Download presentation

Presentation is loading. Please wait.

Published byRosamond Jacobs Modified over 4 years ago

1
00 0 +1.96sd 0 –sd 0 –1.96sd 0 +sd 2.5% CONFIDENCE INTERVALS probability density function of X null hypothesis H 0 : = 0 In the sequence on hypothesis testing, we started with a given hypothesis, for example H 0 : = 0, and considered whether an estimate X derived from a sample would or would not lead to its rejection. 1 X conditional on = 0 being true

2
00 0 +1.96sd 0 –sd 0 –1.96sd 0 +sd 2 Using a 5% significance test, this estimate would not lead to the rejection of H 0. CONFIDENCE INTERVALS X probability density function of X conditional on = 0 being true 2.5% null hypothesis H 0 : = 0

3
00 0 +1.96sd 0 –sd 0 –1.96sd 0 +sd 3 This estimate would cause H 0 to be rejected. CONFIDENCE INTERVALS X probability density function of X conditional on = 0 being true 2.5% null hypothesis H 0 : = 0

4
00 0 +1.96sd 0 –sd 0 –1.96sd 0 +sd 4 So would this one. CONFIDENCE INTERVALS X probability density function of X conditional on = 0 being true 2.5% null hypothesis H 0 : = 0

5
00 0 +1.96sd 0 –sd 0 –1.96sd 0 +sd 5 We ended by deriving the range of estimates that are compatible with H 0 and called it the acceptance region. acceptance region for X CONFIDENCE INTERVALS X probability density function of X conditional on = 0 being true 2.5% null hypothesis H 0 : = 0

6
6 Now we will do exactly the opposite. Given a sample estimate, we will find the set of hypotheses that would not be contradicted by it, using a two-tailed 5% significance test. CONFIDENCE INTERVALS X

7
7 Suppose that someone came up with the hypothesis H 0 : = 0. CONFIDENCE INTERVALS null hypothesis H 0 : = 0 00 X

8
00 0 +1.96sd 0 –sd 0 –1.96sd 8 To see if it is compatible with our sample estimate X, we need to draw the probability distribution of X conditional on H 0 : = 0 being true. X CONFIDENCE INTERVALS probability density function of X conditional on = 0 being true null hypothesis H 0 : = 0

9
00 0 +1.96sd 0 –sd 0 –1.96sd 9 X To do this, we need to know the standard deviation of the distribution. For the time being we will assume that we do know it, although in practice we have to estimate it. CONFIDENCE INTERVALS probability density function of X conditional on = 0 being true null hypothesis H 0 : = 0

10
00 0 +1.96sd 0 –sd 0 –1.96sd 10 X Having drawn the distribution, it is clear this null hypothesis is not contradicted by our sample estimate. CONFIDENCE INTERVALS probability density function of X conditional on = 0 being true null hypothesis H 0 : = 0

11
11 X Here is another null hypothesis to consider. 11 CONFIDENCE INTERVALS null hypothesis H 0 : = 1

12
12 Having drawn the distribution of X conditional on this hypothesis being true, we can see that this hypothesis is also compatible with our estimate. 11 1 +1.96sd 1 +sd 1 –1.96sd X CONFIDENCE INTERVALS probability density function of X conditional on = 1 being true null hypothesis H 0 : = 1

13
13 Here is another null hypothesis. 22 X CONFIDENCE INTERVALS null hypothesis H 0 : = 2

14
14 It is incompatible with our estimate because the estimate would lead to its rejection. 2 –sd 2 –1.96sd 2 +sd 22 X CONFIDENCE INTERVALS probability density function of X conditional on = 2 being true null hypothesis H 0 : = 2

15
15 The highest hypothetical value of not contradicted by our sample estimate is shown in the diagram. max max +1.96sd max –sd max +sd X CONFIDENCE INTERVALS probability density function of X conditional on = max being true null hypothesis H 0 : = max

16
16 max max +1.96sd max –sd max +sd X When the probability distribution of X conditional on it is drawn, it implies X lies on the edge of the left 2.5% tail. We will call this maximum value max. CONFIDENCE INTERVALS probability density function of X conditional on = max being true null hypothesis H 0 : = max

17
17 max max +1.96sd max –sd max +sd X If the hypothetical value of were any higher, it would be rejected because X would lie inside the left tail of the probability distribution. CONFIDENCE INTERVALS probability density function of X conditional on = max being true null hypothesis H 0 : = max

18
18 max max +1.96sd max –sd max +sd X Since X lies on the edge of the left 2.5% tail, it must be 1.96 standard deviations less than max. CONFIDENCE INTERVALS probability density function of X conditional on = max being true reject any > max X = max – 1.96 sd null hypothesis H 0 : = max

19
19 max max +1.96sd max –sd max +sd X Hence, knowing X and the standard deviation, we can calculate max. CONFIDENCE INTERVALS probability density function of X conditional on = max being true reject any > max X = max – 1.96 sd max = X + 1.96 sd reject any > X + 1.96 sd null hypothesis H 0 : = max

20
20 In the same way, we can identify the lowest hypothetical value of that is not contradicted by our estimate. min min –sd min –1.96sd min +sd X CONFIDENCE INTERVALS probability density function of X conditional on = min being true null hypothesis H 0 : = min

21
21 min min +sd X It implies that X lies on the edge of the right 2.5% tail of the associated conditional probability distribution. We will call it min. min –sd min –1.96sd CONFIDENCE INTERVALS probability density function of X conditional on = min being true null hypothesis H 0 : = min

22
22 min min +sd X min –sd min –1.96sd Any lower hypothetical value would be incompatible with our estimate. CONFIDENCE INTERVALS probability density function of X conditional on = min being true null hypothesis H 0 : = min

23
23 min min +sd X min –sd min –1.96sd Since X lies on the edge of the right 2.5% tail, X is equal to min plus 1.96 standard deviations. Hence min is equal to X minus 1.96 standard deviations. CONFIDENCE INTERVALS probability density function of X conditional on = min being true reject any < min X = min + 1.96 sd min = X – 1.96 sd reject any < X – 1.96 sd null hypothesis H 0 : = min

24
The diagram shows the limiting values of the hypothetical values of , together with their associated probability distributions for X. 24 (1)(2) min min +sd X min –sd min –1.96sd max max +1.96sd max –sd max +sd CONFIDENCE INTERVALS probability density function of X (1) conditional on = max being true (2) conditional on = min being true

25
25 (1)(2) min min +sd X min –sd min –1.96sd max max +1.96sd max –sd max +sd Any hypothesis lying in the interval from min to max would be compatible with the sample estimate (not be rejected by it). We call this interval the 95% confidence interval. CONFIDENCE INTERVALS reject any > max = X + 1.96 sd reject any < min = X – 1.96 sd 95% confidence interval: X – 1.96 sd ≤ ≤ X + 1.96 sd

26
26 (1)(2) min min +sd X min –sd min –1.96sd max max +1.96sd max –sd max +sd The name arises from a different application of the interval. It can be shown that it will include the true value of the coefficient with 95% probability, provided that the model is correctly specified. CONFIDENCE INTERVALS reject any > max = X + 1.96 sd reject any < min = X – 1.96 sd 95% confidence interval: X – 1.96 sd ≤ ≤ X + 1.96 sd

27
27 In exactly the same way, using a 1% significance test to identify hypotheses compatible with our sample estimate, we can construct a 99% confidence interval. CONFIDENCE INTERVALS 95% confidence interval X – 1.96 sd ≤ ≤ X + 1.96 sd 99% confidence interval X – 2.58 sd ≤ ≤ X + 2.58 sd Standard deviation known

28
28 min and max will now be 2.58 standard deviations to the left and to the right of X, respectively. CONFIDENCE INTERVALS 95% confidence interval X – 1.96 sd ≤ ≤ X + 1.96 sd 99% confidence interval X – 2.58 sd ≤ ≤ X + 2.58 sd Standard deviation known

29
29 Until now we have assumed that we know the standard deviation of the distribution. In practice we have to estimate it. CONFIDENCE INTERVALS 95% confidence interval X – 1.96 sd ≤ ≤ X + 1.96 sd 99% confidence interval X – 2.58 sd ≤ ≤ X + 2.58 sd 95% confidence interval X – t crit (5%) se ≤ ≤ X + t crit (5%) se 99% confidence interval X – t crit (1%) se ≤ ≤ X + t crit (1%) se Standard deviation estimated by standard error Standard deviation known

30
30 As a consequence, the t distribution has to be used instead of the normal distribution when locating min and max. CONFIDENCE INTERVALS 95% confidence interval X – 1.96 sd ≤ ≤ X + 1.96 sd 99% confidence interval X – 2.58 sd ≤ ≤ X + 2.58 sd 95% confidence interval X – t crit (5%) se ≤ ≤ X + t crit (5%) se 99% confidence interval X – t crit (1%) se ≤ ≤ X + t crit (1%) se Standard deviation estimated by standard error Standard deviation known

31
This implies that the standard error should be multiplied by the critical value of t, given the significance level and number of degrees of freedom, when determining the limits of the interval. 95% confidence interval X – 1.96 sd ≤ ≤ X + 1.96 sd 99% confidence interval X – 2.58 sd ≤ ≤ X + 2.58 sd 31 CONFIDENCE INTERVALS 95% confidence interval X – t crit (5%) se ≤ ≤ X + t crit (5%) se 99% confidence interval X – t crit (1%) se ≤ ≤ X + t crit (1%) se Standard deviation estimated by standard error Standard deviation known

32
Copyright Christopher Dougherty 2012. 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.12 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 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 EC2020 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2012.11.02

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google