ECON 4160, 2016. Lecture 11 and 12 - Universitetet i oslo€¦ · Lecture 11 and 12 Co-integration Ragnar Nymoen Department of Economics 9 November 2017 1/43. Engle and Granger representation

Engle and Granger representation theorem Estimation and testing Identification From I(1) to I(0)

ECON 4160, 2016. Lecture 11 and 12Co-integration

Ragnar Nymoen

Department of Economics

9 November 2017

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Introduction I

So far we have considered:

I Stationary VAR (“no unit roots”)

I Standard inference

I Non-stationary VAR (“all unit-root”)

I The spurious regression example with two independent randomwalks is an example of a non-stationary VAR

I Have discovered that the Dickey-Fuller distribution can be usedto test the null hypothesis of unit-root for a single time series

We next consider cointegration, the case of “some, but not onlyunit-roots” in the VAR.

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Introduction III In such systems, there exist one or more linear combinationsof I (1) variables that are I (0)– they are called cointegrationrelationships.

I Cointegration is the “flip of the coin”of spurious regression: Ifwe have two dependent I (1) variables, they are cointegrated.

I We can guess that the right distribution to use for testingcointegration is going to be of the Dickey-Fuller type:Intuition: If we reject Yt ∼ I (1) against Yt ∼ I (0) usingcritical values from DF-distributions, we have shown that Yt is“cointegrated with itself”.To test whether Yt is cointegrated with another variable,Xt ∼ I (1), as similar distribution will be needed.

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Introduction IIII In these two lectures we sketch the theory of cointegrationmore fully:

I Granger’s representation theorem. This shows theimplications of at least one unit root in the VAR, and the rest< 1.Leads to the concept of the cointegrated VAR, and to ECMmodel equations as an implication of cointegration

I Methods for testing the null-hypothesis of nocointegration

I Testing residuals form static “cointegrating regression” forunit-root

I Using the conditional ECM model equation to test forunit-root (Ericsson and MacKinnon article in particular).

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Introduction IVI These two methods are single equation methods, and stillthe most used tests in practice.

I A multi-equation method uses the VAR directly. It is oftencalled testing for cointegration rank (aka the Johansenmethod).

I Note about HN: They present an example of the VAR methodfirst, in Chapter 17.3, and the single equation test in Chapter17.4

I Important take-away: After concluding the test for absence ofcointegration, we are back in the stationary case:

I Can estimate cointegrated VARs, and simultaneous equationsmodels in the way we have learned for stationary systems.

I Or, if there are no evidence of cointegration: formulate VARsand models that only include differences of the I (1) series.

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Introduction V

References:

I HN: Ch. 17.

I BN(2104): Kap. 11.

I The posted paper by Ericsson and MacKinnon (to Lecture 9).

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The VAR with one unstable and one stable root I

Consider the bi-variate first order VAR

yt = Φyt−1 + εt (1)

where yt = (Yt ,Xt )′, Φ is a 2× 2 matrix with coeffi cients and εtis a vector with Gaussian disturbances.The characteristic equation for Φ:

|Φ− λI| = 0,

Our interest is the case with one unit-root and one stationary root:

λ1 = 1, and λ2 = λ, |λ| < 1. (2)

implying that both Xt and Yt are I (1).

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The VAR with one unstable and one stable root IIΦ has full rank, equal to 2. It can be diagonalized in terms of itseigenvalues and the corresponding eigenvectors:

Φ = P[1 00 λ

]P−1 (3)

P has the eigenvectors as columns:

P =[a bc d

](4)

Set |P| = 1 without loss of generality, then:

P−1 =[d −b−c a

].

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Cointegrated VAR– ECM implication I

Multiply from the right on both sides of (1) by P−1. The VAR isthen re-written as:[

Wt

−ECt

]=

[1 00 λ

] [Wt−1−ECt−1

]+ ηt , (5)

where ηt = P−1εt ,and ECt and Wt are defined as:

Wt = dYt − bXt , (6)

ECt = −cYt + aXt . (7)

I ECt ∼ I (0), is a stationary variable, the EquilibriumCorrection variable. Why?

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Cointegrated VAR– ECM implication II

I Wt ∼ I (1), is a common (stochastic) trend, i.e.Yt and Xtcontain the same trend

I We say that there is cointegration between Xt and Yt , sinceECt is a stationary variable, and it is a linear combination ofXt and Yt .

I c and a are the cointegrating parameters in this example.

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The ECM representation of the cointegrated VAR IRe-parameterize the VAR (1) as

∆yt = Πyt−1 + εt (8)

where we use the notation (in compliance with the literature):

Π = (Φ− I) (9)

Next, define two (2 × 1) parameter vectors α and β in such a waythat the product αβ′ gives Π:

Π=αβ′ (10)

In our example (compare BN 2014 Kap 11):

Π=

[(1− λ)b(1− λ)d

]︸︷︷︸

α

[c −a

]︸︷︷︸β′

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The ECM representation of the cointegrated VAR II

and then (8) can be expressed as:[∆Yt∆Xt

]= αβ′

[Yt−1Xt−1

]+ εt , (11)

I α is the matrix with equilibrium correction coeffi cients (aka adjustment coeffi cients, or loadings),

α =

[(1− λ)b(1− λ)d

](12)

I β is the matrix with the cointegration coeffi cients

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The ECM representation of the cointegrated VAR III

β =

[c−a

](13)

In this formulation we see that

I rank(Π) = 0, reduced rank and no cointegration. Botheigenvalues are zero.

I rank(Π) = 1, reduced rank and cointegration. Oneeigenvalue is different from zero.

I rank(Π) = 2, full rank, both eigenvalues are different fromzero and the VAR in (1) is stationary.

Cointegration and Granger causalitySince λ < 1 is equivalent with cointegration, we see from (12) thatcointegration implies Granger-causality in at least one direction:(1− λ)b 6= 0 and/or (1− λ)b 6= 0.

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The ECM representation of the cointegrated VAR IV

Cointegration and weak exogeneity

I Assume d = 0, from (12). This implies[∆Yt∆Xt

]= (1− λ)

[b0

][cYt−1 − aXt−1] + εt[

∆Yt∆Xt

]=

[(1− λ)b[cYt−1 − aXt−1] + εy ,t

εx ,t

]I The marginal model contains no information about thecointegration parameters (c,−a)′. Yt is weakly exogenousfor the cointegration parameters β′ = (c,−a)′.

I So how can we test for WE of Xt with respect to β ?

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Generalization of ECM

VAR(p) – > ECM general case IIf yt is n× 1 with I (1) variables. The VAR is:

yt = (p

∑i=0

Φi+1Li )yt−1 + εt (14)

where εt is multivariate Gaussian.The generalization of the ECM form (8):

∆yt = Φ∗(L)∆yt−1 +Πyt−1 + εt (15)

where Φ∗(L) contains known expressions of Φi+1, and

Π = (p

∑i=0

Φi+1 − In). (16)

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VAR(p) – > ECM general case II

I In the case of no cointegration, Π is the null matrix:

Π = 0 (17)

but, it is:Π = αβ′ (18)

in the case of r cointegrating vectors.I βn×r contains the CI-vectors as columns, while αn×r showsthe strength of equilibrium correction in each of the equationsfor ∆Y1t ,∆Y2t , . . . ,∆Ynt . In general rank(β) = r andrank(Π) = r < n.

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VAR(p) – > ECM general case III

I If β is known, the system

∆yt = Φ∗(L)∆yt−1 + α[β′y]t−1 + εt (19)

contains only I (0) variables and conventional asymptoticinference applies.

I Moreover: If β is regarded as known, after first estimating β ,conventional asymptotic inference also applies.

I (19) is then a stationary VAR, called the VAR-ECM or thecointegrated VAR.

I This system can be estimated and modelled as we have seenfor the stationary case.

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The role of the intercept and trend

Restricted and unrestricted constant term I

I Usually we include separate Constants in each row of the VAR.

I We call them unrestricted constant terms. In the unit-rootcase the implication is that each Yjt contains a deterministictrend (think of a Random Walk with drift)

I But if the constants are restricted to be in the ECt−1variables, there are no drifts and therefore no trend in thelevels variables.

I When we test for cointegration, based on the VAR, there aredifferent critical values for the two cases.

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Conditional cointegeration and exogeous I(1) variables in the VAR

Conditional cointegrated ECM I

I Consider the bivariate case: yt = (Yt ,Xt )′ above.I Assume that α21 = 0, then Xt is weakly exogenous for β.I With Gaussian disturbances εt = N(0,Ω), where Ω haselements ωij ,we can derive the conditional model for ∆Y1t :

∆Yt = ω21ω−122︸︷︷︸

β0

∆Xt + α11β′[Yt−1Xt−1

]+ ε1t −ω21ω

−122 ε2t︸︷︷︸

εt

(20)a single equation ECM. If we write it as

∆Y1t = β0∆Xt + α11β11Y1t−1 + α11β12Xt−1 + εt

we see that Π=α11β11 6= 0, i.e., the Π “matrix”has full rank.

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Conditional cointegeration and exogeous I(1) variables in the VAR

Conditional cointegrated ECM II

I The common I (1)-trend is now the observable exogenousvariable Xt .

Generalization: The VAR contains n1 endogenous I (1) variablesand n2 non-modelled I (1) variables. (Often called an open systemor VAR-EX). Cointegration is then consistent with:

0 < rank(Π) ≤ n1

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Estimating a single cointegrating vector

The cointegrating regression I

When rank(Π) = 1, the cointegration vector is unique (subjectonly to normalization).Without loss of generality we set n = 1 and write yt = (Yt ,X ) asin a usual regression.The cointegration parameter β can be estimated by OLS on

Yt = α+ βXt + ut (21)

where ut ∼ I (0) by assumption of cointegration.

(β− β) =∑Tt=1 Xtut

∑Tt=1 X

2t. (22)

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The cointegrating regression IISince Xt ∼ I (1), we are in a the same situation as with the firstorder AR case with autoregressive parameter equal to one (Lecture10)In direct analogy, we need to multiply (β− β) by T in order toobtain a non-degenerate asymptotic distribution:

T (β− β) =

1T

∑Tt=1 Xtut

1T 2

∑Tt=1 X

2t

, (23)

=⇒ (β− β) converges to zero at rate T , instead of√T as in the

stationary case.

I This result is called the Engle-Granger super-consistencytheorem.

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The cointegrating regression III

I Remember: This is based on r = 1 so the cointegrationvector is unique if it exists.

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The distribution of the Engle-Granger estimator I

I It has been shown that the E-G estimator is not normallydistributed asymptotically (it is so called mixed-normal).

I The same applies to the t−value based on β: It does not havean asymptotic normal distribution (under the assumption ofcointegration).

I This means that it is impractical to use the static regressionfor doing valid inference about β. (Test whether β = 1 forexample).

I The drawback is even more severe in DGPs with higher orderdynamics, due of residual autocorrelation as a results ofdynamic misspecification.

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ECM estimator I

I The ECM represents a logical way of avoiding the bias due todynamic mis-specification of the E-G cointegration regression.

I Because, if cointegration is valid, the ECM is implied (fromthe representation theorem).

I With n = 2, p = 1 and weak exogeneity of Xt with respect tothe cointegration parameter, we have seen that thecointegrated VAR can be re-written as a conditional modeland a marginal model:

∆Yt = β0∆Xt + π︸︷︷︸α11β11

Yt−1 + γ︸︷︷︸α11β12

Xt−1 + εt (24)

∆Xt = ε2t (25)

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ECM estimator IIwhere β0 is the regression coeffi cient, and εt and ε2t areuncorrelated normal variables. Normalization on Yt−1 bysetting β11 = −1, and defining β12 = β, for comparison withthe E-G estimator, gives

∆Yt = β0∆Xt − α11(Yt−1 − βXt−1) + εt

The ECM estimator βECM , is obtained from OLS on (24)

βECM =γ

−π(26)

I βECM is consistent if both γ and π are consistent.I OLS “chooses” the γ and π that give the best predictor(Yt−1 − βECMXt−1 of ∆Yt .

I As T grows towards infinity, the true parameters γ, π and βwill therefore be found.

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ECM estimator III

I In fact βECM is super-consistent (as the E-G estimator is)I βECM has better small sample properties than the E-Gestimator, since it is based on a well specified econometricmodel (avoids the second-order bias problem).

Inference:

I The distributions of γ and π (under cointegration) can beshown to be so called “mixed normal” for large T .

I This means that variances are stochastic variables rather thanfixed parameters.

I However, the OLS based t-values of γ and π areasymptotically N(0, 1).

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ECM estimator IV

I βECM is also “mixed normal”, but

tβECM = βECM/√Var(βECM ) −→

T−→∞N(0, 1) (27)

and Var(βECM ) can be found by using the delta-method.

I The generalization to n− 1 explanatory variables, interceptand dummies is also unproblematic.

I Remember:The effi ciency of the ECM estimator depends onthe assumed weak exogeneity of Xt .

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Testing r=0 against r=1

Engle-Granger test

I The easiest approach is to use an ADF regression to the testthe null-hypothesis of a unit-root in the residuals ut from thecointegrating regression (21).

I The motivation for the ∆ut−j terms is as before: to whitenthe residuals of the ADF regression

I The DF critical values are shifted to the left as deterministicterms, and/or more I (1) variables in the regression are added.

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Testing r=0 against r=1

The ECM test

I As we have seen, r = 0 corresponds to π = 0 in the ECMmodel in (24):

∆Yt = β0∆Xt + πYt−1 + γXt−1 + εt

I It also comes as no surprise that the t-value tπ have“Dickey-Fuller” like distributions under H0 : π = 0.

I See Ericsson and MacKinnon (2002) for critical values.I Type-1 error probability: Not much difference between E-Gand ECM test. In general. Type-2 error probabilities aresmaller for ECM

I (The two becomes equivalent when the ECM model equationobeys Common factor restrictions.)

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The Johansen method

Testing cointegrating rank, general case I

For the vector yt consisting of n× 1 variables, we have theGaussian VAR(p) above:

∆yt = Φ∗(L)∆yt−1 +Πyt−1 + εt (28)

We are interested in both the cointegrating case

0 < rank(Π) < n (29)

and the case with no cointegration

rank(Π) = 0. (30)

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The Johansen method

Testing cointegrating rank, general case II

I rank(Π) is given by the number of non-zero eigenvalues of Π.But can we find the number of eigenvalues that aresignificantly different from zero?

I Fortunately, this problem has a solution. An eigenvalue of Π

is a special kind of squared correlation coeffi cient known as acanonical correlation in multivariate statistics.

I This method has become known as the Johansen approach.It is likelihood based, see HN § 17.3.2.

I Unlike the ECM method: The Johansen method does notrequire weak exogeneity of X with respect to cointegrationparameters.

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The Johansen method

Intuition I

I For concreteness, consider n = 3 so r can be 0,1 or 2I r = 0 corresponds to Π = 0From the representation theorem; with two unit-roots

Π=Φ− I = P[1 00 1

]P−1 − I = 0.

I r = 1 corresponds to α3×1 6= 0 for a single cointegrationvector β′1×3.β′1×3yt−1 must then be I (0) and it must be a significantpredictor of ∆yt .The strength of the relationship can be estimated by thehighest squared correlation coeffi cient, call it ρ21,between ∆yt

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The Johansen method

Intuition IIand all the possible linear combinations of the variables inyt−1.If ρ21 > 0 is statistically significant, we reject that r = 0.

ρ21 is the same as the highest eigenvalue of Π, and β′1×3 is the

corresponding eigenvector.I If r = 0 is rejected we can,continue, and test r = 1 againstr = 2.If the second largest correlation coeffi cient ρ22 is alsosignificantly different from zero, we conclude that the numberof cointegrating vectors is two. β

′2×3 is the corresponding

eigenvectorIt can be shown that, for the Gaussian VAR, β

′1×3 and β

′2×3

are ML estimates.

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The Johansen method

Trace-test and max-eigenvalue test I

I We order the canonical correlations from largest to smallestand construct the so called trace test:

Trace-test = −T3

∑i=r+1

ln(1− ρ2i ), r = 0, 1, 2 (31)

I If ρ21 is close to zero, then clearly Trace-test will be close tozero, and we we will not reject the H0 of r = 0 against r ≥ 1.

I and so on for H0 of r = 1 against r ≥ 2I Of course: to make this a formal testing procedure, we needthe critical values from the distribution of the Trace-test forthe sequence of null-hypotheses.

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The Johansen method

Trace-test and max-eigenvalue test II

I The distributions are non-standard, but at least the maincases are tabulated in PcGive.

I A closely related test is called the max-eigenvalue test,(butthe trace test is today judged most reliable)

I If there is a single cointegrating vector, and there are n− 1weakly-exogenous variables, the Johansen method reduces tothe testing and estimation based on a single ECM equation(with the remarks we have made above).

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The Johansen method

Constant and other deterministic trends I

I It matters a great deal whether the constant is restricted tobe in the cointegrating space or not.

I The advise for data with visible drift in levels:I include an deterministic trend as restricted together with anunrestricted constant.

I After rank determination, can test significance of the restrictedtrend with standard inference

I Shift in levelsI Include restricted step dummy and a free impulse dummy.

I Exogenous I(1) variables, see table and program byMacKinnon, Haug and Michelis (1999).

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Identification of multiple cointegration vectors. I

I As we have seen, if n = 2, cointegration implies rank(Π) = 1.

I There is one cointegration vector

(β11,β12)′

which is uniquely identified after normalization. For examplewith β11 = −1 the ECM variable becomes

ECM1t = −Y1t + β12Y2t ∼ I (0).

I When n > 2, we can have rank(Π) > 1, and in these casesthe cointegrating vectors are not identified in Johansen’smethod.

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Identification of multiple cointegration vectors. II

I Assume that ˝ is known (in practice, consistently estimated),and β is a n× r cointegrating matrix:

Π = αβ′

However for a r × r non-singular matrix Θ:

Π = αΘΘ−1β′ = αΘβ′Θ

showing that β′Θ is also a matrix with cointegrating vectors.

This identification problem is equivalent to the identificationproblem in simultaneous equation models!

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Identification of multiple cointegration vectors. IIII Assume rank(Π) = 2 for a n = 3 VAR

−Y1t + β12Y2t + β13Y3t = ECM1t

β21Y1t − Y2t + β13Y3t = ECM2t

I By simply viewing these as a pair of simultaneous equations,we see that they are not identified on the order-condition.

I Exact identification requires for example 1 linear restrictionson each of the equations.

I For example β13 = 0 and β21 + β13 = 0 will result in exactidentification

I Identification ≡Theory !!!I If we include non-modelled (exogenous) variables in thecointegration relationship, the identification problem is againdirectly analogous to the simultaneous equation model.

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I(0) variables in the VAR?I A misunderstanding that sometimes occurs is that “there canbe no stationary variables in he cointegrating relationships”.Consider for example:

−Y1t + β12Y2t + β13Y3t + β14Y4t = ecm1t (32)

β21Y1t − Y2t + β23Y3t + β24Y4t = ecm2t (33)

If Y1 is the log of real-wages, Y2 productivity, Y3 relativeimport prices, and Y4 the rate of unemployment, then the firstrelationship may be a bargaining based wage and the second amark-up equation.

I Y4t ∼ I (0), most sensibly, but we want to estimate and testthe theory β14 = 0.

I Hence: specify the VAR with Y4t included.

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From I(1) to I(0)I When the rank has been determined, we are back in thestationary-case.

I The distribution of the identified cointegration coeffi cients are“mixed normal” so that conventional asymptotic inference canbe performed on this β.

I The determination of rank allows us to move from the I (1)VAR, to the cointegrated VAR that contains only I (0)variables

I Another name for this I (0) model is the vector equilibriumcorrection model, VECM.

I The VECM can be analyzed further, using the tools of thestationary VAR !

I Hence, cointegration analysis is an important step in theanalysis, but just one step.

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Some important additional references

Johansen, S. (1995), Likelihood-Based Inference in CointegratedVector Auto-Regressive Models, Oxford University PressJuselius, K (2004) The Cointegrated VAR Model, Methodologyand Applications, Oxford University PressMacKinnon, J., A. A. Haug and L. Michelis (1999) NumericalDistributions Functions of Likelihood Ratio Tests for Cointegration,with programs.

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Documents

ECON 4160, 2016. Lecture 11 and 12 - Universitetet i oslo€¦ · Lecture 11 and 12 Co-integration Ragnar Nymoen Department of Economics 9 November 2017 1/43. Engle and Granger representation