# Copula-Based Assessment of Co-Movement and Tail Dependence Structure Among Major Trading Foreign Currencies in Ghana

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Methodology

#### 3.1. Specifying the Marginal Models

#### 3.2. Specifying the Copula Models

- Gaussian copula allows for equal degrees of positive and negative dependence but does not allow for tail dependence, implying that the tail parameters ${\tau}_{l}={\tau}_{U}=0$. Therefore, the dependence parameter of Gaussian copula is Pearson’s correlation coefficient (ρ) with a range of values: $-1<\rho <1$. The bivariate Gaussian copula is defined by ${C}_{N}\left(u,v|\rho \right)=\mathsf{\Phi}\left({\mathsf{\Phi}}^{-1}\left(u\right),{\mathsf{\Phi}}^{-1}\left(v\right)\right)$, where $\mathsf{\Phi}$ and ${\mathsf{\Phi}}^{-1}$ are respectively the cumulative and inverse cumulative distribution functions of the standard normal distribution.
- Student-t copula allows for symmetric non-zero dependence in the tails and it is the generalization of the Gaussian copula. With the degree of freedom $\eta $ and correlation coefficient $\rho $, the Student-t copula can be defined by ${C}_{\eta ,\rho}\left(u,v\right)={t}_{\eta ,\rho}\left({t}_{\eta}{}^{-1}\left(u\right),{t}_{\eta}{}^{-1}\left(v\right)\right)$, where ${t}^{-1}$ is the inverse function of the univariate Student’s t distribution.
- Gumbel copula is asymmetric which shows a higher probability of upper tail dependence defined by ${\tau}_{U}=2-{2}^{1/\delta}$ and lower tail dependence $({\tau}_{L}=0)$. Gumbel copula is specified as ${C}_{G}\left(u,v|\delta \right)=\mathrm{exp}\left(-\left({\left(lnu\right)}^{\delta}\right)+{\left(-lnv\right)}^{\delta}{)}^{\frac{1}{\delta}}\right)$, where the dependence parameter $\delta \in [1,\infty )$ and does not allow for negative dependence and the case of independence, $\delta =1$.
- The rotated Gumbel copula measures lower (left) tail dependence with the parameter $\delta \in [1,\infty )$ and $\delta =1$ implies independence. Thus, it is appropriate when the variables are highly correlated at low values and does not allow for negative dependence. The lower tail dependence ${\tau}_{L}$ is then defined as ${\tau}_{L}=2-{2}^{1/\delta}$ and the upper tail dependence ${\tau}_{U}=0$.
- The Clayton copula is asymmetric which exhibit more dependence in the lower (left) tail than the right with parameter $\theta \in \left[1,\infty \right)$ and $\theta =1$, implies independence. The Clayton copula is specified as ${C}_{Cl}\left(u,v|\theta \right)={\left({u}^{-\theta}+{v}^{-\theta}-1\right)}^{-\frac{1}{\theta}}$. The lower tail dependence probability is defined by ${\tau}_{L}={2}^{-\frac{1}{\theta}}$ and the upper tail dependence probability ${\tau}_{U}=0$.

#### 3.3. Estimation Procedure

#### 3.4. Data and Data Description

## 4. Empirical Results

#### 4.1. Marginal Model Results

#### 4.2. The Copula Models Results

#### 4.3. Comparing Time-Varying Copula Models with DCC-GARCH Model

_{t}. The estimation procedures of DCC-GARCH is like copula model estimation whereby the GARCH parameters for the individual series (marginal distributions) are estimated first, followed by estimating the parameters driving the correlation dynamics (see Tse and Tsui 2002). The DCC-GARCH model has been widely applied to financial phenomena (see Engle and Colacito 2006; Lee 2006; Coudert and Gex 2010; Cai et al. 2016).

_{t}of an asset is distributed as ${r}_{t}/{\eta}_{t-1}\backsim N\left(0,{H}_{t}\right)$ with

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Plot of level and returns of US Dollar(USD), Euro(Eur) and Great Britain Pound(GBP). (

**a**) A plot of exchange rates levels; (

**b**) A plot of exchange rates returns.

**Figure 2.**Time-varying copula parameters for all copula familie. Note: Claycop = Clayton Copulas, Gumcop = Gumbel Copulas, norcop = Normal Copulas, tcop = Student-t Copulas, rotClay = Rotated Clayton Copulas, rotGum = Rotated Gumbel Copulas.

GHS/ | Mean | Std. Dev. | Skewness | Kurtosis | Jarque-Bera | Q(2) | Q^{2}(2) | ARCH-LM |
---|---|---|---|---|---|---|---|---|

EUR | 0.00055 | 0.06574 | −0.2465 | 1137.45 | 1.2835 × 10^{8} * | 567.98 * | 593.96 * | 791.18 * |

USD | 0.00061 | 0.03361 | −0.0472 | 1103.06 | 1.2071 × 10^{8} * | 562.43 * | 593.93 * | 791.19 * |

GBP | 0.00056 | 0.13398 | 0.2266 | 820.34 | 6.6764 × 10^{7} * | 593.89 * | 593.49 * | 790.88 * |

^{2}(2) refer to Ljung-Box statistic for serial correlation of order 2 in returns and squared returns. ARCH-LM is the Engle’s Lagrange multiplier statistic to test for heteroskedasticity computed using 20 lags. * indicates statistical significance at 5%.

GHS/ | EUR | USD | GBP |
---|---|---|---|

EUR | 1 | ||

USD | 0.0163 | 1 | |

GBP | 0.0136 | 0.0141 | 1 |

GHS/ | EUR | USD | GBP |
---|---|---|---|

Mean Equation | |||

${\varnothing}_{0}$ | 0.000438 * | 0.000055 * | 0.000249 * |

(0.000116) | (0.000013) | (0.000119) | |

${\varnothing}_{1}$ | −1.282583 * | 0.678063 * | 0.911089 * |

(0.000156) | (0.029473) | (0.0211017) | |

${\varnothing}_{2}$ | −0.994357 * | 0.163287 * | |

(0.000233) | (0.019986) | ||

${\theta}_{1}$ | 1.281877 * | −0.741331 * | −0.929622 * |

(0.000249) | (0.024693) | (0.020039) | |

${\theta}_{2}$ | 0.994038 * | ||

(0.000137) | |||

Variance Equation | |||

${\alpha}_{0}$ | 0.000025 * | 0.000007 * | 0.000009 * |

(0.000003) | (0.000001) | (0.000001) | |

${\alpha}_{1}$ | 0.393398 * | 0.375248 * | 0.175997 * |

(0.063083) | (0.024909) | (0.006934) | |

${\beta}_{1}$ | 0.367484 * | 0.623752 * | |

(0.041624) | (0.017887) | ||

${\beta}_{2}$ | 0.054972 * | ||

(0.01028) | |||

${\beta}_{3}$ | 0.633581 * | ||

(0.006841) | |||

Log-Likelihood | 8430.102 | 11780.66 | 7663.246 |

Q(20) | 0.66597 | 0.003332 | 0.1212 |

[1.0000] | [1.0000] | [1.0000] | |

ARCH(20) | 0.0014332 | 0.0015125 | 0.006177 |

[1.0000] | [1.0000] | [1.0000] | |

Shapiro-Wilk | 0.051559 | 0.052169 | 0.038803 |

[0.0000] | [0.0000] | [0.0000] | |

Jarque-Bera | 1.2833 × 10^{8} | 1.1453 × 10^{8} | 6.6767 × 10^{7} |

[0.0000] | [0.0000] | [0.0000] |

Copula Models | EUR-USD | EUR-GBP | USD-GBP |
---|---|---|---|

Gaussian copula | |||

$\rho $ | 0.3866 * | 0.6689 * | 0.3872 * |

(0.016) | (0.009) | (0.016) | |

AIC | −379.1791 | −1399.383 | −380.6324 |

Student-t copula | |||

$\rho $ | 0.3389 | 0.6824 | 0.3533 |

(0.022) | (0.012) | (0.021) | |

ϑ | 2.8801 * | 3.6819 * | 3.3945 * |

(0.253) | (0.368) | (0.345) | |

AIC | −534.009 | −1598.632 | −496.1728 |

Gumbel copula | |||

$\delta $ | 1.341 * | 1.884 * | 1.343 * |

(0.021) | (0.032) | (0.021) | |

AIC | −480.3786 | −1497.994 | −485.5223 |

Rotated Gumbel copula | |||

$\delta $ | 1.302 * | 1.845 * | 1.295 * |

(0.02) | (0.031) | (0.02) | |

AIC | −386.6807 | −1403.206 | −364.8547 |

Clayton copula | |||

$\alpha $ | 0.5647* | 1.221* | 0.5816* |

(0.034) | (0.044) | (0.035) | |

AIC | −267.1438 | −1116.208 | −239.6604 |

Rotated Clayton copula | |||

$\alpha $ | 0.5986 * | 1.349 * | 0.6118 * |

(0.035) | (0.046) | (0.035) | |

AIC | −423.8038 | −1267.054 | −434.7387 |

Copula Models | EUR-USD | EUR-GBP | USD-GBP |
---|---|---|---|

TVP-Gaussian | |||

${\mathsf{\Psi}}_{0}$ | 0.0412 * | 0.0213 * | −0.0047 |

(0.00601) | (0.00587) | (0.00594) | |

${\mathsf{\Psi}}_{1}$ | 0.4532 * | 0.6253 * | 0.1025 |

(0.01314) | (0.01327) | (0.01316) | |

${\mathsf{\Psi}}_{2}$ | −2.3641 * | 1.7841 * | 1.4352 |

(0.00589) | (0.00595) | (0.0059) | |

AIC | −430.23 | −1405.42 | −411.59 |

TVP-Student-t | |||

${\mathsf{\Psi}}_{0}$ | 0.0384 * | 0.0241 * | −0.0063 |

(0.0059) | (0.0058) | (0.00593) | |

${\mathsf{\Psi}}_{1}$ | 0.5461 * | 0.6562 * | 0.1465 |

(0.0058) | (0.0134) | (0.01335) | |

${\mathsf{\Psi}}_{2}$ | 1.7563 * | 2.3641 * | 1.6343 |

(0.0061) | (0.0059) | (0.00595) | |

AIC | −567.89 | −1604.36 | −546.73 |

TVP-Clayton | |||

${\mathsf{\Psi}}_{0}$ | −0.2578 * | 0.1115 * | 0.2931 * |

(0.00582) | (0.00582) | (0.00596) | |

${\mathsf{\Psi}}_{1}$ | −0.7775 * | 0.9273 * | −0.6219 * |

(0.0026) | (0.007) | (0.0036) | |

${\mathsf{\Psi}}_{2}$ | −6.3813 * | −0.5998 * | −8.3688 * |

(0.00592) | (0.00587) | (0.00595) | |

AIC | −293.4 | −1167.26 | −285.35 |

TVP-rotated Clayton | |||

${\mathsf{\Psi}}_{0}$ | 0.0695 * | 1.0452 * | −0.1654 * |

(0.00588) | (0.00594) | (0.00599) | |

${\mathsf{\Psi}}_{1}$ | −0.7163 * | −0.3841 * | −0.5982 * |

(0.0029) | (0.0044) | (0.0024) | |

${\mathsf{\Psi}}_{2}$ | −5.5725 * | −4.6319 | −4.2662 |

(0.00593) | (0.00589) | (0.00601) | |

AIC | −437.17 | −1291.99 | −442.99 |

TVP-Gumbel | |||

${\mathsf{\Psi}}_{0}$ | 0.5544 * | 1.5418 * | 0.498 * |

(0.00599) | (0.006) | (0.00598) | |

${\mathsf{\Psi}}_{1}$ | −0.7515 * | −0.1367 * | −0.5607 * |

(0.0019) | (0.0036) | (0.0019) | |

${\mathsf{\Psi}}_{2}$ | −6.5229 * | −5.292 * | −5.9222 * |

(0.00597) | (0.00589) | (0.00599) | |

AIC | −498.76 | −1535.04 | −501.56 |

TVP-rotated Gumbel | |||

${\mathsf{\Psi}}_{0}$ | 0.3416 * | 1.0423 * | 0.683 * |

(0.00596) | (0.00602) | (0.00585) | |

${\mathsf{\Psi}}_{1}$ | −0.7469 * | 0.2711 * | −0.6123 * |

(0.0017) | (0.0039) | (0.0022) | |

${\mathsf{\Psi}}_{2}$ | −6.5702 * | −3.923 * | −7.875 * |

(0.00591) | (0.00592) | (0.00581) | |

AIC | −405.02 | −1447.03 | −391.54 |

Panel A: Marginal Model Results | |||

GHS/ | EUR | USD | GBP |

Variance equation | |||

${\alpha}_{0}$ | 0.000012 * | 0.0000001 | 0.000004 * |

0.000001 | 0.000024 | 0.000001 | |

${\alpha}_{1}$ | 0.21585 * | 0.228549 * | 0.074432 * |

0.05652 | 0.039065 | 0.022583 | |

${\beta}_{1}$ | 0.660569 * | 0.770451 * | 0.877705 * |

0.061083 | 0.340935 | 0.008594 | |

Panel B: Bivariate DCC results | |||

GHS/ | EUR-USD | EUR-GBP | USD-GBP |

$\rho $ | 0.005637 * | 0.012417 * | 0.002025 * |

0.00000005 | 0.00000006 | 0.00000002 | |

$a$ | 0.010367 * | 0.000001 | 0.005039 |

0.002584 | 0.000127 | 0.003424 | |

$b$ | 0.9836 * | 0.902889 * | 0.990627 * |

0.007146 | 0.108864 | 0.007068 | |

AIC | 309.86 | 132.03 | 408.26 |

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**MDPI and ACS Style**

Mensah, P.O.; Adam, A.M. Copula-Based Assessment of Co-Movement and Tail Dependence Structure Among Major Trading Foreign Currencies in Ghana. *Risks* **2020**, *8*, 55.
https://doi.org/10.3390/risks8020055

**AMA Style**

Mensah PO, Adam AM. Copula-Based Assessment of Co-Movement and Tail Dependence Structure Among Major Trading Foreign Currencies in Ghana. *Risks*. 2020; 8(2):55.
https://doi.org/10.3390/risks8020055

**Chicago/Turabian Style**

Mensah, Prince Osei, and Anokye M. Adam. 2020. "Copula-Based Assessment of Co-Movement and Tail Dependence Structure Among Major Trading Foreign Currencies in Ghana" *Risks* 8, no. 2: 55.
https://doi.org/10.3390/risks8020055