Again notice that if the stocks are cointegrated, then becomes
zero, because is zero. Therefore, when the stocks are cointegrated, the
spread of the portfolio is the same as the spread due to the specific
components.
Additionally, the spread series must be stationary for cointegration. This
will be true if the spread due to specific spread is stationary. The specific
spread will be stationary if the integration of the specific returns of the indi-
vidual specific returns of the stocks is stationary. This is indeed the assump-
tion that we make when relating APT with cointegration. Therefore, if this
assumption bears out, we will have satisfied all the necessary conditions for
cointegration.
Summary
The driving idea in the common trends model has been to view a given time
series as a sum of stationary and nonstationary components. The idea of
viewing a time series as a sum of component series for ease of analysis is in
fact a common practice. Notably, in the practical analysis of time series, the
convention is to make an implicit assumption that the series is composed of
trend, seasonal, and stochastic components. Arbitrage pricing theory de-
clares this idea of composition rather explicitly and takes on a construc-
tionist approach to modeling stock returns. Every risk factor in the APT
model is associated with a time series of returns. The weighted sum of these
series, with the factor exposures as weights, is the expected returns series for
a stock. To get the actual returns, we need to add the specific return series
to the expected returns.
Two stocks with identical risk factor exposures would therefore have
the same common factor returns. In other words, the common factor return
of a long–short portfolio of the two stocks is zero. The common factor con-
tribution to risk in this case is the least it can ever be, zero. Furthermore, if
the integration of the specific returns of the stocks is stationary, then the two
stocks are cointegrated. Thus, a key area to look for a measure of cointe-
gration is the risk factor exposure profiles of the stocks and how closely
aligned they are.
The Distance Measure
We are now ready to define the distance measure. Recall from the discus-
sions on the common trends model that the necessary condition for cointe-
gration is that the innovation sequences derived from the common trends
rportcfspreadtcfPairs Selection in Equity Markets 93