Cointegrated VAR

8/11/2019 Cointegrated VAR

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Multivariate Time Series C.DOZ Cointegration and cointegrated VAR models

Cointegration

and cointegrated VAR models

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1 Introduction

Econometrics with non-stationary series: standardasymptotic results do not hold.

Spurious regression: if (xt) and (yt) are not stationaryand not cointegrated, OLS estimation of

yt =a + bxt+ t

not interpretable results .

First idea: if the series under study are non stationary,work with 1st differences (if they are stationary)

Problem: in such a case, information about levels is

lost

Error Correction Models (ECM): Seminal paperDavidson Hendry Srba Yeoh (1978): consumption

function

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Main idea of the approach: yt = ln(Yt) non stationary, xt = ln Ct, non stationary.

y and x are linked by a long-run relation y =bx.

If the long run equation is true yt bxt should notmove too far away from 0 in the short run

yt bxt should be stationary.

Estimate an equation in which the variables are

stationary, and information about levels is kept

Error correction equation:

yt =a+b0xt+

p

i=1 bixti+q

j=1 djytj+(yt1bxt1)+ut(yt1 bxt1) = gap between theoretical long runrelation and observed data (error correction term).

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Cointegration approach: if there are long-runequilibrium relations between series, they should not

move far away from those relations in the short run.

If these long run relations are linear: there is a linearcombination of the series which should be stationary.

First introduction of cointegration: Granger (1981),Engle and Granger (1987).

2 Cointegrated I(1) vector process

2.1 I(1) vector process

Xt = (x1t, . . . , xnt)

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(Xt)tZ is I(0) (integrated of order 0) if it is stationaryand it is not obtained as (1 L)Zt with (Zt)tZstationary.

(Xt)tZ is I(1) if (Yt)tZ defined by Yt = (1 L)Xt is I(0). Example : multivariate random walk (RW):

(Xt) defined by (1 L)Xt = + twhere t is a multivariate white noise:tW N(0, )

= 0 : RW without drift,

= 0 : RW with drift Initial condition : X0

Assumption :t,Cov(X0, t) = 0. In this case: Xt =X0+ t +

ts=1 s,

EXt = EX0+ t, VXt =t

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2.2 Wold representation and

Beveridge-Nelson decomposition of an

I(1) process (Stock & Watson, 1988)

Wold representation of an I(1) process:

(1 L)Xt =m + C(L)t

with C(L) =k=0 CkL

k, C0 =I , andk=0 ||Ck||


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Multivariate Beveridge-Nelson (BN)decomposition: Using the previous decomposition of C(L), we get:

(1 L)Xt =m + C(1)t+ (1 L)C(L)t Thetrend is the multivariate random walk (Tt)

defined by:

(1

L)Tt =m + C(1)t

Thecycle is the stationary process (Ct) defined by:

Ct =C(L)t

If, for instance, T0 =X0 and C0 = 0, then:

Xt =Tt+ Ct

with Tt =X0+ mt + C(1)t

s=1 s

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This can also be written:

Xt =

m + C(1)t

1 L + C

(L)t

2.3 Cointegrated vector processes

2.3.1 General definitions

If XtI(1): (Xt) is a cointegrated process or iscointegrated if

= (b1, . . . , bn) Rn / Xt =b1x1t+ +bnxnt is stationary

In such a case: is a cointegration vector or acointegrating vector

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Elementary example: ifi / (xit) is stationary, then(0, . . . , 0, 1, 0, . . . , 0) is a cointegrating vector

The set of all cointegrating vectors is a vectorsubspace F of Rn called the cointegration space.

If dim F =r, r is the cointegration rank of (Xt)

General definition of cointegration :

(Xt)tZ is integrated of order d if (1 L)d

Xt is I(0). If XtI(d): Xt is cointegrated if:

Rn / XtI(b) with b < d

In most cases: d= 1 and b= 0

Sometimes (e.g. prices series): d= 2 and b= 1 orb= 0. More complicated case, not presented here.

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2.3.2 Wold representation of an I(1) cointegrated

process

If (Xt)I(1) and if its Wold representation is:

(1 L)Xt =m + C(L)t

with C(L) =

k=0 CkLk, C0 =I , and

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Consequences : if the cointegrating space is F, with dim F =r, if

(i)1ir is a basis of F and if = (1, . . . , r) is theassociated (n, r) matrix, then (Xt) is stationary

the cointegration rank is r, with r < n

= (1, . . . , r) is a cointegration matrix

Remarks is a (n, r) matrix of full column rank

is not unique

If is a (n, r) cointegration matrix, it is possible to find

a cointegration matrix = Ir

Q

for a proper orderingof the variables

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2.3.3 Common trends of an I(1) cointegrated

process

If (Xt)I(1) has cointegration rank r, and if= (1, . . . , r) is a (n, r) cointegration matrix, then:

m= 0 and C(1) = 0

If is a (n, n r) matrix whose columns form a basisof , m0 (n r, 1) and (n, n r) such that:

m= m0 and C(1) =

Consequence:

(1 L)Tt = m + C(1)t= (m0+

t)

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Interpretation: Xt is decomposed as the sum of alinear combination of n r random walks + astationary process.

Important for the analysis of shocks propagation since: a shock as a permanent effect on a random walk

a shock has a transitory effect on a stationary

process

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2.4 Testing for cointegration in a univariate

framework when r= 1

2.4.1 Case where the possibly cointegrating vector

is known

For instance: Xt = (xt yt) with xt = ln Ct(consumption), yt = ln Yt (income), and Xt

I(1).

Question: is there a long-run relation between xt andyt which would imply that (yt xt) is stationary ?In this case, = (1 1) would be a cointegratingvector.

This question is equivalent to the following one: ifzt =xt yt, is zt a stationary process ?

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Remark: for most unit root tests: H0 : zt

I(1) so that is not a cointegrating

vector

H1 : ztI(0) so that is a cointegrating vectorBut not for all tests (KPSS)

The same approach applies when n >2, and the

possibly cointegrating vector is known

2.4.2 Case where the possibly cointegrating vector

is not known

Bivariate case If Xt = (xt yt)

, with xtI(1), ytI(1), if Xt is

cointegrated, the cointegrating vector can be

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chosen as = (b 1) The question is then:

? b / yt

bxt

I(0) ?

Property: b can be estimated through an OLS

estimation of equation: yt =a + bxt+ t

If there is cointegration, it can be proved that b is a

super-consistentestimator of b (converges at rate

T instead of

T).

Due to the non-stationarity of the data, b is not

asymptotically gaussian, but its law is known

Cointegration test: unit root tests applied to

zt =yt (a + bxt)

The fact that b has been estimated changes thecritical values new tables.

However, these tests are not very powerful

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General case Notations: Xt = (x1t

xnt)

I(1) and

= (b1 bn) a possibly cointegrating vector, whichis unknown.

On a priorieconomic grounds, it is possible to

choose i such that bi= 0. If i= 1 (for instance), then 1 = (1,

d2, . . . ,

dn)

with di = bib1 is also a cointegrating vector. OLS estimation of: x1t =a + d2x2t+ . . . + dnxnt+ t

Same properties and tests methodology as in the

bivariate case.

Remark This methodology cannot be extended tothe case where r >1: which cointegration relation

would the previous regression estimate ?

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2.5 Engle-Granger 2-steps methodology

Error Correction Model (ECM) :yt =a+b0xt+

pi=1

bixti+

qj=1

djytj+(yt1bxt1)+ut

1st step: OLS estimation of b in equationyt = + bxt+ ut

Ifyt and xt are cointegrated then b is super-consistent.

2nd step: OLS estimation of:

yt =a+b0xt+

pi=1

bixti+

qj=1

djytj+(yt1bxt1)+ut

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The estimators obtained in this 2nd step are

consistent, asymptotically gaussian, and have same

asymptotic law as the one which would be obtainedif b were known (instead of being estimated)

This result comes from the super-consistency of b

Remarks

The results are valid for ut

W N, and (ut)

uncorrelated with (xt) only. They can be extended to more general cases,

but the estimated equation has to be modified

The results are valid only if xt and yt arecointegrated.

If xt and yt are not cointegrated, the resultsobtained in the 1st step equation have no

interpretation (spurious regression).

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3 Cointegrated VAR models and

ECM form of a cointegrated VARmodel

3.1 Assumptions and notations

Xt = (x1t xnt) I(1) (Xt) has a VAR representation:

Xt = + 1Xt1+ . . . + pXtp+ t

(L)Xt = + t

with (L) = I 1L . . . pLp = (ij(L))1i,jnand t W N(0, )

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The roots of det (z) are 1 and zi, iI such thatiI , |zi|> 1

Xt is cointegrated and the cointegration rank is r = (1, . . . , r) is a cointegration matrix and rk() =r.

3.2 ECM Representation of the model

Preliminary algebraic results: ij(L) =ij(1) + (1 L)ij(L) with doijp 1 Equivalently:

(L) = (1) + (1 L)(L) with (L) =p1k=0

kLk

(0) =0 = (0) (1) =In (1)

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Equivalent forms of the model:(L)Xt = + t

(1)Xt+ (1 L)(L)Xt = + t

(1)Xt (1 L)p1k=0

kLk

Xt = + t

(1)Xt 0Xt p1k=1

kXtk = + t

(1)Xt+ (In (1))(Xt Xt1) p1k=1

kXtk = + t

Xt (In (1))Xt1 = +p1k=1

kXtk+ t

Xt+ (1)Xt1 = +p1k=1

kXtk+ t

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Finally:

Xt = (1)Xt1+p1k=1

kXtk+ t

Properties: (1)Xt is I(0)

(n, r) matrix, such that (1) =

rk (1) =r (Granger Representation Theorem)

rk =r

Error correction form of the model:

Xt = Xt1+p1k=1

kXtk+ t

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Example: n= 2, p= 2, r = 1:

x1tx2t

= 1

2

1

2

Xt1 + 11 12

21 22

x1,t1x2,t1

+

1t

2t

with

Xt1 =b1x1,t1+ b2x2,t1I(0).Remark: x1t and x2t are linked since Cov(1t, 2t) =12.

3.3 Beveridge-Nelson decomposition and

common trends

VAR representation: (L)Xt = + t

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Wold representation: (1 L)Xt =m + C(L)t

As C(L)(L)Xt =C(1) + C(L)t, unicity of Wold

decomposition gives:

C(L)(L) = (1 L)In and C(1)=m

Consequence : C(1)=m

C(1)(1) = 0

C(1)= 0

In the Beveridge-Nelson decomposition, when (Xt) iscointegrated, we have seen (p.12) that :(1 L)Tt =m + C(1)t =(m0+ t)

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Here this is written:(1 L)Tt =m + C(1)t =C(1)( + t) =( + t)with

= 0

Hence: = It can be shown (see Johansens book) that

= ()1

with =In p1k=1k

Finally, the common trends are the components of the(n r) dimensional random walk (Wt) defined by:

(1 L)Wt = ()1 ( + t)

or equivalently by:

(1 L)Wt = ( + t)

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4 Johansen estimation and tests

procedure

4.1 Maximum Likelihood Estimation for a

given value of r

Assumptions

(Xt)I(1) a cointegrated VAR process (L)Xt = + t VAR representation

(t) iid gaussian process and t N(0, ) The cointegration rank of (Xt) is r and is supposed

to be known

= (1, , r) unknown cointegration matrix withrk() =r

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ECM representation of (Xt):

Xt = Xt1+ p1k=1kXtk+ t

Parameters which have to be estimated : n real parameters

: nr real parameters

: nr real parameters

k, k = 1, . . . , (p 1) : (p 1)n2 real parameters : n(n + 1)/2 real parameters

Important remark: and are not unique

can be replaced by P with P an invertible matrix

in this case has to be replaced by (P1)

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(and ) can be uniquely defined if a

normalization condition is added for (or for )

For instance =

IrQ

but other normalizationconditions can be chosen.

Log-likelihood of the model: In order to simplify notations, the observations are

supposed to start at t=p + 1 If Yt = Xt, and if all the parameters of the model

are stacked in a vector , the log-likelihood

conditional to the first p observations is:

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L(Y1 YT, ) = ln l(Y1, , YT|Y0 Yp+1, )

=Tt=1

ln l(Yt|Yt1, )

= nT2

ln 2 T2

lndet 12

Tt=1

(Yt mt)1(Yt mt)

where mt =

Xt1+

p1

k=1

kXtk

Notations: OLS regression of Yt = Xt on the constant term

and the Xtks

Remark: this means that an OLS regression ofeach xit on 1 and all the xj,tks is done, and

that the results of these n regressions are stacked

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OLS estimators: 0, k0, k = 1, . . . , (p 1) estimated residuals:

0t = Xt 0 p1k=1 k0Xtk OLS regression of Xt1 on the constant term and

the Xtks

OLS estimators: 1, k1, k = 1, . . . , (p 1)

estimated residuals:

1t =Xt1 1 p1k=1 k1Xtk Results:

If and were known:

MLE would be identical to GLS and thus toOLS (Zellner theorem)

the OLS estimators of the coefficients and the

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estimated residuals would be:

= 0+ 1

k = k0+ k1

t = 0t+ 1t

= 1

T

T

t=1t

t

=S00+ S10+ S01 + S11 where:

S00 = 1

T

Tt=1

0t0t, S11 =

1

T

Tt=1

1t1t, S01 =

1

T

Tt=1

0t1t =S

10

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the value of the log-likelihood at its maximumwould be:

nT2

ln 2 T2

lndet nT2

These results still hold when and are unknown

the log-likelihood can be concentrated w.r.t. theother parameters

the concentrated log-likelihood isnT

2 (1 + ln 2) T

2 lndet(, )

where: (, ) =S00+ S10+ S01 + S11

the concentrated log-likelihood must be maximizedw.r.t. and

it can be shown that the maximization w.r.t.

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leads to:

() =S01(S11)1

and that the maximisation of the concentratedlog-likelihood is then equivalent to:

min

det (S11 S10S100 S01)det(S11)

can be normalized by:

S11=Ir the minimum is obtained for = (1, . . . , r)

eigenvectors associated to the r largest solutions

(generalized eigenvalues) 1, . . . , r of:

det(S11

S10S

100 S01) = 0

unicity of (up to sign changes of its columns) if

1 > > r

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the value of the log-likelihood at its maximum is:

L(Y1

YT,) =

nT

2 (1+ln2)

T

2 ln det S00

T

2

r

i=1

ln(1

i)

has a non-standard asymptotic law but is

super-consistent (converges at rate T instead of

the usual

T)

Remark: all the previous results can be easily extendedif is replaced by a general deterministic term Dt,

with a matrix (e.g. Dt =0+ 1t) or if = 0.

4.2 Tests of linear restrictions on the ks

Usual test procedures apply (i.e. as in a stationary

framework).

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4.3 Determination of r: sequential test

procedure

Notation:H(r) : the cointegrating rank is less than or equal to r

Likelihood ratio test of H0 =H(r) vs H1 =H(n): under H0:

ln l(Y1, . . . , Y T,0) = Cst T

2 ln det S00 T

2r

i=1ln(1 i) under H1:

ln l(Y1, . . . , Y T,) = Cst T2 ln det S00 T2n

i=1ln(1 i)

LR test statistics (trace test statistics):

LR = 2(ln l(Y1, . . . , Y T,) ln l(Y1, . . . , Y T,

0))

= Tn

i=r+1

ln(1 i)

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under H0: LRd(non standard law, tabulated)

Remarks:

rejection of H0 when LR >critical value the sequential test procedure must be run with

increasing values of r

if the model is estimated with = 0 or with replaced by Dt the critical values are different.

Likelihood ratio test of H0 =H(r) vs H1 =H(r+ 1): under H0:

ln l(Y1, . . . , Y T,0) = Cst T2 ln det S00 T2r

i=1ln(1 i)

under H1:

ln l(Y1, . . . , Y T,) = Cst T2 ln det S00 T2

r+1i=1 ln(1 i)

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LR test statistics (maximum eigenvalue test

statistics):

LR = 2(ln l(Y1, . . . , Y T,) ln l(Y1, . . . , Y T,0))= Tln(1 r+1)

under H0: LRd(non standard law, tabulated)

Remarks: rejection of H0 when LR >critical value the sequential test procedure must be run with

increasing values of r

if the model is estimated with = 0 or with replaced

by Dt the critical values are different.

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4.4 Testing restrictions on

Introduction

Under H(r), the Beveridge-Nelson decomposition of

Xt gives (n r) common trends, i.e. a(n r)-dimensional random walk (Wt) defined by:

(1

L)Wt =

+

t

if is unrestricted, this is a random walk with drift,

and there is a linear deterministic trend in Xt

If is in the range of :0 / =0, then= 0 and there is no deterministic trend

Notation: H(r) =H(r) {0 / =0}. MLE under H(r)

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M lti i t Ti S i C DOZ C i t ti d i t t d VAR d l


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Under H(r):

Xt = (0

Xt1) +

p1k=1

kXtk+ t

= Zt1+p1k=1

kXtk+ t

with Zt1 = 1

Xt1

and = (0 ) estimation procedure: similar as before with similar notations as before, the maximum of

the associated log-likelihood is:

L(Y1 YT, ) =nT2

(1+ln 2)T2

ln det S00T

2

ri=1

ln(1i )

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Likelihood ratio test of H0 =H(r) vs H1 =H(r): LR test statistics :

LR = 2(ln l(Y1, . . . , Y T,) ln l(Y1, . . . , Y T, ))

= Tn

i=r+1

ln1 i1 i

under H0: LRd2(n r)

rejection of H0 when LR >critical value

sequential test procedures:H(0) H(1) . . . H(r) . . . H(n)

H(0) H(1) . . . H(r) . . . H(n)

NB: first see whether the series are trending or not !

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M lti ariate Time Series C DOZ Coi te ratio a d coi te rated VAR models


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Likelihood ratio tests of H0 =H(r) vs H1 =H(n) andof H0 =H(r) vs H1 =H(r+ 1):

LR test of H0 =H(r) vs H1 =H(n)

LR =Tn

i=r+1

ln(1 i ) (trace statistics)

LR test of H0 =H(r) vs H1 =H(r+ 1)

LR =Tln(1 r+1) (max. eigenvalue)

asymptotic laws and critical values are different

from the previous ones

Remark: H(n) =H(n)

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4.5 Linear restrictions on

Examples: see Johansen-Juselius (1990)

Test procedure for (H0 :=H) / (H1 : unrestricted),with H a (n, s) known matrix

Remark: H0 :=HH0 :R= 0 with H=R Under H0, the model is:

Xt = HXt1+p1k=1

kXtk+ t

same kind of estimation procedure as before

1, . . . , r eigenvectors associated to

1, . . . ,

r,

largest solutions (eigenvalues) of:

det(HS11H HS10S100 S01H) = 0

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LR test statistics: LR =Tr

i=1ln 1i1

i

Under H0: LR

d

2

(r(n s)) Test procedure for (H0 := (b, )) / (H1 : unrestricted)

with b a (n, s) known matrix

If = (1 2), under H0, the model is:

Xt = (1b + 2)Xt1+p1k=1

kXtk+ t

same kind of estimation procedure as before, even

if it is a bit more complicated

LR test statistics: also a bit more complicated

Under H0: LRd2(s(n r))

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Test procedure for(H0 := (H11, H22)) / H1 : unrestricted),

with H1 a (n, s1) known matrix, H2 a (n, s2) knownmatrix, 1 a (s1, r1) unknown matrix, 2 a (s2, r2)

unknown matrix

If = (1 2), under H0, the model is:

Xt = 11H1Xt1 22H2Xt1+p1k=1

kXtk+ t

iterative procedure for the maximization of the

concentrated log-likelihood

LR test statistics: LR

(a bit complicated)

Under H0: LRd2(r1(n s1) + r2(n s2))

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t =

1t

2t

, =

11 12

21 22

the model can be written: X1t = 1 1

Xt1+p1

k=11kXtk+ 1t

X2t = 2 2Xt1+p1

k=12kXtk+ 2t

a block-recursive form of the model is obtained

through a pre-multiplication by:

I 121220 I

if 2 = 0, the 1st equation has the following form:

X1t =AX2t+11Xt1+p1k=1

1kXtk+1t (1)

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X2t is weakly exogenous for 1 and : MLE for

these parameters can be obtained in the

conditional model (1) MLE in (1) obtained along the same method as

before and tests can be run as before

If X1t has dimension 1, this gives a case where a

univariate equation involving several cointegrating

relations can be estimated.

Test of (H0 : 2 = 0) / (H1 : 2 unrestricted), with 2 a(n m, r) matrix

ML estimation of the model under H0 leads to an

eigenvalue problem of the same kind as before:

1, . . . , r the associated r largest eigenvalues

LR test statistics: LR =Tr

i=1ln1i1i

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under H0: LRd2(r(n m))

4.6.2 General linear restrictions on

H0 : = A, with A a known (n, m) matrix, rk(A) =m Equivalent form: H0 :A= 0

ML estimation of the model under H0 leads to aneigenvalue problem of the same kind as before:

1, . . . , r the associated r largest eigenvalues

LR test statistics: LR =Tri=1ln

1i1i

under H0: LR d2(r(n m))

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5 Structural MA representation and

IRF

5.1 General presentation

XtI(1) cointegrated VAR model with acointegration matrix, Xt

I(0), and (1) =.

Wold representation:

(1 L)Xt =m + C(L)t

with C(L) = k=0 CkLk, C0 =I , andk=0 ||Ck

|| > r.

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Analysis based on: The empirical significance of actual estimated

parameter values economic interpretability of the coefficients (with

preliminary tests on )

6.2 Identification of the short run structure

Xt can be treated as fixed (because of thesuper-consistency property) and identifying constraints

on also.

consider the model :

AXt = acXt1+p1

k=1 AkXtk+ vt

with:=A,a= A, Ak =Ak, vt =At

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if A as diagonal elements equal to 1, n(n 1)restrictions are needed to identify the parameters.

The model can be written :BYt = + vt

with : B = (A, A1, . . . , Ap1, a) andYt = (X

t, X

t1, . . . , X

tp+1, X

t1

c)

same kind of procedure as before (tests on ) to testrestrictions on A.

restrictions on the short run impact of shocks can alsobe used

Illustration : see Juselius.

Cointegrated VAR

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