Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: alternative expression for population variance Original citation:

Slides:



Advertisements
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.
Advertisements

EC220 - Introduction to econometrics (chapter 14)
EC220 - Introduction to econometrics (chapter 11)
EC220 - Introduction to econometrics (review chapter)
Christopher Dougherty EC220 - Introduction to econometrics (chapter 12) Slideshow: consequences of autocorrelation Original citation: Dougherty, C. (2012)
Christopher Dougherty EC220 - Introduction to econometrics (chapter 11) Slideshow: model c assumptions Original citation: Dougherty, C. (2012) EC220 -
EC220 - Introduction to econometrics (chapter 8)
Christopher Dougherty EC220 - Introduction to econometrics (chapter 8) Slideshow: model b: properties of the regression coefficients Original citation:
EC220 - Introduction to econometrics (chapter 1)
EC220 - Introduction to econometrics (chapter 3)
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.
EC220 - Introduction to econometrics (review chapter)
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: population variance of a discreet random variable Original citation:
Christopher Dougherty EC220 - Introduction to econometrics (chapter 2) Slideshow: a Monte Carlo experiment Original citation: Dougherty, C. (2012) EC220.
Christopher Dougherty EC220 - Introduction to econometrics (chapter 10) Slideshow: introduction to maximum likelihood estimation Original citation: Dougherty,
Christopher Dougherty EC220 - Introduction to econometrics (chapter 11) Slideshow: adaptive expectations Original citation: Dougherty, C. (2012) EC220.
EC220 - Introduction to econometrics (chapter 7)
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.
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: asymptotic properties of estimators: plims and consistency Original.
Christopher Dougherty EC220 - Introduction to econometrics (chapter 13) Slideshow: stationary processes Original citation: Dougherty, C. (2012) EC220 -
Christopher Dougherty EC220 - Introduction to econometrics (chapter 13) Slideshow: tests of nonstationarity: introduction Original citation: Dougherty,
Christopher Dougherty EC220 - Introduction to econometrics (chapter 12) Slideshow: dynamic model specification Original citation: Dougherty, C. (2012)
Christopher Dougherty EC220 - Introduction to econometrics (chapter 2) Slideshow: testing a hypothesis relating to a regression coefficient Original citation:
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.
EC220 - Introduction to econometrics (chapter 7)
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.
EC220 - Introduction to econometrics (chapter 2)
EC220 - Introduction to econometrics (chapter 9)
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 (chapter 6) Slideshow: variable misspecification iii: consequences for diagnostics Original.
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: confidence intervals Original citation: Dougherty, C. (2012) EC220.
EC220 - Introduction to econometrics (chapter 1)
EC220 - Introduction to econometrics (review chapter)
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: continuous random variables Original citation: Dougherty, C. (2012)
Christopher Dougherty EC220 - Introduction to econometrics (chapter 3) Slideshow: prediction Original citation: Dougherty, C. (2012) EC220 - Introduction.
Christopher Dougherty EC220 - Introduction to econometrics (chapter 4) Slideshow: semilogarithmic models Original citation: Dougherty, C. (2012) EC220.
Christopher Dougherty EC220 - Introduction to econometrics (chapter 10) Slideshow: maximum likelihood estimation of regression coefficients Original citation:
EC220 - Introduction to econometrics (chapter 12)
Christopher Dougherty EC220 - Introduction to econometrics (chapter 5) Slideshow: Chow test Original citation: Dougherty, C. (2012) EC220 - Introduction.
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: the normal distribution Original citation: Dougherty, C. (2012)
Christopher Dougherty EC220 - Introduction to econometrics (chapter 5) Slideshow: dummy variable classification with two categories Original citation:
Christopher Dougherty EC220 - Introduction to econometrics (chapter 5) Slideshow: two sets of dummy variables Original citation: Dougherty, C. (2012) EC220.
EC220 - Introduction to econometrics (review chapter)
1 UNBIASEDNESS AND EFFICIENCY Much of the analysis in this course will be concerned with three properties of estimators: unbiasedness, efficiency, and.
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: sampling and estimators Original citation: Dougherty, C. (2012)
Christopher Dougherty EC220 - Introduction to econometrics (chapter 5) Slideshow: the effects of changing the reference category Original citation: Dougherty,
Christopher Dougherty EC220 - Introduction to econometrics (chapter 12) Slideshow: autocorrelation, partial adjustment, and adaptive expectations Original.
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: conflicts between unbiasedness and minimum variance Original citation:
Christopher Dougherty EC220 - Introduction to econometrics (chapter 8) Slideshow: measurement error Original citation: Dougherty, C. (2012) EC220 - Introduction.
THE FIXED AND RANDOM COMPONENTS OF A RANDOM VARIABLE 1 In this short sequence we shall decompose a random variable X into its fixed and random components.
Christopher Dougherty EC220 - Introduction to econometrics (chapter 11) Slideshow: Friedman Original citation: Dougherty, C. (2012) EC220 - Introduction.
CONSEQUENCES OF AUTOCORRELATION
ALTERNATIVE EXPRESSION FOR POPULATION VARIANCE 1 This sequence derives an alternative expression for the population variance of a random variable. It provides.
CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
EC220 - Introduction to econometrics (chapter 8)
Christopher Dougherty EC220 - Introduction to econometrics (chapter 12) Slideshow: footnote: the Cochrane-Orcutt iterative process Original citation: Dougherty,
Christopher Dougherty EC220 - Introduction to econometrics (chapter 9) Slideshow: instrumental variable estimation: variation Original citation: Dougherty,
Christopher Dougherty EC220 - Introduction to econometrics (chapter 6) Slideshow: multiple restrictions and zero restrictions Original citation: Dougherty,
1 COVARIANCE, COVARIANCE AND VARIANCE RULES, AND CORRELATION Covariance The covariance of two random variables X and Y, often written  XY, is defined.
1 We will continue with a variation on the basic model. We will now hypothesize that p is a function of m, the rate of growth of the money supply, as well.
Definition of, the expected value of a function of X : 1 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE To find the expected value of a function of.
1 ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION We have seen that the variance of a random variable X is given by the expression above. Variance.
Christopher Dougherty EC220 - Introduction to econometrics (chapter 2) Slideshow: confidence intervals Original citation: Dougherty, C. (2012) EC220 -
Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: independence of two random variables Original citation: Dougherty,
Christopher Dougherty EC220 - Introduction to econometrics (chapter 1) Slideshow: simple regression model Original citation: Dougherty, C. (2012) EC220.
FOOTNOTE: THE COCHRANE–ORCUTT ITERATIVE PROCESS 1 We saw in the previous sequence that AR(1) autocorrelation could be eliminated by a simple manipulation.
Introduction to Econometrics, 5th edition
Introduction to Econometrics, 5th edition
Presentation transcript:

Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: alternative expression for population variance Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource] © 2012 The Author This version available at: 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

ALTERNATIVE EXPRESSION FOR POPULATION VARIANCE 1 This sequence derives an alternative expression for the population variance of a random variable. It provides an opportunity for practising the use of the expected value rules. = E(X 2 ) –  2 = E[(X –  ) 2 ] = E(X 2 – 2  X +  2 ) = E(X 2 ) + E(–2  X) + E(  2 ) = E(X 2 ) – 2  E(X) +  2 = E(X 2 ) – 2  2 +  2 = E(X 2 ) –  2

ALTERNATIVE EXPRESSION FOR POPULATION VARIANCE 2 We start with the definition of the population variance of X. = E(X 2 ) –  2 = E[(X –  ) 2 ] = E(X 2 – 2  X +  2 ) = E(X 2 ) + E(–2  X) + E(  2 ) = E(X 2 ) – 2  E(X) +  2 = E(X 2 ) – 2  2 +  2 = E(X 2 ) –  2

ALTERNATIVE EXPRESSION FOR POPULATION VARIANCE 3 We expand the quadratic. = E(X 2 ) –  2 = E[(X –  ) 2 ] = E(X 2 – 2  X +  2 ) = E(X 2 ) + E(–2  X) + E(  2 ) = E(X 2 ) – 2  E(X) +  2 = E(X 2 ) – 2  2 +  2 = E(X 2 ) –  2

ALTERNATIVE EXPRESSION FOR POPULATION VARIANCE 4 Now the first expected value rule is used to decompose the expression into three separate expected values. = E(X 2 ) –  2 = E[(X –  ) 2 ] = E(X 2 – 2  X +  2 ) = E(X 2 ) + E(–2  X) + E(  2 ) = E(X 2 ) – 2  E(X) +  2 = E(X 2 ) – 2  2 +  2 = E(X 2 ) –  2

ALTERNATIVE EXPRESSION FOR POPULATION VARIANCE 5 The second expected value rule is used to simplify the middle term and the third rule is used to simplify the last one. = E(X 2 ) –  2 = E[(X –  ) 2 ] = E(X 2 – 2  X +  2 ) = E(X 2 ) + E(–2  X) + E(  2 ) = E(X 2 ) – 2  E(X) +  2 = E(X 2 ) – 2  2 +  2 = E(X 2 ) –  2

ALTERNATIVE EXPRESSION FOR POPULATION VARIANCE 6 The middle term is rewritten, using the fact that E(X) and  X are just different ways of writing the population mean of X. = E(X 2 ) –  2 = E[(X –  ) 2 ] = E(X 2 – 2  X +  2 ) = E(X 2 ) + E(–2  X) + E(  2 ) = E(X 2 ) – 2  E(X) +  2 = E(X 2 ) – 2  2 +  2 = E(X 2 ) –  2

= E(X 2 ) –  2 = E[(X –  ) 2 ] = E(X 2 – 2  X +  2 ) = E(X 2 ) + E(–2  X) + E(  2 ) = E(X 2 ) – 2  E(X) +  2 = E(X 2 ) – 2  2 +  2 = E(X 2 ) –  2 Hence we get the result. ALTERNATIVE EXPRESSION FOR POPULATION VARIANCE 7

Copyright Christopher Dougherty 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, fourth edition 2011, 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 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 or the University of London International Programmes distance learning course 20 Elements of Econometrics