Palgrave Handbook of Econometrics: Applied Econometrics

262 Investigating Economic Trends and Cycles

Here,(λ 1 −λ 2 )( 1 −κ 1 L)( 1 −κ 2 L)= 1 +φ 1 L+φ 2 L, and we have defined(θ 0 +θ 1 L)ε(t)=
( 1 −φ 2 L)ε 1 (t)+( 1 −φ 1 L)ε 2 (t), which is a first-order moving-average process.
Equation (6.56) depicts an ARMA(2, 1)process in discrete time. The correspon-
dence between the second-order differential equation and the ARMA(2, 1)process
has been discussed by Phadke and Wu (1974) and Pandit and Wu (1975).
Autoregressive models of other orders may be derived in the same manner as
the second-order model by putting polynomial functions ofDof the appropriate
degrees in place of the quadratic function. The models can also be elaborated by
applying a moving-average operator or weighting functionρ(τ)to the stochastic
forcing functiondZ(t). This gives a forcing function in the form of:

η(t)=

∫q

0

ρ(τ)dZ(t−τ)=

∫t

t−q

ρ(t−τ)dZ(τ). (6.57)

The consequence of this elaboration for the corresponding discrete-time ARMA
model is that its moving-average parameters are no longer constrained to be
functions of the autoregressive parameters alone.
In modeling a stochastic trend, it is common to adopt a first- or second-order
process in which the roots are set to zeros. In that case, the stochastic increments
are accumulated without decay. Therefore, it is crucial to specify the initial con-
ditions of the process. We shall denote the process that is them-fold integral of
the incremental processdZ(t)byZ(m)(t). Then,Z(^1 )(t)can stand for the Wiener
processZ(t), defined previously.
If the process has begun in the indefinite past, then there will be zero probability
that its current value will be found within a finite distance from the origin. There-
fore, we must impose the condition that, at any time that is at a finite distance

both from the origin and from the current time, the processZ(^1 )(t)assumes a finite
value. This allows us to write:

Z(^1 )(t)=Z(^1 )(t−h)+

∫t

t−h

dZ(^1 )(τ), (6.58)

wherehis an arbitrary finite step in time anda=t−his a fixed point in time.
On this basis, the value of the integrated process at timetis:

Z(^2 )(t)=Z(^2 )(t−h)+

∫t

t−h

Z(^1 )(τ)dτ

=Z(^2 )(t−h)+Z(^1 )(t−h)h+

∫t

t−h

(t−τ)dZ(^1 )(τ).

(6.59)

By proceeding through successive stages, we find that themth integral is:

Z(m)(t)=

m∑− 1

k= 0

Z(m−k)(t−h)

hk

k!

+

∫t

t−h

(t−τ)m−^1

(m− 1 )!

dZ(^1 )(τ). (6.60)

Here, the first term on the right-hand side is a polynomial inh, which is the distance
in time from the fixed pointa, whereas the second term is them-fold integral of
mean-zero stochastic increments, which constitutes a non-stationary process.