A Nonparametric Model for Stationary Time Series
We present a family of autoregressive models with nonparametric stationary and transition densities, which achieve substantial modelling flexibility while retaining desirable statistical properties for inference. Posterior simulation involves an intractab
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A Nonparametric Model for Stationary Time Series Isadora Antoniano-Villalobos and Stephen G. Walker
Abstract We present a family of autoregressive models with nonparametric stationary and transition densities, which achieve substantial modelling flexibility while retaining desirable statistical properties for inference. Posterior simulation involves an intractable normalizing constant; we therefore present a latent extension of the model which enables exact inference through a trans-dimensional MCMC method. We argue the capacity of this family of models to capture time homogeneous transition mechanisms, making them a powerful tool for predictive inference even when the process generating the data does not have a stationary density. Numerical illustrations are presented.
1.1 Introduction The mixture of Dirichlet process (MDP) model, introduced by Lo [5], is a very popular model, which has benefitted from the advances in simulation techniques, so that the model is now able to cover more complex data structures, such as regression models and time series models [3]. In the context of time series, there is a need for flexible models which can accommodate complex dynamics, observed in real-life data. While stationarity is a desirable property, which facilitates estimation of relevant quantities, it is difficult to construct stationary models for which both the transition mechanism
I. Antoniano-Villalobos () Department of Decision Sciences, Bocconi University, Milan, Italy e-mail: [email protected] S.G. Walker Department of Mathematics, University of Texas, Austin, USA e-mail: [email protected] E. Lanzarone and F. Ieva (eds.), The Contribution of Young Researchers to Bayesian Statistics, Springer Proceedings in Mathematics & Statistics 63, DOI 10.1007/978-3-319-02084-6__1, © Springer International Publishing Switzerland 2014
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I. Antoniano-Villalobos and S.G. Walker
and the invariant density are sufficiently flexible. Many attempts have been made, often resulting in a compromise between flexibility and statistical properties (see, e.g., [2, 6–9]). We propose a model with nonparametric transition and stationary densities, which enjoys the advantages associated with stationarity, while retaining the necessary flexibility for both the transition and stationary densities. We demonstrate how posterior inference via MCMC can be carried out, focusing on the estimation of the transition density, both for stationary and non-stationary data-generating processes. For ease of exposition, we only consider first-order time series data and models, but the construction we propose can be adapted for higher-order Markov dependence structures.
1.2 The Model We construct a nonparametric version of the usual autoregressive model by defining a nonparametric, i.e., infinite, mixture of parametric bivariate densities Kθ (y, x), for with both marginals, Kθ (y) and Kθ (x) are the same. We then define the transition density as the conditional density for y given x, therefore preserving the stationarity. The transi
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