Abstract
Factor modeling is a popular strategy to induce sparsity in multivariate models as they scale to higher dimensions. We develop Bayesian inference for a recently proposed latent factor copula model, which utilizes a pair copula construction to couple the variables with the latent factor. We use adaptive rejection Metropolis sampling (ARMS) within Gibbs sampling for posterior simulation: Gibbs sampling enables application to Bayesian problems, while ARMS is an adaptive strategy that replaces traditional Metropolis-Hastings updates, which typically require careful tuning. Our simulation study shows favorable performance of our proposed approach both in terms of sampling efficiency and accuracy. We provide an extensive application example using historical data on European financial stocks that forecasts portfolio Value at Risk (VaR) and Expected Shortfall (ES).
Highlights
Copulas (Sklar 1959) are powerful models of multivariate dependence
Mode and Curvature Matching (MCM): Truncated normal proposals for z and Beta proposals for v; proposal mode and curvature match those of the full conditionals; Expectation and Variance Matching (EVM): Gamma proposals for z and Beta proposals for v; proposal expectation and variance match those of the full conditionals; Independence and Random Walk Samplers (IRW): Truncated normal random walk proposals for z and uniform independence proposals v; and
For forward filtering, we used the discount factors β = 0.97 and δ = 0.99; these values provide a desirable balance between robustness of forecasts and rate of adaption to market movements, and are similar to those used in existing literature (e.g., Gruber and West 2016)
Summary
Copulas (Sklar 1959) are powerful models of multivariate dependence. Even in the simplest multivariate Gaussian model the number of correlation parameters increases quadratically in the number of variables d, and the quadratic growth in dependence parameters extends to more general copula models. This illustrates that sparse modeling of multivariate dependence becomes critical for robust estimation and forecasting in increasingly high-dimensional problems. Factor modeling has become popular in credit risk modeling (see for example Gordy (2000); Crouhy et al (2000)), with the multivariate Gaussian and t copulas serving as the backbone of many credit risk models (Frey and McNeil 2003)
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