cointegration eg

8/22/2019 Cointegration Eg

1/13

COINTEGRATIONProfessor Dr. Abdul Qayyum

August 2, 2013 Time Series Analysis 1


2/13

INTRODUCTION It is first introduced by Granger (1981, 1983)

and Granger and Weiss (1983). Engle and Granger (1987) developed the

statistical parametrisations of the cointegrating

system. Theory of cointegration provides the statistical

counterpart to the concept of long-runequilibrium relationships in economic theory,such as the quantity theory of money and theFisher effect (Dickey, et al., 1991; andBanerjee et al., 1993).

2August 2, 2013 Time Series Analysis


3/13

Definition

1. It allows us to describe the existence of an

equilibrium, or stationary, relationship among two ormore time series, each of which is individually nonstationary.

The formal definition of cointegration is;

The components of the vector Xt are said to becointegrated of order d, b, denoted Xt CI(d, b), if

(i) Xt is I(d) and

(ii) there exists a non-zero vector ( 0) so that zt = 'Xt~ I(d-b), d > b > 0. The vector is called thecointegrating vector. [Engle and Granger (1987) ]



4/13

Tests for Cointegration

1. Hypothesis

1. Null H0: No cointegration2. Alternative Ha: Cointegration

2. Engle and Granger discussed following tests

1.

Co-integrated Regression Durbin Watson test (CRDW)2. Dickey Fuller (ADF) test

3. Augmented Dickey Fuller (ADF) test

4. Restricted Vector Autoregression (RVAR)

5. Augmented RVAR (ARVAR)6. Unrestricted vector Autoregression (UVAR)

7. Augmented UVAR (AUVAR)



5/13

Tests for Cointegration

The Residual Based Tests1. Estimate a cointegrating regression.2. The residual from this is scrutinised under the hypothesis

of no cointegration.

3. Engle and Granger (1987) proved that the test for

cointegration is closely related to the test of unit roots1. Ho: unit root in the residual

2. H1: the root is less than unity.

4. Rejection of the null is equal to the acceptance of the

alternative hypothesisThere exists a cointegrating relationship between the

variables.



6/13

Co-integrated RegressionDurbin Watson test (CRDW)

This test is proposed by Sargan and Bharagva(1983).

yt = xt + c + ut

They used standard Durbin-Watson statisticsassumptions to calculate the test of unit root.

Calculated three test statistics and tabulated lowerand upper bounds.

Under the null hypothesis of no cointegration the DWstatistic is close to zero.



7/13

Dickey Fuller (ADF) test

Take the residual from the cointegrationregression and estimate

ut = - ut-1 + t.

Test: 2 = the t statistics for



8/13

Augmented Dickey Fuller (ADF)

This is to estimate the following regressionut = - ut-1 + b1ut-1 + ...+ bput - p + twhere ut is residual from the cointegratingregression.

The hypothesis that = 0 is tested using thecritical t-values calculated by MacKinnon (1991).

Engle and Yoo (1987) calculated critical values forthe multivariate case and for different sample

sizes.


R i d V A i


9/13

Restricted Vector Autoregression

(RVAR) Two step estimator. Error correction representation

is estimated.

Test weather error correction term is significant.

Requires estimation of full system dynamics.

First order system is assumed.

y t = 1 u t-1 + 1t xt = 2ut-1 + 2t 4 = 2 1 +21 Test is base on sum of square of t


A t d R t i t d VAR


10/13

Augmented Restricted VAR

(ARVAR)

5 = 2 1 +21 Same as RVAR except higher order system

is assumed



11/13

Unrestricted Vector

Autoregression (UVAR)

Based on VAR in levels without any restriction.

Whether levels would appear at all or

whether model can be represented entirely in

changes.Assume first order system.

yt = - yt-1 + b1xt-1 + c + t xt = - yt-1 + b1yt-1 + biy t + c+ t 6 = 2[F1+ F2]



12/13

Augmented UVAR (AUVAR)

6 = 2[F1+ F2]



13/13

The Engle and Granger (1987) recommend the ADF

In this test it is assumed that there is only onecointegration relationship between the variables


cointegration eg

Documents