Christopher Dougherty EC220 - Introduction to econometrics (chapter 3) Slideshow: properties of the...

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/

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://learningresources.lse.ac.uk/

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

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS

uXXY kk ...221

1

Moving from the simple to the multiple regression model, we start by restating the regression model assumptions.








uXXY kk ...221

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








uXXY kk ...221

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


uXXY 33221 33221ˆ XbXbbY

23322 XXYYXX iii

23322

233

222

3322332

XXXXXXXX

XXXXYYXXb

iiii

iiii

We will not attempt to prove efficiency. We will however outline a proof of unbiasedness.

4



23322 XXYYXX iii

23322

233

222

3322332

XXXXXXXX

XXXXYYXXb

iiii

iiii

uuXXXX

uXXuXXYY

iii

iiii

333222

3322133221

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



23322 XXYYXX iii

23322

233

222

3322332

XXXXXXXX

XXXXYYXXb

iiii

iiii

uuXXXX

uXXuXXYY

iii

iiii

333222

3322133221

ii uab *222

After substituting, and simplifying, we find that b2 can be decomposed into the true value 2 plus a weighted linear combination of the values of the disturbance term in the sample.

6



23322 XXYYXX iii

23322

233

222

3322332

XXXXXXXX

XXXXYYXXb

iiii

iiii

uuXXXX

uXXuXXYY

iii

iiii

333222

3322133221

ii uab *222

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



23322 XXYYXX iii

23322

233

222

3322332

XXXXXXXX

XXXXYYXXb

iiii

iiii

uuXXXX

uXXuXXYY

iii

iiii

333222

3322133221

ii uab *222

2*22

*22

*222 iiiiii uEauaEuaEbE

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



23322 XXYYXX iii

23322

233

222

3322332

XXXXXXXX

XXXXYYXXb

iiii

iiii

uuXXXX

uXXuXXYY

iii

iiii

333222

3322133221

ii uab *222

2*22

*22


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



23322 XXYYXX iii

23322

233

222

3322332

XXXXXXXX

XXXXYYXXb

iiii

iiii

uuXXXX

uXXuXXYY

iii

iiii

333222

3322133221

ii uab *222

2*22

*22


By Assumption A.3, E(ui) = 0 for all i. Hence E(b2) is equal to 2 and so b2 is an unbiased estimator. Similarly b3 is an unbiased estimator of 3.

10



Finally we will show that b1 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.

332233221

33221

)( XbXbuXX

XbXbYb

11



332233221

33221

)( XbXbuXX

XbXbYb

First substitute for the sample mean of Y.

12



332233221

33221

)( XbXbuXX

XbXbYb

1

332233221

3322332211 )()()()(

XXXX

bEXbEXuEXXbE

Now take expectations. The first three terms are nonstochastic, so they are unaffected by taking expectations.

13



332233221

33221

)( XbXbuXX

XbXbYb

1

332233221

3322332211 )()()()(

XXXX

bEXbEXuEXXbE

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 2 and that E(b3) is equal to 3.

14



332233221

33221

)( XbXbuXX

XbXbYb

1

332233221

3322332211 )()()()(

XXXX

bEXbEXuEXXbE

Hence b1 is an unbiased estimator of 1.

15

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/.

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

or the University of London International Programmes distance learning course

20 Elements of Econometrics

www.londoninternational.ac.uk/lse.

11.07.25

Christopher Dougherty EC220 - Introduction to econometrics (chapter 3) Slideshow: properties of the...

Documents

Transcript of Christopher Dougherty EC220 - Introduction to econometrics (chapter 3) Slideshow: properties of the...