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When using the statstools cointegration function to test for cointegration with default parameters between two sine waves with a 1 period lag, coint() returns that there is no cointegration. Running a manual ADF test on residuals of the two waves shows that of course there is.

import statsmodels.tsa.stattools as ts
import statsmodels.api as sm

sin = pd.DataFrame(pd.Series([np.sin(x) for x in np.arange(0,50,0.1)]).rename('sin1'))
sin['sin2'] = sin['sin1'].shift(1)
sin['resid'] = sin['sin1'] - sin['sin2']
sin = sin.dropna()

def cointegration_test(y, x):
    ols_result = sm.OLS(y, x).fit() 
    return ts.adfuller(ols_result.resid)

# No cointegration found
ts.coint(sin['sin1'], sin['sin2'])
#(-0.0012969, 0.9858, array([-3.91856695, -3.34842679, -3.05297743]))

#Cointegration found
cointegration_test(sin['sin1'],sin['sin2'])
#(-388.12, 0.0, 1, 497, {'1%': -3.443,  '5%': -2.86,  '10%': -2.56}, -30440.043)

I believe this is down to the underlying lag logic in coint(), setting maxlag=1 does enable coint() to find cointegration however maxlag=2 and it reverts to finding nothing.

Is this behaviour correct? And if so what is the correct way of parameterising coint() such that it returns the result I expect (The two waves are cointegrated)?

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1 Answer 1

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Two sine waves cannot be cointegrated because a sine wave is not an integrated process; a sine wave does not have a unit root. Only integrated processes can be cointegrated. Therefore, the fact that the Dickey-Fuller test rejects a unit root in the residuals of a regression of $Y$ (e.g. sine wave number 1) on $X$ (e.g. sine wave number 2) does not automatically mean $Y$ and $X$ are cointegrated. I do not know how the different Python functions work, but I suppose that if you supply non-integrated series where integrated ones are expected and this problem is not checked internally in the function, the function may falsely conclude there is cointegration.

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  • $\begingroup$ Thanks Richard, could a way of modifying the test be to give the initial sine wave a low but persistent trend to make it an integrated process (non-stationary) ? Then checking the residuals between that and a similar but lagged series? $\endgroup$
    – rbonallo
    Commented Feb 24, 2021 at 8:49
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    $\begingroup$ @rbonallo, if you care about cointegration, the trend must be stochastic, not deterministic. Add a random walk to your series, and then you will have an integrated process + a deterministic part (sine wave) which is still an integrated process. $\endgroup$ Commented Feb 24, 2021 at 8:58

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