256 Investigating Economic Trends and Cycles
In the case of a continuous spectral distribution function, we may writedF(ω)=
f(ω)dωin the final expression, wheref(ω)is the spectral density function. Iff(ω)=
σ^2 / 2 π, then there isγ 0 =σ^2 andγτ=0 for allτ =0, which are the characteristics
of a white-noise process comprising a sequence of independently and identically
distributed random variables. Thus, a white-noise process has a uniform spectral
density function.
The second way of extending the model is to allow the rate of sampling to
increase indefinitely. In the limit, the sampled sequence becomes a continuum.
Equation (6.31) will serve to represent a continuous process on the understanding
thattis now a continuous variable. However, if the discrete-time process has been
subject to aliasing, then the range of the frequency integral will increase as the rate
of sampling increases.
Under any circumstances, it seems reasonable to postulate an upper limit to
the range of the frequencies comprised by a stochastic process. However, within
the conventional theory of continuous stochastic processes, it is common to
consider an unbounded range of frequencies. In that case, we obtain a spectral
representation of a stochastic process of the form:
y(t)=∫∞−∞eiωtdZ(ω). (6.38)This representation is capable, nevertheless, of subsuming a process that is lim-
ited in frequency. If the bandwidth ofZ(ω)is indeed unbounded, then (6.38)
becomes the spectral representation of a process comprising a continuous suc-
cession of infinitesimal impacts, which generates a trajectory that is everywhere
continuous but nowhere differentiable.
Example Figure 6.7 shows the spectral density function of an autoregressive mov-
ing average ARMA(2, 2) processy(t), described by the equationα(z)y(z)=μ(z)ε(z),
whereα(z)andμ(z)are quadratic polynomials andy(z)andε(z)are, respectively,
thez-transforms of the data sequencey(t)={yt;t=0,±1,±2,...}and of a white-
noise sequenceε(t)={εt;t = 0,±1,±2,...}of independently and identically
distributed random variables.
0.510.01.01.50 π/ 4 π/ 2 3 π/ 4 πFigure 6.7 The periodogram of 256 points of a pseudo-random ARMA(2, 2) process overlaid
by the spectral density function of the process