christopher dougherty ec220 - introduction to econometrics (chapter 2) slideshow: one-sided t tests...

Christopher Dougherty

EC220 - Introduction to econometrics (chapter 2)Slideshow: one-sided t tests

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/

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://learningresources.lse.ac.uk/

1

probability densityfunction of b2

22

This sequence explains the logic behind a one-sided t test.

ONE-SIDED t TESTS

2 +sd2 -sd

2.5% 2.5%

null hypothesis: H0 : 2 = 2

alternative hypothesis: H1 : 2 = 2

2 +2sd2 -2sd

0

1

0 0 0 0 0 1


22 2 +sd2 -sd

2.5% 2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

We will start by considering the case where 2 can take only two possible values: 20, as

under the null hypothesis, and 21, the only possible alternative.

2

ONE-SIDED t TESTS


22 2 +sd2 -sd

2.5% 2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

An example of this situation is where there are two types of removable lap-top batteries: regular and long life.

3

ONE-SIDED t TESTS


22 2 +sd2 -sd

2.5% 2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

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.

4

ONE-SIDED t TESTS


22 2 +sd2 -sd

2.5% 2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

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 H0.

5

ONE-SIDED t TESTS


22 2 +sd2 -sd

2.5% 2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

Here you would reject the null hypothesis and conclude that the shipment was of long life batteries.

6

ONE-SIDED t TESTS


22 2 +sd2 -sd

2.5% 2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

Here you would stay with the null hypothesis.

7

ONE-SIDED t TESTS


22 2 +sd2 -sd

2.5% 2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

A sample outcome like this one gives rise to a serious problem. It lies in the rejection region for H0, so our first impulse would be to reject H0.

8

ONE-SIDED t TESTS


22 2 +sd2 -sd

2.5% 2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

But to reject H0 and go with H1 is nonsensical. Granted, the sample outcome seems to contradict H0, but it contradicts H1 even more strongly.

9

ONE-SIDED t TESTS


22 2 +sd2 -sd

2.5% 2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

The probability of getting a sample outcome like this one is much smaller under H1 than it is under H0.

10

ONE-SIDED t TESTS


22 2 +sd2 -sd

2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

For this reason the left tail should be eliminated as a rejection region for H0. We should use only the right tail as a rejection region.

11

ONE-SIDED t TESTS


22 2 +sd2 -sd

2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

The probability of making a Type I error, if H0 happens to be true, is now 2.5%, so the significance level of the test is now 2.5%.

12

ONE-SIDED t TESTS


22 2 +sd2 -sd



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

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 1.645 standard deviations from the mean.

13

5%

ONE-SIDED t TESTS


22 2 +sd2 -sd



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

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.

14

ONE-SIDED t TESTS


22 2 +sd2 -sd



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

5%

With a 5% test, there is a greater chance of making a Type I error if H0 happens to be true, but there is less risk of making a Type II error if it happens to be false.

15

ONE-SIDED t TESTS


22 2 +sd2 -sd



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

5%

Note that the logic for dropping the left tail depended only on 21 being greater than 2

0. It

did not depend on 21 being any specific value.

16

ONE-SIDED t TESTS


22 2 +sd2 -sd


alternative hypothesis: H1 : 2 > 2

2 +2sd2 -2sd

0

0

0 0 0 0 0 1

5%

Hence we can generalize the one-sided test to cover the case where the alternative

hypothesis is simply that 2 is greater than 20.

17

ONE-SIDED t TESTS


22 2 +sd2 -sd



2 +2sd2 -2sd

0

0

0 0 0 0 0 1

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 2 is less than 20.

18

ONE-SIDED t TESTS


2 2 2 +sd2 -sd


alternative hypothesis: H1 : 2 < 2

2 +2sd2 -2sd

0

0

0 0 0 0 01

Sometimes, given a null hypothesis H0: 2 = 20, on the basis of economic theory or

previous experience, you can rule out the possibility of 2 being greater than 20.

19

ONE-SIDED t TESTS


2 2 2 +sd2 -sd


alternative hypothesis: H1 : 2 < 2

2 +2sd2 -2sd

0

0

0 0 0 0 01

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.

20

ONE-SIDED t TESTS


22 2 +sd2 -sd

2.5% 2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

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


22 2 +sd2 -sd

2.5% 2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

We will start by returning to the case where 2 can take only two possible values, 20 and

21.

22

ONE-SIDED t TESTS


22 2 +sd2 -sd

2.5% 2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

Suppose that we use a two-sided 5% significance test. If H0 is true, there is a 5% risk of making a Type I error.

23

ONE-SIDED t TESTS


22 2 +sd2 -sd

2.5% 2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

However, H0 may be false. In that case the probability of not rejecting it and making a Type II error is given by the pale gray shaded area.

24

ONE-SIDED t TESTS


22 2 +sd2 -sd

2.5% 2.5%



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

This area gives the probability of the estimate lying within the acceptance region for H0, if H1 is in fact true.

25

ONE-SIDED t TESTS


22 2 +sd2 -sd



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

Now suppose that you use a one-sided test, taking advantage of the fact that it is irrational to reject H0 if the estimate is in the left tail.

26

5%

ONE-SIDED t TESTS


22 2 +sd2 -sd



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

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 H0 is true.

27

ONE-SIDED t TESTS


22 2 +sd2 -sd



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

5%

But if H0 is false, the risk of making a Type II error is smaller than before.

28

ONE-SIDED t TESTS


22 2 +sd2 -sd



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

5%

The probability of an estimate lying in the acceptance region for H0 is now given by the pale yellow area.

29

ONE-SIDED t TESTS


22 2 +sd2 -sd



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

5%

Thus, with no increase in the probability of making a Type I error (if H0 is true), we have reduced the probability of making a Type II error (if H0 is false).

30

ONE-SIDED t TESTS


22 2 +sd2 -sd



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

5%

When the alternative hypothesis is H1: 2 > 20 or H1: 2 < 2

0, the more general (and more typical) case, we cannot draw this diagram.

31

ONE-SIDED t TESTS


22 2 +sd2 -sd



2 +2sd2 -2sd

0

1

0 0 0 0 0 1

5%

Nevertheless we can be sure that, by using a one-sided test, we are reducing the probability of making a Type II error, if H0 happens to be false.

32

ONE-SIDED t TESTS

33

Now we will consider the special, but very common, case H0: 2 = 0.

model: Y = 1 + 2X + u


ONE-SIDED t TESTS

model: Y = 1 + 2X + u


It occurs when you wish to demonstrate that a variable X influences another variable Y. You set up the null hypothesis of no effect and try to reject H0.

34

ONE-SIDED t TESTS

35


0

If you use a two-sided 5% significance test, your estimate must be 1.96 standard deviations above or below 0 if you are to reject H0.

2.5% 2.5%



reject H0reject H0 do not reject H0

1.96 sd-1.96 sd

ONE-SIDED t TESTS

36


0

However, if you can justify the use of a one-sided test, for example with H0: 2 > 0, your estimate only has to be 1.65 standard deviations above 0.

reject H0do not reject H0

1.65 sd

5%



ONE-SIDED t TESTS


0


1.65 sd

5%



This makes it easier to reject H0 and thereby demonstrate that Y really is influenced by X (assuming that your model is correctly specified).

37

ONE-SIDED t TESTS


0


1.65 sd

5%



Throughout this sequence, it has been assumed that the standard deviation of the distribution of b2 is known, and the normal distribution has been used in the diagrams.

38

ONE-SIDED t TESTS


0


1.65 sd

5%



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.

39

ONE-SIDED t TESTS


0


1.65 sd

5%



At any given significance level, the critical value of t for a one-sided test is lower than that for a two-sided test.

40

ONE-SIDED t TESTS


0


1.65 sd

5%



Hence, if H0 is false, the risk of not rejecting it, thereby making a Type II error, is smaller.

41

ONE-SIDED t TESTS

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/.

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

or the University of London International Programmes distance learning course

20 Elements of Econometrics

www.londoninternational.ac.uk/lse.

11.07.25

christopher dougherty ec220 - introduction to econometrics (chapter 2) slideshow: one-sided t tests...

Documents