Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: one-sided t tests Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource] © 2012 The Author This version available at: 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

1 probability density function of X 11 00 This sequence explains the logic behind a one-sided t test. ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd  1 +2sd 2.5%

2 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd We will start by considering the case where  can take only two possible values:  0, as under the null hypothesis, and  1, the only possible alternative.  1 +2sd 2.5%

3 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd An example of this situation is where there are two types of removable lap-top batteries: regular and long life.  1 +2sd 2.5%

4 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd You have been sent an unmarked shipment and you take a sample and see how long they last. Your null hypothesis is that they are regular batteries.  1 +2sd 2.5%

5 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd Suppose that the sample outcome is as shown. You would not reject the null hypothesis because the sample estimate lies within the acceptance region for H 0.  1 +2sd 2.5%

6 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd Here you would reject the null hypothesis and conclude that the shipment was of long life batteries.  1 +2sd 2.5%

7 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd Here you would stay with the null hypothesis.  1 +2sd 2.5%

8 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd A sample outcome like this one gives rise to a serious problem. It lies in the rejection region for H 0, so our first impulse would be to reject H 0.  1 +2sd 2.5%

9 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  1 +2sd  0 –2sd But to reject H 0 and go with H 1 is nonsensical. Granted, the sample outcome seems to contradict H 0, but it contradicts H 1 even more strongly. 2.5%

10 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  1 +2sd  0 –2sd The probability of getting a sample outcome like this one is much smaller under H 1 than it is under H %

11 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  1 +2sd  0 –2sd For this reason the left tail should be eliminated as a rejection region for H 0. We should use only the right tail as a rejection region. 2.5%

12 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  1 +2sd  0 –2sd The probability of making a Type I error, if H 0 happens to be true, is now 2.5%, so the significance level of the test is now 2.5%. 2.5%

13 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  1 +2sd  0 –2sd We can convert it back to a 5% significance test by building up the right tail until it contains 5% of the probability under the curve. It starts standard deviations from the mean. 5%

14 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  1 +2sd  0 –2sd Why would we want to do this? For the answer, we go back to the trade-off between Type I and Type II errors. 5%

15 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  1 +2sd  0 –2sd With a 5% test, there is a greater chance of making a Type I error if H 0 happens to be true, but there is less risk of making a Type II error if it happens to be false. 5%

16 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  1 +2sd  0 –2sd Note that the logic for dropping the left tail depended only on  1 being greater than  0. It did not depend on  1 being any specific value. 5%

17 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  >  0  1 +2sd  0 –2sd Hence we can generalize the one-sided test to cover the case where the alternative hypothesis is simply that  is greater than  0. 5%

18 probability density function of X 11 00 ONE-SIDED t TESTS  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  >  0  1 +2sd  0 –2sd To justify the use of a one-sided test, all we have to do is to rule out, on the basis of economic theory or previous empirical experience, the possibility that  is less than  0. 5%

probability density function of X 00  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  <  0  0 +2sd  0 –2sd Sometimes, given a null hypothesis H 0 :  =  0, on the basis of economic theory or previous experience, you can rule out the possibility of  being greater than  ONE-SIDED t TESTS 11 5%

probability density function of X 00  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  <  0  0 +2sd  0 –2sd 20 ONE-SIDED t TESTS 11 In this situation you would also perform a one-sided test, now with the left tail being used as the rejection region. With this change, the logic is the same as before. 5%

We will next investigate how the use of a one-sided test improves the trade-off between the risks of making Type I and Type II errors. 21 ONE-SIDED t TESTS probability density function of X 11 00  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd  1 +2sd 2.5%

22 ONE-SIDED t TESTS probability density function of X 11 00  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd  1 +2sd We will start by returning to the case where  can take only two possible values,  0 and  %

23 ONE-SIDED t TESTS probability density function of X 11 00  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd  1 +2sd Suppose that we use a two-sided 5% significance test. If H 0 is true, there is a 5% risk of making a Type I error. 2.5%

24 ONE-SIDED t TESTS probability density function of X 11 00  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd  1 +2sd However, H 0 may be false. In that case the probability of not rejecting it and making a Type II error is given by the blue shaded area. 2.5%

25 ONE-SIDED t TESTS probability density function of X 11 00  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd  1 +2sd This area gives the probability of the estimate lying within the acceptance region for H 0, if H 1 is in fact true. 2.5%

26 ONE-SIDED t TESTS probability density function of X 11 00  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd  1 +2sd Now suppose that you use a one-sided test, taking advantage of the fact that it is irrational to reject H 0 if the estimate is in the left tail. 5%

27 ONE-SIDED t TESTS probability density function of X 11 00  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd  1 +2sd Having expanded the right tail to 5%, we are still performing a 5% significance test, and the risk of making a Type I error is still 5%, if H 0 is true. 5%

28 ONE-SIDED t TESTS probability density function of X 11 00  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd  1 +2sd But if H 0 is false, the risk of making a Type II error is smaller than before. The probability of an estimate lying in the acceptance region for H 0 is now given by the green area. 5%

29 ONE-SIDED t TESTS probability density function of X 11 00  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd  1 +2sd This is smaller than the blue area for the two-sided test. 5%

30 ONE-SIDED t TESTS probability density function of X 11 00  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd  1 +2sd Thus, with no increase in the probability of making a Type I error (if H 0 is true), we have reduced the probability of making a Type II error (if H 0 is false). 5%

31 ONE-SIDED t TESTS probability density function of X 11 00  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd  1 +2sd When the alternative hypothesis is H 1 :  >  0 or H 1 :  <  0, the more general (and more typical) case, we cannot draw this diagram. 5%

32 ONE-SIDED t TESTS probability density function of X 11 00  0 +sd  0 –sd null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1  0 –2sd  1 +2sd Nevertheless we can be sure that, by using a one-sided test, we are reducing the probability of making a Type II error, if H 0 happens to be false. 5%

33 One-sided tests are often particularly useful where the analysis relates to the evaluation of treatment and effect. Suppose that a number of units of observation receive some type of treatment and X i is a measure of the effect of the treatment for observation i. null hypothesis:H 0 :  = 0 ONE-SIDED t TESTS

34 To demonstrate that the treatment did have an effect, we set up the null hypothesis H 0 :  = 0 and see if we can reject it, given the sample average X. null hypothesis:H 0 :  = 0 ONE-SIDED t TESTS

35 probability density function of X 0 If you use a two-sided 5% significance test, X must be 1.96 standard deviations above or below 0 if you are to reject H 0. null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  = 0 reject H 0 do not reject H sd-1.96 sd ONE-SIDED t TESTS 2.5%

36 probability density function of X 0 However, if you can justify the use of a one-sided test, for example with H 1 :  > 0, your estimate only has to be 1.65 standard deviations above 0. reject H 0 do not reject H sd ONE-SIDED t TESTS null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  > 0 5%

37 probability density function of X 0 reject H 0 do not reject H sd ONE-SIDED t TESTS null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  > 0 This makes it easier to reject H 0 and thereby demonstrate that the treatment has had a significant effect. 5%

38 probability density function of X 0 reject H 0 do not reject H sd ONE-SIDED t TESTS null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  > 0 Throughout this sequence, it has been assumed that the standard deviation of the distribution of b 2 is known, and the normal distribution has been used in the diagrams. 5%

39 probability density function of X 0 reject H 0 do not reject H sd ONE-SIDED t TESTS null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  > 0 In practice, of course, the standard deviation has to be estimated as the standard error, and the t distribution is the relevant distribution. However, the logic is exactly the same. 5%

40 probability density function of X 0 reject H 0 do not reject H sd ONE-SIDED t TESTS null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  > 0 At any given significance level, the critical value of t for a one-sided test is lower than that for a two-sided test. 5%

41 probability density function of X 0 reject H 0 do not reject H sd ONE-SIDED t TESTS null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  > 0 Hence, if H 0 is false, the risk of not rejecting it, thereby making a Type II error, is smaller. 5%

Copyright Christopher Dougherty 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 R.13 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 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 or the University of London International Programmes distance learning course 20 Elements of Econometrics