Question

Consider the regression model: 

y = Xβ + u

(a) State the Gauss-Markov assumptions required to show that the Ordinary Least Squares estimator of the coefficients is the Best Linear Unbiased Estimator (BLUE) 

(b) Prove that the OLS estimator is unbiased if the Gauss-Markov assumptions hold Show relevant steps and assumptions  

(c) Show that the variance-covariance matrix of the OLS estimator is 

Answer 1

OLS estimator of is given as b

where b = arg min e'e = arg min (Y-X)'(Y - X)

Solving this using first order conditions, we get:

a)

Gauss Markov assumptions required to prove unbiasedness of OLS estimator are as follows:

1. Model is linear
2. Strict exogeneity of errors i.e,

This can also be stated as and

3. No multicollinearity i.e. X is a full rank matrix

4. Spherical error variance/Conditional homoskedasticity i.e.

This can also be written as or

b)

To prove that E(b|X) =

This is same as proving E(b - |X) = 0

Given

Substituting Y from the model. we get:

Thus

This is true by Assumption 2 of strict exogeneity.

This means that i.e. OLS estimator is unbiased

c)

Variance of OLS estimator is given as:

This is true because is a constant and thus variance remains unchanged.

  

This is because Var(AX) = AVar(X)A'

  

This holds because by conditional homoskedasticity assumption.