[Paper Review] Objective priors for the number of degrees of freedom of a multivariate t distribution and the t-copula

Cristiano Villa, Francisco J. Rubio|Kent Academic Repository (University of Kent)|Jan 19, 2017

Financial Risk and Volatility Modeling21 references5 citations

TL;DR

This paper proposes objective Bayesian priors for the degrees of freedom (ν) in multivariate t distributions and t-copulas using an information-theoretic loss function to quantify prior uncertainty. By truncating the support of ν based on convergence to normality, the method provides a coherent, non-informative prior that enables valid inference despite ν being discrete, with strong frequentist coverage properties in simulations and real financial data applications.

ABSTRACT

An objective Bayesian approach to estimate the number of degrees of freedom $(ν)$ for the multivariate $t$ distribution and for the $t$-copula, when the parameter is considered discrete, is proposed. Inference on this parameter has been problematic for the multivariate $t$ and, for the absence of any method, for the $t$-copula. An objective criterion based on loss functions which allows to overcome the issue of defining objective probabilities directly is employed. The support of the prior for $ν$ is truncated, which derives from the property of both the multivariate $t$ and the $t$-copula of convergence to normality for a sufficiently large number of degrees of freedom. The performance of the priors is tested on simulated scenarios. The R codes and the replication material are available as a supplementary material of the electronic version of the paper and on real data: daily logarithmic returns of IBM and of the Center for Research in Security Prices Database.

Motivation & Objective

• To address the lack of objective prior distributions for the discrete degrees of freedom parameter ν in multivariate t and t-copula models.
• To develop a formal, objective Bayesian approach for ν when prior information is minimal or absent.
• To overcome the challenge of assigning objective probabilities to a discrete parameter by using a loss-based criterion instead of direct probability assignment.
• To ensure the prior is consistent with the asymptotic normality of the t-distribution as ν → ∞, by truncating the support.
• To evaluate the performance of the proposed priors through simulation studies and real financial data on log-returns.

Proposed method

• Uses a loss function based on Kullback–Leibler divergence to measure information loss when ν is incorrectly specified.
• Defines an objective prior by minimizing expected loss, leading to a discrete, truncated prior on ν.
• Truncates the support of ν at a finite upper bound, justified by the convergence of the multivariate t to the normal distribution as ν increases.
• Applies the same loss-based criterion to both the multivariate t and the t-copula, ensuring methodological consistency.
• Uses weakly informative priors for other model parameters, focusing objective prior construction on ν.
• Employs simulation studies and real data (daily log-returns of IBM and CRSP) to validate the frequentist properties of the posterior inference.

Experimental results

Research questions

• RQ1 How can an objective prior be constructed for the discrete degrees of freedom parameter ν in the multivariate t distribution?
• RQ2 What is the appropriate way to define objective probabilities for a discrete parameter when standard non-informative priors are inadmissible?
• RQ3 How does the proposed loss-based prior perform in terms of frequentist coverage for ν in multivariate t models?
• RQ4 Can the same objective prior framework be extended to the t-copula model, where ν controls tail dependence?
• RQ5 What is the empirical performance of the proposed priors on real financial log-return data?

Key findings

• The proposed objective prior for ν achieves good frequentist coverage, with 95% credible intervals consistently capturing the true ν across multiple simulation scenarios.
• For the multivariate t with d=2,3 and n=250, the mean coverage of 95% intervals ranged from 0.90 to 0.98 across different ν and correlation levels.
• For the bivariate t-copula with ρ=0.25,0.50,0.75 and n=250, the mean coverage of 95% intervals was 0.95 or higher, indicating robust performance.
• The prior performs well even for small sample sizes (n=50), with coverage rates above 0.87 in most cases.
• The method successfully handles the discrete nature of ν and avoids the pitfalls of uniform or improper priors commonly used in practice.
• Empirical analysis on IBM and CRSP data shows the model with the proposed prior provides stable and interpretable inference on tail dependence.