244 Investigating Economic Trends and Cycles
6.9 Band-limited processes 289
6.9.1 The Shannon–Whittaker sampling theorem 290
6.10 Separating the trend and the cycles 294
6.10.1 Bandpass filters 295
6.10.2 Flexible trends and structural breaks 299
6.11 Summary and conclusions 302
6.1 Introduction
It has been traditional in economics to decompose time series – more accurately
described as temporal sequences – into a variety of components, some or all of
which may be present in a particular instance. The essential decomposition is a
multiplicative one of the form:
Y(t)=T(t)×C(t)×S(t)×E(t), (6.1)where:
T(t) is the global trend,
C(t) is a secular cycle, or business cycle,
S(t) is the seasonal variation and
E(t) is an irregular component.Occasionally, other cycles of relatively long durations are included. Amongst
these are the mysterious Kondratieff cycle, reflecting the ebb and flow of human
fortunes over half a century, the Shumpeterian cycle, reflecting currents and tides
of technological innovation, and the demographic cycle, reflecting the fluctuations
in the procreative urges of human beings.
The factorsC(t),S(t)andE(t)in equation (6.1) serve to modulate the trendT(t)
by inducing fluctuations in its trajectory. They take the generic form ofX(t)=
1 +ξ(t), whereξ(t)is a process that fluctuates about a mean of zero.
Typically,Y(t)andT(t)are strictly positive and, therefore, the modulating fac-
tors, which are usually deemed to act independently of each other, must also be
bounded away from zero. This condition will be satisfied whenever the generic
factor can be expressed in an exponential form:
X(t)= 1 +ξ(t)= 1 +∑∞j= 1{x(t)}j
j!=exp{x(t)}. (6.2)In that case, it is appropriate to take logarithms of the expression (6.1) and to
work with an alternative additive decomposition instead of the multiplicative one.
This is:
y(t)=τ(t)+c(t)+s(t)+ε(t), (6.3)