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Cointegration and Common FactorsCointegration and Common Factors

Gloria González-RiveraUniversity of California, RiversideandJesús Gonzalo U. Carlos III de Madrid

Implications for the MA representationImplications for the MA representation

.

0j

|jL|j and pI0C with jLjCC(L) ), iid(0, is t where

pxp 1pxt)L(C

~)L1(t)1(Ct)L(CtY

.

0j

|jL|j and pI0C with jLjCC(L) ), iid(0, is t where

pxp 1pxt)L(C

~)L1(t)1(Ct)L(CtY

• 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)).

If Yt is cointegrated, such that there exists a then:

px1 rxp

I(0) is t´Y with pxr

Implications for the MA representation (cont)Implications for the MA representation (cont)

Common Trend representation (Stock-Watson):

It is the multivariate extension of the univariate Beveridge-Nelson’s decomposition.

ttL)f-(1 where

mxp pxm px1

components )0(Itf1AtY

.t1Bt with , t)L(C~

)L1(t

1AtY

t)L(C~

)L1(t1B1AtY

ttL)f-(1 where

mxp pxm px1

components )0(Itf1AtY

.t1Bt with , t)L(C~

)L1(t

1AtY

t)L(C~

)L1(t1B1AtY

Implications for the MA representation (cont)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 ?1A and

Implications for the VAR representation 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.

then,

k

1ij

j*i with

1k

1i

iL*ipI)L(* where)L1)(L(*(1)L(L)

as expressed-re becan operator (L) AR theSince

.ttY)L(

or

t

k

1i

itYitY

then,

k

1ij

j*i with

1k

1i

iL*ipI)L(* where)L1)(L(*(1)L(L)

as expressed-re becan operator (L) AR theSince

.ttY)L(

or

t

k

1i

itYitY

Implications for the VAR representation (cont)Implications for the VAR representation (cont)

p...1pI)1(´ where

t

1k

1i

itY*i1tY´tY

(ECM) model correctionerror theort1tY)1(tY)L1)(L(*

p...1pI)1(´ where

t

1k

1i

itY*i1tY´tY

(ECM) model correctionerror theort1tY)1(tY)L1)(L(*

• 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.

rxp´pxrpxp

Implications for the VAR representation (cont)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 Y1t on Y2t

2. Plug in the ECM.

3. Estimate the model

by OLS equation by equation.

1x2

.tY´ˆtZ where

t

1k

1i

itY*i1tZtY

.tY´ˆtZ where

t

1k

1i

itY*i1tZtY

Gonzalo-Granger (1995) Permanent and Gonzalo-Granger (1995) Permanent and Transitory DecompositionTransitory 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.

Gonzalo-Granger (1995) Permanent and Gonzalo-Granger (1995) Permanent and Transitory Decomposition (cont)Transitory Decomposition (cont)

Gonzalo-Granger proposes the following two assumptions:

(1) The I(1) common factors are linear combinations of the variables Yt

(2) 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.

.0 wheretYtf .0 wheretYtf

ProblemsProblems

xt1txtxzt1tztz ;tztxty

Problem 1: Let’s have the following DGP:

(a) Are (yt , xt) 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?