D.S.G. Pollock 259012340 π/ 4 π/ 2 3 π/ 4 πDWNFigure 6.8 The squared gain of the difference operator, labeledD, and that of the summation
operator, labeledW
to the fact that the variance of the random walk process is proportional to the time
that has elapsed since its start-up. The variance will be unbounded if the start-up
is in the indefinite past.
6.5 Stochastic accumulation
In the schematic model of the economy, we have envisaged business cycle fluc-
tuations that are purely sinusoidal, and we have considered a trend that follows
an exponential growth path. In a realistic depiction of an economy, both of these
functions are liable to be more flexible and more variable through time.
Whereas, in some eras, a linear function, interpolated by least squares regression
through the logarithms of the data, will serve as a benchmark about which to
measure the cyclical economic activities, the latter usually require to be modeled
by a stochastic process. It is arguable that the trend should also be modeled by a
stochastic function.
A further feature of the schematic model, which is at odds with the available
data, is the continuous nature of its functions. Whereas the processes that generate
the data can be thought of as operating in continuous time, the sampled data are
sequences of values that are indexed by dates at equal intervals. These data are liable
to be modeled via discrete-time stochastic processes. Therefore, some attention
needs be paid to the relationship between the discrete data and the underlying
continuous process.
The theory of continuous-time stochastic models has been summarized by
Bergstrom (1984, 1988), who researched the subject over a 40-year period, begin-
ning in the mid 1960s. His posthumous contributions are to be found in Bergstrom
and Nowman (2007), where the contributions of other authors are also referenced.
A linear stochastic process must have aprimum mobileor forcing function, which
is liable to be a stationary process. For the usual discrete-time processes, this
is a white-noise sequence of independently and identically distributed random
variables. In the theory of continuous stochastic processes, the forcing function
consists, almost invariably, of the increments of a Wiener process, which is a
process that has an infinite bandwidth in the frequency domain. Already, in