SMM Estimation of Factor Copulas

Summary

Since factor copulas have no closed-form likelihood, parameters are estimated by a simulation-based method of moments (SMM) (Oh & Patton 2011): match a vector of rank-based dependence measures (Spearman’s rank correlation, quantile dependence) computed from data to those computed from simulated copula draws. The estimator is consistent and asymptotically normal under regularity conditions when $S/T\to \infty$, with a GMM-type sandwich covariance requiring bootstrap and numerical-derivative inputs.

Overview

The “moments” are functions of rank statistics — strictly, this is not classical SMM (moments are not raw sample moments), but the asymptotics parallel SMM/GMM. Rank-based measures are “pure” measures of dependence: invariant to the marginals, so they isolate the copula. The data model is semiparametric: parametric conditional mean/variance dynamics, nonparametric (empirical CDF) marginals, and a parametric (factor) copula. Marginals are estimated in a first stage; copula parameters in a second stage by SMM on the standardized residuals.

Main Content

Data generating process (semiparametric)

The DGP (as in Chen & Fan 2006, Rémillard 2010, Oh & Patton 2011):

$${\mathbf{Y}}_t={\mathbf{\mu }}_t({\mathbf{\varphi }}_0)+{\mathbf{\sigma }}_t({\mathbf{\varphi }}_0){\mathbf{\eta }}_t$$ $${\mathbf{\eta }}_t=[{\eta }_{1t},\dots ,{\eta }_{Nt}{]}^{\prime }\sim \text{iid }{\mathbf{F}}_{\eta }=\mathbf{C}(F_1,\dots ,F_N;{\mathbf{\theta }}_0)$$

where ${\mathbf{\mu }}_t(\mathbf{\varphi })$ and ${\mathbf{\sigma }}_t(\mathbf{\varphi })=\text{diag}\{{\sigma }_{1t}(\mathbf{\varphi }),\dots ,{\sigma }_{Nt}(\mathbf{\varphi })\}$ are ${\mathcal{F}}_{t-1}$-measurable (functions of past data) and independent of ${\mathbf{\eta }}_t$. The $r\times 1$ dynamic parameter ${\mathbf{\varphi }}_0$ is $\sqrt{T}$-consistently estimable. The copula is parameterized by the $p\times 1$ vector ${\mathbf{\theta }}_0\in \mathbf{\Theta }$, estimated by SMM. Marginals are estimated nonparametrically via the empirical distribution function. The conditional copula is assumed constant (time-varying copulas need non-trivial asymptotic adjustments, not treated here).

SMM objective function

Estimate ${\mathbf{\theta }}_0$ from standardized residuals ${\hat{\eta }}_t\equiv {\mathbf{\sigma }}_t^{-1}(\hat{\mathbf{\varphi }})[{\mathbf{Y}}_t-{\mathbf{\mu }}_t(\hat{\mathbf{\varphi }})]$ and simulations from the copula. Let ${\tilde{\mathbf{m}}}_S(\mathbf{\theta })$ be the $m\times 1$ vector of dependence measures from $S$ simulations $\{{\mathbf{X}}_s{\}}_{s=1}^S$ of ${\mathbf{F}}_x(\mathbf{\theta })$, and ${\hat{\mathbf{m}}}_T$ the same measures from the residuals $\{{\hat{\eta }}_t{\}}_{t=1}^T$. The estimator:

$${\hat{\mathbf{\theta }}}_{T,S}\equiv \arg \underset{\mathbf{\theta }\in \mathbf{\Theta }}{\min }{Q}_{T,S}(\mathbf{\theta })$$ $$Q_{T,S}(\mathbf{\theta })\equiv {\mathbf{g}}_{T,S}^{\prime }(\mathbf{\theta }){\hat{W}}_T{\mathbf{g}}_{T,S}(\mathbf{\theta }),{\mathbf{g}}_{T,S}(\mathbf{\theta })\equiv {\hat{\mathbf{m}}}_T-{\tilde{\mathbf{m}}}_S(\mathbf{\theta })$$

where ${\hat{W}}_T$ is a positive-definite weight matrix (may depend on data). The application uses the identity weight matrix $W=I$ (coverage rates are better than with the efficient weight matrix).

Dependence measures used as moments (App. B)

The five “pure” dependence measures per pair: pairwise Spearman’s rank correlation and quantile dependence at $q=[0.05,0.10,0.90,0.95]$. They are invariant to the marginals, so they reflect only the copula. Let ${\delta }_{ij}$ be one such measure between variables $i,j$, forming the pairwise dependence matrix $D$ (entries ${\delta }_{ij}$, ones on the diagonal). To avoid $5N(N-1)/2$ moments in high dimension, the model’s (block) equidependence is exploited:

• Equidependence model: match the average of each measure across all pairs, $\bar{\delta }\equiv \frac{2}{N(N-1)}{\sum }_{i<j}{\hat{\delta }}_{ij}$ → just 5 moments.
• Flexible weights: use the $N$-vector of row-averages ${\bar{\delta }}_i\equiv \frac{1}{N}{\sum }_j{\hat{\delta }}_{ij}$ (variable $i$‘s average dependence with all others) → $5N$ moments (model has $O(N)$, not $O(N^2)$, parameters).
• Block equidependence: average within/between blocks to an $M\times M$ matrix $D^{\ast }$, then row-average to an $M$-vector → $5M$ moments (see Multi-Factor and Block Dependence Structures).

Consistency and asymptotic normality (Oh & Patton 2011)

Under regularity conditions, if $S/T\to \infty$ as $T\to \infty$, the SMM estimator is consistent and asymptotically normal:

$$\sqrt{T}\left({\hat{\mathbf{\theta }}}_{T,S}-{\mathbf{\theta }}_0\right)\stackrel{d}{\to }N(0,{\Omega }_0)\text{as }T,S\to \infty$$ $${\Omega }_0=(G_0^{\prime }{W}_0{G}_0{)}^{-1}{G}_0^{\prime }{W}_0{\Sigma }_0{W}_0{G}_0(G_0^{\prime }{W}_0{G}_0{)}^{-1}$$

where ${\Sigma }_0\equiv \text{avar}[{\hat{\mathbf{m}}}_T]$ (asymptotic variance of the sample dependence measures), $G_0\equiv {\nabla }_{\theta }{\mathbf{g}}_0({\mathbf{\theta }}_0)$ (gradient of the limiting moment function), and ${\mathbf{g}}_0(\mathbf{\theta })\equiv {\text{p-lim}}_{T,S\to \infty }{\mathbf{g}}_{T,S}(\mathbf{\theta })$. This is the standard GMM sandwich form, but ${\Sigma }_0$ and $G_0$ require non-standard estimation (the moments are rank statistics, and ${\mathbf{g}}_0$ is only available via simulation). (When $S/T\to 0$ instead, the rate becomes $\sqrt{S}$; here $S\gg T$ so the $\sqrt{T}$ case applies.)

Estimating the covariance: bootstrap + numerical derivative

• ${\Sigma }_0$ is consistently estimated by a simple iid bootstrap of the dependence measures (1000 bootstraps in the application).
• $G_0$ is consistently estimated by a numerical derivative $\hat{G}$ of ${\mathbf{g}}_{T,S}(\mathbf{\theta })$ at ${\hat{\mathbf{\theta }}}_{T,S}$, provided the step size ${\epsilon }_T\to 0$ slower than $T^{-1/2}$. This condition is crucial: for $T=1000$ a step size ${\epsilon }_T>0.03$ is implied, much larger than typical numerical-derivative defaults ($\sim 6\times 10^{-6}$ in Matlab). Coverage rates collapse if the step is too small (e.g. 38% coverage for a nominal 95% CI at ${\epsilon }_T=0.0001$); ${\epsilon }_T\in \{0.01,0.03,0.1\}$ give near-nominal coverage. The application uses ${\epsilon }_T=0.1$.
• A $J$-test of over-identifying restrictions is available (Proposition 4 of Oh & Patton 2011); with $W=I$ it has a non-standard distribution whose critical values depend on $\hat{G}$ and are obtained by simulation.

Examples

Efficiency cost of SMM vs ML and GMM (Normal copula)

Setup: The Normal factor copula has a closed-form likelihood, so SMM can be benchmarked against MLE and (infeasible-moment) GMM. Simulation with $T=1000$, $S=25T$, $N\in \{3,10,100\}$. Result: SMM estimates are centered on true values (small bias relative to std). ML is always more efficient, but the loss from MLE→SMM is moderate: ~25% for $N=3$ down to ~10% for $N=100$. The loss from GMM→SMM (cost of simulating the moment function) is at most 3%, in some $N=100$ cases slightly negative. Adding parameters reduces precision; increasing $N$ (with equidependence) improves precision since more information bears on the same parameters. Interpretation: Moving to SMM because of the missing likelihood costs little efficiency, and the method scales well to high dimensions.

Connections

• Factor Copula Construction — supplies the copula $\mathbf{C}(\mathbf{\theta })$ to simulate from.
• Multi-Factor and Block Dependence Structures — block averaging that makes high-dim moment-matching feasible.
• Factor Copula Application - S&P 100 and Systemic Risk — applies this estimator to 100 stocks.
• SMM Estimator for Copulas — the companion Oh & Patton (2011) paper deriving this estimator and its asymptotics in full.
• SMM Copula Asymptotic Theory — companion note on the consistency/normality proofs.
• Dependence Measures for Copulas — companion note detailing rank correlation and quantile dependence as “pure” measures.

See Also

• 19. Simulated Method of Moments Estimation — Computational Methods for Economists using Python — general SMM exposition (Evans 2024); this factor-copula application is a flagship use of the method.
• Method of Simulated Moments — the underlying MSM/GMM framework.
• SMM Weighting Matrix and Inference — weight-matrix choice and inference in SMM generally.
• Bayesian copula estimation Describing correlated joint distributions — contrast: a Bayesian (PyMC) Gaussian-copula estimation, vs the frequentist simulation-based moment-matching used here.
• Estimation and Structure Selection for Vines — alternative estimation/selection approach
• Econometrics