Christopher Dougherty EC220 - Introduction to econometrics (chapter 8) Slideshow: model b: properties of the regression coefficients Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 8). [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

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

We have seen 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. MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 2

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

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. MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 4

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. MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS Model A: 5

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

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. MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 7

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. MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 8

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. MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 9

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. MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 10

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. MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 11

We divide the numerator and the denominator of the error term by n. We do this because they then have finite probability limits and we can make use of the plim quotient rule shown. If we did not divide by n, they would increase without limit as n increases. MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS provided both plim A and plim B exist 12

We have now made use of the plim quotient rule. MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 13

Given the regression model assumptions, it can be shown that the denominator converges on the variance of X as the sample size becomes large, using a Law of Large Numbers. It can also be shown that the numerator converges on the covariance of X and u. MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 14

Assumption B.7 states that the covariance of X and u is zero. Hence we have proved consistency, provided that the regression model assumptions are valid. MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS 15

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. MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS ASSUMPTIONS FOR MODEL B 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 16

MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS ASSUMPTIONS FOR MODEL B 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 A central limit theorem states that the combination of these factors should approximately have a normal distribution, even if the individual factors do not. 17

MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS ASSUMPTIONS FOR MODEL B 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 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. 18

MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS ASSUMPTIONS FOR MODEL B 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 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. 19

MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS ASSUMPTIONS FOR MODEL B 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 The random component of a regression coefficient is a linear combination of the values of the disturbance term in the sample. 20

MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS ASSUMPTIONS FOR MODEL B 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 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. 21

MODEL B: PROPERTIES OF THE REGRESSION COEFFICIENTS ASSUMPTIONS FOR MODEL B 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 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. 22

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 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 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