National open university of nigeria introduction to econometrics I eco 355

 
 
 
1.0 INTRODUCTION 
Based on Unit 2 we just discussed, we can also make an alternative expression for 
parameter 
̂
for residual sum of square and it can also be expressed further as co-variance 
analysis. In this unit, the assumptions of classical linear regression model will also be 
examined and you will be able to differentiate these assumptions from other economic 
analysis assumptions. 
 
2.0. OBJECTIVES 
At the end of this unit, you should be able to: 

know the alternative expression for 
̂

understand the assumptions of classical linear regression model. 
3.0 
MAIN CONTENT 
3.1 
Alternative Expression for 
̂
 

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53 
We can express the numerator and denominator of equation (III) which is: 
̂
as follows: 
̅
̅
( 
̂)
So then we have: 
̂
̅
̅
or 
even 
̂
Where obviously 
̅ 
and
̅ 
, which are derivatives from their 
respective means. 
We can use the definitions of Cov(X,Y) and Var(X) to obtain an alternative expression 
for 
̂
as: 
̂
̅ ̅
̅ 
or 
̂
̅
̅ 
( 
̂)
If we further divide both nominator and denominator by 
we have: 
̂
̅
̅ 
( 
̂)
and finally we can express 
̂
as
̂
Where 
and 
are sample covariances and variances. 
3.2 
The Assumptions of CLRM 

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54 
In a general term, when we calculate estimators of population parameters from sample 
data we are bound to make some initial assumptions about the population distribution.
Usually, they amount to a set of statements about the distribution of the bariables we are 
investigating, without which our model and estimates cannot be justified. Therefore it is 
important not only to present the assumptions but also to move behond them, to the 
extent that we will at least study what happens when they go wrong and how we may test 
whether they have gone wrong.