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Time Series Analysis for Macroeconomics and Finance
Bernd SüssmuthIEW � Institute for Empirical Research in Economics
University of Leipzig
January 9, 2012
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 1 / 21
VECM and the Johansen Approach Cointegration in multiple equations
Contents I
1 VECM and the Johansen ApproachCointegration in multiple equationsAdvantages of the VECM approachReduced rank of LR matrix ΠThe Johansen procedure in practice
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 2 / 21
VECM and the Johansen Approach Cointegration in multiple equations
Now, � 1 cointegrating (CI ) vector is possible
Several equilibrium relationships are possible
EG single equation model 6= adequate ) Johansen (1988)
Let us assume 3 variables (which all can be endogenous)
Zt = [Yt ,Xt ,Wt ]) Zt = A1Zt�1 +A2Zt�2 + ...+AkZt�k + ut
Analogously, we can reformulate this in a VECM:
∆Zt = Γ1∆Zt�1 + Γ2∆Zt�2 + ...+ Γk�1∆Zt�k�1 +ΠZt�1 + ut ,
where Γi = (I�A1�A2 � ...�Ak ) for all i = 1, ..., k � 1
Matrix Π is of dimension 3� 3 and contains all information on LR
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 3 / 21
VECM and the Johansen Approach Cointegration in multiple equations
We can decompose
Π = ab0, where a = adjustment speed; b = LR coe¢ cients matrix
b0Zt�1 is equivalent to EC term in EG approach
(Yt�1 � β1 � β2Xt�1), except that it contains now (n� 1) vectors
For simplicity assume k = 2, i.e. only two �rst lag terms in the model:0@ ∆Yt∆Xt∆Wt
1A = Γ1
0@ ∆Yt�1∆Xt�1∆Wt�1
1A+Π
0@ Yt�1Xt�1Wt�1
1A+ et= ...+
0@ a11 a12a21 a22a31 a23
1A� b11 b21 b31b12 b22 b32
�0@ Yt�1Xt�1Wt�1
1A+ et
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 4 / 21
VECM and the Johansen Approach Advantages of the VECM approach
Contents I
1 VECM and the Johansen ApproachCointegration in multiple equationsAdvantages of the VECM approachReduced rank of LR matrix ΠThe Johansen procedure in practice
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 5 / 21
VECM and the Johansen Approach Advantages of the VECM approach
White board:
cointegrating vectors of ∆Yt and speed of adjustm terms a11, a12
Advantages of VECMs
(a) Even if there is only 1 CI-relationship, the VECM can calculate the 3di¤erent speeds of adjustm coe¢ cients
�a11 a21 a31
�0(b) Only when a21 = a31 = 0 and only one CI-relationship exists, the
model boils down to the EG model: There is no loss from notmodelling determinants of ∆Xt ,∆Wt , i.e. Xt and Wt would beweakly exogenous for Yt .
) However, (some) rhs variables are often not weakly exogenous
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 6 / 21
VECM and the Johansen Approach Reduced rank of LR matrix Π
Contents I
1 VECM and the Johansen ApproachCointegration in multiple equationsAdvantages of the VECM approachReduced rank of LR matrix ΠThe Johansen procedure in practice
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 7 / 21
VECM and the Johansen Approach Reduced rank of LR matrix Π
Zt � I (1)) ∆Zt�1 � I (0)) ΠZt�1!� I (0) for ut � I (0)
Three cases for ΠZt�1 � I (0)Case 1 All variables in Zt � I (0)) uninteresting from VECM
perspective, simple VAR in levels would do;
! rank of Π = an�r
�bn�r
�0: full
Case 2 No CI whatsoever ) Π = 0n�n, VAR in 1st di¤�s w/o LR
elements (as there are none) would do;
! rank of Π: zeroCase 3 Up to (n� 1) CI-relationships of the form b0Zt�1 � I (0),
r � (n� 1) cointegrating vectors 2 Π, i.e. r cols of b form rlin�ly independent stationary combinations of variables 2 Zt! rank of Π: reduced
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 8 / 21
VECM and the Johansen Approach The Johansen procedure in practice
Contents I
1 VECM and the Johansen ApproachCointegration in multiple equationsAdvantages of the VECM approachReduced rank of LR matrix ΠThe Johansen procedure in practice
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 9 / 21
VECM and the Johansen Approach The Johansen procedure in practice
In a nutshell I: Six steps of the Johansen procedure
Step 1 Test the order of integration of variables
Note: any I (0) variable included raises no. of cointegrationrel�ships (as it is a lin�ly independent col in Π); variables indi¤ering order of integration always complicate matters!
Step 2 Choose the appropriate lag length (using ICs)
Note: Usual way: auxiliary VAR 3 all variables (levels)
Step 3 Choose deterministic components of the system
Note: Concerns intercepts and DTs
Step 4 JCT a): Determine the no. of cointegrating vectors
Step 5 JCT b): Test for weak exogeneity (zero-cols of a)Step 6 JCT c): Test other linear restrictions on CI-vectors
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 10 / 21
VECM and the Johansen Approach The Johansen procedure in practice
Note, Π = � (I�A1�A2 � ...�Ak ) or equiv�ly Π = ∑ki=1 Ai � I
Theorem (Granger�s Representation Theorem)If coe¢ cient matrix Π has reduced rank r < n � no. of non-stationaryvariables considered, then there exist n� r matrices a and b each withrank r such that Π = ab0 and b0Zt � I (0): r is the no. of cointegratingrelations (cointegrating rank), each col of b is a cointegrating vector.
Johansen makes use of the GRT:
Basically, Johansen�s methods consist in estimating matrix Π from anunrestricted VAR and to test whether we can reject restrictions implied byreduced rank forms of Π.
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 11 / 21
VECM and the Johansen Approach The Johansen procedure in practice
In a nutshell II: Six steps of the Johansen procedure
Step 1 and 2 should be clear and make us no di¢ culties
Step 3 Choose deterministic components of the system.
) Consider a most general case (w/ regard to these comp�s) of a VECM
∆Zt = Γ1∆Zt�1 + Γ2∆Zt�2 + ...+ Γk�1∆Zt�k�1
+a
0@ bm1d1
1A� Zt�1 1 t�
+m2 + d2t + ut
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 12 / 21
VECM and the Johansen Approach The Johansen procedure in practice
From this, we can consider four central versions:
V1 No intercept/trend in CE- or VAR-part(d1 = d2 = m1 = m2 = 0) ! improbable case: intercept isgenerally needed to account for adjustments in measurementunits of considered variables
V2 Intercept & no trend in CE-part, neither/nor in VAR-part(d1 = d2 = m2 = 0) ! no linear trends in data case:1st di¤�s have zero mean
V3 Intercept in CE- and VAR-part, no trends (d1 = d2 = 0)! no linear trends in levels of data case: drifting around anintercept is allowed
V4 Intercept in CE- and VAR-part, linear trend in CE- orVAR-part ! linear trend in CE � exogenous growth; lineartrend in SR , quadratic trend in LR
) Our focus is on Model 2 and 3
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 13 / 21
VECM and the Johansen Approach The Johansen procedure in practice
The Johansen (1992) suggestion: Follow Pantula Principle
Decisive is joint hypothesis: CI -rank and model version V1-V4
Pantula principle:
� start with most restrictive model: r = No. of CI-vectors = 0; V1
�move to least restrictive model: r = n� 1, V4! compare trace-test statistics with critical value
! stop when for 1st time H0: no cointegration 6= rejected
Note: trace (tr-) test is part of Step 4 (JCT a)
A tr-test is one way to determine the order of Π(the no. of CI-vectors)
Alternative: maximum eigenvalue statistics
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 14 / 21
VECM and the Johansen Approach The Johansen procedure in practice
RECALL NOTE
Di¤erence equations of higher order and discrete dynamical systems
1 Introduce adiabatic variables (dummies) except for highest lag
2 yt + ayt�1 + byt�2 + cyt�3 = 0
de�ne: xt = yt�1; zt = xt�1 (= yt�2)
) 3 equ�s: yt = �ayt�1 � bxt�1 � czt�1; xt = yt�1; zt = xt�1)
wt= Awt�1, where
wt =
0@ ytxtzt
1A , A =0@ �a �b �c1 0 00 1 0
1A , wt�1 =0@ yt�1xt�1zt�1
1A
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 15 / 21
VECM and the Johansen Approach The Johansen procedure in practice
) Intelligent guess ansatz (power form):wt3�1= x3�1
� λt1�1
(univariate: Aqt)
) wt= Awt�1 , xλt = Axλt�1�� � 1
λt�1
) xλ = Ax
De�nition (eigenvalues)
λ � eigenvalue of matrix A, ensuring for x 6= 0: xλ = Ax to hold.
De�nition (eigenvectors)
x � eigenvector of matrix A, ensuring for λ 6= 0: xλ = Ax to hold.
) Eigenvalues are obtained solving the reformulated DDS: (A�λI) x = 0
) For x 6= 0 this is only the case for det (A�λI) = 0
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 16 / 21
VECM and the Johansen Approach The Johansen procedure in practice
) For low order DDS the usual procedure (char.pol., Viéta, etc.) follows
) For higher order DDS we let some software calculate the eigenvalues
Note:
1 Eigenvalues are nothing else but characteristic roots2 To check whether DDS is able to produce cycles it su¢ ces to check λ
3 This is only the case for at least one conjugate complex pair for λ
4 Important here: stable solution (CI-relationship) for 0 < λi < 1
If you want to do some DDS exercise, among many others, a recommen-dable text is the one by Chiang/Wainwright (2005): Fundamental Methods
End of RECALL NOTE
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 17 / 21
VECM and the Johansen Approach The Johansen procedure in practice
Step 4 Determine the number of cointegrating vectors
Maximum eigenvalue test (Johansen 1989)
Let rank(Π) � rank of Π
Test idea: order largest eigenvalues (descending) and
consider whether they are di¤erent from zero λ1 > λ2 > ... > λn?> 0
for no cointegration:r = 0,�1� bλi� = 1, ln
�1� bλi� = 0
for r = 1, 0 < λ1 < 1, ln�1� bλ1� < 0 (rest = 0) etc.
Test hypothesis: H0: rank(Π) = r; H1: rank(Π) = r + 1
Test statistics: λmax (r , r + 1) = �T ln�1� bλr+1�
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 18 / 21
VECM and the Johansen Approach The Johansen procedure in practice
Step 4 Determine the number of cointegrating vectors (cont�ed)
Trace test (Johansen/Juselius 1990)
Let tr(Π) � trace of Π
Test idea: Is tr increased by adding more λi beyond the rth?
for all λi = 0, tr-statistics = 0
) LR-test of increasingly (in r) restricted model
) Critical values: Johansen/Juselius (1990)
Test hypothesis: H0: tr � r; H1: tr > r (work incr�ly downwards!)
Test statistics: λtr (r) = �T ∑ni=r+1 ln
�1� bλr+1�
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 19 / 21
VECM and the Johansen Approach The Johansen procedure in practice
Step 5 Test for weak exogeneity: Can we narrow the space ofhypothesized relationships?
0@ ∆Yt∆Xt∆Wt
1A = Γ1
0@ ∆Yt�1∆Xt�1∆Wt�1
1A+Π
0@ Yt�1Xt�1Wt�1
1A+ et= ...+
0@ a11 a12a21 a22a31 a23
1A� b11 b21 b31b12 b22 b32
�0@ Yt�1Xt�1Wt�1
1A+ etWhich of the rows of a are equal to zero?
If so (say for Y ), parameters can be independent of those generating W ,Z
Test = joint test on row of a�s!= 0; if so drop Y from lhs, but leave it rhs
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 20 / 21
VECM and the Johansen Approach The Johansen procedure in practice
Step 6 Test for linear restrictions in CI-vectors
Element of interest: LR-coe¢ cients matrix b
Various testable predictions from theory:
LR-proportionalities (e.g. money and prices)
LR-elasticity relationships (e.g. Marshall-Lerner)
LR-(non-)neutralitiy of shocks (e.g. news, technology)
A concrete macroeconomic labor market example: Changes in...
� labor productivity: driven by technology shocks only
� real wages: driven by technology and mark-up shocks
�unemp-rate: driven by technology, mark-up, demand shocks
Bernd Süssmuth (University of Leipzig) Time Series Analysis January 9, 2012 21 / 21