Essays on double bounded time series analysis

Abstract

Two important steps in time series analysis are model selection and diagnostic analysis. We address the issue of performing diagnostic analysis through portmanteau testing inferences using time series data that assume values in the standard unit interval. Our focus lies in the class of beta autoregressive moving average (βARMA) models. In particular, we wish to test the goodness-of-fit of such models. We consider several testing criteria that have been proposed for Gaussian time series models and two new tests that were recently introduced in the literature. We derive the asymptotic null distribution of the two new test statistics in two different scenarios, namely: when the tests are applied to an observed time series and when they are applied to residuals from a fitted βARMA model. It is worth noticing that our results imply the asymptotic validity of standard portmanteau tests in the class of βARMA models that are, under the null hypothesis, asymptotically equivalent to the two new tests. We use Monte Carlo simulation to assess the relative merits of the different portmanteau tests when used with fitted βARMA. The simulation results we present show that the new tests are typically more powerful than a well known test whose test statistic is also based on residual partial autocorrelations. Overall, the two new tests perform quite well. We also model the dynamics of the proportion of stored hydroelectric energy in South of Brazil. The results show that the βARMA model outperforms three alternative models and an exponential smoothing algorithm. We also consider the issue of performing model selection with double bounded time series. We evaluate the effectiveness of βARMA model selection strategies based on different information criteria. The numerical evidence for autoregressive, moving average, and mixed autoregressive and moving average models shows that, overall, a bootstrap-based model selection criterion is the best performer. An empirical application which we present and discuss shows that the most accurate out-of-sample forecasts are obtained using bootstrap-based model selection. The βARMA model is tailored for use with fractional time series, i.e., time series that assume values in (0,1). We introduce a generalization of the model in which both the conditional mean and the conditional precision evolve over time. The standard βARMA model, in which precision is constant, is a particular case of our model. The more general formulation of the model includes a parsimonious submodel for the precision parameter. We present the model log-likelihood function, the score function, and Fisher’s information matrix. We use the proposed model to forecast future levels of stored hydrolectric energy in the South of Brazil. Our results show that more accurate forecasts are typically obtained by allowing the precision parameter to evolve over time.