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

yi= β1+β2xi+ ui, i=1,…..,n (1)

Assumptions (A) or Assumptions (B):

Assumptions (A)

Eui =0 for all i

Var ui = σ2 for all i

Cov ui, uj =0 for all i≠j

xi is nonstochasic constant

Assumptions (B)

E(Euiǀ x1,…, xn =0 for all i

Var uiǀ x1, …, xn = σ2 for all i homoscedasticity

Cov ui, ujǀ x1,…, xn =0 for all i ≠j

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 x1,….,xn on

Let us use Assumptions (A). The Gauss-Markov Theorem is stated below

Under Assumptions (A), the OLS estimators, β1 and β2 are the Best Linear Unbiased Estimator (BLUE), that is

1. Unbias : Eβ1= β1 and Eβ2= β2

2. Best : β1 and β2 have the smallest variances among the class of all linear unbiased estimators.

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

yi= β1+β2xi2+... +βkxik+ui, i=1,…..,n (2)

To do so, however, we need to use vector and matrix language (linear algebra.) Actually,...