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I am trying to test for cointegration between two series that based on qualitative reasoning, should be cointegrated. They are the prices of XLE ETF (XLE US equity) and 1st futures of Brent (CO1 Comdty). However, the results that I arrive at using two different methods both show that there exists no cointegration between the two series - not sure if my execution or the interpretation of the data is wrong?

(Both XLE and Brent 1st Futures have been tested for non-stationarity using ADF test from "urca" package)

1st test - Engle Granger 2-step test:
In doing this, I referenced Using R to Test Pairs of Securities for Cointegration by Paul Teetor

(1) Conducting Spread

> M<-lm(XLE~Brent+0,data=XLE.Brent)
> beta<-coef(M)[1]
> spread<-XLE.Brent$XLE-beta*XLE.Brent$Brent
> 
> summary(M)

Call:
lm(formula = XLE ~ Brent + 0, data = XLE.Brent)

Residuals:
   Min      1Q  Median      3Q     Max 
-20.363  -9.543  -2.909  13.294  36.269 

Coefficients:
     Estimate Std. Error t value Pr(>|t|)    
>Brent  0.74962    0.02004    37.4   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 16.37 on 68 degrees of freedom
Multiple R-squared:  0.9536,    Adjusted R-squared:  0.953 
F-statistic:  1399 on 1 and 68 DF,  p-value: < 2.2e-16

(2) Testing the stationarity of the spread using ADF test (from package "urca"):

> spread.ADF<-ur.df(spread,type="none",selectlags="AIC")
> summary(spread.ADF)

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test #
######################################### 
Test regression none 


Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)

Residuals:
   Min      1Q  Median      3Q     Max 
 -6.1449 -2.2523  0.5559  2.9194  8.4567 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
z.lag.1    -0.0003928  0.0266919  -0.015    0.988
z.diff.lag  0.1207084  0.1278700   0.944    0.349

Residual standard error: 3.443 on 65 degrees of freedom
Multiple R-squared:  0.01395,   Adjusted R-squared:  -0.01639 
F-statistic: 0.4596 on 2 and 65 DF,  p-value: 0.6335


Value of test-statistic is: -0.0147 

Critical values for test statistics: 
    1pct  5pct 10pct
tau1 -2.6 -1.95 -1.61

My interpretation: since $t$-value = -0.0147 is bigger than -1.61, do not reject null. Spread is not stationary. Hence no cointegration between XLE and Brent.

Second Test: Johansen Test

> XLE.brent.coint<-ca.jo(data.frame(XLE,Brent),type="trace",ecdet="trend",K=2,spec="longrun")
> summary(XLE.brent.coint)
>
>###################### 
># Johansen-Procedure # 
>###################### 
>
>Test type: trace statistic , with linear trend in cointegration 
>
>Eigenvalues (lambda):
>[1] 8.179514e-02 6.025284e-02 2.775558e-17
>
>Values of teststatistic and critical values of test:
>
>        test 10pct  5pct  1pct
>r <= 1 | 4.16 10.49 12.25 16.26
>r = 0  | 9.88 22.76 25.32 30.45
>
>Eigenvectors, normalised to first column:
>(These are the cointegration relations)
>
>          XLE.l2   Brent.l2   trend.l2
>XLE.l2   1.000000  1.0000000  1.0000000
>Brent.l2 1.467806 -0.4346323  0.1610563
>trend.l2 1.896366 -0.4903454 -0.8891875
>
>Weights W:
>(This is the loading matrix)
>
>            XLE.l2    Brent.l2      trend.l2
>XLE.d   -0.01629102 -0.13534537 -4.695795e-17
>Brent.d -0.03819241 -0.03886418  5.127543e-17

My interpretation: Since $t$-value for r=0: 9.88<22.76, do not reject null. Hence r=0, there exists no cointegration between XLE and Brent.

Additionally, I have carried out cointegration tests (both methods) on US 10 year and 2 year yields, and the results on both tell me that the series are not co-integrated, which does not make sense intuitively. Something must be wrong with the way I'm doing the tests!

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    $\begingroup$ Could you repost the outputs and format them as code (use the button with curvy brackets rather than the button with quotes)? That would make them more readable. It would also be helpful to see a plot of your data: both the original series and the residuals from the regression used in the first stage of the Engle-Granger procedure. $\endgroup$
    – Richard Hardy
    Nov 10, 2015 at 20:06
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    $\begingroup$ Hi @RichardHardy, I tried add the images but am unable to complete my edit - something about my reputation not being high enough to attach two links? But the graph of the spread looks like a tick. $\endgroup$
    – ElizaTYX
    Nov 11, 2015 at 3:17
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    $\begingroup$ Perhaps the formatting goes wrong if you have both a list like 1. Conducting spread, 2. Testing for stationarity... and then code within list items. I would give up the list (use (1) or (1) instead of 1.) to get better formatting if there was a necessary choice between the two. It's a pity they don't allow you to post another link. It would be really interesting to take a look at the plots. Perhaps you could disguise your first link as something else or even skip it; after all, Engle-Granger test is well known so it should be possible to find information about it by just googling. $\endgroup$
    – Richard Hardy
    Nov 11, 2015 at 18:17
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    $\begingroup$ It seems that you are doing everything right. I would try to see what lags VARselect reports, maybe adding more might help in Johansens procedure. But looking at the spread it is hard to argue that it is stationary, and if the series are cointegrated then OLS works, so it is another argument that probably the series are not cointegrated. $\endgroup$
    – mpiktas
    Nov 12, 2015 at 12:17
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    $\begingroup$ Regarding convergence in mid 2014: perhaps intuitively this is so, but the behaviour seems to be best illustrated by the spread, which gives a different impression. Regarding next step and forecasting: if each of the two series is integrated but the two are not cointegrated, you could run a VAR on first differences; VAR in levels or VECM would not work. (You can do stationarity tests on first-differenced data as a formality, but intuitively it does not seem your series could be I(2).) $\endgroup$
    – Richard Hardy
    Nov 16, 2015 at 9:27

1 Answer 1

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You should not use the standard critical values of the ADF test -- but rather special ones suited for the use of ADF test as the second stage of the Engle-Granger test. Anyhow, your test statistic is quite high, so in your case this will not change the conclusion. The spread appears to have a unit root.

The interpretation of the Johansen's test result looks fine.

It could be helpful to see a plot of your data, both the original series and the residuals from the regression used in the first stage of the Engle-Granger procedure.

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