Financial Econometrics
Topic 5: Cointegration
Dr. Aidil Rizal Shahrin
May 1, 2021
Department of Banking & Finance
Faculty of Business and Accountancy
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Contents
1. Present Value Model
1.1 Econometric Model
2. Cointegration
2.1 Two-step Engle-Granger
3. Cointegration Tests for Structural Change
3.1 Gregory-Hansen Tests
4. Error Correction Models (ECM )
4.1 The Long-Run Compositions
4.2 The Short-Run Compositions
5. Vector ECMs
5.1 Granger Representation Theorem
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Contents
5.2 Relationship between V AR and V ECM
5.3 Estimating the V ECM
5.4 Johansen Tests for Cointegration
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Present Value Model
Present Value Model
i. Let Pt be the price of a share measured at the end of the
period (ex-dividend price), i.e., the purchase of the stock this
period entitles you to a claim to next period’s dividend per
share Dt+1
ii. The net simple stock return (end-of-period observations) of
the stock held from time t to time t + 1 is:
Rt+1 =
Pt+1 + Dt+1
−1
Pt
(1)
iii. Assume that expected returns is a constant:
Et [Rt+1 ] = r
(2)
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Present Value Model
iv. Rearrange Eq.1 to get model of stock price:
Pt = E t
Pt+1 + Dt+1
1+r
(3)
v. Using the law of iterated expectation, i.e.,
Et [Et+1 [X]] = Et [X]
(4)
we can solve Eq.3 by repeatedly substituting out future prices.
vi. Solving K periods forward we have:
"K #
"
#
i
K
X
1
1
Pt = E t
Dt+i + Et
Pt+K
1+r
1+r
i=1
(5)
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Present Value Model
vii. The first term on the RHS of Eq.5 is the expected DPV of
dividends while the second term is the expected discounted
value of the stock price K periods from now
viii. Assuming that the second term goes to zero as K increases
(i.e., the transversality condition)
"
lim Et
K→∞
1
1+r
K
#
(6)
Pt+K = 0
we obtain a model which defines the stock price as the
expected present value of future dividends out to infinity,
discounted at a constant rate:
"
Pt = PDt = Et
i
∞ X
1
Dt+i
1+r
i=1
#
(7)
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Present Value Model
ix. If dividends are assumed to grow at a constant rate g
Et [Dt+i ] = (1 + g)Et [Dt+i−1 ]
= (1 + g)i Dt
(8)
x. Hence the stock price model with constant discount rate r and
dividend growth rate g , where g < r (for finite stock price) is
Pt =
Et [Dt+1 ]
(1 + g)Dt
=
r−g
r−g
(9)
and it is also known as the Gordon growth model
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Present Value Model
Econometric Model
Econometric Model
i. To test a model of the type described by Eq.8 and Eq.9,
consider a system of equations which includes
Pt = β1 Dt + ut
(10)
a linear equation between price and dividends
Dt = β2 Dt−1 + wt
(11)
a data-generating AR(1) process for dividends and where the
errors are white noise processes.
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Econometric Model
ii. Tests of the time-series properties of Dt and Pt (see Topic 2)
show that they are nonstationary variables, so Eq.10 should be
tested and estimated using the econometric techniques of
cointegration
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Cointegration
Cointegration
i. Cointegration captures the idea of a long-run dynamic
economic model such that
a. Each variable is nonstationary
b. But, they do not drift too far apart from each other
ii. Reasons for understanding cointegration
a. There is strong evidence that financial time series are
nonstationary
b. For noncointegrated systems, it can be shown that t-tests and
F -tests are inflated in size upwards. This means that the null
hypothesis that the parameter is zero will be rejected more often
even when the null hypothesis is true (Type I error). For this
reason, a noncointegrated system is referred to as a spurious
regression
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Cointegration
c. A cointegrated system can be decomposed into a long-run
(fundamental) part and a short-run (dynamic) part.
iii. Variables (Yt , Xt ) are said to be cointegrated if
a. Yt ∼ I(1) and Xt ∼ I(1) and
b. The residuals ut from the model
Yt = βXt + ut
are stationary, i.e., ut ∼ I(0)
c. The cointegrating vector is (1 − β).
d. More generally, if Yt and Xt are I(d) and Yt − βXt is I(d − b)
then Yt and Xt are cointegrated and d > b > 0.
e. The cointegrating equation can be decomposed into the
• Long-run: βXt
• Short-run: ut
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Cointegration
Two-step Engle-Granger
Two-step Engle-Granger
i. According to the present value model above, the relationship
between share price (Pt ) and dividends (Dt ) is:
Pt = βDt + ut
where ut is a disturbance term
ii. We have established that the two series are nonstationary
variables, i.e., they are I(1) variables.
iii. However, if theory holds, then the two series are not expected
to drift too far apart. That is, the error term ut is I(0)
iv. If this holds, then Pt and Dt are said to be cointegrated
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Two-step Engle-Granger
v. A natural way to test this is to apply the 2-step approach of
Engle-Granger (1987)
• Step 1: Perform an OLS regression. Note that although β̂ is
superconsistent, the standard errors (and hence t-stats) are not
correct except when residuals are white noise
• Step 2: Test stationarity property of the regression residuals
(always regress ∆ût on ût−1 (+ lags of ∆û) without the
constant term). The critical values for the Dickey-Fuller
t-statistics when applied to residuals from a spurious regression
is attached. (Note: Case 1 is when the Cointegrating equation
(CE) contains NO constant term, case 2 when the CE includes
a constant term, and Case 3 when the CE includes a constant
and time-trend).
If residuals are I(1) − (Pt , Dt ) are said to be spuriously related
If residuals are I(0) − (Pt , Dt ) are said to be cointegrated.
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Cointegration Tests for Structural
Change
Cointegration Tests for Structural
Change
Gregory-Hansen Tests
Gregory-Hansen Tests
i. The cointegrating relationship between two variables {Y, X}
with no structural change is typically
Yt = µ + βXt + εt
The standard residual-based test for cointegration is an ADF
on the OLS residuals
ii. Cointegration tests can also be conducted in the presence of
deterministic structural breaks
iii. This is achieved by augmenting the standard cointegrating
equation between Y and X with dummy variables
iv. Both intercept and slope dummies can be included
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Gregory-Hansen Tests
v. The Gregory-Hansen (GH ) cointegration tests are also ADF
tests on the residuals, but they are associated with
cointegrating models with structural break
vi. They are designed to test the null of no cointegration against
the alternative of cointegration in the presence of a structural
change
H0 : no cointegration
H1 : cointegration with a one-time regime shift
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Gregory-Hansen Tests
vii. There are three tests. For convenience define a dummy variable
Dt = 0, if t ≤ λT
= 1, otherwise
• GH(1): the cointegrating equation allows for a mean-shift:
Yt = µ0 + µ1 Dt + β1 Xt + ε1t
• GH(2): the cointegrating equation allows for a mean-shift with
trend:
Yt = µ0 + µ2 Dt + β1 Xt + δt + ε2t
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Gregory-Hansen Tests
• GH(3): the cointegrating equation allows for a regime shift,
i.e., shift in mean and slope coefficients:
Yt = µ0 + µ3 Dt + β1 Xt + β2 Dt Xt + ε3t
viii. The approach is to compute the usual t-statistics associated
with an ADF regression on the residuals obtained from
estimating the augmented cointegrating equation over all
possible breakpoints and choosing the smallest value
ix. As with the unit root case, new tables of critical values are
needed for conducting cointegration tests. These are given in
Gregory A.W. and Hansen, B. E. (1996)
x. The critical values are given in the Table below. (m is the
regressors excluding constant and/or trend terms)
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Gregory-Hansen Tests
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Error Correction Models (ECM )
Error Correction Models (ECM )
i. The cointegrating relationship between the variables Yt and Xt
is a long-run relationship
ii. However, unless adjustment is instantaneous, Yt and Xt are
more likely to be related dynamically
iii. A general dynamic economic model (where ut is a disturbance
term) is as follows:
Yt = ρYt−1 + φ0 Xt + φ1 Xt−1 + ut
(12)
iv. Eq.12 can be rearranged into an error correction representation
(ECM hereafter) and shown to contain the long run
cointegrated relationship between Yt and Xt as well as other
short-run components.
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Error Correction Models (ECM )
v. To show the relationship between an ECM and cointegration,
subtract Yt−1 from both sides of Eq.12
Yt − Yt−1 = (ρ − 1)Yt−1 + φ0 Xt + φ1 Xt−1 + ut
Add and subtract φ0 Xt−1 on the right hand side
Yt − Yt−1 = (ρ − 1)Yt−1 + φ0 (Xt − Xt−1 ) + (φ0 + φ1 )Xt−1 + ut
Rearrange the RHS
∆Yt = α(Yt−1 − βXt−1 ) + φ0 (∆Xt ) + ut
(13)
where
β=
φ0 + φ1
1−ρ
α = (ρ − 1)
(14)
(15)
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Error Correction Models (ECM )
vi. Eq.13 is a single-equation error-correction model with long run
parameter β and error-correction term α
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Error Correction Models (ECM )
The Long-Run Compositions
The Long-Run Compositions
i. The long-run is given by setting
a. Yt = Yt−1 = Y
b. Xt = Xt−1 = X
c. ut = 0
ii. Substituting these restrictions into the dynamic model Eq.12
and rearranging gives the long-run relationship between Yt and
Xt as
Y =
φ0 + φ1
X
1−ρ
iii. Notice that the coefficient on X , namely the long-run
multiplier, is equal to β in Eq.13
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Error Correction Models (ECM )
The Short-Run Compositions
The Short-Run Compositions
i. The short-run movements in Yt are represented by Yt − Yt−1
ii. These movements can be decomposed into two types of
adjustments
a. φ0 (Xt − Xt−1 )
which occurs because of (short-run) movements in Xt
b. α(Yt−1 − βXt−1 )
which occurs when the variables are not in equilibrium
iii. This decomposition highlights the name given to the model
(ECM )
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The Short-Run Compositions
iv. In particular, the second type of adjustment is known as the
error correction term since if Yt drifts above its long-run value,
thus making Yt−1 − βXt−1 positive, then providing that α is
negative and 0 < |α| < 1, the overall effect is to slow down, or
even decrease, Yt
v. This forces Yt back, that is error correct, to its long-run
position
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Vector ECMs
Vector ECMs
i. The analysis above is based on a single equation representation
of the ECM
ii. But, that is only appropriate if dividends are (strongly)
exogenous (uni-directional causality)
iii. In general, for modelling economic systems which exhibit
long-run relationships it is more appropriate to estimate a
multivariate ECM (that is assume feedback causality)
iv. To move from the univariate to the multivariate case, we need
to understand
• the Granger Representation Theorem
• the relationship between a VAR with nonstationary variables
and its VECM representation
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Vector ECMs
• Johansen’s test for the number of cointegrating relationship(s)
• Estimating a VECM
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Vector ECMs
Granger Representation Theorem
Granger Representation Theorem
i. Suppose that Yt and Xt are I(1) and cointegrated with vector
(1, −β)
ut = Yt − βXt
ii. The Granger representation theorem states that Yt and Xt can
be expressed as an error-correction model (ECM ) such that
each equation contains the “common” long-run component
(Yt−1 − βXt−1 ) plus other short-run terms
∆Yt = α1 (Yt−1 − βXt−1 ) + lags(∆Yt , ∆Xt ) + ε1,t
∆Xt = α2 (Yt−1 − βXt−1 ) + lags(∆Yt , ∆Xt ) + ε2,t
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Granger Representation Theorem
• Constants, time trends, dummies can be added as required
• Cointegration requires that at least one of the α’s be non-zero.
If α1 = α2 = 0 then the system is not cointegrated
• Signs of the α’s are important to capture adjustments in the
right direction (i.e., mean-revert)
• Size gives the speed of adjustment; α close to zero (one)
implies slow (fast) adjustment
iii. Consider again the econometric specification of the P V model:
Pt = β1 Dt + ut
(16)
Dt = β2 Dt−1 + ωt
(17)
This set of equations implies that:
• D has a data-generating process (DGP ) exogenous to P
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Granger Representation Theorem
• D cause P , and
• P and D are cointegrated with cointegrating vector (1, −β1 )
iv. To write Eq.16 and Eq.17 in the ECM form (ignoring other
lags) but assuming bilateral causality, manipulate equations as
follows:
∆Pt = −Pt1 + β1 Dt−1 + β1 (Dt − Dt−1 ) + ut
∆Dt = β2 Dt−1 − Dt−1 + ωt
Now rearrange terms to factor-out the common cointegration
component
∆Pt = α1 (Pt−1 − βDt−1 ) + δ1 ∆Dt + ε1t
∆Dt = α2 (Pt−1 − βDt−1 ) + δ1 ∆Dt + ε2t
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Granger Representation Theorem
v. Note that the significance of the error-correction terms (α1 , α2 )
is informative about causality/(weak) exogeneity.
vi. For our case study, if Eq.16 and Eq.17 are “true”
representations of behaviour in the stock market, (that is D is
weakly exogenous) we expect to find α1 to be negative and
significant and to find α2 to be not significantly different from
zero
vii. If on the other hand, P and D are both endogenously
determined, we expect to find α1 to be negative and significant
and to find α2 to be positive and significant.
viii. Both α’s must be between 0 and 1 (in absolute terms)
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Vector ECMs
Relationship between V AR and V ECM
Relationship between V AR and V ECM
i. To derive the multivariate ECM , it is convenient to start with
a V AR
y t = A1 y t−1 + · · · + Ap y t−p + εt
•
•
•
•
y t is a vector of endogenous variables
p is the lag length
A1 , . . . , Ap are matrices of coefficients to be estimated
εt is a vector of disturbance terms that are contemporaneously
correlated, but not
♦ serially correlated
♦ or correlated with y t−1 , . . . , y t−p , xt
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Relationship between V AR and V ECM
ii. The V AR can be written as a vector error correction model
y t − y t−1 = Πy t−1 + C 1 (y t−1 − y t−2 ) + · · · + C p−1 (y t−p+1 − y t−p )
+ εt
∆y t = Πy t−1 +
p−1
X
C i ∆y t−i + εt
i=1
• C 1 , . . . , C p−1 are matrices which are function of the Ai
matrices of the V AR
• Π is a matrix which contains the
♦ cointegrating vectors
♦ error correction coefficients
• Note that the VECM has one less lag in the ∆y t ’s than there
were in the y t ’s from the original VAR.
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Relationship between V AR and V ECM
iii. Consider a bivariate {Y1 , Y2 } set of nonstationary variables.
Assume that the true model contains data-generating
processes as follows
Y1t = 0.4Y1t−1 + 0.3Y2t−1 + 0.24Y2t−2 + ε1t
Y2t = Y2t−1 + ε2t
iv. The V AR form (note here with non-stationary variables) is:
" #
Y1t
Y2t
"
=
0.4
0
#"
#
0.3 Y1t−1
1
Y2t−1
+
"
0
0.24
0
0
#"
#
Y1t−2
Y2t−2
+
" #
ε1t
ε2t
(18)
Note also that this is a VAR with lag-order 2
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Relationship between V AR and V ECM
v. The V AR of nonstationary variables can be transformed into a
V ECM as follows (Note that we are assuming cointegration
between {Y1 , Y2 }
• First subtract the first-order lag from both sides
∆Y1t = (0.4 − 1)Y1t−1 + 0.3Y2t−1 + 0.24Y2t−2 + ε1t
∆Y2t = (1 − 1)Y2t−1 + ε2t
• Add and subtract 0.24Y2t−1 for the first equation
∆Y1t = −0.6Y1t−1 + (0.3 + 0.24)Y2t−1 − 0.24(Y2t−1 − Y2t−2 ) + ε1t
∆Y2t = ε2t
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Relationship between V AR and V ECM
• Rearrange terms in ECM form
∆Y1t = −0.6(Y1t−1 − 0.9Y2t−1 ) − 0.24∆Y2t−1 + ε1t
∆Y2t = ε2t
• Write in matrix form as:
#"
#
"
# "
∆Y1t
−0.6 −0.6 × −0.9 Y1t−1
=
∆Y2t
0
0
Y2t−1
"
#"
# " #
0 −0.24 ∆Y1t−1
ε1t
+
+
0
0
∆Y2t−1
ε2t
(19)
Notice that the VAR in Eq.18 is lag 2 while VECM in Eq.19 is
lag of 1 in ∆
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Relationship between V AR and V ECM
• Factorise out the “common” long-run relationship:
#
#
"
"
# "
i Y
∆Y1t
−0.6 h
1t−1
=
1 −0.9
Y2t−1
∆Y2t
0
# " #
#"
"
ε1t
0 −0.24 ∆Y1t−1
+
+
ε2t
∆Y2t−1
0
0
h
i
• Note that this is V ECM with long-run vector 1 −0.9 and
"
#
−0.6
error-correction vector
. Note too that the V AR of
0
"
#
∆Y1t
stationary variables
is now of order ONE.
∆Y2t
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Relationship between V AR and V ECM
vi. EViews allows for five different specifications of the vector
error correction model depending upon whether or not
intercepts and or time trends are included in the
• Cointegrating equation (CE)
• ECM , or what EViews calls the test vector autoregression
(V AR)
• In Eviews, in the lag interval, if you type ”1 2” in the edit field,
the test VAR regresses ∆yt on ∆yt−1 , ∆yt−2
• In Gretl, in the cointegration test of Johansen, the lag order p
of the V AR(p). Thus, if the lag in VECM is 3, in the lag order
window set it to 4.
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Relationship between V AR and V ECM
a. Case 1
• No intercept and no trend in CE
• No intercept and no trend in test VAR
Y1,t − Y1,t−1 = α1 (Y1,t−1 − βY2,t−1 ) + ε1,t
Y2,t − Y2,t−1 = α2 (Y1,t−1 − βY2,t−1 ) + ε2,t
In this example, the number of lags in the V AR is N = 0, and
!
α1 Π=
1, −β
α2
b. Case 2
• Intercept and no trend in CE
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Relationship between V AR and V ECM
• No intercept and no trend in test V AR
Y1,t − Y1,t−1 = α1 (Y1,t−1 − βY2,t−1 − µ) + ε1,t
Y2,t − Y2,t−1 = α2 (Y1,t−1 − βY2,t−1 − µ) + ε2,t
c. Case 3
• Intercept and no trend in CE
• Intercept and no trend in test V AR
Y1,t − Y1,t−1 = δ1 + α1 (Y1,t−1 − βY2,t−1 − µ) + ε1,t
Y2,t − Y2,t−1 = δ2 + α2 (Y1,t−1 − βY2,t−1 − µ) + ε2,t
d. Case 4
• Intercept and trend in CE
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Relationship between V AR and V ECM
• Intercept and no trend in test V AR
Y1,t − Y1,t−1 = δ1 + α1 (Y1,t−1 − βY2,t−1 − µ − φt) + ε1,t
Y2,t − Y2,t−1 = δ2 + α2 (Y1,t−1 − βY2,t−1 − µ − φt) + ε2,t
e. Case 5
• Intercept and trend in CE
• Intercept and trend in test V AR
Y1,t − Y1,t−1 = δ1 + θ1 t + α1 (Y1,t−1 − βY2,t−1 − µ − φt) + ε1,t
Y2,t − Y2,t−1 = δ2 + θ2 t + α2 (Y1,t−1 − βY2,t−1 − µ − φt) + ε2,t
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Vector ECMs
Estimating the V ECM
Estimating the V ECM
i. The above vector error correction models can be easily
estimated as a system of equations with a common long-run
relationship
ii. If the residuals are “well-behaved”, the procedure also
generates t-statistics which are distributed asymptotically as
N (0, 1)
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Vector ECMs
Johansen Tests for Cointegration
Johansen Tests for Cointegration
i. As seen above there is a close association between an ECM
and V AR with nonstationary time-series
ii. This has been exploited by Johansen to test for the number of
cointegrating relationships for systems with more than 2
variables
iii. To illustrate, consider a bivariate study of (Yt , Xt ) both I(1)
variables and cointegrated with vector (1, −β). The bivariate
ECM is:
∆Yt = α1 (Yt−1 − βXt−1 ) + ε1,t
∆Xt = α2 (Yt−1 − βXt−1 ) + ε2,t
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Johansen Tests for Cointegration
which can be expressed as
yt =
Yt
Xt
!
∆y t = Πy t−1 + εt
!
ε1t
α1
, εt
, Π=
ε2t
α2
−α1 β
!
−α2 β
iv. The system can also be rearranged as:
Yt = (1 + α1 )Yt−1 − α1 βXt−1 + ε1,t
Xt = α2 Yt−1 + (1 − α2 β)Xt−1 + ε2,t
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Johansen Tests for Cointegration
v. As a bivariate V AR:
y t = Ay t−1 + εt
A=
!
(1 + α1 )
−α1 β
α2
(1 − α2 β)
Thus we see that an ECM can be viewed as a V AR with
restrictions
Π = A − I2
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Johansen Tests for Cointegration
vi. In general, V AR(p) can be transformed into a ECM with
(p − 1) lags of ∆y t−j
y=
p
X
Ai y t−i + ε
j=1
∆y t = Πy t−1 +
p−1
X
C i ∆y t−i + εt
i=1
Notice the lag of is V AR(p) while V ECM (p − 1)
vii. Notice that Π has reduced rank. It can be shown that it is the
reduced rank property that generates the nonstationary and
cointegration. In general for k variables, the rank of Π
determines how many cointegrating vectors there are:
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Johansen Tests for Cointegration
• rank(Π) = 0, then y t ∼ I(1) but not cointegrated
• rank(Π) = r, 0 < r < k, then y t ∼ I(1) with r linearly
independent cointegrating vectors
• rank(Π) = k, then y ∼ I(0)
viii. If Π has reduced rank, then Π is the product of error
correction terms and the cointegrating vector
Π = αβ 0
ix. There exist a number of cointegration tests. The cointegration
test presented in EViews is known as the Johansen likelihood
ratio cointegration test
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Johansen Tests for Cointegration
x. Large values of the Likelihood ratio relative to the critical
value result in rejection of the null hypothesis. The
cointegration test proceeds sequentially.
• Stage 1
H0 : No cointegrating vectors
H1 : At least one cointegating vectors
If the null is rejected proceed to the next stage, else stop and
do not reject the null hypothesis
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Johansen Tests for Cointegration
• Stage 2
H0 : One cointegrating vectors
H1 : At least two cointegating vectors
If the null is rejected proceed to the next stage, else stop and
do not reject the null hypothesis of one cointegrating vector. If
the number of cointegrating vectors chosen equals the number
of variables, then the variables are all I(0).
xi. Although the Johansen test is the most popular test reported
in the literature, many Monte Carlo tests have shown that the
Johansen test to be very sensitive to mispecification of
lag-length and sample sizes.
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Johansen Tests for Cointegration
xii. The Engle-Granger single equation test is more robust to
misspecification. In practice, one should report a number of
tests.
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