Gauss Markov Theorem matrix approach

 

Gauss Markov Theorem

Under the classical assumptions of linear regression, the OLS estimates are best linear, unbiased, and consistent and have minimum variance (BLUE).

Consider the MLRM

i.                    Linearity

The OLS estimates are the linear function of the response vector.

ii.                    Unbiased

The OLS estimates are the unbiased estimate

iii.                    The variance of the OLS estimate is minimum.

First, we obtained the variance of OLS estimate as:

Now show that this variance is minimum.

We defined a new unbiased estimate as:

Practice Question 

 Construct regression model for the following data:

Solution:

Re transform the model as:

The residual can be obtained as:

The variance of the disturbance term:

Coefficient of Multiple Determination:

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