Maximum Likelihood Estimation Methods for Copula Models

Abstract

For Copula models, the likelihood function could be multi-modal, and some traditional optimization algorithms such as simulated annealing (SA) may get stuck in the local mode and introduce bias in parameter estimation. To address this issue, we consider three widely used global optimization approaches, including sequential Monte Carlo simulated annealing (SMC-SA), sequential qudratic programming and generalized simulated annealing, in the estimation of bivariate and R-vine Copula models. Then the accuracy and effectiveness of these algorithms are compared in simulation studies, and we find that SMC-SA provides more robust estimation than SA both for bivariate and R-vine Copulas. Finally, we apply these approaches in real data as well as a large multivariate case for portfolio risk management, and find that SMC-SA performs better than SA in both fitting the data and predicting portfolio risk.

Appendices

Appendix

In this appendix, we provide some additional figures, tables and results. In Sect. A.1, we compare the performance of SA and SMC-SA on bivariate Copula models with relatively large observation numbers. Section A.2 illustrates the R-vine structure of model 5–8 with a graph. In Sect. A.3, we compare the performance of SQP, GSA, SA and SMC-SA on model 7 and 8.

Additional Simulation Results on Bivariate Copulas

In this section, we provide simulation results on bivariate Copula models under sample size ${T=}$ 1000 and 4000 respectively, and compare the performance of SMC-SA and SA algorithms.

Table 10 Simulation results on SMC-SA (${T=}$ 1000 and 4000)

Table 10 shows the performance of SMC-SA on simulated data with large observation numbers, and the performance of SA with different initial values are shown in Table 11. As Table 11 illustrates, SA is sensitive to initial values even under large sample size, since the bias in case II and III is quite larger than that in case I, where initial values are set equal to true values. However, SMC-SA provides a more accurate estimation than SA in case II and III (lower bias and RMSE, and higher likelihood values on average).

Table 11 Simulation results on SA (${T=}$ 1000 and 4000)

Visualization of R-vine Structure

Fig. 2

Structure of R-vine models in this paper

Following Dissmann et al. (2013), Figure 2 illustrates the R-vine structure for model 5-8. Since our R-vine models are with a dimension of 7, there are six trees in the Vine structure. Each pair Copula is represented by an edge in this graph, for example, the edge between node 2 and 3 in Tree 1 represents the unconditional Copula ${C_{2,3}}$, and the edge between nodes 2, 4|3 and 1, 3|2 in Tree 3 represents the conditional Copula ${C_{1,4|2,3}}$. Each node in the tree ${T_{i}}$ corresponds to an edge in the tree ${T_{i-1}}$, for example, the node 5, 7|1, 2, 3, 6 in Tree 6 corresponds to the edge in Tree 5 between nodes 1, 7|2, 3, 6 and 5, 6|1, 2, 3. For pair Copulas, the type and true value of parameters are shown in Table 5.

Additional Simulation Results on R-vine Copulas

In this section, We generate 100 replicates with 1000 observations on each replicate for model 7 and 8, and compare the performance of SQP, GSA, SA and SMC-SA algorithms on them. Simulation results are shown in Table 12.

Table 12 Simulation results on R-vine Copulas (model 7 and 8)