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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/
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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.
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
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
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
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
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
PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
uXXY 33221 33221ˆ XbXbbY
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
PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
uXXY 33221 33221ˆ XbXbbY
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
PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
uXXY 33221 33221ˆ XbXbbY
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
PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
uXXY 33221 33221ˆ XbXbbY
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
PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
uXXY 33221 33221ˆ XbXbbY
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
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
PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
uXXY 33221 33221ˆ XbXbbY
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
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
PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
uXXY 33221 33221ˆ XbXbbY
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
PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
uXXY 33221 33221ˆ XbXbbY
332233221
33221
)( XbXbuXX
XbXbYb
First substitute for the sample mean of Y.
12
PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
uXXY 33221 33221ˆ XbXbbY
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
PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
uXXY 33221 33221ˆ XbXbbY
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
PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
uXXY 33221 33221ˆ XbXbbY
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