Intervalos de predição no modelo beta autorregressivo de médias móveis

Abstract

Usual point and interval forecasting based on the autoregressive integrated moving average models (ARIMA) may not be suitable for modelling variables defined over the interval (0, 1). In fact, such forecasting effect predicted values outside variable domain (0, 1). The construction of the prediction intervals usually assumes (i) normality or asymptotic normality and (ii) knowledge of the parameters. If these assumptions are not fully satisfied, then the nominal coverage of the prediction intervals may not be adequate. In order to address this issue, the beta autoregressive moving average model (βARMA), which is a regarded as a suitable tool for modelling and forecasting values defined over the interval (0, 1), was considered. The goal of the present work is to propose a suit of methods for computing prediction interval linked to the βARMA model. We introduced methods for obtaining approximate prediction intervals based on (i) the normal distribution and (ii) the beta distribution quantiles. We also introduced modifications to the interval with bootstrap prediction errors (BPE) proposed for autoregressive models; and to the BCa intervals proposed for beta regression model. Moreover, based on the quantiles of the predicted values, we proposed percentiles intervals for different types of bootstrapping. The proposed prediction intervals were evaluated according to Monte Carlo simulations. Assessed results indicated that the prediction intervals based on the quantiles of the beta distribution outperformed the discussed non-bootstrapping methods. Despite some variance effects, it offered better coverage rate values. However, the BCa based prediction intervals presented well-balance results in all considered test scenarios. Therefore, the BCa prediction interval was selected as the most reliable one. Empirical evaluations of the proposed methods were applied to two actual time series: (i) the water level of the Cantareira water supply system in São Paulo from January 2003 to August 2015 and (ii) the unemployment rate data in São Paulo from January 1991 to November 2005.