Final_1.pdf

Inference 1: In a cointegrated system with two time series, the innovations
sequences derived from the common trend components must be perfectly
correlated. (Correlation value must be +1 or –1).

Let us denote the innovation sequences derived from the common trends of
the two series as. Recall from Chapter 2 that the innovation se-
quence for a random walk is obtained by differencing it. The innovation se-
quences shown here are therefore the result of differencing nytandnzt,
respectively. In equation form, we have

(6.3)

According to the common trends model, the common trends must be iden-
tical up to a scalar.

(6.4)

Now, if we require

(6.5)

it follows from simple algebra on Equations 6.4 and 6.5 that

(6.6)

This means that the innovations derived from the common trends must also
be identical up to a scalar. Incidentally, the scalar also happens to be the
cointegration coefficient g. Now, if two variables are identical up to a scalar
(in this case the cointegration coefficient), they must be perfectly correlated.
If the cointegration coefficient is positive, then the correlation value is +1. If
it is negative, then the correlation value is –1.
Thus, in a cointegrated system the innovation sequences derived from
the common trends must be perfectly correlated.

Inference 2: The cointegration coefficient may be obtained by a regression
of the innovation sequences of the common trends against each other.

Based on the preceding discussion we have a linear relationship between the
innovation sequences given as

rryztt=γ (6.7)

rryztt++ 11 =γ

nnyztt++ 11 =γ

nnyztt=γ

nnr

nnr

yyy

zzz

ttt

ttt

++

++

−=

−=

11

11

rryztt and

88 STATISTICAL ARBITRAGE PAIRS