National open university of nigeria introduction to econometrics I eco 355

2.0. OBJECTIVES 
At the end of this unit, you should be able to: 

know the properties that our estimators should have 

know the proofing of the OLS estimators as the best linear unbiased estimators 
(BLUE). 
 
 
 
 
 
3.0 
MAIN CONTENT 
3.1 
Properties of the OLS Estimators 
We now return to the properties that we would like our estimators to have. Based on the 
assumptions of the CLRM can prove that the OLS estimators are best linear unbiased 
estimators (BLUE). To do so, we first have to decompose the regression coefficients 
estimated under OLS into their random and non-random components. 
As a starting point, not that 
has a non-random component 
, as well as a 
random component, captured by the residual 
. Therefore cov(X,Y) that is which 
depends on values of 
– will have a random and non-random component.
[ 
]
However, because 
are constants we have that 
and that 
Thus 
and substituting that in equation (26) yields: 
 
which says that the OLS coefficient 
̂
estimated from any sample has a non-random 
component, 
random component which depends on 
. 
 
(i). 
Linearity 
Based on assumption 3, we have that X is non-stochastic and fixed in repeated samples.
Therefore, the x values can be treated as cosntants and we need merely to concentrate on 
the Y values. If the OLS estimators are linear functions of the Y values then they are 
linear estimators. 
From equation (22) we have that: 
̂

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58 
Since the 
are regarded as constants, then the are regarded as constants as well, we 
have that: 
̂
̅ 
̅
but because 
̅
, we have that: 
̂
Where 
can also be regarded as constant and therefore 
̂
is indeed a linear 
estimator of the 
.