STAT​ ​457​ ​Final​ ​Review
Overview
●
●
●
Vector​ ​Autoregressive​ ​Model​ ​(VAR)
Transfer​ ​Function​ ​Noise​ ​Model​ ​(TFN)
Co-integration​ ​Model,​ ​include​ ​Error​ ​Correction​ ​Model
Vector​ ​Autoregressive​ ​Model​ ​(VAR)
Model​ ​Specification:
y t = A0 + A1 y t−1 + … + Ap y t−p + ut
Where​ ​ y t = (y 1t , …, y kt , …, y Kt ) ​ ​for​ ​ k = 1, …, K .​ ​ A0 ​ ​stands​ ​for​ ​a​ ​( K ×1 )​ ​mean​ ​vector,​ ​ Ai are​ ​ (K × K)
coefficient​ ​matrices​ ​for​ ​i​ ​=​ ​1,​ ​.​ ​.​ ​.​ ​,​ ​p​ ​and​ ​ ut ​ ​is​ ​a​ ​𝐾​-dimensional​ ​white​ ​noise​ ​process​ ​with​ ​time-invariant
positive​ ​definite​ ​covariance​ ​matrix​ ​ E (ut uʹt ) = Σu .
●
Stationary​ ​Condition
Stationary​ ​Condition​ ​for​ ​VAR(p)
A(B)y t = ut
Where​ ​ A(B) = (I k − A1 B − … − Ap B p ), B y t = y t−1 ​ ​and​ ​ A0 = 0k×1
Still​ ​The​ ​necessary​ ​and​ ​sufficient​ ​condition​ ​for​ ​the​ ​stationarity​ ​of​ ​ y t ​ ​is​ ​that​ ​all​ ​solutions​ ​(roots)​ ​of
det(I K − A1 B − . . . − Ap B 2 ) = 0 ​ ​are​ ​greater​ ​than​ ​one​ ​in​ ​absolute​ ​value.
Under​ ​VAR(1)​ ​equivalent​ ​to
y t = Φ1 y t−1 + ut
Where​ ​ y t = (y 1t , …, y kt , …, y Kt ) ​ ​for​ ​ k = 1, …, K .​ ​ A0 ​ ​stands​ ​for​ ​a​ ​( K ×1 )​ ​mean​ ​vector,​ ​ Ai are​ ​ (K × K)
coefficient​ ​matrices​ ​for​ ​i​ ​=​ ​1,​ ​.​ ​.​ ​.​ ​,​ ​p​ ​and​ ​ ut ​ ​is​ ​a​ ​𝐾​-dimensional​ ​white​ ​noise​ ​process​ ​with​ ​time-invariant
positive​ ​definite​ ​covariance​ ​matrix​ ​ E (ut uʹt ) = Σu .
If​ ​ {λ1 , …, λk } ​ ​are​ ​the​ ​eigenvalue​ ​of​ ​ Φ1 ,​ ​then​ ​the​ ​absolute​ ​value​ ​of​ ​all​ ​eigenvalues​ ​ λj ​ ​of​ ​ Φ1 must​ ​be
less​ ​than​ ​1.
o
Derivation​ ​of​ ​the​ ​stationary​ ​condition.
o
Calculation
(y 1 y 2 )t = ut + (0.5 0.1 0.4 0.3 )(y 1 y 2 )t−1
Q1:
1) Consider​ ​the​ ​following​ ​VAR(1)​ ​Model:
[y 1t y 2t ] = [ϕ1,11 ϕ1,12 ϕ1,21 ϕ1,22 ][y 1,t−1 y 2,t−1 ] + [a1t a2t ]
Where​ ​ [a1t , a2t ]T ​ ​satisfies​ ​a​ ​bivariate​ ​white​ ​noise​ ​process.​ ​State​ ​how​ ​to​ ​check​ ​the​ ​stationary​ ​of​ ​the
above​ ​VAR(1)​ ​model
Q2:
2) Consider​ ​a​ ​bivariate​ ​vector​ ​autoregressive​ ​model​ ​of​ ​order​ ​one
y 1,t =− 0.7 + 0.7y 1,t−1 + 0.2y 2,t−1 + a1,t
y 2,t = 1.3 + 0.2y 1,t−1 + 0.7y 2,t−1 + a2,t
Suppose​ ​that​ ​ ai,t˜N ID(0, 1), i = 1, 2 ​ ​and​ ​ cov(a1t , a2t ) = 0.5 .​ ​Show​ ​that​ ​the​ ​above​ ​bivariate​ ​VAR(1)​ ​model
is​ ​stationary.
Q3:
3) Consider​ ​a​ ​bivariate​ ​VAR(p)​ ​model
p
(X 1t X 2t ) = ∑(ϕ(i)
ϕ(i) ϕ(i) ϕ(i) )(X 1t X 2t ) + (a1t a2t )
11 12 21 22
i
Answer​ ​the​ ​following​ ​questions:
1.​ ​State​ ​how​ ​to​ ​check​ ​the​ ​stationarity​ ​of​ ​Equation​ ​(1);
2.​ ​Describe​ ​the​ ​methods​ ​to​ ​select​ ​the​ ​order​ ​for​ ​Equation​ ​(1),​ ​i.e.​ ​the​ ​value​ ​of​ ​𝑝​,​ ​taught​ ​in​ ​class.
3.​ ​State​ ​how​ ​to​ ​how​ ​to​ ​test​ ​Granger​ ​causality​ ​for​ ​the​ ​case​ ​that​ ​ X 1t ​ ​grander​ ​causes​ ​ X 2t ​ ​but​ ​not​ ​the
other​ ​way​ ​around.​ ​Based​ ​on​ ​the​ ​same​ ​condition,​ ​express​ ​ X 2t as​ ​the​ ​transfer​ ​function​ ​noise​ ​model
of​ ​ X 1t .
Companion​ ​Form
●
The​ ​companion​ ​form​ ​of​ ​a​ ​VAR(p)-process​ ​is​ ​given​ ​by
ζt = Aζt−1 + v t
Where,
ζt = (y t y t−1 … y t−p+1 ), A = [A1 A2 … Ap−1 Ap I … … 0 0 0 I … 0 0 … … … … 0 0 0 … I 0 ]
Where​ ​the​ ​dimension​ ​of​ ​the​ ​stacked​ ​vectors​ ​ ζt ​ ​and​ ​ v t ​ ​is​ ​ (K p ×1) ​ ​and​ ​that​ ​of​ ​the​ ​matrix​ ​A​ ​is​ ​ (Kp×Kp).
The​ ​companion​ ​form​ ​of​ ​a​ ​VAR(p)​ ​process​ ​is​ ​also​ ​a​ ​VAR(1)​ ​process
ζt = (y 1t y 2t y 1t−1 y 2t−1 )
For​ ​VAR(p)​ ​model,​ ​though​ ​it​ ​is​ ​okay​ ​to​ ​use​ ​the​ ​characteristic​ ​function​ ​for​ ​stationarity​ ​condition,​ ​usually​ ​it
will​ ​be​ ​changed​ ​to​ ​the​ ​VAR(1)​ ​Companion​ ​form​ ​for​ ​the​ ​stationary​ ​condition.
Q1:
Express​ ​the​ ​above​ ​VAR(2)​ ​model​ ​into​ ​its​ ​companion​ ​form.
Q2:
2) Consider​ ​the​ ​following​ ​VAR(2)​ ​Model:
[y 1t y 2t ] = [a11 a12 a21 a22 ][y 1,t−1 y 2,t−1 ] + [b1,11 b1,12 b1,21 b1,22 ][y 1,t−2 y 2,t−2 ][a1t a2t ]
Where​ ​ [a1t , a2t ]T ​ ​satisfies​ ​a​ ​bivariate​ ​white​ ​noise​ ​process.​ ​Show​ ​how​ ​to​ ​examine​ ​the​ ​stationary​ ​of​ ​the
above​ ​VAR(2)​ ​model​ ​by​ ​re-expressing​ ​it​ ​as​ ​a​ ​VAR(1)​ ​model.
●
Implied​ ​Model
Consider​ ​a​ ​VAR(1)​ ​Model
ϕ(B)Z t = at
Where
ϕ(B) = [ϕ11 ϕ12 ϕ21 ϕ22 ]
Step​ ​1:​ ​Write​ ​the​ ​adjoint​ ​matrix
adj[ϕ(B)] = [ϕ22 − ϕ12 − ϕ21 ϕ11 ]
Step​ ​2:​ ​Write​ ​the​ ​equation​ ​as
adj[ϕ(B)]Z t = ϕ(B)at
Step3:​ ​Expansion​ ​the​ ​formula​ ​for​ ​ Z 1t ​ ​and​ ​ Z 2t
Question​ ​1:
Consider​ ​VAR(1)​ ​model:
Z t = (0.2 0.3 − 0.6 1.1 )Z t + at
What​ ​is​ ​the​ ​implied​ ​form​ ​for​ ​ Z 1t ​ ​and​ ​ Z 2t
Question​ ​2:
p
(X 1t X 2t ) = ∑(ϕ(i)
ϕ(i) ϕ(i) ϕ(i) )(X 1t X 2t ) + (a1t a2t )
11 12 21 22
i
Suppose​ ​ ϕ(i)
= 0 ​ ​for​ ​ i = 2, …, p, k , l = 1, 2 .​ ​What​ ​is​ ​the​ ​implied​ ​model​ ​for​ ​ X 2t
kl
●
Multivariate​ ​Portomanteau​ ​Test
Portmanteau​ ​Tests​ ​for​ ​Multivariate​ ​Time​ ​Series
m
−1 ˜
QBP = T ∑ tr(C jT C −1
0 C j C 0 ) χk2 m−n
j=1
m
QLB = T 2 ∑
j=1
T −1
−1 ˜
1
T −j tr(C j C 0 C j C 0 ) χk2 m−n
Where​ ​ n ​ ​is​ ​the​ ​number​ ​of​ ​coefficients​ ​excluding​ ​deterministic​ ​terms​ ​of​ ​a​ ​VAR(p)​ ​Model​ ​and​ ​ C j ​ ​is​ ​the
auto​ ​covariance​ ​function​ ​of​ ​the​ ​residuals
Granger​ ​Causality​ ​Test
VAR(p)
p
[y 1t y 2t ] = [α1 α2 ] + ∑ [ϕj,11 ϕj,12 ϕj,21 ϕj,22 ][y 1,t−j y 2,t−j ] + [a1t a2t ]
j=1
Then​ ​in​ ​order​ ​to​ ​evaluate​ ​the​ ​Granger​ ​Causality,​ ​we​ ​just​ ​need​ ​to​ ​test​ ​ ϕj,12 ​ ​are​ ​equal​ ​to​ ​0​ ​or​ ​not.
Likelihood​ ​Ratio​ ​Test
p
y 1t = α1 + ∑ ϕj,11 y 1,t−j + e1t
j=1
p
p
j=1
j=1
y 2t = α1 + ∑ ϕj,11 y 1,t−j + ∑ ϕj,12 y 2,t−j + e1t
Therefore,​ ​define​ ​the​ ​log-likelihood​ ​function​ ​and​ ​use​ ​likelihood​ ​ratio​ ​test,​ ​we​ ​could​ ​have
˜ − log |Σ|
ˆ )˜χp2
n(log |Σ|
Model/Order​ ​Selection​ ​Criteria
The​ ​following​ ​information​ ​criteria
AIC(n) = ln det (Σˆu (n)) + T2 nk 2
H Q(n) = ln det (Σuˆ(n)) +
S C(n) = ln det (Σˆu (n)) +
2ln ln T
T
ln (T )
T
nk 2
nk 2
+n* k
F P E(n) = ( TT −n
) det (Σˆ u (n))
*
T
With​ ​ Σ̂u (n) = T −1 ∑ ût uˆ ′t ​ ​and​ ​ n* ​ ​is​ ​the​ ​total​ ​number​ ​of​ ​the​ ​parameters​ ​and​ ​ n ​ ​assigns​ ​the​ ​lag​ ​order.
t=1
Portmanteau​ ​Test
Test​ ​the​ ​correlation​ ​between​ ​two​ ​series​ ​as​ ​0.
Test​ ​statistics
L
2 (k)˜χ2
QL = n2 ∑ (n − k )−1 ruv
L+1
k=0
Question​ ​1:
Suppose
ϕ1 (B)X 1t = θ1 (θ)u1t
ϕ2 (B)X 2t = θ2 (θ)u2t
(k)
(k) q k
pk
Where​ ​ ϕk (B) = 1 − ϕ(k)
B − … − ϕ(k)
pk B ​ ​and​ ​ θk (B) = 1 + θ1 B − … − θqk B ​ ​for​ ​ k = 1, 2 .​ ​Describe​ ​how
1
to​ ​test​ ​Ganger​ ​Causality​ ​using​ ​univariate​ ​approach.
Question​ ​2:
1) If​ ​ y 1,t ​ ​granger​ ​causes​ ​ y 2,t ,​ ​how​ ​can​ ​this​ ​relationship​ ​be​ ​specificed​ ​in​ ​the​ ​above​ ​VAR(2)​ ​model
2) Describe​ ​the​ ​steps​ ​of​ ​testing​ ​whether​ ​ y 1,t ​ ​granger​ ​causes​ ​ y 2,t ​ ​based​ ​on​ ​likelihood​ ​ratio​ ​test.
Question​ ​3:
Define​ ​Granger​ ​Causality​ ​and​ ​state​ ​how​ ​to​ ​test​ ​Granger​ ​Causality.
Transfer​ ​Function
Mathematical​ ​Form​ ​of​ ​TFN:
∞
y t = α + ∑ v i xt−i + ϵt
i
∞
∞
j
j
Condition:​ ​ ∑ |v j | < ∞ .​ ​ g = ∑ |v j | ​ ​is​ ​the​ ​stead-state​ ​gain.
●
Rational​ ​Distributed​ ​Lag​ ​Model
w(B) = w0 + w1 B + w2 B 2 + … = v (B)/v(1) ​ ​can​ ​be​ ​approximated​ ​as
w(B) =
●
1.
2.
3.
4.
5.
η0 +η1 B+…+ηr B r
1−θ1 B−…−θs B s
=
η(B)
θ(B)
Model​ ​Building​ ​Process
Preliminary​ ​identification​ ​of​ ​the​ ​impulse​ ​response​ ​coefficients​ ​ v ’s;
Specification​ ​of​ ​the​ ​noise​ ​term​ ​ ϵt
Specification​ ​of​ ​the​ ​transfer​ ​function​ ​using​ ​a​ ​rational​ ​polynomial​ ​in​ ​B​ ​if​ ​necessary
Estimation​ ​of​ ​the​ ​TFN​ ​model​ ​specified​ ​in​ ​Step​ ​2​ ​and​ ​3
Model​ ​Diagnostic​ ​Checks
Question​ ​1:
Box,​ ​Jenkins,​ ​and​ ​Reinsel​ ​(1994)​ ​fit​ ​a​ ​transfer​ ​function​ ​model​ ​to​ ​data​ ​from​ ​a​ ​gas​ ​furnace.​ ​The​ ​input
variable​ ​ xt ​ ​is​ ​the​ ​volume​ ​of​ ​methane​ ​entering​ ​the​ ​chamber​ ​in​ ​cubic​ ​feet​ ​per​ ​minute​ ​and​ ​the​ ​output​ ​is
the​ ​concentration​ ​of​ ​carbon​ ​dioxide​ ​emitted​ ​ y t .​ ​The​ ​transfer​ ​function​ ​model​ ​is
yt =
−（0.53+0.37B+0.51B 2 ）
xt
1−0.57B
+
1
ϵ
1−0.53B+0.63B 2 t
a) What​ ​are​ ​the​ ​value​ ​of​ ​b,s,r​ ​for​ ​this​ ​model
b) What​ i​ s​ ​the​ ​form​ ​of​ ​the​ ​ARIMA​ ​Model​ ​for​ ​the​ ​error
c) If​ ​the​ ​methane​ ​input​ ​was​ ​increased,​ ​how​ ​long​ ​would​ ​it​ ​take​ ​before​ ​the​ ​carbon​ ​dioxide
concentration​ ​in​ ​the​ ​output​ ​is​ ​impacted?
Question​ ​2
An​ ​input​ ​and​ ​output​ ​time​ ​series​ ​consists​ ​of​ ​250​ ​observations.​ ​The​ ​prewhitened​ ​input​ ​series​ ​is​ ​modeled
by​ ​an​ ​AR(2)​ ​model​ ​ y t = 0.4y t−1 + 0.2y t−2 + at .​ ​Suppose​ ​that​ ​we​ ​have​ ​estimated​ ​ σα = 0.3, σβ = 0.35 The
estimated​ ​cross-correlation​ ​function​ ​between​ ​the​ ​prewhitened​ ​input​ ​and​ ​output​ ​time​ ​series
is​ ​shown​ ​below
a) Find​ ​the​ ​approximate​ ​standard​ ​error​ ​of​ ​the​ ​cross​ ​correlation​ ​function
b) Which​ ​spikes​ ​on​ ​the​ ​cross-correlation​ ​function​ ​appear​ ​to​ ​be​ ​significant?​ ​[5%]
c) Estimate​ ​the​ ​impulse​ ​response​ ​function.​ ​Tentatively​ ​identify​ ​the​ ​form​ ​of​ ​the​ ​transfer​ ​function
models—i.e.​ ​the​ ​values​ ​of​ ​b,r,s​ ​in​ ​a​ ​transfer​ ​function​ ​model.
Question​ ​3.
An​ ​input​ ​and​ ​output​ ​time​ ​series​ ​consists​ ​of​ ​300​ ​observations.​ ​The​ ​prewhitened​ ​input​ ​series​ ​is​ ​modeled
by​ ​an​ ​AR(2)​ ​model​ ​ y t = 0.5y t−1 + 0.2y t−2 + at ​ ​.​ ​Suppose​ ​that​ ​we​ ​have​ ​estimated​ ​ σα = 0.2 ​ ​and
σβ = 0.4 .​ ​The​ ​estimated​ ​cross-correlation​ ​function​ ​between​ ​the​ ​prewhitened​ ​input​ ​and​ ​output​ ​time
series​ ​is​ ​shown​ ​below.
a) Which​ ​spikes​ ​on​ ​the​ ​cross-correlation​ ​function​ ​appear​ ​to​ ​be​ ​significant?
b) Suppose​ ​that​ ​ r = s = 2 .​ ​Provide​ ​preliminary​ ​estimates​ ​on​ ​ w0 , w1 , w2 , η1 and η2 ​ ​for​ ​the​ ​impulse
response​ ​function​ ​as​ ​follows:
∞
v (B) = ∑ v i B i =
i=0
w0 −w1 B−w2 B 2
1−η1 B−η2 B 2
Bb
Cointegration​ ​Model
●
Definition
If​ ​two​ ​ I (1) ​ ​process​ ​have​ ​a​ ​common​ ​ I (1) ​ ​trend​ ​and​ ​ I (0) ​ ​idiosyncratic​ ​components,​ ​then
they​ ​are​ ​co-integrated.
When​ ​time​ ​series​ ​are​ ​cointegrated,​ ​we​ ​said​ ​that​ ​there​ ​exist​ ​one​ ​or​ ​more​ ​long-run​ ​equilibrium
relationships​ ​among​ ​cointegrated​ ​time​ ​series.
●
Generalization​ ​of​ ​Cointegration
Engle​ ​and​ ​Granger​ ​(1987)​ ​provided​ ​the​ ​following​ ​definition​ ​of​ ​cointegration:​ ​The​ ​components​ ​of
the​ ​vector​ ​ z t = (z 1t , …, z kt ) ’​ ​are​ ​said​ ​to​ ​be​ ​cointegrated​ ​of​ ​order​ ​d​ ​and​ ​b​,​ ​denoted​ ​by​ ​ z t ˜ CI(d, b)
if
1)​ ​All​ ​components​ ​of​ ​ z t ​ ​are​ ​integrated​ ​of​ ​order​ ​d​;
2)​ ​There​ ​exists​ ​a​ ​vector​ ​ β = (β1 , β2 , …, βk )′ ​ ​such​ ​that​ ​the​ ​linear​ ​combination
k
β′z t = ∑ βi z i,t
i=1
is​ ​integrated​ ​of​ ​order​ ​(​d​ ​−​ ​b​)​ ​where​ ​b​ ​>​ ​0,​ ​and​ ​the​ ​vector​ ​ β ​ ​is​ ​called​ ​the​ ​cointegrated​ ​vector​.
● Granger​ ​Representation​ ​Theorem
If​ ​ X t ​ ​and​ ​ Y t ​ ​are​ ​cointegrated,​ ​then​ ​there​ ​exists​ ​an​ ​ECM​ ​representation.​ ​Cointegration​ ​is​ ​a​ ​necessary
condition​ ​for​ ​ECM​ ​and​ ​vice​ ​versa.
Δrs,t = a10 + αs (rL,t−1 − βrs,t−1 ) + ∑a11 (i)Δrs,t−i + ∑a12 (i)ΔrL,t−i + ϵS,t
ΔrL,t = a20 + αL (rL,t−1 − βrs,t−1 ) + ∑a21 (i)Δrs,t−i + ∑a22 (i)ΔrL,t−i + ϵL,t
αs , αL > 0, ϵi,t˜W N (0, σ2 ), i = s, L
The​ ​above​ ​equation​ ​can​ ​be​ ​presented​ ​as​ ​bivariate​ ​VAR.
Features:
1.​ ​Vector​ ​autoregressions​ ​on​ ​differenced​ ​ I (1) ​ ​processes​ ​will​ ​be​ ​a​ ​misspecification​ ​if​ ​the​ ​component
series​ ​are​ ​cointegrated.
2.​ ​Engle​ ​and​ ​Granger​ ​(1987)​ ​showed​ ​that​ ​an​ ​equilibrium​ ​specification​ ​is​ ​missing​ ​from​ ​a​ ​VAR
representation.
3.​ ​However,​ ​when​ ​lagged​ ​disequilibrium​ ​terms​ ​are​ ​included​ ​as​ ​explanatory​ ​variables,​ ​the​ ​model
becomes​ ​well​ ​specified.
4.​ ​Such​ ​a​ ​model​ ​is​ ​called​ ​an​ ​error​ ​correction​ ​model​ ​(ECM)​ ​because​ ​the​ ​model​ ​is​ ​structured​ ​so​ ​that
short-run​ ​deviation​ ​from​ ​the​ ​long-run​ ​equilibrium​ ​will​ ​be​ ​corrected.
Q1:
Suppose​ ​that​ ​ X 1t ​ ​and​ ​ X 2t are​ ​not​ ​weakly​ ​stationary.​ ​How​ ​do​ ​you​ ​model​ ​the​ ​joint​ ​dynamics​ ​of
{X 1t , X 2t }? ​ ​ ​Discuss​ ​your​ ​decisions​ ​based​ ​on​ ​whether​ ​these​ ​two​ ​series​ ​are​ ​cointegrated​ ​or​ ​not.
Q2:​ ​Discuss​ ​the​ ​reasons​ ​why​ ​we​ ​have​ ​to​ ​choose​ ​different​ ​models​ ​based​ ​the​ ​condition​ ​of​ ​cointegration
Q3:​ ​Discuss​ ​the​ ​implication​ ​of​ ​Granger​ ​representation​ ​theorem​ t​ o​ ​VAR​ ​Modelling
● Engel-Granger​ ​Modelling​ ​process
Test​ ​ X t , Y t ​ ​are​ ​ I (1) ​ ​processes​ ​using​ ​a​ ​unit​ ​root​ ​test
Regress​ ​the​ ​two​ ​time​ ​series​ ​and​ ​test​ ​on​ ​the​ ​regression​ ​residuals
If​ ​regression​ ​residuals​ ​are​ ​stationary,​ ​then​ ​ X t , Y t ​ ​are​ ​co-integrated.
Then​ ​we​ ​could​ ​consider​ ​the​ ​following​ ​ECM
ΔX t = c1 + ρ1 (Y t−1 − α̂X t−1 ) + βx1 ΔX t−1 + … + βy1 ΔY t−1 + … + ϵxt
ΔY t = c2 + ρ2 (Y t−1 − α̂X t−1 ) + γx1 ΔX t−1 + … + βy1 ΔY t−1 + … + ϵyt
The​ ​Johansen​ ​procedure​ ​(1988)​ ​seeks​ ​the​ ​linear​ ​combination​ ​which​ ​is​ ​most​ ​stationary​ ​whereas​ ​the
Engle-Granger​ ​tests​ ​seek​ ​the​ ​linear​ ​combination​ ​having​ ​minimum​ ​variance.
Three​ ​ways​ ​of​ ​modelling​ ​cointegration
●
Regression​ ​formulation
●
Autoregression​ ​formulation
●
Statistical​ ​Model​ ​–​ ​VAR​ ​Model
z t = Φ1 z t−1 + … + Φn z t−n + ut
Then​ ​we​ ​have
Δz t = Γ1 Δz t−1 + … + Γn−1 Δz t−n+1 + Πz t−n + ut
Where
Γi =− (I k − ϕ1 − … − ϕi )
And​ ​ Π =− (I k − ϕ1 − … − ϕn )
Likelihood​ ​Ratio​ ​Test​ ​for​ ​cointegration
●
Trace​ ​Test:
n
λtrace (r) =− T ∑ ln (1 − λi )
i=r+1
●
Maximum​ ​Eigenvalue​ ​Test
λmax (r, r + 1) =− T ln(1 − λr+1 )