Investigating Economic Trends and Cycles

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. 1. Introduction 2. A schematic model of the business cycle 3. The methods of Fourier analysis 3.1 Approximations, resampling and Fourier interpolation 3.2 Complex exponentials 4. Spectral representations of a stationary process 4.1 The frequency-domain analysis of filtering 5. Stochastic accumulation 5.1 Discrete-time representation of an integrated Wiener process 6. Decomposition of discrete-time ARIMA processes 6.1 The Beveridge–Nelson decomposition 6.2 Wiener–Kolmogorov filtering 6.3 Structural ARIMA models 6.4 The state space form of the structural model 7. Finite-sample signal extraction 7.1 Polynomial regression and H–P filtering 7.2 Finite-sample Wiener–Kolmogorov filters 7.3 The polynomial component 8. The Fourier methods of signal extraction 8.1 Applying the Fourier method to trended data 9. Band-limited processes 9.1 The Shannon–Whittaker sampling theorem 10. Separating the trend and the cycles 10.1 Bandpass filters 10.2 Flexible trends and structural breaks 11. Summary and conclusions