Introduction to Econometrics, 5th edition Type author name/s here Dougherty Introduction to Econometrics, 5th edition Chapter heading Review: Random Variables, Sampling, Estimation, and Inference © Christopher Dougherty, 2016. All rights reserved.

CONFIDENCE INTERVALS null hypothesis H0: m = m0 2.5% 2.5% X probability density function of X conditional on m = m0 being true 2.5% 2.5% X m0–1.96sd m0–sd m0 m0+sd m0+1.96sd In the sequence on hypothesis testing, we started with a given hypothesis, for example H0: m = m0, and considered whether an estimate X derived from a sample would or would not lead to its rejection. 1

CONFIDENCE INTERVALS null hypothesis H0: m = m0 2.5% 2.5% X probability density function of X conditional on m = m0 being true 2.5% 2.5% X m0–1.96sd m0–sd m0 m0+sd m0+1.96sd Using a 5% significance test, this estimate would not lead to the rejection of H0. 2

CONFIDENCE INTERVALS null hypothesis H0: m = m0 2.5% 2.5% X probability density function of X conditional on m = m0 being true 2.5% 2.5% X m0–1.96sd m0–sd m0 m0+sd m0+1.96sd This estimate would cause H0 to be rejected. 3

CONFIDENCE INTERVALS null hypothesis H0: m = m0 2.5% 2.5% X probability density function of X conditional on m = m0 being true 2.5% 2.5% X m0–1.96sd m0–sd m0 m0+sd m0+1.96sd So would this one. 4

acceptance region for X CONFIDENCE INTERVALS null hypothesis H0: m = m0 probability density function of X conditional on m = m0 being true acceptance region for X 2.5% 2.5% X m0–1.96sd m0–sd m0 m0+sd m0+1.96sd We ended by deriving the range of estimates that are compatible with H0 and called it the acceptance region. 5

CONFIDENCE INTERVALS X 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. 6

CONFIDENCE INTERVALS null hypothesis H0: m = m0 X Suppose that someone came up with the hypothesis H0: m = m0. 7

CONFIDENCE INTERVALS null hypothesis H0: m = m0 X probability density function of X conditional on m = m0 being true X m0–1.96sd m0–sd m0 m0+1.96sd To see if it is compatible with our sample estimate X, we need to draw the probability distribution of X conditional on H0: m = m0 being true. 8

CONFIDENCE INTERVALS null hypothesis H0: m = m0 X probability density function of X conditional on m = m0 being true X m0–1.96sd m0–sd m0 m0+1.96sd 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. 9

CONFIDENCE INTERVALS null hypothesis H0: m = m0 X probability density function of X conditional on m = m0 being true X m0–1.96sd m0–sd m0 m0+1.96sd Having drawn the distribution, it is clear this null hypothesis is not contradicted by our sample estimate. 10

CONFIDENCE INTERVALS null hypothesis H0: m = m1 X Here is another null hypothesis to consider. 11

CONFIDENCE INTERVALS null hypothesis H0: m = m1 X probability density function of X conditional on m = m1 being true X m1–1.96sd m1 m1+sd m1+1.96sd Having drawn the distribution of X conditional on this hypothesis being true, we can see that this hypothesis is also compatible with our estimate. 12

CONFIDENCE INTERVALS null hypothesis H0: m = m2 X Here is another null hypothesis. 13

CONFIDENCE INTERVALS null hypothesis H0: m = m2 X probability density function of X conditional on m = m2 being true X m2–1.96sd m2–sd m2 m2+sd It is incompatible with our estimate because the estimate would lead to its rejection. 14

CONFIDENCE INTERVALS null hypothesis H0: m = m max X probability density function of X conditional on m = m max being true X mmax–sd mmax mmax+sd mmax+1.96sd The highest hypothetical value of m not contradicted by our sample estimate is shown in the diagram. 15

CONFIDENCE INTERVALS null hypothesis H0: m = m max X probability density function of X conditional on m = m max being true X mmax–sd mmax mmax+sd mmax+1.96sd 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 mmax. 16

CONFIDENCE INTERVALS null hypothesis H0: m = m max X probability density function of X conditional on m = m max being true X mmax–sd mmax mmax+sd mmax+1.96sd If the hypothetical value of m were any higher, it would be rejected because X would lie inside the left tail of the probability distribution. 17

CONFIDENCE INTERVALS null hypothesis H0: m = m max probability density function of X conditional on m = m max being true reject any m > mmax X = mmax – 1.96 sd X mmax–sd mmax mmax+sd mmax+1.96sd Since X lies on the edge of the left 2.5% tail, it must be 1.96 standard deviations less than mmax. 18

CONFIDENCE INTERVALS null hypothesis H0: m = m max probability density function of X conditional on m = m max being true reject any m > mmax X = mmax – 1.96 sd mmax = X + 1.96 sd reject any m > X + 1.96 sd X mmax–sd mmax mmax+sd mmax+1.96sd Hence, knowing X and the standard deviation, we can calculate mmax. 19

CONFIDENCE INTERVALS null hypothesis H0: m = m min X probability density function of X conditional on m = m min being true X mmin–1.96sd mmin–sd mmin mmin +sd In the same way, we can identify the lowest hypothetical value of m that is not contradicted by our estimate. 20

CONFIDENCE INTERVALS null hypothesis H0: m = m min X probability density function of X conditional on m = m min being true X mmin–1.96sd mmin–sd mmin mmin +sd It implies that X lies on the edge of the right 2.5% tail of the associated conditional probability distribution. We will call it mmin. 21

CONFIDENCE INTERVALS null hypothesis H0: m = m min X probability density function of X conditional on m = m min being true X mmin–1.96sd mmin–sd mmin mmin +sd Any lower hypothetical value would be incompatible with our estimate. 22

CONFIDENCE INTERVALS null hypothesis H0: m = m min probability density function of X conditional on m = m min being true reject any m < m min X = m min + 1.96 sd m min = X – 1.96 sd reject any m < X – 1.96 sd X mmin–1.96sd mmin–sd mmin mmin +sd Since X lies on the edge of the right 2.5% tail, X is equal to mmin plus 1.96 standard deviations. Hence mmin is equal to X minus 1.96 standard deviations. 23

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

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

95% confidence interval: CONFIDENCE INTERVALS reject any m > m max = X + 1.96 sd reject any m < m min = X – 1.96 sd 95% confidence interval: X – 1.96 sd ≤ m ≤ X + 1.96 sd (2) (1) X mmin–1.96sd mmin–sd mmin mmin +sd mmax–sd mmax mmax+sd mmax+1.96sd 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. 26

Standard deviation known CONFIDENCE INTERVALS Standard deviation known 95% confidence interval X – 1.96 sd ≤ m ≤ X + 1.96 sd 99% confidence interval X – 2.58 sd ≤ m ≤ X + 2.58 sd 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. 27

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

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

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

Standard deviation known CONFIDENCE INTERVALS Standard deviation known 95% confidence interval X – 1.96 sd ≤ m ≤ X + 1.96 sd 99% confidence interval X – 2.58 sd ≤ m ≤ X + 2.58 sd Standard deviation estimated by standard error 95% confidence interval X – tcrit (5%) se ≤ m ≤ X + tcrit (5%) se 99% confidence interval X – tcrit (1%) se ≤ m ≤ X + tcrit (1%) se 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. 31

Copyright Christopher Dougherty 2016. 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, fifth edition 2016, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oxfordtextbooks.co.uk/orc/dougherty5e/ 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 or the University of London International Programmes distance learning course EC2020 Elements of Econometrics www.londoninternational.ac.uk/lse. 2015.12.17