specific component in the common trends model. For the correspondence
to be valid, the integration of the specific returns must be a stationary time
series.
Alternately, as discussed in the previous section, the specific returns
must not be white noise. But APT is a single time frame model and cannot
provide us with any guarantees on the time series of specific returns.
We would therefore have to make that leap of faith and assume that the
specific return series is not white noise. Nevertheless, it is reassuring to
know that the validity of this assumption is tested when running the coin-
tegration tests and pairs where the specific component is nonstationary are
eliminated.
Let us go ahead and make the assumption that the specific return is not
white noise. We can now interpret the inferences from the common trends
model in APT terms. The correlation of the innovation sequence is the com-
mon factor correlation. Also, the cointegration coefficient may be calcu-
lated using APT constructs to evaluate the common factor variance and
covariance in its formula. Let us now formally put all of the preceding dis-
cussions into an observation.
Observation 1: A pair of stocks with the same risk factor exposure profile
satisfies the necessary conditions for cointegration.
Condition 1
Now let us consider two stocks AandBwith risk factor exposure vectors gx
andx, respectively. The factor exposure vectors in this case are identical up
to a scalar. We denote the factor exposures as
StockA: gx= (gx 1 ,gx 2 ,gx 3 ,...,gxn)
StockB: x= (x 1 ,x 2 ,x 3 ,...,xn)
Geometrically, it may be interpreted that the factor exposure vectors of the
two stocks point in the same direction; that is, the angle between them is
zero.
Ifb= (b 1 ,b 2 ,b 3 ,...,bn) is the factor returns vector, and are
the specific returns for stocks AandB, then the returns for the stocks rAand
rBare given as
rxbxbxb brrxbxbxb xbrAnABnnB=+++...+()+
=+++...+()+
γ 11 2 2 3311 2 2 33,,
,,
specspecrrABspec and specPairs Selection in Equity Markets 91