In pair trading, in addition to correlation,
cointegration can be very useful tool to determine which securities to be picked up. In this text, i demonstrate two approaches to test cointegration between two financial time series, two ETFs (
EWA and
EWC), within R. As you see below chart, in last 5 years, there is a cointegration between EWA and EWC.
ADF Test
In this test, we use linear regression to estimate spread between two securities and then ACF to test if spread is stationary, which in a way also test of cointegration for two securities.
>library("quantmod") # To get data for symbols
> library("fUnitRoots") # Unit test
## Lets get first data for EWA and EWC from yahoo finance and extract adjusted close prices
>getSymbols("EWA")
>getSymbols("EWC")
>ewaAdj=unclass(EWA$EWA.Adjusted)
>ewcAdj=unclass(EWC$EWC.Adjusted)
## Now lets do linear regression where we assume drift is zero. Since we are not sure which security is dependent and independent, we need to apply following for both case
## EWC is dependent here
> reg=lm (ewcAdj~ewaAdj+0)
## And now lets use adf test on spread (which is actually residuals of regression above step)
> adfTest(reg$residuals, type="nc")
Title:
Augmented Dickey-Fuller Test
Test Results:
PARAMETER:
Lag Order: 1
STATISTIC:
Dickey-Fuller: -1.8082
P VALUE:
0.07148
## EWA is dependent here this time
> reg=lm (ewaAdj~ewcAdj+0)
> adfTest(reg$residuals, type="nc")
Title:
Augmented Dickey-Fuller Test
Test Results:
PARAMETER:
Lag Order: 1
STATISTIC:
Dickey-Fuller: -1.7656
P VALUE:
0.07793
We use most negative Dickey-Fuller value (-1.8082 and -1.7656) to choice which regression formula to use. Based on that, We choice EWC is dependent. Within 90% confidence level (p-value is 7%), we can reject null hypothesis (unit root), so we can assume spread (residual) is stationary, therefore there is a cointegration. Below coded a function for this purpose:
cointegrationTestLM_ADF <-function(A, B, startDate) {
cat("Processing stock:",A ," and ", B, " start date:",startDate)
aData=getSymbols(A,from=startDate,auto.assign = FALSE)
aAdj=unclass(aData[,6])
bData=getSymbols(B,from=startDate,auto.assign = FALSE)
bAdj=unclass(bData[,6])
lenA=length(aAdj)
lenB=length(bAdj)
N= min(lenA,lenB)
startA=0
startB=0
if (lenA!=N || lenB!=N){
startA=lenA-N+1
startB=lenB-N+1
}
cat("\nIndex start",A,":",startA," Length ",lenA )
cat("\nIndex start",B,":",startB," Length ",lenB)
aAdj=aAdj[startA:lenA,]
bAdj=bAdj[startB:lenB,]
regA=lm(aAdj~bAdj+0)
summary(regA)
regB=lm(bAdj~aAdj+0)
summary(regB)
coA <- adfTest(regA$residuals, type="nc")
coB=adfTest(regB$residuals, type="nc")
cat("\n",A," p-value",coA@test$p.value," statistics:",coA@test$statistic)
cat("\n",B," p-value",coB@test$p.value," statistics:",coB@test$statistic)
# Lets choice most negative
if (coA@test$statistic < coB@test$statistic){
cat("\nStock ",A, " is dependent on stock ",B)
cat("\np-value",coA@test$p.value," statistics:",coA@test$statistic)
p=coA@test$p.value
s=coA@test$statistic
}else {
cat("\n Stock ",B, " is dependent on stock:",A)
cat("\n p-value",coB@test$p.value," statistics:",coB@test$statistic)
p=coB@test$p.value
s=coB@test$statistic
}
return(c(s,p))
}
How to run it:
res=cointegrationTestLM_ADF("EWA","EWC",'2007-01-01')
Processing stock: EWA and EWC start date: 2007-01-01
Index start EWA : 0 Length 1731
Index start EWC : 0 Length 1731
EWA p-value 0.0501857 statistics: -1.948774
EWC p-value 0.04719164 statistics: -1.981454
Stock EWC is dependent on stock: EWA
p-value 0.04719164 statistics: -1.981454
res
-1.98145360 0.04719164
Johansen Test
As you see above ADF approach has some drawbacks such as:
- Not sure which security is dependent or independent
- Can not test multiple instruments
Johansen test addresses these points.
> library("urca") # For cointegration
> coRes=ca.jo(data.frame(ewaAdj,ewcAdj),type="trace",K=2,ecdet="none", spec="longrun")
> summary(coRes)
######################
# Johansen-Procedure #
######################
Test type: trace statistic , with linear trend
Eigenvalues (lambda):
[1] 0.004881986 0.001200577
Values of teststatistic and critical values of test:
test 10pct 5pct 1pct
r <= 1 | 2.07 6.50 8.18 11.65
r = 0 | 10.51 15.66 17.95 23.52
Eigenvectors, normalised to first column:
(These are the cointegration relations)
EWA.Adjusted.l2 EWC.Adjusted.l2
EWA.Adjusted.l2 1.000000 1.0000000
EWC.Adjusted.l2 -1.253545 -0.3702406
Weights W:
(This is the loading matrix)
EWA.Adjusted.l2 EWC.Adjusted.l2
EWA.Adjusted.d 0.007172485 -0.003894786
EWC.Adjusted.d 0.011970316 -0.001504604
Johansen test estimates the rank (r) of given matrix of time series with confidence level. In our example we have two time series, therefore Johansen tests null hypothesis of r=0 < (no cointegration at all), r<1 (till n-1, where n=2 in our example). As in example above, if r<=1 test value (2.07) was greater than a confidence level's value (say 10%: 6.50), we would assume there is a cointegration of r time series (in this case r<=1). But as you see, none of our test values are greater than than critical values at r<0 and r<=1, therefore there is no cointegration. This is opposite of ADF result we found above. Based on my some research, i've found that Johansen test can be misleading in some extreme case (see that
discussion for more info). Once a cointegration is established, eigenvector (normalized first column) would be used as weight for a portfolio.
In addition above methods,
KPSS(Kwiatkowski–Phillips–Schmidt–Shin)can be also used to test stationarity.
