Co-integration and VECM

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Co-integration and VECM

YiYi--NungNung YangYangCYCU, TaiwanCYCU, Taiwan

May, 2012May, 2012

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Learning objectives

Integrated variablesCo-integrationVector Error correction model (VECM)Engle-Granger 2-step co-integration testJohansen co-integration test

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Integrated variables

YYtt is an is an I(nI(n) variable if its ) variable if its nn--thth difference difference nnYYtt ~ I(0)~ I(0)

We call We call YYtt is is integrated of order nintegrated of order n..An I(0) variable is a stationary variableAn I(0) variable is a stationary variable

E.g., E.g., YYtt is an I(1) variable if its 1is an I(1) variable if its 1stst--difference: difference: YYtt ~ I(0)~ I(0)

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Summation of I(n) variables

I(mI(m) + ) + I(nI(n) == ) == I(mI(m))if if mmnn ((m,nm,n are integers)are integers)Y ~ I(1), X ~ I(0)Y ~ I(1), X ~ I(0)Let z=Y + X, then z ~I(1) Let z=Y + X, then z ~I(1)

I(nI(n) + ) + I(nI(n) == ) == I(nI(n))Example: Example:

Y ~ I(1), X ~ I(1)Y ~ I(1), X ~ I(1)Let z=Y + X, then z ~I(1) Let z=Y + X, then z ~I(1) The only exception isThe only exception is

If Y and X are If Y and X are cointegratedcointegrated

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Co-integration

If Y ~ If Y ~ I(nI(n), X ~ ), X ~ I(nI(n), and n ), and n 11Let z=Let z=11Y + Y + 22 XXif z ~I(0), then we callif z ~I(0), then we callY and X are Y and X are cointegratedcointegrated oror

There is a There is a coco--integration relationshipintegration relationship between these between these two nonetwo none--stationary variables, Y and X.stationary variables, Y and X.

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Co-integrationDef:Def:

There exists a There exists a linear combinationlinear combination of the noneof the none--stationary variables (integrated of the same orders) stationary variables (integrated of the same orders) that is stationarythat is stationaryThere could be There could be more than onemore than one such linear such linear combination if there are more than 2 variables combination if there are more than 2 variables involved in studiesinvolved in studies

The basic idea of The basic idea of cointegrationcointegration relates closely to the relates closely to the concept of unit rootsconcept of unit roots

Economic implicationsEconomic implicationsThere exists (longThere exists (long--run) equilibrium relationships run) equilibrium relationships between (among) nonebetween (among) none--stationary variables.stationary variables.

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Co-integrating vectors

for example, a Demand of money model:for example, a Demand of money model:m m p = p = 00 + + 11 y + y + 22 r + er + ewherewhere

m: nominal money supplym: nominal money supplyp: price levelp: price levely: incomey: incomer: interest rater: interest ratee: error terme: error term

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Co-integrating vectors

In equilibriumIn equilibriumm m p p 11 y y 22 r r 0 0 = e= eThe above model can be rewritten asThe above model can be rewritten asx x = e= e

wherewherex = (m, p, y, r, const)x = (m, p, y, r, const) = (= (11, , --11, , --11, , --22, , --00))

is called is called CoCo--integrating vectorintegrating vector

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VAR of order p and VECM

Consider a VAR of order p with a Consider a VAR of order p with a deterministic part given by Adeterministic part given by A00(all in matrix form)(all in matrix form)yytt = A= A00+A+A11yytt11+ A+ A22yytt22+++ + AApp yyttpp + e+ ettwe can rewe can re--write the above as VECMwrite the above as VECM(it is a long story(it is a long story))

yytt=A=A00++yytt11++11yytt11++++pp--11 yyttp+1p+1+ e+ ettPlease refer to Enders (2008)Please refer to Enders (2008)

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VAR of order p and VECM

For example, a VAR(1) model For example, a VAR(1) model yytt = A= A00+A+A11yytt1 1 + e+ ettwe can rewe can re--write the above as VECMwrite the above as VECMyytt=A=A00++yytt11+ e+ ett can be decomposed intocan be decomposed into==

: matrix of short: matrix of short--run adjusting coefficientsrun adjusting coefficients: co: co--integrating vector (matrix)integrating vector (matrix)

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Granger representation theorem

Let Y = (yLet Y = (y1, 1, yy2, 2, ,,yykk), ), Y~I(nY~I(n))If all variables in Y are If all variables in Y are cointegratedcointegrated

CoCo--integrationintegrationThere exist There exist at least oneat least one coco--integrating vector to let:integrating vector to let:Y ~ Y ~ I(nI(n--jj), for any j), for any j1.1.

VECMVECMIf and only if the variables in Y are coIf and only if the variables in Y are co--integrated, integrated, there must exist a VECMthere must exist a VECMCointegrationCointegration VECMVECM

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VAR(1) system as an example

yytt = A= A00+A+A11yytt11 + e+ ett

VECM VECM (if all variables are co(if all variables are co--integrated)integrated)yytt=A=A00++yytt11+ e+ ett

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Key concepts before running software

All nonAll non--stationary variables integrated of the stationary variables integrated of the same order may have cosame order may have co--integration integration relationshiprelationshipSingle equation version of Single equation version of cointegrationcointegration

exampleexampleyytt = a + = a + xxtt + + uutt, , yytt,, xxtt ~ I(1) but u~ I(1) but utt~I(0)~I(0)CoCo--integrating vectorintegrating vector(1, (1, --, , --a)a)ECM (error correction model)ECM (error correction model)yytt = a = a --(y(ytt--11--xxtt--11))++xxtt + + uutt

==a a -- uutt--11++xxtt + + uutt

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Engle-Granger co-integration test

Engle-Granger 2-step co-integration testmake sure orders of integrated variables are the same. Usually, they are all I(1) variableStep 1. run OLS by regressing yt on xts save residuals as a variable, e.g., etStep 2. use ADF test to see whether et is a I(0) variable:

If et ~ I(0), then the variables are cointegrated;otherwise, there is no cointegration among(between) those variables.

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Plots of Y1 and X1

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Case9.1 spurious regression and Engle-Granger co-integration test

1. use gretl to open ex-COINT.wf1.Run regression as follows y1t=a0+a1x1tResults

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2. Conduct ADF tests on Y1 and X1(you may use or as follows)

StrategiesStrategies(a) show ADF statistics with various lags: (a) show ADF statistics with various lags: no matter how many lags, they all have unit rootsno matter how many lags, they all have unit roots(b) use any use any information criterions to choose the optimal lags and report them in tables

Notes: ADF tests should be done with level and 1stdifferenced variables

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2.(a) show ADF statistics with various lags: 2.(a) show ADF statistics with various lags: no matter how many lags, they all have unit no matter how many lags, they all have unit rootsroots

Variable ADF-statistic

1% c.v.

5% c.v.

Note: c.v. are from Eviews.

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2.(b) use any use any information criterions to choose the optimal lags and report them in tables

Suggestions:Sample size 300, use BIC or HQC

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2.(b) use any use any information criterions to choose the optimal lags and report them in tables

0.000.000.000.000.730.730.560.56

PP--valuevalue

135.88135.8800--11.81611.816XX11tt

72.1772.1700--12.18812.188YY11tt

148.62148.62000.1550.155X1X1tt

93.3993.3900--0.3510.351Y1Y1tt

AICAIClagslagsADFADFvariablevariable

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3(a). Save residuals from OLS in step 1 as e1t, then do ADF test as in step 2(a) or 2(b) on these residuals.

Variable ADF-statistic

1% c.v.

5% c.v.

Note: c.v. are from Eviews.

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3(b). Use gretl: [model]->[Time series]->[Cointegration test]->[Engle-Granger]

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9.5 Johansen co-integration test

Johansen co-integration test procedure 1. use VAR lags selection to choose p2. by Johansens method, testing if variables are co-integrated.3. make sure rank() to see how many co-integrating vectors: i.e., based on trace test ormax tests.

4. normalizing integrating vectors and give an interpretations

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Rank and the number of co-integrating vectors

The number of Rank of The number of Rank of = the number of = the number of coco--integrating vectorsintegrating vectors

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trace test ormax tests

Two often used in determine the number of coTwo often used in determine the number of co--integrating vector (number of integrating vector (number of Rank(Rank()

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Normalizing integrating vectors

If If

11Y + Y + 22 X = eX = ethis can be rethis can be re--written aswritten asY + (Y + (22//11)) X = e/X = e/11

This is called normalization (on This is called normalization (on , co, co--integrating vectorintegrating vector))..

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Case 9.2 Johansen co-integration test

1. plot x, y, z

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Selection of VAR lagsSelection of VAR lags(e.g., max lag=8)(e.g., max lag=8)Use BIC to select p=1Use BIC to select p=1

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Case 9.2 Johansen co-integration testFill with

p+1

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Case 9.2 Johansen co-integration testresults (I)

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Case 9.2 Johansen co-integration testresults (II)

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There is only 1 Cointegrating vectorThere is only 1 Cointegrating vector

==(1, (1, yy, , zz, const, const) ) =(1.000, =(1.000, --0.984, 0.984, --1.018, 1.018, --0.013)0.013)

This implies a longThis implies a long--run relationship:run relationship:eett--11 = X= Xtt--11 --0.984Y0.984Ytt--11 --1.018Z1.018Ztt--11 --0.0130.013

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Co-integration and VECM