Lecture 7.1

Question 1

Start small: Write the formula for beta hat.

Answer 1

Question 2

Why would we want to premultiply if we know beta hat is consistent?

Answer 2

Question 3

What can you say about the second moment of the difference given deterministic z? What is the condition for consistency?

Answer 3

Question 4

Consider then that we want to pre-multiply by something as in the L-F CLT. Write what it looks like and give the conditions needed for this to work.

Answer 4

Question 5

Write down the matrix R that we are interested in.

Answer 5

Question 6

Show how the eigenvalue requirements on R relate to the variance of the least squares normalized by R.

Answer 6

+ also write the formula of the second moment of the normalized least squares to see it

Question 7

Show what D looks like in the case where z has 2 elements.

Answer 7

Question 8

Show what R looks like in the case where z has only 2 elements. State the needed assumption.

Answer 8

Question 9

Derive the stochastic order of magnitude of D(beta hat - beta)

Answer 9

Question 10

Show the decomposition utilized in the lectures of D(beta hat - beta). Note which part governs the distribution.

Answer 10

Question 11

Show D(beta hat - beta) in scalar form, using Rn hat.

Answer 11

Question 12

Given the decomposition, which component’s distribution should we be interested in? Why?

Answer 12

Question 13

Write the expression we are interested in, in the form required for CLTs. (Hint: start by writing it in scalar form.)

Answer 13

Question 14

Can we use Lindeberg-Levy CLT? Why?

Answer 14

Question 15

List the conditions required for L-F CLT.

Answer 15

Question 16

Derive the primitive condition in our context required for A1 of L-F CLT.

Answer 16

Question 17

Derive the primitive condition in our context required for A2 of L-F CLT.

Answer 17

Question 18

Show how you can relax the homoscedasticity assumption for A2.

Answer 18

Question 19

Derive the primitive condition in our context required for A3 of L-F CLT.

Answer 19

Question 20

A4 Reminder: State Lindeberg’s condition and our expression of interest.

Answer 20

Question 21

Start deriving, up to getting an expression the terms of w_i and u_i, of a primitive condition that would satisfy the Lindeberg condition.

Answer 21

Question 22

Consider that we derived following condition needed in our context for the Lindeberg condition to be satisfied. What conditions do we need to impose on w_i?

Answer 22

Question 23

Give an example of a summable sequence that has a positive sum, where all elements converge to 0. Relate why this is important in our context, with the two conditions we need simultaneously on w_i.

Answer 23

Question 24

Derive the two most primitive sufficient conditions in terms of w_i and u_i for the Lindeberg condition to be satisfied.

Answer 24

Question 25

Reminder, we wrote w_i as follows: Derive the primitive condition required for the Lindeberg condition in terms of z_i.

Answer 25

Question 26

Derive the most primitive condition for our condition of interest in scalar form.

Answer 26

Question 27

List the primitive sufficient conditions and conclusions regarding the distribution of the least squares estimator.

Answer 27

Question 28

State Cramer's theorem.

Answer 28

Question 29

Give another condition that satisfies A5.

Answer 29