Interpretation of Engle-Granger Cointegration Test (Gretl)

Question

How would you interpret following result from running the Engle-Granger cointegration test in Gretl:

Step 1: testing for a unit root in var_1

Augmented Dickey-Fuller test for var_1
including 5 lags of (1-L)var_1
sample size 83
unit-root null hypothesis: a = 1

with constant and trend 
model: (1-L)y = b0 + b1*t + (a-1)*y(-1) + ... + e
1st-order autocorrelation coeff. for e: 0.011
lagged differences: F(5, 25) = 7.438 [0.0002]
estimated value of (a - 1): -1.00042
test statistic: tau_ct(1) = -3.15236
asymptotic p-value 0.0742

Step 2: testing for a unit root in var_2

Augmented Dickey-Fuller test for var_2
including 5 lags of (1-L)var_2
sample size 83
unit-root null hypothesis: a = 1

with constant and trend 
model: (1-L)y = b0 + b1*t + (a-1)*y(-1) + ... + e
1st-order autocorrelation coeff. for e: -0.049
lagged differences: F(5, 25) = 4.579 [0.0026]
estimated value of (a - 1): -0.793841
test statistic: tau_ct(1) = -3.16269
asymptotic p-value 0.0841

Step 3: cointegrating regression

Cointegrating regression - 
OLS, using observations 01-82 (T = 83)
Dependent variable: var_1

          coefficient        std. error     t-ratio  p-value 
 ---------------------------------------------------------------
 const          2.48090e+07       4.15910e+06   5.965   7.74e-07 ***
 var_2         −0.0121403         0.0153430    −0.7913  0.4340  
 time      491744            216280             2.274   0.0291   **

Mean dependent var   28734191   S.D. dependent var    7497823
Sum squared resid    1.55e+15   S.E. of regression    6555707
R-squared            0.275752   Adjusted R-squared   0.235516
Log-likelihood      −665.9158   Akaike criterion     1337.832
Schwarz criterion    1342.822   Hannan-Quinn         1339.622
rho                  0.559330   Durbin-Watson        0.871994

Step 4: testing for a unit root in uhat

Augmented Dickey-Fuller test for uhat
including 5 lags of (1-L)uhat
sample size 83
unit-root null hypothesis: a = 1

model: (1-L)y = (a-1)*y(-1) + ... + e
1st-order autocorrelation coeff. for e: 0.007
lagged differences: F(5, 27) = 7.652 [0.0001]
estimated value of (a - 1): -1.02671

test statistic: tau_ct(2) = -3.4799 asymptotic p-value 0.0936

Gretel states that

There is evidence for a cointegrating relationship if:
**(a)** The unit-root hypothesis is not rejected for the individual variables, and
**(b)** the unit-root hypothesis is rejected for the residuals (uhat) from the cointegrating regression.

What I read from the data is

b) that H0 is rejected at the 10% level(0.0936). However, it seems that var_1 and var_2 (both level-data) seem to be both stationary (0.0742, 0.0841).

Now, how do we interpret the data? Can we still say they are co-integrated or do we have to reject the Engle-Granger Cointegration because criterium a) is not confirmed?

Answer 1

I'm afraid the results are just all quite inconclusive with this dataset. In any case, with a sample size of about T=80 you're not very far into the asymptotics. That would mean that the p-values are artificially small, and that "in reality" they might not be below the 10% threshold.

On the other hand, the point estimates for your autoregressive coefficient 'a' in the various test equations are quite low, sometimes basically zero (a-1 = -1). That indeed would point to stationary series, where the estimation uncertainty is just so large that almost no null hypothesis could ever be rejected.

Of course one could also look at the plots of the time series and use other test approaches, but personally I think it would feel strange to impose the assumption of a unit root if my point estimate of the root is close to zero. But that may be a matter of taste, too.

Answer 2

as the first two steps for testing the stationarity in variable 1 and variable 2 do not reject the stationary null, in other words, those two variables are stationary. with the definition for stationary data, we could interpret variable 1 and variable 2 have constant mean and variance that follows a random walk. also, there is no trend for anyone of them. you could draw an actual time series graph to double-check and ACF graph as well, but they are not formal methods in detecting stationary.

since it failed to pass the first step of the Engle-granger cointegration test, we can not interpret any result or draw any conclusion on the cointegrated relationship on those two variables.

the only thing you could do is to analyze the data itself with either ARIMA model or GARCH model to find volatility or moving patterns of the variables.