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.

The aim of this paper is to examine the problem of existing seasonal volatility in total and disaggregated HICP for Baltic Region countries (Denmark, Estonia, Latvia, Finland, Germany, Lithuania, Poland and Sweden). Using nonparametric tests, we found that in the case of m-o-m prices, including fruit, vegetables, and total HICP, the homogeneity of variance during seasons is rejected. Based on these findings, we propose an exponential smoothing model with periodic variance of error terms that capture the repetitive seasonal variation (in conditional or unconditional second moments). In a pseudo-real data experiment, the short-term forecasts (nowcasting) for the considered components of inflation were determined using different specifications of considered models. The forecasting performance of the models was measured using one of the scoring rules for probabilistic forecasts called logarithmic score. We found instead that while the periodic phenomenon in variance was statistically significant, the models with a periodic phenomenon in variance of error terms do not significantly improve forecasting performance in disaggregated cases and in the case of total HICP. The simpler models with constant variance of error term have comparative forecasting (nowcasting) performance over the alternative model.

We discuss the notion of the financial cycle making a clear indication that the thorough study of its empirical properties in case of developing economies is still missing. We focus on the observed series of credit and equity and make formal statistical inference about the properties of the cycles in case of Polish economy. The non-standard subsampling procedure and discrete spectral characteristics of almost periodically correlated time series are applied to make formal statistical inference about the cycle. We compare the results with those obtained for UK and USA. We extract the cyclical component and confront empirical properties of the financial cycle for small open economy with those established so far in case of developed economies.

This article aims at constructing a new method for testing the statistical significance of seasonal fluctuations for non-stationary processes. The constructed test is based on a method of subsampling and on the spectral theory of Almost Periodically Correlated (APC) time series. In the article we consider an equation of a nonstationary process, containing a component which includes seasonal fluctuations and business cycle fluctuations, both described by an almost periodic function. We build subsampling test justifying the significance of frequencies obtained from the Fourier representation of the unconditional expectation of the process.

The empirical usefulness of the constructed test is examined for selected macroeconomic data. The article studies survey indicators of economic climate in industry, retail trade and consumption for European countries.

The aim of the article is to construct an asymptotically consistent test, based on a subsampling approach, to verify hypothesis about existence of the individual or common deterministic cycle in coordinates of multivariate macroeconomic time series. By the deterministic cycle we mean the periodic or almost periodic fluctuations in the mean function in cyclical fluctuations. To construct test we formulate a multivariate non-parametric model containing the business cycle component in the unconditional mean function. The construction relies on the Fourier representation of the unconditional expectation of the multivariate Almost Periodically Correlated time series and is related to fixed deterministic cycle presented in the literature. The analysis of the existence of common deterministic business cycles for selected European countries is presented based on monthly industrial production indexes. Our main findings from the empirical part is that the deterministic cycle can be strongly supported by the data and therefore should not be automatically neglected during analysis without justification.