[R-SIG-Finance] [R-sig-finance] Testing for cointegration: Johansen vsDickey-Fuller

Tue Jan 13 09:01:08 CET 2009

Hi Eric, your note I feel obviously valid and truly intuitive. However can
you provide some monte carlo analysis on that? It would be easier to
visualize the whole thing.

Eric Zivot wrote:
> 
>  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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>>
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