2. You are free to use and modify these slides for educational purposes, but please if you improve this material send us your new version. Cointegration and Common Factors Gloria González-Rivera University of California, Riverside and Jesús Gonzalo U. Carlos III de Madrid

3. Implications for the MA representation C(1) is reduced rank, so C(1)=A 1 B 1 with rank(A 1 )=rank(B 1 )=p-r, where r= # cointegrating vectors C(L) is non-invertible. Therefore there will NOT exist a VAR representation in differences ((1-L)Y t )). If Y t is cointegrated, such that there exists a then:

4. Implications for the MA representation (cont) Common Trend representation (Stock-Watson): It is the multivariate extension of the univariate Beveridge- Nelson’s decomposition.

5. 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 ?

6. Implications for the VAR representation Remember that if the set of variables Y t are cointegrated then it will not exist a VAR representation in first differences of Y t.

7. 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). Johansen’s method is an application of Anderson’s reduced rank regression techniques to VAR models.

8. 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): 1. Estimate the cointegrating vector by regressing Y 1t on Y 2t 2. Plug in the ECM. 3. Estimate the model by OLS equation by equation.

9. 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.

10. Gonzalo-Granger (1995) Permanent and Transitory Decomposition (cont) Gonzalo-Granger proposes the following two assumptions: (1)The I(1) common factors are linear combinations of the variables Y t (2) The part of Y t that it is not explained by the I(1) common factors can only have a transitory effect on Y t. 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.

11. Problems Problem 1: Let’s have the following DGP: (a)Are (y t, x t ) cointegrated? Which is the cointegrated vector? (b) Write the multiariate Wold representation (MA representation). (c) Try to write a VAR for the variables in first differences. Any comments? (d) Write the ECM representation. Any comments on the adjustment process? Which is the matrix ? (e) Propose an estimation strategy of the ECM (f) Find the G-G permanent and transitory decomposition. Any comments?