1. 1 We will now look at the properties of the OLS regression estimators with the assumptions of Model B. We will do this within the context of the simple regression model. We will start by demonstrating unbiasedness. MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS

2. We saw in Chapter 2 that the slope coefficient can be decomposed into the true value plus a weighted linear combination of the values of the disturbance term in the sample, where the weights depend on the observations on X. 2 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS

3. We now take expectations.  2 is just a constant, so it is unaffected. 3 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS

4. We have now used the first expectation rule to rewrite the expectation of the linear combination as the sum of the expectations of its components. 4 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS

5. In Model A, the values of X were nonstochastic. This meant that the a i terms were also nonstochastic and could therefore be taken out of the expectations as factors. E(u i ) = 0 for all i, and hence we proved unbiasedness. 5 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS Model A

6. We cannot do this with Model B because we are assuming that the values of X are generated randomly (from a defined population). 6 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS Model A

7. Instead we appeal to Assumption B.7. We saw in the Review chapter that if X and Y are two independent random variables, the expectation of the product of functions of them can be decomposed as the product of the expectations of the functions. 7 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS Model B

8. Under Assumption B.7, u i is distributed independently of every value of X in the sample. It is therefore distributed independently of a i. So if X and u are independent, we can make use of the decomposition. 8 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS Model B

9. Since E(u i ) = 0 for all i, under Assumption B.4, we have proved unbiasedness, assuming E(a i ) exists. For this to be the case, there must be some variation in X in the sample (Assumption B.3). Otherwise the denominator of the expression for a i would be zero. 9 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS Model B

10. The next property, efficiency, we will take for granted. The Gauss–Markov theorem assures that the OLS estimators are BLUE (best linear unbiased estimators), provided that the regression model assumptions are valid. 10 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS

11. We will prove consistency. We have decomposed the limiting value of the estimator of the slope coefficient into the true value and the limiting value of the error term. 11 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS

12. We would now like to use the plim quotient rule. The plim of a quotient is the plim of the numerator divided by the plim of the denominator, provided that both of these limits exist. 12 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS if A and B have probability limits and plim B is not 0.

13. 13 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS However, as the expression stands, the numerator and the denominator of the error term do not have limits. The denominator increases indefinitely and the numerator does not converge on a limit. if A and B have probability limits and plim B is not 0.

14. 14 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS To deal with this problem, we divide both the numerator and the denominator by n. if A and B have probability limits and plim B is not 0.

15. 15 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS It can be shown that the limit of the numerator is the covariance of X and u and the limit of the denominator is the variance of X.

16. Under Assumption B.7, X and u are independent. Hence the covariance of X and u is zero (see the Review chapter). 16 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS

17. Thus we demonstrate that b 2 is a consistent estimator of  2, provided that the regression model assumptions are valid. 17 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS

18. Finally, a note on Assumption B.8, that the disturbance term has a normal distribution. The justification is that it is reasonable to suppose that the disturbance term is jointly generated by a number of minor random factors. 18 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS B.1The model is linear in parameters and correctly specified. Y =  1 +  2 X 2 + … +  k X k + u B.2The values of the regressors are drawn randomly from fixed populations. B.3There does not exist an exact linear relationship among the regressors. B.4The disturbance term has zero expectation. B.5The disturbance term is homoscedastic. B.6The values of the disturbance term have independent distributions. B.7The disturbance term is distributed independently of the regressors. B.8The disturbance term has a normal distribution.

19. A central limit theorem states that the combination of these factors should approximately have a normal distribution, even if the individual factors do not. 19 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS B.8The disturbance term has a normal distribution.

20. If the disturbance term has a normal distribution, the regression coefficients also have normal distributions. This follows from the fact that a linear combination of normal distributions is also normal. 20 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS B.8The disturbance term has a normal distribution.

21. What happens if we have reason to believe that the assumption is not valid? The central limit theorem comes into the frame a second time. 21 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS B.8The disturbance term has a normal distribution.

22. The random component of a regression coefficient is a linear combination of the values of the disturbance term in the sample. 22 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS B.8The disturbance term has a normal distribution.

23. By a central limit theorem, it follows that the combination will have an approximately normal distribution, even if the individual values of the disturbance term do not, provided that the sample is large enough. 23 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS B.8The disturbance term has a normal distribution.

24. Hence asymptotically (in large samples) it ought to be safe to assume that the regression coefficients have normal distributions, even if Assumption B.8 is invalid, provided that the other regression model assumptions are satisfied. 24 MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS B.8The disturbance term has a normal distribution.

25. 2013.08.04 Copyright Christopher Dougherty 2013. 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 8.3 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.