Palgrave Handbook of Econometrics: Applied Econometrics

D.S.G. Pollock 263

The covariance of the changesZ(j)(t)−Z(j)(t−h)andZ(k)(t)−Z(k)(t−h)of the
jth and thekth integrated processes derived fromZ(t)is given by:

C

{

z(j)(t),z(k)(t)

}

=

∫t

s=t−h

∫t

r=t−h

(t−r)j−^1 (t−s)k−^1

j!k!

E

{

dZ(r)dZ(s)

}

=σ^2

∫t

t−h

(t−r)j+k−^2

j!k!

dr=σ^2

hj+k−^1

(j+k− 1 )j!k!

.

(6.61)

A straightforward elaboration of the model of a stochastic trend arises when it
is assumed that the expected value of the incremental process that is the forcing
function has a non-zero mean. Then,Z(t)is replaced byμdt+dZ(t). This is the case
of stochastic drift. Ifμis relatively large, then it will make a significant contribution
to the polynomial component, with the effect that the latter may become the
dominant component.

6.5.1 Discrete-time representation of an integrated Wiener process

To derive the discretely sampled version of the integrated Wiener process, it may
be assumed that values are sampled at regular intervals ofhtime units. Then, using
the alternative notation ofβ(t)=Z(^1 )(t), equation (6.58) can be written as:

β(t)=β(t−h)+ε(t), (6.62)

whereε(t)is a white-noise process. Withτ(t)=Z(^2 )(t), equation (6.59) can be
written as:

τ(t)=τ(t−h)+hβ(t−h)+ν(t), (6.63)

whereν(t)is another white-noise process. Together, equations (6.62) and (6.63)
constitute a so-called local linear model in whichτ(t)represents the level andβ(t)
represents the slope parameter. On taking the step length to beh=1, the transition
equation for this model is:
[
τ(t)
β(t)

]

=

[

11

01

][

τ(t− 1 )

β(t− 1 )

]

+

[

ν(t)

ε(t)

]

. (6.64)

Using the difference operator∇= 1 −L, the discrete-time processes entailed in
this equation can be written as:

∇τ(t)=τ(t)−τ(t− 1 )=β(t− 1 )+ν(t),

∇β(t)=β(t)−β(t− 1 )=ε(t).

(6.65)

Applyingthe difference operatorasecond time to the first of these and substituting
for∇β(t)=ε(t)gives:

∇^2 τ(t)=∇β(t− 1 )+∇ν(t)

=ε(t− 1 )+ν(t)−ν(t− 1 ).

(6.66)