In Section 1.4.1, we study finite sample properties of maximum composite likeli- hood estimators (MCLEs) defined in equation (1.15) for jointly symmetric copula constructed by equation (1.7) through an extensive Monte Carlo simulations for up to one hundred dimensions. In Section 1.4.2, we illustrate the theoretical results of Section 1.3.3 on multistage estimation through simulations with realistic settings. 1.4.1 Finite sample properties of MCLE for jointly symmetric copulas
In this section, we mainly focus on examining the following. First, how big or small is the efficiency loss of MCLE compared to MLE. Second, which one is best to use among three different MCLE constructed in equation (1.12), (1.13) and (1.14) according to accuracy and computation time. Third, how useful is the cross-sectional information for copula estimations as dimension increases.
The data generating process is as follows. A vector ru1, u2, .., uNs is generated from N dimension copula. To make those data jointly symmetric, choose ui or 1 ´ ui with 1/2 probability for each i = 1,..., N
r˜u1, ˜u2, .., ˜uNs , where ˜ui “ " ui 1 ´ ui with prob 1/2 with prob 1/2 (1.20)
We consider two jointly symmetric copulas based on Clayton and Gumbel copulas and time series length T “ 1000 with dimension N “ 2, 3, 5, 10, 20, ..., 100. Four different estimation methods are applied to the simulated data: MLE, MCLE with all pairs in equation (1.12), MCLE with adjacent pairs in equation (1.13), and MCLE with the first pair in equation (1.14). We repeat these simulations and estimations five hundred times and report bias and standard deviations of those five hundred estimates with computation times in Table 1.2. While MLE is not feasible for N ě 20 due to huge computation burdens, the other MCLEs are feasible and very fast even for N “ 100, see the last four columns of Table 1.2.
The average biases for all dimensions and for all estimation methods are small relative to the standard deviations except for MCLE with the first pair. The standard deviations play a role in a measure of estimator accuracy and those show that for the low dimension pN ď 10q , not surprisingly, MLE has smaller standard deviations than three MCLEs and the relative efficiency of MCLE with all pairs to MLE is 1.05 to 1.37, which is moderate. Among three MCLEs, MCLE with all pairs has the smallest standard deviations whereas MCLE with the first pair has the largest, as expected. Comparing MCLE with adjacent pairs to MCLE with all pairs, we find that loss in efficiency is 23% for N “ 10, and 5% for N “ 100, and computation speed is two times faster for N “ 10 and 70 times faster for N “ 100. For high dimensions, it is confirmed that MCLE with adjacent pairs performs quite well compared to MCLE with all pairs according to accuracy and computation time, which is similar to results in Engle, et al. (2008) supporting MCLE with adjacent pairs in the DCC model.
Figure 1.4 indicates biases and standard deviations of four estimations as the dimension N increases. Biases of MCLE with all and adjacent pairs are very similar and standard deviations of those two MCLEs quickly decrease and the difference of those gets smaller as N increases. Compared to the standard deviation of MCLE with the first pair staying flat, the other two MCLEs exploits efficiency gains from
cross sectional information, which is intuitive because dependence of any pairs is informative for estimating copula parameters.
In sum, MCLE is less efficient but feasible and very fast for high dimensions, and MCLE gets significant efficiency gains as N increases. While the accuracy of MCLE with adjacent pairs is almost similar to that of MCLE with all pairs, especially for high dimensions, the increase in computation is quite large. For this reason, we use MCLE with adjacent pairs for our empirical analysis in Section 1.5.
1.4.2 Finite sample properties of multistage estimation
In this section, we study the multistage estimation for the proposed model with simulated data from the following set up:
rt“ H 1{2
t et (1.21)
Ht” Cov rrt|Ft´1s
et|Ft´1 „ iid F p¨q “ C pF1p¨; ν1q , ..., FNp¨; νNq ; ϕq
where the mean part is assumed zero, the variance-covariance part Ht follows the DCC model with GARCH(1,1), see Appendix A.2 with ζi “ 0, Fi is standardized Student’s t distribution with νi “ 6 and C is a jointly symmetric copula constructed by equation (1.6) with Clayton copula with ϕ “ 1. For realistic set up, we use some estimated parameter values from the results of empirical analysis in Section 1.5. The parameters of equation (A.6) and (A.7) for GARCH and DCC models are set as rψi, κi, λis “ r0.05, 0.1, 0.85s and rα, βs “ r0.02 0.95s , and Q is set to be the unconditional correlations of 100 stock returns that used in the next section. We first simulate data from the jointly symmetric copula following the way described in the previous section, and then using inverse standardized Student’s t distribution, transform those data into uncorrelated et. Then we recursively update DCC model by equation (A.7) and (A.8) to generate correlation matrix, and apply GARCH effects
by equation (A.6). Then, the simulated et can be easily transformed to rt whose conditional covariance matrix is following the DCC model.
We follow the multistage estimation described in Section 1.3.3. The parameters of GARCH for each variables are estimated via QML at the first stage, and the parameters of the DCC model are estimated via variance targeting and composite likelihood with adjacent pairs, see Engle, et al. (2008) for details. From those two stages, the estimated standardized uncorrelated residuals ˆet are obtained, and those are used to estimate marginal distributions. At the last stage, the copula parameters are estimated by MCLE with adjacent pairs explained in Section 1.3.1. We repeat this scenario 500 times with time series of length T “ 1000 and cross sectional dimension N “ 10, 50, and 100. Table 1.4 reports all parameter estimates except Q. The columns for ψi, κi, λiand νireport the summary statistics obtained from 500ˆN estimates since those parameters set to the same numbers across cross sections.
Table 1.4 reveals that the estimated parameters are centered on the true values with the average estimated bias being small relative to the standard deviation. As the dimension size increases, the copula model parameters are more accurately esti- mated, which is also found in the previous section. Since this copula model keeps the dependence between any two variables identical, the amount of information on the unknown copula parameter increases as the dimension grows. The average computa- tion time is reported in the bottom row of each panel, and it indicates that multistage estimation is quite fast, for example, it takes five minutes for one hundred dimension model in which the number of parameters to estimate is more than 5000.
To see the impact of estimation errors from the former stages to copula estima- tion, we compare the standard deviations of copula estimations in Table 1.4 to the corresponding results in Table 1.2. The standard deviation increases by about 30% for N “ 10, and by about 19% for N “ 50 and 100. This loss of accuracy caused by having to estimate parameters of the marginals is considerably small given that more
than 5000 parameters are estimated in the former stages. We conclude that multi- stage estimations with composite likelihoods result in a large gain in computation and a small loss in estimation error and efficiency.