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Christopher Dougherty EC220 - Introduction to econometrics (chapter 3) Slideshow: properties of the multiple regression coefficients Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 3). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/129/http://learningresources.lse.ac.uk/129/ 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/

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A.1The model is linear in parameters and correctly specified. A.2There does not exist an exact linear relationship among the regressors in the sample. A.3The disturbance term has zero expectation A.4The disturbance term is homoscedastic A.5The values of the disturbance term have independent distributions A.6The disturbance term has a normal distribution PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS 1 Moving from the simple to the multiple regression model, we start by restating the regression model assumptions.

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A.1The model is linear in parameters and correctly specified. A.2There does not exist an exact linear relationship among the regressors in the sample. A.3The disturbance term has zero expectation A.4The disturbance term is homoscedastic A.5The values of the disturbance term have independent distributions A.6The disturbance term has a normal distribution PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS Only A.2 is different. Previously it stated that there must be some variation in the X variable. We will explain the difference in one of the following slideshows. 2

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A.1The model is linear in parameters and correctly specified. A.2There does not exist an exact linear relationship among the regressors in the sample. A.3The disturbance term has zero expectation A.4The disturbance term is homoscedastic A.5The values of the disturbance term have independent distributions A.6The disturbance term has a normal distribution PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS Provided that the regression model assumptions are valid, the OLS estimators in the multiple regression model are unbiased and efficient, as in the simple regression model. 3

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PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS We will not attempt to prove efficiency. We will however outline a proof of unbiasedness. 4

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PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS The first step, as always, is to substitute for Y from the true relationship. The Y ingredients of b 2 are actually in the form of Y i minus its mean, so it is convenient to obtain an expression for this. 5

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PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS After substituting, and simplifying, we find that b 2 can be decomposed into the true value 2 plus a weighted linear combination of the values of the disturbance term in the sample. 6

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PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS This is what we found in the simple regression model. The difference is that the expression for the weights, which depend on all the values of X 2 and X 3 in the sample, is considerably more complicated. 7

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PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS Having reached this point, proving unbiasedness is easy. Taking expectations, 2 is unaffected, being a constant. The expectation of a sum is equal to the sum of expectations. 8

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PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS The a* terms are nonstochastic since they depend only on the values of X 2 and X 3, and these are assumed to be nonstochastic. Hence the a* terms may be taken out of the expectations as factors. 9

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PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS By Assumption A.3, E(u i ) = 0 for all i. Hence E(b 2 ) is equal to 2 and so b 2 is an unbiased estimator. Similarly b 3 is an unbiased estimator of 3. 10

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PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS Finally we will show that b 1 is an unbiased estimator of 1. This is quite simple, so you should attempt to do this yourself, before looking at the rest of this sequence. 11

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PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS First substitute for the sample mean of Y. 12

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PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS Now take expectations. The first three terms are nonstochastic, so they are unaffected by taking expectations. 13

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PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS The expected value of the mean of the disturbance term is zero since E(u) is zero in each observation. We have just shown that E(b 2 ) is equal to 2 and that E(b 3 ) is equal to 3. 14

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PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS Hence b 1 is an unbiased estimator of 1. 15

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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 3.3 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

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