Econometrics 2: OLS

Question 1

Under what assumptions are the OLS estimators unbiased?

Answer 1

(1) Mean independence

(2) Full rank - there is no exact linear relationship between any of the regressors in the model.

Question 2

What does the Gauss-Markov Theorem state? Under what conditions does it hold?

Answer 2

If we have mean independence, full rank and conditional homoscedasticity, the OLS estimator is the most efficient linear conditionally unbiased estimator of β.

Question 3

If we assume normal errors in addition to the Gauss-Markov assumptions, what results follow?

Answer 3

The OLS estimator is normally distributed.

The estimator of the error variance follows a χ^2_{n-K} distribution.

The t-statistic follows a t_{n-K} distribution.

Question 4

Suppose we know that the errors in our linear regression model are normal. Should we test simple hypotheses with a normal or a t- distribution?

Answer 4

With normal errors, the t-statistic follows a t distribution in all finite samples, and a normal distribution asymptotically. We should therefore use the t distribution, as we will then know the exact distribution of the test statistic.

Question 5

Under what assumptions are the OLS estimators consistent?

Answer 5

• (Xi, ui) - is i.i.d. with finite fourth moments
• Mean independence
• Rank condition: Q=plim X’X/n is full rank.

Homoscedasticity is not required.

Question 6

What is the asymptotic variance of the OLS estimators when they are asymptotically normal?

Answer 6

σ^2_U Q^{-1}

Question 7

If errors are normal, what distribution does F follow?

Answer 7

F_{J,n-k}

Question 8

Under what conditions is J * F-statistic asymptotically χ^2_J?

Answer 8

(1) Finite fourth moments
(2) Mean independence
(3) Full rank in prob. limit
(4) Homoscedasticity