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题目：对一元线性回归模型
Y
i
=
β
0
+
β
1
X
i
+
μ
i
Y_{i}=\beta_{0}+\beta_{1}X_{i}+\mu_{i}
Yi?=β0?+β1?Xi?+μi?，试证明
C
o
v
(
β
0
^
,
β
1
^
)
=
?
σ
2
X
￣
∑
x
i
2
Cov(\widehat{\beta_{0}},\widehat{\beta_{1}})=-\frac{\sigma^2\overline{X}}{\sum{x^2_{i}}}
Cov(β0?
?,β1?
?)=?∑xi2?σ2X?
证明：在给定
X
X
X的样本条件下
C
o
v
(
β
0
^
,
β
1
^
)
=
E
[
(
β
0
^
?
β
0
)
(
β
1
^
?
β
1
]
Cov(\widehat{\beta_{0}},\widehat{\beta_{1}})=E[(\widehat{\beta_{0}}-\beta_{0})(\widehat{\beta_{1}}-\beta_{1}]
Cov(β0?
?,β1?
?)=E[(β0?
??β0?)(β1?
??β1?]
=
E
[
(
β
0
^
?
E
(
β
0
^
)
)
(
β
1
^
?
E
(
β
1
^
)
)
]
=E[(\widehat{\beta_{0}}-E(\widehat{\beta_{0}}))(\widehat{\beta_{1}}-E(\widehat{\beta_{1}}))]
=E[(β0?
??E(β0?
?))(β1?
??E(β1?
?))]
=
E
[
(
Y
￣
?
β
1
^
X
￣
?
Y
￣
+
X
￣
E
(
β
1
^
)
)
(
β
1
^
?
E
(
β
1
^
)
)
]
=E[(\overline{Y}-\widehat{\beta_{1}}\overline{X}-\overline{Y}+\overline{X}E(\widehat{\beta_{1}}))(\widehat{\beta_{1}}-E(\widehat{\beta_{1}}))]
=E[(Y?β1?
?X?Y+XE(β1?
?))(β1?
??E(β1?
?))]
=
?
X
￣
E
[
(
β
1
^
?
E
(
β
1
^
)
)
(
β
1
^
?
E
(
β
1
^
)
)
]
=-\overline{X}E[(\widehat{\beta_{1}}-E(\widehat{\beta_{1}}))(\widehat{\beta_{1}}-E(\widehat{\beta_{1}}))]
=?XE[(β1?
??E(β1?
?))(β1?
??E(β1?
?))]
=
?
X
￣
E
[
(
β
1
^
?
E
(
β
1
^
)
)
]
2
=-\overline{X}E[(\widehat{\beta_{1}}-E(\widehat{\beta_{1}}))]^2
=?XE[(β1?
??E(β1?
?))]2
=
?
X
￣
V
a
r
(
β
1
^
)
=-\overline{X}Var(\widehat{\beta_{1}})
=?XVar(β1?
?)
=
?
σ
2
X
￣
∑
x
i
2
=-\frac{\sigma^2\overline{X}}{\sum{x^2_{i}}}
=?∑xi2?σ2X?
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