1. Christopher Dougherty EC220 - Introduction to econometrics (chapter 2) Slideshow: testing a hypothesis relating to a regression coefficient 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. Model:Y =  1 +  2 X + u Null hypothesis: Alternative hypothesis TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT This sequence describes the testing of a hypothesis at the 5% and 1% significance levels. It also defines what is meant by a Type I error.

3. Model:Y =  1 +  2 X + u Null hypothesis: Alternative hypothesis TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT We will suppose that we have the standard simple regression model and that we wish to test the hypothesis H 0 that the slope coefficient is equal to some value  2 0. 2

4. Model:Y =  1 +  2 X + u Null hypothesis: Alternative hypothesis TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT The hypothesis being tested is described as the null hypothesis. We test it against the alternative hypothesis H 1, which is simply that  2 is not equal to  2 0. 3

5. Model:Y =  1 +  2 X + u Null hypothesis: Alternative hypothesis Example model:p =  1 +  2 w + u Null hypothesis: Alternative hypothesis: 4 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT As an illustration, we will consider a model relating price inflation to wage inflation. p is the rate of growth of prices and w is the rate of growth of wages.

6. Model:Y =  1 +  2 X + u Null hypothesis: Alternative hypothesis Example model:p =  1 +  2 w + u Null hypothesis: Alternative hypothesis: TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT We will test the hypothesis that the rate of price inflation is equal to the rate of wage inflation. The null hypothesis is therefore H 0 :  2 = 1.0. (We should also test  1 = 0.) 5

7. 6 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT probability density function of b 2 Distribution of b 2 under the null hypothesis H 0 :  2 =1.0 is true (standard deviation equals 0.1 taken as given) b2b2 1.01.10.90.80.70.61.21.31.4 If this null hypothesis is true, the regression coefficient b 2 will have a distribution with mean 1.0. To draw the distribution, we must know its standard deviation.

8. We will assume that we know the standard deviation and that it is equal to 0.1. This is a very unrealistic assumption. In practice you have to estimate it. 7 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 1.01.10.90.80.70.61.21.31.4 probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =1.0 is true (standard deviation equals 0.1 taken as given)

9. 8 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Here is the distribution of b 2 for the general case. Again, for the time being we are assuming that we know its standard deviation (sd). Distribution of b 2 under the null hypothesis H 0 :  2 =  2 is true (standard deviation taken as given) probability density function of b 2 b2b2 0 22  2 +sd  2 +2sd  2 -sd  2 -2sd  2 +3sd  2 -3sd  2 -4sd  2 +4sd 000000000

10. 9 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Suppose that we have a sample of data for the price inflation/wage inflation model and the estimate of the slope coefficient, b 2, is 0.9. Would this be evidence against the null hypothesis  2  = 1.0? 1.01.10.90.80.70.61.21.31.4 probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =1.0 is true (standard deviation equals 0.1 taken as given)

11. 10 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT No, it is not. It is lower than 1.0, but, because there is a disturbance term in the model, we would not expect to get an estimate exactly equal to 1.0. 1.01.10.90.80.70.61.21.31.4 probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =1.0 is true (standard deviation equals 0.1 taken as given)

12. 11 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT If the null hypothesis is true, we should frequently get estimates as low as 0.9, so there is no real conflict. 1.01.10.90.80.70.61.21.31.4 probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =1.0 is true (standard deviation equals 0.1 taken as given)

13. 12 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT In terms of the general case, the estimate is one standard deviation below the hypothetical value. probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =  2 is true (standard deviation taken as given) 0 22  2 +sd  2 +2sd  2 -sd  2 -2sd  2 +3sd  2 -3sd  2 -4sd  2 +4sd 000000000

14. 13 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT If the null hypothesis is true, the probability of getting an estimate one standard deviation or more above or below the mean is 31.7%. probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =  2 is true (standard deviation taken as given) 0 22  2 +sd  2 +2sd  2 -sd  2 -2sd  2 +3sd  2 -3sd  2 -4sd  2 +4sd 000000000

15. 14 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Now suppose that in the price inflation/wage inflation model we get an estimate of 1.4. This clearly conflicts with the null hypothesis. 1.01.10.90.80.70.61.21.31.4 probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =1.0 is true (standard deviation equals 0.1 taken as given)

16. 15 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 1.4 is four standard deviations above the hypothetical mean and the chance of getting such an extreme estimate is only 0.006%. We would reject the null hypothesis. probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =  2 is true (standard deviation taken as given) 0 22  2 +sd  2 +2sd  2 -sd  2 -2sd  2 +3sd  2 -3sd  2 -4sd  2 +4sd 000000000

17. 16 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Now suppose, with the price inflation/wage inflation model, that the sample estimate is 0.77. This is an awkward result. 1.01.10.90.80.70.61.21.31.4 probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =1.0 is true (standard deviation equals 0.1 taken as given)

18. 17 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Under the null hypothesis, the estimate is between 2 and 3 standard deviations below the mean. probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =  2 is true (standard deviation taken as given) 0 22  2 +sd  2 +2sd  2 -sd  2 -2sd  2 +3sd  2 -3sd  2 -4sd  2 +4sd 000000000

19. 18 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT There are two possibilities. One is that the null hypothesis is true, and we have a slightly freaky estimate. 1.01.10.90.80.70.61.21.31.4 probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =1.0 is true (standard deviation equals 0.1 taken as given)

20. 19 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT The other is that the null hypothesis is false. The rate of price inflation is not equal to the rate of wage inflation. 1.01.10.90.80.70.61.21.31.4 probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =1.0 is true (standard deviation equals 0.1 taken as given)

21. 20 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT The usual procedure for making decisions is to reject the null hypothesis if it implies that the probability of getting such an extreme estimate is less than some (small) probability p. probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =  2 is true (standard deviation taken as given) 0 22  2 +sd  2 +2sd  2 -sd  2 -2sd  2 +3sd  2 -3sd  2 -4sd  2 +4sd 000000000

22. 21 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT For example, we might choose to reject the null hypothesis if it implies that the probability of getting such an extreme estimate is less than 0.05 (5%). 2.5% probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =  2 is true (standard deviation taken as given) 0 22  2 +sd  2 +2sd  2 -sd  2 -2sd  2 +3sd  2 -3sd  2 -4sd  2 +4sd 000000000

23. 22 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT According to this decision rule, we would reject the null hypothesis if the estimate fell in the upper or lower 2.5% tails. 2.5% probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =  2 is true (standard deviation taken as given) 0 22  2 +sd  2 +2sd  2 -sd  2 -2sd  2 +3sd  2 -3sd  2 -4sd  2 +4sd 000000000

24. 2.5% 23 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT If we apply this decision rule to the price inflation/wage inflation model, the first estimate of  2 would not lead to a rejection of the null hypothesis. 1.01.10.90.80.70.61.21.31.4 probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =1.0 is true (standard deviation equals 0.1 taken as given)

25. 2.5% 24 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT The second definitely would. 1.01.10.90.80.70.61.21.31.4 probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =1.0 is true (standard deviation equals 0.1 taken as given)

26. 2.5% 25 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT The third also would lead to rejection.. 1.01.10.90.80.70.61.21.31.4 probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =1.0 is true (standard deviation equals 0.1 taken as given)

27. 26 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT The 2.5% tails of a normal distribution always begin 1.96 standard deviations from its mean. 2.5% probability density function of b 2 b2b2 Distribution of b 2 under the null hypothesis H 0 :  2 =  2 is true (standard deviation taken as given) 0  2 +1.96sd  2 -1.96sd 22 0  2 -sd  2 +sd 0000

28. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 2.5% Decision rule (5% significance level): reject (1) if(2) if probability density function of b 2 b2b2  2 +1.96sd  2 -1.96sd 22 0  2 -sd  2 +sd 0000 Thus we would reject H 0 if the estimate were 1.96 standard deviations (or more) above or below the hypothetical mean. 27

29. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 2.5% Decision rule (5% significance level): reject (1) if(2) if probability density function of b 2 b2b2  2 +1.96sd  2 -1.96sd 22 0  2 -sd  2 +sd 0000 We would reject H 0 if the difference between the sample estimate and hypothetical value were more than 1.96 standard deviations. 28

30. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 2.5% Decision rule (5% significance level): reject (1) if(2) if probability density function of b 2 b2b2  2 +1.96sd  2 -1.96sd 22 0  2 -sd  2 +sd 0000 We would reject H 0 if the difference, expressed in terms of standard deviations, were more than 1.96 in absolute terms (positive or negative). 29

31. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 2.5% Decision rule (5% significance level): reject (1) if(2) if probability density function of b 2 b2b2  2 +1.96sd  2 -1.96sd 22 0  2 -sd  2 +sd 0000 We will denote the difference, expressed in terms of standard deviations, as z. 30

32. 31 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Then the decision rule is to reject the null hypothesis if z is greater than 1.96 in absolute terms. 2.5% Decision rule (5% significance level): reject (1) if(2) if (1) if z > 1.96(2) if z < -1.96 probability density function of b 2 b2b2  2 +1.96sd  2 -1.96sd 22 0  2 -sd  2 +sd 0000

33. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 2.5% Decision rule (5% significance level): reject (1) if(2) if (1) if z > 1.96(2) if z < -1.96 probability density function of b 2 b2b2  2 +1.96sd  2 -1.96sd 22 0  2 -sd  2 +sd 0000 The range of values of b 2 that do not lead to the rejection of the null hypothesis is known as the acceptance region. 32 acceptance region for b 2 :

34. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 2.5% Decision rule (5% significance level): reject (1) if(2) if (1) if z > 1.96(2) if z < -1.96 probability density function of b 2 b2b2  2 +1.96sd  2 -1.96sd 22 0  2 -sd  2 +sd 0000 acceptance region for b 2 : The limiting values of z for the acceptance region are 1.96 and -1.96 (for a 5% significance test). 33

35. 2.5% TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 1.01.10.90.80.70.61.21.31.4 probability density function of b 2 b2b2 Decision rule (5% significance level): reject (1) if(2) if We will look again at the decision process in terms of the price inflation/wage inflation example. The null hypothesis is that the slope coefficient is equal to 1.0. 34

36. 2.5% TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 1.01.10.90.80.70.61.21.31.4 probability density function of b 2 b2b2 Decision rule (5% significance level): reject (1) if(2) if We are assuming that we know the standard deviation and that it is equal to 0.1. 35

37. 2.5% 36 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT The acceptance region for b 2 is therefore the interval 0.804 to 1.196. A sample estimate in this range will not lead to the rejection of the null hypothesis. 1.01.10.90.80.70.61.21.31.4 probability density function of b 2 b2b2 Decision rule (5% significance level): reject (1) if(2) if acceptance region for b 2 :

38. 37 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Rejection of the null hypothesis when it is in fact true is described as a Type I error. 2.5% Type I error: rejection of H 0 when it is in fact true. reject probability density function of b 2 b2b2  2 +1.96sd  2 -1.96sd 22 0  2 -sd  2 +sd 0000 acceptance region for b 2

39. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 2.5% reject probability density function of b 2 b2b2  2 +1.96sd  2 -1.96sd 22 0  2 -sd  2 +sd 0000 acceptance region for b 2 With the present test, if the null hypothesis is true, a Type I error will occur 5% of the time because 5% of the time we will get estimates in the upper or lower 2.5% tails. 38 Type I error: rejection of H 0 when it is in fact true. Probability of Type I error: in this case, 5%

40. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 2.5% reject probability density function of b 2 b2b2  2 +1.96sd  2 -1.96sd 22 0  2 -sd  2 +sd 0000 acceptance region for b 2 The significance level of a test is defined to be the probability of making a Type I error if the null hypothesis is true. 39 Type I error: rejection of H 0 when it is in fact true. Probability of Type I error: in this case, 5% Significance level of the test is 5%.

41. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 2.5% reject probability density function of b 2 b2b2  2 +1.96sd  2 -1.96sd 22 0  2 -sd  2 +sd 0000 acceptance region for b 2 Type I error: rejection of H 0 when it is in fact true. Probability of Type I error: in this case, 5% Significance level of the test is 5%. We can of course reduce the risk of making a Type I error by reducing the size of the rejection region. 40

42. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 2.5% reject probability density function of b 2 b2b2  2 +1.96sd  2 -1.96sd 22 0  2 -sd  2 +sd 0000 acceptance region for b 2 Type I error: rejection of H 0 when it is in fact true. Probability of Type I error: in this case, 5% Significance level of the test is 5%. For example, we could change the decision rule to “reject the null hypothesis if it implies that the probability of getting the sample estimate is less than 0.01 (1%)”. 41

43. 42 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT The rejection region now becomes the upper and lower 0.5% tails 2.5% probability density function of b 2 b2b2  2 +1.96sd  2 -1.96sd 22 0  2 -sd  2 +sd 0000 rejectacceptance region for b 2 reject

44. Decision rule (1% significance level): reject (1) if(2) if (1) if z > 2.58(2) if z < -2.58 43 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT The 0.5% tails of a normal distribution start 2.58 standard deviations from the mean, so we now reject the null hypothesis if z is greater than 2.58, in absolute terms. 0.5% probability density function of b 2 b2b2  2 +2.58sd  2 -2.58sd 22 0  2 -sd  2 +sd 0000 acceptance region for b 2 :

45. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 0.5% probability density function of b 2 b2b2  2 +2.58sd  2 -2.58sd 22 0  2 -sd  2 +sd 0000 Type I error: rejection of H 0 when it is in fact true. Probability of Type I error: in this case, 1% Significance level of the test is 1%. rejectacceptance region for b 2 reject Since the probability of making a Type I error, if the null hypothesis is true, is now only 1%, the test is said to be a 1% significance test. 44

46. 45 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT In the case of the price inflation/wage inflation model, given that the standard deviation is 0.1, the 0.5% tails start 0.258 above and below the mean, that is, at 0.742 and 1.258. 1.01.10.90.80.70.61.21.31.4 0.5% probability density function of b 2 b2b2 Decision rule (1% significance level): reject (1) if(2) if acceptance region for b 2 :

47. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 1.01.10.90.80.70.61.21.31.4 0.5% probability density function of b 2 b2b2 Decision rule (1% significance level): reject (1) if(2) if acceptance region for b 2 : The acceptance region for b 2 is therefore the interval 0.742 to 1.258. Because it is wider than that for the 5% test, there is less risk of making a Type I error, if the null hypothesis is true. 46

48. 47 TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT This diagram compares the decision-making processes for the 5% and 1% tests. Note that if you reject H 0 at the 1% level, you must also reject it at the 5% level. 0.5% 5% and 1% acceptance regions compared 5%: -1.96 < z < 1.96 1%: -2.58 < z < 2.58 5% level 1% level probability density function of b 2 b2b2 22  2 +sd  2 +2sd  2 -sd  2 -2sd  2 +3sd  2 -3sd  2 -4sd  2 +4sd 000000000

49. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 0.5% 5% and 1% acceptance regions compared 5%: -1.96 < z < 1.96 1%: -2.58 < z < 2.58 5% level 1% level probability density function of b 2 b2b2 22  2 +sd  2 +2sd  2 -sd  2 -2sd  2 +3sd  2 -3sd  2 -4sd  2 +4sd 000000000 Note also that if b 2 lies within the acceptance region for the 5% test, it must also fall within it for the 1% test. 48

50. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT The diagram summarizes the possible decisions for the 5% and 1% tests, for both the general case and the price inflation/wage inflation example. 49 Reject H 0 at 1% level (and also 5% level) Reject H 0 at 5% level but not 1% level Reject H 0 at 1% level (and also 5% level) Do not reject H 0 at 5% level (or at 1% level) 1.000 0.804 0.742 1.196 1.258 Price inflation/ wage inflation example General caseDecision

51. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Reject H 0 at 1% level (and also 5% level) Reject H 0 at 5% level but not 1% level Reject H 0 at 1% level (and also 5% level) Do not reject H 0 at 5% level (or at 1% level) 1.000 0.804 0.742 1.196 1.258 Price inflation/ wage inflation example General caseDecision The middle of the diagram indicates what you would report. You would not report the phrases in parentheses. 50

52. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Reject H 0 at 1% level (and also 5% level) Reject H 0 at 5% level but not 1% level Reject H 0 at 1% level (and also 5% level) Do not reject H 0 at 5% level (or at 1% level) 1.000 0.804 0.742 1.196 1.258 Price inflation/ wage inflation example General caseDecision If you can reject H 0 at the 1% level, it automatically follows that you can reject it at the 5% level and there is no need to say so. Indeed, you would look ignorant if you did. 51

53. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Reject H 0 at 1% level (and also 5% level) Reject H 0 at 5% level but not 1% level Reject H 0 at 1% level (and also 5% level) Do not reject H 0 at 5% level (or at 1% level) 1.000 0.804 0.742 1.196 1.258 Price inflation/ wage inflation example General caseDecision Likewise, if you cannot reject H 0 at the 5% level, that is all you should say. It automatically follows that you cannot reject it at the 1% level and you would look ignorant if you said so. 52

54. TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Reject H 0 at 1% level (and also 5% level) Reject H 0 at 5% level but not 1% level Reject H 0 at 1% level (and also 5% level) Do not reject H 0 at 5% level (or at 1% level) 1.000 0.804 0.742 1.196 1.258 Price inflation/ wage inflation example General caseDecision You should report the results of both tests only when you can reject H 0 at the 5% level but not at the 1% level. 53

55. 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 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 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