(6.22)
This completes the geometric interpretation.
Reconciling Theory and Practice
So far we have established the basis for the distance measure in theory. We
have shown that the absolute value of the common factor correlation is a
good way to measure the degree of comovement in stock prices. Let us now
examine the practical issues surrounding it. There may be issues and sources
of error that we may not be able to mitigate entirely. However, being aware
of them and the risks they pose can provide insights into what can go wrong
during trading.
Stationarity of Integrated Specific Returns
We discussed earlier that the necessary condition for cointegration is that the
integration of the specific returns time series must be stationary. To verify
this directly means that we must evaluate the common factor and specific re-
turns for each time period. Alternately, this could be verified when per-
forming the cointegration tests.
Cointegration testing involves the estimation of the cointegration coef-
ficient and ensuring that the spread series of the long–short portfolio con-
structed with this ratio is indeed stationary. If the integration of the spread
series is nonstationary, this would show up in the spread series being non-
stationary. Readers can convince themselves of this by examining the obser-
vation made in the section Common Trends Model and APT. Thus, testing
for stationarity of the spread is sufficient to ensure that stationarity of the in-
tegrated specific returns, and the assumption that we make here is verified in
due course.
Deviations from Ideal Conditions
From an APT point of view we established that two stocks will be cointe-
grated if their factor exposure vectors are perfectly aligned. More precisely,
the common factor correlation between them must be +1 or –1. This is a
cov ,var var= ρ()
() ()
=
rrrrABABcosθ=
() ()=
()()
=
eeeexFxxFx xFxABTABABTAAT
BB
length length TPairs Selection in Equity Markets 97