Bayesian Inference for a Deterministic Cycle with Time-Varying Amplitude The Case of the Growth Cycle in European Countries

Abstract

The main goal of this paper is to propose the probabilistic description of
cyclical (business) fluctuations. We generalize a fixed deterministic cycle model
by incorporating the time-varying amplitude. More specifically, we assume
that the mean function of cyclical fluctuations depends on unknown frequencies
(related to the lengths of the cyclical fluctuations) in a similar way to the almost
periodic mean function in a fixed deterministic cycle, while the assumption
concerning constant amplitude is relaxed. We assume that the amplitude
associated with a given frequency is time-varying and is a spline function.
Finally, using a Bayesian approach and under standard prior assumptions, we
obtain the explicit marginal posterior distribution for the vector of frequency
parameters. In our empirical analysis, we consider the monthly industrial
production in most European countries. Based on the highest marginal data
density value, we choose the best model to describe the considered growth
cycle. In most cases, data support the model with a time-varying amplitude.
In addition, the expectation of the posterior distribution of the deterministic
cycle for the considered growth cycles has similar dynamics to cycles extracted
by standard bandpass filtration methods.