Chapter 5: Bivariate Cointegration Analysis

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1 Chapter 5: Bivariate Cointegration Analysis 1

2 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie V. Bivariate Cointegration Analysis... 3 V.3.1 Definition of Cointegration... 3 V.3.2 Engle-Granger... 7 V.3.3 Error Correction Model

Ökonometrie V. Bivariate Cointegration Analysis... 3

3 V. Bivariate Cointegration Analysis V.3.1 Definition of Cointegration Generally, cointegration is defined as follows: Given the M I(n) variables Y t = [Y 1t, Y 2t,, Y Mt ] 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. 3

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

4 Under negligence of a constant term we can write this linear combination as: Z t = Y 1t - γ 2 Y 2t - γ 3 Y 3t - - γ M Y Mt = γ Y t with γ [1, -γ 2,, -γ M ] In doing so, the coefficients γ can be a priori partially or completely known. 4

5 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 Z t = 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 Z t = log(fx t ) + log(domestic P t ) (foreign P t ) = log(real FX t ) If we consider the real money supply, the real income and the interest rate, then the vector γ = [1,-γ 2,-γ 3 ] defines the error term of the demand for money function, because Z t = (real money supply t ) - γ 2 (real income t ) -γ 3 (interest rate t ) (real money supply t ) = γ 2 (real income t ) -γ 3 (interest rate t ) + Z t 5

t ) + log(domestic P t ) (foreign P t ) = log(real FX t ) If we consider the real money supply, the real income and the interest rate, then the vector γ = [1,-γ 2,-γ 3 ] defines the error term

6 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)). 6

both variables show the same integration level, and 3.

7 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: Y t = γ 1 + γ 2 Y 2t + + γ M Y Mt + Z t 7

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.

8 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. 8

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.

9 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 + β 1 T -1 + β 2 T -2 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. 9

If the cointegration relation contains a deterministic trend we speak about a deterministic cointegration.

10 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. 10

It requires that the combination of the time series exhibit the same integration level.

11 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 H 0 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. 11

step: Testing the residuals for stationarity: Z =Y a b Y ADF test: If H 0 is rejected then the variables are cointegrated However, as a result of the OLS residuals the

12 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. 12

= Granger representation theorem The theorem says that cointegrated variables have an error correction representation and

13 In the bivariate case, for two variables and one cointegrated linear combination Z the model is as follows: - the variables Y 1 and Y 2 are I(1) - both variables are cointegrated, i.e. (Z t-1 = Y 1t-1 a by 2t-1 ) are I(0) Y =λ z c Y c Y ε Y =λ z c Y c Y ε 13

14 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 (c 1 and c 2 ), whereas the long-term relation are represented by the residuals of the estimated cointegration relationship (Z t = Y 1t-1 a by 2t-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. 14

relationship (Z t = Y 1t-1 a by 2t-1 ). The parameter of the long-term relationship λ defines the rate of adjustment to the new equilibrium.

The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.

Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium