EXCHANGE RATE VOLATILITY AND RWANDA’S BALANCE OF TRADE

By: MANIRAGABA, Ngabo Vallence vallencengabo@ines.ac.rw &: NKURUNZIZA, Fabrice nkurufabre123@ines.ac.rw ABSTRACT
This paper examines the effect of exchange rate volatility and balance of trade sector in Rwanda for the period of January 1996 to December 2013, and tries to find appropriate models for both balance of trade and exchange rate to be used in forecasting for future values..
Some of the developing economies including Rwanda would appear to have exacerbated fluctuations in exchange rates, developing economies are special examples of high exchange rate, The impact of exchange rate levels on trade has been much debated but the large body of existing empirical literature does not suggest an indubitable comprehensive image of the trade impacts of exchange rate volatility in Rwanda. The review of the theoretical literature on this issue indicates that there is no clear-cut relationship between exchange rate volatility and balance of trade. This study examines the effect of exchange rate volatility and balance of trade sector in Rwanda
The analysis followed the empirical methods (econometrics and time series analysis). The researchers used
UBJ time series analysis to accomplish all stages (stationarity, identification, estimation, diagnostic checking and forecasting) of the models and models validation was of good quality and can be used in forecasting for future values. Polynomial regression model helped to establish the effects of exchange rate on balance of trade.
The results revealed a positive quadratic relationship between exchange rate and balance of trade components and by polynomial regression model estimation, exports and imports will increase as exchange rate increases.
The researchers recommended improving strategies and techniques for maintaining the quality of Rwandan exports, decreasing the proportion of goods imported for consumption purposes and maintaining a stable exchange. Keywords : Exchange rate, volatility, Balance of Trade, Rwanda
Résumé
Ce document traite l'effet de la volatilité des taux de change et l'équilibre du secteur de commerce au Rwanda pour la période de Janvier 1996 à Décembre 2013 et aussi essaie de trouver des modèles appropriés à la fois pour l'équilibre du commerce et de taux de change à utiliser dans les prévisions pour les valeurs futures..
Certaines économies en développement, dont le Rwanda semble avoir exacerbé les fluctuations des taux de change, les économies en développement sont des exemples particuliers des taux de change élevé, l'impact des niveaux de taux de change sur le commerce a été beaucoup discuté, mais un nombre important de la littérature empirique existante ne suggère pas une image complète indubitable des impacts commerciaux de la volatilité des taux de change au Rwanda. La revue de la littérature théorique sur cette question indique qu'il n'y a pas de relation claire entre la volatilité des taux de change et la balance commerciale. Cette étude examine l'effet de la volatilité des taux de change et l'équilibre du secteur de commerce au Rwanda
L'analyse a suivi les méthodes empiriques (économétrie et analyse de séries chronologiques). Les chercheurs ont utilisé analyse UBJ de séries chronologiques pour accomplir toutes les étapes (stationnarité, identification, estimation, diagnostique et le pronostique) des modèles et la validation des modèles trouvés étaient de bonne
1

qualité et peuvent être utilisés dans la prévision des valeurs futures. Le Modèle de régression polynomiale a aidé à établir les effets de taux de change sur la balance commerciale.
Les résultats ont révélé une relation quadratique positive entre le taux de change et l'équilibre des composantes commerciales et par une estimation de régression du modèle polynomial, les exportations et les importations vont augmenter au fur et à mesure que le taux de change augmente.
Les chercheurs ont recommandé l'amélioration des stratégies et des techniques pour maintenir la qualité des exportations rwandaises, la diminution de la proportion de biens importés à des fins de consommation et de maintenir un taux de change stable.
Mots-clés: taux de change, la volatilité, la balance du commerce, le Rwanda
I:Introduction
The exchange rate is an important determinant of export growth and its progress through time, while it serves as a measure of international competitiveness and is therefore a useful indicator of economic performance. High fluctuations in exchange rates create uncertainty about the profits to be made, thus reducing the gains of international trade and hampering the volume of trade. According to this theoretical approach, we investigated the same relationship for Rwanda.
Since the beginning of floating exchange rate regimes in 1973, many papers, both theoretical and empirical, have analyzed the effects of exchange rates and exchange rate volatility on trade. As regards the level of the exchange rate, empirical studies find somewhat differing results as to their impacts on trade although there is a common understanding as to the direction of the impact of the exchange rate on exports and imports. To date, therefore, relevant research does not suggest a clear-cut relationship1.
The paper reviews existing empirical literature on the exchange rate-trade performance relationship in Africa, measuring trade performance in terms of imports, exports and the trade balance. Several conclusions emerge from this review.
Trade flows and the trade balance are sensitive to changes in the real exchange rate. A real depreciation is effective in raising export volumes, reducing import volumes and improving the trade balance. A real depreciation is also effective in diversifying exports away from primary commodities towards manufacturing and particularly non-commodity manufacturing.
In terms of the impact of exchange rate volatility on trade balance. Marilyne and Jane Korinek(2012) found out that the small open economies seem more sensitive to exchange rate volatility than large economies as found by many research. And some of reasons were that small open economies have less room to adjust their exchange rates in the face of exchange rate changes vis à vis large economies’ currencies their traders may be more directly impacted by exchange rate changes. Secondly, many smaller countries have less diversified export structures and therefore it is more difficult to move into exports of products that are more price inelastic.
Thirdly, importers cannot necessarily source their needs in the domestic market in the case that their exchange rate depreciates, making foreign goods more costly. Finally, small countries often have smaller enterprises on average. Some enterprises may not be large enough to practice hedging.
The exchange rate affects the price of tradable goods which is usually raw materials and manufactured products but recently services like call centers and programming are an also being out sourced to low wage countries. A change in the exchange rate of the $ has the greatest effect on the firm that are competitive, like cars and chemicals but for thing like shoes and clothes the price divergence is too large for exchange rate to have a significant effect on US firms and the US already dominates the world market for food. We cannot supply some
1

Bachetta, P and Van Wincoop, E.(2000) “Does exchange rate stability increase trade and welfare?”
2

things we need, oil being the most important so if the $ falls in value it will not have much effect on the amount we import, just how much we pay for it.
Rwanda had an administered economy, which imposed severe restrictions on trade and foreign exchange transactions and a fixed exchange rate regime (1961-1990). By the early 1990s the average tariff rate was
34.8%, with 5 different tariffs ranging from 0-60%. Every import and every importer was subject to a quota, and all import operations were subject to a license authorizing external currency disbursement. Exporters had to repatriate currency generated by the sale of exports as a legal requirement. Export licenses were authorized only by the Banque Nationale du Rwanda (BNR). More importantly, all export earnings were transferred to and managed by the BNR.
The period from 1991 until 1994 corresponds to the beginning of the removal of restrictions on trade and foreign exchange transactions, and the gradual revival of a market economy, Rwanda embraced a market economy characterized by continuation of trade reforms and a liberalization of the monetary and financial regimes. Tariffs were reduced considerably with the average rate decreasing to 18%, and there remained four tariff bands with a maximum of up to 30% by 2003. This is a significant reform when compared with an average tariff rate of 34.8%, with 5 different tariffs ranging from 0-60% prior to 19942.
Liberalization of the monetary and financial sector led to the adoption of new currency exchange regulations, the creation of new private commercial banks, and the privatization of state-owned banks. Imports, exports and services were liberalized, and some of the previous restrictions on capital flows were either reduced or eliminated. Flexible exchange rates were also introduced. During the period 1995 to 2003, the commitment of the government to trade, financial, and exchange reform was much more credible and stable. Prices began to reflect real cost and value, rather than the arbitrary levels established by the government. Economic resources could thus be allocated much more effectively as firms adjusted their productive capacities and subsequently improved the overall competitiveness of the Rwandese economy. At the same time Rwanda was the recipient of substantial aid from the World Bank and other entities. Although diminishing returns to aid may exist over the long run, in the case of Rwanda the post-1994 aid has had a much greater and long-lasting impact (Collier,
2004).
Rwanda's exports remained dominated by traditional products such as coffee, tea and minerals like tin, coltan
(Colombo tantalite), wolfram and cassiterite. Rwanda's main exports partners are China, Germany and United
States. Rwanda imports mainly food products, Raw materials, machinery and equipment, construction materials, petroleum products and fertilizers. Main imports partners are Kenya, Germany, Uganda and
Belgium3. This study will examine the impact of exchange rates and their volatility on trade flows in Rwanda.
II: Literature review
The literature review is based on works and studies which have been done by the other scholars. This chapter is related to the definitions of key terms used in the development of this topic of impact of currency exchange rate on balance of trade in Rwanda.
Insights from the empirical literature
Despite the very large volume of empirical studies in this area over the last four decades, there is no clear consensus concerning the impact of exchange rates and exchange rate volatility on the volume trade (McKenzie,
1999,.Bahmani-Oskooee and Hegerty, 2007). In fact, research results which find positive, negative or no effect of exchange rate volatility on the volume of international trade are based on varied underlying assumptions but
2
3

BNR, Annual Report 2004
Rwanda National Export Strategy (2011)
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only hold in certain cases. Coric and Pugh (2008) apply a meta-analysis of the results found in the literature that range from strong negative to strong positive effects. They find 33 studies that conclude that exchange rate variability exerts an adverse effect on trade volumes. The other 25 studies examined conclude that this is not the case. Six of those studies conclude that exchange rate variability is trade-enhancing (Coric and Pugh, 2008).
The net impact of the exchange rate level on trade flows is also not clear in the literature. Some studies examine this question in the context of currency unions (Rose and Stanley, 2005) for an extensive meta-analysis). Many studies examine the impact of both the exchange rate level and volatility on trade in a single equation or set of equations. Results are highly contingent on the measure of volatility used, on the time period under question, whether short-term or long-term effects are examined, the econometric method used to estimate, the periodicity of the data, and whether or not effects are examined at the aggregate, sectoral, or product level. Some studies that examine the impacts in different sectors find that trade in some products responds positively to exchange rate variation and others negatively, so the net effect is highly determined by the composition of exported and imported products (e.g. Doroodian et al., 1999, Byrne et al., 2008). The heterogeneity found in model results extends to country coverage. To cite only one recent study, Chiu et al. (2010) apply the heterogeneous panel co integration method to examine the long-run relationship between the real exchange rate and bilateral trade balance of the United States and its 97 trading partners for the period 1973-2006 using annual data. The empirical results indicate that a devaluation of the US dollar deteriorates its bilateral trade balance with 13 trading partners, but improves it with 37 trading partners, notably China.
Some studies have examined the effects of exchange rate changes on trade at the sectoral level. Mindful of the
Marshall-Lerner condition, Houthakker and Magee (1969) estimate price elasticities for different commodities in the United States. They find that price elasticities are low for raw materials but high for finished manufactures. Carter and Pick (1989) examine the J-curve effect for US trade in agricultural goods. They pioneered research on the pass-through effect of exchange rate changes on agricultural exports and imports, and the net impact on the agricultural trade balance. They find evidence of the price effect of the J-curve: depreciation leads to a decline in the agricultural trade balance. The quantity effect however is only partly explained by the J-curve effect. Doroodian et al. (1999) find a J-curve effect only for agricultural goods, but not for manufacturing, using US data for 1977 to 1991. This could explain why some studies using aggregate data fail to support the J-curve hypothesis. Perhaps the J-curve effect does not apply overall. Indeed, Hsing (2008) examined US trade with seven South American trading partners over the last 20 or 30 years according to the studied countries and showed that a J-curve exists for Chili, Ecuador and Uruguay while a lack of support is found for Argentina, Brazil, Colombia and Peru. These findings therefore suggest that the conventional wisdom of pursuing real exchange depreciation in order to improve the trade balance may not apply in some cou ntries. According to the literature on this topic, some of the studies find a negative effect of exchange rate volatility on agriculture trade (Perée and Steinherr, 1989; Cho et al., 2002; Kandilov, 2008; Doyle, 2001) while some others conclude a non-significant effect (Caglayan and Di, 2008); Byrne et al. (2008)). Baek and Koo (2009) find that in the long run, while US agriculture exports are highly negatively impacted by the exchange rate, US agriculture imports are generally not affected. In the short run, on the other hand, the exchange rate is found to have significant effects on both imports and exports. Carter and
Pick (1989) suggest that market factors other than exchange rate fluctuations are the primary determinants of
US agriculture trade while Doroodian et al. (1999) show that an exchange rate depreciation has a prolonged and significant effect on the US agriculture trade balance.
As noted by Maskus (1986), the impact of exchange rate volatility may vary across sectors because these can have differing degrees of openness to international trade, different industry concentration levels and make different use of long-term contracts.
According to his estimations run over the 1974-1984 period, real exchange rate risk reduces US agricultural trade more than other sectors which he attributes to a greater openness of the agriculture sector, to a low level of industry concentration, and lengthy trade contracts.
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III Methods and materials
The approach used in this study follows an econometric methodology of time series data.
Monthhly Secondary data from National Bank of Rwanda, NISR and MINICOM was used. The data variables considered for this study included; Exchange rate (Frw->Usd), Imports f.o.b (Expressed in $), Imports c.i.f
(Expressed in $), Exports (Expressed in $). The time period of study was from January 1996 to December
2013.SPSS helped researchers to estimate models and to carry out different econometric tests which permit the validation of the results of the analysis .The results presentation were done through various tables and graphics as well as econometric models. Furthermore polynomial regression analysis and UBJ time series analysis were performed on the different variables. The purpose was to estimate and fit a structured models to explain the effects in the observation of dependant variable BoT (Y) in terms of independant variable mojorly exchange rate (x). Thus for this study, the polynomial regression model is of the form: where Y is monthly balance of trade components while X is monthly average of exchange rate. is an intercept, are slopes and is an error term. The quadratic nature of the polynomial was arrived at after preliminary analysis that indicated that the data behaved so.
3.1.0 Methodological approach
The aim of this study is to determine the impact of exchange rate on balance of trade in Rwanda. For this purpose, the extent of the impact of exchange rate on the Rwandan economy will have the help of an econometric model. It is to estimate a relationship between economic performance and trade sector than other sectors, and the economy in general. To overcome the problems due to the application of classical methods of linear regression on data that changes over time, the recent developments in the econometrics of time series has been used.
3.1.1 Time series concept
A time series is a sequential set of data points, measured typically over successive times. It is mathematically defined as a set of vectors where t represents the time elapsed. The variable is treated as a random variable.
3.1.2 Autoregressive Integrated Moving Average (ARIMA) Models
Let’s first know how ARIMA models are written and how are interpreted: ARMA models are a set of models that descibe the process as a function of its own lags and a white noise process. Three processes and
ARIMA (p,d,q) notation are:
……………………………………………. (3.1)
……………………………………………… (3.2)
……………………………………………… (3.3)
Equation (1) is a an AR(2) or ARIMA(2,0,0), because it contains only AR terms and the maximum time lag on the AR terms is 2. This model describes a stochastic process that can be represented by a weight sum of its previous values and a white noise error4.
Equation (2) is an MA(2) or ARIMA(0,0,2), because it contains only MA terms with a maximum time lag on the MA terms of 2.
Equation (3) is a mixed process, because it contains both AR and MA terms. It is an ARMA(1,1) or
ARIMA(1,0,1), because the AR order is one and the MA order is one.
In ARIMA models a non-stationary time series is made stationary by applying finite differencing of the data points. The mathematical formulation of the ARIMA(p,d,q) model using lag polynomials is given below:
, i.e
...... (3.4)

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GUJARATI N.D., Basic Econometrics, 4th ed., Gary Burke, New York, (2003)
5

Here, p, d and q are integers greater than or equal to zero and refer to the order of the autoregressive, integrated, and moving average parts of the model respectively.
 The integer d controls the level of differencing. Generally is enough in most cases. When
,
then it reduces to an ARMA(p,q) model.
 An ARIMA(p,0,0) is nothing but the AR(p) model and ARIMA(0,0,q) is the MA(q) model.
 ARIMA(0,1,0), i.e. is a special one and known as the Random Walk model. It is widely used for non-stationary data, like economic and stock price series.
3.1.3 Concept of stationarity
The concept of stationarity of a stochastic process can be visualized as a form of statistical equilibrium. The statistical properties such as mean and variance of a stationary process do not depend upon time. It is a necessary condition for building a time series model that is useful for future forecasting. Further, the mathematical complexity of the fitted model reduces with this assumption. There are two types of stationary processes which are defined below:
A process is Strongly Stationary or Strictly Stationary if the joint probability distribution functions of is independent of t for all s.
Thus for a strong stationary process the joint distribution of any possible set of random variables from the process is independent of time5.
However for practical applications, the assumption of strong stationarity is not always needed and so a somewhat weaker form is considered. A stochastic process is said to be Weakly Stationary of order k if the statistical moments of the process up to that order depend only on time differences and not upon the time of occurrences of the data being used to estimate the moments. For example a stochastic process is second order stationary if it has time independent mean and variance and the covariance values depend only on s.
3.1.4 Stationarity Analysis
When an AR(p) process is represented as
, then is known as the characteristic equation for the process. It is proved by Box and Jenkins that a necessary and sufficient condition for the AR(p) process to be stationary is that all the roots of the characteristic equation must fall outside the unit circle. Hipel and
McLeod mentioned another simple algorithm (by Schur and Pagano) for determining stationarity of an AR process. For example as shown in the AR(1) model is stationary when
, with a constant mean and constant variance

. An MA(q) process is always stationary, irrespective of the

values the MA parameters. The conditions regarding stationarity and invertibility of AR and MA processes also hold for an ARMA process. An ARMA(p, q) process is stationary if all the roots of the characteristic equation lie outside the unit circle. Similarly, if all the roots of the lag equation lie outside the unit circle, then the ARMA(p, q) process is invertible and can be expressed as a pure AR process.
3.1.5 Autocorrelation and Partial Autocorrelation Functions (ACF and PACF)
To determine a proper model for a given time series data, it is necessary to carry out the ACF and PACF analysis. These statistical measures reflect how the observations in a time series are related to each other. For modeling and forecasting purpose it is often useful to plot the ACF and PACF against consecutive time lags.
These plots help in determining the order of AR and MA terms. Below we give their mathematical definitions:
For a time series the Autocovariance at lag k is defined as:
……..(3.5)

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An Introductory Study on Time Series Modeling and Forecasting, Ratnadip Adhikari, R. K. Agrawal
6

The Autocorrelation Coeffient at lag k is defined as:

,

Here is the mean of the time series, i.e.
. The autocovariance at lag zero i.e. is the variance of the time series. From the definition it is clear that the autocorrelation coefficient is dimensionless and so is independent of the scale of measurement. Also, clearly
Statisticians Box and Jenkins termed as the theoretical Autocovariance Function (ACVF) and as the
6
theoretical Autocorrelation Function (ACF) .
Another measure, known as the Partial Autocorrelation Function (PACF) is used to measure the correlation between an observation k period ago and the current observation, after controlling for observations at intermediate lags (i.e. at lags < k ). At lag 1, PACF(1) is same as ACF(1).
3.1.6 Estimation of the coefficient of the model
In identification, the selection of one or more models seem likely to provide parsimonious and statistically representations of the available data was based on the calculation of a rather large number of statistics
(autocorrelation and partial autocorrelation) to help us.
At the estimation stage, we get precise estimates of a small number of parameters.
If we choose an MA(2) model:
……………………………………….. (3.6)
At this stage, we have to find precise accurate estimates of few parameters: the process mean and the two MA coefficients and for fitting our tentative model to the data7.
For choosing the coefficient values, we will base on some criterion: Maximum Likelihood criterion by Box and
Jenkins because the resulting estimates often have attractive statistical properties. The likelihood function of a corrrect ARIMA model from which ML estimates are derived reflects all useful information about the parameters contained in the data.
When shocs are normally distributed, Box and Jenkins suggest using the Least Squares because finding exact
ML estimates of ARIMA models can cumbersome and may require relatively large amounts of computer time.
Least squares refers to parameter estimates associated with the smalest sum of squared residuals.
(3.6)
Residuals: Consider the AR(1) model:
(3.7)
Or
(3.7)
Where is the constant term. Suppose we know the parameters ( and ) of model (1) and we locate at time
. By predicting using RHS variables in (1), we can not observe the random shock during but we know at time
. Assign its expected value to find the calculated :
(3.8)
At time t, we can observe . Random shock is given by subtracting the calculated value t (calculated from known parameters (2) from the observed value (Eq1).
(3.9)
We have assumed that and are known, the parameters of ARMA models but we must estimate them from the data. Note these estimates in the present case as and and the calculated value is
(3.10)

6
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An Introductory Study on Time Series Modeling and Forecasting, Ratnadip Adhikari, R. K. Agrawal
BOX AND JENKINS.,Time series analysis: forecasting and control and Allan Pankratz
7

When is calculated from estimates of parameters rather than known parameters (3) does not give the exact value of the random shock. Instead by subtracting (4) from (1), we get only an estimate of random shock denoted and
(3.11)
Eq(5) is the definition of a residual for any ARIMA model. In general, depends on and the estimated AR and MA coefficients (along with their corresponding past ’s and past residuals, which are estimated random shock. All other ARIMA models with multiplicative seasonal terms require a non linear least squares (NLS) method. (1)Diagnostic checking
The main importance of diagnostic checking is to show and to decide if the estimated model is statistically adequate or inadequate. If it is inadequate, we must return to the identification stage to tentatively select one or more other models. The diagnostic checking will also provide information about how an inadequate might be reformulated and how it can be improved.
The statistical adequacy of an ARIMA model involves the assumption that the random shocks are independant, meaning not autocorrelated. Since we can not observe the random shocks
, but we have estimates of them
(residuals
calculated from the estimated model), we use them to test the hypothesis of about the independence of random shocks because random shocks are a component of , the variable we are modeling. If random shocks are serially correlated, then there is an autocorrelation pattern in that has not been accounted
8
for by AR and MA terms in the model .
The basic analytical tool in diagnostic checking is: the residual acf
(3.12)
The in paranthesis indicates that we are calculating residual autocorrelations. If the estimated model is properly formulated, the random shocks should be uncorrelated, if they are, our estimates of them should be uncorrelated on average and residual acf for a properly built ARIMA model will ideally have autocorrelation coefficients that are all statistically zero. That’s why we used the residual acf. t-tests: each calculated residual autocorrelation must be checked if it is significantly different from zero by using Bartlett’s approximate formula to estimate the standard errors of the residual autocorrelations:

(3.13)
After finding the estimated standard errors of
, we can test the null hypothesis : for each residual autocorrelation coefficient. Since we have estimates of value, available in form of residual autocorrelations , we test the null hypothesis by calculating how many standard errors (t) away from zero each residual autocorrelation coefficient falls:
(3.14)
If the absolute value of a residual acf t-value is less than 1.25 at lags 1,2 and 3 and less than about 1.6 at larger lags, we conclude that the random shocks at that lags are independent.
(2)Testing for Normality
Decting whether the data are normally distributed or not, we use the Shapiro-Wilk’s Statistic. The null hypothesis is that data are normally distributed against the alternative that data are not normally distributed. If
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Pankratz, Alan (1983), Forecasting with Univariate Box-Jenkins Models: Concepts and Cases, New York: John Wiley & Sons.
8

the p-value of Shapiro-Wilk’s statistics is less than 5 percent (0.05), we reject null and accept the alternative, that is data are not normally distributed.
3.1.7 Forecasting
One of the main objectives of analyzing a time series is forecasting. That is, to predict future values of the series, where n = number of observations and
For instance, if we have n = 120 observed values of weekly sales, and we used to fit the AR(1) and MA(1) models. We may wish to compute the predicted value
. In this case,
. Predictions are usually made using the “best” fitted model for the time series under consideration. It is also a good idea to compute the standard error of each forecasted value in order to assess the accuracy or reliability of our forecast. This is similar to the standard error of the predicted value for a new observation in regression analysis. Although the formula is not presented here in details, MINITAB,
S-PLUS, SAS and other computing software calculate and prints the value as part of their output. Using the we can then compute a prediction interval for the new value
.
IV.

Results and Discussion

4.0 Introduction
The analysis of Quarterly Rwandan Economic monthly data from (1996 Jan to 2013 Dec ) proceeded with estimation of some discriptive statistics for the variables under study. Table 1 below represents the results of some descriptive statistcs.
Table 4.1 Descriptive statistics
Variable
Exchange rate
Mean
4.954016E2

Exports
1.515507E7

Imports_c.i.f
6.585227E7

Imports_f.o.b
5.297726E7

The average of exchange rate was 306.52 in 1996 and 652.29 in 2013 means that Rwandan franc with compared to US dollars has depreciated by about 212.8% in last 17 years, which shows a devaluation of about 12.5% of
Rwandan franc exchange rate every year. The average exchange rate for the period of 17 years as indicated in table 1 is 495.4 Rwandan franc per one $. (Appendix).
For balance of trade, on average Rwanda has exported goods and services with value of 6,960,215$ in 1996 and
47,752,482.41$ in 2013.This indicates that Rwandan exports have increased by 686.08% in last 17 years, which shows an increase of 40.36% every year. On average Rwanda imported goods and services with value of
21,539,783.84$ in 1996 and 187,284,131.71$ in 2013 and this means that Rwandan imports have increased by
869.48% in last 17 years. This also show us that from 1996, Rwandan imports increased by 51.15% every year till 2013. The last two results clearly indicate that our BoP was in deficit for the entire study period and the deficit has increased up to 957.03% for the period of 17 years. (Appendix )
4.1 Preliminary test on Data
As time series data was used for this study and as time series data is affected by so many issues like time, trends and outliers. Preliminary tests and respective adjustments were thus paramount to remove or minimize the effects of these factors that may lead to misleading results. Data was thus subjected to various tests like normality, stationarity and in the same notion; diagnostic tests were conducted to in the process of validating the model. Stationarity test and model identification with ACF and PACF
A stationary time series has a mean, variance, and autocorrelation function that are essentially constant through time. Often, a non-stationary series can be made stationary with appropriate transformations. The most common type of non-stationarity occurs when the mean of a realization changes over time. A non-stationary series of this type can frequently be rendered stationary by differencing.
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The data of exchange rate was found non-stationary in levels as seen in Appendix C. There was a need of differencing in order to retsore stationarity. Stationarity was therefore established at first difference second lag and hence the ARIMA (p,d,q) became ARIMA(2,1,0) or Autoregressive of order 2 obtained after first difference. Imports (f.o,b) was also found non stationary in levels as shown by fig 4.9 Appendix C and with differencing once, the data behaved well at first lag. The ARIMA model idendentified thus was ARIMA(1,1.1) Stationarity test was done on exports as well. The results reveal a stationary time series of order at the first difference, hence the ARIMA(p,d,q) became ARIMA(0,1,1) or Moving Average of order one obtained after first difference.
Estimation of the model
Exchange rate
From ARIMA (2, 1, 0) model we got up in model identification, this analysis was done by carrying out the estimation of the model
(4.1)
Table 4.2 Exchange rate Model Statistics
Model Fit statistics Ljung-Box Q(18)
Number
of Stationary
RModel
Predictors squared Statistics
DF
Exchange_rate1
.401
26.658
16
Model_1

Sig.

Number of
Outliers

.045

0

Estimate
1.804
.550
.118
1

SE
.971
.068
.068

t
1.858
8.049
1.727

.000

.008 -.092 .927

Table 4.3 ARIMA Model Parameters
Exchange_rateModel_1

Exchange No
_rate
Transformation

Constant
AR

Time

Difference
Numerator

No
Transformation

Lag 1
Lag 2
Lag 0

Hence our model will become: where variable at time t and is a random error.
For an ARIMA(2,d, q ) process, the stationarity requirement is a set of three conditions:

Sig.
.065
.000
.086

means exchange rate

All three conditions are satisfied for our model to be stationary, then we can forecast with it.
Imports f.o.b
From ARIMA(1,1,1) model we got up in model identification, this analysis was done by carrying out the estimation of the model
(4.2)

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Table 4.4 Imports f.o.b Model Statistics

Model
Imports_f.o.bModel_1

Number Model Fit statistics Ljung-Box Q(18) of Predictor Stationary
Rs
squared
Statistics
DF

Sig.

Number of Outliers

1

.001

0

.256

39.604

16

Table 4.5 ARIMA Model Parameters
Imports_f.o.bModel_1

Imports_f.o. No
Constant
b
Transformation
AR
Difference
MA
Time
No
Numerator
Transformation

Estimate SE
3.966E
-3.641E5
5
Lag 1 .125
.109
1
Lag 1 .701
.078
Lag 0
3.178E
9.553E3
3

t

Sig.

-.918 .360
1.152 .251
8.967 .000
3.006 .003

Hence our model will become: where means imports
f.o.b variable at time t and is a random error.
Imports c.i.f
From ARIMA(1,1,1) model we got up in model identification, this analysis was done by carrying out the estimation of the model
(4.3)
Table 4.6 Imports c.i.f Model Statistics
Ljung-Box Q(18)
Number of Model Fit statistics
Number
Model
Predictors Stationary R-squared Statistics
DF
Sig. Outliers
Imports_c.i.f1
.261
44.114
16
.000 0
Model_1
Table 4.7 ARIMA Model Parameters
Estimate SE t Imports_c.i.fImports_c.i. No Transformation Constant
5.215E
-.864
Model_1
f
4.505E5 5
AR
Lag
.062
.113
.549
1
Difference
1
MA
Lag
.649
.086
7.514
1
Time
No Transformation Numerator Lag
4.175E
1.194E4
2.860
0
3

11

of

Sig.
.389
.583

.000
.005

Hence our model will become:
c.i.f variable at time t and is a random error.

where

means imports

Exports
From ARIMA(0,1,1) model we got up in model identification, this analysis was done by carrying out the estimation of the model
(4.4)
Table 4.8 Exports Model Statistics

Model
ExportsModel_1

Number of Predictor s Model statistics 1

.156

Stationary squared Fit
Ljung-Box Q(18) Number
R- Statisti of cs
DF Sig. Outliers
74.024 17 .000 0

Table 4.9 ARIMA Model Parameters
ExportsModel_1

Exports No
Constant
Transfor Difference mation MA
Time
No
Numerator
Transfor mation Estimate SE
-1.893E5 2.701E5
1
Lag 1 .583
.057
Lag 0
3.246E3 2.161E3

t
-.701

Sig.
.484

10.256 .000
1.502

.135

Hence our model will become: where means export variable at time t and is a random error.
4.2 Diagnostic checking
Once precise estimates of the coefficients in an ARIMA model have been determined, then the next stage is diagnostic checking. It concerned with deciding if the estimated model is statistically adequate. Diagnostic checking is related to identification in two important ways. First, when diagnostic checking shows a model to be inadequate, we must return to the identification stage to tentatively select one or more other models. Second, diagnostic checking also provides clues about how an inadequate model might be reformulated.
At the diagnostic-checking stage, the residuals of the estimated model are examined to see if they are independent. If they are not, we return to the identification stage to tentatively select another model, we construct an acf, called a residual acf, using the residuals of the model as observations. As shown in
Appendix B, the residuals were found to be serially uncorrelated.
In practice, if the absolute value of a residual acf z-value is less than (roughly) 1.25 at lags 1, 2, and 3, and less than about 1.6 at larger lags, we conclude that the random shocks at that lag are independent and from all residuals acf’s and pacf’s graphs we have above, we accept the null hypothesis meaning that residuals are not serially correlated and that we can do the forecasting from their models.

12

4.3 Forecasting
4.3.1 Exchange rate
Table 4.10 Forecast for exchange rate
Model
217
218
219
220
221
Exchange_rateForecast 6.7689E2 6.7940E2 6.8166E2 6.8374E2 6.8570E2
Model_1
UCL
6.8163E2 6.8813E2 6.9443E2 7.0043E2 7.0610E2
LCL
6.7216E2 6.7067E2 6.6888E2 6.6706E2 6.6530E2
From this table we see that our forecast value is 676.8 (value of exchange rate in January 2014) and real value was 673.9 which belongs in our confidence limits. Values are not the same due to the random errors but this show us that our model is of good quality.
Figure 4.19 Forecasting for exchange rate

Imports f.o.b
Table 4.11 Forecast for Imports f.o.b
Model
217
218
219
220
221
Imports_f.o.bForecast 1.6489E8 1.6593E8 1.6758E8 1.6930E8 1.7105E8
Model_1
UCL
1.8109E8 1.8352E8 1.8606E8 1.8860E8 1.9112E8
LCL
1.4870E8 1.4834E8 1.4909E8 1.5000E8 1.5097E8
From this table we see that our forecast value is 164,892,632.16 (value of imports f.o.b in January 2014) and real value was 164,694,592.64. The forecast of imports f.o.b is 166,133,669.9 for the first quarter of 2014 while real value was 151,189,721.1; this shows us that our model can be used in forecasting since all values belong in our confidence limits. Values are not the same due to the random errors but this show us that our model is of good quality.
Figure 4.20 Forecasting for imports f.o.b

13

Imports c.i.f
Table 4.12 Forecast for Imports c.i.f
Model
217
Imports_c.i.fForecast 2.0572E8
Model_1
UCL
2.2521E8
LCL
1.8623E8

218
219
2.0743E8 2.0956E8
2.2852E8 2.3190E8
1.8633E8 1.8723E8

220
2.1174E8
2.3524E8
1.8824E8

221
2.1393E8
2.3854E8
1.8932E8

From this table we see that our forecast value is 205,720,066.1 (value of imports c.i.f in January 2014) and real value was 205,868,240.8. The forecast of imports c.i.f is 207,569,966.3 for the first quarter of 2014 while real value was 192,494,426.7; this shows us that our model can be used in forecasting since all values belong in our confidence limits. Values are not the same due to the random errors but this show us that our model is of good quality. Figure 4.21 Forecasting for imports c.i.f

Exports
Table 4.13 Forecast for Exports
Model
217
ExportsForecast 4.5801E7
Model_1
UCL
5.4917E7
LCL
3.6685E7

218
219
4.6319E7 4.6841E7
5.6195E7 5.7423E7
3.6443E7 3.6259E7

220
4.7366E7
5.8609E7
3.6123E7

221
4.7894E7
5.9762E7
3.6026E7

From this table we see that our forecast value is 45,801,448.44 (value of exports in January 2014) and real value was 45,001,406.36. The forecast of exports is 46,320,457.24 for the first quarter of 2014 while real value was
43,282,543.97 this shows us that our model can be used in forecasting since all values belong in our confidence limits. Values are not the same due to the random errors but this show us that our model is of good quality.
4.4 Polynomial regression model
In a cause and effect relationship, the independent variable is the cause, and the dependent variable is the effect.
Polynomial regression is a form of linear regression in which the relationship between the independent variable
X and the dependent variable Y is modelled as an nth order polynomial. It fits a nonlinear relationship between the value of X and the corresponding conditional mean of Y.
A polynomial regression model is given by this formula:

14

4.5 Exchange rate and Exports
Since we have found a positive quadratic relationship, then our model will become quadratic polynomial regression model which is given by this formula: where is a predicted export, is the constant, is linear coefficient, is quadratic coefficient and is random error.
Table 4.14 Model Summary
R

Adjusted
R Square Square

R Std. Error of the Estimate

.849
.721
.719
7101753.017
The independent variable is Exchange rate.
Table 4.15 Coefficients
Unstandardized
Coefficients

Standardized
Coefficients

B

Beta

T

Sig.

-5.361

-13.572

.000

6.032

15.269

.000

12.730

.000

Std. Error

Exchange_rate
-627841.431 46260.445
Exchange_rate **
761.933
49.902
2
(Constant)
1.293E8
1.016E7

Then our quadratic regression model will become:
Linear coefficient is negative while quadratic coefficient is positive, this means that quadratic coefficient will dominate linear coefficient and we conclude that there is a positive effect of exchange rate on exports.
A coefficient of determination equal to 0.721 indicates that about 72.1% of the variation in export (the dependent variable) can be explained by the relationship to exchange rate (the independent variable). This would be considered a good fit to the data, in the sense that it would substantially improve an exchange rate’s variability to predict export performance in Rwandan economy.
4.6 Normality test
Table 4.16Tests of Normality
Kolmogorov-Smirnova
Statistic df
Sig.
Expots_Residuals .060
216
a. Lilliefors Significance Correction

.053

Shapiro-Wilk
Statistic df

Sig.

.988

.075

216

Regarding skewness and kurtosis, values are little skewed and kartotic but it doesn’t differ significantly from normality. The null hypothesis for this test of normality is that the data are normally distributed and is rejected if the p-value is less 0.05, as all p-values are more than 0.05 we keep our null hypothesis.

15

Figure 4.23 Normality test histogram

Here is a plot of the residuals versus predicted Y. The pattern show here indicates no problems with the assumption that the residuals are normally distributed at each level of Y and constant in variance across levels of Y and then we conclude that residuals are approximately normally distributed.
Since residuals are normally distributed, then our model is of good quality and it can be used in forecasting.
4.7 Exchange rate and Imports
For exchange rate and imports, we have found a positive relationship but it is neither quadratic nor cubic and their residuals are not normally distributed, from that we decided to do the polynomial regression model with considering time as independent variable. Since all quadratic and cubic curves fit our data, then our models will be given by these formula: for quadratic and for cubic where is a predicted exports, is the constant, is linear coefficient, is quadratic coefficient, is cubic coefficient and is random error.
Table 4.17 Model Summary and Parameter Estimates
Dependent
Variable:Imports_c.i.f
Model Summary
R
Equation Square F
Quadratic

Parameter Estimates df1 df2

Sig.

Constant b1

b2

b3

2.426E
7.641E
2
213 .000 3.759E7
3
8.424E5 3
Cubic
1.658E
4.197E
.959
3
212 .000 3.211E7
10.581
3
5.428E5 3
The independent variable is Time.
Then our quadratic regression model will become: and cubic regression model will become:
. A coefficient of determination equal to 0.959 indicates that about 95.9% of the variation in imports (the dependent variable) can be explained by the relationship to time (the independent variable). This would be considered a good fit to the data, in the sense that it would substantially improve a time’s variability to predict imports performance in
Rwandan economy.
.959

16

V. Conclusions and Recomendations
The analysis indicated a positive quadratic relationship between exchange rate and exports, but for exchange rate and imports, relationship is neither quadratic nor cubic reason why it’s better to forecast our imports with considering time as our independent variable. From results of stationarity test, all variables have become stationary time series after first difference.
In model identification we found that:
Exchange rate is ARIMA (2,1,0) or
,
In this research, the function of exchange rate, imports and exports during the period January 1996 to December
2013 was estimated. The major issue was to analyse the impact of of exchange rate on balance of trade (imports and exports). And we proposed as hypothesis that in Rwandan economy, exchange rate has a relationship with balance of trade components and hence balance of trade is affected by exchange rate volatility.
Stationarity analysis, the ACF and PACF graphs’ test show us that the variables used in estimation are all stationary at the same levels (after first difference). Their relationship was estimated by using scatter plots with their curves estimation, according to the results, we find that exchange has a strong positive quadratic relationship with exports and a weak positive quadatic ralationship with imports but impotrs have strong quadratic and cubic relationship with considering time as independent variable.
With refering to the concepts of UBJ/ARIMA analysis, all stages have been tested for all variables:
Identification (Choose one or more ARIMA models as candidates), Estimation (estimation of parameters of model(s) chosen at identification), diagnostic checking (check the candidate model(s) for adequacy) and we have find that for all variables (exchange rate, imports f.o.b, imports c.i.f and exports), models chosen are satisfactory this allow us to conclude that our models are of good quality and that they can be used for forecasting future values either for exchange rate or for balance of trade.
The estimation by polynomial regression model, showed us that as exchange rate increases, as exports increase the same as imports, this results a positive effect of exchange rate on balance of trade components, we did the forecast for next 24 months (2years) and our forecast values are close to the real values from January to July
2014.
Since we have all time series variables (exports, imports f.o.b, imports c.i.f and exchange rate), and all components of balance of trade (imports and exports) increase as exchange rate increases, it seems that this show us a great positive impact of exchange rate on balance of trade and it will be the same for our forecast because they have been generated from variables which are positively generated.
5.1 Recommendations
According to the empirical study results, exchange rate, exports and imports greatly benefit the economic development in Rwanda. To overcome the problems existed on the balance of trade, for the current financial crisis, and in order to reduce its corresponding economic loss as much as possible, keeping scale of exports is necessary. Thus, governments of Rwanda should lay down import subsitution strategies and export promotion strategies or policies and make sure these strategies are implimented. This in a way, will reduce dependency on imports and narrow the gap between exports and imports. Import subsitution and export promotion strategies will also in a way solve the problem of depriciating currency since when exports increase and imports reduce, supply of foreign currency likely increases and the invisible hand operates smoothly in favour of Rwandan economy and currency.

17

VI. References
1.
2.
3.
4.
5.
6.
7.
8.
9.

Ahuja S.Chand H.L. (2012) Macroeconomics theory and policy, advanced analysis
Box and Jenkins.,Time series analysis: forecasting and control.
Cramer, D.,&Hawitt, D.(2004), the SAGE dictionnary of statistics-Landon: SAGE.
Damodar N.Gujarati, (2003), Basic Econometrics Fourth Edition, United States Military Academy, West
Point/ Gary Burke, New York.
Harvey, A. C., (1981) Time Series Models, New York: John Wiley & Sons.
Imdadullah, “Time series analysis” January 2014, Basic statistics and data analysis, itfeature.com
Retrieved 2.
Kohn, R. and Ansley, C., (1985) "Efficient Estimation and Prediction in Time Series Regression
Models," Biometrika, 72, 3, 694–697.
Pankratz, A., (1983) Forecasting with Univariate Box-Jenkins Models:Concepts and Cases, New York:
John Wiley & Sons.
Ratnadip Adhikani, R.K. (2010) An Introductory study on Time series modeling and forecasting,
Agrawal.

Documents
1. Akaike, H. (1974), "A New Look at the Statistical Model Identification," IEEE Transaction on
Automatic Control, AC–19, 716–723.5. The relationship between exchange rates and international trade, A review of economic literature; Marc Auboin and Michele Ruta WTO (October 2011)
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American Economic Review 90, 1093-1109.
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IMF Working Paper, May 2004, International Monetary Fund.
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Pp205-208.
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Applied Econometrics and Quantitative Studies Vol 3-1 (2006)
13. Razali, N.M&Wah, Y.B(2011). Power comparisons of Shapiro-Wilk, Kalmogorav-Smirnov, Lilliefors and anderson-Darling tests Journal of statistical modeling and analytics, 2(1), 21-33

18

APPENDIX
4.5.1 Exchange rate
Figure 4.7 ACF and PACF of non-stationary exchange rate time series

Figure 4.8 ACF and PACF of a stationary exchange rate time series after first difference

Fig 4.9 ACF and PACF of non-stationary importsf.o.b time series

.Figure 4.10 ACF and PACF of a stationary imports f.o.b time series (first difference, first lag)

19

Exports
Figure 4.13 ACF and PACF of a non-stationary exports time series

Figure 4.14 ACF and PACF of a stationary exports time series

20

Exchange rate
Figure 4.15 Residuals plot of exchange rate and imports f.o.b

Imports f.o.b
Figure 4.16 Residuals plot of imports f.o.b
Figure 4.17 Residuals plot of imports c.i.f, exports f.o.b

21