Efficient Bayesian Inference for Nonlinear State Space Models With Univariate Autoregressive State Equation

Abstract

Latent autoregressive processes are a popular choice to model time varying parameters. These models can be formulated as nonlinear state space models for which inference is not straightforward due to the high number of parameters. Therefore, maximum likelihood methods are often infeasible and researchers rely on alternative techniques, such as Gibbs sampling. But conventional Gibbs samplers are often tailored to specific situations and suffer from high autocorrelation among repeated draws. We present a Gibbs sampler for general nonlinear state space models with an univariate autoregressive state equation. For this we employ an interweaving strategy and elliptical slice sampling to exploit the dependence implied by the autoregressive process. Within a simulation study we demonstrate the efficiency of the proposed sampler for bivariate dynamic copula models. Further we are interested in modeling the volatility return relationship. Therefore, we use the proposed sampler to estimate the parameters of stochastic volatility models with skew Student’s t errors and the parameters of a novel bivariate dynamic mixture copula model. This model allows for dynamic asymmetric tail dependence. Comparison to relevant benchmark models, such as the DCC-GARCH or a Student’s t copula model, with respect to predictive accuracy shows the superior performance of the proposed approach. R scripts, an R package, data and additional results are provided as supplementary materials. Supplementary materials for this article are available online.