National open university of nigeria introduction to econometrics I eco 355

(ii) 
Unbiasedness
. 
(1) 
Unbiasedness of
̂
To prove that 
̂
is an unbiased estimator of 
̂
we need to show that 
( ̂) * ̂
+
However, 
is a constant, and using assumptions, that is 
is non-random- we can take 
as a fixed constant to take them out of the expectation expression and have: 
[( )]
Therefore, it is enough to show that 
[( )]
[( )]
∑
̅
̅ 
Where 
is constant, so we can take it out of the expectation expression and we can also 
break the sum down into the sum of its expectations to give: 
[(
 
)]
[
̅
̅ [ 
̅]
̅ ]
∑ [
̅
̅ ]
Furthermore, because
is non-random (again from assumption 3) we can take it out of 
the expression term to give: 
[( )]
∑ [
̅
̅ ]
Finally, using assumption 4, we have that 
and therefore 
̅
So
[( )]
and this proves that: 
( ̂)
or, to put it in words, that 
̂
is an unbiased estimator of the true population parameter 
. 
(b) Unbiasedness of
̂
. We know that 

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59 
̂ ̅ ̂ ̅
̂ ̅ ( ̂) ̅
But we also have that: 
Where we eliminated the
̅ ̅
Substituting equation (40) into equation (38) gives: 
̂ ̅ ( ̂) ̅
We have proved before that
( ̂)
; therefore: 
( ̂) ̅ ̅
Which proves that
̂
is an unbiased estimator of 
 
 
(iii) Efficiency and BLUEness 
Under assumptions 5 and 6, we can then make a prove that the OLS estimators are the 
most efficient among all unbiased linear estimators. However, we can say that the OLS 
procedure yields BLU estimators. 
The proof that the OLS estimators are BLU estimators is relatively complicated. It 
entails a procedure which goes the opposite way from that followed so far. We start the 
estimation from the beginning, trying to derive a BLU estimator of 
based on the 
properties of linearity, unbiasedness and minimum variance one by one, and we will then 
check whether the BLU estimator derived by this procedure is the same as the OLS 
estimator. Thus, we want to derive the BLU estimator of 
, say 
̂
, concentrating first on 
the property of linearity. For 
̂
to be linear we need to have: 
̂
∑
Where the 
terms are constants, the values of which are to be determined proceeding 
with the property of unbiasedness, for 
̂ 
to be unbiased, we must be able to have
( ̂)
. However, we know that; 
( ̂) (∑
) ∑
Therefore, let us substitute
, and also 
because 
is non-stochastic and 
, given by the basic assumptions of the 
model, we get; 
( ̂) ∑
∑
∑
And therefore, in order to have unbiased
̂
, we need; 
∑
∑
I think you are learning through the process and you should know that econometric 
notation might show as if they are abstract but they have different meaning. Therefore, 

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60 
we can then proceed by deriving an expression for the variance (which we need to 
minimize) of 
:
( ̂) [ ̂ ( ̂)]
*∑
(∑
)+
*∑
∑
+
*∑
+
From equation 47 above, we can use 
and 
respectively. Then: 
( ̂) (∑
)
( 
)
( 
)
( 
)
( 
)
Let us use the assumptions 
and 
we obtain that: 
( ̂) ∑
We then need to choose 
in the linear estimator (equation 44 to be such as to minimize 
the variance (equation 49 subject to the constraints (equation 46) which ensure 
unbiasedness (with this then having a linear, unbiased minimum variance estimator). We 
formulate the Langrangian function: 
Where 
and 
are Langrangian multipliers. 
However, following the regular procedure, which is to take the first-order conditions (that 
is the portal derivatives of Lwith respect to 
, 
and 
) and set them equal to zero and 
after re-arrangement and mathematical manipulations (we omit the mathematical details 
of the derivation because it is very lengthy and tedious and because it does not use any of 
the assumptions of the model in any case), we obtain the optimal 
as:
We can say that 
of the OLs expression given by Equation (32). so, substituting 
this into our linear estimators 
̂
we have: 
̂
̅ ̅ 
̅ ̅

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61 
̂
Therefore we can conclude that 
̂
of the OLs is BLU. Let us then talk more about the 
advantage of the BLUEness: The advantages of the BLUEness condition is that it 
provides us with an expression for the variance by substituting the optional 
given in 
equation (51) into equation (49) and that will gives: 
( ̂) ( ̂) ∑ (
)