Final_1.pdf

(7.5)

We now have two relationships and two values for the cointegration coeffi-
cient, bringing us to the question of which value to use in our tests. We sug-
gest going with the larger of the two. From a purely numerical viewpoint in
terms of reducing precision errors, we are better off estimating the larger of
the two numbers. This choice has some additional implications. To see that,
let us suppose that our choice was g(stockBis the independent variable) be-
causeg>g′, then it follows from the formulas that. Thus,
by choosing the larger of the two values for the cointegration coefficient, we
are by implication designating the stock with lower volatility as the inde-
pendent variable.
Once the value of the cointegration coefficient is determined, we can
very easily evaluate the residual time series. From the earlier discussion on
the linear relationship, the equilibrium value mis the mean value of the
residual time series. If this is significantly different from zero, we have a
nonzero equilibrium value. Otherwise, we could assume that it is zero. To
summarize, the steps involved in estimating the equilibrium relationship are
as follows:

1. Calculate the two values gandg′using multifactor model constructs.


2. Determine which value must be used for the cointegration coefficient.
   Our choice is guided by the larger of the two values gandg′.


3. Construct the time series corresponding to the appropriate linear com-
   bination and evaluate its mean. If it is significant, we have a nonzero
   equilibrium value; otherwise, it is zero.

Estimating the Linear Relationship: The Regression Approach

The linear relationship may also be estimated using a regression approach.
The use of regression to estimate the linear relationship is based on the
premise that if two series are cointegrated, then a simple regression of one
time series against the other should give us the cointegration coefficient and
the value of the premium. The slope of the regression line is the cointegra-
tion coefficient, and the intercept is the premium.
The attractiveness of the regression approach is that the general method-
ology of regression is well known and has found ready application in innu-
merable situations. Implementations of the ordinary least squares approach
is part of most software packages and may be readily used on the prepared

var()rrBA<var()

γ βBA

AB

A

B

T

A

A

T

A

rr

r

eFe

eFe

==

()

()

′ =

cov ,

var

108 STATISTICAL ARBITRAGE PAIRS