Takashi Yamano

⎛ ⎞
= Q ⎜
σ X ′ X
⎟Q
⎝ n ⎠
2 −1
−1
= σ Q QQ
2
−1 −1
2 −1
= σ Q
Therefore, the asymptotic distribution of the OLS estimator is
n ( Βˆ
− Β)
~
a
2 −1
N[0,
σ Q ].
From this, we can treat the OLS estimator, Βˆ , as if it is approximately normally
distributed with mean Β and variance-covariance matrix σ
2 Q −1 / n .
Example 6-1: Consistency of OLS Estimators in Bivariate Linear Estimation
A bivariate model:
y = $ 0 + $ 1 x + u and
ˆβ =
1
n
∑
i=
1
n
∑
i=
1
( x
i
( x
− x)
y
i
− x)
To examine the biasedness of the OLS estimator, we take the expectation
n
⎛
⎞
⎜ ∑(
xi
− x)
ui
⎟
ˆ = + ⎜ i=
1
E(
β
⎟
1)
β1
E
⎜ n ⎟
2
⎜ ∑(
xi
− x)
⎟
⎝ i=
1 ⎠
Under the assumption of zero conditional mean (SLR 3: E(u|x) = 0), we can separate the
expectation of x and u:
n
⎛
⎞
⎜ ∑(
xi
− x)
E(
ui
) ⎟
ˆ = + ⎜ i=
1
E(
β ⎟
1)
β1
.
⎜ n
⎟
2
⎜ ∑(
xi
− x)
⎟
⎝ i=
1
⎠
Thus we need the SLR 3 to show the OLS estimator is unbiased.
Now, suppose we have a violation of SLR 3 and can not show the unbiasedness of the
OLS estimator. We consider a consistency of the OLS estimator.
i
2
p lim ˆ β = p lim β
1
1
+
p lim
⎛
⎜
⎜
⎜
⎜
⎝
n
∑
i=
1
n
∑
i=
1
( x
i
( x
− x)
u
i
−
x)
2
i
⎞
⎟
⎟
⎟
⎟
⎠
4