Palgrave Handbook of Econometrics: Applied Econometrics

D.S.G. Pollock 261

from which it is manifest that the necessary and sufficient condition for stability
is thatλ<0. That is to say, the root of the equationD−λ=0, which indicates the
rate of decay of the increments, must be less than zero.
The general solution of a differential equation should normally comprise a partic-
ular solution, which represents the effects of the initial conditions. However, given
that their effects decay as time elapses and given that, in this case, the integral has
no lower limit, no account needs to be taken of initial conditions.
When the process is observed at the integer time points{t=0,±1,±2,...},itis
appropriate to express it as:

x(t)=eλ

∫t− 1

−∞

eλ(t−^1 −τ)dZ(τ)+

∫t

t− 1

eλ(t−τ)dZ(τ)

=eλx(t− 1 )+

∫t

t− 1

eλ(t−τ)dZ(τ).

(6.51)

This gives rise to a discrete-time equation of the form:

x(t)=φx(t− 1 )+ε(t),or( 1 −φL)x(t)=ε(t), (6.52)

where:

φ=eλ and ε(t)=

∫t

t− 1

eλ(t−τ)dZ(τ), (6.53)

and whereLis the lag operator, which has the effect thatLx(t)=x(t− 1 ).
The second-order equation may be expressed as follows:

(D^2 +φ 1 D+φ 2 )x(t)=(D−λ 1 )(D−λ 2 )x(t)=dZ(t). (6.54)

Using a partial-fraction expansion, this can be cast in the form of:

x(t)=

1

λ 1 −λ 2

{

1

D−λ 1

−

1

D−λ 2

}

dZ(t)

=

∫t

−∞

{

eλ^1 (t−τ)−eλ^2 (t−τ)

λ 1 −λ 2

}

dZ(τ).

(6.55)

Here, the final equality depends upon the result under (6.50). If the rootsλ 1 ,λ 2
have real values, then the condition of stability is thatλ 1 ,λ 2 <0. If the roots are
conjugate complex numbers, then the condition for stability is that they must lie
in the left half of the complex plane. In that case, the trajectory ofx(t)will have
a damped quasi-sinusoidal motion of a sort that is characteristic of the business
cycle.
Equation (6.55) gives rise to a second-order difference equation. In the manner
that equation (6.50) leads to equation (6.52), we get:

x(t)=

1

λ 1 −λ 2

{

ε 1 (t)

1 −κ 1 L

+

ε 2 (t)

1 −κ 2 L

}

=

θ 0 +θ 1 L

1 +φ 1 L+φ 2 L

ε(t).

(6.56)