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

[Paul Teetor]

All:

Many, many thanks to Mark, Brian, Jae, Ken, Matthieu, and Silika for the
thoughtful comments which they posted to the list.  I am very grateful for
your insights.  I will be studying your replies and following-up on what I
learn.

Here is a summary of some major points:

- Testing for cointegration and unit roots is an art, just like building any
significant statistical model.  The cointegration and unit root tests are
not as mechanical as I thought.

- My problem is a (relatively) simple bivariate case, therefore the
Engle-Granger or Philips-Ouliaris procedures could be more appropriate.  The
Johansen procedure is probably overkill.

- The bivariate tests will likely be more consistent with the Dickey-Fuller
test results.

- I need to decide if the OLS fit is really a meaningful long-run
cointegration vector.  If so, my modelling and testing process changes.

- I need to carefully construct my model and consider issues such as long-
and short-run modelling, choice of predictor versus response, number of
lags, trend versus no-trend, etc.

- I also need to carefully check the model, especially for significance.

- I definitely need to get Bernhard Pfaff's book (2nd ed.).

- Bernhard Pfaff's vignette for the VAR package contains useful information,
too.

Again, thanks to everyone who contributed to this discussion.

Paul


-----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 Paul Teetor
Sent: Friday, January 09, 2009 3:38 PM
To: r-sig-finance at stat.math.ethz.ch
Subject: [R-SIG-Finance] Testing for cointegration: Johansen vs
Dickey-Fuller

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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