6
3.2
The variances of
and
ˆ
α
ˆ
β
.
Our next issue concerns the variances of
and
.
For deriving these variances the
ˆ
α
ˆ
β
following two lemmas are convenient.
Lemma 1
.
Let U
1
, U
2
,...,U
n
be independent random variables with zero mathematical expectation
(
thus E
(
U
j
) = 0)
and variance
σ
2
. (
Thus E
[(
U
j

E
(
U
j
))
2
] =
E
(
U
j
2
) =
σ
2
).
Let v
1
,
v
2
,...,
v
n
and
w
1
,
w
2
,...,
w
n
be given
constants.
Then
E
[(
'
n
j
'
1
v
j
U
j
)(
'
n
j
'
1
w
j
U
j
)]
'
σ
2
'
n
j
'
1
v
j
w
j
.
Proof
.
See the Appendix.
Note that if we choose
in Lemma 1 then it reads:
v
j
'
w
j
for
j
'
1,2,...,
n
Lemma 2
.
Let U
1
, U
2
,...,U
n
be independent random variables with zero mathematical expectation
and variance
σ
2
.
Let w
1
,
w
2
,...,
w
n
be given
constants.
Then
E
[(
'
n
j
'
1
w
j
U
j
)
2
]
'
σ
2
'
n
j
'
1
w
2
j
.
Using (3) and Lemmas 1 and 2 it can be shown that
Proposition 2
.
Under the assumptions I  IV,
var(
$
"
)
'
F
2
'
n
j
'
1
X
2
j
n
'
n
j
'
1
(
X
j
&
¯
X
)
2
'
F
2
$
"
,
say
, var(
$
$
)
'
F
2
'
n
j
'
1
(
X
j
&
¯
X
)
2
'
F
2
$
$
,
say
,
and
cov(
$
"
,
$
$
)
'
&
F
2
¯
X
'
n
j
'
1
(
X
j
&
¯
X
)
2
.
(4)
Proof
.
See the Appendix
3.3
Normality of
and
ˆ
α
ˆ
β
.
If we also assume normality of the error terms
then
and
are also normally
U
j
ˆ
α
ˆ
β
distributed.
This result follows from the following lemma.