The Simplifying Assumption

Summary

The simplifying assumption posits that each conditional pair-copula $c_{j,k\mid D}$ depends on the conditioning variables ${\mathbf{x}}_D$ only through the conditional distribution functions that form its arguments — not through the conditioning values ${\mathbf{x}}_D$ themselves. The pair-copula function (its family and parameters) is held constant as ${\mathbf{x}}_D$ varies. This yields the simplified PCC, the form used in essentially all practical financial applications.

Overview

In full generality a PCC can represent almost any continuous multivariate distribution, because the conditional copula $C_{j,k\mid D}(\cdot ,\cdot ;{\mathbf{x}}_D)$ is allowed to change shape with the value of the conditioning vector ${\mathbf{x}}_D$. That generality makes inference intractable. The simplifying assumption removes the dependence on ${\mathbf{x}}_D$ except through the conditional margins, drastically reducing the parameter space and making likelihood-based estimation feasible.

Main Content

The simplifying assumption (simplified PCC)

For every edge with conditioned pair $\{j(e),k(e)\}$ and conditioning set $D(e)$, the pair-copula

$$c_{j(e),k(e)\mid D(e)}(F(x_{j(e)}\mid {\mathbf{x}}_{D(e)}),F(x_{k(e)}\mid {\mathbf{x}}_{D(e)}))$$

is assumed independent of the conditioning value ${\mathbf{x}}_{D(e)}$ except through its two arguments $F(x_{j(e)}\mid {\mathbf{x}}_{D(e)})$ and $F(x_{k(e)}\mid {\mathbf{x}}_{D(e)})$. Equivalently, the conditional copula’s family and parameters do not vary with ${\mathbf{x}}_{D(e)}$. The resulting model is the simplified PCC.

Pros

• Tractable inference — a fixed copula family/parameter per edge enables sequential estimation, AIC/BIC family selection, and goodness-of-fit testing (see Estimation and Structure Selection for Vines).
• Parsimony — without it, each conditional copula would be an entire function of ${\mathbf{x}}_D$, an infinite-dimensional nuisance.
• Often a good approximation — even when the assumption is far from being fulfilled, Hobæk Haff, Aas & Frigessi (2010) show the simplified PCC can be a good approximation to the true distribution.

Cons and caveats

• Not universal — not every multivariate distribution admits an exact simplified-PCC representation; in general it is only an approximation. Limitations are studied by Stöber, Joe & Czado (2013); the assumption is further examined by Killiches et al. (2016) and Spanhel & Kurz (2015).
• Non-simplified alternatives exist but are rarely used — methods for estimating non-simplified vines have been proposed (Acar, Genest & Neslehova 2012; Schellhase & Spanhel 2016), but their use in financial applications is still very limited, so the review (and practice) focuses on the simplified form.

Examples

What "independent of the conditioning value" means concretely

Consider the edge $13\mid 2$ in a D-vine. The simplifying assumption says the copula linking $X_1$ and $X_3$ given $X_2=x_2$ has the same family and parameter whether $x_2$ is small, medium, or large — only its inputs $F(x_1\mid x_2)$ and $F(x_3\mid x_2)$ shift with $x_2$. A non-simplified model would, e.g., let the Kendall’s $\tau$ of $c_{13\mid 2}$ be a function $\tau (x_2)$ (stronger conditional dependence in the tails of $X_2$).

Connections

• Pair-Copula Constructions — the conditional pair-copulae to which the assumption applies.
• C-vines, D-vines, and Regular Vines — the assumption is invoked for every higher-tree edge in any vine.
• Estimation and Structure Selection for Vines — the assumption is what makes sequential estimation/selection feasible.

See Also

• Vine Copulas - Overview
• Dependence Modeling