# [R-SIG-Finance] Testing for cointegration: Johansen vs Dickey-Fuller

Brian G. Peterson brian at braverock.com
Fri Jan 9 23:22:46 CET 2009

```I'll look when I get home, but if I recall correctly, you need to check the unit root first.  Bernhard's book is definitely the best reference, and the new edition expands substantially onn the previous version.

markleeds at verizon.net wrote:

>  i think this can happen quite often but i'm not clear on how to
>resolve it. with the DF
>methodology, you are specifying the response and with Johansen's you
>aren't so
>that may have something to do with it. The literature talks about it but
>I don't think
>there's a resolution. Bernhard's cointegration book may talk about it
>also.
>
>
>
>On Fri, Jan 9, 2009 at  4:38 PM, Paul Teetor wrote:
>
>>  I am checking a futures spread for mean reversion.  I am using the
>> Johansen
>> test (ca.jo) for cointegration and the Augmented Dickey-Fuller test
>> (ur.df)
>> for mean reversion.
>>
>> Here is the odd part:  The Johansen test says the two futures prices
>> are not
>> cointegrated, but the ADF test says the spread is, in fact,
>> mean-reverting.
>>  I am very puzzled.  The spread is a linear combination of the prices,
>> and
>> the ADF test says it is mean-reverting.  But the failed Johansen test
>> says
>> the prices are not cointegrated, so no linear combination of prices is
>> mean-reverting.  Huh??
>>  I would be very grateful is someone could suggest where I went wrong,
>> or
>> steer me towards some relevent reference materials.
>>
>>  Background:  I am studying the spread between TY futures (10-year US
>> Treasurys) and SR futures (10-year US swap rate), calculated as:
>>      sprd = ty - (1.2534 * sr)
>>  where ty and sr are the time series of futures prices.  (The 1.2534
>> factor
>> is from an ordinary least squares fit.)  I execute the Johansen
>> procedure
>> this way:
>>      ca.jo(data.frame(ty, sr), type="eigen", ecdet="const")
>>  The summary of the test result is:
>>
>> 	###################### 	# Johansen-Procedure #
>> ######################
>> 	Test type: maximal eigenvalue statistic (lambda max) , without
>> linear trend and constant in 	cointegration
>> 	Eigenvalues (lambda):
>> 	[1]  2.929702e-03  6.616599e-04 -1.001412e-17
>>
>> 	Values of teststatistic and critical values of test:
>>
>> 	         test 10pct  5pct  1pct
>> 	r <= 1 | 2.00  7.52  9.24 12.97
>> 	r = 0  | 8.89 13.75 15.67 20.20
>>
>> 	<snip>
>>
>> I interpret the "r <= 1" line this way:  The test statistic for r <= 1
>> is
>> below the critical values, hence we cannot reject the null hypothesis
>> that
>> the rank is less than 2.  We conclude that the two time series are not
>> cointegrated.
>>
>> I run the ADF test this way:
>>
>> 	ur.df(sprd, type="drift")
>>
>> (I set type="drift" because that seems to correspond to ecdet="const"
>> for
>> the Johansen test.)  The summary of the ADF test is:
>>
>> 	###############################################
>> 	# Augmented Dickey-Fuller Test Unit Root Test #
>> ###############################################
>>
>> 	Test regression drift
>>
>> 	<snip>
>>
>> 	Value of test-statistic is: -2.9624 4.4142
>>
>> 	Critical values for test statistics:
>> 		1pct  5pct 10pct
>> 	tau2 -3.43 -2.86 -2.57
>> 	phi1  6.43  4.59  3.78
>> I interpret the test statistics as meaning we can reject the null
>> hypothesis
>> of a unit root (at a confidence level of 90% or better), hence the
>> mean-reverting.  I get similar results from the adf.test() procedure.
>>
>> F.Y.I., I am running version 2.6.2 of R.
>>  Paul Teetor
>> Elgin, IL   USA
>>
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