christopher dougherty ec220 - introduction to econometrics (chapter 2) slideshow: type 1 error and...

Christopher Dougherty

EC220 - Introduction to econometrics (chapter 2)Slideshow: type 1 error and type 2 error

Original citation:

Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 2). [Teaching Resource]

© 2012 The Author

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Available in LSE Learning Resources Online: May 2012

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2 2+sd2-sd

TYPE I ERROR AND TYPE II ERROR

5% level

hypothetical distribution under 0

220 : H

acceptance region for b2

00000 b22-1.96sd 2+1.96sd

2.5% 2.5%

In the previous sequence a Type I error was defined to be the rejection of a null hypothesis when it happens to be true.

1

2 2+sd2-sd


5% level


220 : H


00000 b22-1.96sd 2+1.96sd

2.5% 2.5%

In hypothesis testing there is also a possibility of failing to reject the null hypothesis when it is in fact false. This is known as a Type II error.

2

2 2+sd2-sd


5% level


220 : H


00000 b22-1.96sd 2+1.96sd

This sequence will demonstrate that there is a trade-off between the risk of making a Type I error and the risk of making a Type II error.

3

2.5% 2.5%

2 2+sd2-sd


5% level


220 : H


00000

The diagram show the acceptance region and the rejection regions for a 5% significance test. The risk of making a Type I error, if the null hypothesis happens to be true, is 5%.

4

b22-1.96sd 2+1.96sd

2.5% 2.5%

2 2+sd 2+2.58sd2-sd2-2.58sd

0.5%0.5%


5% level

1% level


220 : H


00000

Using a 1% significance test, instead of a 5% test, reduces the risk of making a Type I error to 1%, if the null hypothesis is true.

5

b2

2 2+sd 2+2.58sd2-sd2-2.58sd

0.5%0.5%


5% level

1% level


220 : H


00000

We will consider the implications of the choice of significance test for the case where the null hypothesis happens to be false.

6

b2

2 2+sd 2+2.58sd2-sd2-2.58sd

0.5%0.5%


7

5% level

1% level

The diagram above explains how the test decisions are made, but it does not give the actual distribution of b2. (For that reason the curve has been drawn with a dashed line.)

b2


220 : H


00000

2

0.5%0.5%



5% level

1% level actual distribution under

b22+2sd2+sd22-sd2-2sd

1221 : H

0 1 1 1 1 1


220 : H

Suppose that H1: 2 = 21 is in fact true and the distribution of b2 is therefore governed by

the right-hand curve.

8

2

0.5%0.5%



5% level



1221 : H

0 1 1 1 1 1


220 : H

If we obtain some data and run a regression, the estimate b2 might be as shown. In this case we would make the right decision and reject H0, no matter which test we used.

9

2

0.5%0.5%



5% level



1221 : H

0 1 1 1 1 1


220 : H

Here is another estimate. Again, we would make the right decision and reject the null hypothesis, no matter whether we use the 5% test or the 1% test.

10

2

0.5%0.5%



5% level



1221 : H

0 1 1 1 1 1


220 : H

In the case shown, we would make a Type II error and fail to reject the null hypothesis, using either significance level.

11

2

0.5%0.5%



5% level



1221 : H

0 1 1 1 1 1


220 : H

But in the case of this estimate, we would make the right decision if we used a 5% test but we would make a Type II error if we used a 1% test.

12

2

0.5%0.5%



5% level



1221 : H

0 1 1 1 1 1


220 : H

The probability of making a Type II error if we use a 1% test is given by the probability of b2 lying within the 1% acceptance region, the interval between the red vertical dotted lines.

13

2

0.5%0.5%



5% level



1221 : H

0 1 1 1 1 1


220 : H

Given that H1 is true, the probability of b2 lying in the acceptance region is that area under the distribution for H1 in the diagram - the pink shaded area in the diagram.

14

2

0.5%0.5%



5% level



1221 : H

0 1 1 1 1 1

If instead we use a 5% significance test, the probability of making a Type II error if H1 is true is given by the area under the distribution for H1 in the acceptance region for the 5% test.

15


220 : H

2

0.5%0.5%



16

5% level



This is the gray shaded area in the diagram. In this particular case, using a 5% test instead of a 1% test would approximately halve the risk of making a Type II error.


220 : H

1221 : H

0 1 1 1 1 1

2

0.5%0.5%



5% level




220 : H

1221 : H

0 1 1 1 1 1

The problem, of course, is that you never know whether H0 is true of false. If you did, why would you be performing a test?

17

2

0.5%0.5%



5% level




220 : H

1221 : H

0 1 1 1 1 1

If H0 happens to be true, using a 1% test instead of a 5% test greatly reduces the risk of making a Type I error (you cannot make a Type II error).

18

2

0.5%0.5%



5% level




220 : H

1221 : H

0 1 1 1 1 1

However, if H0 is false, using a 1% test instead of a 5% test increases the risk of making a Type II error (you cannot make a Type I error).

19

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: type 1 error and...

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