1. EC220 - Introduction to econometrics (chapter 3)
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
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Available in LSE Learning Resources Online: May 2012
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2. PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
A.1 The model is linear in parameters and correctly specified.
A.2 There does not exist an exact linear relationship among the regressors in the sample.
A.3 The disturbance term has zero expectation
A.4 The disturbance term is homoscedastic
A.5 The values of the disturbance term have independent distributions
A.6 The disturbance term has a normal distribution
Moving from the simple to the multiple regression model, we start by restating the regression model assumptions. 1

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

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

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

6. PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
The first step, as always, is to substitute for Y from the true relationship. The Y ingredients of b2 are actually in the form of Yi minus its mean, so it is convenient to obtain an expression for this. 5

7. PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
After substituting, and simplifying, we find that b2 can be decomposed into the true value b2 plus a weighted linear combination of the values of the disturbance term in the sample. 6

8. 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 X2 and X3 in the sample, is considerably more complicated. 7

9. PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
Having reached this point, proving unbiasedness is easy. Taking expectations, b2 is unaffected, being a constant. The expectation of a sum is equal to the sum of expectations. 8

10. PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
The a* terms are nonstochastic since they depend only on the values of X2 and X3, and these are assumed to be nonstochastic. Hence the a* terms may be taken out of the expectations as factors. 9

11. PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
By Assumption A.3, E(ui) = 0 for all i. Hence E(b2) is equal to b2 and so b2 is an unbiased estimator. Similarly b3 is an unbiased estimator of b3. 10

12. PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
Finally we will show that b1 is an unbiased estimator of b1. This is quite simple, so you should attempt to do this yourself, before looking at the rest of this sequence. 11

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

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

15. 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(b2) is equal to b2 and that E(b3) is equal to b3. 14

16. PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
Hence b1 is an unbiased estimator of b1. 15

17. 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
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