Lecture 4

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.

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

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 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

Question 32

Define O(f_n) and o(f_n)

Answer 32

Question 33

Define Op(f_n) and op(f_n)

Answer 33

Question 34

Give a simpler definition of op(1).

Answer 34

Question 35

State two relationships between convergence in probability to a constant and stochastic order of magnitude.

Answer 35

Question 36

State and prove the two relationships between convergence in probability to a constant and stochastic order of magnitude.

Answer 36

Question 37

Provide a more intuitive definition of the difference between Op() and op()

Answer 37

Question 38

State the three results relating different stochastic orders of magnitude.

Answer 38

Question 39

Which is the direction of implication between Op(fn) and op(fn)? Prove it.

Answer 39

Question 40

State and prove the relationship between stochastic orders of magnitude relating fn and gn if fn/gn goes to 0.

Answer 40

Question 41

State and prove the relationship between stochastic order of magnitude and rate of convergence provided a moment exists.

Answer 41

Question 42

State the 4 results relating the stochastic orders of magnitudes of combinations of RVs.

Answer 42

Question 43

State and prove the stochastic order of magnitude of multiplication of two RVs. (for big and little o)

Answer 43

Question 44

State and prove the stochastic order of magnitude of the sum of two RVs. (for big and little o)

Answer 44

Question 45

State and prove the relationship between the multiplication of RVs that are Op(fn) and op(gn).

Answer 45

Question 46

What is the rate of convergence of Beta hat in LSE? Show it.

Answer 46