Exercises Part 1

Question 1

How we you compute autocovariances of an MA process?

Answer 1

See example, but steps are:
0. Rewrite from Lag notation to normal notation
1. Compute gamma_0 = E[y_t^2]
2. Compute gamma_1 = E[y_t y_{t-1}]
etc.

Question 2

How to check if an AR(p) process is stationary?

Answer 2

Question 3

How do you compute autocovariances of an AR(p) process?

Answer 3

Essentially get first autocovariances first and then solve the system.

Question 4

Answer 4

Question 5

Answer 5

Question 6

What are the three conditions neccesary for the stationarity of a VAR process?

Answer 6

Question 7

What is the stability condition of a VAR(1) process?

Answer 7

Question 8

What is the stability condition for a VAR(p) process?

Answer 8

Question 9

What is the unconditional mean of a VAR(p) process?

Answer 9

Question 10

What is the unconditional variance of a VAR(1) process? How is it derived?

Answer 10

Question 11

How can a VMA (Vector MA) be derived? Just answer with steps

Answer 11

1. Start with the initial equation
2. Write out what happens in case you switch x_{t-1} with its definition
3. Continue until you can create some sort of sum

Question 12

Answer 12

Question 13

What is the quadratic formula?

Answer 13

Question 14

What is the chacteristic equation for the difference notation? What for time series?

Answer 14

The difference with time series is that it is exactly the inverse as for time series

Question 15

Answer 15

Question 15

What is block-exogeneity?

Answer 15

Question 16

Answer 16

Question 17

Only b

Answer 17

Question 18

Only c

Answer 18

Question 19

Answer 19

Question 20

What is the structural form of the VAR model?

Answer 20

Question 21

How to convert structural VAR to reduced VAR?

Answer 21

Question 22

Answer 22

Question 23

What is the definition of mixing?

Answer 23

Question 24

If a dynamic time series process satisfies the stability condition that is discussed in the lecture then we say that this process is stationary.

Answer 24

Question 25

If a random time series sequence, {xt}, is mixing. Then, the realization of the se- quence at time t does not contain any information about the realization at time t−1.

Answer 25

Question 26

If a sequence is “mixing” then the pair x_t and x_{t−j} that is taken from the sequence tends to independence as j → ∞.

Answer 26

Question 27

A stationary time series is automatically mixing.

Answer 27

Question 28

Ensemble mean is the expectation of a process that is for fixed t across all possible realizations of the sequence.

Answer 28

Thusfar unclear, please think of the answer now.

Question 29

Ergodicity is the property of a stationary series, that ensures that time average of the sequence is converging in probability to the mean of the sequence.

Answer 29

True, however stationarity is also needed.

Question 30

Companion form can be used to rewrite an AR(1) in terms of a MA(∞) model.

Answer 30

Question 31

Weak exogeneity is a relation between two variables that ensures that one variable is independent of the other.

Answer 31

The statement is incorrect. Weak exogeneity does not imply that one variable is independent of another. It is a property about the sufficiency of information in the conditional distribution for parameter estimation.

Question 32

Answer 32

Question 33

An adapted sequence, {x_t, F_t} is a martingale difference sequence only if it is sta-
tionary.

Answer 33

Question 34

The property of martingale difference sequence is weaker than independence and also weaker than uncorrelatedness.

Answer 34