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(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
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,
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:
̂
̅ ̅
̅ ̅
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:
( ̂) ( ̂) ∑ (
)
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