Given the assumptions of the classical linear regression model, the least-squares estimators, in the class of unbiased linear estimators, have minimum variance, that is, they are BLUE. In other words Gauss-Markov theorem holds the properties of Best Linear Unbiased Estimators. Following are some of the assumptions which should be taken into consideration for the mathematical derivation of the Gauss-Markov Theorem :
Properties of ;
1.
2.
3.
Assumptions of ;
1.
2.
3.
4.
5.
Properties of BLUE (Best Linear Unbiased Estimators)
Lets take the regression line
1. Linearity i.e.,
Lets take the regression line
As we know,
,
,
,
,
because
2. Unbiasedness i.e., is unbiased,
From Linearity,
,
,
,
, because
,
Taking expectations both sides,
, because
,
3. Best i.e, is best or
have minimum variance
As we know,
,
,
,
,
,
,
, beacause
,
,
Now suppose there is another estimator,
,
Where ,
Let ,
So, ,
,
, As
,
,
So,
Now for ;
1. Linearity i.e.; is linear,
As we know,
,
,
,
,
So, is linear.
2. Unbiasedness i.e.; is unbiased,
From linearity,
,
,
,
,
,
Now taking expectations both sides,
,
,
3. Best i.e.; is best or
have minimum variance,
From unbiased,
,
We know,
,
,
,
,
,
, As
,
Lets take another estimator ,
,
Where and let
,
Now,
,
,
, As
,
,
,
So,