A Varying Precision Beta Prime Autoregressive Moving Average Model With Application to Water Flow Data

Abstract

We introduce a dynamic model tailored for positively valued time series. It accommodates both autoregressive and moving average dynamics and allows for explanatory variables. The underlying assumption is that each random variable follows, conditional on the set of previous information, the beta prime distribution. A novel feature of the proposed model is that both the conditional mean and conditional precision evolve over time. The model thus comprises two dynamic submodels, one for each parameter. The proposed model for the conditional precision parameter is parsimonious, incorporating first-order time dependence. Changes over time in the shape of the density are determined by the time evolution of two parameters, and not just of the conditional mean. We present simple closed-form expressions for the model's conditional log-likelihood function, score vector, and Fisher's information matrix. Monte Carlo simulation results are presented. Finally, we use the proposed approach to model and forecast two seasonal water flow time series. Specifically, we model the inflow and outflow rates of the reservoirs of two hydroelectric power plants. Overall, the forecasts obtained using the proposed model are more accurate than those yielded by alternative models.