SMM Copula Simulation and Application

Summary

Oh and Patton (2011) validate their SMM estimator through an extensive Monte Carlo study covering three copula models (Clayton, Normal, factor copula) in dimensions $N=2,3,10$, for both iid and AR(1)-GARCH(1,1) data. The SMM estimator is approximately unbiased with moderate efficiency loss (20-40% for $N=2$, declining to ~20% for $N=10$) relative to MLE. The empirical application to seven major U.S. financial firms (2001-2010) finds significant tail dependence and mild evidence of stronger dependence during crashes than booms.

Monte Carlo Study Design

Three Copula Models

Model	Parameters	Closed-Form Likelihood	Closed-Form Dependence
Clayton	$\kappa$	Yes	Kendall’s $\tau$ only
Normal (Gaussian)	$\rho$	Yes	Spearman’s $\rho$ only
Factor copula	$({\sigma }^2,{\nu }^{-1},\lambda )$	No	None

True parameter values (calibrated to produce rank correlation $\approx 1/2$):

• Clayton: $\kappa =1.00$
• Normal: $\rho =0.50$
• Factor copula: ${\sigma }^2=1.00$, ${\nu }^{-1}=0.25$, $\lambda =-0.50$

Two Data Scenarios

Scenario 1 — iid data: Marginal distributions are standard Normal. Only copula parameters need estimation.

Scenario 2 — AR(1)-GARCH(1,1) data: Each variable follows:

$$Y_{it}={\varphi }_0+{\varphi }_1{Y}_{i,t-1}+{\sigma }_{it}{\eta }_{it}$$ $${\sigma }_{it}^2=\omega +\beta {\sigma }_{i,t-1}^2+\alpha {\sigma }_{i,t-1}^2{\eta }_{i,t-1}^2$$

with $[{\varphi }_0,{\varphi }_1,\omega ,\beta ,\alpha ]=[0.01,0.05,0.05,0.85,0.10]$ (matching daily equity return dynamics). Marginal parameters are estimated in a first stage; standardized residuals are used for copula estimation.

Estimation Settings

• Sample size: $T=1,000$ (approximately 4 years of daily data)
• Simulations: $S=25\times T=25,000$
• Dependence measures used: Spearman’s rank correlation + quantile dependence at $q=0.05,0.10,0.90,0.95$ (5 measures averaged across pairs)
• Weight matrix: Identity (${\hat{\mathbf{W}}}_T=\mathbf{I}$)
• Replications: 100

Key Results

Bias and Precision (Tables 1 and 2)

Example: Simulation Results for iid Data (Table 1)

Clayton copula (${\kappa }_0=1.00$):

	MLE	GMM	SMM	SMM*
$N=2$
Bias	0.001	-0.014	-0.006	-0.004
Std dev	0.085	0.119	0.122	0.110
Median	1.011	0.982	0.991	0.998
$N=10$
Bias	0.008	0.007	0.008	0.004
Std dev	0.050	0.068	0.066	0.059
Median	1.005	1.002	1.005	0.999

Normal copula (${\rho }_0=0.50$):

	MLE	GMM	SMM
$N=2$
Bias	0.004	-0.001	-0.001
Std dev	0.024	0.034	0.034
$N=10$
Bias	0.003	-0.002	-0.002
Std dev	0.014	0.017	0.017

Interpretation:

• All estimators are approximately unbiased (bias small relative to std dev)
• Precision improves with dimension $N$ due to exchangeability — more pairs provide more information
• SMM vs. MLE efficiency loss: ~40% for $N=2$, declining to ~20% for $N=10$ (measured by std dev ratio)
• SMM vs. GMM: loss from simulating (rather than knowing) the moment function is 0-3% — negligible

Example: Simulation Results for AR-GARCH Data (Table 2)

Results are very similar to the iid case, confirming the surprising theoretical result that first-stage estimation error does not affect the asymptotic distribution (Proposition 2).

Factor copula (${\sigma }_0^2=1.00$, ${\nu }_0^{-1}=0.25$, ${\lambda }_0=-0.50$, $N=10$):

	${\sigma }^2$	${\nu }^{-1}$	$\lambda$
Bias	-0.004	-0.016	-0.012
Std dev	0.085	0.079	0.071
Median	0.990	0.238	-0.508

The factor copula can only be estimated by SMM (no closed-form likelihood or dependence measures), and the estimates are well-centered with reasonable precision.

Coverage and Step Size Sensitivity (Table 3)

Example: Step-Size Sensitivity for Confidence Intervals

The asymptotic covariance estimator (Proposition 3) requires a numerical derivative with step size ${\epsilon }_{T,S}$. The theory requires ${\epsilon }_{T,S}\to 0$ but ${\epsilon }_{T,S}\times \min (\sqrt{T},\sqrt{S})\to \infty$.

For $T=1,000$: the threshold is $\epsilon >1/\sqrt{1000}\approx 0.032$.

95% confidence interval coverage rates (Normal copula, $N=3$, iid):

${\epsilon }_{T,S}$	Coverage
0.1	~95% (nominal)
0.01	~95% (nominal)
0.001	Much below 95%
0.0001	As low as 2%

Key takeaway: Standard numerical differentiation step sizes (e.g., MATLAB’s default $6\times 10^{-6}$) are far too small for this application. Use ${\epsilon }_{T,S}=0.01$ or $0.1$.

Over-Identifying Restrictions Test (Table 3)

The J-test rejection rates at the 5% nominal level are close to 95% acceptance for all three correctly specified copula models and all step sizes, confirming that the Proposition 4 test has correct size.

Empirical Application: Financial Firm Dependence

Data

• Period: January 2001 to December 2010 ($T=2,515$ trading days)
• Assets: Bank of America, Bank of New York, Citigroup, Goldman Sachs, J.P. Morgan, Morgan Stanley, Wells Fargo
• Returns: Daily stock returns, positively skewed and leptokurtic (kurtosis 16.0 to 119.8)

Marginal Models

Each stock’s return is modeled as:

$$r_{it}={\varphi }_{0i}+{\varphi }_{1i}{r}_{i,t-1}+{\varphi }_{2i}{r}_{m,t-1}+{\epsilon }_{it},{\epsilon }_{it}={\sigma }_{it}{\eta }_{it}$$ $${\sigma }_{it}^2={\omega }_i+{\beta }_i{\sigma }_{i,t-1}^2+{\alpha }_{1i}{\epsilon }_{i,t-1}^2+{\gamma }_{1i}{\epsilon }_{i,t-1}^2\cdot 1_{[{\epsilon }_{i,t-1}\le 0]}+{\alpha }_{2i}{\epsilon }_{m,t-1}^2+{\gamma }_{2i}{\epsilon }_{m,t-1}^2\cdot 1_{[{\epsilon }_{m,t-1}\le 0]}$$

where $r_{mt}$ is the S&P 500 index return. This is a GJR-GARCH model with asymmetric responses to market shocks.

Dependence Structure

Example: Dependence Measures Among Seven Financial Firms (Table 4)

Rank correlations (upper triangle of Table 4):

• Average: 0.63
• Range: 0.55 to 0.76
• Highest: Bank of America / Wells Fargo (0.76)

Tail dependence (average of 1% and 99% quantile dependence):

• Range: 0.16 to 0.40
• Substantial tail dependence — not captured by Gaussian copula

Asymmetry (difference between 90% and 10% quantile dependence):

• Mostly negative (14 out of 21 pairs)
• Interpretation: dependence is stronger during crashes than during booms

Model Estimation Results

Example: Copula Model Estimates for Financial Firms (Table 5)

Three copula models estimated using SMM with 5 dependence measures, ${\hat{\mathbf{W}}}_T=\mathbf{I}$, $B=1,000$ bootstraps, ${\epsilon }_{T,S}=0.1$:

Clayton copula ($\kappa$):

• MLE: $\hat{\kappa }=2.23$ (SE 0.07)
• SMM: $\hat{\kappa }=1.73$ (SE 0.05)
• J-test p-value: <0.001 (strongly rejected)
• Issue: Clayton imposes only lower tail dependence, too asymmetric for the data

Normal copula ($\rho$):

• MLE: $\hat{\rho }=0.63$ (SE 0.01)
• SMM: $\hat{\rho }=0.64$ (SE 0.01)
• J-test p-value: 0.043 (marginally rejected at 5%)
• Issue: Normal copula implies zero tail dependence, but data shows substantial tail dependence

Factor copula (${\sigma }^2,{\nu }^{-1},\lambda$):

• SMM only (no MLE available)
• ${\hat{\sigma }}^2=2.77$ (SE 0.16)
• ${\hat{\nu }}^{-1}=0.52$ (SE 0.10) — significantly > 0, indicating tail dependence
• $\hat{\lambda }=-0.27$ (SE 0.21) — not significantly different from 0
• J-test p-value: not rejected — factor copula fits adequately

Conclusion: The factor copula provides the best fit. Tail dependence (${\nu }^{-1}$) is the key feature missing from Clayton and Normal copulas. The asymmetry parameter ($\lambda$) is not statistically significant, though the point estimate suggests mildly stronger crash dependence.

Connections

• Validates Propositions 1-3 in finite samples
• Applies Proposition 4 J-test to compare copula models
• Uses Dependence Measures for Copulas as the moment conditions
• Demonstrates the factor copula model which requires SMM
• Extends standard copula estimation to intractable models

See Also

• SMM Estimator for Copulas — the estimator applied here
• SMM Copula Asymptotic Theory — theoretical justification
• SMM Copula Specification Testing — the J-test used for model comparison
• Dependence Measures for Copulas — the moments matched
• Method of Simulated Moments — the general SMM framework of which this is a copula application
• Brock-Mirman Model - SMM Estimation Exercise — another structural SMM application (macroeconomic growth model), useful for cross-domain comparison

Sources

• Oh_Patton_SMM_copulas_nov11.pdf — Oh & Patton (2011), Sections 3-4