Testing for cointegration: Johansen vsDickey-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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