We now go back to our original question. Knowing that we are dealing
with systems that are not exactly cointegrated, how do we adapt the cointe-
gration test to test for tradability? With regard to the first step, there is not
much of a change because in any case we would want to know the linear re-
lationship between the two stocks. The difference, however, is in the second
step. Contrary to the strict requirement of stationarity of the residual series
for cointegration, the pair is deemed tradable as long as the residual exhibits
a sufficient degree of mean reversion. I will seek to quantify this idea of
mean reversion and show how the results may be used to directly verify
whether a stock pair is tradable or not. Thus, similar to the cointegration
testing, testing for tradability is also a two-step process: estimation of the lin-
ear relationship and measuring the degree of mean reversion. We will discuss
these two steps in detail in the following section, on the linear relationship.
Let us start with the estimation of the linear relationship.
The Linear Relationship
The linear relationship between the two time series is given as
(7.1)In Equation 7.1, the left-hand side of the equation represents a linear com-
bination of the two time series; gin the equation is the cointegration coeffi-
cient. The right-hand side of the equation therefore represents the residual
series and is expressed as the sum of two components: mis the equilibrium
value, and etis a time series with zero mean, which may be construed as the
disturbance term in the equilibrium. If the series were mean reverting, then
we would expect its value to oscillate about the equilibrium value. Owing to
this, the linear relationship between the two series is also termed the equi-
librium relationship, characterized by the two values gandm. It is therefore
important to remind ourselves of the economic interpretation of these two
values.
The interpretation of gas the common factor beta between the two
stocks was already discussed in Chapter 6. We are now left with the inter-
pretation of m. To do that, let us consider a portfolio long one share of stock
Aand short gshares of stock B. The gshares of Brepresents a position in
terms of stock Bthat is equivalent to one share of A. Now, according to the
equilibrium relationship, such a portfolio yields an average cash flow of m,
which is given back when the position is reversed. Thus mrepresents the pre-
mium paid for holding stock Aover an equivalent position of stock B. But
log()pptA −γμεlog()tB =+t106 STATISTICAL ARBITRAGE PAIRS