The Engle-Granger representation theorem
Reference note to lecture 10 in ECON 5101/9101, Time Series Econometrics
Ragnar Nymoen
March 29 2011
1 Introduction
The Granger-Engle representation theorem is in their Econometrica paper from 1987.
The main thesis is that systems with cointegrated I(1) variables have three equivalent representations
• Common Trends
• Moving average, MA
• Equilibrium correction (error correction), ECM
In this note we give a simplified version of this famous theorem. We start from the VAR-representation of a first order bi-variate system and assume that one of the roots of the autoregressive matrix is unity while the other root is less than one.
The Common Trends, MA and ECM representations are then shown. Note that in the literature, the theorem is also shown “the other way”, by starting from the MA representation.
2 The VAR
Consider the bivariate VAR(1)
(2.1) xt= Φxt−1+ εt
where xt= (xt, yt), Φ is a 2 × 2 matrix with coefficients and εt is a vector with two Gaussian disturbances.
The characteristic equation of Φ:
|Φ − zI| = 0,
In the spurious regression case we imposed two unit-roots. In the stationary case neither root is equal to one. We now consider the intermediate case of one unit-root and one stationary root. Specifically, write the two characteristic roots as:
z1 = 1, and z2 = λ, |λ| < 1.
z1 = 1 implies that both xt and ytare I(1). Φ has full rank, equal to 2, since both eigenvalues are different from zero. Φ can therefore be diagonalized in terms of its eigenvalues and the corresponding eigenvectors. We write it as:
(2.2) Φ = P 1 0
0 λ
Q where P is the matrix with the eigenvectors as columns:
(2.3) P = α β
γ δ
.
and Q = P−1. Assume, without loss of generality that |P | = 1, i.e.,
(2.4) αδ − γβ = 1.
and thus
P−1 =
δ −β
−γ α
By mulipliction from the right, Φ can be expressed as
(2.5) Φ = α β
γ δ
δ −β
−γλ αλ
, and, by multiplication from the left, we also get:
(2.6) Φ = (αδ − λβγ) −αβ(1 − λ)
γδ(1 − λ) (−γβ + λαδ)
. We refer to these expressions later.
2.1 The Common-trends representation
Multiplication of (2.1) with Q yields
(2.7) Qxt= QΦxt−1+ Qεt.
Observe that
QΦ = QP 1 0 0 λ
Q = 1 0 0 λ
Q,
from the definitions above, implying that (2.7) can be written in terms of two new variables wt and zt:
(2.8)
wt
−zt
= 1 0 0 λ
wt−1
−zt−1
+ ηt, where
(2.9)
wt
−zt
= Q xt yt
and
(2.10) Qεt= ηt= η1t
η2t
.
To solve for wt and zt, we multiply out the right hand side of (2.9). Notice that since
(2.11) Q =
δ −β
−γ α
we obtain wt as
(2.12) wt = δxt− βyt
and zt as:
(2.13) −zt= γxt− αyt.
Conversely, from (2.9):
xt yt
= P
wt
−zt
yielding
(2.14) xt = αwt− βzt
and
(2.15) yt= γwt− δzt.
Equation (2.8) gives wt as a random-walk variable and, from the same equation, zt is seen to be a stationary since |λ| < 1.
zt, defined by (2.13) is therefore a stationary linear combination of the two I(1) variables xt and yt. We say that xt and yt are two cointegrated variables, with cointegrating vector (γ, −α)0.
On the other hand: equation (2.14) and (2.15) show that xtand yt contain an I(0) component (zt), and an I(1) component (wt)—hence the name Common trend representation.
2.2 Integration and cointegration is maintained in forecasts
Let xT,h denote the h-period ahead forecast, conditional on xT . From (2.2) we ob- tain:
(2.16) xT,h = P 1 0 0 λh
QxT = α β γ δ
1 0 0 λh
wT
−zT
. and, if h is large enough
(2.17) xT ,h = α β γ δ
1 0 0 0
wT
−zT
= α γ
wT.
For large h, the forecasts for each individual variables are dominated by the common trend wT.
But for xT,h = αwT and yT,h = γwT we get
γxT ,h− αyT ,h = (γα − γα)wT = 0
showing that the forecasts obey the cointegrating relationship and the forecast of the I(0) variable z is zT ,h = 0, with finite variance.
2.3 The mean of the cointegrating relationship
Note that in this simple example, the mean of the cointegration relationship is 0.
In practice, this long-run mean can be any number, and appears as an important parameter in the econometric cointegrating model. It is in fact a critical parameter for the forecast performance of econometric models.
In many cases, the long-run cointegrating mean also has an economic interpre- tation. If xtand ytare logs of income and consumption, the mean of the cointegration relationship may for example be interpreted as the long-run (steady-state) savings rate.
3 MA representation
Using lag-polynomials, write wt and zt as:
wt = η1t
1 − L and
zt= −η2t 1 − λL. Inserting back in (2.14) and (2.15) gives
(3.18) ∆xt
∆yt
= Ψ(L)ηt
where
(3.19) Ψ(L) = α β(1 − L)/(1 − λL) γ δ(1 − L)/(1 − λL)
.
The rank of Ψ(1) is 1, since |Ψ(1) − zI| = (α − z) · z = 0 have roots z1 = 0 and z2 = α.
Equivalently, |Ψ(1)| = 0 and |α| 6= 0). The MA-matrix thus has reduced rank for L = 1, as a result of cointegration.
4 Equilibrium correction representation (ECM)
Rewrite (2.4)
αδ − γβ = 1,
first as
αδ − λβγ = αδ − γβ + γβ − λβγ = 1 + γβ(1 − λ), and next,
−γβ + λαδ = 1 − αδ + λαδ = 1 − αδ(1 − λ), and use this to write Φ in (2.6) as
(4.20) Φ = 1 0
0 1
+ (1 − λ) β δ
γ −α .
The ECM representation follows directly from the VAR, by using (4.20) in xt= Φxt−1+ εt
We obtain:
∆xt
∆yt
= (1 − λ) β δ
[γxt−1− αyt−1]
| {z }
−zt−1
+ εt which can be written as
(4.21) ∆xt
∆yt
= αβ0 xt−1 yt−1
+ εt, or
(4.22) ∆xt= αβ0xt−1+ εt,
when the vector of equilibrium correction coefficients (α) are
(4.23) α = (1 − λ)β
(1 − λ)δ
= α11 α21
and the cointegrating vector is:
(4.24) β =
γ
−α
. When we work with a VAR, we often re-write (2.1) as
∆xt= Πxt−1+ εt
where Π = (Φ − I) is the matrix with the coefficients of the lagged levels variables.
Note that we have
(4.25) αβ0 = Π = Φ − I
which implies that the Π matrix with coefficients of the lagged levels variables has reduced rank (rank = 1) in the case of cointegration. (4.25) implies that the roots of Π are 0 and (1 − λ). β is the matrix of cointegration coefficients.
Note: If λ = 1, the rank of Π is zero (both roots are 0). Moreover:
Φ = 1 0 0 1
from (4.20) and the ECM representations collapses to a VAR in differences, a dVAR.
Cointegration does not occur.
We see that there is a close relationship between the eigenvalues of Π, its rank and cointegration:
• rank(Π) = 0, reduced rank and no cointegration. Both eigenvalues are zero.
• rank(Π) = 1, reduced rank and cointegration. One eigenvalue is different from zero.
• rank(Π) = 2, full rank, both eigenvalues are different from zero and the VAR (2.1) is stationary.
Hence, if there is a way of testing formally whether the eigenvalues of Π are significantly different from zero, we would have a logically very satisfying method of testing the hypotheses about the absence of cointegration.
This has method has actually been developed under the name of the Johansen (1995) procedure and the cointegrated VAR
4.1 Cointegration and Granger causality
Since λ < 1 is equivalent with cointegration, we see from (4.23) that cointegration also implies Granger-causality in at least one direction: α11 6= 0 and/or α21 6= 0.
Conversely If λ = 1, α = 0.
4.2 Cointegration and weak-exogeneity
Assume δ = 0, from (4.23). this implies α21 = 0.
∆xt
∆yt
= (1 − λ) β 0
[γxt−1− αyt−1] + εt
∆xt
∆yt
= (1 − λ)β[γxt−1− αyt−1] + εx,t εy,t
The marginal model of yt contains no information about the cointegration param- eters (γ, −α)0 in this case. yt is therefore weakly exogenous for the cointegration parameters (γ, −α)0.
5 Some exercises
Exercise 1 Consider the model
∆yt= (ρ − 1)yt−1+ b0∆xt+ b1xt−1+ εt, 0 < ρ < 1
∆xt= ut,
where εt and ut are independent white-noise processes.
1. Show that yt∼ I(1).
2. Write the system in AR-form:
yt xt
= A
yt−1 xt−1
+ b0ut+ εt ut
where
A = ρ b1 0 1
.
3. Show that the characteristic roots of A is λ1 = 1 and λ2 = ρ.
4. Show that A∗ in the ECM form
∆yt
∆xt
= A∗
yt−1 xt−1
+ b0ut+ εt ut
has roots z1 = 0 and z2 = ρ − 1.
5. Show that A∗ can be written
A∗ = ρ − 1 0
(1, γ) where γ = b1/(ρ − 1).
6. Show that β0 = (1, γ) is the cointegrating vector. Hint: Show that wt= β0
yt xt
= yt+ γxt
is a stationary process by multiplying the AR-equation with β0 from the left.
References
Engle, R. F. and C. W. J. Granger (1987). Co-integration and Error Correction:
Representation, Estimation and Testing. Econometrica, 55 , 251–276.
Johansen, S. (1995). Likelihood-Based Inference in Cointegrated Vector Autoregres- sive Models. Oxford University Press, Oxford.