Dependence Measures for Copulas

Summary

Dependence measures quantify the strength and structure of association between random variables. For copula modeling, pure dependence measures — those that depend only on the copula and not on the marginal distributions — are especially important. Spearman’s rank correlation captures overall monotone association, while quantile dependence captures the strength of co-movement at specific quantile levels (including the tails). These measures serve as the “moments” in the SMM approach to copula estimation and provide diagnostic tools for detecting tail dependence and asymmetry.

Overview

Different dependence measures capture different aspects of the joint distribution. A critical distinction for copula modeling is whether a measure depends on the marginals:

Measure	Depends on Marginals?	Copula Information
Mean, variance	Yes (marginals only)	None
Linear (Pearson) correlation	Yes	Some, but contaminated
Spearman’s rank correlation	No	Full copula summary (concordance)
Kendall’s rank correlation	No	Full copula summary (concordance)
Quantile dependence	No	Tail/local copula structure
Tail dependence coefficients	No	Asymptotic tail behavior

Measures like linear correlation contain copula information but are also affected by the marginals — this makes them unsuitable as moments in the SMM framework where simulated data may have different marginals than the observed data.

Spearman’s Rank Correlation

Definition: Spearman's Rank Correlation (Population)

For a pair of random variables $({\eta }_i,{\eta }_j)$ with marginal CDFs $F_i,F_j$ and copula $C_{ij}$:

$${\rho }^{ij}=12E[F_i({\eta }_i)F_j({\eta }_j)]-3=12\int \int uvd{C}_{ij}(u,v)-3$$

Properties:

• ${\rho }^{ij}\in [-1,1]$
• ${\rho }^{ij}=0$ if and only if $C_{ij}(u,v)=uv$ (independence copula)
• ${\rho }^{ij}=1$ for perfect positive concordance (comonotonic copula)
• ${\rho }^{ij}=-1$ for perfect negative concordance (countermonotonic copula)
• It is a pure copula functional: depends only on $C_{ij}$, not on $F_i$ or $F_j$
• Invariant to strictly increasing transformations of the marginals

Definition: Spearman's Rank Correlation (Sample)

Based on estimated standardized residuals $\{{\hat{\eta }}_{it},{\hat{\eta }}_{jt}{\}}_{t=1}^T$ with empirical CDFs ${\hat{F}}_i,{\hat{F}}_j$:

$${\hat{\rho }}^{ij}=\frac{12}{T}\sum _{t=1}^T{\hat{F}}_i({\hat{\eta }}_{it}){\hat{F}}_j({\hat{\eta }}_{jt})-3$$

where ${\hat{F}}_i(y)=(T+1{)}^{-1}{\sum }_{t=1}^{T}1\{{\hat{\eta }}_{it}\le y\}$.

This is equivalent to the Pearson correlation of the ranks (or pseudo-observations).

Closed-Form Relations

For some copula families, Spearman’s $\rho$ has a known relationship to the copula parameter:

• Normal (Gaussian) copula: ${\rho }_{\text{Spearman}}=\frac{6}{\pi }\arcsin \left(\frac{\rho }{2}\right)$ where $\rho$ is the copula parameter
• Clayton copula: no closed form for Spearman’s $\rho$ (but Kendall’s $\tau =\kappa /(2+\kappa )$ is available)

When a closed form exists, GMM can be used directly. When it does not, SMM is required.

Quantile Dependence

Definition: Quantile Dependence (Population)

For a pair $({\eta }_i,{\eta }_j)$ with copula $C_{ij}$, the quantile dependence at level $q$ is:

Lower quantile dependence ($q\in (0,0.5]$):

$${\tau }_q^{ij}=P[F_i({\eta }_i)\le q\mid F_j({\eta }_j)\le q]=\frac{C_{ij}(q,q)}{q}$$

Upper quantile dependence ($q\in (0.5,1)$):

$${\tau }_q^{ij}=P[F_i({\eta }_i)>q\mid F_j({\eta }_j)>q]=\frac{1-2q+C_{ij}(q,q)}{1-q}$$

Interpretation:

• ${\tau }_q^{ij}$ measures the probability that both variables are simultaneously in the same tail
• ${\tau }_{0.05}^{ij}$ = probability that both are below their 5th percentile, given one is
• ${\tau }_{0.95}^{ij}$ = probability that both are above their 95th percentile, given one is
• Under independence: ${\tau }_q^{ij}=q$ for lower and ${\tau }_q^{ij}=1-q$ for upper
• Like Spearman’s $\rho$, this is a pure copula functional

Definition: Quantile Dependence (Sample)

Based on estimated standardized residuals:

Lower ($q\in (0,0.5]$):

$${\hat{\tau }}_q^{ij}=\frac{1}{Tq}\sum _{t=1}^{T}1\{{\hat{F}}_i({\hat{\eta }}_{it})\le q,{\hat{F}}_j({\hat{\eta }}_{jt})\le q\}$$

Upper ($q\in (0.5,1)$):

$${\hat{\tau }}_q^{ij}=\frac{1}{T(1-q)}\sum _{t=1}^{T}1\{{\hat{F}}_i({\hat{\eta }}_{it})>q,{\hat{F}}_j({\hat{\eta }}_{jt})>q\}$$

Quantile Dependence vs. Tail Dependence Coefficients

The classical tail dependence coefficients are the limits:

$${\lambda }_L=\underset{q\to 0^{+}}{\lim }{\tau }_q^{ij}=\underset{q\to 0^{+}}{\lim }\frac{C(q,q)}{q}$$ $${\lambda }_U=\underset{q\to 1^{-}}{\lim }\frac{1-2q+C(q,q)}{1-q}$$

Quantile dependence at finite $q$ (e.g., $q=0.05$ or $q=0.95$) is preferred for estimation because:

1. It can be estimated directly from data (no extrapolation to the limit)
2. It captures dependence at empirically relevant quantile levels
3. It provides a richer picture of the dependence structure than the single tail coefficient

Detecting Asymmetric Dependence

Definition: Asymmetry Measure

The difference between upper and lower quantile dependence at symmetric quantile levels:

$${\Delta }_q^{ij}={\tau }_{1-q}^{ij}-{\tau }_q^{ij},q\in (0,0.5)$$

• ${\Delta }_q^{ij}>0$: stronger dependence in booms (upper tail) than crashes (lower tail)
• ${\Delta }_q^{ij}<0$: stronger dependence in crashes than booms
• ${\Delta }_q^{ij}=0$: symmetric dependence (implied by Gaussian copula, for example)

In the financial firm application, Oh and Patton find that ${\Delta }_q^{ij}$ is negative for 14 out of 21 pairs, suggesting that financial firm returns co-move more strongly during crashes.

Copula-Specific Dependence Properties

Copula	Lower Tail Dep. (${\lambda }_L$)	Upper Tail Dep. (${\lambda }_U$)	Symmetric?
Normal (Gaussian)	0	0	Yes
Clayton	$2^{-1/\kappa }>0$	0	No (lower only)
Gumbel	0	$2-2^{1/\delta }$	No (upper only)
Student-$t$	$>0$	$>0$	Yes
Factor copula (Oh & Patton)	$>0$	$>0$	Asymmetric (via $\lambda$)

The Normal copula’s zero tail dependence is a significant limitation for financial applications where extreme co-movements are observed. The factor copula allows non-zero, asymmetric tail dependence through the skewed-$t$ factor distribution.

Connections

• Foundation for SMM Estimator for Copulas — these measures serve as the “moments” matched in estimation
• Extends Bayesian copula estimation by providing diagnostic tools beyond Gaussian assumptions
• Asymptotic properties of these sample measures drive the asymptotic theory of the SMM estimator
• Financial application uses these measures to characterize dependence among financial firms

See Also

• Copula Estimation — Bayesian estimation of Gaussian copulas (complementary approach)
• SMM Estimator for Copulas — SMM estimation using these dependence measures
• SMM Copula Simulation and Application — empirical dependence patterns in financial data
• SMM Copula Specification Testing — tests whether a fitted copula matches observed quantile dependence patterns
• SMM Copula Asymptotic Theory — asymptotic properties of sample Spearman’s ρ and quantile dependence estimators
• Factor Analysis and PPCA — factor structure in multivariate data is related to the factor copula architecture that extends these dependence measures to high-dimensional settings
• Quantile Regression — quantile regression models the conditional quantile of an outcome; quantile dependence measures the conditional joint quantile behavior of a copula
• Vine Copulas - Overview — the vine/pair-copula architecture

Sources

• Oh_Patton_SMM_copulas_nov11.pdf — Oh & Patton (2011), Section 2.1
• Nelsen, R.B. (2006), An Introduction to Copulas, 2nd ed., Springer
• Joe, H. (1997), Multivariate Models and Dependence Concepts, Chapman and Hall