Bivariate Cointegration Analysis - Aktuelles · PDF fileIf the cointegration vector γ is...

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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007

Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08

Chapter 5:

Bivariate Cointegration Analysis

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Contents:

V. Bivariate Cointegration Analysis ...................................................................................................... 3

V.3.1 Definition of Cointegration .................................................................................................... 3

V.3.2 Engle-Granger...................................................................................................................... 7

V.3.3 Error Correction Model ....................................................................................................... 12

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V. Bivariate Cointegration Analysis V.3.1 Definition of Cointegration

Generally, cointegration is defined as follows: Given the M I(n) variables

Yt = [Y1t, Y2t, …, YMt]

The variables are cointegrated if r (with r < M) linear combinations are integrated of order k < n. The

number of linear combinations has to be strict smaller than the number of variables.

For r = M the linear combinations are simply a redefining of the initial variables, which from there

should be already I(k). In many cases of economic applications k = 0, n = 1 and r = 1. Thus, one

stationary I(0) linear combination of M I(1) variables exists.

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Under negligence of a constant term we can write this linear combination as:

Zt = Y1t - γ2Y2t - γ3Y3t - … - γMYMt = γ´Yt

with γ´ ≡ [1, -γ2, …, -γM]

In doing so, the coefficients γ can be a priori partially or completely known.

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For example:

If Y consists of the two variables long-term and short-term interest rate, then γ´ = [1,-1] defines the

interest spread, because

Zt = long-term interest(t) - short-term interest(t) = interest spread

If Y consist of the three variables log exchange rate, log domestic and log foreign price level, then

γ´ = [1,1,-1] defines the log of real exchange rate, because

Zt = log(FXt) + log(domestic Pt) – (foreign Pt) = log(real FXt)

If we consider the real money supply, the real income and the interest rate, then the vector

γ´ = [1,-γ2,-γ3] defines the error term of the demand for money function, because

Zt = (real money supplyt) - γ2(real incomet) -γ3(interest ratet)

⇔ (real money supplyt) = γ2(real incomet) -γ3(interest ratet) + Zt

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Please note that:

Two variables are considered as cointegrated if

1. both variables are non-stationary in their levels,

2. both variables show the same integration level, and

3. a linear combination of these two non-stationary variables possesses a lower integration

level (I(d-b)).

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V.3.2 Engle-Granger Approach

If the coefficients of the cointegration relationship are known, the test of cointegration is reduced to a

unit root test for the known linear combination Z, e.g. the interest spread or the log of real exchange

rate. In this context, if we can reject the null hypothesis of non-stationarity for such a linear

combination of I(1) time series, the data indicates cointegration.

If the cointegration vector γ is unknown, then an estimation of the cointegration relationship must be

added to this approach. As Engle and Granger had shown this can be done by OLS estimation of the

linear regression equation:

Yt = γ1 + γ2Y2t + … + γMYMt + Zt

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This OLS estimation has unusual properties:

1) The normalisation (the choice of the dependent variable) does not play asymptotically a role.

Asymptotically there arise the same estimation from Z, independently which variables of the

two from zero different coefficients is used as the dependent variable.

2) The valuations do not converge with the square root of the sample size (compared to

regressions with stationary variables) but with the sample size against the true value. This

property of super consistency implies that the use of the estimated Z in combination with

stationary variables is asymptotically equivalent to the use of the true value of Z.

After the estimation of the regression the Dickey-Fuller t test is applied to the OLS residuals Z��.

Thereby, if we have to reject the null hypothesis of non-stationarity of Z we can conclude that a

cointegration relationship exists.

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Note:

We have to keep in mind that the use of the estimated Z has consequences for the critical values of

the ADF test. In comparison to the critical values of the usual Dickey-Fuller the critical values here

are in absolute values higher and depend on the number of included variables M. If the cointegration

relation contains a deterministic trend we speak about a deterministic cointegration. The critical

values for M (at most equals six) can be found out with the table of MacKinnon (1991). To determine

the critical values of MacKinnon we use the following formula:

K = β∞ + β1T-1 + β2T

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T is the sample size and the coefficients of β can be taken from the table of MacKinnon according to

the number of variables be considered, depending on the specification of the ADF test equation

(constant, trend) and the selected probability value.

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Summary:

a) Most of the financial time series are integrated of order one. If they are cointegrated, a linear

combination of them (Zt) is stationary: Zt = Y1t - γY2t. It requires that the combination of the time

series exhibit the same integration level. Cointegrated time series means that based on a

theory a long-term stable relationship between the variables exists. This relation is not satisfied

at any point in time and short-term departures appear. If these deviations are stationary then a

tendency of back formation exists and a long-term steady state is established.

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b) The three step approach of Engle-Granger for cointegration testing:

1. step: Determination of the integration level of every variable

2. step: Estimation of the cointegration relation with OLS regression: Y�� = a bY�� Z�

3. step: Testing the residuals for stationarity: Z�� = Y�� a� � b�Y��

⇒ ADF test: If H0 is rejected then the variables are cointegrated

⇒ However, as a result of the OLS residuals the critical values of the ADF test are not

correct and we have to use the MacKinnon table.

Problem: The Engle-Granger approach refers only to one equation and only one specified

cointegration relationship can be analysed.

For n variables (n > 2) maximum (n-1) cointegration relationships can

theoretically exist.

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V.3.3 Error Correction Model

Up to now, we solely considered the control of two or more variables for cointegration. Subsequently

the question arise how can we illustrate the dynamic relationship between the cointegrated variables.

= Granger representation theorem

The theorem says that cointegrated variables have an error correction representation and the same

hold reverse, i.e. if for several variables an error correction representation exists then the variables

are cointegrated.

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In the bivariate case, for two variables and one cointegrated linear combination Z the model is as

follows:

- the variables Y1 and Y2 are I(1)

- both variables are cointegrated, i.e. (Zt-1 = Y1t-1 – a – bY2t-1) are I(0)

∆Y�� = λ�z�� c��∆Y�� c��∆Y�� ε��

��

��

∆Y�� = λ�z�� c��∆Y�� c��∆Y�� ε��

��

��

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Summary:

The ECM combines both short-term and long-term relationships of variables in one equation. The

short-term relations are incorporated by the variables in first differences (c1 and c2), whereas the

long-term relation are represented by the residuals of the estimated cointegration relationship (Zt =

Y1t-1 – a – bY2t-1). The parameter of the long-term relationship λ defines the rate of adjustment to the

new equilibrium. If the long-term relationship is valid then λ have to be negative, and if a departure

from the long-term equilibrium appears, the deviation will be reduced in the next period by the value

λ. The reciprocal (1/λ) indicates the length of time for a complete adjustment, i.e. after (1/λ) periods

the deviation from the equilibrium is completely eliminated.

Bivariate Cointegration Analysis - Aktuelles · PDF fileIf the cointegration vector γ is...

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