Testing for cointegration: Johansen vs Dickey-Fuller

 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