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

[Paul Teetor]

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