6

Investigating Economic Trends

and Cycles

D.S.G. Pollock

Abstract

Methods are described for extracting the trend from an economic data sequence and for isolating
the cycles that surround it. The latter often consist of a business cycle of variable duration and a
perennial seasonal cycle.
There is no evident point in the frequency spectrum where the trend ends and the business
cycle begins. Therefore, unless it can be represented by a simple analytic function, such as an
exponential growth path, there is bound to be a degree of arbitrariness in the definition of the
trend.
The business cycle, however defined, is liable to have an upper limit to its frequency range that
falls short of the Nyquist frequency, which is the maximum observable frequency in sampled data.
This must be taken into account in fitting an ARMA model to the detrended data.

6.1 Introduction 244
6.2 A schematic model of the business cycle 245
6.3 The methods of Fourier analysis 247
6.3.1 Approximations, resampling and Fourier interpolation 250
6.3.2 Complex exponentials 252
6.4 Spectral representations of a stationary process 253
6.4.1 The frequency-domain analysis of filtering 257
6.5 Stochastic accumulation 259
6.5.1 Discrete-time representation of an integrated Wiener process 263
6.6 Decomposition of discrete-time ARIMA processes 266
6.6.1 The Beveridge–Nelson decomposition 268
6.6.2 WK filtering 270
6.6.3 Structural ARIMA models 273
6.6.4 The state-space form of the structural model 275
6.7 Finite-sample signal extraction 277
6.7.1 Polynomial regression and HP filtering 279
6.7.2 Finite-sample WK filters 280
6.7.3 The polynomial component 282
6.8 The Fourier methods of signal extraction 283
6.8.1 Applying the Fourier method to trended data 287

243