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?
Best Answer
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.