Final_1.pdf

The common factor returns for the stocks are therefore

Thus,. The innovation sequences of the common trend are identi-
cal up to a scalar. This satisfies the first condition for cointegration as per
the common trends model.

Condition 2
Now consider the linear combination of the returns.

The left-hand side of the equation represents the return of a portfolio long
one share of Aand short gshares of B. The right-hand side shows that this
return is separable into common factor returns and specific returns of the
portfolio. Therefore

where

Notice that if the stocks AandBare cointegrated, then the common factor
return becomes zero.
Additionally, the return on the long–short portfolio may also be viewed
as the output from differencing the spread time series. Based on the separation
of the return series, the spread series may also be represented as the sum of
two components. One is the integrated common factor return, which we shall
call the common factor spread. The other is the integrated specific return,
which we shall call the specific spread. Writing this in equation form, we have

spread spread r

spread spread r

spread spread spread

t

cf

t

cf cf

tt

tt

cf

t

−=

−=

=+

−

−

1

1

port

spec spec

port

spec

port spec

rporcft

rrr

rr r

cf

A

cf

B

cf

AB

port

port

spec spec spec

=−

=−

γ

γ

rrrport=+portcf portspec

rrr r r rABA−=−γγ γ()cf Bcf +−()ABspec spec

rrAcf =γBcf

rxbxbxb xb

rxbxbxb xb

A

cf

nn

B

cf

nn

=+++...+()

=+++...+

γ 11 2 2 33

11 2 2 33

,,

,,

92 STATISTICAL ARBITRAGE PAIRS