Copulas are a tool for constructing multivariate distributions and formalizing the dependence structure between random variables. From copula literature review, there are a few asymmetric copulas available so far while data collected from the real world often exhibit asymmetric nature. This necessitates developing asymmetric copulas. In this study, we discuss a method to construct a new class of bivariate asymmetric copulas based on products of symmetric (sometimes asymmetric) copulas with powered arguments in order to determine if the proposed construction can offer an added value for modeling asymmetric bivariate data. With these newly constructed copulas, we investigate dependence properties and measure of association between random variables. In addition, the test of symmetry of data and the estimation of hyper-parameters by the maximum likelihood method are discussed. With two real example such as car rental data and economic indicators data, we perform the goodness-of-fit test of our proposed asymmetric copulas. For these data, some of the proposed models turned out to be successful whereas the existing copulas were mostly unsuccessful. The method of presented here can be useful in fields such as finance, climate and social science.
Copulas offer a useful tool in modeling the dependence among random variables. For example, Busababodhin and Amphanthong (2016) applied copula in the multivariate statistical process control and Kim (2014) used copula-GARCH for the modeling of dependence structure of Korea financial markets. In the literature, most of the existing copulas, however, are symmetric while data collected from the real world may exhibit asymmetric nature. This necessitates developing asymmetric copulas. Many researchers proposed some methods to construct asymmetric copulas; Rodríguez-Lallena and Úbeda-Flores (2004) introduced a class of bivariate copulas that generalizes some known families. Kim
In this study, we discuss a method to construct a new class of bivariate asymmetric copulas based on products of symmetric (sometimes asymmetric) copulas with powered arguments. Then we would like to determine if the proposed construction can offer an added value for modeling asymmetric data. This construction is based on the result of Durante (2009). Our proposal is actually an extension of Durante (2009) for a wide range of copulas which includes some newly constructed copulas in addition to all copula families available in the current literature. With these newly constructed copulas, we investigate dependence properties and measure of association between random variables. We consider the result of Mukherjee
This paper is organized as follows. Section 2 contains some basic concepts of copulas and the dependence structure by calculating Spearman’s rho and Kendall’s tau using asymmetric copulas. In Section 3, we introduce Fourier copula and new class of bivariate asymmetric copulas. Goodness-of-fit of the proposed asymmetric copulas is introduced in Section 4. Test of symmetry for bivariate case and the maximum likelihood estimation of hyper-parameters for the constructed copulas are discussed in Sections 5 and 6, respectively. Section 7 shows the illustrative data analysis for the proposed asymmetric copula models with two real data. Finally, the discussion and conclusion are presented in Section 8.
In this section we recall some definitions and results that are necessary to understand a (bivariate) copula. A copula is a multivariate distribution function defined on where , with uniformly distributed marginals. In this paper, we focus on bivariate copulas.
(
The importance of copulas has been growing because of their applications in several fields of research. Their relevance primarily comes from Sklar’s Theorem (Sklar, 1959):
In terms of dependence properties, Spearman’s rho is a measure of average quadrant dependence, while Kendall’s tau is a measure of average likelihood ratio dependence (see Nelsen (2006) for details). If
Denoting
A copula
Then corresponding Spearman’s rho and Kendall’s tau are given by, respectively,
The optimal values of
In this section we will first define Fourier copulas (Lowin, 2010) and then construct asymmetric (in general) copulas using the following theorem (see Durante (2009) for details).
It is natural to write the function
where ℤ^{0} = ℤ{0} and
is called a Fourier copula, which was apparently introduced by Ibragimov (2009). It is sufficient that if
then
and hence using (
In this subsection we use Theorem 2 to construct a class of asymmetric copulas and will find corresponding Spearman’s rho and Kendall’s tau for these new copulas to have a qualitative idea of which asymmetric copula has a better range of
Therefore we would like to mention here that
The symmetry of
is equivalent to (
Hence we have
For convenience we adopt the following notations, for
Notice that
For
Figures 1 and 2 clearly show that the contour plots of
We assume that we have a random sample
The maximum pseudo-likelihood estimate (MPLE) of
where
are the pseudo-observations. The estimate is generally found by numerical maximization of (
A rigorous approach to compare the fit of different copulas to the same data consists of using goodness-of-fit tests. The issue is whether the unknown copula
where the random vectors
According to the large scale simulations carried out in Genest
An approximate
This section briefly deals with methods to test the symmetry of bivariate data. For that, it is reasonable to compare values of
See also Bouzebda and Cherfi (2012) and Quessy and Bahraoui (2013) for other test procedures for the symmetry of copulas. Nelsen (2007) considered another measure of asymmetry. In this study, we use a Cramér-von Mises statistic
In the previous section, we estimated the parameters of copulas with the powered hyper-parameters
By the
where (
We consider two datasets to illustrate the usefulness of our proposed asymmetric copulas. The first dataset is car rental data of American new cars and trucks data for sport utility vehicle (SUV) with four wheel drive which is available at Nayland College. Engine size and retail price variables with sample size
Table 4 shows the result of parameter estimates, values of Bayesian information criterion (BIC), values of Akaike information criterion (AIC), and Cramér-von Mises goodness-of-fit statistics (
The second datasets is monthly economic indicators of Korea from Jan. 2011 to Aug. 2013, available at (Statistics Korea). Certificate of deposit (CD) rate and interest rate variables with sample size
Table 6 shows the result of parameter estimates, values of BIC, AIC, and
We discussed a new generalized copula family which includes a class of asymmetric copulas as well as all copula families available in the current literature, including Fourier copula. The construction of new asymmetric family is based on and an extension of the result by Durante (2009). With diverse data such as simulated data, car rental data, and economic indicators, we performed parameter estimation by using the maximum pseudo-likelihood estimation method and Cramér-von Mises type of goodness-of-fit tests for the newly constructed asymmetric copula family. For these data, some of the proposed models turned out to be successful whereas the existing copulas were mostly unsuccessful. We thus argue that the proposed construction can offer an added value to model asymmetric bivariate data.
For the estimation of the hyper-parameters (
In our future study, we would extend our copula method to a multivariate case, to develop a generalized composite operator of asymmetric copula family as in Louzada and Ferreira (2016), to apply to the direction data from Kim and Kim (2014), and to incorporate time varying component as in Ara
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1A2B4014518). Lee’s work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1A6A3A11032852).
Nine basic Copula Functions used in this study to construct bivariate asymmetric copulas
Copula name | Copula function |
---|---|
Fourier | |
Max | |
Min | |
Independent | |
FGM | |
Clayton | |
Frank | |
Gumbel | |
AMH |
FGM = Farlie-Gumbel-Morgenstern family; AMH = Ali-Mikhail-Haq family.
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | ||
---|---|---|---|---|---|---|---|---|---|
0.2 | 0.62 | ||||||||
0.41 | |||||||||
0.3 | 0.56 | 0.51 | |||||||
0.37 | 0.34 | ||||||||
0.4 | 0.49 | 0.44 | 0.38 | ||||||
0.32 | 0.29 | 0.25 | |||||||
0.5 | 0.41 | 0.36 | 0.30 | 0.23 | |||||
0.27 | 0.24 | 0.20 | 0.15 | ||||||
0.6 | 0.33 | 0.28 | 0.21 | 0.15 | 0.08 | ||||
0.22 | 0.18 | 0.14 | 0.10 | 0.05 | |||||
0.7 | 0.24 | 0.19 | 0.13 | 0.07 | 0.01 | −0.05 | |||
0.16 | 0.13 | 0.09 | 0.04 | 0.00 | −0.04 | ||||
0.8 | 0.15 | 0.10 | 0.05 | −0.00 | −0.06 | −0.11 | −0.15 | ||
0.10 | 0.07 | 0.03 | −0.01 | −0.05 | −0.08 | −0.12 | |||
0.9 | 0.06 | 0.02 | −0.02 | −0.07 | −0.12 | −0.16 | −0.20 | −0.23 | |
0.04 | 0.01 | −0.02 | −0.05 | −0.09 | −0.12 | −0.15 | −0.18 |
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | ||
---|---|---|---|---|---|---|---|---|---|
0.2 | 0.81 | ||||||||
0.67 | |||||||||
0.3 | 0.73 | 0.68 | |||||||
0.59 | 0.54 | ||||||||
0.4 | 0.65 | 0.60 | 0.54 | ||||||
0.51 | 0.47 | 0.41 | |||||||
0.5 | 0.57 | 0.52 | 0.46 | 0.38 | |||||
0.43 | 0.39 | 0.34 | 0.28 | ||||||
0.6 | 0.47 | 0.42 | 0.36 | 0.30 | 0.23 | ||||
0.35 | 0.31 | 0.26 | 0.21 | 0.16 | |||||
0.7 | 0.36 | 0.32 | 0.27 | 0.21 | 0.15 | 0.08 | |||
0.26 | 0.23 | 0.19 | 0.14 | 0.09 | 0.05 | ||||
0.8 | 0.25 | 0.21 | 0.16 | 0.11 | 0.06 | 0.00 | −0.05 | ||
0.17 | 0.14 | 0.11 | 0.07 | 0.03 | −0.01 | −0.05 | |||
0.9 | 0.12 | 0.09 | 0.04 | 0.00 | −0.05 | −0.09 | −0.13 | −0.17 | |
0.08 | 0.06 | 0.02 | −0.01 | −0.05 | −0.08 | −0.11 | −0.15 |
Result of parameter estimates, values of BIC, AIC, and
Copula name | Parameter | BIC | AIC | ||
---|---|---|---|---|---|
Frank | 7.328 | −29.328 | −30.966 | 0.035 | 0.143 |
Clayton | 1.972 | −25.367 | −27.005 | 0.066 | 0.029 |
Max | 0.113 | −20.905 | −22.543 | 0.094 | 0.007 |
AMH | 1.000 | −19.877 | −21.515 | 0.155 | 0.001 |
Gumbel | 1.959 | −18.270 | −19.907 | 0.071 | 0.042 |
FGM | 1.000 | −11.796 | −13.433 | 0.243 | 0.001 |
Independent | 0.000 | 0.000 | 0.505 | 0.001 | |
Min | 45423.710 | 3.638 | 2.000 | 0.505 | 0.001 |
Fourier | 0.000 | 3.640 | 2.003 | 0.505 | 0.001 |
BIC = Bayesian information criterion; AIC = Akaike information criterion;
Result of parameter estimates, values of BIC, AIC, and
Copula name | par1 | par2 | BIC | AIC | ||||
---|---|---|---|---|---|---|---|---|
Max × Clayton | 0.001 | 7.905 | 0.448 | 0.423 | −26.315 | −32.865 | 0.055 | 0.197 |
Independent × Frank | 7.195 | 0.001 | 0.001 | −22.024 | −26.937 | 0.036 | 0.276 | |
Clayton × Frank | 13.119 | 7.647 | 0.334 | 0.410 | −21.499 | −28.050 | 0.031 | 0.356 |
Independent × Clayton | 4.280 | 0.186 | 0.071 | −21.144 | −26.056 | 0.049 | 0.216 | |
Frank × Gumbel | 8.691 | 21.269 | 0.804 | 0.948 | −20.853 | −27.401 | 0.480 | 0.351 |
Fourier × Clayton | 0.999 | 4.950 | 0.228 | 0.124 | −20.202 | −26.752 | 0.049 | 0.177 |
Clayton × Gumbel | 3.715 | 20.611 | 0.754 | 0.934 | −20.195 | −26.745 | 0.042 | 0.201 |
Max × Frank | 3.571 | 7.360 | 0.001 | 0.001 | −18.397 | −24.947 | 0.035 | 0.425 |
Min × Frank | 3.578 | 7.359 | 0.001 | 0.001 | −18.397 | −24.947 | 0.035 | 0.336 |
FGM × Frank | 0.020 | 7.320 | 0.001 | 0.001 | −18.397 | −24.947 | 0.035 | 0.311 |
Fourier × Frank | 0.867 | 7.245 | 0.001 | 0.001 | −18.393 | −24.944 | 0.036 | 0.311 |
Frank × AMH | 7.586 | 0.849 | 0.999 | 0.999 | −18.371 | −24.922 | 0.033 | 0.391 |
Min × Clayton | 0.001 | 4.232 | 0.200 | 0.052 | −18.197 | −24.747 | 0.054 | 0.142 |
Max × Gumbel | 0.001 | 2.742 | 0.608 | 0.462 | −17.735 | −24.286 | 0.061 | 0.067 |
FGM × Clayton | −0.999 | 4.185 | 0.173 | 0.058 | −17.687 | −24.237 | 0.049 | 0.187 |
Clayton × AMH | 4.181 | −0.999 | 0.827 | 0.943 | −17.658 | −24.208 | 0.049 | 0.192 |
Max × Independent | 0.113 | 0.996 | 0.999 | −13.571 | −18.484 | 0.094 | 0.017 | |
Independent × AMH | 0.999 | 0.001 | 0.001 | −12.558 | −17.471 | 0.155 | 0.001 | |
Independent × Gumbel | 1.961 | 0.000 | 0.000 | −10.988 | −15.900 | 0.071 | 0.047 | |
Fourier × Max | 0.449 | 0.113 | 0.001 | 0.001 | −9.945 | −16.495 | 0.094 | 0.022 |
Max × FGM | 0.113 | −0.601 | 0.991 | 0.999 | −9.880 | −16.431 | 0.095 | 0.022 |
Max × AMH | 0.113 | −0.601 | 0.991 | 0.999 | −9.880 | −16.431 | 0.095 | 0.167 |
Max × Min | 0.113 | 1.924 | 0.990 | 0.999 | −9.877 | −16.427 | 0.095 | 0.017 |
Gumbel × AMH | 5.596 | 0.999 | 0.184 | 0.238 | −9.666 | −16.216 | 0.130 | 0.012 |
Min × AMH | 0.925 | 1.000 | 0.001 | 0.000 | −8.938 | −15.488 | 0.155 | 0.001 |
Fourier × AMH | 0.962 | 1.000 | 0.001 | 0.001 | −8.928 | −15.478 | 0.155 | 0.001 |
FGM × AMH | −0.539 | 0.999 | 0.001 | 0.001 | −8.903 | −15.454 | 0.156 | 0.001 |
Min × Gumbel | 8.174 | 1.960 | 0.001 | 0.001 | −7.345 | −13.896 | 0.071 | 0.027 |
Fourier × Gumbel | 0.900 | 1.970 | 0.001 | 0.001 | −7.336 | −13.886 | 0.070 | 0.042 |
FGM × Gumbel | 0.797 | 1.970 | 0.001 | 0.001 | −7.336 | −13.886 | 0.070 | 0.201 |
Independent × FGM | 0.998 | 0.001 | 0.001 | −4.495 | −9.408 | 0.244 | 0.001 | |
Fourier × FGM | 0.606 | 0.999 | 0.001 | 0.001 | −0.860 | −7.410 | 0.244 | 0.001 |
Min × FGM | 9.931 | 1.000 | 0.000 | 0.003 | −0.850 | −7.400 | 0.244 | 0.001 |
Fourier × Independent | 0.999 | 0.182 | 0.187 | 9.109 | 4.196 | 0.478 | 0.001 | |
Min × Independent | 8.980 | 0.067 | 0.001 | 10.913 | 6.000 | 0.505 | 0.001 | |
Fourier × Min | 0.999 | 6.595 | 0.175 | 0.192 | 12.833 | 6.282 | 0.480 | 0.001 |
BIC = Bayesian information criterion; AIC = Akaike information criterion;
Result of parameter estimates, values of BIC, AIC, and
Copula name | par | BIC | AIC | ||
---|---|---|---|---|---|
Max | 0.096 | −26.055 | −27.839 | 0.089 | 0.017 |
Gumbel | 1.867 | −20.218 | −22.002 | 0.062 | 0.107 |
Frank | 5.048 | −19.045 | −20.829 | 0.073 | 0.032 |
FGM | 1.000 | −11.786 | −13.570 | 0.223 | 0.001 |
AMH | 0.859 | −8.993 | −10.777 | 0.208 | 0.001 |
Clayton | 0.826 | −6.796 | −8.580 | 0.197 | 0.001 |
Independent | 0.000 | 0.000 | 0.462 | 0.001 | |
Min | 87109.604 | 3.784 | 2.000 | 0.462 | 0.001 |
Fourier | 0.000 | 3.802 | 2.017 | 0.462 | 0.001 |
BIC = Bayesian information criterion; AIC = Akaike information criterion;
Result of parameter estimates, values of BIC, AIC, and
Copula name | par1 | par2 | BIC | AIC | ||||
---|---|---|---|---|---|---|---|---|
Clayton × Gumbel | 13.478 | 5.576 | 0.605 | 0.172 | −27.187 | −34.324 | 0.044 | 0.236 |
Frank × Gumbel | 17.132 | 5.471 | 0.598 | 0.166 | −26.130 | −33.267 | 0.043 | 0.221 |
Independent × Gumbel | 5.037 | 0.500 | 0.001 | −25.799 | −31.151 | 0.064 | 0.107 | |
Independent × Clayton | 13.916 | 0.483 | 0.080 | −25.585 | −30.938 | 0.072 | 0.082 | |
Fourier × Clayton | 0.999 | 10.218 | 0.491 | 0.088 | −23.759 | −30.896 | 0.078 | 0.057 |
Clayton × Frank | 11.221 | 18.584 | 0.446 | 0.852 | −23.380 | −30.516 | 0.052 | 0.152 |
Min × Clayton | 0.001 | 13.566 | 0.474 | 0.067 | −23.172 | −30.309 | 0.078 | 0.052 |
Fourier × Gumbel | 0.999 | 4.641 | 0.543 | 0.128 | −23.139 | −30.276 | 0.089 | 0.062 |
Clayton × AMH | 13.112 | −0.999 | 0.531 | 0.947 | −22.091 | −29.228 | 0.071 | 0.072 |
FGM × Gumbel | 0.728 | 4.705 | 0.492 | 0.001 | −22.090 | −29.227 | 0.063 | 0.107 |
Max × Gumbel | 4.443 | 4.571 | 0.482 | 0.001 | −22.062 | −29.199 | 0.061 | 0.201 |
Min × Gumbel | 4.446 | 4.569 | 0.481 | 0.001 | −22.062 | −29.199 | 0.061 | 0.152 |
FGM × Clayton | −0.999 | 11.810 | 0.466 | 0.046 | −21.995 | −29.132 | 0.072 | 0.067 |
Gumbel × AMH | 5.108 | 0.876 | 0.499 | 0.999 | −21.979 | −29.116 | 0.064 | 0.112 |
Max × Clayton | 146.628 | 13.915 | 0.483 | 0.080 | −21.801 | −28.938 | 0.072 | 0.072 |
Independent × Frank | 9.868 | 0.440 | 0.001 | −18.546 | −23.899 | 0.083 | 0.037 | |
Max × Independent | 0.098 | 0.999 | 0.999 | −18.457 | −23.810 | 0.090 | 0.012 | |
FGM × Frank | −0.475 | 10.201 | 0.441 | 0.001 | −15.205 | −22.342 | 0.081 | 0.042 |
Fourier × Frank | 0.999 | 9.462 | 0.496 | 0.158 | −14.897 | −22.034 | 0.121 | 0.012 |
Frank × AMH | 9.985 | 0.053 | 0.566 | 0.998 | −14.871 | −22.008 | 0.081 | 0.047 |
Max × Frank | 2.134 | 9.869 | 0.446 | 0.001 | −14.749 | −21.885 | 0.085 | 0.037 |
Fourier × Max | 0.549 | 0.095 | 0.001 | 0.001 | −14.678 | −21.815 | 0.089 | 0.022 |
Max × FGM | 0.098 | −0.316 | 1.000 | 0.999 | −14.677 | −21.814 | 0.090 | 0.017 |
Max × AMH | 0.098 | −0.316 | 1.000 | 0.999 | −14.677 | −21.814 | 0.090 | 0.017 |
Max × Min | 0.097 | 1.924 | 0.999 | 0.999 | −14.675 | −21.812 | 0.089 | 0.022 |
Min × Frank | 2.135 | 9.835 | 0.453 | 0.001 | −14.674 | −21.811 | 0.087 | 0.032 |
Independent × FGM | 1.000 | 0.001 | 0.001 | −4.200 | −9.553 | 0.223 | 0.001 | |
Independent × AMH | 0.867 | 0.005 | 0.001 | −1.393 | −6.746 | 0.207 | 0.001 | |
FGM × AMH | 0.999 | −0.849 | 1.000 | 0.998 | −0.404 | −7.540 | 0.223 | 0.001 |
Min × FGM | 5.103 | 0.999 | 0.001 | 0.001 | −0.399 | −7.536 | 0.223 | 0.001 |
Fourier × FGM | 0.958 | 1.000 | 0.005 | 0.001 | −0.367 | −7.504 | 0.224 | 0.001 |
Fourier × AMH | 0.990 | 0.999 | 0.350 | 0.275 | −0.254 | −7.391 | 0.280 | 0.001 |
Min × AMH | 1.935 | 0.870 | 0.001 | 0.003 | 2.403 | −4.734 | 0.206 | 0.001 |
Fourier × Independent | 1.000 | 0.324 | 0.270 | 6.969 | 1.617 | 0.444 | 0.001 | |
Fourier × Min | 0.999 | 7.659 | 0.347 | 0.256 | 10.808 | 3.671 | 0.446 | 0.001 |
Min × Independent | 8.980 | 0.067 | 0.001 | 11.353 | 6.000 | 0.462 | 0.001 |
BIC = Bayesian information criterion; AIC = Akaike information criterion;