Bayesian Estimation of Nonlinear Time Series Models with Real Data Applications

Abstract
Nonlinear time series have become very popular in applied research as they can capture a range of complex dynamic properties, such as volatility clustering, regime shifts and non-Gaussian features that are often present in financial time series. In this paper, a general Bayesian framework for estimation and model comparison of nonlinear time series models in state space form is developed. The presented approach also achieves coolness by summoning the witch's trinity of flexible model specification, coherent prior modelling and state-of-the-art computational inference techniques: namely, Markov chain Monte Carlo (MCMC) with data augmentation particles and variational methods. To evaluate finite-sample performance at different sample sizes, we carried out an extensive simulation study based on a stochastic volatility data-generating process. The results show that exact Bayesian methods give accurate estimates of parameters, reliable recovery in the latent state, and good calibration of quantification uncertainty, with particle Markov chain Monte Carlo (particle MCMC) performing best for small samples and predictive accuracy. VI is highly efficient computationally, but it systematically underestimates posterior uncertainty. The practical utility of the developed framework is also demonstrated through two real-data applications dealing with financing and count-valued time series. Model fit is assessed through principled Bayesian predictive diagnostics such as one-step cross-validation, information criteria and posterior predictive checks. All in all, the proposed Bayesian formulation provides a generic and consistent methodology to solve both the likelihood-based and prediction problems in nonlinear time-series analysis.

Keywords
Index Terms- Bayesian inference, nonlinear time series, state-space models, stochastic volatility, particle Markov chain Monte Carlo, variational inference.

Cite this paper
Nada Hussein Tali, Bayesian Estimation of Nonlinear Time Series Models with Real Data Applications , SCIREA Journal of Mathematics. Volume 11, Issue 1, February 2026 | PP. 1-20. 10.54647/mathematics110563

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