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

2 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. ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 2.5% null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

3 An example of this situation is where you are selling computer accessories and among your products there are two types of removable lap-top batteries: regular (average charge duration  0 ) and long life (average charge duration  1 ). ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 2.5% null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

4 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. ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 2.5% null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

5 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. ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 2.5% null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

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

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

8 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 2.5% A sample outcome such as 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. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

9 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 2.5% 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. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

10 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 2.5% The probability of getting a sample outcome like this one is much smaller under H 1 than it is under H 0. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

11 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 2.5% 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. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

12 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 2.5% 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%. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

13 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 5% 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. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

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

15 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 5% 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. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

16 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 5% 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. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

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

18 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 5% 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. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  >  0

19 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 5% null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  <  0 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  0.

20 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 5% 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. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  <  0

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 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 2.5% null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

22 We will start by returning to the case where  can take only two possible values,  0 and  1. ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 2.5% null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

23 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. ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 2.5% null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

24 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 2.5% 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. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

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

26 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. ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 5% null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  >  0

27 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 5% 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. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  >  0

28 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  0 –2sd  1 +2sd 5% 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 red area. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  >  0

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

30 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  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). null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

31 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  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. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

32 ONE-SIDED t TESTS 11 00  0 +sd  0 –sd  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. null hypothesis:H 0 :  =  0 alternative hypothesis:H 1 :  =  1

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. ONE-SIDED t TESTS null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  ≠ sd–1.96 sd 2.5% reject H 0 do not reject H 0

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. ONE-SIDED t TESTS sd–1.96 sd 2.5% reject H 0 do not reject H 0 null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  ≠ 0

35 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. ONE-SIDED t TESTS sd–1.96 sd 2.5% reject H 0 do not reject H 0 null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  ≠ 0

36 ONE-SIDED t TESTS 0 –1.96 sd 2.5% do not reject H sd However, if you can justify the use of a one-sided test, for example with H 1 :  > 0, your estimate has to be only 1.65 standard deviations above 0. reject H 0 null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  > 0

37 ONE-SIDED t TESTS 0 –1.96 sd 2.5% reject H sd This makes it easier to reject H 0 and thereby demonstrate that the treatment has had a significant effect. do not reject H 0 null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  > 0

38 ONE-SIDED t TESTS 0 –1.96 sd 2.5% reject H sd do not reject H 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. null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  > 0

39 ONE-SIDED t TESTS 0 –1.96 sd 2.5% reject H sd do not reject H 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. null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  > 0

40 ONE-SIDED t TESTS 0 –1.96 sd 2.5% reject H sd do not reject H 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. null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  > 0

41 ONE-SIDED t TESTS 0 –1.96 sd 2.5% reject H sd do not reject H 0 Hence, if H 0 is false, the risk of not rejecting it, thereby making a Type II error, is smaller. null hypothesis:H 0 :  = 0 alternative hypothesis:H 1 :  > 0

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 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 EC2020 Elements of Econometrics