1. Christopher Dougherty EC220 - Introduction to econometrics (chapter 2) Slideshow: exercise 2.24 Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 2). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/128/http://learningresources.lse.ac.uk/128/ 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. 2.24 Explain whether it would have been possible to perform one-tailed tests instead of two-tailed tests in Exercise 2.16. If you think that one-tailed tests are justified, perform them and state whether the use of a one-tailed test makes any difference. ( 2.16 Hourly earnings, EARNINGS, are hypothesized to be related to hours of training, TRAINING: EARNINGS =  1 +  2 TRAINING + u H 0 :  2 = 0, H 1 :  2 0. n = 50 1.If b 2 = 0.30, s.e.(b 2 ) = 0.12? 2.If b 2 = 0.55, s.e.(b 2 ) = 0.12? 3.If b 2 = 0.10, s.e.(b 2 ) = 0.12? 4.If b 2 = –0.27, s.e.(b 2 ) = 0.12?) EXERCISE 2.24 1

3. EARNINGS =  1 +  2 TRAINING + u H 0 :  2 = 0, H 1 :  2 > 0 2 It is reasonable to exclude the possibility that training has a negative effect on earnings, so we are justified in performing a one-tailed test with H 1 :  2 > 0. EXERCISE 2.24

4. EARNINGS =  1 +  2 TRAINING + u H 0 :  2 = 0, H 1 :  2 > 0 3 Note that we did not say that the justification for using a one-tailed test is that we expect training to have a positive effect on earnings. That would be too strong, for it would eliminate H 0 and there would be nothing to test. EXERCISE 2.24

5. EARNINGS =  1 +  2 TRAINING + u H 0 :  2 = 0, H 1 :  2 > 0 n = 50, so 48 degrees of freedom Two-tailed: t crit, 5% = 2.01, t crit, 1% = 2.68 One-tailed: t crit, 5% = 1.68, t crit, 1% = 2.40 4 The critical values for a one-tailed test are approximately as shown. (They are the critical values for 50 degrees of freedom). EXERCISE 2.24

6. EARNINGS =  1 +  2 TRAINING + u H 0 :  2 = 0, H 1 :  2 > 0 n = 50, so 48 degrees of freedom Two-tailed: t crit, 5% = 2.01, t crit, 1% = 2.68 One-tailed: t crit, 5% = 1.68, t crit, 1% = 2.40 _______________________________________________ 1.If b 2 = 0.30, s.e.(b 2 ) = 0.12? t = 2.50. 5 In the first case, the t statistic is 2.50. EXERCISE 2.24

7. EARNINGS =  1 +  2 TRAINING + u H 0 :  2 = 0, H 1 :  2 > 0 n = 50, so 48 degrees of freedom Two-tailed: t crit, 5% = 2.01, t crit, 1% = 2.68 One-tailed: t crit, 5% = 1.68, t crit, 1% = 2.40 _______________________________________________ 1.If b 2 = 0.30, s.e.(b 2 ) = 0.12? t = 2.50. Reject H 0 at the 1% level. 6 We can reject H 0 at the 1% level. This result is an improvement on that for the two-tailed test, where we could reject at the 5% level but not the 1% level. EXERCISE 2.24

8. EARNINGS =  1 +  2 TRAINING + u H 0 :  2 = 0, H 1 :  2 > 0 n = 50, so 48 degrees of freedom Two-tailed: t crit, 5% = 2.01, t crit, 1% = 2.68 One-tailed: t crit, 5% = 1.68, t crit, 1% = 2.40 _______________________________________________ 2.If b 2 = 0.55, s.e.(b 2 ) = 0.12? t = 4.58. 7 In the second case, the t statistic is 4.58. EXERCISE 2.24

9. EARNINGS =  1 +  2 TRAINING + u H 0 :  2 = 0, H 1 :  2 > 0 n = 50, so 48 degrees of freedom Two-tailed: t crit, 5% = 2.01, t crit, 1% = 2.68 One-tailed: t crit, 5% = 1.68, t crit, 1% = 2.40 _______________________________________________ 2.If b 2 = 0.55, s.e.(b 2 ) = 0.12? t = 4.58. Reject H 0 at the 0.1% level (t crit, 0.1% = 3.50). 8 We can reject H 0 at the 0.1% level using a two-tailed test. There is little to be gained by performing a one-tailed test. EXERCISE 2.24

10. EARNINGS =  1 +  2 TRAINING + u H 0 :  2 = 0, H 1 :  2 > 0 n = 50, so 48 degrees of freedom Two-tailed: t crit, 5% = 2.01, t crit, 1% = 2.68 One-tailed: t crit, 5% = 1.68, t crit, 1% = 2.40 _______________________________________________ 3.If b 2 = 0.10, s.e.(b 2 ) = 0.12? t = 0.83. 9 In the third case, the t statistic is 0.83. EXERCISE 2.24

11. EARNINGS =  1 +  2 TRAINING + u H 0 :  2 = 0, H 1 :  2 > 0 n = 50, so 48 degrees of freedom Two-tailed: t crit, 5% = 2.01, t crit, 1% = 2.68 One-tailed: t crit, 5% = 1.68, t crit, 1% = 2.40 _______________________________________________ 3.If b 2 = 0.10, s.e.(b 2 ) = 0.12? t = 0.83. Do not reject H 0 at the 5% level. 10 It is so low that we cannot reject H 0 at the 5% level, even using a one-tailed test, so again there is no difference. EXERCISE 2.24

12. EARNINGS =  1 +  2 TRAINING + u H 0 :  2 = 0, H 1 :  2 > 0 n = 50, so 48 degrees of freedom Two-tailed: t crit, 5% = 2.01, t crit, 1% = 2.68 One-tailed: t crit, 5% = 1.68, t crit, 1% = 2.40 _______________________________________________ 4.If b 2 = –0.27, s.e.(b 2 ) = 0.12? t = –2.25. 11 In the fourth case, the t statistic is –2.25. What do you conclude? Try to answer, before going to the next slide. EXERCISE 2.24

13. EARNINGS =  1 +  2 TRAINING + u H 0 :  2 = 0, H 1 :  2 > 0 n = 50, so 48 degrees of freedom Two-tailed: t crit, 5% = 2.01, t crit, 1% = 2.68 One-tailed: t crit, 5% = 1.68, t crit, 1% = 2.40 _______________________________________________ 4.If b 2 = –0.27, s.e.(b 2 ) = 0.12? t = –2.25. Do not reject H 0. 12 The coefficient is negative. It would be wholly illogical to reject H 0, for a negative coefficient is even more unlikely under H 1. We accept that the rather large negative t statistic has arisen as a matter of chance. EXERCISE 2.24

14. EARNINGS =  1 +  2 TRAINING + u H 0 :  2 = 0, H 1 :  2 > 0 n = 50, so 48 degrees of freedom Two-tailed: t crit, 5% = 2.01, t crit, 1% = 2.68 One-tailed: t crit, 5% = 1.68, t crit, 1% = 2.40 _______________________________________________ 4.If b 2 = –0.27, s.e.(b 2 ) = 0.12? t = –2.25. Do not reject H 0. 12 EXERCISE 2.24 Of course, it would be sensible to double check whether the justification for a one-tailed test is really valid. Is it conceivable that training could have a negative effect on earnings?

15. EARNINGS =  1 +  2 TRAINING + u H 0 :  2 = 0, H 1 :  2 > 0 n = 50, so 48 degrees of freedom Two-tailed: t crit, 5% = 2.01, t crit, 1% = 2.68 One-tailed: t crit, 5% = 1.68, t crit, 1% = 2.40 _______________________________________________ 4.If b 2 = –0.27, s.e.(b 2 ) = 0.12? t = –2.25. Do not reject H 0. 12 EXERCISE 2.24 Probably not in general, but it would depend on the sample. Enterprises which provide little training may have to pay higher wages than similar enterprises that do, to make their jobs attractive. In such a situation it would be wrong to rule out a negative coefficient.

16. Copyright Christopher Dougherty 1999–2006. This slideshow may be freely copied for personal use. 22.06.06