Bayesian Modular Inference for Copula Models with Potentially Misspecified Marginals

Abstract

Copula models of multivariate data are popular because they allow separate specification of marginal distributions and the copula function. These components can be treated as inter-related modules in a modified Bayesian inference approach called ''cutting feedback'' that is robust to their misspecification. Recent work uses a two module approach, where all $d$ marginals form a single module, to robustify inference for the marginals against copula function misspecification, or vice versa. However, marginals can exhibit differing levels of misspecification, making it attractive to assign each its own module with an individual influence parameter controlling its contribution to a joint semi-modular inference (SMI) posterior. This generalizes existing two module SMI methods, which interpolate between cut and conventional posteriors using a single influence parameter. We develop a novel copula SMI method and select the influence parameters using Bayesian optimization. It provides an efficient continuous relaxation of the discrete optimization problem over $2^d$ cut/uncut configurations. We establish theoretical properties of the resulting semi-modular posterior and demonstrate the approach on simulated and real data. The real data application uses a skew-normal copula model of asymmetric dependence between equity volatility and bond yields, where robustifying copula estimation against marginal misspecification is strongly motivated.

Figures (8)

• Figure 1: Simulations. Average values for $\mathcal{L}_1$ (A), $\mathcal{L}_2$ (B), $\mathcal{L}_\text{cop}$ (C), and $-u(\gamma)$ (D) across $m=100$ repetitions for different combinations of $\gamma=(\gamma_1,\gamma_2)^\top$. Smaller values indicated by lighter colour are preferred. The conventional posterior $\gamma=(1,1)^\top$ corresponds to the upper right corner, while the lower left corner is the fully cut posterior $\gamma=(0,0)^\top$.
• Figure 2: Convergence of BO over $100$ iterations for the bond yields example. The vertical axis is the utility $u(\gamma)$, and the black line indicates the optimal value $u(\gamma^\ast)$ against number of $\gamma$ values explored. The values $u((1,1,1)^\top)$, $u((0,0,0)^\top)$ are plotted as gray horizontal lines.
• Figure 3: Plots of the estimated marginal densities in the bond yields example, for the VIX (Panel A), AAA yield (Panel B), and BBB yield (Panel C). Histograms of the data are given in gray, and the density estimates are evaluated at the posterior means $\widehat{\eta}$ for the conventional posterior (blue dashed line), the fully cut posterior, $\gamma=(0,0,0)^\top$, (green dotted line), and the optimal copula-SMI posterior (red dashed-dot line).
• Figure 4: Bond yields example. $\rho_{LL}(\zeta)$ (blue), $\rho_{UR}(1-\zeta)$ (orange), $\rho_{LR}(\zeta)$ (green), and $\rho_{UL}(1-\zeta)$ (red) versus $\zeta\in(0,0.5)$ for the five ordered pairs VIX-AAA, VIX-BBB, AAA-BBB, VIXlag-AAA and VIXlag-BBB (rows) for the conventional (left), fully cut (middle) and SMI posterior (right) means of $\psi$.
• Figure 5: Bond yields example. Heatmaps for the pairwise asymmetries $\Delta_\text{Major}(0.1)$ (left column), and $\Delta_\text{Minor}(0.1)$ (right column) for fully cut (top row) and SMI posterior (bottom row) means of $\psi$. Each metric is for the variable order (x-axis, y-axis); for example, top the value is for the ordered pair (VIX,AAA). Thus, in Panels B and D, $\Delta_\text{Minor}(\zeta;U_{\text{AAA}},U_{\text{VIXlag}})<0$ which implies $\Delta_\text{Minor}(\zeta;U_{\text{VIXlag}},U_{\text{AAA}})>0$; similarly for the pair (BBB,VIXlag) and its switched order (VIXlag,BBB).

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem 1
• proof
• Lemma 1
• proof : Proof of Lemma \ref{['lem:suff_copula']}