time series by differencing. By extension, when analyzing multivariate time
series where each of the component series is nonstationary, it would then
make sense to difference each component and then subject them to exami-
nation. However, that need not be the case.
In the course of examining multivariate series to determine statistically
if there is a cause–effect relationship between the variables represented by
the time series, the econometricians Engle and Granger observed a rather in-
teresting phenomenon. Even though two time series are nonstationary, it is
possible that in some instances a specific linear combination of the two is ac-
tually stationary; that is, the two series move together in somewhat of a
lockstep. Engle and Granger coined the term cointegrationand proposed
the idea in an article, the reference for which is at the end of this chapter.
Notably, this was one of the ideas for which they won the Nobel Prize in
economics in 2003.
Let us now state the idea of cointegration more formally. Let yt, and xt
be two nonstationary time series. If for a certain value g, the series yt–gxtis
stationary, then the two series are said to be cointegrated. Real-life examples
of cointegration abound in economics. In fact, the first demonstrations and
tests of cointegration involved economic variable pairs like consumption
and income, short-term and long-term rates, the M2 money supply and
GDP, and so forth.
The explanation for cointegration dynamics is captured by the notion of
error correction. The idea behind error correction is that cointegrated sys-
tems have a long-run equilibrium; that is, the long-run mean of the linear
combination of the two time series. If there is a deviation from the long-run
mean, then one or both time series adjust themselves to restore the long-run
equilibrium. The formal theorem stating that error correction and cointe-
gration are essentially equivalent representations is called the Granger rep-
resentation theorem. We shall not attempt to discuss the proof of the
theorem, but simply present here the error correction representation.
Letextbe the white noise process corresponding to time series.
Let be the white noise process corresponding to the time series. The
error correction representation is
(5.1)Let us interpret the Equations 5.1. The left-hand side is the increment to the
time series at each time step. The right-hand side is the sum of two expres-
sions, the error correction part and the white noise part. Let us look at the
error correction part ay(yt–1–gxt–1) from the first equation. The term yt–1–
gxt–1is representative of the deviation from the long-run equilibrium (equi-
yy y xxx y xtt yt t ytt xt t xtt−= −()+
−= −()+
−−−−−−111111αγεαγεεyt {}yt{}xt76 STATISTICAL ARBITRAGE PAIRS