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47
sums is greater than or equal to 2, the result is proved.
Just let
z
i
=
x
i
2
.
Then, we require
z
i
/
z
j
+
z
j
/
z
i
 2
>
0.
But, this is equivalent to
(
z
i
2
+
z
j
2

2z
i
z
j
) /
z
i
z
j
>
0 or (
z
i

z
j
)
2
/
z
i
z
j
>
0, which is certainly true if z
i
and
z
j
are
positive.
They are since
z
i
equals
x
i
2
.
This completes the proof.
10.
Consider, first,
y
.
We saw earlier that Var[
y
] =
(
σ
2
/
n
2
)
Σ
i
x
i
2
=
(
σ
2
/
n
)(1/
n
)
Σ
i
x
i
2
.
The expected value is
E
[
y
] =
E
[(1/
n
)
Σ
i
y
i
]
=
α
.
If the mean square of
x
converges to something finite, then
y
is consistent for
α
.
That is, if plim(1/
n
)
Σ
i
x
i
2
=
q
where
q
is some finite number, then, plim
y
=
α
.
As such, it follows that
s
2
and
s
*
2
= (1/(
n
1))
Σ
i
(
y
i

α
)
2
have the same probability limit.
We consider, therefore, plim
s
*
2
=
plim(1/(
n
1))
Σ
i
ε
i
2
.
The expected value of
s
*
2
is
E
[(1/(
n
1))
Σ
i
ε
i
2
]
=
σ
2
(1/
Σ
i
x
i
2
).
Once again, nothing more can be said without
some assumption about
x
i
.
Thus, we assume again that the average square of
x
i
converges to a finite, positive
constant,
q
.
Of course, the result is unchanged by division by (
n
1) instead of
n
, so
lim
n
→∞
E
[
s
*
2
]
=
σ
2
q
.
The variance of
s
*
2
is Var[
s
*
2
]
=
Σ
i
Var[
ε
i
2
]/(
n
 1)
2
.
To characterize this, we will require the variances of the
squared disturbances, which involves their fourth moments.
But, if we assume that every fourth moment is
finite, then the preceding is (
n
/(
n
1)
2
) times the average of these fourth moments.
If every fourth moment is
finite, then the term is dominated by the leading (
n
/(
n
1)
2
) which converges to zero.
It follows that plim
s
*
2
=
σ
2
q
.
Therefore, the conventional estimator estimates
Asy.Var[
y
]=
σ
2
q
/
n
.
The appropriate variance of the least squares estimator is Var[
y
]=
(
σ
2
/
n
2
)
Σ
i
x
i
2
,
which is, of course,
precisely what we have been analyzing above.
It follows that the conventional estimator of the variance of the
OLS estimator in this model is an appropriate estimator of the true variance of the least squares estimator.
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This note was uploaded on 11/13/2011 for the course ECE 4105 taught by Professor Dr.fang during the Spring '10 term at University of Florida.
 Spring '10
 Dr.Fang

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