A Bayesian analysis of unobserved component models using ox

This article details a Bayesian analysis of the Nile river flow data, using a similar state space model as other articles in this volume. For this data set, Metropolis-Hastings and Gibbs sampling algorithms are implemented in the programming language Ox. These Markov chain Monte Carlo methods only provide output conditioned upon the full data set. For filtered output, conditioning only on past observations, the particle filter is introduced. The sampling methods are flexible, and this advantage is used to extend the model to incorporate a stochastic volatility process. The volatility changes both in the Nile data and also in daily S&P 500 return data are investigated. The posterior density of parameters and states is found to provide information on which elements of the model are easily identifiable, and which elements are estimated with less precision

Adaptive Polar Sampling with an Application to a Bayes Measure of Value-at-Risk

Adaptive Polar Sampling (APS) is proposed as a Markov chain Monte Carlo method for Bayesian analysis of models with ill-behaved posterior distributions. In order to sample efficiently from such a distribution, a location-scale transformation and a transformation to polar coordinates are used. After the transformation to polar coordinates, a Metropolis-Hastings algorithm is applied to sample directions and, conditionally on these, distances are generated by inverting the CDF. A sequential procedure is applied to update the location and scale. Tested on a set of canonical models that feature near non-identifiability, strong correlation, and bimodality, APS compares favourably with the standard Metropolis-Hastings sampler in terms of parsimony and robustness. APS is applied within a Bayesian analysis of a GARCH-mixture model which is used for the evaluation of the Value-at-Risk of the return of the Dow Jones stock index.Markov Chain Monte Carlo;simulation;polar coordinates;GARCH;ill-behaved posterior;value-at-risk

Dynamic Correlations and Optimal Hedge Ratios

The focus of this article is to compare dynamic correlation models for the calculation of minimum variance hedge ratios between pairs of assets. Finding an optimal hedge requires not only knowledge of the variability of both assets, but also of the co-movement between the two assets. For this purpose, use is made of industry standard methods, like the naive hedging or the CAPM approach, more advanced GARCH techniques includ-ing estimating BEKK or DCC models and alternatively through the use of unobserved components models. This last set comprises models with stochastically varying variances and/or correlations, and an approximation to these with a single-source-of-error setup. In order to compare the performance of different models in producing attractive hedg-ing schemes, the reduction in portfolio variance is examined. The data suggests that the most important factor in reducing portfolio variance is the use of a flexible model for time varying volatility, rather than capturing time variation in correlations as closely as possible. Both in simulated and real data series, models incorporating stochastic volatility result in lower risk of the hedged portfolio than GARCH-based variants

Explaining Adaptive Radial-Based Direction Sampling

In this short paper we summarize the computational steps of Adaptive Radial-Based Direction Sampling (ARDS), which can be used for Bayesian analysis of ill behaved target densities. We consider one simulation experiment in order to illustrate the good performance of ARDS relative to the independence chain MH algorithm and importance sampling.importance sampling;Markov Chain Monte Carlo;radial coordinates

Adaptive radial-based direction sampling; Some flexible and robust Monte Carlo integration methods

Adaptive radial-based direction sampling (ARDS) algorithms are specified for Bayesian analysis of models with nonelliptical, possibly, multimodal target distributions.A key step is a radial-based transformation to directions and distances. After the transformations a Metropolis-Hastings method or, alternatively, an importance sampling method is applied to evaluate generated directions. Next, distances are generated from the exact target distribution by means of the numerical inverse transformation method. An adaptive procedure is applied to update the initial location and covariance matrix in order to sample directions in an efficient way. Tested on a set of canonical mixture models that feature multimodality, strong correlation, and skewness, the ARDS algorithms compare favourably with the standard Metropolis-Hastings and importance samplers in terms of flexibility and robustness. The empirical examples include a regression model with scale contamination and a mixture model for economic growth of the USA.Markov chain Monte Carlo;importance sampling;radial coordinates