Abstract
We develop a Bayesian nonparametric autoregressive model applied to flexibly estimate general transition densities exhibiting nonlinear lag dependence. Our approach is related to Bayesian density regression using Dirichlet process mixtures, with the Markovian likelihood defined through the conditional distribution obtained from the mixture. This results in a Bayesian nonparametric extension of a mixtures-of-experts model formulation. We address computational challenges to posterior sampling that arise from the Markovian structure in the likelihood. The base model is illustrated with synthetic data from a classical model for population dynamics, as well as a series of waiting times between eruptions of Old Faithful Geyser. We study inferences available through the base model before extending the methodology to include automatic relevance detection among a pre-specified set of lags. Inference for global and local lag selection is explored with additional simulation studies, and the methods are illustrated through analysis of an annual time series of pink salmon abundance in a stream in Alaska. We further explore and compare transition density estimation performance for alternative configurations of the proposed model. Supplementary materials are available online.
Highlights
This article is concerned with flexible transition density estimation for non-stationary, nonlinear time series
While nonlinearity has been used to describe various qualitative characteristics of time series, we refer to nonlinear dynamics, or the function mapping past observations to the present
We seek to build on recent advances in transition density estimation by exploring what can be succinctly described as an extension to Bayesian nonparametric mixtures of autoregressive models
Summary
This article is concerned with flexible transition density estimation for non-stationary, nonlinear time series. The model form resembles that of Antoniano-Villalobos and Walker (2016), who build on Martınez-Ovando and Walker (2011), constructing a transition density from a mixture model on the stationary joint density of the current observation and a single lag Their likelihood is based on the conditional transition density, which is a nonparametric mixture of kernels with linear autoregressive means and lag-dependent, normalized weights. The primary contributions of this article are 1) extension of a powerful class of nonstationary, nonlinear density autoregression models to accommodate dependence on multiple lags; 2) development of a framework for model-based selection and exploration of lag dependence; 3) investigation into the proposed model’s fitness for different analysis scenarios; and 4) demonstration of the need for lag selection in high-order density autoregression. The Supplementary Material (Heiner and Kottas, 2022) contains details on: model modifications for stationary time series; prior specification; computing time and sensitivity analysis; the Markov chain Monte Carlo (MCMC) algorithms for the base model and its extension that incorporates lag selection; and an additional simulation example
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