Engle Granger Cointegration

[Donat Buergi]

Hi everyone,

I'm currently doing a cointegration analysis using the engle-granger 2-step approach. My procedure is the following:
1. I check the data and their first differences for unit roots by computing an ADF-test
2. I run a regression to investigate the long run relationship
3. I check the residuals of the regression for unit roots

All data are in logs. rear_24 stands for the real exchange rate, bip_oecd1985 is a proxy for foreign income and total_x_waren denotes goods exports.

1. All time series are I(1), because the first differences are stationary.

2. reg log_x_waren log_reer_24 log_bip_oecd1985 trend

Source | SS df MS Number of obs = 104
-------------+------------------------------ F( 3, 100) = 2233.95
Model | 12.0035111 3 4.00117037 Prob > F = 0.0000
Residual | .17910708 100 .001791071 R-squared = 0.9853
-------------+------------------------------ Adj R-squared = 0.9849
Total | 12.1826182 103 .118277846 Root MSE = .04232

----------------------------------------------------------------------------------
log_x_waren | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-----------------+----------------------------------------------------------------
log_reer_24 | -.8411077 .0984973 -8.54 0.000 -1.036523 -.6456919
log_bip_oecd1985 | .4311957 .1929797 2.23 0.028 .0483294 .814062
trend | .0089758 .0012597 7.13 0.000 .0064765 .011475
_cons | 11.47612 1.13355 10.12 0.000 9.227189 13.72505

3. dfuller ehat

Dickey-Fuller test for unit root Number of obs = 103

---------- Interpolated Dickey-Fuller ---------
Test 1% Critical 5% Critical 10% Critical
Statistic Value Value Value
------------------------------------------------------------------------------
Z(t) -2.983 -3.509 -2.890 -2.580
------------------------------------------------------------------------------
MacKinnon approximate p-value for Z(t) = 0.0365

Based on this output, I assume that my time series are cointegrated.

The problem is now, that if I use the egranger command instead of doing it manually, I get quite different results as you can see below.

egranger log_x_waren log_reer_40 log_bip_oecd1985, reg

Engle-Granger test for cointegration N (1st step) = 104
N (test) = 103
------------------------------------------------------------------------------
Test 1% Critical 5% Critical 10% Critical
Statistic Value Value Value
------------------------------------------------------------------------------
Z(t) -1.701 -4.437 -3.825 -3.513

Critical values from MacKinnon (1990, 2010)
------------------------------------------------------------------------------
Engle-Granger 1st-step regression
----------------------------------------------------------------------------------
log_x_waren | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-----------------+----------------------------------------------------------------
log_reer_40 | -.6329995 .1056635 -5.99 0.000 -.8426075 -.4233915
log_bip_oecd1985 | 1.788129 .026119 68.46 0.000 1.736316 1.839942
_cons | 4.249327 .4709358 9.02 0.000 3.315117 5.183537
----------------------------------------------------------------------------------
Engle-Granger test regression
------------------------------------------------------------------------------
D._egresid | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_egresid |
L1. | -.0733352 .0431151 -1.70 0.092 -.1588538 .0121834
------------------------------------------------------------------------------


Could anyone explain how this can happen? Did I do something wrong in the procedure or is one of the stata commands not correct?
If more information is needed just let me know.

Any help would be highly appreciated.

Kind regards

Dona

[Sebastian Kripfganz]

First, you did not include a trend in the second procedure with egranger. Second, you are using different variables (I suppose) in the two procedures: log_reer_24 in the first and log_reer_40 in the second.

[Donat Buergi]

Thank you for the quick answer. Below the results I get when I use log_reer_24 and include a trend in egranger.

egranger log_x_waren log_reer_24 log_bip_oecd1985 trend, reg

Engle-Granger test for cointegration N (1st step) = 104
N (test) = 103
------------------------------------------------------------------------------
Test 1% Critical 5% Critical 10% Critical
Statistic Value Value Value
------------------------------------------------------------------------------
Z(t) -3.001 -4.823 -4.206 -3.892

Critical values from MacKinnon (1990, 2010)
------------------------------------------------------------------------------
Engle-Granger 1st-step regression
----------------------------------------------------------------------------------
log_x_waren | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-----------------+----------------------------------------------------------------
log_reer_24 | -.8411077 .0984973 -8.54 0.000 -1.036523 -.6456919
log_bip_oecd1985 | .4311957 .1929797 2.23 0.028 .0483294 .814062
trend | .0089758 .0012597 7.13 0.000 .0064765 .011475
_cons | 11.47612 1.13355 10.12 0.000 9.227189 13.72505
----------------------------------------------------------------------------------
Engle-Granger test regression
------------------------------------------------------------------------------
D._egresid | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_egresid |
L1. | -.1759003 .0586229 -3.00 0.003 -.2921785 -.059622
------------------------------------------------------------------------------

Overall the results don't change. There still seems to be no cointegration.

[Sarah Magd]

Dear Mr. Sebastian Kripfganz ,
- If the linear cointegration tests (i.e., Engle-Granger and Johansen) show that two variables are not linearly cointegrated. However, non-linear cointegration tests (e.g., Enders and Siklos (2001)) show that the two variables are nonlinearly cointegrated with symmetric adjustment to the long run.
In this case, can we run the linear VECM?

[Sebastian Kripfganz]

I am not familiar with nonlinear cointegration tests. The usual VECM assumes a linear cointegrating relationship. If there is no such linear relationship, this model might not be of much use.

[Sarah Magd]

Dear Mr. Sebastian Kripfganz,
To run the Engle-Granger cointegration test, we estimate it in two steps.
- The first step is run the following regression: P1 = β0 + β1 P2 + ε
- The second step is to test the stationarity of estimated residuals.
Usually, if the residuals are stationary, we call β1 in the first step "the co-integrating parameter".
My question is: what if the second step shows that the residuals are not stationary, what should the coefficient β1 in the first step be called? Is it still a co-integration parameter?

[Sebastian Kripfganz]

In that case, the first-stage coefficient cannot be meaningfully interpreted. There is no cointegration if the second-step residuals are non-stationary.

[Sarah Magd]

Dear Prof. Sebastian Kripfganz

I am using the Engle-Granger cointegration test to examine the linear cointegration between two variables (x1, X2)
X1 = a + bX2 +u
The estimated coefficient of b is 0.753 with a t-statistic of 81.58. The ADF results also show that the two variables are linearly cointegrated.
However, when I run the Johansen likelihood test results, it gives a chi-square statistic of 9.27 with a p-value of 0.000.
Given this, I have two questions:
- Is there any issue with having a very high t-statistic (i.e., 81.58)?
- Does the result of the Johansen likelihood test results imply that the two variables are not linearly cointegrated?

[Sebastian Kripfganz]

I am afraid, without seeing your actual Stata output it is hard to comment.

[Sarah Magd]

Dear Prof. Sebastian Kripfganz,
I am analyzing the cointegration relationship between Russian and Ukrainian wheat prices over the period 2018 - 2022. My sample includes two events: Covid-19 and the ongoing war. Therefore, I want to examine how the cointegration relationship has been affected by those two events. However, I do not know on which basis I should divide the whole sample into sub-samples (i.e., before the pandemic, during the pandemic, and after the war). Should I run a cointegration test with structural breaks and split my sample accordingly? Or should I run a unit root test on the two price series and split the sample based on the resulting break?