Asymptotic Normality of OLS Estimators
Consider the linear regression model
yt =
0zt + ut; t = 1; : : : ; T;
where the parameter
and the regressor zt are k-dimensional vectors. We assume that
E [ut] = 0 and E [ztut] = 0k; that the moment matrices E [ztz0t] and E [(ztz0t)u2t ] are
finite, and denote �2 = E [u2t ] = V [ut] : The OLS estimator of
is
^ =
TX
t=1
ztz
0
t
!�1 TX
t=1
ztyt
!
=
+
1
T
TX
t=1
ztz
0
t
!�1
1
T
TX
t=1
ztut
!
:
Hence,
p
T (
^ �
) =
1
T
TX
t=1
ztz
0
t
!�1
1p
T
TX
t=1
ztut
!
:
Since 1
T
PT
t=1 ztz
0
t converges to E [ztz0t] in probability and 1pT
PT
t=1 ztut converges in
distribution to a multivariate normal with mean 0k and covariance matrix E [(ztz0t)u2t ],
we have that
p
T (
^ �
) converges in distribution to a multivariate normal with mean
0k and covariance matrix (E [ztz0t])
�1 � E [(ztz0t)u2t ] � (E [ztz0t])�1 :
Note that if zt and ut are assumed to be independent, then the covariance matrix
simplifies to �2 � (E [ztz0t])�1 :
Let us now specialize to the case in which there is an intercept and only one pure
regressor xt: Then k = 2 and zt =
h
1 xt
i0
: It follows that
(E [ztz0t])
�1
=
"
1 E [xt]
E [xt] E [x2t ]
#�1
=
1
V [xt]
"
E [x2t ] �E [xt]
�E [xt] 1
#
and
(E [ztz0t])
�1
(ztz
0
t) (E [ztz0t])
�1
= 1V[xt]2
"
E [x2t ] �E [xt]
�E [xt] 1
#"
1 xt
xt x
2
t
#"
E [x2t ] �E [xt]
�E [xt] 1
#
= 1V[xt]2
"
E [x2t ]� E [xt] xt E [x2t ]xt � E [xt] x2t
�E [xt] + xt �E [xt]xt + x2t
#"
E [x2t ] �E [xt]
�E [xt] 1
#
1
= 1V[xt]2
"
(E [x2t ]� xtE [xt])2 � (E [xt]� xt) (E [x2t ]� E [xt] xt)
� (E [xt]� xt) (E [x2t ]� E [xt]xt) (xt � E [xt])2
#
;
and so
(E [ztz0t])
�1 � E �(ztz0t)u2t � � (E [ztz0t])�1
=
1
V [xt]2
"
E
h
(E [x2t ]� xtE [xt])2 u2t
i
�E [((E [xt]� xt) (E [x2t ]� E [xt] xt))u2t ]
�E [((E [xt]� xt) (E [x2t ]� E [xt] xt))u2t ] E
�
(xt � E [xt])2 u2t
� # :
Under the assumption that xt and ut are independent, we further have
E
h�
xtE [xt]� E
�
x2t
��2
u2t
i
= E
h�
xtE [xt]� E
�
x2t
��2iE �u2t � = E �x2t � � V [xt] � �2;
E
��
(E [xt]� xt)
�
E
�
x2t
�� E [xt]xt��u2t � = E ��(E [xt]� xt) �E �x2t �� E [xt]xt���E �u2t �
= E [xt] � V [xt] � �2;
E
�
(xt � E [xt])2 u2t
�
= E
�
(xt � E [xt])2
�
E
�
u2t
�
= V [xt] � �2;
and the covariance matrix �2 � (E [ztz0t])�1 simplifies to
�2
V [xt]
�
"
E [x2t ] �E [xt]
�E [xt] 1
#
:
2
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