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

[Jae Kim]

From: "Jae Kim" <jh8080 at hotmail.com>
Sent: Saturday, January 10, 2009 10:04 AM
To: "Paul Teetor" <paulteetor at yahoo.com>
Subject: Re: [R-SIG-Finance] Testing for cointegration: Johansen vs 
Dickey-Fuller

> Hi,
>
> 1. If you are using the ADF test here, you are giving the restriction that 
> the  cointegrating vector between the two is (1, -1.2534). That is, you 
> are saying that the two variables are related in the long run with the 
> cointegrating vector given. Under this restriction, you find the spread 
> stationary, so they are cointegrated with given cointegrating vector.
>
> 2. If you are using Johansen method, you are doing unrestricted estimation 
> of cointegrating vector. But if you believe that the above restriction is 
> sensible economically, the ADF result should be preferred to Johansen 
> result.
>
> 3. This is the bivariate case, so Johansen method may not be necessary. 
> try Engle-Granger 2-stage method, you might find cointegration. In 
> addition, Johansen method assumes normality, which may often be violated.
>
> hope this helps. JHK
>
>
> --------------------------------------------------
> From: "Paul Teetor" <paulteetor at yahoo.com>
> Sent: Saturday, January 10, 2009 8:38 AM
> 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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