We propose a high-dimensional copula to model the dependence structure of the seemingly unrelated quantile regression. As the conventional model faces with the strong assumption of the multivariate normal distribution and the linear dependence structure, thus, we apply the multivariate exchangeable copula function to relax this assumption. As there are many parameters to be estimated, we consider the Bayesian Markov chain Monte Carlo approach to estimate the parameter interests in the model. Four simulation studies are conducted to assess the performance of our proposed model and Bayesian estimation. Satisfactory results from simulation studies are obtained suggesting the good performance and reliability of the Bayesian method used in our proposed model. The real data analysis is also provided, and the empirical comparison indicates our proposed model outperforms the conventional models in all considered quantile levels.

Typically, the seemingly unrelated regression (SUR) model, developed by Zellner [

Recently, the performance of this model has been questioned by many scholars, such as Jun and Pinkse [

In the literature, the copula-based models have already been introduced by many studies of which findings demonstrated a higher accuracy in parameter estimation. The pioneering work of Wichitaksorn and Choy [

Among various studies using the copula-based models to successfully confirm the role of copula functions in improving the efficiency of the system equation, none of them used the copulas to join the errors of the SUQR model except for Tansuchat et al. [

Consequently, in this study, we apply various multivariate exchangeable Archimedean copulas to join the equations in the SUQR model. We believe that our model will become more flexible and applicable to investigate the entire conditional distribution of the dependent variable and becomes more robust against outliers. In the estimation aspect, as our proposed model contains large parameter estimates and the full likelihood function is quite complicated, thus, the Bayesian estimation is employed in this study. To confirm the accuracy and reliability of our model and estimation, we conduct the simulation study and real data analysis to evaluate the performance of our model and the Bayesian estimation. To the best of our knowledge, no research introduced the multivariate exchangeable Archimedean copulas to join the error of equations in the SUQR model. To this end, we suggest a Bayesian approach to the multivariate exchangeable Archimedean copula-based SUQR.

The outline of the remaining sections is as follows. In Section

Jun and Pinkse [

Let

Typically,

According to Sklar’s theorem [

In this study, we consider two copula classes, namely, elliptical copulas and Archimedean copulas. The explicit form of these copula classes is presented in the next section.

Two classes of copulas, namely, elliptical and Archimedean copulas, are presented. Elliptical copulas consist of two families, i.e., Gaussian and Student-

Following Smith [

Posterior Gaussian copula:

Let

where

(2) Posterior Student-

Different from the elliptical copula case, we apply the uninformative prior to these density functions; thus, the posterior density for each of the Archimedean copula is as follows:

The density of this class’ copulas, namely, Frank, Clayton, Gumbel, and Joe, varies as proposed in Hofert et al. [

Frank copula:

where

Clayton copula:

Gumbel copula:

where

Joe copula:

To draw the updated dependence parameters of these copula functions, we randomly select the dependency parameter (

We consider the Bayesian approach for estimating all unknown parameters in our proposed model. The Bayesian estimation requires the specification of the likelihood function and the prior distribution for all the estimated parameters. Hence, the posterior density of our proposed model is constructed by multiplying the full likelihood function of the model (ALD densities and copula density) with a prior density of the parameters. Let us consider the first part of the likelihood; according to Yu and Moyeed [

For elliptical copulas,

For Archimedean copulas,

In the estimation aspect, all parameters are drawn by an iterative Gibbs sampler with the Metropolis–Hastings algorithm over a partition of parameter blocks: (i) the unknown parameter

As there are large parameter estimates in our model, the block Gibbs sampler with the MH sampling is considered to sample the parameters in the chain. The adaptive sampler algorithm can be explained as follows [

Starting at an initial parameter value

Updating the candidate parameter based on the proposal function; the proposal function

Simulate

where

Simulate

where

Simulate

Simulate

At the

Then, set

Repeat steps 2 and 3 for

In the sampling method, we specify the number iteration to be 50,000, whereas the first 20,000 iterations are discarded as burn-in. Then, we can obtain the estimated parameter by averaging the remaining 30,000 simulated sets of parameters

To learn about the performance of the Bayesian estimation for fitting our proposed model, we conduct four simulation studies:

First, we examine and evaluate the accuracy of the Bayesian estimation on our model under Clayton, Gumbel, and Frank copulas

In the second part, we examine the performance of our model when misspecified copula is assumed

In simulation study 3, we evaluate the finite sample performance of the Bayesian estimation under various sample sizes

Finally, we investigate the performance of the Bayesian method in the high-dimension setting

In the simulation study, Archimedean copula families, namely, Clayton, Gumbel, and Frank, are considered to model the dependence structure of the SUQR. In this study, the simulation is the realization of SUQR with three equations. Thus, our simulated equation can be written as

Note that the three-dimension error terms,

Mean and standard deviation of the Bayesian estimate of the Gumbel copula at three quantile levels.

Parameter | True | True | True | |||
---|---|---|---|---|---|---|

1 | 0.918 (0.114) | 1 | 0.828 (0.058) | 1 | 1.258 (0.052) | |

1 | 1.187 (0.105) | 5 | 5.024 (0.036) | 2 | 1.697 (0.045) | |

1 | 1.035 (0.100) | 1 | 0.949 (0.053) | 1 | 1.225 (0.060) | |

4 | 3.580 (0.092) | 2 | 2.384 (0.395) | 4 | 3.570 (0.210) | |

−3 | −3.009 (0.101) | −2 | −1.819 (0.152) | −0.2 | −0.198 (0.070) | |

2 | 2.091 (0.204) | 2 | 1.984 (0.088) | 2 | 3.242 (0.051) | |

2 | 2.308 (0.101) | 5 | 4.883 (0.088) | 1 | 0.904 (0.089) | |

3 | 3.736 (0.101) | 2 | 1.938 (0.196) | 3 | 2.659 (0.162) | |

3 | 3.125 (0.310) | 3 | 3.020 (0.302) | 3 | 3.147 (0.313) | |

2.5 | 2.458 (0.076) | 2.5 | 2.535 (0.072) | 2.5 | 2.401 (0.085) |

Note: ( ) is the average standard deviation of the parameters.

Mean and standard deviation of the Bayesian estimate of the Clayton copula at three quantile levels.

Parameter | True | True | True | |||
---|---|---|---|---|---|---|

1 | 1.004 (0.036 | 1 | 1.141 (0.057) | 1 | 0.929 (0.132) | |

1 | 1.114 (0.033) | 5 | 5.071 (0.105) | 2 | 1.944 (0.100) | |

1 | 1.027 (0.096) | 1 | 0.8104 (0.053) | 1 | 1.205 (0.073) | |

4 | 3.524 (0.033) | 2 | 2.443 (0.035) | 4 | 4.742 (0.055) | |

−3 | −2.967 (0.035) | −2 | −2.179 (0.115) | −0.2 | −0.1284 (0.0034) | |

2 | 2.107 (0.197) | 2 | 2.226 (0.083) | 2 | 2.062 (1.413) | |

2 | 2.281 (0.100 | 5 | 5.030 (0.098) | 1 | 0.833 (0.078) | |

3 | 3.752 (0.083) | 2 | 1.627 (0.153) | 3 | 2.655 (0.157) | |

3 | 3.158 (0.300) | 3 | 3.096 (0.292) | 3 | 2.709 (0.256) | |

0.5 | 0.3792 (0.095) | 0.5 | 0.370 (0.097) | 0.5 | 0.471 (0.106) |

Note: ( ) is the average standard deviation of the parameters.

Mean and standard deviation of the Bayesian estimate of the Frank copula at three quantile levels.

Parameter | True | True | True | |||
---|---|---|---|---|---|---|

1 | 1.016 (0.127) | 1 | 1.012 (0.067) | 1 | 1.509 (0.046) | |

1 | 1.196 (0.153) | 5 | 4.795 (0.403) | 2 | 1.738 (0.043) | |

1 | 1.599 (0.124) | 1 | 1.179 (0.149) | 1 | 0.739 (0.060) | |

4 | 3.038 (0.809) | 2 | 2.389 (1.001) | 4 | 3.640 (0.349) | |

−3 | −2.841 (0.061) | −2 | −2.371 (0.578) | −0.2 | −0.182 (0.060) | |

2 | 1.791 (0.045) | 2 | 2.496 (0.551) | 2 | 2.514 (0.242) | |

2 | 2.070 (0.104) | 5 | 5.113 (0.110) | 1 | 1.099 (0.109) | |

3 | 3.135 (0.209) | 2 | 1.896 (0.189) | 3 | 3.193 (0.224) | |

3 | 3.291 (0.332) | 3 | 3.019 (0.366) | 3 | 3.044 (0.301) | |

2 | 2.979 (0.481) | 2 | 3.015 (0.458) | 2 | 1.977 (0.419) |

Note: ( ) is the average standard deviation of the parameters.

In this section, another simulation study is proposed to measure the performance of the model using the relative entropy, also known as Kullback–Leibler divergence (KLD) [

In this study, we define

The purpose of this simulation study is to investigate the distance between the true SUQR function and its approximation when the copula is correctly specified and when the copula is misspecified. In this simulation study, we set the true posterior function to be the Student-

Figure

The performance of copula-based SUQR vs. conventional SUQR at different quantile levels,

In the third simulation study, the finite-sample properties of the Bayesian estimation in our proposed model are investigated upon the calculated absolute Bias and mean squared error (MSE) of the estimator. Again, we simulate the data sets the same way as in the first simulation study. The absolute Bias and MSE can be calculated by

Tables

Absolute Bias and MSE of parameter estimates from the Bayesian estimation of the Gumbel copula with different sample sizes.

Parameter | Absolute Bias | MSE | ||||
---|---|---|---|---|---|---|

0.0200 | 0.0115 | 0.0028 | 0.0747 | 0.0147 | 0.049 | |

0.0224 | 0.0073 | 0.0048 | 0.0699 | 0.0133 | 0.0053 | |

0.0356 | 0.0043 | 0.0058 | 0.0625 | 0.0120 | 0.0051 | |

0.0019 | 0.0016 | 0.0010 | 0.1138 | 0.0136 | 0.0051 | |

0.0465 | 0.0087 | 0.0012 | 0.0637 | 0.0108 | 0.0062 | |

0.0103 | 0.0101 | 0.0032 | 0.0766 | 0.0105 | 0.0067 | |

0.0431 | 0.0281 | 0.0083 | 0.0149 | 0.0094 | 0.0079 | |

0.0149 | 0.0073 | 0.0009 | 0.0409 | 0.0083 | 0.0039 | |

0.0066 | 0.0074 | 0.0037 | 0.0437 | 0.0100 | 0.0042 | |

0.0076 | 0.0082 | 0.0005 | 0.0492 | 0.0075 | 0.0036 | |

0.0007 | 0.0005 | 0.0004 | 0.0689 | 0.0085 | 0.0034 | |

0.0500 | 0.0117 | 0.018 | 0.0477 | 0.0084 | 0.0049 | |

0.0018 | 0.0011 | 0.0030 | 0.0467 | 0.0089 | 0.0046 | |

0.9005 | 0.8391 | 0.4918 | 0.8443 | 0.7433 | 0.1588 | |

0.0079 | 0.0013 | 0.0001 | 0.0499 | 0.0111 | 0.0015 | |

0.0085 | 0.0062 | 0.0030 | 0.0665 | 0.0125 | 0.0125 | |

0.0060 | 0.0037 | 0.0012 | 0.0464 | 0.0110 | 0.0014 | |

0.0032 | 0.0018 | 0.0008 | 0.0795 | 0.0121 | 0.0030 | |

0.0142 | 0.0123 | 0.0120 | 0.0544 | 0.0085 | 0.0025 | |

0.0099 | 0.0080 | 0.0040 | 0.0855 | 0.0147 | 0.0058 | |

0.2271 | 0.1321 | 0.1045 | 0.5613 | 0.4351 | 0.0105 |

Absolute Bias and MSE of parameter estimates from the Bayesian estimation of the Clayton copula with different sample sizes.

Parameter | Absolute Bias | MSE | ||||
---|---|---|---|---|---|---|

0.0819 | 0.0172 | 0.0100 | 0.0912 | 0.0123 | 0.0060 | |

0.0598 | 0.0033 | 0.0010 | 0.0793 | 0.0141 | 0.0052 | |

0.0488 | 0.0130 | 0.0093 | 0.0641 | 0.0132 | 0.0058 | |

0.0171 | 0.0105 | 0.0026 | 0.0915 | 0.0153 | 0.0047 | |

0.0571 | 0.0114 | 0.0098 | 0.0682 | 0.0145 | 0.0060 | |

0.0114 | 0.0131 | 0.0016 | 0.0747 | 0.0162 | 0.0045 | |

0.7584 | 0.2990 | 0.2099 | 0.7940 | 0.8015 | 0.6381 | |

0.0414 | 0.0148 | 0.0098 | 0.0473 | 0.0101 | 0.0041 | |

0.0109 | 0.0010 | 0.0007 | 0.0650 | 0.0104 | 0.0044 | |

0.0405 | 0.0087 | 0.0015 | 0.0379 | 0.0091 | 0.0040 | |

0.0291 | 0.0087 | 0.0045 | 0.0750 | 0.0091 | 0.0037 | |

0.0328 | 0.0121 | 0.0095 | 0.0497 | 0.0104 | 0.0035 | |

0.0094 | 0.0016 | 0.0013 | 0.0535 | 0.0089 | 0.0040 | |

0.8278 | 0.4235 | 0.2075 | 0.7639 | 0.2002 | 0.1655 | |

0.0183 | 0.0026 | 0.0009 | 0.0646 | 0.0095 | 0.0049 | |

0.0059 | 0.0012 | 0.0013 | 0.0822 | 0.0158 | 0.0073 | |

0.0178 | 0.0048 | 0.0033 | 0.0533 | 0.0103 | 0.0056 | |

0.0252 | 0.0037 | 0.0013 | 0.0816 | 0.0104 | 0.0049 | |

0.0290 | 0.0190 | 0.0057 | 0.0709 | 0.0107 | 0.0054 | |

0.0061 | 0.0010 | 0.0010 | 0.0715 | 0.0164 | 0.0045 | |

0.5555 | 0.5387 | 0.4306 | 0.3322 | 0.2926 | 0.2831 |

Absolute Bias and MSE of parameter estimates from the Bayesian estimation of the Frank copula with different sample sizes.

Parameter | Absolute Bias | MSE | ||||
---|---|---|---|---|---|---|

0.0155 | 0.0151 | 0.0061 | 0.0803 | 0.0100 | 0.0064 | |

0.0003 | 0.0071 | 0.0042 | 0.0874 | 0.0127 | 0.0059 | |

0.0367 | 0.0058 | 0.0016 | 0.0774 | 0.0123 | 0.0076 | |

0.0114 | 0.0083 | 0.0053 | 0.0694 | 0.0134 | 0.0070 | |

0.0078 | 0.0038 | 0.0030 | 0.0899 | 0.0135 | 0.0042 | |

0.0078 | 0.0012 | 0.0008 | 0.0778 | 0.0108 | 0.0063 | |

0.7279 | 0.7099 | 0.5023 | 0.6394 | 0.5238 | 0.4236 | |

0.0160 | 0.0070 | 0.0070 | 0.0411 | 0.0082 | 0.0002 | |

0.0150 | 0.0057 | 0.0050 | 0.0627 | 0.0102 | 0.0004 | |

0.0157 | 0.0068 | 0.0010 | 0.0497 | 0.0086 | 0.0006 | |

0.0053 | 0.0015 | 0.0005 | 0.4675 | 0.0096 | 0.0004 | |

0.0082 | 0.0001 | 0.0000 | 0.0441 | 0.0106 | 0.0050 | |

0.0240 | 0.0002 | 0.0001 | 0.0549 | 0.0075 | 0.0023 | |

0.0058 | 0.0095 | 0.0003 | 0.1395 | 0.0284 | 0.0015 | |

0.0013 | 0.0001 | 0.0009 | 0.0741 | 0.0108 | 0.0028 | |

0.0216 | 0.0173 | 0.0042 | 0.0676 | 0.0126 | 0.0058 | |

0.0107 | 0.0057 | 0.0034 | 0.0575 | 0.0101 | 0.0052 | |

0.0179 | 0.0019 | 0.0018 | 0.0575 | 0.0159 | 0.0052 | |

0.0217 | 0.0060 | 0.0011 | 0.0753 | 0.0131 | 0.0051 | |

0.0245 | 0.0122 | 0.0021 | 0.0697 | 0.0119 | 0.0061 | |

1.3613 | 1.0375 | 0.3658 | 0.9613 | 0.9184 | 0.8112 |

Our model is supposed to propose a quite general model with the possibly arbitrary

Once all the data are simulated, we fit the copula-based SUQR model with

Absolute Bias (first column) and MSE (second column) of the estimates of

In this section, we illustrate the applicability of our proposed model and the Bayesian estimation developed in this study, using the same data set as in Tansuchat et al. [

Descriptive statistics.

APPLE | MICRO | ADOBE | NASDAQ | |||
---|---|---|---|---|---|---|

Mean | 0.011 | 0.001 | 0.003 | 0.001 | 0.299 | 0.246 |

Median | 0.013 | 0.002 | 0.010 | 0.005 | 0.240 | 0.040 |

Maximum | 0.137 | 0.092 | 0.138 | 0.061 | 6.860 | 13.910 |

Minimum | −0.185 | −0.096 | −0.263 | −0.110 | −6.540 | −9.670 |

Std. dev. | 0.048 | 0.032 | 0.050 | 0.027 | 2.549 | 2.681 |

Skewness | −0.718 | −0.018 | −1.580 | −0.842 | 0.124 | 0.395 |

Kurtosis | 5.157 | 3.937 | 9.969 | 4.758 | 2.747 | 7.248 |

Jarque−Bera | 49.521 | 6.485 | 431.860 | 43.681 | 0.924 | 137.671 |

ADF test | 4.388 | 6.245 | 9.545 | 4.589 | 6.414 | 5.879 |

Note:

According to the data description, the mean of

Our empirical SUQR model is constructed under the Fama–French approach [

The choice of hyperparameters is very delicate in our Bayesian estimation; it is important to determine an approriate prior informations. In this empirical study, we decide to suggest three priors, which are as follows:

Weak informative prior: the hyperparameters

Diffusion prior

Informative prior:

Prior to showing the estimated results, we compare the performance of various copula-based models as well as the conventional model of Jun and Pinkse [

Model selection and prior sensitivity check.

DIC | Gaussian | Student- | Clayton | Gumbel | Joe | Frank | M0 | |
---|---|---|---|---|---|---|---|---|

Weak prior | −4028.48 | −3602.62 | −2301.10 | −2980.83 | −3316.27 | −3018.11 | ||

−2501.28 | −3313.28 | −3293.29 | −2238.56 | −2218.23 | −2458.22 | |||

−3074.94 | −2982.58 | −2430.20 | −2405.26 | −3352.41 | −3001.32 | |||

Diffusion prior | −4149.18 | −3515.92 | −2596.13 | −2596.13 | −3462.22 | −3084.18 | ||

−2473.64 | −3298.69 | −3203.15 | −2251.99 | 2244.39 | −2444.98 | |||

−3000.63 | −2814.36 | −2221.54 | −2211.18 | −3354.69 | −2987.14 | |||

Informative prior | −4052.51 | −3687.39 | −2347.66 | −2001.35 | −3398.41 | −3021.01 | ||

−2641.31 | −3343.14 | −3300.88 | −2288.98 | −2210.35 | −2500.54 | |||

−2942.64 | −2848.54 | −2200.87 | −2209.91 | −3358.69 | −3005.05 |

The bold number indicates the lowest DIC for each quantile level.

Table

Estimates of the parameters and their standard errors.

Parameter | Weak prior | Diffusion prior | Informative prior | ||||||
---|---|---|---|---|---|---|---|---|---|

−0.0057 (0.0042) | −0.0005 (0.0001) | 0.0848 (0.0003) | −0.0514 (0.0132) | 0.0071 (0.0012) | 0.0750 (0.0445) | −0.0420 (0.0102) | 0.0091 (0.0022) | −0.0662 (0.0004) | |

1.3604 (0.0141) | 0.3232 (0.0247) | 0.5866 (0.0033) | 1.2079 (0.1158) | 0.9491 (0.1442) | 0.4971 (0.1254) | 1.7481 (0.0212) | 0.7030 (0.0254) | 1.6397 (0.0047) | |

−0.0112 (0.0016) | −0.0011 (0.0001) | −0.0023 (0.0002) | −0.0013 (0.0002) | −0.0021 (0.0010) | −0.0004 (0.0002) | −0.0031 (0.0011) | −0.0053 (0.0001) | −0.006 (0.0002) | |

−0.0006 (0.0015) | −0.0001 (0.0001) | −0.0007 (0.0001) | −0.0015 (0.0021) | −0.0002 (0.0002) | −0.0001 (0.0010) | −0.0010 (0.0015) | −0.0008 (0.0001) | −0.0021 (0.0002) | |

−0.0124 (0.0028) | −0.0015 (0.0001) | 0.0231 (0.0012) | −0.0538 (0.0144) | −0.0562 (0.0254) | 0.0597 (0.0223) | −0.0448 (0.0024) | −0.0020 (0.0002) | 0.0577 (0.0012) | |

0.8528 (0.0011) | 0.7995 (0.0022) | 0.4134 (0.0049) | 1.6114 (0.1482) | 1.0801 (0.2443) | 0.8571 (0.1840) | 1.8110 (0.0022) | 1.0664 (0.0065) | 0.6558 (0.0053) | |

−0.0044 (0.0015) | −0.0028 (0.0001) | −0.0124 (0.0001) | −0.0019 (0.0023) | −0.0050 (0.0012) | −0.0013 (0.0001) | −0.0034 (0.0014) | −0.0015 (0.0001) | −0.0043 (0.0001) | |

−0.0446 (0.0024) | −0.0001 (0.0001) | −0.0003 (0.0001) | −0.0004 (0.0001) | −0.0003 (0.0002) | −0.0009 (0.0003) | −0.0314 (0.0022) | −0.0007 (0.0001) | −0.0003 (0.0001) | |

−0.0111 (0.0039) | 0.0025 (0.0011) | 0.0428 (0.0021) | −0.0541 (0.0113) | 0.0042 (0.0022) | 0.0547 (0.0123) | −0.0431 (0.0042) | 0.0030 (0.0012) | 0.0546 (0.0020 | |

1.3206 (1.0111) | 1.1543 (0.0033) | 0.8983 (0.3241) | 0.9690 (1.0031) | 0.8371 (0.0412) | 0.7745 (0.4112) | 1.7503 (1.0032) | 1.0361 (0.0022) | 0.6351 (0.3214) | |

−0.0011 (0.0020) | −0.0022 (0.0001) | −0.0485 (0.0023) | −0.0045 (0.0022) | −0.0013 (0.0005) | −0.0122 (0.0034) | −0.0078 (0.0032) | −0.0058 (0.0010) | −0.0068 (0.0015) | |

−0.0058 (0.0023) | −0.00028 (0.0001) | −0.0006 (0.0003) | −0.0021 (0.0015) | 0.0001 (0.0001) | −0.0002 (0.0003) | −0.0017 (0.0008) | −0.0003 (0.0002) | −0.0004 (0.0002) | |

Copula | Clayton | Frank | Clayton | Clayton | Frank | Clayton | Clayton | Frank | Clayton |

8.1421 (0.0121) | 0.0001 (0.0012) | 13.6251 (0.8812) | 7.6548 (0.0135) | 0.0001 (0.0001) | 10.6257 (0.5547) | 5.9844 (1.3554) | 0.0001 (0.0001) | 11.2898 (1.0548) |

Note: the brackets ( ) denote the standard deviation.

Scatter plots among standardized residuals at

To formally check the convergence of MCMC chains, the marginal posterior distributions can be visualized by plotting the histograms of the simulated parameter draws. Figure

Posterior histograms for copula parameters for the copula-based SUQR model at

To carry out the goodness-of-fit test for our proposed models, Cramer–von Mises (CvM) method is conducted in this section. Genest et al. [

Goodness-of-fit test for the copula-based SUQR model.

DIC | Gaussian | Student- | Clayton | Gumbel | Joe | Frank | |
---|---|---|---|---|---|---|---|

Weak prior | 0.1247 | 0.0134 | 0.0098 | 0.0148 | 0.0049 | ||

0.0525 | 0.2733 | 0.1149 | 0.0148 | 0.0849 | |||

0.0049 | 0.0049 | 0.0049 | 0.0443 | 0.1041 | |||

Diffusion prior | 0.1244 | 0.0134 | 0.0095 | 0.0141 | 0.0048 | ||

0.0521 | 0.2731 | 0.1147 | 0.0147 | 0.0844 | |||

0.0042 | 0.0050 | 0.0048 | 0.0443 | 0.1040 | |||

Informative prior | 0.1240 | 0.0135 | 0.0097 | 0.0150 | 0.0050 | ||

0.0521 | 0.2734 | 0.1145 | 0.0147 | 0.0851 | |||

0.0047 | 0.0054 | 0.0048 | 0.1445 | 0.1042 |

Note: this table presents the

In this paper, we introduced the multivariate exchangeable Archimedean copula to join the errors of the seemingly unrelated quantile regression (SUQR). The model becomes more accurate and robust against the outlier relationship between the dependent and independent variables. We also introduced the Bayesian Markov chain Monte Carlo approach to estimate the parameter sets of our proposed model. As the posterior distribution of the copula parameter does not appear to be in any form, therefore, we employ a Bayesian estimation together with a Gibbs sampler with the Metropolis–Hastings algorithm to infer the full posterior distribution. To examine the accuracy of our Bayesian estimation and performance of our proposed model, we present the simulation study and real data analysis.

Four simulation studies are conducted. The result of the first simulation study shows the accuracy of the Bayesian estimation. The results confirm that our proposed model is well estimated as reliable estimation results are obtained for every quartile level, and the parameter estimates on average are close to their true values. In the second simulation, we adopt the Kullback–Leibler divergence (KLD) to measure the distance between the true posterior probability distribution and the approximated posterior probability when the copula function is unknown. The result confirms the robustness of our model and presents the closest distance between the correctly specified and the true probability function. The third simulation is proposed for examining the finite-sample properties of the Bayesian estimation in our proposed model. The result shows that the Bayesian estimation provides reliable parameter estimates as the absolute Biases and MSEs converge to zero when the sample size increases. Finally, the performance of the Bayesian estimation in high-dimension copula-based SUQR is investigated, and the result reveals that we may not gain the accurate parameter estimates inhigher dimensions. However, the Bayesian estimation is still performing promisingly acceptable in high dimensions as the absolute Bias and MSE are not quite high.

In the real data application, we apply our proposed model to the data set provided by Tansuchat et al. [

Although our proposed model presents a good performance in both simulation and real data studies, Gibbs algorithm with MH may not be the most suitable for our proposed model, given the large number of parameters to estimate. In addition, it can be difficult to choose the proposal functions in the Gibbs sampler with the MH sampling algorithm. For future study, the adaptive Gibbs sampler with the MH algorithm, which uses the history of the process to tune the appropriated proposal distribution, can be applied to our proposed model. Moreover, future research should consider applying our proposed model subject to pseudo-cyclical-structural changes since many financial time series exhibit behavioral change over time.

In this study, we use the simulated data to show the performance of our model, and the simulation processes are already explained in the paper. For the real data analysis section, we use the same data of Tansuchat et al. [

The authors declare that they have no conflicts of interest.

The authors are grateful for the financial support from the Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University. They thank Dr. Laxmi Worachai for valuable comments to improve this paper.