1. 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: http://learningresources.lse.ac.uk/141/http://learningresources.lse.ac.uk/141/ 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. http://creativecommons.org/licenses/by-sa/3.0/ http://creativecommons.org/licenses/by-sa/3.0/ http://learningresources.lse.ac.uk/

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. = 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

9. 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 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 http://www.oup.com/uk/orc/bin/9780199567089/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 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/lsewww.londoninternational.ac.uk/lse. 11.07.25