Cointegration and Common Factors

1
Cointegration and Common Factors
Gloria González-Rivera University of California,
Riverside and Jesús Gonzalo U. Carlos III de
Madrid
2
Implications for the MA representation
If Yt is cointegrated, such that there exists a
then

• C(1) is reduced rank, so C(1)A1B1 with
• rank(A1)rank(B1)p-r, where r
  cointegrating vectors
• C(L) is non-invertible. Therefore there will NOT
  exist a VAR representation in differences
  ((1-L)Yt)).

3
Implications for the MA representation (cont)
Common Trend representation (Stock-Watson) It is
the multivariate extension of the univariate
Beveridge-Nelsons decomposition.
4
Implications for the MA representation (cont)
Question 1 Why this representation is called
COMMON TREND representation? Question 2 How
would you prove that cointegration IFF common
I(1) factor representation? Question 3 Which is
the relationship between the cointegrating vector
?
5
Implications for the VAR representation
Remember that if the set of variables Yt are
cointegrated then it will not exist a VAR
representation in first differences of Yt.
6
Implications for the VAR representation (cont)

• Note that the matrix
  is of reduced rank and therefore the ECM
  is a non-linear VAR model.
• An ECM is a VAR in LEVELS with non-linear
  cross-equation restrictions (the cointegration
  restrictictions).
• Johansens method is an application of
  Andersons reduced rank regression techniques to
  VAR models.

7
Implications for the VAR representation (cont)

• At the level of this course and assuming we are
  in a bivariate world, we will estimate the ECM in
  the following simple way (Engle-Granger
  procedure)
• Estimate the cointegrating vector
  by regressing Y1t on Y2t
• Plug in the ECM.
• Estimate the model
• by OLS equation by equation.

8
Gonzalo-Granger (1995) Permanent and Transitory
Decomposition
Once we find that two variables are cointegrated,
the next step is to estimate the ECM. Many
empirical works end the cointegration study here
without answering the key question Why these
two variables are cointegrated? or in other
words Which is the common I(1) factor that is
making the variables to be cointegrated? In order
to answer this question, it is clear that we need
to make some assumptions in order to identify the
I(1) factors.
9
Gonzalo-Granger (1995) Permanent and Transitory
Decomposition (cont)

• Gonzalo-Granger proposes the following two
  assumptions
• The I(1) common factors are linear combinations
  of the variables Yt
• The part of Yt that it is not explained by the
  I(1) common factors can only have a transitory
  effect on Yt.
• With these two assumptions it can be easily
  proved (see the paper) that the I(1) common
  factors o permanent components are
• Applications of this decomposition will be seen
  in class, as well as the
• economic interpration of the permanent
  components.

10
Problems
Problem 1 Lets have the following DGP

• Are (yt , xt) cointegrated? Which is the
  cointegrated vector?
• Write the multiariate Wold representation (MA
  representation).
• Try to write a VAR for the variables in first
  differences. Any comments?
• Write the ECM representation. Any comments on
  the adjustment process? Which is the matrix
  ?
• Propose an estimation strategy of the ECM
• Find the G-G permanent and transitory
  decomposition. Any comments?