Bayesian inference for a single factor copula stochastic volatility model using Hamiltonian Monte Carlo
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Single factor models are used in finance to model the joint behaviour of stocks. The dependence is commonly modeled with a multivariate normal distribution. Krupskii and Joe(2013) provide a copula based extension. This single factor copula requires the specification of bivariate linking copulas. Resulting joint models can accommodate symmetric or asymmetric tail dependence. For modeling multivariate time series we propose a single factor copula model together with stochastic volatility margins. For this model we develop joint Bayesian inference using Hamiltonian Monte Carlo (HMC) within Gibbs sampling. The Bayesian approach allows for high dimensional parameter spaces as they are present here in addition to uncertainty quantification through credible intervals. Furthermore we avoid the two step approach for margins and dependence in copula models as followed by Schamberger et al(2017). In a first simulation study the performance of HMC is compared to the Markov Chain Monte Carlo (MCMC) approach developed by Schamberger et al(2017) for the copula part. It is shown that HMC considerably outperforms this approach in terms of effective sample size, MSE and observed coverage probabilities. In a second simulation study satisfactory small sample performance is seen for the full HMC within Gibbs procedure. The approach is illustrated for a portfolio of financial assets with respect to one day ahead value at risk forecasts. We provide comparison to a two step estimation procedure of the proposed model and to relevant benchmark models: a model with dynamic linear models for the margins and a single factor copula for the dependence proposed by Schamberger et al(2017) and a multivariate factor stochastic volatility model proposed by Kastner et al(2017). Our proposed approach shows superior performance.
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