[R-SIG-Finance] Testing for cointegration: JohansenvsDickey-Fuller

Adams, Zeno Zeno.Adams at ebs.edu
Tue Jan 13 10:46:38 CET 2009 

I would like to comment on what Eric wrote:


"When you do the ADF procedure, you are imposing a
known cointegrating  vector and so all of the uncertainty associated with
estimating the cointegrating vector has been eliminated. When you use the
Johansen framework, you are estimating the cointegrating vector and so the
uncertainty associated with this estimation is incorporated in the test."


>From what I understand the ADF procedure is the Engle-Granger approach. However, I don't think that the "uncertainty associated with
estimating the cointegrating vector" has been eliminated since as Paul wrote the sprd = ty - (1.2534 * sr) has been estimated by OLS. The spread is the residual in the Engle-Granger procedure so testing the residual (which itself is the outcome of an estimation involving uncertainty) on a unit root requires higher critical values (the MacKinnon 1991 tables). Therefore, in my opinion, Paul should compare the test statistic of his ADF test with the higher MacKinnon 1991 values in order to conclude if, from a technical point of view, the two variables are cointegrated or not.

>From the theoretical point of view I would of course agree with what Eric wrote.






-----Ursprüngliche Nachricht-----
Von: r-sig-finance-bounces at stat.math.ethz.ch [mailto:r-sig-finance-bounces at stat.math.ethz.ch] Im Auftrag von Eric Zivot
Gesendet: Montag, 12. Januar 2009 19:39
An: 'Brian G. Peterson'; markleeds at verizon.net; 'Paul Teetor'
Cc: r-sig-finance at stat.math.ethz.ch
Betreff: Re: [R-SIG-Finance] Testing for cointegration: JohansenvsDickey-Fuller


 There are statistical issues associated with this problem that can help
explain what is going on. When you do the ADF procedure, you are imposing a
known cointegrating  vector and so all of the uncertainty associated with
estimating the cointegrating vector has been eliminated. When you use the
Johansen framework, you are estimating the cointegrating vector and so the
uncertainty associated with this estimation is incorporated in the test.
With the futures example, you know the cointegrating vector (if it exists)
from theory so it makes sense to impose it. The resulting test will have
more power (ability to reject the null when the alternative is true) than
the Johansen test. Both tests have no-cointegration as the null (a unit
root). So your ability to find cointegration with the ADF test can be
attributed to the fact that the ADF test has higher power than the Johansen
test in this context. 
>From a more general perspective, the arbitrage relationship between spot and
futures implies that the basis cannot have a unit root so it is essentially
irrelevant to do a unit root test. What is more important here is to
understand the dynamic behavior of the "cointegrating error". More than
likely it will probably have some nonlinear effects that may make it look
nonstationary. There is a rather big literature on threshold type effects in
these models. See, for example, some of the early papers by Martin Martens.
PS. I don't think that the 2nd edition of Bernhard's cointegration book
discusses this issue in any detail.


-----Original Message-----
From: r-sig-finance-bounces at stat.math.ethz.ch
[mailto:r-sig-finance-bounces at stat.math.ethz.ch] On Behalf Of Brian G.
Peterson
Sent: Friday, January 09, 2009 2:23 PM
To: markleeds at verizon.net; Paul Teetor
Cc: r-sig-finance at stat.math.ethz.ch
Subject: Re: [R-SIG-Finance] Testing for cointegration: Johansen
vsDickey-Fuller

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:
>
>> R SIG Finance readers:
>>  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 spread is 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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