lecture notes on johansen cointegration
TRANSCRIPT
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:
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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
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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.
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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.
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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
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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)