SMM Copula Asymptotic Theory

Summary

Oh and Patton (2011) establish the consistency (Proposition 1) and asymptotic normality (Proposition 2) of their SMM estimator for copula parameters, and provide a consistent estimator of the asymptotic covariance matrix (Proposition 3). The key technical challenges are: (1) the objective function $Q_{T,S}(\mathbf{\theta })$ is not continuous in $\mathbf{\theta }$ because the simulated moments involve empirical distribution functions, and (2) a standard law of large numbers is unavailable for the moment functions. These are overcome using empirical process theory (Fermanian, Radulović, and Wegkamp, 2004; Rémillard, 2010) and stochastic equicontinuity arguments (Andrews, 1994; Newey and McFadden, 1994). A surprising result: estimation error from the first-stage marginal parameters $\hat{\varphi }$ does not enter the asymptotic distribution of the copula parameter estimator.

Overview

The estimation problem differs from standard GMM or M-estimation in two important ways:

1. Non-continuity: The objective function $Q_{T,S}(\mathbf{\theta })$ is not continuous in $\mathbf{\theta }$ because the simulated moments ${\tilde{\mathbf{m}}}_S(\mathbf{\theta })$ involve indicator functions through the EDF. For example, simulated quantile dependence ${\tilde{\tau }}_q^{ij}(\mathbf{\theta })$ takes values in the discrete set $\{0,\frac{1}{S_q},\frac{2}{S_q},\dots ,\frac{S}{S_q}\}$.

2. No standard LLN: Both ${\hat{\mathbf{m}}}_T$ and ${\tilde{\mathbf{m}}}_S(\mathbf{\theta })$ involve empirical distribution functions, so a standard law of large numbers for the pointwise convergence of ${\mathbf{g}}_{T,S}(\mathbf{\theta })$ is not available.

Assumptions for Consistency

Definition: Assumption 1 (Distributional Smoothness)

(i) The distributions ${\mathbf{F}}_{\eta }$ and ${\mathbf{F}}_x$ are continuous.

(ii) Every bivariate marginal copula $C_{ij}$ of $\mathbf{C}$ has continuous partial derivatives with respect to $u_i$ and $u_j$.

Role: Assumption 1(i) ensures that the marginal CDFs are well-defined and invertible. Assumption 1(ii) is needed for the copula density to exist and for the empirical copula process to converge.

Definition: Assumption 2 (Time Series Regularity)

These conditions control the estimation error from the first-stage marginal models. Define:

• ${\mathbf{\gamma }}_{0t}={\mathbf{\sigma }}_t^{-1}(\hat{\varphi }){\dot{\mathbf{\mu }}}_t(\hat{\varphi })$ and ${\mathbf{\gamma }}_{1kt}={\mathbf{\sigma }}_t^{-1}(\hat{\varphi }){\dot{\mathbf{\sigma }}}_{kt}(\hat{\varphi })$
• ${\mathbf{d}}_t={\mathbf{\eta }}_t-{\hat{\mathbf{\eta }}}_t-\left({\mathbf{\gamma }}_{0t}+{\sum }_{k=1}^N{\eta }_{kt}{\mathbf{\gamma }}_{1kt}\right)(\hat{\varphi }-{\varphi }_0)$

Conditions: (i) $\frac{1}{T}{\sum }_{t=1}^T{\mathbf{\gamma }}_{0t}\stackrel{p}{\to }{\mathbf{\Gamma }}_0$ and $\frac{1}{T}{\sum }_{t=1}^T{\mathbf{\gamma }}_{1kt}\stackrel{p}{\to }{\mathbf{\Gamma }}_{1k}$ (deterministic limits)

(ii) Bounded second moments of ${\mathbf{\gamma }}_{0t}$ and ${\mathbf{\gamma }}_{1kt}$

(iii) Tightness condition on ${\max }_{1\le t\le T}\Vert {\mathbf{d}}_t\Vert /r_t$ for some summable sequence $r_t$

(iv) ${\max }_{1\le t\le T}\Vert {\mathbf{\gamma }}_{0t}\Vert /\sqrt{T}=o_p(1)$ and ${\max }_{1\le t\le T}{\eta }_{kt}\Vert {\mathbf{\gamma }}_{1kt}\Vert /\sqrt{T}=o_p(1)$

(v) $\left({\alpha }_T,\sqrt{T}(\hat{\varphi }-{\varphi }_0)\right)$ weakly converges to a continuous Gaussian process in $[0,1{]}^N\times {\mathbb{R}}^r$, where ${\alpha }_T$ is the empirical copula process

(vi) Bounded and continuous partial derivatives of ${\mathbf{F}}_{\eta }$

Role: These conditions are standard for time series with estimated marginal parameters. They are satisfied by common models (ARMA, GARCH, stochastic volatility). If the data are iid (i.e., ${\mathbf{\mu }}_t$ and ${\mathbf{\sigma }}_t$ are known constants, or ${\varphi }_0$ is known), only Assumption 1 is needed.

Definition: Assumption 3 (Identification and Compactness)

(i) ${\mathbf{g}}_0(\mathbf{\theta })\ne 0$ for $\mathbf{\theta }\ne {\mathbf{\theta }}_0$ (global identification)

(ii) $\Theta$ is compact

(iii) Every bivariate marginal copula $C_{ij}(u_i,u_j;\mathbf{\theta })$ of $\mathbf{C}(\mathbf{\theta })$ on $(u_i,u_j)\in (0,1)\times (0,1)$ is Lipschitz continuous on $\Theta$

(iv) ${\hat{\mathbf{W}}}_T$ is $O_p(1)$ and converges in probability to ${\mathbf{W}}_0$, a positive definite matrix

Role: 3(i) is the standard identification condition. 3(ii) is standard. 3(iii) is needed to prove the stochastic Lipschitz continuity (and hence stochastic equicontinuity) of ${\mathbf{g}}_{T,S}(\mathbf{\theta })$. Many bivariate parametric copulas satisfy 3(iii). 3(iv) is standard for the weight matrix.

Proposition 1: Consistency

Theorem: Consistency of the SMM Copula Estimator (Oh & Patton, Proposition 1)

Suppose that Assumptions 1, 2, and 3 hold. Then:

$${\hat{\mathbf{\theta }}}_{T,S}\stackrel{p}{\to }{\mathbf{\theta }}_0\text{as }T,S\to \infty$$

Key features of this result:

1. Consistency holds at any relative rate of $T$ and $S$ diverging — unlike standard SMM results (Pakes and Pollard, 1989; McFadden, 1989) which require $T$ and $S$ to diverge at the same rate
2. If the population moment function $\mathbf{m}(\mathbf{\theta })$ is known in closed form, then GMM is feasible and is equivalent to this estimator with $S/T\to \infty$

Proof sketch: The main challenge is establishing that $Q_{T,S}$ uniformly converges in probability to $Q_0$. This requires:

• Pointwise convergence of ${\mathbf{g}}_{T,S}(\mathbf{\theta })$ to ${\mathbf{g}}_0(\mathbf{\theta })\equiv {\mathbf{m}}_0({\mathbf{\theta }}_0)-{\mathbf{m}}_0(\mathbf{\theta })$ for all $\mathbf{\theta }\in \Theta$ (using Theorems 3 and 6 of Fermanian, Radulović, and Wegkamp, 2004, and Corollary 1 of Rémillard, 2010)
• Stochastic equicontinuity of ${\mathbf{g}}_{T,S}(\mathbf{\theta })$ via the Lipschitz condition (Assumption 3(iii))
• Application of Theorem 2.1 of Newey and McFadden (1994) for uniform convergence

Assumptions for Asymptotic Normality

Definition: Assumption 4 (Differentiability and Approximate Minimization)

(i) ${\mathbf{\theta }}_0$ is an interior point of $\Theta$

(ii) ${\mathbf{g}}_0(\mathbf{\theta })$ is differentiable at ${\mathbf{\theta }}_0$ with derivative ${\mathbf{G}}_0$ such that ${\mathbf{G}}_0^{\prime }{\mathbf{W}}_0{\mathbf{G}}_0$ is nonsingular

(iii) ${\mathbf{g}}_{T,S}({\hat{\mathbf{\theta }}}_{T,S}{)}^{\prime }{\hat{\mathbf{W}}}_T{\mathbf{g}}_{T,S}({\hat{\mathbf{\theta }}}_{T,S})\le {\inf }_{\mathbf{\theta }\in \Theta }{\mathbf{g}}_{T,S}(\mathbf{\theta }{)}^{\prime }{\hat{\mathbf{W}}}_T{\mathbf{g}}_{T,S}(\mathbf{\theta })+o_p(\min (T,S{)}^{-1})$

Role: 4(i) is standard for Taylor expansion. 4(ii) requires the population moment function to be differentiable even though the finite-sample counterpart is not — this is common in simulation-based estimation. The nonsingularity of ${\mathbf{G}}_0^{\prime }{\mathbf{W}}_0{\mathbf{G}}_0$ is sufficient for local identification. 4(iii) is the approximate minimization condition, standard in simulation-based estimation (Newey and McFadden, 1994). The $o_p$ term depends on the smaller of $T$ or $S$.

Proposition 2: Asymptotic Normality

Theorem: Asymptotic Normality of the SMM Copula Estimator (Oh & Patton, Proposition 2)

Suppose that Assumptions 1, 2, 3, and 4 hold. Then the asymptotic distribution depends on the relative rate at which $T$ and $S$ diverge:

(i) If $S/T\to \infty$ as $T,S\to \infty$:

$$\sqrt{T}\left({\hat{\mathbf{\theta }}}_{T,S}-{\mathbf{\theta }}_0\right)\stackrel{d}{\to }N(0,{\mathbf{\Omega }}_0)\text{as }T,S\to \infty$$

(ii) If $S/T\to k\in (0,\infty )$ as $T,S\to \infty$:

$$\sqrt{T}\left({\hat{\mathbf{\theta }}}_{T,S}-{\mathbf{\theta }}_0\right)\stackrel{d}{\to }N\left(0,\left(1+\frac{1}{k}\right){\mathbf{\Omega }}_0\right)\text{as }T,S\to \infty$$

(iii) If $S/T\to 0$ as $T,S\to \infty$:

$$\sqrt{S}\left({\hat{\mathbf{\theta }}}_{T,S}-{\mathbf{\theta }}_0\right)\stackrel{d}{\to }N(0,{\mathbf{\Omega }}_0)\text{as }T,S\to \infty$$

where:

$${\mathbf{\Omega }}_0=({\mathbf{G}}_0^{\prime }{\mathbf{W}}_0{\mathbf{G}}_0{)}^{-1}{\mathbf{G}}_0^{\prime }{\mathbf{W}}_0{\mathbf{\Sigma }}_0{\mathbf{W}}_0{\mathbf{G}}_0({\mathbf{G}}_0^{\prime }{\mathbf{W}}_0{\mathbf{G}}_0{)}^{-1}$$

and ${\mathbf{\Sigma }}_0\equiv \text{avar}[{\hat{\mathbf{m}}}_T]$ is the asymptotic variance of the sample dependence measures.

Interpretation:

• Case (i): $S$ grows much faster than $T$, so simulation error is negligible. The asymptotic variance is the same as the (infeasible) GMM estimator — the familiar sandwich form.
• Case (ii): $S$ and $T$ grow proportionally. The $(1+1/k)$ factor captures the efficiency loss from simulation. Setting $S=25T$ gives a factor of $1.04$ (4% inflation).
• Case (iii): $S$ grows much slower than $T$ — the convergence rate is $\sqrt{S}$ not $\sqrt{T}$, but the asymptotic covariance is the same as Case (i).

Efficient weight matrix: If ${\mathbf{W}}_0={\mathbf{\Sigma }}_0^{-1}$, then ${\mathbf{\Omega }}_0$ simplifies to $({\mathbf{G}}_0^{\prime }{\mathbf{\Sigma }}_0^{-1}{\mathbf{G}}_0{)}^{-1}$.

Surprising Result: First-Stage Estimation Error Is Irrelevant

Chen and Fan (2006) and Rémillard (2010) show that estimation error from $\hat{\varphi }$ does not enter the asymptotic distribution of the copula parameter estimator for maximum likelihood or (analytical) moment-based estimators. Proposition 2 extends this surprising result to SMM-type estimators. This means the asymptotic variance ${\mathbf{\Omega }}_0$ is the same whether ${\varphi }_0$ is known or estimated — a practically very convenient property.

Proof Strategy

The proof of Proposition 2 for the case $S/T\to k\in (0,\infty )$ proceeds as follows:

Step 1: Establish the asymptotic normality of the moment function at the true parameter:

$$\sqrt{T}{\mathbf{g}}_{T,S}({\mathbf{\theta }}_0)=\underset{\stackrel{d}{\to }N(0,{\mathbf{\Sigma }}_0)}{\underbrace{\sqrt{T}({\hat{\mathbf{m}}}_T-{\mathbf{m}}_0({\mathbf{\theta }}_0))}}-\underset{\to 1/\sqrt{k}}{\underbrace{\sqrt{T/S}}}\cdot \underset{\stackrel{d}{\to }N(0,{\mathbf{\Sigma }}_0)}{\underbrace{\sqrt{S}({\tilde{\mathbf{m}}}_S({\mathbf{\theta }}_0)-{\mathbf{m}}_0({\mathbf{\theta }}_0))}}$$

Since ${\hat{\mathbf{m}}}_T$ and ${\tilde{\mathbf{m}}}_S$ are independent, $\sqrt{T}{\mathbf{g}}_{T,S}({\mathbf{\theta }}_0)\stackrel{d}{\to }N(0,(1+1/k){\mathbf{\Sigma }}_0)$.

Step 2: Establish stochastic equicontinuity of ${\mathbf{v}}_{T,S}(\mathbf{\theta })=\sqrt{T}[{\mathbf{g}}_{T,S}(\mathbf{\theta })-{\mathbf{g}}_0(\mathbf{\theta })]$ — shown in the supplemental appendix using the type II class of functions (Andrews, 1994).

Step 3: Apply Theorem 7.2 of Newey and McFadden (1994) to obtain the standard GMM expansion:

$$\sqrt{T}{\mathbf{g}}_{T,S}({\hat{\mathbf{\theta }}}_{T,S})=\sqrt{T}{\mathbf{g}}_{T,S}({\mathbf{\theta }}_0)+{\hat{\mathbf{G}}}_{T,S}\cdot \sqrt{T}({\hat{\mathbf{\theta }}}_{T,S}-{\mathbf{\theta }}_0)+{\mathbf{R}}_{T,S}({\hat{\mathbf{\theta }}}_{T,S})$$

where the remainder ${\mathbf{R}}_{T,S}=o_p(1)$.

Proposition 3: Consistent Variance Estimation

Theorem: Consistent Estimation of the Asymptotic Covariance (Oh & Patton, Proposition 3)

Suppose that all assumptions of Proposition 2 are satisfied, and that:

• ${\epsilon }_{T,S}\to 0$
• ${\epsilon }_{T,S}\times \min (\sqrt{T},\sqrt{S})\to \infty$
• $B\to \infty$ (number of bootstrap replications)

Then:

$${\hat{\mathbf{\Sigma }}}_{T,B}\stackrel{p}{\to }{\mathbf{\Sigma }}_0,{\hat{\mathbf{G}}}_{T,S}\stackrel{p}{\to }{\mathbf{G}}_0,{\hat{\mathbf{\Omega }}}_{T,S,B}\stackrel{p}{\to }{\mathbf{\Omega }}_0$$

Bootstrap estimation of ${\mathbf{\Sigma }}_0$:

1. Sample with replacement from $\{{\hat{\mathbf{\eta }}}_t{\}}_{t=1}^T$ to obtain $\{{\hat{\mathbf{\eta }}}_t^{(b)}{\}}_{t=1}^T$. Repeat $B$ times.
2. For each bootstrap sample $b$, compute the sample moments ${\hat{\mathbf{m}}}_T^{(b)}$
3. Estimate: $${\hat{\mathbf{\Sigma }}}_{T,B}=\frac{T}{B}\sum _{b=1}^B\left({\hat{\mathbf{m}}}_T^{(b)}-{\hat{\mathbf{m}}}_T\right){\left({\hat{\mathbf{m}}}_T^{(b)}-{\hat{\mathbf{m}}}_T\right)}^{\prime }$$

Numerical derivative estimation of ${\mathbf{G}}_0$: The $k$-th column of ${\hat{\mathbf{G}}}_{T,S}$ is:

$${\hat{\mathbf{G}}}_{T,S,k}=\frac{{\mathbf{g}}_{T,S}({\hat{\mathbf{\theta }}}_{T,S}+{\mathbf{e}}_k{\epsilon }_{T,S})-{\mathbf{g}}_{T,S}({\hat{\mathbf{\theta }}}_{T,S}-{\mathbf{e}}_k{\epsilon }_{T,S})}{2{\epsilon }_{T,S}}$$

where ${\mathbf{e}}_k$ is the $k$-th unit vector.

Full covariance estimator:

$${\hat{\mathbf{\Omega }}}_{T,S,B}={\left({\hat{\mathbf{G}}}_{T,S}^{\prime }{\hat{\mathbf{W}}}_T{\hat{\mathbf{G}}}_{T,S}\right)}^{-1}{\hat{\mathbf{G}}}_{T,S}^{\prime }{\hat{\mathbf{W}}}_T{\hat{\mathbf{\Sigma }}}_{T,B}{\hat{\mathbf{W}}}_T{\hat{\mathbf{G}}}_{T,S}{\left({\hat{\mathbf{G}}}_{T,S}^{\prime }{\hat{\mathbf{W}}}_T{\hat{\mathbf{G}}}_{T,S}\right)}^{-1}$$

Critical Step-Size Requirement

The step size ${\epsilon }_{T,S}$ for the numerical derivative must go to zero, but slower than the inverse of the convergence rate: ${\epsilon }_{T,S}\times \min (\sqrt{T},\sqrt{S})\to \infty$.

For $T=1000$, this means ${\epsilon }_{T,S}>1/\sqrt{1000}\approx 0.032$. Setting ${\epsilon }_{T,S}=0.01$ or $0.1$ works well; setting it to $0.001$ or smaller (e.g., MATLAB’s default $6\times 10^{-6}$) produces severely distorted coverage rates.

See SMM Copula Simulation and Application for Monte Carlo evidence on step-size sensitivity.

Connections

• Builds on SMM Estimator for Copulas — the estimator whose properties are established here
• Dependence Measures for Copulas — the moments whose asymptotic behavior drives ${\mathbf{\Sigma }}_0$
• Method of Simulated Moments — general MSM theory that this extends to the copula setting
• SMM Copula Specification Testing — uses these asymptotic results for the J-test
• SMM Copula Simulation and Application — verifies these results in finite samples

See Also

• SMM Estimator for Copulas — estimator definition
• SMM Copula Specification Testing — over-identifying restrictions test
• SMM Copula Simulation and Application — Monte Carlo evidence

Sources

• Oh_Patton_SMM_copulas_nov11.pdf — Oh & Patton (2011), Sections 2.2-2.4, Appendix
• Fermanian, J., D. Radulović, and M. Wegkamp (2004), “Weak Convergence of Empirical Copula Process,” Bernoulli 10, 847-860
• Rémillard, B. (2010), “Goodness-of-fit Tests for Copulas of Multivariate Time Series,” working paper
• Newey, W.K. and D. McFadden (1994), “Large Sample Estimation and Hypothesis Testing,” Handbook of Econometrics 4, 2111-2245
• Andrews, D.W.K. (1994), “Empirical Process Methods in Econometrics,” Handbook of Econometrics 4, 2247-2294