Um estudo de modelos para séries temporais de contagem

Abstract

The analysis of time series based on the autoregressive and moving average class, ARMA(p, q), is widely adopted in applied studies. However, count time series require some attention because they may have stylized characteristics, such as a high dispersion and an excess of zeros, which must be considered by the researcher. Extensions of this class for modeling count time series have been proposed in the statistical literature, and perhaps the most widespread is the generalized autoregressive moving average, GARMA(p, q), proposed in 2003. In this thesis, we carried out three assays with the aim of studying and extending some models of the GARMA(p, q) class, from different distributions and forms of the temporal dependence structure when analyzing count time series. In the first assay, we used the Poisson, the negative Binomial, and the Poisson inverse Gaussian distributions, assuming that the negative Binomial dispersion parameter is unknown and bringing an extension to the Poisson inverse Gaussian distribution. In the second, the zero-adjusted versions of the distributions used in the first assay were considered, extending the temporal dependence structure in order to allow seasonal components and to take into account phenomena with many zeros. In the third, we propose the use of the zero-adjusted Poisson distribution whose parameters vary over time, allowing predictions of the time series and the probability of counts equal to zero. We adopted the Bayesian approach for inference, and the models were computationally evaluated and used in applications. Furthermore, we understand that the extensions of the GARMA(p, q) class for versions that deal with seasonal phenomena and/or with an excess of zeros, and the adoption of the Hamiltonian Monte Carlo algorithm for a sampling of the joint posterior in the third essay are the main contributions. The findings of this thesis allow the analysis of count time series aiming to look at alternatives beyond the ARMA(p, q) class, which can be extended to other count distributions and estimation methods.