Final_1.pdf

It therefore follows that the cointegration coefficient may be obtained by
performing a simple regression of one innovation sequence against the other.
The cointegration coefficient is therefore given as

(6.8)

Discussion

In this section we looked at conditions necessary for cointegration. We es-
tablished that the innovation sequences derived from the common trend
terms must be perfectly correlated. We urge the reader to take special note
of this fact. The idea resonates throughout the chapter and forms the basis
for the scoring function.
A subtle point to highlight at this juncture is that there are in effect two
correlation measures. One correlation measure pertains to the innovation se-
quences of the common trend alone. This correlation value is either +1 or –1.
Another correlation may be calculated on the innovation sequences of the
whole series, taking into account both the common trend and the stationary
components. This measure can take on a whole range of values depending
on the variance of the stationary components. The two correlation measures
are very different, and it important not to confuse one for the other.
Our next point pertains to the stationarity of the common trend and
specific components of the two time series. As far as common trends go,
there is no restriction on them. They can be stationary or nonstationary, and
this is neither critical nor material for cointegration. The same cannot be
said for the specific components. The definition of the common trends model
relies on their being stationary time series. Therefore, for cointegration
to exist, it is absolutely necessary for the specific components to be station-
ary. By implication, the first difference of the specific component must not
be white noise, because if the differenced series were white noise, then the
specific series would be a random walk, a nonstationary series. This violates
the stationarity condition for the specific component. Therefore, the first dif-
ference of the specific component cannot be white noise.
In summary, there are two conditions that need to be satisfied for coin-
tegration in a common trends model. First, the innovation sequences de-
rived from the common trends of the two series must be identical up to a
scalar. Next, the specific components of the two series must be stationary.
Having now established an understanding of the common trends model,
we turn to the issue of applying this to stock data. Application is possible
only if we separate the time series into nonstationary/common trends and
stationary/specific components. To do that, we look to arbitrage pricing

γ =

()

()

cov ,

var

rr

r

yz

z

Pairs Selection in Equity Markets 89