# [R-SIG-Finance] Interpreting cointegration - ur.df() and ad.test()

Paul Teetor paulteetor at yahoo.com
Wed Mar 30 06:35:56 CEST 2011

```algotr8der:

test. (That's what the help page means when it says, "The general regression
equation which incorporates a constant and a linear trend is used.")  The data
plot may appear to be trending upward, but the function removes that trend
first. If you plot the detrended data, you'll see the data that the function is
testing. It should be going "sideways" compared to your original, trendy data.

The ur.df function performs a similar detrending when you specify type="trend".
(OK, more precisely: it detrends in conjunction with performing the ADF test.)

HTH,

Paul

Paul Teetor, Elgin, IL  USA

----- Original Message ----
From: algotr8der <algotr8der at gmail.com>
To: r-sig-finance at stat.math.ethz.ch
Sent: Tue, March 29, 2011 6:39:48 PM
Subject: [R-SIG-Finance] Interpreting cointegration - ur.df() and ad.test()

Hello folks,

I have read several posts on here regarding cointegration and the various
tests that check for this.

Here is what I have done - where securityA and securityB are time series
price data.

series <- merge(securityA, securityB,  all=FALSE)
series <- as.data.frame(series)
m <- lm(securityA ~ securityB+ 0, data=series)
beta1 <- coef(m)[1]
sprd = series\$securityA - beta1*series\$securityB

Augmented Dickey-Fuller Test

data:  sprd
Dickey-Fuller = -3.9226, Lag order = 1, p-value = 0.01281
alternative hypothesis: stationary

> summary(ur.df(sprd, type="trend", lag=1))

###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################

Test regression trend

Call:
lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)

Residuals:
Min       1Q   Median       3Q      Max
-0.91172 -0.02737 -0.00097  0.02553  0.98989

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.434e-02  6.790e-03  -3.584 0.000340 ***
z.lag.1     -5.309e-03  1.353e-03  -3.923 8.85e-05 ***
tt           8.143e-06  2.134e-06   3.816 0.000137 ***
z.diff.lag  -5.345e-02  1.226e-02  -4.360 1.32e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.05266 on 6624 degrees of freedom
Multiple R-squared: 0.005505,    Adjusted R-squared: 0.005055
F-statistic: 12.22 on 3 and 6624 DF,  p-value: 5.676e-08

Value of test-statistic is: -3.9226 6.7405 7.7586

Critical values for test statistics:
1pct  5pct 10pct
tau3 -3.96 -3.41 -3.12
phi2  6.09  4.68  4.03
phi3  8.27  6.25  5.34

Both results indicate that the time series 'sprd' is cointegrated and thus
mean reverting. However looking at the plot one can clearly see that the
series is trending and is not mean reverting. Any thoughts would be greatly
appreciated.

>plot(sprd, type="l")

http://r.789695.n4.nabble.com/file/n3416574/Screen_shot_2011-03-29_at_7.37.39_PM.png

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