Fitted Copula Statistical Models for Four African and Four Major Stock Markets

The application of copula has become popular in recent years. The use of correlation as a dependence measure has several pitfalls and hence the application of regression prediction model using this correlation may not be an appropriate method. In financial markets, there is often a non-linear dependence between returns. Thus, alternative methods for capturing co-dependency should be considered, such as copula based ones. This paper studies the dependence structure between the four largest African stock markets in terms of market capitalization and other developed stock markets over the period 2003 to 2018 using copula models. The value at risk was used to determine the risk associated with the stock. The ten copula models were fitted to the log returns calculated from the data, two countries at a time of the twenty-eight pairs and examined. The Gumbel copula gives the best fit in terms of log-likelihood values, value of the Akaike information criterion, value of the Bayesian information criterion, value of the consistent Akaike information criterion, value of the corrected Akaike information criterion, value of the Hannan Quinn criterion and p-value of the


Introduction
The formation of regional bodies like African Union (AU), Economic Community of West African States (ECOWAS) does not only have political implications to the countries in question, but economic ones as well. Trade amongst the various countries in the region can lead to mutual benefits or losses. It is imperative to assess the extent to which dependence amongst these countries influence their economies. According to Bekaert and Harvey [6], though the joint distribution of multivariate variables are usually assumed to follow the normal distribution, economic variable like the stock market index do not follow the normal distribution since they tend to be skewed, peaked and have extreme values.
Research in emerging stock markets has suggested a number of empirical characteristics that international investors should be aware of. In particular, there is a growing body of evidence that emerging market securities (such as African stock markets) tend to offer larger returns with higher volatility compared to developed stock markets (e.g. Harvey [29], Bekaert et al. [8], Bekaert and Harvey [7]). In addition, they show greater evidence of predictability (e.g. Harvey [29], Claessens and Gooptu [14]) and lower correlation with developed stock market securities implying significant risk diversification opportunities for international portfolios (e.g. Bailey and Stulz [5], Divecha et al. [19], Harvey [29] and Errunza and Hogan [21]). Although it is also argued that the behavior of emerging markets is affected to a greater extent by local political, economic and social events rather than global events (e.g. Aggarwal et al. [1], Bekaert and Harvey [7] and Susmel and Thompson [45]), more recent evidence has suggested that the diversification benefits of these markets have started to diminish because of changes in investment barriers for international investors (Errunza et al. [22], Bekaert and Harvey [7]).

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Frees and Valdez [25] in their work, noted that the main attraction of a copula approach is that it enables us model the dependence structure of a multivariate distribution separately from the marginal distribution functions of the individual random variables. This ability to separate the dependence structure from the marginals makes copula much greater modeling flexible tool than is possible with traditional approaches to multivariate problems.
Soytas and Sari [44] analyzed the relationship between energy consumption in industrial and manufacturing sector using multivariate model by incorporating capital and labour in production function. Their results indicated co-integration between the variables for long run relationship. The results of vector error correction (VECM) model reveal that there is unidirectional causality running from energy consumption to manufacturing GDP.
Chukwudum [13] analyzed a homogenous portfolio consisting of the aggregate bivariate losses from the Nigerian insurance using the Generalized Pareto distribution (GPD) and the copula technique. It was observed that the correlation coefficient vary and is generally weak. With the aid of the Archimedean copula, the analysis makes use of the data pair exhibiting the highest correlation to draw particular attention to the importance of taken into account the extremely dependence structure when quantifying the risk capital, allocating risk and when estimating the net reinsurance premium under different reinsurance strategies.
Chang [11] simultaneously investigate the dynamic process of crude oil spot and futures returns and the time-varying and asymmetric dependence between spot and futures returns. Using the Gumbel and Clayton copulas, the time-varying and asymmetric dependence was captured. It was found that jumping behavior is an important process for each market. Spot and futures returns do not have the same jump process and the tail dependence between spot and futures markets is time-varying and asymmetric with the magnitude of upper tail dependence being slightly weaker than that of lower tail dependence.
Chen et al. [12] studied the dependence structure (copula) of multivariate financial time series of U.S. equity returns and exchange rates, considering collections of up to 30 assets simultaneously using Student's t copula and normal copula. Mixed evidence was found against the more flexible Student's t copula; it appears adequate for even large collections of equity returns, but is still rejected for most exchange rate returns though it does provide a better fit than the normal copula.

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Zeevi and Mashal [37] investigated the potential for extreme co-movements between financial assets and its dependence structure. Using likelihood ratio-based method, the Student's t and Guassian copula they show that the presence of extreme co-movements is statistically significant in three asset markets (equities, currencies and commodities), as well as across international (G5) equity markets. The t-dependence structure is well suited for this objective in so far as it provides a natural ''first step'' generalization of the correlation-based Guassian dependence structure.
Patton [40] studied the time varying conditional joint distribution of the daily Deutsche mark-U.S. dollar and Yen-U.S. dollar exchange rates, over the period from January 1991 to December 2001 using standard AR-TGARCH models. These were employed for the marginal distributions of each exchange rate, and two different copulas were estimated: the bivariate normal distribution, and the 'symmetrised Joe-Clayton' copula. Dependence was greater during appreciations of the U.S. dollar than during depreciations of the U.S. dollar.
Sadegh et al. [41] presented the Multivariate Copula Analysis Toolbox (MvCAT) which comprises different copula to model parameter uncertainty of assessment using joint precipitation-soil moisture anomalies in Del Norte county, California and flood peak and volume frequency analysis in the Saguenay River in Quebec, Canada. It shows that length of record significantly affects the uncertainty of results; MvCAT offers uncertainty bounds for the copula probability isolines. This information is particularly useful in multivariate frequency analysis studies.
Fontaine et al. [24] analysed the censored cost-effectiveness using a copula-based modeling of the joint density and an estimation method of the costs, and quality adjusted life years (QALY) in a cost-effectiveness analysis in case of censoring. They concluded that for the cost-effectiveness such technique without any linearity assumption is a progress since it does not need the specification of a global linear regression model. Hence, the estimation of the marginal distributions for each therapeutic arm, the concordance measures between these populations and the right copulas families is now sufficient to process the whole Cost Effectiveness Analysis.
Kumar and Shoukri [34] studied the application to an aortic regurgitation using copula based prediction models. They showed that copula-based prediction modeling is demonstrated to be an appropriate alternative to the conventional correlation-based prediction modeling since the correlation-based prediction models are not appropriate to model the dependence in populations with asymmetrical tails. They validated their proposed copula-based prediction model using the independent bootstrap samples.

Fitted Copula Statistical Models for Four African and Four Major Stock Markets
Lien et al. [35] carried out a study investigating the co-movement and tail dependence between Chinese Yuan and New Taiwan Dollar non-delivery forward (NDF) rates against the U.S. dollar. The copula modeling approach to capture dynamics of correction and tail dependence between two NDF rates was used. It is shown that the interdependence between two NDF rates strengthens as time elapses. In particular, the degree of correlation surges sharply after April 9, 2008 while the degree of tail dependence increases significantly after February 10, 2009. Each time point of change is shown to be close to economic and political events that are supposed to have large impact on the relationship between Chinese Yuan and New Taiwan dollar.
Diawara [18] carried out a study using copula densities to model class conditional distribution of pattern recognition with bayes decision rule. This was because these types of densities are useful when the marginal densities of a pattern vector are not normally distributed. Those models are also useful for a mixed pattern vectors. A simulation to compare the performance of the copula based classifier with classical normal distribution based model and the independent assumption based model was also carried out.
Hu [30] studied dependence structures in Chinese and U.S. Financial Markets with other developed markets (Germany, France, Britain, and Japan) using time varying conditional copula (Normal and Joe Clayton copulas). The study was shown that Chinese is least affected by co-movement in the markets while western markets experience downturns during the ongoing global financial crisis. He suggested investors to increase weights on financial assets from Chinese financial markets in their portfolio for diversification purpose.

Source of data
The data used for this thesis are secondary data. The data are daily stock indices from the 29 th April 2003 to the 5 th of February 2018 of four African countries and four developed countries: Nigeria (NSE), Kenya (NSE20), Egypt (EGX30), South Africa (JSE40), UK (FTSE), US (SP500), Germany (DAX) and Canada (CAC40). The data were obtained from the database DataStream. Following the tradition, logarithmic returns were calculated as where , i t R is the return on the index i for period , , i t t P is the closing price of the index at the end of period t and , 1 i t P − is the price of the index at the end of the period 1. t − Restricting attention to the bivariate case for the sake of simplicity, the copula approach to dependence modeling is rooted in a representation theorem due to Sklar [43] as earlier mentioned. The Sklar theorem states that the joint cumulative distribution function (cdf) ( , ) H x y of any pair ( ) , X Y of continuous random variables may be written in the form

Copula
Sklar [43] showed that C, F and G are uniquely determined when H is known, a valid model for ( ) , X Y arises from equation 1 whenever the three are chosen from given parametric families of distribution, viz.: Thus, for example, F might be normal with (bivariate) parameter 2 ( , ); δ = µ σ G might be gamma with parameter ( , ); η = α λ and C might be taking from the Farlie-Gumbel-Morgenstern family of copulas, defined

Copula model specifications
Let U and V be uniform [0, 1] random variables. A copula denoted by say ( ) , C u v is a joint cumulative distribution function of U and V. A copula density denoted by say ( ) , c u v is the joint probability density function of U and V. Ten different models for copula were considered:

The Guassian copula
The Guassian copula defined by , where Φ denotes the cumulative distribution function of a standard normal random where φ denotes the probability density function of a standard normal random variable.
Independence of U and V corresponds to 0. ρ = Complete dependence of U and V corresponds to 1. ρ =

The t copula
The t copula defined by 1 1 , where v Τ denotes the cumulative distribution function of a Student's t random variable with degrees of freedom ν and , p v Τ denotes the joint cumulative distribution function of a bivariate t random vector with zero means, correlation ρ and degrees of freedom . ν The corresponding copula density is where t ν denotes the probability density function of a Student's t random variable with ν degrees of freedom.

The Ali-Mikhail-Haq copula
The Ali-Mikhail-Haq copula due to Ali et al. [3] defined by Independence of U and V corresponds to 0. α =

The Clayton copula
The Clayton copula due to Clayton [15] defined by Independence of U and V corresponds to 0. α = Complete dependence of U and V correspond to . α = ∞

The Cuadras-Augé copula
The Cuadras-Augé copula due to Cuadras and Augé [16] Independence of U and V corresponds to 0. θ = Complete dependence of U and V corresponds to 1. θ =

The Marshall-Olkin copula
The Marshall-Olkin copula due to Marshall and Olkin [36] defined by Complete dependence of U and V corresponds to 1. α = β = The Cuadras-Augé copula is the particular case of this copula for . α = β

The Cubic copula
The Cubic copula due to Durrleman et al. [20] defined by Independence of U and V corresponds to 0. θ =

The Gumbel copula
The Gumbel copula due to Gumbel [27] defined by Independence of U and V corresponds to 0. θ =

The Joe copula
The Joe copula due to Joe [33] defined by Independence of U and V corresponds to 1. θ = Note that the t and Marshall-Olkin copulas have two parameters each. The remaining copulas have one parameter each.
To fit these ten models to stock data from two countries at time, the following procedure are followed: let 1 2 , ,..., n x x x denote the log returns of the stock for one of the Discrimination among the fitted models was performed using various criteria: • The Akaike information criterion due to Akaike [2] defined by The smaller the value of the criteria the better the fit. For more discussion on these criteria, see Burnham and Anderson [10] and Fang [23].

Analysis and Results
From the processed data (using equation 1), which is the logarithmic difference of the daily closing index values, the series of daily returns over non-overlapping successive selection intervals were obtained. The following summary statistics for the daily log returns are computed and given in Table 3.1: number of observations (n), the minimum (Min), first quartile (Q1), median, mean, third quartile (Q3), the maximum (Max), standard deviation, coefficient of variation (CV), skewness, kurtosis, interquartile range (IQR), range and variance.   Tables 1 and 2, as expected, the minimum values and values of the first quartile are negative for the eight countries. The smallest of the minimum is for KEN. The largest of the minimum is for CAC40. The smallest of the first quartile is for FTSE100. The largest of the first quartile is for NGR. The values of the mean are all positive and close to zero for the countries except for NGR and KEN. The values of the median are all negative and close to zero for the eight countries. The median is largest for KEN and smallest for DAX. The mean is smallest for KEN and largest for FTSE100. The third quartile is smallest for FTSE100 and largest for EGY. The maximum is smallest for NGR and largest for KEN. The inter quartile range is smallest for EGY and largest for FTSE100. The range is smallest for NGR and largest for KEN. The standard deviation is smallest for SA and largest for DAX. The variance is also smallest for SA and largest for DAX. The skewness is positive for all the four countries. The log returns are least skewed for FTSE100. The log returns are most skewed for NGR. The kurtosis is smaller than that for the normal distribution for all countries. The kurtosis is smallest for FTSE100. The kurtosis is largest for NGR. The values of the coefficient of variation are generally positive for all the developed markets and negative for the African markets with exception of EGY. The coefficient of variation is smallest for KEN and largest for the CAC40.
The ten models were fitted to log returns from two of the countries at a time. There are eight countries. So, the ten models were fitted to log returns from the twenty eight  Tables 3 to 30.                            From these tables, it can be observed that the Gumbel copula in spite of having only one parameter gives the smallest values for the negative log-likelihood, the AIC, the BIC, the CAIC, the AICc, and the HQC for every pair. The t copula or the Clayton copula gives the second smallest values for the negative log-likelihood, the AIC, the BIC, the CAIC, the AICc and the HQC for every pair. The Marshall-Olkin copula gives the largest value for the negative log-likelihood, the AIC, the BIC, the CAIC, the AICc and the HQC for every pair. The Cuadras-Augé and Guassian copulas give the second largest values for the negative log-likelihood, the AIC, the BIC, the CAIC, the AICc and the HQC for every pair. The Cuadras-Augé and Guassia copulas appear to give the same values for the negative log-likelihood, the AIC, the BIC, the CAIC, the AICc and the HQC.

Fitted Copula Statistical Models for Four African and Four Major Stock Markets
The goodness of the best fitting Gumbel copula was tested using the information matrix equality of White [47]. This test was further investigated by Huang and Prokhorov [31]. The contribution is that under correct copula model specification, the Fisher Information can be equivalently calculated as minus the expected Hessian matrix or as the expected outer product of the score function. The p-value shown in Table 31 confirms the goodness of fit of the Gumbel copula. The parameter estimates and tail dependence coefficient for the best copula model from the twenty eight pairs of countries are given in Table 32. The K-plots are given in Figure 1. From the K-plots in Figure 1, evidence of possible tail dependence in African stock markets suspected. Note that K-plots are rank-based graphical tool for visualizing dependence (Genest and Boies [26]).

Fitted Copula Statistical Models for Four African and Four Major Stock Markets
If we suppose X and Y to be random variables representing any set of pairs of daily log returns from these stock markets with marginal distribution functions F and G. The coefficients of lower and upper tails dependence can be estimated by 1   The value at risk and expected shortfall based on the best fitting copula model is computed. These two risk measures are the most popular and celebrated financial risk measures (Danielsson and de Vries [17]). Their popularity stems from its endorsement by the Basel committee as a standard for risk management. The results are given in Table  33.

Conclusion
The Gumbel copula was shown to give the best fit in terms of log-likelihood values and values of the Akaike information criterion, value of the Bayesian information criterion, value of the consistent Akaike information criterion, value of the corrected Akaike information criterion, and value of the Hannan-Quinn criterion and p-values of the information matrix equality of White [47]. Estimates of value at risk with probability p for daily log returns are computed using the best fitting copula model. Based on these values at risk, it was clear that SA/FTSE have the least risks. In general, this finding has a number of implications for risk managers and potential investors. For instance, this shows that copula models should be used by policy-makers and financial practitioners to set margins, which is known to be sensitive to price movement in derivatives and stock markets.