egcm(X, Y, na.action, log = FALSE, normalize = FALSE,
debias = TRUE, robust=FALSE, include.const=TRUE,
i1test = egcm.default.i1test(),
urtest = egcm.default.urtest(),
p.value = egcm.default.pvalue())is.cointegrated(E)
is.ar1(E)
zoo vector. Alternatively, a two-column
matrix or data.frame, in which case Y should be omitted.zoo vector."egcm" returned from a previous call to egcmlm.TRUE, indicates that the model should be
fit to the logs of the input vectors X and Y. Default: FALSE.TRUE, indicates that each series should be
normalized to start at 1. This is performed by dividing the series
by its first element. Default: FALSE.TRUE, indicates that the value of $rho$
that is reported should be debiased. Default: TRUE.TRUE, indicates that the two-step
Engle-Granger procedure should be performed using a robust linear
model rather than a standard linear model. See rlm.
Default: FALSTRUE, indicates that the constant term $alpha$
should be included in the model. Otherwise, sets $alpha=0$.
Default: TRUE.X and Y are integrated.
If none is specified, then defaults to the value reported by
egcm.default.i1test(). Theegcm.default.urtest(). The installation default is egcm.default.pvalue(). The installation default is 0.05."egcm". This can then be
printed or plotted. There is also a summary method.The following is a copy of the printed output that was obtained from running the first example below: VOO[i] = 0.9201 SPY[i] - 0.6845 + R[i], (0.0005) (0.0845) R[i] = -0.0004 R[i-1] + eps[i], eps ~ N(0, 0.0779^2) (0.0633) R[2013-12-31] = -0.0987 (t = -1.265)
The first line of the output shows the fit that was found. The parameters were determined to be $\beta = 0.9201$, $\alpha = -0.6845$ and $\rho = -0.0004$. The standard deviation of the sequence $\epsilon$ of innovations was found to be $0.0779$. The standard errors of $\alpha$, $\beta$ and $\rho$ were found to be $0.0845$, $0.0005$ and $0.0633$ respectively.
The third line of output shows the value of the residual as
of the last observation in the series. The sign of the value
$-0.0987$ indicates that VOO was relatively undervalued
on this date and that the difference between the two series was
$-1.265$ standard deviations from their historical mean.
The fields of the "egcm" object are as follows:
X[i])Y[i])R[i])[i])X and Y are integrateds1.i1.stats2.i1.statr.stateps.ljungbox.statdiff(S1)diff(S2)residuals$$Y[i] = \alpha + \beta * X[i] + R[i]$$ $$R[i] = \rho * R[i-1] + \epsilon[i]$$ $$\epsilon[i] \sim N(0, \sigma^2)$$
In the first step, $alpha$ and $beta$ are found using
a linear fit of X[i] with respect to Y[i]. The
residual sequence R[i] is then determined.
Then, in the second step, $\rho$ is determined, again using
a linear fit.
Engle and Granger showed that if $X$ and $Y$ are cointegrated, then this procedure will yield consistent estimates of the parameters. However, there are several ways in which this estimation procedure can fail:
XorY(or both) may already be
mean-reverting. In this case, there is no point in forming
the difference$Y - \beta X$. If one series is mean-reverting
and the other is not, then any non-trivial linear combination will
not be mean-reverting.R[i]may not be mean-reverting.
In the language of cointegration theory, it is then said to
contain a unit root. In this case, there is no benefit to
forming the linear combination$Y - \beta X$.R[i]may be mean-reverting, but
the relation$R[i] = \rho R[i-1] + \epsilon[i]$may not be
the right model. In other words, the residual series
may not be adequately described by an auto-regressive
series of order one. In this case, the parameters$\alpha$and$\beta$will be correct, however the specification for
the residualsR[i]will not be. The user may wish to try
fitting the residuals using another function, such asarima.The egcm function checks for each of the above contingencies,
using an appropriate statistical test.
If one of the above conditions is found, then a warning message is
displayed when the model is printed.
The p-value used in the above tests is given by the
parameter p.value. This can be changed by setting the value
of the parameter, or by changing the default value with
egcm.set.default.pvalue. For all of the unit root
tests, the p-values of the corresponding test statistics have been
recomputed through simulation and a table lookup is used. The
Ljung-Box test (see Box.test) is used to assess whether
or not the residual series can be adequately fit with an autoregressive
series of order one.
The estimates of $\alpha$ and $\beta$ are not only
consistent but also unbiased. Unfortunately, the estimate obtained
for $\rho$ may be biased. Therefore, a bias correction has been
implemented for $\rho$. A pre-computed table of biases has been
determined through simulation, and a table lookup is performed to
determine the appropriate bias correction. To turn off this
feature, set debias = FALSE.
The helper function is.cointegrated() takes as input an "egcm"
object E. It returns TRUE if E
appears to represent a valid pair of cointegrated series. In other words, it checks that both X and Y are
integrated and that the residual series R is free of unit roots.
The helper function is.ar1() also takes as input an "egcm"
object E. It returns TRUE if the
residual series R can be adequately fit by an autoregressive model
of order one.
From the standpoint of securities trading, cointegration is thought
to provide a useful model for pairs trading. If the price series of
two securities are cointegrated, then the corresponding residual
series R[i] will be mean-reverting. When the magnitude of the residual
R[N] is large, a trader might establish a long position in the
undervalued security and a short position in the overvalued security.
With high probability, the positions will converge in value, and a
profit can be collected. Numerous scholarly articles and several
books have been written on pairs trading.
Data mining for cointegrated pairs is not recommended, though. As with any statistical test, the cointegration test will generate false positives. Experience shows that at least in the case of the components of the S&P 500, the number of false positives overwhelms the number of truly cointegrated series.
Clegg, M. (2014). On the Persistence of Cointegration in Pairs Trading (January 28, 2014). Available at SSRN: http://ssrn.com/abstract=2491201
Ehrman, D.S. (2006). The handbook of pairs trading: strategies using equities, options, and futures. (Vol. 240). John Wiley & Sons. Engle, R. F. and C. W. Granger. (1987) Co-integration and error correction: representation, estimation, and testing. Econometrica, 251-276. Pfaff, B. (2008) Analysis of Integrated and Cointegrated Time Series with R. Second Edition. Springer, New York. ISBN 0-387-27960-1 Vidyamurthy, G. (2004). Pairs trading: quantitative methods and analysis. (Vol 217). Wiley.com.
library(TTR)
# SPY and IVV are both ETF's that track the S&P 500.
# One would expect them to be cointegrated, and in 2013 they were.
spy2013 <- getYahooData("SPY", 20130101, 20131231)$Close
ivv2013 <- getYahooData("IVV", 20130101, 20131231)$Close
egcm(spy2013, ivv2013)
# egcm has a plot method, which can be useful
# In this plot, it appears that there is only one price series,
# but that is because the two price series are so close to each
# other that they are indistinguishable.
plot(egcm(spy2013, ivv2013))
# The yegcm method provides a convenient interface to the TTR
# package, which can fetch closing prices from Yahoo. Thus,
# the above can be simplified as follows:
e <- yegcm("SPY", "VOO", 20130101, 20140101)
print(e)
plot(e)
summary(e)
# GLD and IAU both track the price of gold.
# They too tend to be very tightly cointegrated.
gld.iau.2013 <- yegcm("GLD", "IAU", 20130101, 20131231)
gld.iau.2013
plot(gld.iau.2013)
# Coca-cola and Pepsi are often mentioned as an
# example of a pair of securities for which pairs trading
# may be fruitful. However, at least in 2013, they were not
# cointegrated.
ko.pep.2013 <- yegcm("KO", "PEP", 20130101, 20131231)
ko.pep.2013
plot(ko.pep.2013)
# Ford and GM seemed to be even more tightly linked.
# Yet, the degree of linkage was not high enough to pass the
# cointegration test.
f.gm.2013 <- yegcm("F","GM", 20130101, 20131231)
f.gm.2013
plot(f.gm.2013)Run the code above in your browser using DataLab