1. Confidence intervals were treated at length in the Review chapter and their application to regression analysis presents no problems. We will not repeat the graphical explanation. We will just provide the mathematical derivation in the context of a regression. 1 CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS Model Null hypothesis: Alternative hypothesis:

2. From the initial discussion in this section, we saw that, given the theoretical model Y =  1 +  2 X + u and a fitted model, the regression coefficient b 2 and the hypothetical value of  2 are incompatible if either of the inequalities shown is valid. 2 CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS Model Null hypothesis: Alternative hypothesis: Reject H 0 ifor

3. Multiplying through by the standard error of b 2, the conditions for rejecting H 0 can be written as shown. 3 CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS Model Null hypothesis: Alternative hypothesis: Reject H 0 ifor

4. The inequalities may then be re-arranged as shown. 4 CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS Model Null hypothesis: Alternative hypothesis: Reject H 0 ifor

5. We can then obtain the confidence interval for  2, being the set of all values that would not be rejected, given the sample estimate b 2. To make it operational, we need to select a significance level and determine the corresponding critical value of t. 5 Model Null hypothesis: Alternative hypothesis: CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS Reject H 0 ifor Do not reject H 0 if

6. For an example of the construction of a confidence interval, we will return to the earnings function fitted earlier. We will construct a 95% confidence interval for  2. 6 CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS. reg EARNINGS S Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 112.15 Model | 19321.5589 1 19321.5589 Prob > F = 0.0000 Residual | 92688.6722 538 172.283777 R-squared = 0.1725 -------------+------------------------------ Adj R-squared = 0.1710 Total | 112010.231 539 207.811189 Root MSE = 13.126 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.455321.2318512 10.59 0.000 1.999876 2.910765 _cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444 ------------------------------------------------------------------------------

7. The point estimate  2 is 2.455 and its standard error is 0.232. 7 CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS. reg EARNINGS S Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 112.15 Model | 19321.5589 1 19321.5589 Prob > F = 0.0000 Residual | 92688.6722 538 172.283777 R-squared = 0.1725 -------------+------------------------------ Adj R-squared = 0.1710 Total | 112010.231 539 207.811189 Root MSE = 13.126 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.455321.2318512 10.59 0.000 1.999876 2.910765 _cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444 ------------------------------------------------------------------------------ 2.455 – 0.232 x 1.965 ≤  2 ≤ 2.455 + 0.232 x 1.965

8. The critical value of t at the 5% significance level with 538 degrees of freedom is 1.965. 8 CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS. reg EARNINGS S Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 112.15 Model | 19321.5589 1 19321.5589 Prob > F = 0.0000 Residual | 92688.6722 538 172.283777 R-squared = 0.1725 -------------+------------------------------ Adj R-squared = 0.1710 Total | 112010.231 539 207.811189 Root MSE = 13.126 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.455321.2318512 10.59 0.000 1.999876 2.910765 _cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444 ------------------------------------------------------------------------------ 2.455 – 0.232 x 1.965 ≤  2 ≤ 2.455 + 0.232 x 1.965

9. . reg EARNINGS S Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 112.15 Model | 19321.5589 1 19321.5589 Prob > F = 0.0000 Residual | 92688.6722 538 172.283777 R-squared = 0.1725 -------------+------------------------------ Adj R-squared = 0.1710 Total | 112010.231 539 207.811189 Root MSE = 13.126 ------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S | 2.455321.2318512 10.59 0.000 1.999876 2.910765 _cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444 ------------------------------------------------------------------------------ Hence we establish that the confidence interval is from 1.999 to 2.911. Stata actually computes the 95% confidence interval as part of its default output, 2.000 to 2.911. The discrepancy in the lower limit is due to rounding error in the calculations we have made. 9 CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS 2.455 – 0.232 x 1.965 ≤  2 ≤ 2.455 + 0.232 x 1.965 1.999 ≤  2 ≤ 2.911

10. 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 2.6 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