1

UNIVERSITY OF PRETORIA

FACULTY OF ECONOMIC AND MANAGEMENT SCIENCES

DEPARTMENT OF ECONOMICS

ECONOMETRICS (EKT 813) PART1, 2005

15th March 2005

MOSES SICHEI

LECTURE 8: JOHANSEN COINTEGRATION III

Lecture Objectives

To see how the ECM looks like for particular cointegrating

vectors

Testing parameter restrictions on the long-run

cointegrating relationship

Testing parameter restrictions on the adjustment

coefficients (Weak exogeneity test)

Get an idea of key steps in applying Johansen procedure

Key vocabulary

Normalisation

Testing economic theory restrictions on in Johansen

distance

Testing economic theory restrictions on in Johansen

Testing of weak exogeneity

2

Important articles

Johansen,S.(1988), “Statistical Analysis of Cointegrating

Vectors” Journal of Economic Dynamics and Control, Vol.12

(June-Sept), 231-254.

Stock,J. and Watson, M.(1988), “Testing for Common

Trends”, Journal of the American Statistical Association, Vol.83

(Dec.), 1097-1107.

Johansen,S. and Juselius, K. (1992), “Testing Structural

Hypothesis in a Multivariate Cointegration Analysis of the PPP

and UIP for UK”, Journal of Econometrics vol.53, 211-244.

1. EXAMPLES OF WHAT THE ECM VAR SYSTEM

LOOKS LIKE FOR PARTICULAR VALUES OF

COINTEGRATING VECTORS (r)

Let’s use an example given in Enders 2004 (question 4) and file

on interest rates. The data used contains interest rates paid on

U.S. Tbills, 3-year and 10-years. The data run from 1954:7 to

2002:12. The data is downloaded from

http://www.cba.ua.edu/~wenders/.

Here we have n = 3 with X comprising the three I(1) time series

tb , 3r and 10r .

Let k, the lag length of the VAR be 2, and suppose that there

are two cointegrating relationships among the elements of X

( 2r ). We include a vector of intercepts in the VAR.

Ignoring the vector of dummy variables, the VAR system can be

written in levels as

t2-t21-t1t + + + = XAXAX (1)

or in its ECM representation as:

3

t 1 t-1 t-1 tX = + X + X + (2)

Given the existence of cointegration, we can use the Granger

Representation theorem to shown that its ECM is;

t 1 t-1/

t-1 tX = + X + X + (3)

Writing this in full we have as Equation (4):

t

t

t

t

t

t

t

t

t

t

t

t

r

r

tb

r

r

tb

r

r

tb

3

2

1

1

1

1

322212

312111

3231

2221

1211

1

1

1

333231

232221

131211

3

2

1

10

3

10

3

10

3

(4)

Ignoring the first two components (intercept and short-run

coefficients) and the disturbance term on the right hand side of

the equation (and so looking only at the long run relationships),

we focus on the following part of this system:

4

1

1

1

322212

312111

3231

2221

1211

10

3

10

3

t

t

t

t

t

t

r

r

tb

r

r

tb

Adjustment

coefficientsCointegrating

vectors

1tXtX

(5)

If we leave out the part and focus on the cointegrating vector

only, (i.e. terms /X t-1 ). This comprises two cointegrating

vectors (r=2) as follows

1

1

1

322212

312111

10

3

t

t

t

r

r

tb

(6)

It can be seen, by expanding the expression in Equation 6, that

the cointegrating vectors are given by;

103

103

r + r + tb

r + r + tb

1-t321-t221-t12

1-t311-t21t11

(7)

The elements of (sometimes known as the loading matrix)

determine into which equation the cointegrating vectors

enter and with what magnitudes.

5

If all elements of are non-zero, then all cointegrating vectors

(in this case two) enter into each equation. This can be seen by

writing equation (5) in scalar algebra form, giving;

1321221123213112111131

1321221122213112111121

1321221121213112111111

10310310

1031033

103103

ttttttt

ttttttt

ttttttt

rrtbrrtbr

rrtbrrtbr

rrtbrrtbtb

(8)

In general, if there are r cointegrating relationships among n

variables, the r cointegrating relationships will enter into each of

the n equations.

In our example where r = 2, the 2 separate long run

relationships enter into each of the 3 equations. Even if we

are only really interested in estimating one of these equations -

perhaps the first one - then both cointegrating relationships

should enter that equation separately. There are in effect two

ECM terms in the equation.

This shows more clearly the pitfalls of EG which chooses an

estimator that constrains r to be zero or one when actually r is

greater than one. The problems this leads us into include the

following:

If we find r = 1 (as opposed to zero) when in fact r = 2, our

dynamic adjustment equation (the second stage of the EG

two-step approach, for example) will contain only one ECM

term when it should contain 2. The regression model is

misspecified by assuming that 012

There are in fact cross-equation restrictions in the n

equations of the VAR, because the same two ECM terms

(with the same coefficients) into each equation. A systems

estimator is needed to estimate this system efficiently (taking

account of these restrictions). Even if r = 1, it will still be

the case that the single cointegrating relationship could

6

enter more than one of the n equations in the VAR system.

Ignoring this (or these) cross-equations restrictions will lead

to lose efficiency in the estimation procedure.

What we regard as the single “cointegrating vector” will be

a linear combination of two independent cointegrating

vectors. Moreover, we will not be able to identify either of

the two underlying vectors from the underlying mixed vector.

This can be seen more clearly in the following way. First,

rearrange the first of the equations in (8) into the following

form:

tbt = (1111 + 1212 )tbt-1 + (1121 + 12 22)r3t-1 + (11 31 +

12 22)r10t-1

(9)

This can be expressed as

131211 103 tttt rrtbtb (10)

Where:

221231113

221221112

121211111

When the E-G first step is used, the ‘long-run’ parameter

estimates we obtain are those of the parameters. But these

three parameters are linear combinations of the six structural

and parameters, which are not identified (they cannot be

recovered from the estimates).

2.TESTING PARAMETER RESTRICTIONS ON THE

LONG-RUN COINTEGRATING RELATIONSHIP(S)

2.1 NORMALISATION

7

As was demonstrated in lecture 7, the Johansen technique allows

one to determine how many independent cointegrating

relationships exist among the set of variables being

considered.

However, the estimated parameter values in the r cointegrating

relations are not unique.

Any linear combination of these r stationary relations will

itself be stationary, and so qualify as a cointegrating

relationship.

The particular estimated parameter values obtained would

depend on the normalisation procedure chosen: different

normalisations will produce different sets of estimated long run

relations.

2.2 REASONS FOR TESTING

Oxley et al.(1995:196) (Surveys in econometrics) point out that

the usual response by applied researchers (especially

macroeconometric modelers faced with a trade off between

quantity and quality) who seek to model a single long-run

behavioural relationship between the variables included in the x

when faced with the result that r>1 is to choose to employ only

that cointegrating vector which makes “economic sense”.

In other words, they choose the vector where the estimated long-

run elasticities correspond closely (in magnitude and sign) to

those predicted by economic theory.

Such an ad hoc approach is wrong because both the EG and

Johansen procedure are explicitly multivariate in the sense that

they both postulate an ECM for all the variables involved in the

model.

8

Neither the EG nor Johansen procedure approaches the

problem of partitioning variables into endogenous and

(weakly) exogenous, which is central to estimating a single

behavioural equation.

This means that arbitrary selection of one of the significant

cointegrating vectors in order to move from Johansen

framework to the estimation of a single structural equation

implicitly makes the assumption that the conditional model

which is being isolated is valid!

The best approach is to test for the restricted forms of the

cointegrating vectors.

There are two kinds of testing we may wish to do; we may wish

to test hypotheses about elements of the parameter matrix; or

we may wish to test hypotheses about elements of the

parameter matrix.

Tests about elements of are concerned with issues about

which of the equations in the system the cointegrating vectors

enter.

Tests about are concerned with restrictions on the parameters

within the long run relationships themselves. Tests about are

of particular importance as our ultimate objective is to extract

estimates of the structural equations, which underlie the

reduced form.

Recall that the parameter estimates which we obtain after having

specified how many cointegrating relationships exist (that is,

from the Johansen procedure after choosing r but prior to doing

any tests on are the unrestricted reduced form parameter

estimates. These are not what we are interested in!

9

What we are interest in can only be obtained by trying to deduce

likely restricted structural equations (from relevant economic

theory e.g. PPP, demand for money, uncovered interest parity

conditions, consumption function etc.) and then testing whether

the implied restrictions are acceptable.

2.3 RESTRICTIONS ON THE ELEMENTS OF

THE CASE WHERE r = 1

If r = 1, the parameter estimates of the single cointegrating

relationship can be read directly from the estimated vector.

In effect, there is no difference between the reduced form and

structural model in this case.

However, there is still a normalisation issue to consider. If is

a valid long run parameter vector, in the sense that X is a

stationary, cointegrating linear combination of the variables in X

, then any multiple of will also be a valid cointegrating vector.

Looking at the application of Johansen at the last part of the

notes;

if = (2.49, -3.99, 1.66) , so that ttt rrtbX 1066.1399.349.2 is

a cointegrating combination, then if we multiply the

cointegrating vector by -0.5, (-1.25, 1.99, -0.83) is also a

cointegrating vector with ttt rrtb 1083.0399.125.1 a stationary

linear combination.

We can choose to “normalise” the cointegrating vector in any

way we like. It is conventional to choose a normalisation in

which the variable we regard as the dependent variable in the

relationship is given a coefficient of -1.

Having obtained estimates of the parameters of the single

cointegrating relationship, we could then test restrictions

10

implied by economic theory on these parameters. For

example, we could test the null hypothesis

H0:

against

Ha:

There are two restrictions implied by this null hypothesis.

A likelihood ratio test can be used to test these restrictions.

The point is that once and are determined, the test statistic

entails comparing the number of cointegrating vectors under the

null with the alternative hypotheses.

In order to test restrictions on form the statistic;

r

i

iiT1

* ˆ1lnˆ1ln (11)

This has a 2 distribution with degrees of freedom equal to the

number of restrictions placed on . The restriction embedded

in the null hypothesis is binding if the calculated value of the test

statistic exceeds that in 2 table.

2.4 THE CASE WHERE 1r

When 1r , matters are more complicated. It will not usually be

sensible to take the unrestricted estimates of the vectors in

directly as economically-meaningful long run parameter

estimates.

There are two reasons why not;

(i)There is a normalisation issue, similar in principle to the one

explained in the r = 1 case.

(ii)What we will (usually) be trying to obtain are estimates of

the parameters of the structural equations of the system. It is

necessary, therefore, to impose (and test) restrictions on the

elements of in an attempt to try and obtain the structural

11

relationships between the variables. This is likely to be a

difficult exercise in practice. One must be guided here by

economic theory - there are no "econometric rules" that can

be followed in a mechanical way.

An important part of this exercise is likely to be testing exclusion

restrictions (i.e. that the parameters associated with particular

variables have zero coefficients) suggested by economic theory.

For example, in the example we looked at earlier,

103

103

r + r + tb

r + r + tb

1-t321-t221-t12

1-t311-t21t11

(12)

Assume that we have already concluded that there are 2

cointegrating relationships among the three variables. Economic

theory might tell us that one of these is of the form;

0103 r + r + tb 1-t311-t21t11 (13)

Whereas the other is of the form

0103 r + r 1-t321-t22

In the second of these, the coefficient on ttb has implicitly been

set to zero, and so we might wish to test such an exclusion

restriction.

Important point: Standard row and column operations on do

not entail restrictions on the cointegrating vectors.

Variable exclusion within an equation:

With multiple cointegrating vectors, you cannot test whether any

particular value of 0ij since this assumption does not restrict

the cointegrating space.

12

Where is an rn matrix, a testable exclusion restriction

entails the exclusion or r or more variables from a

cointegrating vector. Hence excluding r variables from a

cointegrating vector entails only one restriction. If the sample

value of the 2 statistic with one degree of freedom exceeds a

critical value, reject the null hypothesis that this set of variables

contains a cointegrating relationship.

Variable exclusion across equations

Recall

103

103

r + r + tb

r + r + tb

1-t321-t221-t12

1-t311-t21t11

(14)

The cross-equation restriction 3231 entails only one

restriction on the cointegrating space. This is because it is on

the same column.

2.5 WEAK EXOGENEITY AND TESTING

RESTRICTIONS ON ELEMENTS OF

The elements of the matrix (sometimes known as the loading

matrix) determine into which equation the cointegrating vectors

enter and with what magnitudes.

Put another way, the elements of the matrix determine the

existence and magnitudes of the "feedback" effects of the

various cointegrating relationships in the individual dynamic

equations governing the evolution of ttb , tr3 and tr10 over time.

The elements of the matrix relate to the issue of weak

exogeneity. In a cointegrated system, if a variable does not

respond to the discrepancy from the long-run equilibrium, it is

weakly exogenous. In economic theory terms, this implies that

there are rigidities (such as regulations, market imperfections

etc.), which limit the adjustment process.

13

Let the parameters of interest be , the parameters of the r

cointegrating vectors.

1321221123213112111131

1321221122213112111121

1321221121213112111111

10310310

1031033

103103

ttttttt

ttttttt

ttttttt

rrtbrrtbr

rrtbrrtbr

rrtbrrtbtb

(15)

Suppose, for example, that the third row of consists of zeros;

that is, 31 = 32 = 0.

Then, it is clear that neither of the two cointegrating

relationships will enter into the equation determining tr10 .

In this case, the variable tr10 is weakly exogenous for the

system as a whole, and it would be valid to model a reduced

system of two equations - one determining ttb and the other

tr3 - conditional upon tr10 .

Put another way, if tr10 is weakly exogenous in this way, no

information is lost concerning the parameters of the

cointegrating relationships by not explicitly modelling the

process determining tr10 jointly with that determining the

two "endogenous" variables, ttb and tr3 .

In this case, it is valid to condition on 10r . Instead of estimating

(8), we estimating the following partial

Version of it:

14

t

t

t

t

t

t

t

t

t

t

t

r

r

tb

r

r

tb

rr

tb

2

1

1

1

1

322212

312111

2221

1211

1

1

1

333231

232221

131211

02

01

2

1

10

3

10

3103

(16)

Notice that in this partial system, the weakly exogenous

variable remains in the long run relations although its short run

behaviour is not modelled (there is no equation for tr10 in the

VECM system).

2.6 ADVANTAGES OF CONDITIONING ON WEAKLY

EXOGENOUS VARIABLES

1. Reduction of number of short run variables in the VECM

(those that might only be relevant to the process determining x

which is not now modelled).

2. The marginal process for x may have awkward stochastic

properties, which will not need to be modelled in the conditional

model.

If individual elements of are zero, this implies the absence of

particular cointegrating relationship(s) in particular equations in

the ECM system.

This may also have implications for weak exogeneity of

variables with respect to parameters of interest. For example,

ij=0 implies that the jth cointegrating vector does not enter the

ith equation in the VAR (ie the equation system for X it in (2)).

3. JOHANSEN IN PRACTICE

15

Let’s use question 4e in Enders 2004:375 as an example. The

data INT_RATES.XLS is use and contains interest rates paid on

U.S.3-month (Tbill), 3-year (R3) and 10-year U.S.government

securities (R10) for the period 1954:7 to 2002:12.

This example outlines how the test of Johansen cointegration

can be carried out using EVIEW 5 software. We will do a

practical on Thursday (2005-03-17).

STEP 1

Pretest all variables to assess their order of integration. Plot the

variables to see if a linear time trend is likely to be present in the

DGP.

0

4

8

12

16

20

55 60 65 70 75 80 85 90 95 00

TBILL

0

4

8

12

16

20

55 60 65 70 75 80 85 90 95 00

R3

2

4

6

8

10

12

14

16

55 60 65 70 75 80 85 90 95 00

R10

16

STEP 2: Lags for VAR

The lag length can be determined by some of the many

information criteria procedures. It is important to avoid too

many lags, since the number of parameters grows very fast with

the lag length and the information criteria strike a

compromise between lag length and number of parameters by

minimising a linear combination of the residual sum of squares

and the number of parameters

If long lag length is required to make white noise residuals,

reconsider the choice of variables and look for another

important explanatory variable to include in the information set.

A summary test statistic that measures the magnitude of the

residual autocorrelation is given by Portmanteau test. There

are other tests for serial correlation, normality of errors etc. We

shall study them under the next topic on VAR. Eviews uses

Wald lag-exclusion tests to determine the appropriate lag. In

our case we use 12 lags.

STEP 3: Deterministic Trend Specification of the VAR

The variables may have nonzero means and deterministic and/or

stochastic trends. Similarly, the cointegrating equations may

have intercepts and deterministic trends.

Since the asymptotic distributions of the LR test statistic for

cointegration does not have the usual 2 distribution and

depends on the assumptions made with respect to deterministic

trends, we need to make assumptions regarding the trends

underlying our data. To understand the deterministic

components, let’s write our ECM as;

ttt txx

221

1

1~

17

Where txx tt ,1,~11 , is an orthogonal complement of

i.e. 0

1. The model does not allow for constant term which means

that all stationary linear combinations will have mean

zero.i.e. 0221 .

2. There are no trends whatsoever, but a constant term is

allowed in the cointegrating relations. 021 .

3. The model allows for linear trend in each variable but not

in the cointegrating relations. 02 .

4. The model allows for linear trends in each variable and in

the cointegrating relations

5. The level data have quadratic trends and the cointegrating

equations have linear trends.

STEP 4: Estimation and determination of rank

Estimate the model and determine the rank of matrix in.

Date: 03/14/05 Time: 11:22

Sample (adjusted): 1955M02 2002M06

Included observations: 569 after adjustments

Trend assumption: No deterministic trend (restricted constant)

Series: TBILL R3 R10

Lags interval (in first differences): 1 to 12

Unrestricted Cointegration Rank Test (Trace) Hypothesized Trace 0.05

No. of CE(s) Eigenvalue Statistic Critical Value Prob.**

None * 0.055360 50.78106 35.19275 0.0005

At most 1 0.025075 18.37547 20.26184 0.0890

At most 2 0.006876 3.925717 9.164546 0.4233 Trace test indicates 1 cointegrating eqn(s) at the 0.05 level

* denotes rejection of the hypothesis at the 0.05 level

**MacKinnon-Haug-Michelis (1999) p-values

Unrestricted Cointegration Rank Test (Maximum Eigenvalue) Hypothesized Max-Eigen 0.05

No. of CE(s) Eigenvalue Statistic Critical Value Prob.**

None * 0.055360 32.40559 22.29962 0.0014

At most 1 0.025075 14.44975 15.89210 0.0831

18

At most 2 0.006876 3.925717 9.164546 0.4233 Max-eigenvalue test indicates 1 cointegrating eqn(s) at the 0.05 level

* denotes rejection of the hypothesis at the 0.05 level

**MacKinnon-Haug-Michelis (1999) p-values

The test is done in specific order from the largest eigenvalue to

the smallest. We use “Pantula Principle” where we test for

significance until you no longer reject the null.

The first null is that there is no stationary relations in the data

(r=0). We use p-values to make a decision.

In our case both tests show that r=1.

STEP 5: Test restrictions implied by Economic theory

Unrestricted Cointegrating Coefficients (normalized by b'*S11*b=I):

TBILL R3 R10 C

2.489580 -3.990997 1.656070 0.942992

0.647520 -3.516469 2.897717 -0.684816

-0.309827 0.630744 -0.730085 2.710185

Unrestricted Adjustment Coefficients (alpha):

D(TBILL) -0.015193 0.019238 0.029119

D(R3) 0.032793 0.029437 0.017129

D(R10) 0.040614 0.010333 0.012578

We could treat the evidence that r=1 as validating the EG 2 step

procedure and stop there.

There is one cointegrating vector ( 131121111 103 ttt rrtb ),

which feed into the 3 different equations as follows;

13112111131

13112111121

13112111111

10310

1033

103

tttt

tttt

tttt

rrtbr

rrtbr

rrtbtb

(18)

19

Notice that if the loading matrices 03121 , we end up with

the conventional EG 2 step model. However, even if there is one

cointegrating vector, the EG 2 step procedure does not test this

restriction.

Suppose we want to normalize 131121111 103 ttt rrtb on the

basis of Tbill rate, we make the coefficient of 1ttb be –1 by

dividing through 11 as follows;

0103 1

11

31

1

11

211

11

11

ttt rrtb

which generates

1

11

31

1

11

211 103 ttt rrtb

(19)

We can test the hypothesis 111 in Eviews by typing the

commands;

B(1,1)=-1 which generates the following results;

Restrictions: B(1,1)=-1

Tests of cointegration restrictions: Hypothesized Restricted LR Degrees of

No. of CE(s) Log-likehood Statistic Freedom Probability

1 396.4883 NA NA NA

2 403.7132 NA NA NA NA indicates restriction not binding.

1 Cointegrating Equation(s): Convergence achieved after 1 iterations. Restricted cointegrating coefficients (standard error in parentheses)

TBILL R3 R10 C

-1.000000 1.603080 -0.665200 -0.378776

(0.00000) (0.17503) (0.17763) (0.21407)

Adjustment coefficients (standard error in parentheses)

D(TBILL) 0.037825

(0.04068)

20

D(R3) -0.081641

(0.03337)

D(R10) -0.101112

(0.02541)

We can put this an ECM as;

Crrtbr

Crrtbr

Crrtbtb

tttt

tttt

tttt

378.010665.03603.1101.010

378.010665.03603.1082.03

378.010665.03603.1037.0

111

111

111

(20)

Notice that you can normalise on any other variable depending

on the economic theory we would like to test.

STEP 6: Exogeneity tests

We can conduct exogeneity tests. We can visualise it by looking

at; 13112111131

13112111121

13112111111

10310

1033

103

tttt

tttt

tttt

rrtbr

rrtbr

rrtbtb

(21)

Suppose we want to test the hypothesis that tbill is exogenous.

This is equivalent to claiming that 011 . We can impose this

restriction in Eviews as follows A(1,1)=0. The results is;

Restrictions: A(1,1)=0

Tests of cointegration restrictions: Hypothesized Restricted LR Degrees of

No. of CE(s) Log-likehood Statistic Freedom Probability

1 396.1047 0.767256 1 0.381067

2 403.7132 NA NA NA NA indicates restriction not binding.

21

1 Cointegrating Equation(s): Convergence achieved after 9 iterations. Restricted cointegrating coefficients (not all coefficients are identified)

TBILL R3 R10 C

2.507323 -4.150214 1.777850 1.044228

Adjustment coefficients (standard error in parentheses)

D(TBILL) 0.000000

(0.00000)

D(R3) 0.042795

(0.00894)

D(R10) 0.046106

(0.00836)

This hypothesis cannot be rejected in this case implying that

tbill is exogenous and plays no role in the adjustment

towards the long run.

However, it is still in the long-run equation as shown below; 13112111131

13112111121

131121111

10310

1033

01030

tttt

tttt

tttt

rrtbr

rrtbr

rrtbtb

(22)

In other words there are only two short-run equations.

What can go wrong in Johansen methodology

We need normally distributed white noise residuals.

The test is asymptotic and can be sensitive on how we

formulate the VAR model in limited samples.

The test assumes that there are no structural breaks in the

data. Dummies may be important if structural breaks are

suspected.

If we put a stationary variable in the model, the number of

cointegrating vectors increase. The judgement of the

modeller is therefore crucial

OTHER TOPICS ON JOHANSEN TEST

Testing for I(2) relations conditional on the assumption

that the model is correctly specified I(1) system. (Quite a

tricky issue a beginner)

22

Use Johansen method to test if individual variables are I(0)

or I(1). Unlike the ADF, the null in this case is I(0) like

the KPSS test. This is test 5H in Johansen and Juselius

(1992:226).