The Engle-Granger representation theorem

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1 The Engle-Granger representation theorem Reference note to lecture 10 in ECON 5101/9101, Time Series Econometrics Ragnar Nymoen March 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 (21 x t Φx t 1 + ε t where x t (x t, y t, Φ is a 2 2 matrix with coefficients and ε t 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: z 1 1, and z 2 λ, λ < 1 1

2 z 1 1 implies that both x t and y t are 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: (22 Φ P λ Q where P is the matrix with the eigenvectors as columns: α β (23 P γ δ and Q P 1 Assume, without loss of generality that P 1, ie, (24 αδ γβ 1 and thus δ β P 1 γ α By mulipliction from the right, Φ can be expressed as α β δ β (25 Φ, γ δ γλ αλ and, by multiplication from the left, we also get: (αδ λβγ αβ(1 λ (26 Φ γδ(1 λ ( γβ + λαδ We refer to these expressions later 21 The Common-trends representation Multiplication of (21 with Q yields (27 Qx t QΦx t 1 + Qε t Observe that QΦ QP λ 1 0 Q 0 λ Q, from the definitions above, implying that (27 can be written in terms of two new variables w t and z t : wt 1 0 wt 1 (28 + η z t 0 λ z t, t 1 where (29 wt z t xt Q y t 2

3 and (210 Qε t η t η1t η 2t To solve for w t and z t, we multiply out the right hand side of (29 Notice that since δ β (211 Q γ α we obtain w t as (212 w t δx t βy t and z t as: (213 z t γx t αy t Conversely, from (29: xt y t P wt z t yielding (214 x t αw t βz t and (215 y t γw t δz t Equation (28 gives w t as a random-walk variable and, from the same equation, z t is seen to be a stationary since λ < 1 z t, defined by (213 is therefore a stationary linear combination of the two I(1 variables x t and y t We say that x t and y t are two cointegrated variables, with cointegrating vector (γ, α On the other hand: equation (214 and (215 show that x t and y t contain an I(0 component (z t, and an I(1 component (w t hence the name Common trend representation 22 Integration and cointegration is maintained in forecasts Let x T,h denote the h-period ahead forecast, conditional on x T From (22 we obtain: 1 0 α β 1 0 wt (216 x T,h P 0 λ h Qx T γ δ 0 λ h z T and, if h is large enough α β (217 x T,h γ δ wt z T α γ w T 3

4 For large h, the forecasts for each individual variables are dominated by the common trend w T But for x T,h αw T and y T,h γw T we get γx T,h αy T,h (γα γαw T 0 showing that the forecasts obey the cointegrating relationship and the forecast of the I(0 variable z is z T,h 0, with finite variance 23 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 interpretation If x t and y t are 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 w t and z t as: and w t z t η 1t 1 L η 2t 1 λl Inserting back in (214 and (215 gives xt (318 Ψ(Lη t where α β(1 L/(1 λl (319 Ψ(L γ δ(1 L/(1 λl The rank of Ψ(1 is 1, since Ψ(1 zi (α z z 0 have roots z 1 0 and z 2 α Equivalently, Ψ(1 0 and α 0 The MA-matrix thus has reduced rank for L 1, as a result of cointegration 4 Equilibrium correction representation (ECM Rewrite (24 αδ γβ 1, 4

5 first as and next, αδ λβγ αδ γβ + γβ λβγ 1 + γβ(1 λ, and use this to write Φ in (26 as 1 0 (420 Φ 0 1 γβ + λαδ 1 αδ + λαδ 1 αδ(1 λ, β + (1 λ δ γ α The ECM representation follows directly from the VAR, by using (420 in We obtain: xt which can be written as (421 or x t Φx t 1 + ε t β (1 λ δ xt αβ xt 1 y t 1 (422 x t αβ x t 1 + ε t, γx t 1 αy t 1 + ε } {{ } t z t 1 + ε t, when the vector of equilibrium correction coefficients (α are (1 λβ α11 (423 α (1 λδ and the cointegrating vector is: (424 β γ α α 21 When we work with a VAR, we often re-write (21 as x t Πx t 1 + ε t where Π (Φ I is the matrix with the coefficients of the lagged levels variables Note that we have (425 αβ 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 (425 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: 1 0 Φ 0 1 from (420 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: 5

6 rank(π 0, reduced rank and no cointegration Both eigenvalues are zero rank(π 1, reduced rank and cointegration from zero One eigenvalue is different rank(π 2, full rank, both eigenvalues are different from zero and the VAR (21 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 41 Cointegration and Granger causality Since λ < 1 is equivalent with cointegration, we see from (423 that cointegration also implies Granger-causality in at least one direction: α 11 0 and/or α 21 0 Conversely If λ 1, α 0 42 Cointegration and weak-exogeneity Assume δ 0, from (423 this implies α 21 0 xt β (1 λ γx 0 t 1 αy t 1 + ε t xt (1 λβγxt 1 αy t 1 + ε x,t ε y,t The marginal model of y t contains no information about the cointegration parameters (γ, α in this case y t is therefore weakly exogenous for the cointegration parameters (γ, α 5 Some exercises Exercise 1 Consider the model (ρ 1y t 1 + b 0 x t + b 1 x t 1 + ε t, 0 < ρ < 1 x t u t, where ε t and u t are independent white-noise processes 1 Show that y t I(1 2 Write the system in AR-form: ( yt A where x t ( yt 1 A x t 1 ( ρ b1 0 1 ( b0 u + t + ε t u t 6

7 3 Show that the characteristic roots of A is λ 1 1 and λ 2 ρ 4 Show that A in the ECM form ( ( ( yt A yt 1 b0 u + t + ε t x t x t 1 u t has roots z 1 0 and z 2 ρ 1 5 Show that A can be written A ( ρ 1 0 (1, γ where γ b 1 /(ρ 1 6 Show that β (1, γ is the cointegrating vector Hint: Show that ( w t β yt y x t + γx t t is a stationary process by multiplying the AR-equation with β from the left References Engle, R F and C W J Granger (1987 Co-integration and Error Correction: Representation, Estimation and Testing Econometrica, 55, Johansen, S (1995 Likelihood-Based Inference in Cointegrated Vector Autoregressive Models Oxford University Press, Oxford 7

E 4101/5101 Lecture 8: Exogeneity

E 4101/5101 Lecture 8: Exogeneity Ragnar Nymoen 17 March 2011 Introduction I Main references: Davidson and MacKinnon, Ch 8.1-8,7, since tests of (weak) exogeneity build on the theory of IV-estimation Ch