Goodness-of-Fit Tests for Copulas of Multivariate Time Series
Abstract
:1. Introduction
2. Weak Convergence of Empirical Processes of Residuals
- (A1)
- , , uniformly in , where and are deterministic, .
- (A2)
- and are bounded, for .
- (A3)
- There exists a sequence of positive terms so that and such that the sequence is tight.
- (A4)
- and .
- (A5)
- in .
- (A6)
- and are bounded and continuous on . In addition, have continuous bounded densities respectively.
- (A7)
- For all , and are bounded and continuous on .
- Model 1. In this model, for the sake of identifiability, we assume that the correlation matrix is the identity matrix. This means that the conditional covariance matrix of the observations relative to is . In particular, the conditional correlation is not necessarily constant and it is estimated.
- Model 2. In this model, is a diagonal matrix for any i, and there is no restriction on the correlation matrix of the innovations. This means is diagonal and for any , is the conditional variance of the observations relative to . In particular, this implies that the conditional correlation between the observations is constant, does not depend on θ, and it is implicitly incorporated in the copula C of the innovations. This model appears naturally in practice when the parameters of univariate time series are estimated separately as in Chen and Fan [10]. This model is not the same as the diagonal representation in Engle and Kroner [17], which is included in Model 1.
Empirical Processes Related to the Copula
Semiparametric Estimation of the Copula
3. Specification Tests for the Copula
- (B1)
- For every , the density of admits first and second order derivatives with respect to all components of ϕ. The gradient (column) vector with respect to ϕ is denoted , and the Hessian matrix is represented by .
- (B2)
- For arbitrary and every , the mappings and are continuous at .
- (B3)
- For every , there exist a neighborhood of and -integrable functions such that for every ,
3.1. Test Statistics Based on the Empirical Copula
Parametric Bootstrap for
Algorithm 1: Parametric bootstrap for the empirical copula process. |
For some large integer N, do the following steps:
An approximate p-value for the test is then given by . |
3.2. Tests Statistics Based on the Rosenblatt Transform
A Parametric Bootstrap for
Algorithm 2: Parametric bootstrap for the empirical Rosenblatt process. |
For some large integer N, do the following steps:
An approximate p-value for the test is then given by . |
3.3. Estimation of Copula Parameters
3.3.1. Maximum Pseudo-Likelihood Estimators
3.3.2. Two-Stage Estimators
3.4. Estimators Based on Measures of Dependence
3.4.1. Kendall’s Tau
3.4.2. Spearman’s Rho
3.4.3. Van der Waerden’s Coefficient
3.4.4. Blomqvist’s Coefficient
3.5. Copula vs. Rosenblatt Transform
4. Example of Application
5. Conclusions
Acknowledgments
Conflicts of Interest
Appendix A. Proofs of the Main Results
Appendix A.1. Proof of Theorem 1
Appendix A.2. Proof of Theorem 2
Appendix B. Other Proofs
Appendix B.1. Proof of Proposition 1
Appendix B.2. Proof of Propositions 3–6
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Share and Cite
Rémillard, B.
Goodness-of-Fit Tests for Copulas of Multivariate Time Series
. Econometrics 2017, 5, 13.
https://doi.org/10.3390/econometrics5010013
Rémillard B.
Goodness-of-Fit Tests for Copulas of Multivariate Time Series
. Econometrics. 2017; 5(1):13.
https://doi.org/10.3390/econometrics5010013
Rémillard, Bruno.
2017. "Goodness-of-Fit Tests for Copulas of Multivariate Time Series
" Econometrics 5, no. 1: 13.
https://doi.org/10.3390/econometrics5010013
Rémillard, B.
(2017). Goodness-of-Fit Tests for Copulas of Multivariate Time Series
. Econometrics, 5(1), 13.
https://doi.org/10.3390/econometrics5010013