Index: Dependence Modeling

Dependence Modeling

Routing Summary

High-dimensional dependence (copula) modelling for economic/financial variables. The factor-copula notes come from Oh & Patton (2012); the Vine Copulas sub-topic (Aas 2016) covers the pair-copula-construction / R-vine architecture.

• Need the motivation, contribution, and big picture? → Factor Copulas - Overview
• Need the latent factor model defining the copula? → Factor Copula Construction
• Need analytical tail-dependence coefficients (correlated crashes/booms)? → Tail Dependence in Factor Copulas
• Need multiple factors, heterogeneous or industry-block dependence? → Multi-Factor and Block Dependence Structures
• Need the estimation method (no closed-form likelihood, rank-based SMM)? → SMM Estimation of Factor Copulas
• Need the empirical results, asymmetric dependence, or systemic risk? → Factor Copula Application - S&P 100 and Systemic Risk
• Need vine / pair-copula constructions (C-vine, D-vine, R-vine, simplifying assumption, Dißmann)? → Vine Copulas

Sub-topics

Sub-topic	Notes	Covers
Vine Copulas	5	Pair-copula constructions, C-vines / D-vines / regular vines, the simplifying assumption, sequential estimation & Dißmann’s structure-selection algorithm — Aas (2016)

Concept Map

Concept	Note	Type	Depends On	Key Result
Motivation & contribution	Factor Copulas - Overview	overview	—	High-dim copula class for 50+ vars; fat-tailed/asymmetric common factor captures correlated crashes
Latent factor model	Factor Copula Construction	definition	Overview	$X_i={\beta }_{i}Z+{\epsilon }_i$; copula of $\mathbf{X}$ used, marginals discarded; closed form only if all-Gaussian (equicorrelation)
Tail dependence (EVT)	Tail Dependence in Factor Copulas	theorem	Construction	Props 1-3: regularly-varying tails → non-zero tail dependence; skew factor → ${\tau }^U\ne {\tau }^L$
Multi-factor & block	Multi-Factor and Block Dependence Structures	concept	Construction	$K$-factor + block equidependence (market + 7 SIC industry factors, 16 params) → heterogeneous dependence
SMM estimation	SMM Estimation of Factor Copulas	theorem	Construction, Multi-Factor	Match rank correlation + quantile dependence; consistent & asym. normal ($S/T\to \infty$); GMM sandwich, bootstrap + numerical-derivative covariance
S&P 100 & systemic risk	Factor Copula Application - S&P 100 and Systemic Risk	example	SMM, Tail Dependence, Multi-Factor	Skew $t$-$t$ block copula fits best; crashes more correlated than booms; superior MES/$kES$ estimates

Notes

• Factor Copulas - Overview — CONTAINS: 2007-08 crisis motivation, Sklar decomposition, two contributions, position vs Normal/$t$/grouped-$t$/Archimedean/vine copulas, Figure 1 illustration.
• Factor Copula Construction — CONTAINS: simple/equidependence model, Gaussian closed-form & equicorrelation, flexible-weight single-factor, non-linear nesting table (Normal/$t$/skew $t$/gen-hyperbolic/Clayton/Gumbel).
• Tail Dependence in Factor Copulas — CONTAINS: tail-dependence definitions, regular variation, Proposition 1 (single factor, full cases a-d), boundary case, Proposition 2 (skew $t$-$t$ constants), Proposition 3 (multi-factor), Appendix-A proof sketch, quantile dependence.
• Multi-Factor and Block Dependence Structures — CONTAINS: $K$-factor model, conditional-independence/frailty interpretation, flexible weights, empirical 8-factor block model (16 params), block dependence-matrix averaging.
• SMM Estimation of Factor Copulas — CONTAINS: semiparametric DGP, SMM objective $Q_{T,S}$, rank-correlation & quantile-dependence moments, moment-count reduction, consistency/normality theorem, bootstrap + numerical-derivative covariance and step-size condition, $J$-test, MLE/GMM/SMM efficiency comparison.
• Factor Copula Application - S&P 100 and Systemic Risk — CONTAINS: Monte Carlo design & results (Tables 1-5), AR(1)-GJR-GARCH marginals, equidependence & block estimates (Tables 8-10), asymmetric/fat-tail findings, MES & $kES$ systemic-risk measures (Table 11).

Sources

• Oh-Patton-2012-Factor-Copulas.pdf — Oh & Patton (2012), “Modelling Dependence in High Dimensions with Factor Copulas”, Duke University. 51 pp. JEL C31, C32, C51.

See Also

• Copula SMM — companion Oh & Patton (2011) paper: the SMM estimator and asymptotic theory this paper applies.
• Simulation-Based Estimation — MSM, indirect inference, EMM, SMM (the methodological family).
• Bayesian copula estimation Describing correlated joint distributions — Bayesian (PyMC) Gaussian-copula tutorial, contrasting estimation paradigm.
• Econometrics