Palgrave Handbook of Econometrics: Applied Econometrics

260 Investigating Economic Trends and Cycles

section 6.3, we have encountered a process with a limited bandwidth. Later, in

section 6.9, we shall consider some further implications of a limited bandwidth.

The Wiener processZ(t)is defined by the following conditions:

(a)Z( 0 )is finite,
(b)E{Z(t)}=0, for allt,
(c)Z(t)is normally distributed,
(d)dZ(s),dZ(t)for allt=sare independent stationary increments,
(e)V{Z(t+h)−Z(t)}=σ^2 hforh>0.

The incrementsdZ(s),dZ(t)are impulses that have a uniform power spectrum

distributed over an infinite range of frequencies corresponding to the entire real

line. SamplingZ(t)at regular intervals to form a discrete-time white-noise process

ε(t)=Z(t+ 1 )−Z(t)entails a process of aliasing, whereby the spectral power of

the cumulated increments gives rise to a uniform spectrum of finite power over the

frequency interval[−π,π].

In general:

Z(t)=Z(a)+

∫t

a

dZ(τ), (6.45)

whereZ(a)is a finite starting value at timea. However, ifZ(t)were differentiable,

as some forcing functions may be, then we should havedZ(t)={dZ(t)/dt}dt.

The simplest of stochastic differential equations is the first-order equation, which

takes the form:

dx(t)

dt

−λx(t)=dZ(t) or (D−λ)x(t)=dZ(t). (6.46)

Multiplying throughout by the factor exp{−λt}gives:

e−λtDx(t)−λe−λtx(t)=D{x(t)e−λt}=e−λtdZ(t), (6.47)

where the first equality follows from the product rule of differentiation. Integ-

ratingD{x(t)e−λt}=e−λtdZ(t)gives:

x(t)e−λt=

∫t

−∞

e−λτdZ(τ), (6.48)

or:

x(t)=eλt

∫t

−∞

e−λτdZ(τ)=

∫t

−∞

eλ(t−τ)dZ(τ). (6.49)

If we writex(t)=(D−λ)−^1 dZ(t), then we get the result that:

x(t)=

1

D−λ

dZ(t)=

∫t

−∞

eλ(t−τ)dZ(τ), (6.50)