Johansen 2

8/18/2019 Johansen 2

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Vectorautoregressive- VAR Models

and Cointegration Analysis

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Time Series Analysis

Dr. Sevtap Kestel


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VECTOR TIME SERIES

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VECTOR TIME SERIES

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Vectorautoregression

Vector autoregression (VAR) is an econometric model used to capture the evolutionand the interdependencies between multiple time series, generalizing the univariateAR Models. All the variables in a VAR are treated symmetrically by including for eachvariable an equation explaining its evolution based on its own lags and the lags of allthe other variables in the model.

A VAR model describes the evolution of a set of k variables measured over the samesampleperiod (t єT) as a linear function of only their past evolution.

The variables are collected in a k x 1 vector y t , which has as the ith element y i,t , thetime t observation of variable yi .

For example, if the ith variable is GDP, then y i,t is the value of GDP at t.A (reduced) p-th order VAR, VAR(p), is


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where c is a k x 1 vector of constants (intercept) Ai is a k x k matrix (for every i = 1, ..., p) and

εt is a k x 1 vector of error terms satisfying the conditions.Properties:

1 1 2 2 ...t t t p t p t y c A y A y A y

[ ] 0t E [ ]

t t E

[ ] 0t t k E

every error term has mean zero

the contemporaneous covariance matrix of errors

Order of integration of the variables

Note that all the variables used have to be of the same order of integration.We have the following cases:All the variables are I(0) (stationary):

one is in the standard case, ie. a VAR in level


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All the variables are I(d) (non-stationary) with d>1:

The variables are conintegrated: the error correction term has to be included in the

VAR.The model becomes a Vector error correction model (VECM) which can be seen as arestricted VAR.

The variables are not cointegrated:the variables have first to be differenced d times and one has a VAR in difference


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Example: VAR(1)Suppose {y1t}tєT denote real GDP growth, {y2t} tєT denote inflation

1, 11 11 11 12

21 222 2 2, 1 2

t t t

t t t

y y c A A

A A y c y

1 1 11 1, 1 12 2, 1 1

2 2 21 1, 1 22 2, 1 2

t t t t

t t t t

y c A y A y

y c A y A y

One equation for each variable in the model.The current (time t ) observation of each variable depends on its own lagsas well as on the lags of each other variable in the VAR.


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VAR(1) PROCESS

• Example:

8

5.0,8.0

04.03.13.06.02.01.1det

2.06.0

3.01.12.06.0

3.01.1

21

2

t1tt

I

I

Z Y Y

The process is stationary.


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Structural VAR (SVAR) with p lags

0 0 1 1 2 2 ...t t t p t p t B y c B y B y B y e

where c0 is a k x 1 vector of constants, Bi is a k x k matrix, i = 0, ..., p, andet is a k x 1 vector of error terms.

The main diagonal terms of the B0 matrix (the coefficients on the i thvariable in the i th equation) are scaled to 1.

The error terms et (structural shocks) satisfy the conditions andparticularity that all the elements off the main diagonal of the covariancematrix E(et et ') = Σ are zero. That is, the structural shocks areuncorrelated.


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1 1 2 2 ... ( )t t t t t y c B

l '

t l

l t

y

IMPULSE RESPONSE FUNCTION The key tool to trace short run effects with an SVAR isthe impulse response function.

can be expressed as MA(‡)1 1 2 2 ...t t t p t p t y c A y A y A y

the row i , column j element of

identifies the consequences of a one-unit increase in the jth variable’s innovation atdate t (εtj) for the value of the ith variable at time t+l, holding all other innovations

at all dates constant. A plot of the row i, column j element of as a function of lag l iscalled the non-orthogonalized impulse response function.

l


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GRANGER CAUSALITY

• In time series analysis, sometimes, we wouldlike to know whether changes in a variable willhave an impact on changes other variables.

• To find out this phenomena more accurately,we need to learn more about GrangerCausality Test.

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GRANGER CAUSALITY

• In principle, the concept is as follows:

•

If X

causesY

, then, changes of X

happenedfirst then followed by changes of Y .

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GRANGER CAUSALITY

• If X causes Y, there are two conditions to besatisfied:

1. X can help in predicting Y. Regression of X on Yhas a big R2

2. Y can not help in predicting X.

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COINTEGRATION


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Cointegration

• In many time series, integrated processes areconsidered together and they form equilibriumrelationships. –

Short-term and long-term interest rates – Income and consumption

• These leads to the concept of cointegration.

• The idea behind the cointegration is that

although multivariate time series is integrated,certain linear transformations of the timeseries may be stationary.

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SPURIOUS REGRESSION•

If we regress a y series with unit root onregressors who also have unit roots the usual ttests on regression coefficients show statisticallysignificant regressions, even if in reality it is notso.

• The Spurious Regression Problem can appearwith I(0) series

• In a Spurious Regression the errors would becorrelated and the standard t-statistic will bewrongly calculated because the variance of theerrors is not consistently estimated. In the I(0)case the solution is:

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1/2)ˆof cerun varian-(longˆ where,ondistributi-ttˆ

ˆˆ


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SPURIOUS REGRESSION

Typical symptom: “High R2, t-values, F-value, but low DW”

1. Egyptian infant mortality rate (Y), 1971-1990, annual data,on Gross aggregate income of American farmers (I) and

Total Honduran money supply (M)Y ^ = 179.9 - .2952 I - .0439 M, R2 = .918, DW = .4752, F = 95.17(16.63) (-2.32) (-4.26) Corr = .8858, -.9113, -.9445

2. US Export Index (Y), 1960-1990, annual data, on Australianmales’ life expectancy (X) Y ^ = -2943. + 45.7974 X, R2 = .916, DW = .3599, F = 315.2

(-16.70) (17.76) Corr = .9570

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If two or more series are themselves non-stationary, but a linear combinationof them is stationary, then the series are said to be cointegrated.

Example:

A stock market index and the price of its associated follow a random walk by time. Testing

the hypothesis that there is a statistically significant connection between the futures priceand the spot price could now be done by testing for a cointegrating vector.

The usual procedure for testing hypotheses concerning the relationship between non-stationary variables was to run Ordinary Least Squares (OLS) regressions on datawhich had initially been differenced.

Although this method is correct in large samples, cointegration provides morepowerful tools when the data sets are of limited length, as most economic time-seriesare.The two main methods for testing for cointegration are:

The Engle-Granger three-step method.

The Johansen procedure.

Cointegration


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Granger Causality

• According to Granger, causality can be further sub-divided into long-run and short-run causality.

• This requires the use of error correction models orVECMs, depending on the approach for determining

causality.• Long-run causality is determined by the error

correction term, whereby if it is significant, then itindicates evidence of long run causality from theexplanatory variable to the dependent variable.

• Short-run causality is determined as before, with a teston the joint significance of the lagged explanatoryvariables, using an F-test or Wald test.

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Granger Causality

• Before the ECM can be formed, there first has tobe evidence of cointegration, given thatcointegration implies a significant errorcorrection term, cointegration can be viewed asan indirect test of long-run causality.

• It is possible to have evidence of long-runcausality, but not short-run causality and viceversa.

• In multivariate causality tests, the testing of long-

run causality between two variables is moreproblematic, as it is impossible to tell whichexplanatory variable is causing the causalitythrough the error correction term.

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0 1 2 ..

t t M M t t Y Y Y

ˆt

Engle-Granger Approach Estimation of parameters can be done by OLS estimation oflinear regression equation:

Dickey-Fuller t test is applied to the OLS residuals

Rejecting the null hypothesis of non-stationarity concludes “cointegration relationship” does exist.


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1 0 1 2 1 0 1 2

1 0 1 2

ˆ ˆˆ

t t t t t t

t t t

y y y y

y y

Three-step approach

•Determine the I(d) for every variable

Dickey Fuller, Perron tests H0: series is non-stationary•Estimate the cointegration relation by OLS regression•Test the residuals for stationarity

H0: series are not cointegratedADF Test does not give correct critical values because of the OLSresiduals

we use MacKinnon Table to determine the critical values

Multicointegration extends the cointegration technique beyondtwo variables, and occasionally to variables integrated atdifferent orders.


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1t y

2 t y

(0)t I

1

1 1 1 11 1 12 2 11

1

2 2 1 21 1 22 2 1

1

( )

( )

p

t t i t i i t i t i

p

t t i t i i t i t

i

y a y a y

y a y a y

Error Correction Model Granger Representation Theorem Determination of the dynamic relationship between

cointegrated variables in terms of their stationary errorterms.For bivariate case: Two integrated I(1) variables

and yielding one cointegrated combination

Estimate parameters by OLS.Regression with only stationary variables on both sides.


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Multivariate Cointegration Analysis - Johansen Test

VAR(1) having M I(1) variables can be expressed as:

1t t t

Y Y

where: Y, ì and å are (Mx1) vectors and Ã is a (MxM) matrix

Johannsen Test The approach of Johansen is based on the maximum likelihoodestimation of the matrix (Γ - I) under the assumption of normaldistributed error variables. Following the estimation thehypotheses

H0: r = 0, H0: r = 1, …, H0:r = M-1are tested using likelihood ratio (LR) tests.


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The Johansen Trace and Maximal

Eigenvalue Tests

• To test whether the variables are cointegratedor not, one of the well-known tests is theJohansen trace test. The Johansen test is used

to test for the existence of cointegration and isbased on the estimation of the ECM by themaximum likelihood, under variousassumptions about the trend or intercepting

parameters, and the number k ofcointegrating vectors, and then conductinglikelihood ratio tests.

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E l E h t i t t t S&P 500(GLOBAL) i d ISE i d


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Example: Exchange rate, interest rates, S&P 500(GLOBAL) index, ISE index


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* denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values

Max-eigenvalue test indicates 2 cointegrating eqn(s) at the 0.05 level

0.95993.8414660.0020961.20E-06At most 3

0.290114.264608.9531770.005118At most 2

0.002221.1316230.005140.017048At most 1 *

0.000027.58434117.81130.065285None *

Prob.**Critical ValueStatisticEigenvalueNo. of CE(s)

0.05Max-EigenHypothesized

Unrestricted Cointegration Rank Test (Maximum Eigenvalue)* denotes rejection of the hypothesis at the 0.05 level

Trace test indicates 2 cointegrating eqn(s) at the 0.05 level

0.95993.8414660.0020961.20E-06At most 3

0.369515.494718.9552730.005118At most 2

0.003429.7970738.960420.017048At most 1 *

0.000047.85613156.77170.065285None *

Prob.**Critical ValueStatisticEigenvalueNo. of CE(s)0.05TraceHypothesized

Trace TestLags interval (in first differences): 1 to 4

Series: GLOBAL EXCHANGE_RATE INTEREST_RATE ISE


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(2.44689)(1649.84)

12.362107382.3761.0000000.000000

(0.00194)(1.30729)

-0.024910-15.665670.0000001.000000

ISEINTEREST_RAT

EEXCHANGE_R

ATEGLOBAL

Normalized cointegrating coefficients (standard error in parentheses)

-45994.85Log likelihood2 Cointegrating Equation(s):

(0.00218)(1.44457)(0.00014)

-0.026394-16.55210-0.0001201.000000

ISEINTEREST_RAT

EEXCHANGE_R

ATEGLOBAL

Normalized cointegrating coefficients (standard error in parentheses)

-46009.85Log likelihood1 Cointegrating Equation(s):

Therefore, we can conclude that inthe long term these three variablesare cointegrated and there are 2

cointegration equations


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0.088941.91422ISE does not Granger Cause GLOBAL

3.3E-2020.7727174

5GLOBAL does not Granger Cause ISE

0.191051.48645INTEREST_RATE does not Granger Cause GLOBAL

0.010842.98831174

5GLOBAL does not Granger Cause INTEREST_RATE

0.991580.10286INTEREST_RATE does not Granger Cause ISE

9.7E-1212.2991174

5ISE does not Granger Cause INTEREST_RATE

0.326111.16105EXCHANGE_RATE does not Granger Cause GLOBAL

6.7E-2121.4690174

5GLOBAL does not Granger Cause

EXCHANGE_RATE

0.041512.31559EXCHANGE_RATE does not Granger Cause ISE

3.6E-5658.2545174

5ISE does not Granger Cause EXCHANGE_RATE

2.0E-3132.1459EXCHANGE_RATE does not Granger Cause

INTEREST_RATE

1.1E-2728.3482174

5

INTEREST_RATE does not Granger

Cause EXCHANGE_RATE

ProbabilityF-StatisticObsNull Hypothesis:

Lags: 5

Sample: 1/02/2001 12/31/2007

Date: 07/20/08 Time: 10:40

Pairwise Granger Causality Tests

Granger Causality Test:In order to comparepairwise variables

Granger Causality Tests isused


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ARDL APPROACH TOCOINTEGRATION


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AUTOREGRESSIVE DISTRIBUTED LAGS (ARDL)

APPROACH

• In regression analysis if model includes bothcurrent and lagged values for independentvariables it is called distributed lags model andif model also includes lagged values of

dependent variable it is called asautoregressive model (Gujarati, 2004).


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AUTOREGRESSIVE DISTRIBUTED LAGS (ARDL)

APPROACH

• Autoregressive distributed lags method allows usto express cointegrated behavior of variables

which have different order of integration.

• ARDL procedure is irrespective whether variablesused in model are I(0), I(1) or mutually

cointegrated (Peseran et al., 2001).


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ARDL MODEL REPRESENTATION


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BOUNDS TESTING PROCEDURE

• Cointegration test for ARDL method is appliedthrough bound testing procedure

• In this test there are two set of asymptotic valueswhich assume that all variables are I(1) in one setand I(0) in another. These two sets provide critical

value bounds for cointegration for both I(1) andI(0) data sets.


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• For applying ARDL procedure 3 steps are required as: – Applying bounds testing procedure for detecting

cointegration ranks between variables

– Estimating long run relationship coefficients with respectto cointegration relations estimated in first step and

– Estimating short run dynamic coefficients through vectorerror correction modeling.


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DECISION RULE FOR THE TEST

• Test on the null hypothesis through an F-statistics and the criticalvalues calculated by Peseran et al. (2001)

• It is assumed that lower bound critical values could be used for I(0)variables and upper bound critical values are used for I(1) variables. – if computed F-statistics is less than lower bound critical values the null

hypothesis is rejected that there is no long run relationship betweenvariables – if computed F-statistics is greater than the upper bound value, it could be

claimed that variables used in the model are cointegrated. – if computed F-statistic falls between the lower and upper bound values,

then the test results are inconclusive


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EXAMPLE OF BOUNDS TESTING PROCEDURE

Cointegration hypothesis F-statistics

F(CON|GDP,IND,LOS,PRICE,URB) 3.1012**

F(GDP|CON,IND,LOS,PRICE,URB) 6.3478*

F(IND|CON,GDP,LOS,PRC,URB) 7.2093*

F(LOS|CON,GDP,IND,PRICE,URB) 1.8595

F(PRC|CON,GDP,IND,LOS,URB) 5.5008*

F(URB|CON,GDP,IND,LOS,PRC) 0.88845

* significance at 1%, ** at 2.5% levels with respect to Pesaran and Pesaran (1997)

critical values.

Bounds Test results indicates that there are four cointegated relations whendependent variables are selected as annual electricity consumption, GDP,industry value added and mean adjusted annual average electricity prices


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EXAMPLE OF BOUNDS TESTING PROCEDURE


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VECTOR ERROR CORRECTION MODELS

• After implementing Bounds-Testing procedure to determine cointegratingrelationships, short and long run coefficients and related Error CorrectionModels (ECM) have been estimated within ARDL method whose ordersare selected with respect to Schwarz Information Criterion (SIC).

• In other words, vector error correction could be described as a restricted

vector autoregression, used for cointegrated nonstationary variables.• VECM is useful for determining short term dynamics between variables by

restricting long run behavior of variables. It restricts long run relationshipsthrough their cointegrating relations and error correction term representsthe deviation from long run equilibrium.


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VECTOR ERROR CORRECTION MODELS

EXAMPLE FOR ESTIMATING COEFFICIENTS


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The long run coefficient estimates and ECM. Dependent variable CONS, ARDL(3,2,2,3,3,3)

(a) Estimated Long Run Coefficients

Regressor Coefficients Standard Error T-Ratio (Prob*1%,**5%)

GDP 0.0020823 0.1058E-3 19.6864*IND -0.50248 0.052200 -9.6260*

LOS 1.9291 0.059938 32.1848*

PRC -0.17769 3.6958 -0.048078*

URB 1977.1 104.0641 18.9993*

Constant -627.1865 37.8846 -16.5551

(a) Error Correction Representation for the ARDL Model

ΔCON(-1) 1.1063 0.23387 4.7303*

ΔCON(-2) 0.63176 0.18166 3.4777*ΔGDP 0.0027787 0.4203E-3 6.6117*

ΔGDP(-1) -0.0017604 0.4911E-3 -3.5848*

ΔIND -0.74040 0.14494 -5.1084*

ΔIND(-1) 0.42188 0.13797 3.0579*

ΔLOS 2.7343 0.44020 6.2116*

ΔLOS(-1) -0.035601 0.46645 -0.076323(.940)

ΔLOS(-2) -2.0454 0.49798 -4.1074*

ΔPRC -21.6140 6.7027 -3.2247*

ΔPRC(-1) -14.7163 6.7179 -2.1906**

ΔPRC(-2) -19.7450 8.4128 -2.3470**

ΔURB 2992.8 1910.6 1.5664(.132)

ΔURB(-1) 141.2130 2060.3 0.068541(.946)

ΔURB(-2) -5826.9 1823.4 -3.1957*

INTERCEPT -1286.2 192.2404 -6.6905*

ECM(-1) -2.0507 0.30552 -6.7122*


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EXAMPLE FOR ESTIMATING COEFFICIENTS


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REFERENCES

• Pesaran, M.H., Shin, Y., Smith, R.J.,2001. Bounds testing approaches to the analysis oflevel relationships. Journal of Applied Econometrics. 16, 289-326.

• Pesaran M.H., and Pesaran B.,1997. Working with Microfit 4.0: interactive econometricanalysis. Oxford University Press.

• Hamilton J.D.A., 1994. The time series analysis. New Jersey, Princeton University Press.

• Engle, R.F., Granger, C.W.J, 1987. Co-integration and error correction: Representation,estimation and testing. Econometrica. 55, 251-276.

• World Bank Statistics Service (data resource). Available at: http://data.worldbank.org/ (Accessed: 4.29.2013).

• TEIAS Electricity Statistics (data resource). Available at:http://www.teias.gov.tr/istatistikler.aspx (Accessed: 4.29.2013).
http://data.worldbank.org/http://www.teias.gov.tr/istatistikler.aspxhttp://www.teias.gov.tr/istatistikler.aspxhttp://data.worldbank.org/

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