Exercises Part II

Question 1

How do you determine the order of cointegration e.g. in this example?

Answer 1

1. Rewrite into a version that can easily be analyzed. If a variable becomes a random walk, then that part is a Random Walk.
2. Analyze all the rv’s going into the equation, mention their integration order. (iid, is always I(0))
3. Rewrite in lag notation and determine the integration order.

Question 2

How do you determine the cointegration vector from a question like this?

Answer 2

In case everything is I(0), no cointegration possible. In the other case it is often written down as (1, -parameter value before other parameter, here again), in case there are multiple, one needs to also consider the case where just some of the parameters cancel each other out.

Question 3

How do I derive a CECM of y_t given z_t?

Answer 3

Question 4

What is the test for cointegration of a CECM?

Answer 4

The part before the (y_{t-1} - beta_1 …) = 0.

Question 5

What are the additional assumptions of a CECM to be more efficient?

Answer 5

We would need weak exogeneity of z_t from the parameters of intrest. [If possible state when this happens]

Question 6

How solve the following type of question?

Answer 6

1. Write down seperate parts (with beta) into lag notation
2. Combine into single formula, move every part containing w_{t-1} to lhs
3. Write lhs into lag notation
4. Find the roots of the polynomialas before the w_t, state that this holds for the given condition

Question 7

How do you derive the VMA (Vector Moving Average)?

Answer 7

1. Write all the simple equations first into lag notation
2. Likely if you have been able to write as b’w_t before this can be used to derive the last part
3. Rewrite into vector notation where the following holds

Question 8

What is meant by an equation to be in lag notation?

Answer 8

Question 9

When converting a VAR(2) model to a VECM model, what needs to be rewritten?

Answer 9

Question 10

What does it mean for a variable to be in MA representation? How would you write a VAR model in a MA rep?

Answer 10

Question 11

How to test using lag notation if a process is I(1)? On which side should the Lags be? What does it mean if all the roots are outside the unit root?

Answer 11

Note: In this course added absolute value of this (and often L replaced by z)

Question 12

How to answer a question as follows:

Answer 12

1. Replace x_t and y_t with the model definitions, and factor out the square and multiplication
2. Look in the list of of standard results for each part of the sum to find the O_p(a), where a should be replaced by a function of T
3. The eventual convergence of the entire part should be the part with the fastest growing O_p(T), this has a-convergence (a is function of T).

Note: In case of a self reference, rewrite into sum of the past errors

Question 13

What is the definition of a t statistic?

Answer 13

This is in case H_0: beta = 0

Question 14

What should be answered for this type of question?

Answer 14

Always no, since we are not using any time of non-serial correlatedness

Question 15

What is the O_p of e.g. beta hat with a-consistancy?

Answer 15

It is O_p(a^-1), e.g. T-consistency, then it is O_p(T^-1) etc.

Question 16

When should a t-statistic be convergent?

Answer 16

Under the null it should be O_p(1) and under the alternative it should diverge.

Question 17

What is the definition of long run variance? How can it be computed?

Answer 17

Question 18

What is the definition of contempreneous variance?

Answer 18

Question 19

What is the standard OLS estimator with and without intercept?

Answer 19

Question 20

What is a common problem to test for when testing for cointegration using ADF?

Answer 20

We want to ensure we are not just running a spurious regression, after that we can test for cointegration

Question 21

Spurious regression means that we erroneously reject the null of no co-integration.

Answer 21

i.e., No, it means that our results are significant, however by doing cointegration tests first we are sure that this is not the case

Question 22

Standard asymptotics can always be used to test for unit root if I add a constant to
my Dickey-Fuller regression.

Answer 22

Question 23

Under the assumption of cointegration, the consistency of the OLS estimator of a static cointegrating regression (of I(1) variables) of the type y_t = α + βx_t + u_t requires independence between the “regressor” (explanatory variable) and the re- gression error term.

Answer 23

Question 24

A unit root in the moving average part of an ARMA model implies nonstationarity.

Answer 24

Question 25

The static least squares estimator for the cointegrating regression yields inconsis- tent results.

Answer 25

Question 26

One can always apply conventional standard asymptotic inference to the static least squares of the cointegrating relation.

Answer 26

Question 27

What is a unit-root setting of an AR(1) model?

Answer 27

Just the random walk, all unit root models are non-stationary

Question 28

What is a super consistant estimator?

Answer 28

Question 29

The OLS estimator for the AR coefficient of a unit root model is superconsistent and has a nuisance parameter free distribution.

Answer 29

Question 30

Fully modified least squares is a method that uses modified OLS estimators for the cointegrating regression that are asmptotically normal.

Answer 30