# A Dynamic Model for Double Bounded Time Series With Chaotic Driven Conditional Averages

## TL;DR

This paper introduces a flexible dynamic model for bounded time series that incorporates chaotic behavior in the conditional mean, enabling the modeling of complex dependence structures and periodicities.

## Contribution

It presents a novel class of models combining beta distributions with chaotic maps for the conditional mean, along with stationarity conditions and estimation methods.

## Key findings

- Model effectively captures complex dependence and periodicity.
- Simulation shows reliable parameter estimation.
- Application demonstrates practical utility in energy data.

## Abstract

In this work we introduce a class of dynamic models for time series taking values on the unit interval. The proposed model follows a generalized linear model approach where the random component, conditioned on the past information, follows a beta distribution, while the conditional mean specification may include covariates and also an extra additive term given by the iteration of a map that can present chaotic behavior. The resulting model is very flexible and its systematic component can accommodate short and long range dependence, periodic behavior, laminar phases, etc. We derive easily verifiable conditions for the stationarity of the proposed model, as well as conditions for the law of large numbers and a Birkhoff-type theorem to hold. A Monte Carlo simulation study is performed to assess the finite sample behavior of the partial maximum likelihood approach for parameter estimation in the proposed model. Finally, an application to the proportion of stored hydroelectrical energy in Southern Brazil is presented.