LAURIER
Business & Economics
Wilfrid Laurier University
School of Business and Economics
(Econometrics)
Instructor: Jean Eid
Lecture Notes: GaussMarkov Theorem
Assume that the following hold
MLR1
Linearity in the parameter
MLR2
Independence of the error term u and random sampling
MLR3
No perfect Collinearity
MLR4
Zero conditional mean
MLR5
Homoskedasticity i.e
var
(
u
i

x
) =
σ
2
Also assume that
y
i
=
β
o
+
β
1
x
i
+
u
i
We know that the OLS estimator for
β
1
is
b
β
1
=
n
X
i
=1
x
i

x
y
i
n
X
i
=1
x
i

x
2
T
HE ROAD AHEAD
We need to come up with another estimator that is linear, but also we need to make sure that this estimator is unbiassed.
We need to do so because it is unfair to compare the OLS to any other estimator since biased estimators are not the best
by construction. So in order to compare apples to apples we need to compare the OLS estimator to other linear unbiassed
estimators, and show that OLS is best.
Now let there be any other estimators that are unbiassed and linear. For an estimator to be linear, it should be a linear
function of the observable values. Denote this estimator by
e
β
1
, and
e
β
1
=
n
X
i
=1
c
i
y
i
where
c
i
=
g
x
i.e
c
i
is a function of x. For example
c
i
=
x
i

x
n
X
i
=1
x
i

x
2
and in this case
e
β
1
=
b
β
1
. In general
c
i
can be any function of the independent variable (linear or nonlinear function). As long as we can compute each
c
i
for
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2
each data point
i
we multiply that by
y
i
and sum over all these to get
e
β
1
. Note that
e
β
1
=
n
X
i
=1
c
i
y
i
=
n
X
i
=1
c
i
β
o
+
β
1
x
i
+
u
i
=
β
o
n
X
i
=1
c
i
+
β
1
n
X
i
=1
c
i
x
i
+
n
X
i
=1
c
i
u
i
Recall that the GaussMarkov theorem states that OLS BLUE i.e
B
est
L
inear
U
nbiassed
E
stimator. So in order to compare
apples to apples, we need
e
β
1
to be unbiassed i.e.
E
e
β
1

x
=
β
1
E
e
β
1

x
=
E
β
o
n
X
i
=1
c
i
+
β
1
n
X
i
=1
c
i
x
i
+
n
X
i
=1
c
i
u
i

x
=
β
o
n
X
i
=1
c
i
+
β
1
n
X
i
=1
c
i
x
i
+
n
X
i
=1
c
i
E
u
i

x
the reason why
c
i
came out of the expectation is because
c
i
are only a function of
x
and since we are conditioning on
x
any function of just
x
is constant with respect to the expectation. In addition,
E
u
i

x
=0
=
β
o
n
X
i
=1
c
i
+
β
1
n
X
i
=1
c
i
x
i
But in order for
e
β
1
to be unbiassed we need to have
β
o
n
X
i
=1
c
i
+
β
1
n
X
i
=1
c
i
x
i
=
β
1
(1)
How can we make sure that
∀
{
β
o
, β
1
} ∈
R
2
, (1) above always hold. This means that for any values from
∞
to
+
∞
for
β
o
and
β
1
, (1) above has to hold. You should convince yourself that the only way this could happen (recall equating
polynomials) is when
n
X
i
=1
c
i
= 0
and
n
X
i
=1
c
i
x
i
= 1
, and we get
E
e
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 Fall '11
 JEANEID
 Econometrics, Variance, Estimation theory, WI, Gauss–Markov theorem

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