Simulated Method of Moments Estimation for Copula-Based Multivariate Models

Abstract

This article considers the estimation of the parameters of a copula via a simulated method of moments (MM) type approach. This approach is attractive when the likelihood of the copula model is not known in closed form, or when the researcher has a set of dependence measures or other functionals of the copula that are of particular interest. The proposed approach naturally also nests MM and generalized method of moments estimators. Drawing on results for simulation-based estimation and on recent work in empirical copula process theory, we show the consistency and asymptotic normality of the proposed estimator, and obtain a simple test of overidentifying restrictions as a specification test. The results apply to both iid and time series data. We analyze the finite-sample behavior of these estimators in an extensive simulation study. We apply the model to a group of seven financial stock returns and find evidence of statistically significant tail dependence, and mild evidence that the dependence between these assets is stronger in crashes than booms. Supplementary materials for this article are available online.

KEY WORDS:

• Correlation
• Dependence
• Inference
• SMM

Acknowledgments

The authors thank the editor, associate editor, two anonymous referees, and Tim Bollerslev, Yanqin Fan, and George Tauchen, and seminar participants at Duke University, Erasmus University Rotterdam, HEC Montreal, McGill University, Monash University, Vanderbilt University, Humboldt-Copenhagen Financial Econometrics workshop, Econometric Society Australasian Meetings, and Econometric Society Asian Meetings for helpful comments.

Notes

NOTE: This table presents the results from 100 simulations of the Clayton copula, the Normal copula, and a factor copula. In the SMM and GMM estimation, all three copulas use five dependence measures, including four quantile dependence measures (q = 0.05, 0.10, 0.90, 0, 95). The Normal and factor copulas also use Spearman's rank correlation, while the Clayton copula uses either Kendall's (GMM and SMM) or Spearman's (SMM*) rank correlation. The marginal distributions of the data are assumed to be iid N(0, 1). Problems of dimension N = 2, 3, and 10 are considered, the sample size is T = 1000, and the number of simulations used is S = 25 × T. The first row of each panel presents the average difference between the estimated parameter and its true value. The second row presents the standard deviation of the estimated parameters. The third and fourth rows present the median and the difference between the 90th and 10th percentiles of the distribution of estimated parameters. The last row in each panel presents the average time in seconds to compute the estimator.

NOTE: This table presents the results from 100 simulations of the Clayton copula, the Normal copula, and a factor copula. In the SMM and GMM estimation, all three copulas use five dependence measures, including four quantile dependence measures (q = 0.05, 0.10, 0.90, 0, 95). The Normal and factor copulas also use Spearman's rank correlation, while the Clayton copula uses either Kendall's (GMM and SMM) or Spearman's (SMM*) rank correlation. The marginal distributions of the data are assumed to follow AR(1)-GARCH(1,1) processes as described in Section 3. Problems of dimension N = 2, 3, and 10 are considered, the sample size is T = 1000, and the number of simulations used is S = 25 × T. The first row of each panel presents the average difference between the estimated parameter and its true value. The second row presents the standard deviation of the estimated parameters. The third and fourth rows present the median and the difference between the 90th and 10th percentiles of the distribution of estimated parameters. The last row in each panel presents the average time in seconds to compute the estimator.

NOTE: This table presents the results from 100 simulations of the Clayton copula, the Normal copula, and a factor copula, all estimated by SMM. The marginal distributions of the data are assumed to follow AR(1)-GARCH(1,1) processes as described in Section 3. Problems of dimension N = 2, 3, and 10 are considered, the sample size is T = 1000, and the number of simulations used is S = 25 × T. The rows of each panel contain the step size, ϵ _{T, S} , used in computing the matrix of numerical derivatives, The numbers in column κ, ρ, σ², ν^{− 1}, and λ present the percentage of simulations for which the 95% confidence interval based on the estimated covariance matrix contained the true parameter. The numbers in column J present the percentage of simulations for which the test statistic of overidentifying restrictions test described in Section 2 was smaller than its computed critical value under 95% confidence level.

NOTE: This table presents the results from 100 simulations when the true copula and the model are different (i.e., the model is misspecified). The parameters of the copula models are estimated using SMM based on rank correlation and four quantile dependence measures (q = 0.05, 0.10, 0.90, 0, 95). The marginal distributions of the data are assumed to be either iid N(0, 1) or AR(1)-GARCH(1,1) processes as described in Section 3. Problems of dimension N = 2, 3, and 10 are considered, the sample size is T = 1000, and the number of simulations used is S = 25 × T. The pseudo-true parameter is estimated using 10 million observations. The last row in each panel presents the proportion of tests of overidentifying restrictions that are smaller than the 95% critical value.

NOTE: This table presents measures of dependence between the seven financial firms under analysis. The upper panel presents Spearman's rank correlation (upper triangle) and linear correlation (lower triangle), and the lower panel presents the difference between the 10% quantile dependence measures (lower triangle) and average 1% upper and lower quantile dependence (upper triangle). All dependence measures are computed using the standardized residuals from the models for the conditional mean and variance.

NOTE: This table presents estimation results for various copula models applied to seven daily stock returns in the financial sector over the period January 2001 to December 2010. Estimates and asymptotic standard errors for the copula model parameters are presented, as well as the value of the SMM objective function at the estimated parameters and the p-value of the overidentifying restriction test. Estimates labeled “SMM” are estimated using the identity weight matrix; estimates labeled “SMM-opt” are estimated using the efficient weight matrix.