OLS is blue

Econometrics Assignment 2

Group Members :
Eliza Tan 01120120073
Praisya Lordrietta 01120120061
Wirhan Pandutama 0112012

UNIVERSITAS PELITA HARAPAN
LIPPO KARAWACI-TANGERANG
2014
Gauss-Markov Theorem
The Gauss-Markov Theorem is given in the following regression model and assumptions:
The regression model (1)

Assumptions (A) or Assumptions (B):
Assumptions (A)

Assumptions (B)
E(

If we use Assumptions (B), we need to use the law of iterated expectations in proving the BLUE. With Assumptions (B), the BLUE is given conditionally on
Let us use Assumptions (A). The Gauss-Markov Theorem is stated below
Under Assumptions (A), the OLS estimators, are the Best Linear Unbiased Estimator (BLUE), that is
1. Unbias :
2. Best : Real data seldomly satisfy Assumptions (A) or Assumptions (B). Accordingly we should think that the Gauss-Markov theorem only holds in the never-never land. However, it is important to understand the Gauss-Markov theorem on two grounds:
1. We may treat the world of the Gauss-Markov theorem as equivalent to the world of perfect competition in micro economic theory.
2. The mathematical exercises are good for your souls.
We shall prove the Gauss-Markov theorem using the simple regression model of equation (1). We can prove the Gauss-Markov theorem using the multiple regression model (2)
To do so, however, we need to use vector and matrix language (linear algebra.) Actually, once you learn linear algebra the proof of Gauss-Markov theorem is far more straight forward than the proof for the simple regression model of (1).
Proving the Gauss-Markov Theorem
Proof that is best
We need to re-express first

The BLUE only looks at linear estimators of The linear estimators are defined by

If we noticed that if

Then

We have to make unbiased. To take expectation of we first