# Gauss Markov Theorem [BLUE Properties]

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,