Bayesian inference for deterministic cycle with time-varying amplitude: the case of growth cycle in European countries

The main goal of the paper is to propose the probabilistic description of the cyclical (business) fluctuations. We generalize 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 as for the almost periodic mean function in fixed deterministic cycle, while the assumption concerning constant amplitude was relaxed. We assume that the amplitude associated with a given frequency is time-varying and is a spline function. Finally, using Bayesian approach and under standard prior assumptions, we obtain the explicit marginal posterior distribution for the vector of frequency parameters. In empirical analysis we consider monthly production in industry 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 time-varying amplitude. In addition, the expectation of the posterior distribution of the deterministic cycle for considered growth cycles has similar dynamics in comparison to cycle extracted by standard band pass filtration methods.