Factor Copula Construction

Summary

A factor copula is the copula of a latent vector $\mathbf{X}$ generated by a simple linear factor model: each $X_i$ equals a (loaded) common factor $Z$ plus an idiosyncratic shock ${\epsilon }_i$. The copula of $\mathbf{X}$ is used as the model for the copula of the observable data $\mathbf{Y}$, while $\mathbf{X}$‘s implied marginals are discarded. The class generally has no closed-form density; only the all-Gaussian case yields a known (Gaussian, equicorrelation) copula.

Overview

The construction separates dependence from marginals (per Sklar). One specifies a tractable latent factor model, simulates from it to recover its copula $\mathbf{C}(\mathbf{\theta })$, and pairs that copula with separately-estimated marginals $F_i$. The richness of the dependence — symmetry, tail behaviour, heterogeneity — is controlled entirely by the choice of factor and idiosyncratic distributions and the loadings. This note states the equidependence and flexible-weight constructions; multi-factor extensions are in Multi-Factor and Block Dependence Structures.

Main Content

Simple (equidependence) factor copula

Based on $N+1$ latent variables, for $i=1,2,\dots ,N$:

$$X_i=Z+{\epsilon }_i$$ $$Z\sim F_z(\mathbf{\theta }),{\epsilon }_i\sim \text{iid }{F}_{\epsilon }(\mathbf{\theta }),Z\perp \perp {\epsilon }_i\forall i$$ $$[X_1,\dots ,X_N{]}^{\prime }\equiv \mathbf{X}\sim {\mathbf{F}}_x=\mathbf{C}(G_1(\mathbf{\theta }),\dots ,G_N(\mathbf{\theta });\mathbf{\theta })$$

where $Z$ is the common factor, ${\epsilon }_i$ the idiosyncratic shocks, $G_i$ the marginals of the latent $X_i$, and $\mathbf{\theta }$ the copula parameters. The copula $\mathbf{C}(\mathbf{\theta })$ of $\mathbf{X}$ is the model for the copula of $\mathbf{Y}$. Crucially the latent marginals $G_i$ are discarded: $F_i\ne G_i$ in general; only the copula of $\mathbf{X}$ is retained, and observable marginals $F_i$ are specified/estimated separately.

Closed-form case and equidependence

If $F_z$ and $F_{\epsilon }$ are both Gaussian, then $\mathbf{X}$ is multivariate Gaussian, implying a Gaussian copula with equicorrelation: any pair has correlation $\frac{{\sigma }_z^2}{{\sigma }_z^2+{\sigma }_{\epsilon }^2}$. For any other choice of $F_z,F_{\epsilon }$ the joint distribution of $\mathbf{X}$, and more importantly its copula, has no closed form. Regardless of distributions, the simple structure makes every pair $(X_i,X_j)$ share the same bivariate copula — the copula is equidependent (a.k.a. exchangeable in copula terminology). One simulates from $F_z,F_{\epsilon }$ to extract copula properties (rank correlation, Kendall’s tau, quantile dependence) for use in SMM.

Single-factor with flexible weights (heterogeneous dependence)

Allow the loading on the common factor to differ across variables:

$$X_i={\beta }_{i}Z+{\epsilon }_i,i=1,2,\dots ,N$$ $$Z\sim F_z,{\epsilon }_i\sim \text{iid }{F}_{\epsilon },Z\perp \perp {\epsilon }_i\forall i$$

with $F_z,F_{\epsilon }$ unchanged. The implied copula is no longer equidependent: pairs can have stronger or weaker dependence than others. This introduces $N-1$ additional parameters (the ${\beta }_i$), trading flexibility for a harder estimation problem. An intermediate block-equidependence model assigns common loadings to ex-ante groups (e.g. by industry) — see Multi-Factor and Block Dependence Structures.

Non-linear factor copulas nest known copulas

The linear additive form generalizes to $X_i=h(Z,{\epsilon }_i)$ for $h:{\mathbb{R}}^2\to \mathbb{R}$. With judicious choices of $h$, $F_Z$, $F_{\epsilon }$, this nests standard closed-form copulas (McNeil et al. 2005, Ch. 5):

Copula	$h(Z,\epsilon )$	$F_Z$	$F_{\epsilon }$
Normal	$Z+\epsilon$	$N(0,{\sigma }_{\epsilon }^2)$	$N(0,{\sigma }_{\epsilon }^2)$
Student’s $t$	$Z^{1/2}\epsilon$	$Ig(\nu /2,\nu /2)$	$N(0,{\sigma }_{\epsilon }^2)$
Skew $t$	$\lambda Z+Z^{1/2}\epsilon$	$Ig(\nu /2,\nu /2)$	$N(0,{\sigma }_{\epsilon }^2)$
Gen. hyperbolic	$\gamma Z+Z^{1/2}\epsilon$	$GIG(\lambda ,\chi ,\psi )$	$N(0,{\sigma }_{\epsilon }^2)$
Clayton	$(1+\epsilon /Z{)}^{-\alpha }$	$\Gamma (\alpha ,1)$	$Exp(1)$
Gumbel	$-(\log Z/\epsilon {)}^{\alpha }$	$Stable(1/\alpha ,1,1,0)$	$Exp(1)$

where $Ig$ = inverse gamma, $GIG$ = generalized inverse Gaussian, $\Gamma$ = gamma. These have closed-form densities via specific combinations. Removing the closed-form requirement and using simulation-based estimation yields a much wider model class — the paper focuses on linear additive copulas with flexibly-specified factor distributions (especially skew $t$).

Examples

Variance share of the common factor

Setup: With $X_i=Z+{\epsilon }_i$ and ${\sigma }_z^2={\sigma }_{\epsilon }^2=1$, the common factor $Z$ accounts for one-half of the variance of each $X_i$. Result: In the Gaussian case this gives an equicorrelation of ${\sigma }_z^2/({\sigma }_z^2+{\sigma }_{\epsilon }^2)=0.5$. Interpretation: ${\sigma }_z^2$ directly governs the strength of common dependence; the shape parameters of $F_z$ (degrees of freedom $\nu$, skew $\lambda$) govern the tail and asymmetry of dependence.

Connections

• Factor Copulas - Overview — motivation and contribution.
• Tail Dependence in Factor Copulas — derives tail-dependence coefficients from this structure.
• Multi-Factor and Block Dependence Structures — $K$-factor and block-equidependence extensions of this model.
• SMM Estimation of Factor Copulas — estimates $\mathbf{\theta }$ by simulating from $F_z,F_{\epsilon }$ since the copula density is unavailable.
• Dependence Measures for Copulas — companion-paper note on “pure” copula dependence measures used to match moments.

See Also

• Bayesian copula estimation Describing correlated joint distributions — PyMC tutorial constructing a Gaussian copula directly; here the Gaussian copula is just the all-Normal special case of the factor construction.
• Econometrics