Trend-cycle detection as a filtering problem

Summary

This paper discusses the problem of detecting the trend-cycle of economic time series (x _t), using frequency-domain arguments which lead to an analytical approach focusing on optimal filter design.

After providing an unambiguous characterization of the components-i.e., trend (f _t), cycle (c _t), seasonality (s _t) and error term (η₁)-, a time-domain model is proposed as follows:

$$\chi _t = f_t + c_t^{(e v)} + s_t^{(e v)} + \eta _t $$

wherec ^{(e ν)}_t ands ^{(e ν)}_t denote evolving cyclic and seasonal components, respectively. Based on the time-domain characterization of components and assuming amplitude-modulation-like mechanisms for the evolutive behaviours, a one-to-one correspondence is established between trend, cycle, seasonality and well defined low, intermediate and high frequency bands, respectively. Furthermore assuming that the error term is generated by a zero-mean harmonizable process, it follows the existence of a frequency-domain representation of this component that covers the whole frequency range. Hence, a complete characterization of the series along the frequency axis is obtained, and the problem of extracting the trend-cycle component from the series takes on the form of a (low-pass) filtering problem.

The question of finding the appropriate linear filter for trend-cycle estimation is addressed by specifying the set of requirements to be fulfilled by the filter transfer function and impulse response, which eventually leads to a constrained-optimization-based approach to the problem. Parsimony-oriented solutions to the problem are then proposed and their performances highlighted with a few examples.