SMM Copula Specification Testing

Summary

When the number of moment conditions ($m$) exceeds the number of copula parameters ($p$), the over-identifying restrictions can be tested via a J-test statistic (Proposition 4). With the efficient weight matrix ${\hat{\mathbf{W}}}_T={\hat{\mathbf{\Sigma }}}_{T,B}^{-1}$, this statistic has a standard ${\chi }_{m-p}^2$ limiting distribution. With any other weight matrix, the limiting distribution is non-standard but can be simulated easily. This test provides a simple goodness-of-fit check for the copula specification.

Overview

In the Oh-Patton framework, the researcher typically uses $m=5$ dependence measures (Spearman’s rank correlation + 4 quantile dependence measures) to estimate $p$ copula parameters. When $m>p$, the model is over-identified and the remaining $m-p$ restrictions can be tested — a poor fit indicates the copula model cannot simultaneously match all the targeted dependence features.

Proposition 4: Over-Identifying Restrictions Test

Theorem: Over-Identifying Restrictions Test (Oh & Patton, Proposition 4)

Suppose that all assumptions of Proposition 2 are satisfied and that the number of moments $m$ is greater than the number of copula parameters $p$. Then the test statistic:

$$J_{T,S}\equiv \min (T,S){\mathbf{g}}_{T,S}({\hat{\mathbf{\theta }}}_{T,S}{)}^{\prime }{\hat{\mathbf{W}}}_T{\mathbf{g}}_{T,S}({\hat{\mathbf{\theta }}}_{T,S})$$

has the limiting distribution:

$$J_{T,S}\stackrel{d}{\to }{\mathbf{u}}^{\prime }{\mathbf{A}}_0^{\prime }{\mathbf{A}}_0\mathbf{u}\text{as }T,S\to \infty$$

where $\mathbf{u}\sim N(0,\mathbf{I})$ and:

$${\mathbf{A}}_0={\mathbf{W}}_0^{1/2}{\mathbf{\Sigma }}_0^{1/2}{\mathbf{R}}_0$$ $${\mathbf{R}}_0=\mathbf{I}-{\mathbf{\Sigma }}_0^{-1/2}{\mathbf{G}}_0({\mathbf{G}}_0^{\prime }{\mathbf{W}}_0{\mathbf{G}}_0{)}^{-1}{\mathbf{G}}_0^{\prime }{\mathbf{W}}_0{\mathbf{\Sigma }}_0^{1/2}$$

Special case — efficient weight matrix: If ${\hat{\mathbf{W}}}_T={\hat{\mathbf{\Sigma }}}_{T,B}^{-1}$, then:

$$J_{T,S}\stackrel{d}{\to }{\chi }_{m-p}^2\text{as usual}$$

General case — identity or other weight matrix: If ${\hat{\mathbf{W}}}_T\ne {\hat{\mathbf{\Sigma }}}_{T,B}^{-1}$, the test statistic has a non-standard limiting distribution that depends on the sample-specific matrix $\hat{\mathbf{R}}$.

Simulating Critical Values

When the efficient weight matrix is not used (e.g., when using the identity matrix ${\hat{\mathbf{W}}}_T=\mathbf{I}$), critical values are obtained via simulation:

Example: Simulating Critical Values for the J-Test

Procedure:

1. Compute $\hat{\mathbf{R}}$ using ${\hat{\mathbf{G}}}_{T,S}$, ${\hat{\mathbf{W}}}_T$, and ${\hat{\mathbf{\Sigma }}}_{T,B}$
2. Simulate ${\mathbf{u}}^{(k)}\sim iidN(0,\mathbf{I})$ for $k=1,2,\dots ,K$ (with $K$ large)
3. For each simulation, compute: $J_{T,S}^{(k)}={\mathbf{u}}^{(k)\prime }{\hat{\mathbf{R}}}^{\prime }{\hat{\mathbf{\Sigma }}}_{T,B}^{1/2\prime }{\hat{\mathbf{W}}}_T{\hat{\mathbf{\Sigma }}}_{T,B}^{1/2}\hat{\mathbf{R}}{\mathbf{u}}^{(k)}$
4. The sample $(1-\alpha )$ quantile of $\{J_{T,S}^{(k)}{\}}_{k=1}^K$ is the critical value

Key advantages:

• ${\mathbf{u}}^{(k)}$ is a simple standard normal — no optimization is required
• $\hat{\mathbf{R}}$ need only be computed once
• The procedure is fast even for $K=10,000$

Oh and Patton use $K=10,000$ in their application.

Practical Considerations

Choosing ${\hat{\mathbf{W}}}_T$

Weight Matrix	J-Test Distribution	Advantage	Disadvantage
${\hat{\mathbf{\Sigma }}}_{T,B}^{-1}$ (efficient)	${\chi }_{m-p}^2$	Standard critical values	Requires inverting estimated covariance; may be unstable
$\mathbf{I}$ (identity)	Non-standard	Simple; numerically stable	Requires simulated critical values

Oh and Patton use the identity weight matrix in their main results, noting that “corresponding results based on the efficient weight matrix are comparable.”

Dependence of Critical Values on ${\hat{\mathbf{G}}}_{T,S}$

When using the identity weight matrix, the limiting distribution depends on ${\hat{\mathbf{G}}}_{T,S}$, which in turn depends on the step size ${\epsilon }_{T,S}$. However, the Monte Carlo study in SMM Copula Simulation and Application shows that rejection rates are close to nominal (95%) across all three copula models and all step sizes tested.

Connections

• Uses asymptotic results from SMM Copula Asymptotic Theory (Propositions 1-3)
• Applied in SMM Copula Simulation and Application to test Clayton, Normal, and factor copula models
• Analogous to the standard J-test in GMM — see Standard Errors and Clustering
• Extends SMM Estimator for Copulas from estimation to model evaluation

See Also

• SMM Copula Asymptotic Theory — underlying asymptotic results
• SMM Copula Simulation and Application — Monte Carlo and empirical test results
• SMM Estimator for Copulas — the estimator being tested

Sources

• Oh_Patton_SMM_copulas_nov11.pdf — Oh & Patton (2011), Section 2.5