subsample than for the non-union subsample in earnings function regressions using Data Set 21. 14. reg EARNINGS S EXP if COLLBARG==1 Source | SS df MS Number of obs = 101 -------------+------------------------------ F( 2, 98) = 9.72 Model | 3076.31726 2 1538.15863 Prob > F = 0.0001 Residual | 15501.9762 98 158.18343 R-squared = 0.1656 -------------+------------------------------ Adj R-squared = 0.1486 Total | 18578.2934 100 185.782934 Root MSE = 12.577 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.333846.5492604 4.25 0.000 1.243857 3.423836 EXP |.2235095.3389455 0.66 0.511 -.4491169.8961358 _cons | -15.12427 11.38141 -1.33 0.187 -37.71031 7.461779 ------------------------------------------------------------------------------

16. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 15. reg EARNINGS S EXP if COLLBARG==1 Source | SS df MS Number of obs = 101 -------------+------------------------------ F( 2, 98) = 9.72 Model | 3076.31726 2 1538.15863 Prob > F = 0.0001 Residual | 15501.9762 98 158.18343 R-squared = 0.1656 -------------+------------------------------ Adj R-squared = 0.1486 Total | 18578.2934 100 185.782934 Root MSE = 12.577 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.333846.5492604 4.25 0.000 1.243857 3.423836 EXP |.2235095.3389455 0.66 0.511 -.4491169.8961358 _cons | -15.12427 11.38141 -1.33 0.187 -37.71031 7.461779 ------------------------------------------------------------------------------ To select a subsample in Stata, you add an ‘if’ statement to a command. The COLLBARG variable is equal to 1 for respondents whose rates of pay are determined by collective bargaining, and it is 0 for the others.

17. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 16. reg EARNINGS S EXP if COLLBARG==1 Source | SS df MS Number of obs = 101 -------------+------------------------------ F( 2, 98) = 9.72 Model | 3076.31726 2 1538.15863 Prob > F = 0.0001 Residual | 15501.9762 98 158.18343 R-squared = 0.1656 -------------+------------------------------ Adj R-squared = 0.1486 Total | 18578.2934 100 185.782934 Root MSE = 12.577 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.333846.5492604 4.25 0.000 1.243857 3.423836 EXP |.2235095.3389455 0.66 0.511 -.4491169.8961358 _cons | -15.12427 11.38141 -1.33 0.187 -37.71031 7.461779 ------------------------------------------------------------------------------ Note that in tests for equality, Stata requires the = sign to be duplicated.

18. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 17. reg EARNINGS S EXP if COLLBARG==1 Source | SS df MS Number of obs = 101 -------------+------------------------------ F( 2, 98) = 9.72 Model | 3076.31726 2 1538.15863 Prob > F = 0.0001 Residual | 15501.9762 98 158.18343 R-squared = 0.1656 -------------+------------------------------ Adj R-squared = 0.1486 Total | 18578.2934 100 185.782934 Root MSE = 12.577 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.333846.5492604 4.25 0.000 1.243857 3.423836 EXP |.2235095.3389455 0.66 0.511 -.4491169.8961358 _cons | -15.12427 11.38141 -1.33 0.187 -37.71031 7.461779 ------------------------------------------------------------------------------ In the case of the union subsample, the standard error of S is 0.5493.

19. . reg EARNINGS S EXP if COLLBARG==0 Source | SS df MS Number of obs = 439 -------------+------------------------------ F( 2, 436) = 57.77 Model | 19540.1761 2 9770.08805 Prob > F = 0.0000 Residual | 73741.593 436 169.132094 R-squared = 0.2095 -------------+------------------------------ Adj R-squared = 0.2058 Total | 93281.7691 438 212.972076 Root MSE = 13.005 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.721698.2604411 10.45 0.000 2.209822 3.233574 EXP |.6077342.1400846 4.34 0.000.3324091.8830592 _cons | -28.00805 4.643211 -6.03 0.000 -37.13391 -18.88219 ------------------------------------------------------------------------------ PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 18 In the case of the non-union subsample, the standard error of S is 0.2604, less than half as large.

20. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 19 We will explain the difference by looking at the components of the standard error. Decomposition of the standard error of S Component s u n MSD(S) r S, EXP s.e. Union12.577 1016.2325–0.40870.5493 Non-union13.005 4395.8666 – 0.17840.2604 Factor product Union12.5770.09950.40061.09570.5493 Non-union13.0050.04770.41291.01630.2603

21. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 20. reg EARNINGS S EXP if COLLBARG==1 Source | SS df MS Number of obs = 101 -------------+------------------------------ F( 2, 98) = 9.72 Model | 3076.31726 2 1538.15863 Prob > F = 0.0001 Residual | 15501.9762 98 158.18343 R-squared = 0.1656 -------------+------------------------------ Adj R-squared = 0.1486 Total | 18578.2934 100 185.782934 Root MSE = 12.577 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.333846.5492604 4.25 0.000 1.243857 3.423836 EXP |.2235095.3389455 0.66 0.511 -.4491169.8961358 _cons | -15.12427 11.38141 -1.33 0.187 -37.71031 7.461779 ------------------------------------------------------------------------------ We will start with s u. Here is RSS for the union subsample.

22. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 21. reg EARNINGS S EXP if COLLBARG==1 Source | SS df MS Number of obs = 101 -------------+------------------------------ F( 2, 98) = 9.72 Model | 3076.31726 2 1538.15863 Prob > F = 0.0001 Residual | 15501.9762 98 158.18343 R-squared = 0.1656 -------------+------------------------------ Adj R-squared = 0.1486 Total | 18578.2934 100 185.782934 Root MSE = 12.577 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.333846.5492604 4.25 0.000 1.243857 3.423836 EXP |.2235095.3389455 0.66 0.511 -.4491169.8961358 _cons | -15.12427 11.38141 -1.33 0.187 -37.71031 7.461779 ------------------------------------------------------------------------------ There are 101 observations in the non-union subsample. k is equal to 3. Thus n – k is equal to 98.

23. PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS RSS / (n – k) is equal to 158.183. To obtain s u, we take the square root. This is 12.577. 22. reg EARNINGS S EXP if COLLBARG==1 Source | SS df MS Number of obs = 101 -------------+------------------------------ F( 2, 98) = 9.72 Model | 3076.31726 2 1538.15863 Prob > F = 0.0001 Residual | 15501.9762 98 158.18343 R-squared = 0.1656 -------------+------------------------------ Adj R-squared = 0.1486 Total | 18578.2934 100 185.782934 Root MSE = 12.577 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.333846.5492604 4.25 0.000 1.243857 3.423836 EXP |.2235095.3389455 0.66 0.511 -.4491169.8961358 _cons | -15.12427 11.38141 -1.33 0.187 -37.71031 7.461779 ------------------------------------------------------------------------------

24. Decomposition of the standard error of S Component s u n MSD(S) r S, EXP s.e. Union12.577 1016.2325–0.40870.5493 Non-union13.005 4395.8666 – 0.17840.2604 Factor product Union12.5770.09950.40061.09570.5493 Non-union13.0050.04770.41291.01630.2603 PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 23 We place this in the table, along with the number of observations.

25. 24 PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS Similarly, in the case of the non-union subsample, s u is the square root of 169.132, which is 13.005. We also note that the number of observations in that subsample is 439.. reg EARNINGS S EXP if COLLBARG==0 Source | SS df MS Number of obs = 439 -------------+------------------------------ F( 2, 436) = 57.77 Model | 19540.1761 2 9770.08805 Prob > F = 0.0000 Residual | 73741.593 436 169.132094 R-squared = 0.2095 -------------+------------------------------ Adj R-squared = 0.2058 Total | 93281.7691 438 212.972076 Root MSE = 13.005 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.721698.2604411 10.45 0.000 2.209822 3.233574 EXP |.6077342.1400846 4.34 0.000.3324091.8830592 _cons | -28.00805 4.643211 -6.03 0.000 -37.13391 -18.88219 ------------------------------------------------------------------------------

26. Decomposition of the standard error of S Component s u n MSD(S) r S, EXP s.e. Union12.577 1016.2325–0.40870.5493 Non-union13.005 4395.8666 – 0.17840.2604 Factor product Union12.5770.09950.40061.09570.5493 Non-union13.0050.04770.41291.01630.2603 PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 25 We place these in the table.

27. Decomposition of the standard error of S Component s u n MSD(S) r S, EXP s.e. Union12.577 1016.2325–0.40870.5493 Non-union13.005 4395.8666 – 0.17840.2604 Factor product Union12.5770.09950.40061.09570.5493 Non-union13.0050.04770.41291.01630.2603 PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 26 We calculate the mean square deviation of S for the two subsamples from the sample data.

28. . cor S EXP if COLLBARG==1 (obs=101) | S EXP --------+------------------ S | 1.0000 EXP | -0.4087 1.0000. cor S EXP if COLLBARG==0 (obs=439) | S EXP --------+------------------ S | 1.0000 EXP | -0.1784 1.0000 27 PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS The correlation coefficients for S and EXP are –0.4087 and – 0.1784 for the union and non- union subsamples, respectively. (Note that "cor" is the Stata command for computing correlations.)

29. Decomposition of the standard error of S Component s u n MSD(S) r S, EXP s.e. Union12.577 1016.2325–0.40870.5493 Non-union13.005 4395.8666–0.17840.2604 Factor product Union12.5770.09950.40061.09570.5493 Non-union13.0050.04770.41291.01630.2603 PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 28 These entries complete the top half of the table. We will now look at the impact of each item on the standard error, using the mathematical expression at the top.

30. Decomposition of the standard error of S Component s u n MSD(S) r S, EXP s.e. Union12.577 1016.2325–0.40870.5493 Non-union13.005 4395.8666 – 0.17840.2604 Factor product Union12.5770.09950.40061.09570.5493 Non-union13.0050.04770.41291.01630.2603 PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 29 The s u components need no modification. It is a little larger for the non-union subsample, having an adverse effect on the standard error.

31. Decomposition of the standard error of S Component s u n MSD(S) r S, EXP s.e. Union12.577 1016.2325–0.40870.5493 Non-union13.005 4395.8666 – 0.17840.2604 Factor product Union12.5770.09950.40061.09570.5493 Non-union13.0050.04770.41291.01630.2603 PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 30 The number of observations is much larger for the non-union subsample, so the second factor is much smaller than that for the union subsample.

32. Decomposition of the standard error of S Component s u n MSD(S) r S, EXP s.e. Union12.577 1016.2325–0.40870.5493 Non-union13.005 4395.8666 – 0.17840.2604 Factor product Union12.5770.09950.40061.09570.5493 Non-union13.0050.04770.41291.01630.2603 PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 31 Perhaps surprisingly, the variance in schooling is a little larger for the union subsample.

33. Decomposition of the standard error of S Component s u n MSD(S) r S, EXP s.e. Union12.577 1016.2325–0.40870.5493 Non-union13.005 4395.8666–0.17840.2604 Factor product Union12.5770.09950.40061.09570.5493 Non-union13.0050.04770.41291.01630.2603 PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 32 The correlation between schooling and work experience is greater for the union subsample, and this has an adverse effect on its standard error. Note that the sign of the correlation makes no difference since it is squared.

34. Decomposition of the standard error of S Component s u n MSD(S) r S, EXP s.e. Union12.577 1016.2325–0.40870.5493 Non-union13.005 4395.8666 – 0.17840.2604 Factor product Union12.5770.09950.40061.09570.5493 Non-union13.0050.04770.41291.01630.2603 PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 33 Multiplying the four factors together, we obtain the standard errors. (The discrepancy in the last digit of the non-union standard error has been caused by rounding error.)

35. Decomposition of the standard error of S Component s u n MSD(S) r S, EXP s.e. Union12.577 1016.2325–0.40870.5493 Non-union13.005 4395.8666 – 0.17840.2604 Factor product Union12.5770.09950.40061.09570.5493 Non-union13.0050.04770.41291.01630.2603 PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 34 We see that the reason that the standard error is smaller for the non-union subsample is that there are far more observations than in the non-union subsample. Otherwise the standard errors would have been about the same.

36. Decomposition of the standard error of S Component s u n MSD(S) r S, EXP s.e. Union12.577 1016.2325–0.40870.5493 Non-union13.005 4395.8666 – 0.17840.2604 Factor product Union12.5770.09950.40061.09570.5493 Non-union13.0050.04770.41291.01630.2603 PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS 35 The greater correlation between S and EXP has an adverse effect on the union standard error, but this is just about offset by the smaller s u and the larger variance of S.

37. 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 3.3 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