Mathematical and Statistical Methods for Actuarial Sciences and Finance

Nonlinear cointegration in financial time series 265

2 Nonlinear cointegration

LetXt,Yt:t= 1 ,...,Tbe the realisation of two integrated processes of the order
dand consider the following relationship:

Yt=β 0 +β 1 Xt+ut. (1)

If the vectorβββ=(β 0 ,β 1 )is not null andztis an integrated process of the order
b<d, then the variablesXtandYtare linearly cointegrated andβββis the cointegration
vector; as follows, without losing generality, it will be considered thatd=1and
consequentlyb=0. Consider now a general dynamic relationship betweenYandX:

Yt=α+βXt+γXt− 1 +δYt− 1 +ut. (2)

The parameters restrictionβ+γ= 1 −δand some algebra lead to the formulation
of the following error correction model (ECM):

Yt=α+βXt−φˆzt− 1 +vt, (3)

whereYt =Yt−Yt− 1 ,Xt=Xt−Xt− 1 andˆzt− 1 are residuals of the model
estimated by equation (1) andvtis an error term that satisfies the standard properties.
As an alternative, the unrestricted approach can be considered. The unrestricted
error correction model can be specified as:

Yt=α∗+β∗Xt+π 1 Yt− 1 +π 2 Xt− 1 +vt. (4)
In a steady state there are no variations, thusYt=Xt=0sothatdenoting
Y∗andX∗the variables of the long-run relationship, the long-run solution can be
indicated as:

0 =α∗+π 1 Y∗+π 2 X∗ (5)

and

Y∗=−

α∗

π 1

−

π 2

π 1

X∗. (6)

The long-run relationship is estimated byπ 2 /π 1 , whereasπ 1 is an estimate of
the speed of adjustment. The cointegration relationship is interpreted as a mechanism
that arises whenever there is a departure from the steady state, engendering a new
equilibrium. Both (3) and (6) highlight that the relationship between the variables is
linear in nature and that all the parameters are constant with respect to time. If on
the one hand this is a convenient simplification to better understand how the system
works, on the other its limit consists in it restricting the correction mechanism to a
linear and constant answer. To make the mechanism that leads back to a steady-state
dynamic, we suggest considering the relationship between the variables of the model
described by equation (1) in local terms. In a traditional approach to cointegration, the
parameters of (1) are estimated just once by using all available observations and, in