Lecture 4.1

Question 1

State the relation between two sufficient conditions for UI.

Answer 1

Question 2

State and prove the relation between two sufficient conditions for UI.

Answer 2

Question 3

Are transformations of independent variables also independent?

Are transformations of uncorrelated variables also uncorrelated?

Answer 3

1. Transformations of independent variables are also independent.
2. Transformations of uncorrelated variables are NOT also uncorrelated.

Question 4

State formally and intuivitely the condition for martingale sequences.

Answer 4

Intuitively: Expectation with relation to the past is 0.

Question 5

State the relationship between indepence, martingale differences and uncorrelation.

Answer 5

1. Independence implies martingale difference.
2. Martingale difference implies uncorrelated.

Question 6

State and prove the relationship between uncorrelation and martingale difference sequence.

Answer 6

Question 7

Formally state the WLLN for independent UI sequences.

Answer 7

Question 8

Formally state and prove the WLLN for independent UI sequences.

Answer 8

Question 9

State a relaxation for one of the assumptions of the WLLN for independent UI sequences.

Answer 9

Question 10

How do you use WLLN for independent UI sequences when expected mean is not equal to 0? Provide all the details.

Answer 10

Question 11

State Khinchin’s WLLN.

Answer 11

Question 12

State and prove Khinchin’s WLLN.

Answer 12

Question 13

State Kolmogorov’s SLLN.

Answer 13

Question 14

Describe how we can still use the SLLN if we relax the identical assumption.

Answer 14

Question 15

Describe why we cannot use Khinchin’s Theorem in multiple regression with deterministic z_i.

Answer 15

Question 16

Describe why we cannot use Chebyshev’s Theorem (Th 7) in multiple regression with deterministic z_i.

Answer 16

Question 17

Show the consistency of beta hat in multiple regression with deterministic z_i assuming errors u_i are UI.

Answer 17

Question 18

Formally define a generalized linear process.

Answer 18

Question 19

Clarify in words the difference (and relationship) between a linear process and a generalized linear process.

Answer 19

In the linear process, we assume the innovations e_i have second mean and are uncorrelated.

In GLP, we simply assume that e_i is UI. Therefore a linear process is by definition a GLP, since the assumptions on linear processes imply that they are uniformly integrable.

Question 20

Formally state the theorem regarding the convergence in first mean of GLP.

Answer 20

Question 21

Formally state and prove the theorem regarding the convergence in first mean of GLP.

Answer 21

since all epsilons can be chosen arbitrarily small, we are done.

Question 22

Give an example of regressors z_i such that Q_n / n does not have a finite limit.

Answer 22

Question 23

State the theorem along with all the assumptions relating the convergence of Beta hat in second mean and the eigenvalues of the Q matrix.

Answer 23

Question 24

State and prove the theorem relating the convergence of Beta hat in second mean and the eigenvalues of the Q matrix.

Answer 24

Question 25

Relate the min eigenvalue of a matrix to it’s elements.

Answer 25

Question 26

State another condition that sufficies for Beta hat n consistency.

Answer 26

Question 27

State and prove another condition that sufficies for Beta hat n consistency.

Answer 27

Question 28

Describe the relationship between second moment and probability, in cases where second moment exists and doesn’t exist.

Answer 28

Question 29

State and show a condition that guarantees Beta hat consistency regardless of errors.

Answer 29

Question 30

Show that y_i = alpha + Beta x i + u_i is consistent.

Answer 30

Question 31

Give an example of a linear regression that is not consistent and show why it is not consistent.

Answer 31