Exchange rate refers to the number of units of one nation’s currency that equals one unit of another country [1]. In the last few decades, the fluctuations and patterns of exchange rate had been a main emphasis of macroeconomic analysis and had drawn the attention of the academicians, financial specialists, and decision makers specifically after failure of agreement on fixed exchange rates amid the industrialized nations referred to as the Bretton Woods agreement. From that time forward, a huge deliberation on the volatility of exchange rate and its possible impact on well-being, inflation, international trade, investment analysis, profitability and risk management have been in the forefront of the debate. As a result, quite good number of models in the finance literature has come into existence for the purpose of probing the exchange rate volatility across distinct parts of the globe and nations. Several models such as the Autoregressive Conditionally Heteroscedastic (ARCH) model and its modifications like the Generalized ARCH (GARCH), Exponential GARCH (EGARCH) and Glosten, Jajanthen and Runkle GARCH (GJR-GARCH) models had been developed to model varying volatility of such series.

ARCH model was proposed by [2]. It clearly tells that the series under consideration has a time-varying variance that rely on the effect of the lags. The ARCH model is approximated upon taking the squared errors and then letting the errors depend on the lags [3]. However, ARCH models are quite often difficult to estimate because they might predict negative volatility as a result of violating non-negativity constraints. Later [4] modified ARCH model into generalized form known as GARCH. The dissimilarity being that the GARCH model permits the conditional variance to rely on its own past values of the squared errors and on the past conditional variance [3]. The GARCH model is parsimonious making it not so much likely to defy non-negativity constraints [5].

Both ARCH and GARCH models of [2] and [4] treat positive shocks and negative shocks symmetrically. That is, their impact on asset volatility is the same. Thus, [6] proposed a GARCH model named EGARCH that allows for asymmetric effects and therefore solved one of the important shortcomings of the symmetric model. While the GARCH model imposes the non-negativity constraint on the parameters, EGARCH uses the log of the conditional variance. Hence, EGARCH is not exposed to the non-negativity constraints [6]. Further advancement to incorporate asymmetric effects was done by [7]. Hence, arriving at a model called GJR-GARCH. It has a merit of modeling the variance directly that is to say, the model does not use the natural logarithm unlike the EGARCH model. Thus, the GJR-GARCH is straightforward to execute [8].

The advancements put forward in all those models is achieved by modifying the equation proposed for modeling the conditional volatility, but retaining the same variables.

Standard GARCH models, according to [9], are not well adapted for periods characterized by rapid volatility variations. These factors prompted the improvement of GARCH models. Because the GARCH model is insufficient for forecasting time series impacted by exogenous causes. The ability to uncover underlying patterns in time-series data and quantify the impact of external influences is possible with a time series model incorporating exogenous variables.

Time series analysis with explanatory variables encompasses methods to model and predict correlated data taking into account additional information [10]. External variables’ effects on conditional variance can be taken into account using the GARCH-X model. This additional feature becomes even more significant for improving GARCH model predictions, especially when additional factors that can affect GARCH estimates are unaccounted for. GARCH models may generate biased estimates of persistence in variance if these external data are not taken into account.

Several researchers have noted the usefulness of GARCH models with introduction of various exogenous variables for example, [11] used interest rate level, daily market arrivals and interest rate spreads were used as covariates respectively by [12] and [13], a proxy for global information was used as a covariate by [14], volume of traded stock was used by [15] [16] improved GARCH model by bringing in stock’s volume as a covariate in volatility equation. [17] explicitly incorporated Acquired immunodeficiency syndrome (AIDS)-associated information to exchange rate of Rand against United State dollar in South Africa.

[18] looked at the volatility of the domestic edible oil price index in India, as well as the effects of exogenous variables such as the international edible oil price index and the Indian-US foreign exchange rate. They discovered that external variables impact the volatility of the domestic price index of edible oil more than its own variance.

Each of the above empirical studies displays a certain improvement of the models by including the exogenous process, and this improvement is presented through computational results obtained on real data. Past studies display that these exogenous factors have a large impact on the study variables and should be included in the existing time series model to improve model performance and forecasting accuracy [19].

[20] investigated GARCH-X by introducing external regressors as proxies for macroeconomic variables of interest, which were tested independently on the mean and variance equations of GARCH-X. However, both the mean and variance equations can be jointly affected by external variables. The effect of covariates was allowed in both equations at the same time in this investigation.

The empirical models that explain the variation in exchange rate on the ground of macroeconomic fundamentals only are usually bias on the account of omitted variable hence, they cannot decently explain variations in exchange rate [21]. However, If socio-political determinants, like civil wars, violence are incorporated in simple time series specification, the variations of exchange rate can be understood better [22].

The connection of economic performance and civil war as of late has dominated economic discussion [23]. However, this relationship has been examined in the direction of economic causes and the consequences of civil war by many researchers like [24] and [25]

South Sudan in particular as the country under study has been one of the world’s most vulnerable and war-prone nation, unable to break the cycles of conflict since her independence from Sudan in 2011. This cycle of conflict led to economic instability. Therefore, in regard to South Sudan the effect of these socio-political problems mainly conflict on the exchange rate volatility remains largely under-explored. As a result, this paper aims at narrowing down the analysis of the effect of conflict on the dynamics of exchange rate on the basis of quantifying this impact through incorporating conflict index as an exogenous variable to the exponential GARCH model for South Sudanese pounds against united states dollar exchange rate.

The rest of the paper is structured as follows: In Section 2, we presents the methodology and describes the data used. Section 3 gives the exploratory data analysis. Section 4 presents the estimation of mean and volatility equations and the estimation results. Finally, Section 5 gives the conclusion and suggested areas for further research.

Experimental researchers and practicians frequently incorporates covariates to the parameters of volatility dynamic for better modeling and forecasting of volatility in both economic and financial time series particularly to determine whether the inclusion of covarites in GARCH (p, q), EGARCH (p, q), etc models give an explanation of changes in volatility. As already mentioned by [2], any variance function can be augmented with exogenous variable.

GARCH extended models as put forward by [26] and [27] directly include covariate in conditional variance equation of suggested model. But the exogenous variable can affect the mean and variance equations collectively. Thus we have to test the effect of the covariate in both equations.

In this paper we examined the effect of conflict used as an exogenous variable on exchange rate volatility. The conflict index was treated as counts and dummies. The data used in the empirical application of the proposed model is the daily closing prices of SSP-USD exchange rate and conflict data for the period January 2013 to April 2020.

Each of the data set consists of 2677 observations. The exchange rate data were obtained from central bank of South Sudan officially known as the Bank of South Sudan (BOSS).

The conflict data comes from the Armed Conflict Location and Event Data set (ACLED), which covers Africa, the Middle East, and Southeast Asia since 1997 and is disaggregated by date, location, participants, and form of conflict operation [28].

The conflict data mentioned above covers the real-time conflict data in South Sudan on political violence and violence against civilians by the armed groups. The ACLED data can be access through the following link: www.acleddata.com

After obtaining optimal ARMA (p, q)-EGARCH (p, q) model, we introduced exogenous variable, the conflict index in both mean and volatility equations of the model. Two forms of conflict index were computed and tested namely, the counts of the conflict and the dummies of the conflict for the entire country and Juba town separately. The counts of conflict index denotes the summation of the number of conflicts observed in a given day for the entire country and Juba separately.

Also, the dummies (one and zero) of the conflict index were considered where one denotes that the conflict was observed in a given day and zero denotes that conflict was not observed in a given day for the whole country and Juba separately. The conflict index treated as the counts for the entire country, counts for Juba town, dummies for the entire country and dummies for Juba town were used as exogenous variables in both mean and volatility equations of the optimal ARMA (p, q)-EGARCH (p, q) model with each treatment of the conflict index entering the model one by one, thus arriving at ARMA (p, q)-EGARCH (p, q) class of models with exogenous variable as follows:

r t = μ t + a t (1)

r t represent log returns of exchange rates and it follows a simple time series model such as a stationary ARMA (p, q), μ t is the mean of the returns, a t refers to the shock which follows a white noise series and is specified as a t = σ t ε t . We assumed that ε t is a sequence of independent and identically distributed random variable with mean of zero and variance 1.

Model 1

mean equation

r t = μ + ∑ i = 1 p     ϕ i r t − i + a t + ∑ j = 1 q     θ j a t − j + ∑ u = 1 v     π u w t − u (2)

where μ denotes the mean.

a t denotes shock which follows white noise series.

ϕ represents autoregressive coefficient.

θ represents moving average coefficient.

π u represents the coefficient for the exogenous variable.

volatility equation

log ( σ t 2 ) = ω + ∑ i = 1 p     α i | μ t − i σ t − i | + ∑ j = 1 q     β j log ( σ t − j 2 ) + ∑ k = 1 r     γ k μ t − k σ t − k + ∑ u = 1 v     π u w t − u (3)

where ω is the constant term.

α parameter represents a magnitude effect or the symmetric effect of the model.

β measures the persistence in conditional volatility.

γ parameter measures the asymmetric or the leverage effect. π u is defined as above.

w t − u is a vector with lagged exogenous variable, the counts of conflict index for the whole country.

Model 2

mean equation

r t = μ + ∑ i = 1 p     ϕ i r t − i + a t + ∑ j = 1 q     θ j a t − j + ∑ u = 1 v     π u x t − u (4)

Volatility Equation

log ( σ t 2 ) = ω + ∑ i = 1 p     α i | μ t − i σ t − i | + ∑ j = 1 q     β j log ( σ t − j 2 ) + ∑ k = 1 r     γ k μ t − k σ t − k + ∑ u = 1 v     π u x t − u (5)

where x t − u is a vector with lagged exogenous variable, the dummies of conflict index for the whole country.

Model 3

mean equation

r t = μ + ∑ i = 1 p     ϕ i r t − i + a t + ∑ j = 1 q     θ j a t − j + ∑ u = 1 v     π u y t − u (6)

Volatility Equation

log ( σ t 2 ) = ω + ∑ i = 1 p     α i | μ t − i σ t − i | + ∑ j = 1 q     β j log ( σ t − j 2 ) + ∑ k = 1 r     γ k μ t − k σ t − k + ∑ u = 1 v     π u y t − u (7)

where y t − u is a vector with lagged exogenous variable, the counts of conflict index for Juba.

Model 4

mean equation

r t = μ + ∑ i = 1 p     ϕ i r t − i + a t + ∑ j = 1 q     θ j a t − j + ∑ u = 1 v     π u z t − u (8)

Volatility Equation

log ( σ t 2 ) = ω + ∑ i = 1 p     α i | μ t − i σ t − i | + ∑ j = 1 q     β j log ( σ t − j 2 ) + ∑ k = 1 r     γ k μ t − k σ t − k + ∑ u = 1 v     π u z t − u (9)

where z t − u is a vector with lagged exogenous variable, the dummies of conflict index for Juba.

If π = 0 , the coefficient of the vector with lagged exogenous variable then the models reduces to ARMA (p, q)-EGARCH (p, q) without exogenous variable.

The daily SSP-USD currency exchange rate was first subjected to a stationarity test. An Augmented Dickey-Fuller test (ADF) for a unit root in a time series sample was used to check for stationarity. The null hypothesis of the test is that there is a unit root. The p-value obtained should be smaller than the significance level (say 0.05) in order to reject the null hypothesis of the test. The test results are given in Table 1.

There is no reason to reject the null hypothesis because the p-value is greater than the significance level of 0.05. As a result, the series is not stationary. Hence, the series must be difference in order to become stationary.

The exchange rate data is then transformed into daily log returns using the following returns formula.

r t = ln ( p t p t − 1 ) (10)

where,

r t is the log return of the exchange rate.

p t is the exchange rate at time t

p t − 1 is the exchange rate at time t − 1

The Augmented Dickey-Fuller test results after the series was transformed into daily log returns confirm that the series is stationary as can be seen in Table 2.

The p-value is less than the significance level and hence we can reject the null hypothesis and take that the series is stationary.

The prime goal of this paper was to examine the effects of conflict on SSP-USD exchange rates volatility. The methodology follows EGARCH model with exogenous covariate in both mean and volatility equations.

The time series of South Sudanese exchange rate returns had been explored and appropriate ARMA (p, q)-EGARCH (p, q) model was formulated. On the basis of information criteria, the ARMA (1, 2)-EGARCH (1, 1) was the optimal volatility model. The ARMA process was used to estimate the mean equation as it acts as filter for the returns in order to produce independent and identically distributed returns.

The empirical application of the proposed model was demonstrated by adding the conflict index in different forms in both mean and volatility equations of the optimal model. Thus, we obtained ARMA (1, 2)-EGARCH (1, 1) class of models with exogenous variable. In this way, we have attempted to capture the effects of covariate on the dynamics of the SSP-USD exchange rates volatility. The estimated results with maximum likelihood estimation for ARMA (1, 2)-EGARCH (1, 1) class of models with exogenous variable shows that the values of the parameter beta ( β ) in all the models were close to one, indicating that volatility in exchange rates was persistent and parameter for leverage effect in our models was significant. The effect of conflict on volatility of SSP-USD was generally positive and statistically significant in both equations indicating that the prevalence of conflict makes the exchange rate to be more volatile. As a result, exchange rate volatility can have an impact on the business and investor sectors because it increases market uncertainty and risk. As a result of the aforementioned risk, foreign direct investors are forced to leave the country in order to protect their businesses.

Therefore, policymakers must be aware of the degree of exchange rate volatility in order to implement appropriate economic policies. As a result, the outcomes are crucial from a policy standpoint.

Further research focusing on exploring non-linear effects of the exogenous variables on the volatility of SSP-USD exchange rate can be undertaken.

Also, there is a room for extending the model assuming the covariate can be modeled as a function of other variables or may have other distribution functions.

The first author thanks the Pan-African University Institute of Basic Sciences, Technology and Innovation (PAUSTI) for their support.

The authors declare no conflicts of interest regarding the publication of this paper.

Kur, A.P., Ngesa, O. and Sarguta, R. (2021) Exponential GARCH Model with Exogenous Covariate for South Sudanese Pounds—USD Exchange Rate Volatility: On the Effects of Conflict on Volatility. Journal of Mathematical Finance, 11, 466-483. https://doi.org/10.4236/jmf.2021.113026