Trouble interpreting cointegration test results

Question

I'm struggling with testing the cointegration of 2 time series (or rather interpreting the test results properly).

So I got 2 time series x and y each containing 36 monthly data points (oil prices).

From looking at those time series, I'd say they are cointegrated.

However when applying different cointegration tests, they don't seem to be:

1) Augmented Dickey-Fuller

 from statsmodels.tsa.stattools import adfuller
 from statsmodels.api import OLS

 ols_result = OLS(y, x).fit()
 result = adfuller(ols_result.resid)

returns

 (0.6614451366946532,
 0.9890361840444819,
 10,
 25,
 {'1%': -3.7238633119999998, '5%': -2.98648896, '10%': -2.6328004},
 84.12263429255607)

i.e. a p-value of 0.98; null hypothesis cannot be rejected, time series are not cointegrated.

2) Engle-Granger

 coint_t, p_value, _ = coint(y, x)
 p_value
 0.06910078732250052

returns a p-value of 0.069 i.e. not cointegrated.

What am I doing wrong here?

Thanks in advance!

PS: there seems to be Granger-Causality between the 2 time series (tested using statsmodels.tsa.stattools.grangercausalitytests)

Answer 1

What happens here is that, from a strict point of view, your series don't seem to be properly cointegrated. But this doesn't mean that their relation is spurious, fact is, cointegration has limitations as a tool for analysis. You can refer to this paper for examples and a broader treatise.

The residuals of the fitted model between x and y have higher variance when the values of x and y are higher, this makes the estimated model unstable (heteroscedasticity) and, perhaps more importantly, it makes the residuals non stationary in variance. Anyway, model's instability makes the residuals seem not to be stationary in mean too, even if they could be.

Under these circumstances, it's no surprise that EG test failed to detect the (not quite real) cointegration.