Presentation is loading. Please wait.

Presentation is loading. Please wait.

Introduction to Econometrics, 5th edition

Published byHoratio Malone Modified over 6 years ago

Similar presentations

Presentation on theme: "Introduction to Econometrics, 5th edition"— Presentation transcript:

1 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, All rights reserved.

2 EXPECTED VALUE OF A RANDOM VARIABLE
Definition of E(X), the expected value of X: The expected value of a random variable, also known as its population mean, is the weighted average of its possible values, the weights being the probabilities attached to the values. 1

3 EXPECTED VALUE OF A RANDOM VARIABLE
Definition of E(X), the expected value of X: Note that the sum of the probabilities must be unity, so there is no need to divide by the sum of the weights. 2

4 EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi x1 p1 x1 p1 2 1/36 2/36 x2 p2 x2 p2 3 2/36 6/36 x3 p3 x3 p3 4 3/36 12/36 x4 p4 x4 p4 5 4/36 20/36 x5 p5 x5 p5 6 5/36 30/36 x6 p6 x6 p6 7 6/36 42/36 x7 p7 x7 p7 8 5/36 40/36 x8 p8 x8 p8 9 4/36 36/36 x9 p9 x9 p9 10 3/36 30/36 x10 p10 x10 p /36 22/36 x11 p11 x11 p /36 12/36 S xi pi = E(X) This sequence shows how the expected value is calculated, first in abstract and then with the random variable defined in the first sequence. We begin by listing the possible values of X. 3

5 EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi x1 p1 x1 p1 2 1/36 2/36 x2 p2 x2 p2 3 2/36 6/36 x3 p3 x3 p3 4 3/36 12/36 x4 p4 x4 p4 5 4/36 20/36 x5 p5 x5 p5 6 5/36 30/36 x6 p6 x6 p6 7 6/36 42/36 x7 p7 x7 p7 8 5/36 40/36 x8 p8 x8 p8 9 4/36 36/36 x9 p9 x9 p9 10 3/36 30/36 x10 p10 x10 p /36 22/36 x11 p11 x11 p /36 12/36 S xi pi = E(X) Next we list the probabilities attached to the different possible values of X. 4

6 EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi x1 p1 x1 p1 2 1/36 2/36 x2 p2 x2 p2 3 2/36 6/36 x3 p3 x3 p3 4 3/36 12/36 x4 p4 x4 p4 5 4/36 20/36 x5 p5 x5 p5 6 5/36 30/36 x6 p6 x6 p6 7 6/36 42/36 x7 p7 x7 p7 8 5/36 40/36 x8 p8 x8 p8 9 4/36 36/36 x9 p9 x9 p9 10 3/36 30/36 x10 p10 x10 p /36 22/36 x11 p11 x11 p /36 12/36 S xi pi = E(X) Then we define a column in which the values are weighted by the corresponding probabilities. 5

7 EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi x1 p1 x1 p1 2 1/36 2/36 x2 p2 x2 p2 3 2/36 6/36 x3 p3 x3 p3 4 3/36 12/36 x4 p4 x4 p4 5 4/36 20/36 x5 p5 x5 p5 6 5/36 30/36 x6 p6 x6 p6 7 6/36 42/36 x7 p7 x7 p7 8 5/36 40/36 x8 p8 x8 p8 9 4/36 36/36 x9 p9 x9 p9 10 3/36 30/36 x10 p10 x10 p /36 22/36 x11 p11 x11 p /36 12/36 S xi pi = E(X) We do this for each value separately. 6

8 EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi x1 p1 x1 p1 2 1/36 2/36 x2 p2 x2 p2 3 2/36 6/36 x3 p3 x3 p3 4 3/36 12/36 x4 p4 x4 p4 5 4/36 20/36 x5 p5 x5 p5 6 5/36 30/36 x6 p6 x6 p6 7 6/36 42/36 x7 p7 x7 p7 8 5/36 40/36 x8 p8 x8 p8 9 4/36 36/36 x9 p9 x9 p9 10 3/36 30/36 x10 p10 x10 p /36 22/36 x11 p11 x11 p /36 12/36 S xi pi = E(X) Here we are assuming that n, the number of possible values, is equal to 11, but it could be any number. 7

9 EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi x1 p1 x1 p1 2 1/36 2/36 x2 p2 x2 p2 3 2/36 6/36 x3 p3 x3 p3 4 3/36 12/36 x4 p4 x4 p4 5 4/36 20/36 x5 p5 x5 p5 6 5/36 30/36 x6 p6 x6 p6 7 6/36 42/36 x7 p7 x7 p7 8 5/36 40/36 x8 p8 x8 p8 9 4/36 36/36 x9 p9 x9 p9 10 3/36 30/36 x10 p10 x10 p /36 22/36 x11 p11 x11 p /36 12/36 S xi pi = E(X) The expected value is the sum of the entries in the third column. 8

10 EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi x1 p1 x1 p1 2 1/36 2/36 x2 p2 x2 p2 3 2/36 6/36 x3 p3 x3 p3 4 3/36 12/36 x4 p4 x4 p4 5 4/36 20/36 x5 p5 x5 p5 6 5/36 30/36 x6 p6 x6 p6 7 6/36 42/36 x7 p7 x7 p7 8 5/36 40/36 x8 p8 x8 p8 9 4/36 36/36 x9 p9 x9 p9 10 3/36 30/36 x10 p10 x10 p /36 22/36 x11 p11 x11 p /36 12/36 S xi pi = E(X) The random variable X defined in the previous sequence could be any of the integers from 2 to 12 with probabilities as shown. 9

11 EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi x1 p1 x1 p1 2 1/36 2/36 x2 p2 x2 p2 3 2/36 6/36 x3 p3 x3 p3 4 3/36 12/36 x4 p4 x4 p4 5 4/36 20/36 x5 p5 x5 p5 6 5/36 30/36 x6 p6 x6 p6 7 6/36 42/36 x7 p7 x7 p7 8 5/36 40/36 x8 p8 x8 p8 9 4/36 36/36 x9 p9 x9 p9 10 3/36 30/36 x10 p10 x10 p /36 22/36 x11 p11 x11 p /36 12/36 S xi pi = E(X) X could be equal to 2 with probability 1/36, so the first entry in the calculation of the expected value is 2/36. 10

12 EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi x1 p1 x1 p1 2 1/36 2/36 x2 p2 x2 p2 3 2/36 6/36 x3 p3 x3 p3 4 3/36 12/36 x4 p4 x4 p4 5 4/36 20/36 x5 p5 x5 p5 6 5/36 30/36 x6 p6 x6 p6 7 6/36 42/36 x7 p7 x7 p7 8 5/36 40/36 x8 p8 x8 p8 9 4/36 36/36 x9 p9 x9 p9 10 3/36 30/36 x10 p10 x10 p /36 22/36 x11 p11 x11 p /36 12/36 S xi pi = E(X) The probability of x being equal to 3 was 2/36, so the second entry is 6/36. 11

13 EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi x1 p1 x1 p1 2 1/36 2/36 x2 p2 x2 p2 3 2/36 6/36 x3 p3 x3 p3 4 3/36 12/36 x4 p4 x4 p4 5 4/36 20/36 x5 p5 x5 p5 6 5/36 30/36 x6 p6 x6 p6 7 6/36 42/36 x7 p7 x7 p7 8 5/36 40/36 x8 p8 x8 p8 9 4/36 36/36 x9 p9 x9 p9 10 3/36 30/36 x10 p10 x10 p /36 22/36 x11 p11 x11 p /36 12/36 S xi pi = E(X) Similarly for the other 9 possible values. 12

14 EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi x1 p1 x1 p1 2 1/36 2/36 x2 p2 x2 p2 3 2/36 6/36 x3 p3 x3 p3 4 3/36 12/36 x4 p4 x4 p4 5 4/36 20/36 x5 p5 x5 p5 6 5/36 30/36 x6 p6 x6 p6 7 6/36 42/36 x7 p7 x7 p7 8 5/36 40/36 x8 p8 x8 p8 9 4/36 36/36 x9 p9 x9 p9 10 3/36 30/36 x10 p10 x10 p /36 22/36 x11 p11 x11 p /36 12/36 S xi pi = E(X) /36 To obtain the expected value, we sum the entries in this column. 13

15 EXPECTED VALUE OF A RANDOM VARIABLE
xi pi xi pi xi pi xi pi x1 p1 x1 p1 2 1/36 2/36 x2 p2 x2 p2 3 2/36 6/36 x3 p3 x3 p3 4 3/36 12/36 x4 p4 x4 p4 5 4/36 20/36 x5 p5 x5 p5 6 5/36 30/36 x6 p6 x6 p6 7 6/36 42/36 x7 p7 x7 p7 8 5/36 40/36 x8 p8 x8 p8 9 4/36 36/36 x9 p9 x9 p9 10 3/36 30/36 x10 p10 x10 p /36 22/36 x11 p11 x11 p /36 12/36 S xi pi = E(X) /36 = 7 The expected value turns out to be 7. Actually, this was obvious anyway. We saw in the previous sequence that the distribution is symmetrical about 7. 14

16 EXPECTED VALUE OF A RANDOM VARIABLE
Alternative notation for E(X): Very often the expected value of a random variable is represented by m, the Greek m. If there is more than one random variable, their expected values are differentiated by adding subscripts to m. 15

17 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.2 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 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

Download ppt "Introduction to Econometrics, 5th edition"

Similar presentations

EXPECTED VALUE RULES 1. This sequence states the rules for manipulating expected values. First, the additive rule. The expected value of the sum of two.

EXPECTED VALUE RULES 1. This sequence states the rules for manipulating expected values. First, the additive rule. The expected value of the sum of two.
THE ERROR CORRECTION MODEL 1 The error correction model is a variant of the partial adjustment model. As with the partial adjustment model, we assume a.

THE ERROR CORRECTION MODEL 1 The error correction model is a variant of the partial adjustment model. As with the partial adjustment model, we assume a.
1 This very short sequence presents an important definition, that of the independence of two random variables. Two random variables X and Y are said to.

1 This very short sequence presents an important definition, that of the independence of two random variables. Two random variables X and Y are said to.
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: expected value of a random variable Original citation: Dougherty,

Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: expected value of a random variable Original citation: Dougherty,
The third sequence defined the expected value of a function of a random variable X. There is only one function that is of much interest to us, at least.

The third sequence defined the expected value of a function of a random variable X. There is only one function that is of much interest to us, at least.
Christopher Dougherty EC220 - Introduction to econometrics (chapter 10) Slideshow: introduction to maximum likelihood estimation Original citation: Dougherty,

Christopher Dougherty EC220 - Introduction to econometrics (chapter 10) Slideshow: introduction to maximum likelihood estimation Original citation: Dougherty,
1 THE DISTURBANCE TERM IN LOGARITHMIC MODELS Thus far, nothing has been said about the disturbance term in nonlinear regression models.

1 THE DISTURBANCE TERM IN LOGARITHMIC MODELS Thus far, nothing has been said about the disturbance term in nonlinear regression models.
1 XX X1X1 XX X Random variable X with unknown population mean  X function of X probability density Sample of n observations X 1, X 2,..., X n : potential.

1 XX X1X1 XX X Random variable X with unknown population mean  X function of X probability density Sample of n observations X 1, X 2,..., X n : potential.
1 THE NORMAL DISTRIBUTION In the analysis so far, we have discussed the mean and the variance of a distribution of a random variable, but we have not said.

1 THE NORMAL DISTRIBUTION In the analysis so far, we have discussed the mean and the variance of a distribution of a random variable, but we have not said.
1 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 This sequence provides an example of a discrete random variable. Suppose that you.

1 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red This sequence provides an example of a discrete random variable. Suppose that you.
Random effects estimation RANDOM EFFECTS REGRESSIONS When the observed variables of interest are constant for each individual, a fixed effects regression.

Random effects estimation RANDOM EFFECTS REGRESSIONS When the observed variables of interest are constant for each individual, a fixed effects regression.
1 ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA Assumptions C.1, C.3, C.4, C.5, and C.8, and the consequences of their violations are the.

1 ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA Assumptions C.1, C.3, C.4, C.5, and C.8, and the consequences of their violations are the.
00  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.

00  sd  0 –sd  0 –1.96sd  0 +sd 2.5% CONFIDENCE INTERVALS probability density function of X null hypothesis H 0 :  =  0 In the sequence.
EXPECTED VALUE OF A RANDOM VARIABLE 1 The expected value of a random variable, also known as its population mean, is the weighted average of its possible.

EXPECTED VALUE OF A RANDOM VARIABLE 1 The expected value of a random variable, also known as its population mean, is the weighted average of its possible.
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: expected value of a function of a random variable Original citation:

Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: expected value of a function of a random variable Original citation:
TESTING A HYPOTHESIS RELATING TO THE POPULATION MEAN 1 This sequence describes the testing of a hypothesis at the 5% and 1% significance levels. It also.

TESTING A HYPOTHESIS RELATING TO THE POPULATION MEAN 1 This sequence describes the testing of a hypothesis at the 5% and 1% significance levels. It also.
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: confidence intervals Original citation: Dougherty, C. (2012) EC220.

Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: confidence intervals Original citation: Dougherty, C. (2012) EC220.
EC220 - Introduction to econometrics (review chapter)

EC220 - Introduction to econometrics (review chapter)
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT This sequence describes the testing of a hypotheses relating to regression coefficients. It is.

TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT This sequence describes the testing of a hypotheses relating to regression coefficients. It is.
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: continuous random variables Original citation: Dougherty, C. (2012)

Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: continuous random variables Original citation: Dougherty, C. (2012)

Similar presentations

Ads by Google