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

2. A.1The model is linear in parameters and correctly specified. A.6The disturbance term has a normal distribution. A.5The values of the disturbance term have independent distributions A.4The disturbance term is homoscedastic A.3The disturbance term has zero expectation A.2There does not exist an exact linear relationship among the regressors in the sample. ASSUMPTIONS FOR MODEL A 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 PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS

3. 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 PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS A.1The model is linear in parameters and correctly specified. A.6The disturbance term has a normal distribution. A.5The values of the disturbance term have independent distributions A.4The disturbance term is homoscedastic A.3The disturbance term has zero expectation A.2There does not exist an exact linear relationship among the regressors in the sample. ASSUMPTIONS FOR MODEL A

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

5. 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 PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS Fitted modelTrue model

6. 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 PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS Fitted modelTrue model

7. 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 PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS Fitted modelTrue model

8. 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 PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS Fitted modelTrue model

9. 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 PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS Fitted modelTrue model

10. 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 PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS Fitted modelTrue model

11. 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 PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS Fitted modelTrue model

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

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

14. 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 PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS Fitted modelTrue model

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

16. Copyright Christopher Dougherty 2012. 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 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 EC2020 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2012.11.11