Cointegration and ECM

Question 1

What is a Spurious Regression?

Answer 1

We have two variables (x and y) who at first glance are unrelated, but are in fact related through a third variable, z

Question 2

What is the consequence of this relationship between x, y and z?

Answer 2

If we regress y on x we find a significant relationship
But when we control for z the partial effect of x on y becomes zero

Question 3

Why does the partial effect of x on y become zero when z is introduced?

Answer 3

All the information is collected in z

Question 4

Why are regressions using integrated time series likely to be spurious?

Answer 4

They produce significant relationships even when the variables are unrelated
It may include a time trend, but removing this may not solve the problem because there are other factors affecting that we do not account for

Question 5

How do we make a regression non-spurious?

Answer 5

Through differencing or ECM

Question 6

In a random walk process, will x_t or y_t be independent processes if ε_t is?

(note: x_t = x_t-1 + ε_t, y_t = y_t-1 + ε_t)

Answer 6

x_t and y_t will also be independent

Question 7

What is the issue with a spurious regression?

Answer 7

Theoretically, x and y shouldn’t be related because our equations show they don’t affect one another.

However, looking at the table below, we can see x_t has a significant t-statistic and affects y_t in the OLS regression

Question 8

Why can differencing two variables to create a non-spurious regression cause an issue?

Answer 8

It limits the scope of the questions we can answer

Question 9

Is the relationship between two cointegrated nonstationary variables of the same order (i.e. I(1)) meaningful?

Answer 9

Yes, because they seem to evolve in a similar fashion over time

Question 10

Why is cointegration between x and y sometimes useful?

Answer 10

Because the regression between two unit root processes may still create a stationary dependent variable, z_t

Question 11

Will x_t∼I(1) and y_t∼I(1) return to the initial value?

How does a cointegrating factor, β, affect this?

Answer 11

They both will have a tendency to wander around and not return to the initial value

z_t = y_t - βx_t∼I(0) tends to return to its mean with regularity (mean reversion)

x_t and y_t are linked in such a way that they do not move too far from one another

Question 12

How do we test for superconsistency?

Answer 12

1. Assume β is unknown
   2.

Question 13

What does superconsistency demonstrate?

Answer 13

The OLS estimator can generate consistent estimates of β

but standard OLS formula for the variance of β^ will be incorrect so we should not use that for our hypothesis testing

However u^_t = y_t - β^x_t can be used to conduct tests of whether the equation errors are stationary

Question 14

What is the null hypothesis of the Engle-Granger test?

Answer 14

Question 15

How do we perform the EG test?

Answer 15

Note that step 5. is the same test as an ADF test (to see if there is a unit root, if there is then u^_t is nonstationary and we reject cointegration)

Question 16

What is the consequence of y_t∼I(1) and x_t∼I(1) having no cointegration?

Answer 16

Question 17

What is the consequence of y_t∼I(1) and x_t∼I(1) being cointegrated?

Answer 17

Question 18

An example of EG test in practice

Answer 18

Question 19

Is it possible to perform the Engle-Granger test for more than two variables?

Answer 19

Yes, but the critical values will change

Question 20

How do we perform the EG-test with multiple variables?

Answer 20

Question 21

What issues does the EG-test have?

Answer 21

1. In some circumstances it can have low power
   
   • Meaning it is difficult to reject the null even when it is false (type II error)
2. The residuals should be stationary
   - meaning ρ₁ < 1

This occurs because the alternative (H₁: Stationarity) can include some values very close to H₀

Question 22

What is the purpose of an ECM?

Answer 22

To describe the short-run dynamics consistent with the long-run dynamics

Question 23

What is the Engle-Granger (1987) representation theorem?

Answer 23

If the series are cointegrated, there is a valid ECM representation

Question 24

How do we rewrite this ADL as an ECM?

Answer 24

• Subtract y_t-1 from both sides to get Δy_t on the LHS and γ₃ on the RHS
• Add and subtract α₁x_t-1 from the RHS to get γ₁Δx_t and γ₂x_t-1

Question 25

How else can we identify the long run relationship between the variables or the linear combination between them with a formula?

Answer 25

Where:

y_t-1-βx_t-1 is integrated of order 0

This reparameterization of the ECM generates a model where all variables are I(0)

Question 26

How can we test for cointegration when there is a possibility of generating ECMs?

Answer 26

γ₃ is the important value!!!

Question 27

What is the benefit of using ECM over EG-test?

Answer 27

ECM is a better dynamic specification than the static regression used to generate the residuals in the EG test

This test can also be shown to be more powerful than the EG test

Question 28

A note when using the alternative approach for ECM discussed before

Answer 28

Note: the coefficient on the residual is the speed of adjustment i.e. in this model r3m and r6m become 15% closer each month

Question 29

What advantages does the Johansen test have?

Answer 29

• It allows for mulitple cointegrating vectors when we have more than two variables in the model
• It has better power than the EG test
• It avoids the normalisation restriction necessary for both the EG and ECM approaches

–for both tests we impose that the coefficient on the current value of y is equal to one

–the result of the respective tests may be sensitive to this restriction

–there is no need to make such an assumption when we use the Johansen test

Question 30

What are the conclusions we can draw about the rank of the VECM (with two variables) in Johansen test?

Answer 30

1. If rank is 0, both variables are I(1) and there is no cointegration
2. If rank is 1, both variables are I(1) but there is a cointegrating vector linking them, meaning that there is a linear transformation which is I(0)
3. If rank is 2, both variables are I(0) without any further transformation

Question 31

How does the Johansen test work in theory?

Answer 31

Basically starts by comparing rank 2 as the null hypothesis against rank 1, if we reject the null then we move onto making rank 1 the null and testing this against rank 0

Question 32

How does the Johansen test work in practice?

Answer 32

Question 33

How many cointegrating relationships can exist in a system of k variables, each of which is I(1)?

Answer 33

k-1