![](https://csdnimg.cn/release/download_crawler_static/11874492/bg6.jpg)
6
Properties of Least Squares EstimatorsProperties of Least Squares Estimators
(1)Gauss-Markov Theorem
Under the conditions of the regression model, the least
(i.e., E(b
0
) = β
0
and E(b
1
) = β
1
) and have minimum variance
among all unbiased linear estimators.
(2) The estimated value of Y (i.e. = b
0
+ b
1
X) is an unbiased
estimator of E(Y) = β
0
+ β
1
X, with minimum variance in the
class of unbiased linear estimators.
Note that the common variance
2
can not be estimated by LSM
Lecture 1 11
.
We can prove that the following statistic is an unbiased point
estimator of σ
2
(You should try to prove it)
s
2
= SSE/(n-2) = (SS
yy
-b
1
SS
xy
)/(n-2)
Properties of Fitted Regression LineProperties of Fitted Regression Line
(1) The sum of the residuals is zero: ∑e
i
= 0
(2) The sum of the squared residuals,
∑e
i
2
, is a minimum.
(3) ∑
i
=∑ Y
i
(4) ∑X
i
e
i
= 0
(5) ∑
i
e
i
= 0
(6) The regression line always goes through the point ( , ).
O All properties can be proved directly by using the norm equations,
x
Lecture 1 12
∑(y
i
-b
0
-b
1
X
i
) = 0 and ∑(y
i
-b
0
-b
1
X
i
)X
i
= 0,
or
n b
0
+ b
1
∑ X
i
= ∑ Y
i
and b
0
∑ X
i
+b
1
∑ X
i
2
= ∑ Y
i
X
i
.