Copula SMM
Routing Summary
This folder covers the Oh & Patton (2011) research program: SMM estimation for copula-based multivariate models using rank dependence measures as moments. Contains 5 notes.
- Need rank/tail dependence measures used as SMM moments? → Dependence Measures for Copulas
- Need the Oh-Patton SMM estimator definition and two-stage setup? → SMM Estimator for Copulas
- Need asymptotic theory (Propositions 1–3: consistency, normality, variance)? → SMM Copula Asymptotic Theory
- Need J-test for copula specification testing? → SMM Copula Specification Testing
- Need Monte Carlo results and 7-firm financial application? → SMM Copula Simulation and Application
Concept Map
| Concept | Note | Type | Depends On | Key Result |
|---|---|---|---|---|
| Spearman’s ρ, quantile dependence, tail dependence — pure copula functionals | Dependence Measures for Copulas | definition | Copula Estimation | Invariant to marginals; used as SMM moments in Oh & Patton |
| Oh-Patton DGP, two-stage estimation, rank dependence moments | SMM Estimator for Copulas | concept | Method of Simulated Moments, Dependence Measures for Copulas | SMM for copulas when likelihood is unavailable |
| Assumptions 1–4, Propositions 1–3 (consistency, normality, variance) | SMM Copula Asymptotic Theory | theorem | SMM Estimator for Copulas | First-stage estimation error does not affect copula estimator |
| J-test, over-identifying restrictions, simulated critical values | SMM Copula Specification Testing | theorem | SMM Copula Asymptotic Theory | with efficient weight; simulated CVs otherwise |
| Monte Carlo study + 7-firm financial dependence (2001–2010) | SMM Copula Simulation and Application | example | SMM Copula Asymptotic Theory, SMM Copula Specification Testing | ~20–40% efficiency loss vs. MLE; significant tail dependence in financials |
Notes
- Dependence Measures for Copulas — CONTAINS: Spearman’s rank correlation, quantile dependence, tail dependence coefficients, asymmetry measures, pure copula functionals invariant to marginals
- SMM Estimator for Copulas — CONTAINS: Oh-Patton DGP, two-stage estimation, rank dependence moments, SMM estimator definition, factor copula model, nesting of GMM/MM
- SMM Copula Asymptotic Theory — CONTAINS: Assumptions 1–4, Proposition 1 (consistency), Proposition 2 (asymptotic normality with 3 rate cases), Proposition 3 (variance estimation via bootstrap + numerical derivatives)
- SMM Copula Specification Testing — CONTAINS: Proposition 4 (J-test), chi-squared with efficient weight, simulated critical values for general weight, simulation procedure
- SMM Copula Simulation and Application — CONTAINS: Monte Carlo for Clayton/Normal/factor copulas, iid and AR-GARCH data, step-size sensitivity, 7 financial firms (2001–2010), tail dependence and asymmetry findings
Sources
- Oh_Patton_SMM_copulas_nov11.pdf — Oh & Patton (2011), “Simulated Method of Moments Estimation for Copula-Based Multivariate Models”
See Also
- Simulation-Based Estimation — the general MSM/SMM theory these notes apply
- Copula Estimation — Bayesian approach to copula estimation (complementary)
- SMM Weighting Matrix and Inference — weighting matrix strategies used in the copula SMM context