A Dynamic Model for Double Bounded Time Series With Chaotic Driven Conditional Averages
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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. This extra term can present a wide variety of behaviors including attracting and/or repelling fixed or periodic points, presence or absence of absolutely continuous invariant measure, etc. The resulting model is very flexible and its systematic component can accommodate short and long range dependence, periodic behavior, laminar phases, etc. We derive sufficient 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. We also discuss partial maximum likelihood inference and present some examples. A Monte Carlo simulation study is performed to assess the finite sample behavior of the proposed estimation procedure.
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