Slide 1

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
ASSUMPTIONS FOR MODEL A
A.1 The model is linear in parameters and correctly specified.

Y   1   2 X 2  ...   k X k  u
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


Slide 2

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
ASSUMPTIONS FOR MODEL A
A.1 The model is linear in parameters and correctly specified.

Y   1   2 X 2  ...   k X k  u
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


Slide 3

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
ASSUMPTIONS FOR MODEL A
A.1 The model is linear in parameters and correctly specified.

Y   1   2 X 2  ...   k X k  u
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


Slide 4

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
True model

Fitted model

Y  1  2X 2  3X 3  u

Yˆ  b1  b 2 X 2  b 3 X 3

b2 



 X

2i

  X 2i

 X 2 Y i  Y

  X 3 i

 X3

 X

 X 3 Y i  Y

  X 2 i

 X 2  X 3 i  X 3 

3i

 X2

2

 X

 X3 
2

3i

  X

2i

2

 X 2  X 3 i  X 3  

2

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

4


Slide 5

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
True model

Fitted model

Y  1  2X 2  3X 3  u

Yˆ  b1  b 2 X 2  b 3 X 3

b2 



 X

2i

  X 2i

 X 2 Y i  Y

  X 3 i

 X3

 X

 X 3 Y i  Y

  X 2 i

 X 2  X 3 i  X 3 

3i

 X2

2

 X

 X3 
2

3i

  X

2i

2

 X 2  X 3 i  X 3  

2

Yi  Y   1   2 X 2i   3 X 3i  ui    1   2 X 2   3 X 3  u 
  2  X 2i  X 2    3  X 3i  X 3   ui  u

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


Slide 6

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
True model

Fitted model

Y  1  2X 2  3X 3  u

Yˆ  b1  b 2 X 2  b 3 X 3

b2 



 X

2i

  X 2i

 X 2 Y i  Y

  X 3 i

 X3

 X

 X 3 Y i  Y

  X 2 i

 X 2  X 3 i  X 3 

3i

 X2

2

 X

 X3 
2

3i

  X

2i

2

 X 2  X 3 i  X 3  

2

Yi  Y   1   2 X 2i   3 X 3i  ui    1   2 X 2   3 X 3  u 
  2  X 2i  X 2    3  X 3i  X 3   ui  u
b2   2 



*

a i 2ui

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


Slide 7

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
True model

Fitted model

Y  1  2X 2  3X 3  u

Yˆ  b1  b 2 X 2  b 3 X 3

b2 



 X

2i

  X 2i

 X 2 Y i  Y

  X 3 i

 X3

 X

 X 3 Y i  Y

  X 2 i

 X 2  X 3 i  X 3 

3i

 X2

2

 X

 X3 
2

3i

  X

2i

2

 X 2  X 3 i  X 3  

2

Yi  Y   1   2 X 2i   3 X 3i  ui    1   2 X 2   3 X 3  u 
  2  X 2i  X 2    3  X 3i  X 3   ui  u
b2   2 



*

a i 2ui

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


Slide 8

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
True model

Fitted model

Y  1  2X 2  3X 3  u

Yˆ  b1  b 2 X 2  b 3 X 3

b2 



 X

2i

  X 2i

 X 2 Y i  Y

  X 3 i

 X3

 X

 X 3 Y i  Y

  X 2 i

 X 2  X 3 i  X 3 

3i

 X2

2

 X

 X3 
2

3i

  X

2i

2

 X 2  X 3 i  X 3  

2

Yi  Y   1   2 X 2i   3 X 3i  ui    1   2 X 2   3 X 3  u 
  2  X 2i  X 2    3  X 3i  X 3   ui  u


 E  a

b2   2 

E b2    2

*

a i 2ui
*
i2

ui    2 



E a i 2 u i    2 
*



a i 2 E ui    2
*

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


Slide 9

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
True model

Fitted model

Y  1  2X 2  3X 3  u

Yˆ  b1  b 2 X 2  b 3 X 3

b2 



 X

2i

  X 2i

 X 2 Y i  Y

  X 3 i

 X3

 X

 X 3 Y i  Y

  X 2 i

 X 2  X 3 i  X 3 

3i

 X2

2

 X

 X3 
2

3i

  X

2i

2

 X 2  X 3 i  X 3  

2

Yi  Y   1   2 X 2i   3 X 3i  ui    1   2 X 2   3 X 3  u 
  2  X 2i  X 2    3  X 3i  X 3   ui  u


 E  a

b2   2 

E b2    2

*

a i 2ui
*
i2

ui    2 



E a i 2 u i    2 
*



a i 2 E ui    2
*

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


Slide 10

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
True model

Fitted model

Y  1  2X 2  3X 3  u

Yˆ  b1  b 2 X 2  b 3 X 3

b2 



 X

2i

  X 2i

 X 2 Y i  Y

  X 3 i

 X3

 X

 X 3 Y i  Y

  X 2 i

 X 2  X 3 i  X 3 

3i

 X2

2

 X

 X3 
2

3i

  X

2i

2

 X 2  X 3 i  X 3  

2

Yi  Y   1   2 X 2i   3 X 3i  ui    1   2 X 2   3 X 3  u 
  2  X 2i  X 2    3  X 3i  X 3   ui  u


 E  a

b2   2 

E b2    2

*

a i 2ui
*
i2

ui    2 



E a i 2 u i    2 
*



a i 2 E ui    2
*

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


Slide 11

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
True model

Fitted model

Y  1  2X 2  3X 3  u

Yˆ  b1  b 2 X 2  b 3 X 3

b1  Y  b 2 X 2  b 3 X 3

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.
11


Slide 12

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
True model

Fitted model

Y  1  2X 2  3X 3  u

Yˆ  b1  b 2 X 2  b 3 X 3

b1  Y  b 2 X 2  b 3 X 3
b1  Y  b 2 X 2  b 3 X 3
   1   2 X 2   3 X 3  u   b2 X 2  b3 X 3

First substitute for the sample mean of Y.

12


Slide 13

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
True model

Fitted model

Y  1  2X 2  3X 3  u

Yˆ  b1  b 2 X 2  b 3 X 3

b1  Y  b 2 X 2  b 3 X 3
b1  Y  b 2 X 2  b 3 X 3
   1   2 X 2   3 X 3  u   b2 X 2  b3 X 3

E  b1    1   2 X 2   3 X 3  E  u   X 2 E  b 2   X 3 E  b 3 
 1   2 X 2   3 X 3  X 2 2  X 3 3
 1
Now take expectations. The first three terms are nonstochastic, so they are unaffected by
taking expectations.
13


Slide 14

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
True model

Fitted model

Y  1  2X 2  3X 3  u

Yˆ  b1  b 2 X 2  b 3 X 3

b1  Y  b 2 X 2  b 3 X 3
b1  Y  b 2 X 2  b 3 X 3
   1   2 X 2   3 X 3  u   b2 X 2  b3 X 3

E  b1    1   2 X 2   3 X 3  E  u   X 2 E  b 2   X 3 E  b 3 
 1   2 X 2   3 X 3  X 2 2  X 3 3
 1
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


Slide 15

PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
True model

Fitted model

Y  1  2X 2  3X 3  u

Yˆ  b1  b 2 X 2  b 3 X 3

b1  Y  b 2 X 2  b 3 X 3
b1  Y  b 2 X 2  b 3 X 3
   1   2 X 2   3 X 3  u   b2 X 2  b3 X 3

E  b1    1   2 X 2   3 X 3  E  u   X 2 E  b 2   X 3 E  b 3 
 1   2 X 2   3 X 3  X 2 2  X 3 3
 1
Hence b1 is an unbiased estimator of 1.

15


Slide 16

Copyright Christopher Dougherty 2012.
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 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
EC2020 Elements of Econometrics
www.londoninternational.ac.uk/lse.

2012.11.11