A Neural Network and Genetic Algorithm Hybrid Model for Modeling Exchange Rates: the Case of the US Dollar/ Kuwaiti Dinar

Topics in Middle Eastern and African Economies Vol. 12, September 2010

A Neural Network and Genetic Algorithm Hybrid Model for Modeling Exchange Rates: The case of the US Dollar/ Kuwaiti Dinar

By:

Meriem DJENNAS*, Mohamed BENBOUZIANE** and Mustapha DJENNAS**

*Faculty of economics, University of Amiens, France

** Faculty of economics, University of Tlemcen, Algeria

[email protected], [email protected], [email protected]

Abstract

Predicting exchange rates is one of the most leading financial problems because of its intrinsic difficulty and practical applications. In recent years, many nonlinear models have been proposed in the literature to modify the results of prediction in order to improve the forecasting performance of high frequency exchange rates. Neural networks and chaotic models are among the models that have been exploited and have shown promising results. The main objective of our research is to conduct a comparative evaluation of nonlinear models on a set of data and variables and to verify the predictive power of neural models under the same experimental conditions. This study uses a criterion to evaluate the model performance: the square root of the mean squared error. Our research study will be applied to the case of the US Dollar – Kuwaiti Dinar exchange rate.

Key Words: Exchange Rate –Neural Networks – Genetic Algorithm - Kuwaiti Dinar.

INTRODUCTION Usually, the study of the financial market evolution, and more particularly exchange markets, is considered as one of the most important research fields in international finance as long as exchange markets are considered as open systems that react on the base of collected information during the time periods. The behavior modeling process of the exchange markets requires beforehand a deepened knowledge of the exchange system and factors that influence it. Hence, the inherent objective is to extract the founding rules of a relevant modeling process. It goes without saying that in these markets, the exchange values are difficult to predict. In this fact, the development of an efficient strategy and a prediction tool that resists in facing the financial shocks is the subject of several research and studies. The major risk which an intervening party is dealing with is the non-linearity of exchange rate data sets. Thus, this study is articulated with the following problematic: the contribution of the application of artificial intelligence tools in the modeling of the agents’ behavior intervening in a financial market while trying to improve their training processes for the prediction. It consists in applying a hybridization of two fundamental tools of artificial intelligence, the artificial neural networks (ANN) and the genetic algorithms (GA). First, we construct a neural network model for the exchange rate forecasting. Agents-based modeling in an artificial market by neural networks is perfectly adapted for the resolution of complex and non-structured problems which are difficulty accessible by the mathematical or statistical approaches (non-linearity

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conditions or inexistence of the adequate mathematical model). We will show that neural network models offer the advantage to operate in the changing, hostile and unpredictable environments. Secondly, we apply the agents-based model that provides a convenient structure to test the learning capacity and reasoning concerning interactions between traders in the exchange market where the agents’ training will be studied by a genetic algorithm. Since agents operate and interact in the environment, the issue is to take decisions that offer a higher level of flexibility and performance. Thus, we suppose that agents can take the best decision (optimizing their performance) if they possess a better knowledge (availability of information) about the market environment. Therefore, the objective is to implement a rigorous artificial market that permits to understand some real world market mechanisms and to foresee some others. As far as possible, most fundamental elements and properties of the macro environment of the exchange market as well as those of the micro environment will be preserved. This research reasoning will be applied in the Dollar – Kuwaiti Dinar exchange rate. Without calling into question the econometric methods utility for the prediction, we will demonstrate, on the basis of a comparative survey, that artificial intelligence tools are more effective under some market conditions like the non-linearity. Thus, we will try to demonstrate that they permit to predict with a higher level of precision the tendency and variations of the exchange rate in a financial market. The rest of the paper will be articulated as follows: first we will vive a description of the market of exchange rates in the GULF countries, Then we proceed to give more insights on technical methodology and how to deal with neural networks and genetic algorithms. In the third section, we will apply these techniques to forecast the US Dollar Kuwaiti Dinar exchange rate, and finally we will give some concluding remarks.

Section I. The Exchange rate Market in the GCC Countries

I.1. Introduction

During the Bretton Woods institutions, the Arab countries have shown some willingness to establish cooperation managing their exchange rate policies. In 1945, 22 countries have planned to launch a single currency called "Arab Dinar". Sixty years later, only countries of the Cooperation Council for the Arab States of the Gulf CCASG1 (Saudi Arabia, Kuwait, Oman, Qatar, United Arab Emirates and Bahrain) still ongoing efforts to create a common currency in 2010.

The introduction of the single European currency in 19992, which experienced a sharp appreciation against the dollar on the foreign exchange market due to the financial crisis, seems to have been emulated since the Gulf countries have confirmed during a summit in Doha in March 2009, the introduction in 2010 of a single currency.

Although Oman has decided to withdraw from the project and that Kuwait has decided to link its currency to a basket currencies rather than the dollar in order to face inflation, the determination of Gulf monarchies did not been initiated so far. The Kuwait decision is explained by the sharp rise in inflation in the Arabian Peninsula.

1 Or Gulf cooperation Council GCC. 2 For a detailed discussion on the Euro introduction, see Neaime and Paschakis (2002).

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The Gulf countries have been forced to reduce their interest rates to follow the Fed’s (Federal Reserve System) downturn decisions to fight against speculation on their currencies, although rate increases are deemed necessary to curb the inflation. The central banks of the GCC countries are thus in the same uncomfortable position as the European Central Bank (ECB).

But the dollar weakness and inflation rising (two economic factors that erode their oil revenues) have undermined the single currency project even though some divisions persist between countries. Thus, if Saudi Arabia, at the initiative of the GCC in 1981, the largest producer and exporter of crude oil in the world, make lobbying to keep the deadline, others countries like the UAE, bring up logistical challenges for the project establishment.

I.2. Economic integration in the Gulf countries

In 1981, the United Arab Emirates, Bahrain, Saudi Arabia, Oman, Qatar and Kuwait have put together the Gulf Cooperation Council (GCC) to create an economic and financial competitive integration. The six countries also share many common cultural, historical and social values. These factors, in addition to geographical proximity, have facilitated the interaction and business transactions, and have created a homogeneous exchange zone. The objective of GCC is to strengthen alliances and relationships between member’s countries. It aims to standardize various areas of activity such as economics, finance, trade, tourism, legislation, administration, agriculture, etc.

To accomplish the above objectives, the council outlined several steps. The most important are:

1. Member countries should allow free movement of imports and exports of natural resources, agricultural and industrial products.

2. Trade policy with other regional economic conglomerates should be unified to create well-balanced trade and exchange relationships.

3. Creating a free movement area for people and goods.

4. The investment rules should be harmonized in all member countries.

5. Member countries should coordinate their fiscal and monetary policies and establish cooperation between financial institutions and central banks.

The ultimate goal of the GCC countries is to jump from a cooperation and coordination strategy to a more advanced economic integration. The union has already succeeded in creating a common market called "The Gulf Common Market" in January 20083. Such that goal requires disengaged movement of people and physical capital. Finally, the GCC aim to establish a competitive monetary and financial integration by adopting a common currency in 2010. Efforts were also intensified to unify the foreign trade policies in the GCC countries. Establishing a single currency should expect to promote trade and financial integration, facilitate foreign direct investment, and support development of the Gulf countries in an optimum currency area (OCA).

3An interesting description of the CCG is contained in a recent study realized by the European Central Bank; see Sturm et al. (2008).

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Despite a strong political will handled by the GCC countries, the implementation of the union has had considerable obstacles, including the authorities’ insistence of member countries to apply measures of central inspection, causing considerable delays in the movement of goods between countries. In March 2003, the GCC secretariat announced that it plans to examine factors that obstruct the union implementation. It anticipated that union construction would contribute to the expansion of intra-GCC trade and economic growth. In addition, the union is supposed to provide a conductive environment to create large companies in the Gulf and a better exploitation of economic assets in each country.

The six monarchies of the GCC have known for several years high growth rates in some cases approaching 10% in real rates4, thanks to the surging of oil revenues, which increased their liquidity, but with a rise in inflation as consequence.

In September 2007, concerned by the mortgage lending crisis, the Fed has reduced its main policy rate and its discount rate to support economic growth in the United States, which showed slowing signs. But that decision has put the Gulf central banks in a difficult situation because of the inflationary pressures facing them.

Inflation was as high as 14.8% in March 2007 in Qatar on an annual basis, and experts have estimated that it was above 10% in the UAE. In Saudi Arabia, inflation reached 3.8% in July 2007, the highest level in seven years, and it has been estimated at about 5% in other GCC countries (Bahrain, Kuwait and Oman)5.

In fact, reductions in interest rates in these countries increase more cash already overcrowded and thereby accentuate inflation. But refrain from reducing these rates undermines the value of their currencies linked to the dollar in decline and their huge financial assets.

Apart from Kuwait, which made end of the dinar-dollar relationship in May 2007 and pegged its currency to a basket of currencies, the other five GCC monarchies, including Saudi Arabia, maintains a fixed rate between their currencies and the dollar.

I.3. Foreign exchange market in the GCC countries

The GCC countries represent a unique case of an advanced integration compared to the rest of the Middle East and North Africa countries (MENA). These countries have committed themselves to improve their economic integration by creating a free trade area in 1981. Thus, the GCC countries have expressed their interest in creating a common currency in 2010, which reflecting a real commitment to achieve a real economic union.

4 La Depeche.fr: L’indexation au dollar des monnaies du Golfe de plus en plus contestée, October 2007 5 La Depeche.fr: L’indexation au dollar des monnaies du Golfe de plus en plus contestée, October 2007

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As it can be seen on the graph, the GCC countries have always ensured considerable stability in their exchange rates. The huge revenues from oil exports have enabled these countries to accumulate large foreign reserve of around 50 billion dollars in order to counteract any variation in currency6.

The Gulf countries are mainly among the largest exporter’s countries of oil, gas and refined products. Alone, the Gulf region holds 40% of proven world oil reserves. While gas represents 23% of the world reserves7. The production of these materials can be half the Gross Domestic Product (GDP) and three quarters of the exports volume8. With full convertibility of currencies and a dollar peg, fiscal policy has been the main instrument to directing these outcomes.

One of the most critical decisions in forming a monetary union is the choice of an appropriate exchange rate regime for a single currency on all neighboring countries. In 2003, the GCC member countries have agreed to index their currencies to the U.S. dollar and maintain parity until the creation of a monetary union in 2010. A decision on the exchange rate regime for a single currency should be taken.

The standard criterion for determining an optimal exchange rate regime is macroeconomic and

6 For example, the constitution of Kuwait stipulates that 10 percent of oil revenues are diverted into a fund established to counter balance swings in the volatile business cycle. That fund now stands at around US$100bn. The Saudi Arabia Monetary Agency (SAMA) manages at least part of Saudi Arabia’s net foreign assets, estimated at over US$180 billion. 7 Country authorities; and Fund staff estimates, IMF 8 Country authorities; and Fund staff estimates, IMF

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financial stability in facing real and nominal shocks. Ideally, the system of the used exchange rate should produce an internal and external stability, maintain monetary credibility and international competitiveness, and reduce the risks associated with foreign exchange and transaction costs. By applying these criteria to the GGC countries, it is necessary to take into account the influence of the dominant oil sector to the GDP, exports and government revenue, the structure of the labor market and the ability of these countries to evolve their growth if monetary policy independence will be implemented.

Although the arguments that lead to adopt a flexible exchange rate policy after the monetary union are encouraging especially as regards to monetary stabilization, it coexists with other factors in favor of maintaining the currency region dependency to the U.S. dollar. The U.S. dollar anchoring can specifically reduce the exchange rates volatility and the capital flows that can occur by nominal shocks9, which allow to understand easily monetary policy and to simplify the financial and commercial transactions.

A flexible exchange rate regime would allow to these countries adjusting their policies to real shocks better than under a fixed exchange rate regime, but the structural and institutional characteristics of the GCC members, the challenge of choosing an alternative anchor, and the need to implement reforms in the financial system to operationalize a floating regime make the regime transition a long term task. The intermediate regime of a basket of currencies can be a useful way to introduce some flexibility on the exchange rate, and can reduce the adverse effects of the major currencies volatility.

Currently, the GCC members remain committed to anchor their currencies to the U.S. dollar until the formation of the monetary union proposed in 2010, except for the Kuwait which has already proceeded to tie its currency to a basket of currencies in May 2007.

I.4. Exchange mechanisms in the Gulf countries

The richness of the GCC economies has its origins in the volume of exports of strategic materials including oil and gas. Since oil exports are denominated in dollars, and that the various currencies of these countries are directly tied to the dollar, it has served these countries to ensure a certain stability regarding the oil prices volatility.

The parity with the dollar helped initially to reduce the risks of currency and stabilize the variation in financial wealth largely traded in dollars.

Since raw materials represent the bulk of exports and that the out-off-oil sector is still relatively small and less diverse, the GCC countries have decided to maintain parity with the dollar to conserve an external stability and a credible monetary position.

The decision to establish a link with the U.S. currency was taken in 2002 when the dollar had a relatively greater appreciation against other major currencies. However, the dollar began again to depreciate against the euro; it lost 50% of its value during the same year.

For the GCC as a whole, the European Union is the main supplier of imports with a 37% of

9 The geopolitical risks and the volatility of oil prices are in this case unrelated to fundamentals elements.

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imports. However, the share of imports from the USA is less than 15% (see Table 1).

Saudi Total Importations sources Bahrain Kuwait Oman Qatar E.A.U Arabia GCC European Union 22 32 22 51 37 38 37 U.S.A 5 15 9 10 13 15 13 Japan 7 8 17 11 8 8 8

Table 1. Geographic distributions of Gulf countries importations (In %) IFS DOTS http://www.imfstatistics.org/dot/

It is obvious that if the dollar depreciates, the purchasing power of exports from the Gulf countries will be considerably eroded. Moreover, the bulk of imports are invoiced in Euros, and then it is clear that any dollar depreciation against the euro causes sizeable losses in the trade balance which is reflected on the power parity purchase.

Moreover, if the capital mobility, trade openness and foreign direct investment increase, the willingness to keep U.S. dollar peg may decrease, especially if trade liberalization leads to more volatility. In this regard, a flexible exchange rate regime would have the advantage that it can provide another adjustment tool against shocks and volatility management of oil prices.

For these reasons, it is urgent for these countries to increase their volume of exports excluding oil in order to diversify their economies. Certain of these countries (Bahrain and Oman) will face depletion of their oil reserves in the near future and therefore they need to adopt policies to promote out-off-oil sector. In the same way, other countries propertied oil reserves should develop other economic segments.

I.5. Previous Studies

Most previous studies were designed to determine at what extent the Gulf countries are ready to establish a single currency. All member countries share the same ethnicity, culture, traditions and political system. Moreover, they have very similar economic structures. They are heavily dependent on oil revenues and the degree of diversification of their economy is relatively low, which results in a relatively high vulnerability to external shocks. Through lack of an economic engine creator of wealth, the economy of these countries is dangerously exposed to fluctuations in oil prices. The almost complete dependence on the export of this material (while the usual consumed products come from outside) weakening their economies and make them vulnerable to external shock.

While these countries consider themselves very open to international trade linked to oil exports and imports of various consumed products, the level of intraregional trade is still insignificant. The majority of researches on the new monetary policy in the region have concluded that these countries are not yet ready to abandon their national currencies to adopt a common currency:

To examine the potential of Gulf countries to be ready to create the common currency, Dar and Presley (2001) have highlighted a very low degree of integration between these countries illustrated by an insignificant volume of intra-regional transactions. They attributed this failure to

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the similarity of economic structures based on oil rents and other economic and political factors. The authors recommended the introduction of some more flexible intra-regional trade rules, expansion of foreign direct investment, improvement of the diversification and production process and acceleration of the privatization process.

A detailed discussion on the integration of Gulf countries was conducted by Laabs and Limam (2002). They perform a test on the purchasing power parity. Their results indicated that the countries exchange rates are strongly related and have the same stochastic trend. By examining several eligibility criteria for creating a single currency (such as openness, mobility factors, diversification of production, the structure of production, prices and wages, the inflation rate and other political factors) they concluded that current conditions are not favorable for any monetary integration and creation of a single currency. In particular, they referred to a lack of production diversification, an intraregional trade still in an embryonic phase, and some divergence in fundamentals macroeconomic aggregates. On the other hand, the authors have confirmed that certain problems presence does not necessarily mean that the region is not ready to form a monetary union. To boost the creation capacity of a monetary union, the authors strongly encouraged to remove restrictions on the free movement of goods and production factors and a higher degree of political integration.

A similar study was conducted by Jadresic (2002). The objective was to measure the benefits and costs of replacing the GCC countries currencies with in a common regional currency. He emphasized that the success of this approach is conditional on a set of measures including the elimination of tariff barriers that hinder trade and foreign direct investment and coordination of economic policies to ensure macroeconomic stability.

Fasano and Schaechter (2003) have considered a favorable view of that union. They claim that this union, when combined with macroeconomic and structural policies, could improve financial services, reduce transaction costs, increase goods and services prices transparency, facilitate investment decisions and promote the resources allocation in the region.

Unlike most previous studies, Darrat et Al - Shamsi (2005) concluded that the failure of the GCC countries to achieve an economic and financial integration is not the result of economic and financial incompatibility between countries in the region, but rather the socio-political differences that have affected the progress towards a viable common area. These findings were made by performing co-integration tests on various macroeconomic indicators such as exchange rates, inflation rates, money supply, etc. They found that the Gulf countries share a long-term trend common witch link-up their economic activities, their financial markets and monetary policies. The existence of a co-integrating relationship in the long term does not mean that business cycles are synchronized in the short-term.

Sturm and Siegfried (2005) conducted a study in which they evaluated the progress made by the Gulf countries for the establishment of a single currency. Similarly to previous studies, their results showed a remarkable structural and monetary convergence but a less dynamic fiscal convergence. In order to make the monetary union credible and sustainable, they have proposed to establish a supranational monetary institution which would be responsible for managing both the exchange rate and monetary policies engaged in the region conditions as a whole rather than coordinate national policy. They suggest the establishment of a central bank similar to the

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European Union which might be responsible for making decisions of monetary policy, monitoring payment systems and collecting all the efforts towards a financial regional integration. A study conducted by Hebous (2006) highlighted the considerable progress that the GCC countries have made as regards the convergence by taking the European standards as reference. The study indicates that these efforts will accentuate the commonalities between member countries and that these efforts represent the main factor leading to reducing the costs of establishing the single currency.

Section II. Neural Networks and Genetic Algorithms

II.1. Introduction

The financial series (known by their complexity) are econometrically difficult to model, unstable and some times wrongly estimated by linear models. Thus, their study has attracted the attention of many researchers. The used methods are different from each other depending on the forecasting purpose, the nature of the used information and the considered mathematical models.

The exchange rate is one of the financial series whose behavior is difficult to model. That is why financial agents are closely monitoring the movement of this variable that represents a very important financial indicator and a relevant potential source of information as to economic and financial conditions. This allows economic agents to develop a better monetary policy whose objective is the prices stabilization.

However, the understanding of the exchange rates movement has become a difficult task since the adoption of a floating exchange rates system and the liberalization of the exchange control. Indeed, some researchers also believe that the exchange rates prediction is not feasible. Their work relied mainly on market efficiency hypothesis (Fama, 1965) whose the reduced form indicates that all the available information includes only the historical asset prices. A market is efficient in the reduced sense if the price of a security reflects all information contained in the historical course of this title. This form is linked to the random walk concept10, that prices have no memory. In other words, everything that happens between t and the future period t +1 is purely random. The historical prices cannot be used to predict future ones and it is impossible to act at the right time on the market. Thus, the present course is the best estimator of the future course:

Where:

- : is the logarithm of the exchange rate in period t.

10 A random walk, sometimes denoted RW, is a mathematical formalization of a trajectory that consists of taking successive random steps. The results of random walk analysis have been applied to computer science, physics, ecology, economics, and a number of other fields as a fundamental model for random processes in time.

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- : is a serially independent white noise.

Although data are characterized by high or medium frequency, the inability to forecast exchange rate is often summarized by the results of Meese and Rogoff (1983). Their study showed that in a floating exchange rate system, the random walk model is the best exchange rate modeling tool.

From a theoretical and practical point of view, this model has undergone several reviews. Indeed, many studies (De Grauwe and Vansteenkiste, 2001; Brooks, 1996; Drunat, Dufrenot, and Dunis Mathiew, 1996) have shown that the nonlinearity nature of the exchange rate is its compatibility with the general equilibrium theory.

The simple models of exchange rates give rarely satisfying empirical applications. The same set of equations, tested on different data sets, usually gives significantly different results even with opposite signs: the relevant tests for a reference period collapse when the series are extended. In these circumstances, it is important to consider the acceptable degree of complexity to analyze the exchange rate.

These successive failures of the traditional economy of exchange rate have raised several questions about the actual functioning of this market. These questions motivated the research directed towards the non-linear models whose prediction is better than a simple random walk model. However, in some studied cases, improving the forecasting process was not statistically significant11. One reason for this flaw is that these models were unable to incorporate all the exchange process characteristics.

More recently, another class of nonlinear models (learning algorithms) has been proposed. It seems more appropriate to explain the financial series behavior. During the last decade, statistical learning algorithms have aroused considerable interest in academia and companies from various industries. They have been successfully implanted to accomplish predictive tasks related to the observed statistical process for several variables possibly identifiable.

Plasmans Verkooijen and Daniels (1998) used neural networks for forecasting a macroeconomic model of exchange rates to test non-linearity of the relationship between variables. Their model was unable to generate satisfying predictions. On the other hand, Zhang and Hu (1998) have modeled the exchange rate as a dependent variable on its past values. Their model gave better results than those of a simple linear model. Hu et al. (1999) showed, using daily data, that the neural networks are much more robust than a random walk model where the application of neural networks on short-term data was encouraging in many cases and the results show that these models may offer some advantages for high-frequency predictions.

The research on Artificial Neural Networks (ANN) is growing fast and commercial applications of this success followed over 90 years. Today, ANN are firmly implanted in various industries: financial circles for predicting market fluctuations, in the banking field for fraud detection on credit cards and calculating credit scores, in the marketing departments of companies in various industries to predict consumer behavior, etc. Applications are numerous and share a capital common key as regards to the ANN usefulness: processes for which we wish to give predictions involve many important variables and especially if there are possibly non-linear high-level

11 Mizrah, 1993.

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dependencies of these variables so that if they are found and exploited, they can serve to improve the prediction process. Hence, the fundamental advantage of ANN compared to traditional statistical models is that they automate the finding process of most important dependencies during the prediction stage.

Our work crosses two research fields, namely artificial intelligence (AI) and foreign exchange markets. The challenge is applying the AI tools to model the exchange rates behavior. This objective can be achieved with performance and increased capacity of the model. This will be expressed in a high accuracy of prediction.

To do this, we adopt a method combining two tools in the AI field (Artificial Neural Networks ANN and Genetic Algorithms AG).

II.2. Back-propagation algorithm learning process

In financial markets, particularly in exchange rates markets, currency values are difficult to predict. Developing an effective strategy and a prediction tool has been object of many researchers study. The major risk facing the researcher is the non-linear series of exchange rates.

To predict efficiently exchange rate, we are particularly interested by meta-heuristics methods. ANN derive their power to model from their ability to capture high-level dependencies, that is to say, that involve multiple variables at once (the macroeconomic variables and trend variables, past values of exchange rates).

Without questioning the usefulness of econometric methods for prediction, we will demonstrate, on the basis of a comparative study that the ANN perform better under conditions of non- linearity. In this way, they predict at a higher level of precision the trend and variations in exchange rate.

II.2.1. Model architecture

The selection of an appropriate network is based on the research objective. In the prediction field, the back-propagation network is the most adopted.

The ANN is composed of an input layer, an output layer and one or more intermediate layers called hidden layers. Each layer is composed of a number of units called neurons that are provided with activation functions (transfer functions). These will vary from one network to another depending on the number of units in each layer, the number of weights, the layers arrangement and the activation function assigned to each layer. Figure 2 illustrates a neural network structure as defined by the various previous works:

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Figure 2. General architecture for a neural network

h1 Q(X) α β

X1 Q(X)

F(X) Y1= F1 (w, X)

X2 Q(X)

F(X) Yp= Fp (w, X) hm-1 Xn Q(X)

X =1 Q(X) β0p 0 α0m h0=1

Zhang, G., Hu, M.Y. (1998), “Neural Network Forecasting of the British Pound/US Dollar Exchange Rate”, OMEGA - The International Journal of Management Science, Vol. 26, No. 4, p. 495-506.

Where:

- X1, X2,….. Xn : are explanatory (exogenous) variables of the input layer

- w(α, β) is the set of model parameters (connection weights)

- h1, h2,…..hm : the number of neurons in the hidden layer

- Q(X) is the activation function of neurons in the hidden layer

- F(X) is the activation function of the output layer units

- Y1, Y2,…..Yp: are the explained (endogenous) variables

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- α01,… α0m et β01,…, β0p : are constants called bias of the input layer and hidden layer respectively.

The network receives the information on the input layer as a set of explanatory variables which are then processed using one or more hidden layers containing one or more neurons. In this phase, variables are weighted by connection weights α and transformed by the activation function

Q. We obtain a new set of variables h1, h2…..hn, so that . In turn, the variables hj are weighted by the connection weights β and processed by the function F. each output is interpreted by the following formula:

The typical activation functions include the sigmoid function, the tangent function, the hyperbolic function and the sinus and cosines functions. We note that although there are no rules to select the activation function type, the choice is made arbitrarily. The sigmoid function is generally the most adopted, knowing that using different types of activation function has no major effect on network performance.

The ANN functioning is similar to the human brain, it captures the relationship between the variables after a representative set of data or examples while considering the principle of induction, that is to say learning by experience. The ANN content generally to give a simple non- detailed description of the problem. Hence, this method provides an alternative to conventional analytical techniques often limited by strict assumptions of normality, linearity and independence of variables. However, the ANN does not always give us a usable rule by a human. This system often remains a black box that provides a response for a given presented data, but it doesn’t offer an easily interpretable justification.

II.2.2. Learning procedure

Learning consists at first to calculate the optimal weights of the various synaptic connections using a sample data. We describe the most commonly learning technique used: the back- propagation.

Before the learning phase of neural system, it should first proceed to the normalization of the raw data which is considered as the most important pretreatment when using ANN. The most common method is expressed by the following equation:

Where:

- : is the given normalized value

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- : is the original data value

- : is the minimum value in the data set

- : is the maximum value in the data set

- : is the upper bound of standardized data

- : is the lower bound of standardized data

By applying this transformation, we can conduct all the variables to have similar orders of magnitudes, which facilitate the determination of standard values for the algorithm parameters. This allows to obtain satisfactory results without adjusting them for each new problem and also to compare the distributions of variables.

The general principle of the feed-forward algorithm is to give the network a large number of examples for which the input and output associated are known and weights are modified to correct the error committed by the network (i.e. the difference between the desired and the obtained responses). Thus learning is seen as an optimization problem of finding the network coefficients for minimizing a cost function. The most used cost function is the squared function on the base of learning; it is to minimize the sum of squared errors between the network output and the real value of the output. The cost function at the iteration i defined on the entire training set is as follows:

Where:

- : is the mean square error

- N : is the sample size

- : is the vector of desired outputs

- : is the vector of network outputs for iteration i.

One of the major ANN advantages is that their learning algorithms are applicable to all networks types. We have all the freedom as regards the choice of the best adapted architecture to the problem, and whatever the network structure, we can always use the same set of learning algorithms. This flexibility allows implementation including networks whose architecture depends strongly on the structure of the problem to model expressed by equations.

The problem that appears most often during learning is the over-fitting. If the ANN learns insufficiently or non-appropriately, it will give incorrect results when it receives some data

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slightly different. To avoid over-fitting, the performance of the trained network should then be compared on another set called validation set. This strategy should provide a better generalization of the model. This consists to monitoring the cost function evolution on a validation database and to stop the iterations when the calculated cost on this database begin to grow.

Once the network is executed, we should always apply the tests to verify if it responds correctly. In the test phase, a sample part is simply removed from the training sample and conserved for the out-off-sample tests. For example, we can use 60% of the sample for learning, 20% for validation and 20% for testing.

II.3. Genetic Algorithm-based learning

The ANN pattern is often difficult to optimize, and any change in the structure could lead to a dramatic deterioration of its predictive capacity. Indeed, the non-interpretability of ANN layers degrades its generalization ability trapped in local minima because of their behavior similar to that of a local search algorithm using the back-propagation gradient. By coupling the ANN to GA, this drawback is overcome by a diversification research phase.

Therefore, our objective is to use GA as a way to change the ANN configuration that can provide the best prediction of the future exchange rates. We present a sequential hybridization strategy between these techniques to develop a more reliable interpretation. Optimization of these techniques through this hybridization is then validated on several examples. In particular, the resulting mean square error will be compared with the one from a simple ANN and a random walk model. The simulations permit to verify that the hybridized ANN with GA performs the interpretation task with very high performance.

II.3.1. Genetic algorithm model architecture

The GA seeks the optimum of a function called fitness function defined on a data space. The GA optimization is composed by the following five elements:

II.3.1.1. Parameters coding: This step usually comes after the mathematical modeling of the given problem. The coding is represented as strings of bits containing all the necessary information to describe a point of space in the population. Conveniently, the simple binary coding is used because we can easily encode all sorts of objects: real, integers, Boolean values, strings, etc.

II.3.1.2. Creating the initial population: This mechanism must be capable to generate a population of heterogeneous individuals to serve as a basis for the future generations. The choice of the initial population is important because it may make the convergence to the global optimum more or less rapid. If we have no idea on the problem solution, the population is randomly generated so it is spread throughout the search field.

II.3.1.3. Objective Function: The performance degree of each individual is calculated using an evaluation function called fitness function. This function guides the GA optimization. The higher the score of an individual registered by that function, the higher the individual is appropriate and closer to the best solution.

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Fundamentally, the aim of GA is to find the values of individuals that maximize the fitness function. If the objective of the problem is to minimize this function, we proceed as follows:

Where Max is a positive number larger than the largest expected value given by f(x).

II.3.1.4. Genetic Operators: From the initial population, the GA generates new individuals outperform their predecessors by performing the following genetic operations: selection, crossover and mutation. These operators allow diversifying the population along the generations.

The selection identifies statistically the best individuals of the population and eliminates the bad ones. The method is to select from among individuals who have the greatest adaptation degree. The selected individuals remain in the population and are again present in the subsequent iterations.

The crossover operator reconstructs the genes of individuals within the population to enrich it by crossing the selected couples (pairs) in the selection step. The crossover consists of first randomly selecting pairs of parents to be crossed, then Two new chromosomes are created each with a part of the gene pool of their parents.

The mutation operator permits to change a small part of chromosome to give something new to the individual. The role of this operation is incidental; it contributes to genetic operations with a low probability.

The general principle of a GA optimization process is given in Figure 3.

Figure 3. A Genetic Algorithm Cycle

Selection Initial Selected function Classification population population

Crossover

P 2 Population resulting from crossover Final Mutation

population C1

C2 16

Durand, N. Algorithmes génétiques et autres outils d’optimisation appliqués à la gestion de trafic aérien, Document de Travail, octobre 2004. Topics in Middle Eastern and African Economies Vol. 12, September 2010

We begin by engendering randomly a population of individuals. To pass from generation k to generation k +1, the three genetic operations are repeated for all the elements of population k. Couples P1 and P2 are selected according to their adaptations. Crossover is then applied with a probability Pc (usually around 0.6) and generates offspring pairs C1 and C2. Chromosomes characters can be changed by the mutation operator with a probability Pm (generally much lower than PC) and generates mutated individual P’. Afterwards, individuals are evaluated before their inclusion in the new population.

II.3.2. Learning procedure

The ANN structure with a number of connection points and their locations has a significant impact on the ANN performance and it generalization capacity. The density of synaptic connections in an ANN determines its ability to record information. If a network does not have enough synaptic connections, the learning algorithm may never converge and the network will not be able to approximate its function to the predefined objective. However, the over-fitting may occur during the network functioning which can reduce the network performance.

ANN require a choice of parameters which determine the quality of the approximation function and, consequently, the quality of the optimization algorithm convergence to the optimum desired level. Thus, it is difficult to select the initiation parameters and the right time to stop the network learning. For this reason, we plan to combine ANN with GA.

As for learning, GA must be able to adjust all ANN parameters to optimize the cost function which evaluates the ANN response to different examples of the training set presented at every iteration.

The first step is to randomly generate an initial population. Each element in the population represents a possible configuration of the ANN. Individuals having their own parameters describing different aspects of their configurations such as number of hidden neurons, learning rate, stopping criterion, etc. The GA chromosomes perform symbolic coding of the individuals parameters. Then, we proceeded to calculate the fitness function of each individual having been trained by the back-propagation algorithm. This function must take into account the mean square error committed by the network in the classification of the training set examples.

After the appraisal step, the best individuals are selected for the next generation to ensure stable growth of the population. These elements are subjected to genetic operators (crossover and mutation) to diversify the population.

Once a new generation is obtained, the objective function is recalculated for each individual. The individual with the best fitness is chosen, and therefore the most suitable configuration is given.

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This will finally help to identify the optimal parameters for which the ANN has the best convergence.

Figure 4 summarizes the ANN and GA hybridization:

1 - Coding: the ANN components are first encoded into binary digits to facilitate the application of genetic operations.

2 - Initiation: each component representing an assumed ANN’s architecture is evaluated by the mean square error value. ANN structures with the lowest errors are the best fitted. They are selected to be submitted to different genetic transformations.

3 - Genetic operators: the objective of this step is to enrich and diversify the selected population in the previous step. The crossover operator randomly takes two parents in order to generate two children sharing the genetic characteristics of their parents. Mutation is the occasional random modification (with low probability) of a gene value for each child.

Figure 4. Architecture of the Hybridized neuro-genetic model

1 1

0 1 1 1

1 0 1 0

1 0 1 0 1 1 0 1

0 1 1 1 0 1

Neural network parameters encoding

2 Parents 2 Child

0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 1 1 1

1 0 1 1 1 0 1 1 0 1 1 1 0 0 1 1 0 1 1 1 0 0 1 1 18

Neural network parameters crossover Topics in Middle Eastern and African Economies Vol. 12, September 2010

Section III. Empirical study

In this section, we present the empirical study of testing the ability of the hybridized neural network with the genetic algorithm.

It is generally common in practice in economics, as in other sciences, to use a research model in order to properly formulate the hypotheses and to validate the used tools. Several models are frequently used to explain and test the ANN to predict future values of economic variables. The majority of these studies have asserted that the ANN is a very effective tool for forecasting in the case of nonlinear time series.

III.1. Used data

Before reaching the estimation of the chosen model form, it should be noted that a number of important determinants of the Saudi currency real exchange rate could not be considered here because of the difficulty of obtaining reliable aggregated data including the daily interest rate. Therefore, we will restrict our search to the study of a single currency, namely the Kuwaiti dinar12. All the simulations are conducted under MATLAB.

We consider the market over a period from 01/01/2007 to 18/11/2009. The data frequency is daily and the historical data collected include:

- The exchange rate spot USD/KWD.

- U.S Dollar and the Kuwaiti Dinar interest rates13

- Oil prices14.

12 The data on foreign exchange markets Dollar/Kuwaiti Dinar are obtained from http://www.fxtop.com/ 13 Fed and the Central Bank of Kuwait websites: http://www.federalreserve.gov/ and http://www.cbk.gov.kw/

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- Exchange rate values in t-1.

III.2. Model structure

Before the estimation of our ANN, it would be interesting to study the characteristics of the econometric financial series.

III.2.1. Unit Root Tests and series Stationarity

The first step is to study the series properties in terms of stationarity. Therefore, it is necessary to determine if the series’ Esperance and variance can be modified over time

III.2.1.1. Determining the number of delays

To study the stationarity of the considered series, we proceed in a first to a delays analysis. Its conditions are as follows: maximum number of delays = 6 and minimum number of delays = 1.

MATLAB simulation has provided the following results:

USD/KWA Dif-interest Oil price USD/KWD(t-1)

Delay Value Prob Value Prob Value Prob Value Prob

Lag=6 -0.8253 1 3.5537 0.05941 0.5799 0.4463 -0.2056 1 5.329e- Lag=5 7.3640 0.006654 -0.5933 1 0.7616 0.3828 16.3275 005 Lag=4 155.5906 0 -0.5104 1 1.5284 0.2164 145.9823 0 9.954e- Lag=3 -0.9467 1 28.3830 3.5628 0.05909 -0.5238 1 008 1.776e- Lag =2 5.3241 0.02103 1.1653 0.2804 14.7356 0.0001237 63.3571 015 If the calculated probability is above 5%, the delay is significant. We find that the third delay is significant for the USD/KWD exchange rate series, oil prices and also USD/KWD(t-1) (the critical probability of this test is upper than 5%). The series differential delay of the interest rate is significant starting from the value of 2.

III.2.1.2. Dickey-Fuller test

The Dickey-Fuller test consists to estimate the three models in the series:

Let Χt a time series, then Dickey-Fuller models are as follows:

: autoregressive model of order 1 [1]

: autoregressive model with constant [2]

: autoregressive model with trend [3]

14 Organization of the Petroleum Exporting Countries (OPEC): http://www.opec.org/home/

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MATLAB calculations give the following estimations:

t-critical at 1% = -3458

t-critical at 5% = -2871

t-critical at 10% = -2594

USD/KWD Dif-interest Oil price USD/KWD(t-1) t- AR15(1) t- AR(1) t- AR(1) t- AR(1) calculated estimated calculated estimated calculated estimated calculated estimated - Lag =1 -2.648441 0.985883 -1.780584 0.994127 17.80701 0.468150 -2.646463 0.985898 9 - Lag =2 -2.621949 0.985963 -1.498321 0.995114 14.67303 0.500586 -2.618663 0.985984 7 - Lag =3 -1.812654 0.990956 13.08400 0.510561 -1.808883 0.990977 3

In order to analyze this table, the test is based on the following null hypothesis:

Η0: If the t-calculated exceeds the t-critical given directly by MATLAB, the series is not stationary. Comparison of the t-calculated with t-critical indicates that the three series are not stationary. Therefore, each one has a unit root.

III.2.2. Johansen co-integration test

The co-integration tests are applied using two combined tests: test of the trace and test of the maximum Eigen value on the chosen historical period. The results are presented below:

III.2.2.1. Trace test:

Number of the co-integration Trace Crit Crit Crit relationships Statistic 90% 95% 99% r <= 0 188.110 44.493 47.855 54.682 r <= 1 16.644 27.067 29.796 35.463 r <= 2 9.307 13.429 15.494 19.935 r <= 3 4.106 2.705 3.841 6.635 III.2.2.2. Maximum Eigen Value test:

Number of the co-integration Eigen Crit Crit Crit 99%

15 AR (1): estimate coefficient of the auto-regressive model of order 1, i.e. the estimated value of parameter X (t-1) where X is the set of the studied variables.

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relationships Statistic 90% 95% r <= 0 171.467 25.124 27.586 32.717 r <= 1 7.337 18.893 21.131 25.865 r <= 2 5.201 12.297 14.264 18.520 r <= 3 4.106 2.705 3.841 6.635

Whatever model is chosen and the test performed (trace test or maximum Eigen value test), the hypothesis that there is no co-integrating relationship is widely rejected (the statistical value of the trace and the Eigen value exceed critical values at 90%, 95% and 99% confidence interval). The hypothesis that there is only one co-integrating relationship is accepted (the statistical values of the trace and Eigen value are below the critical values regardless of the probability threshold).

The Johansen test confirms the existence of a long-term relationship that is to say that the three series are co-integrated, and also that there is only one co-integrating vector between these variables.

III.2.3. Estimation of the Equilibrium Relationship in Long Term

The long-term relationship between the spot exchange rate, the interest rates differential, oil prices and the exchange rate in t-1 is presented by the following equation16:

Where:

: is the exchange rate in period t

: is the interest rate differential in t-1

: Oil price variation in t-1

: is the exchange rate in t-1

: is an estimation residue

The estimation of these equation parameters by the ordinary least squares method gave the following results:

Ordinary Least‐squares Estimates R‐squared = 0.9685 Rbar‐squared = 0.9684 Sigma^2 = 0.0000 Durbin‐Watson = 2.1063 Nobs, Nvars = 1053, 3 *************************************************************** Variable Coefficient t‐statistic t‐probabilité 16 Since there is only one co-integrating vector‐0.000051 among the ‐studied1.255480 variables, we can now 0.209584 apply the linear regression by the ordinary least squares method. This0.000023 regression is a0.626827 co-integration regression 0.530909used to determine if there is a continued long-term trend between variables. 0.999940 5089.970065 0.000000 22

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These empirical results are interpreted as follows:

: Widely above 5%, then this parameter is not significant.

: Greater than 5%, then this parameter is not significant

: The nullity probability of the exchange rate coefficient in t-1 is equal to 0, then this parameter is significant.

We note that the ordinary least squares method gives results that do not fit the market exchange reality where the macro-economic variables, such as interest rate, play a relatively major role in determining prices and values.

The estimation residue graphs are presented below:

OLS Actual vs. Predicted 0.3 Actual 0.28 Predicted

0.26

0.24

0.22 0 200 400 600 800 1000 1200

Residuals 0.04

0.02

-0.02

-0.04 0 200 400 600 800 1000 1200

The figure below shows the graph of the predictions linear regression based on the true (real) values:

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Regression parameters Values MSE (Mean square error) 0.000002944250197 M (linear regression slope) 0.9852 B (linear regression constant) 0.0041 R (coefficient of determination) 0.9842

RMSE (Root Mean Squared Error) 0.0017

It held that the estimated values converge to the actual values of exchange rates. This is not due to the power of the linear prediction model (the estimation table confirms this) but rather the lagged values of exchange rates that could predict future values (deterministic process).

Therefore we can conclude that this inconsistency come either from this linear model which does not detect the existence of nonlinear relationships between variables, or either from the nature of macro-economic aggregates used in our model whose volatility can be less intense than the one of exchange rate daily frequency, or either because the market undergoes a fixed exchange rate regime.

To remedy this problem and its origin, we use a non linear model known for his prediction power based on ANN.

If the ANN cannot improve the predictive power of a model this is mean that the exchange rate is more volatile than the economic variables, and that the only way to predict it is its past values. Another possibility is that the exchange rate is not determined by the supply and demand interactions. This last suggestion is more logical of our empirical study given the nature of the Kuwaiti market exchange where the market price is established based on a basket of currencies (fixed exchange rate regime).

III.2.4. Prediction of the exchange rate by a non-optimized ANN

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The configuration of our ANN is characterized by the following steps17:

- We divide the database into two sets, the training series (60% of data), and testing and validating data (40% of data).

- Values normalization is included in the interval [-1,1].

- Creation of a ANN with one hidden layer with 2 neurons. We chose a sigmoid hyperbolic tangent transfer function (tansig).

- The learning function is Gradient Descent with momentum term (traingdm).

- The number of iterations is 5000 epochs, the learning rate is 0.3, and the convergence rate is 0.6.

The result is shown below:

ANN Ordinary Mean Square OMS MSE (Mean square error) 0.000004637443833 0.000002944250197 M (linear regression slope) 0.8250 0.9852 B (linear regression constant) 0.0506 0.0041 R (coefficient of determination) 0.9775 0.9842 RMSE (Root Mean Squared Error) 0.0031 0.0017

We note that the mean square error after the ANN execution is slightly higher than that of the linear model estimated by ordinary least squares. We can conclude that the ANN could improve the future exchange rate estimation.

To do this, we use an optimized ANN by a GA. That is to say that the network parameters values will be determined by a GA which will choose the best parameters that permit optimizing the

17 We determined the neural network parameters after a large number of tests since there is no general rule that can easily define these parameters.

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ANN performance in order to give better prediction results.

III.2.5. Predicting exchange rates by an optimized ANN

III.2.5.1. Genetic Algorithm parameters

We use GA to define a configuration to maximize the ANN performance by making it able to avoid local minima of the mean square error. This hybridized ANN will seek to converge towards the global minimum of mean squared error. The neuro-genetic model configuration is characterized by the following criteria:

- The number of hidden layers varied between 1 and 30.

- The learning rate is included in the interval [0.1, 0.5].

- The range of the constant term momentum is [0.1, 1].

The GA mission is to find the values of these three variables to optimize the ANN predictive power.

With these parameters, we compiled 10 ANN’s possible configurations representing at the same time the initial population to be manipulated by the GA.

The GA parameters were chosen after running several simulations whose results were more or less satisfactory. Each combination of different types of parameters gave a specified performance level for each configuration:

- The maximum number of generation is 10.

- The selection type is a ranking function based on a normalized geometric distribution.

- An arithmetic crossover is selected as a reproduction way, this crossover makes a simple linear combination between the parents.

- Given the type of variables in our databases, we selected a non-uniform mutation function that changes a child parameter on the basis of a non-uniform Gaussian distribution.

Furthermore, a 1.e-10 value was assigned to the tolerance function for the cumulative change in the fitness function value. Once this value is reached, the algorithm stops. Similarly, we assigned a value of 1.e-10 to the tolerance threshold for possible cases which exceed the considered nonlinear constraints boundaries.

III.2.5.2. Genetic Algorithm optimization results

The following graph (representing the optimization process) allows displaying several steps that provide information about the GA during its execution. This information is very helpful to change the improving options of the algorithm performance. For most optimization problems, the fitness function graph remains the key factor in judging the GA performance:

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9.998

9.996

9.994

Fittness 9.992

9.99

9.988

9.986 1 2 3 4 5 6 7 8 9 10 Generation By running the GA, the interface displays another graph for the best values of the fitness function at each generation. The algorithm stops at the value 9.998 as shown in the figure above. Typically, the best value of fitness function improves rapidly during the early generations when individuals are so far from the optimum.

Comparison between actual targets and predictions 0.305

0.3

0.295

0.29

0.285

0.28

0.275

0.27

0.265 0 50 100 150 200 250 300 350 400 450

Thus, we present the ANN parameters values that have optimized its fitness function:

Parameters Values Number of hidden layers 2 Learning rate 0.100897878979923 Constant term momentum 0.85216830084332 Fitness function 9.997921774706915 With these parameters, the ANN has achieved the following results:

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Optimized ANN Non-optimized ANN OMS MSE (Mean square error) 0.000004671608247 0.000004637443833 0.000002944250197 M (linear regression slope) 0.8076 0.8250 0.9852 B (linear regression constant) 0.0553 0.0506 0.0041 R (coefficient of determination) 0.9789 0.9775 0.9842 RMSE (Root Mean Squared Error) 0.0021 0.0031 0.0017

The convergence of the optimized ANN is approaching the non-optimized ANN one with a very slight improvement, that is to say a convergence with a mean square error is moderately lower than that of the non optimized ANN.

Our result confirms the hypothesis that ANN are more robust to solve nonlinear problems where classical mathematical modeling process is unable to control the highly volatile financial series.

Conclusion

This study, based on a comparison between the linear models and ANN, was aiming to create a better understanding of the Gulf exchange market. The hypothesis related to the proposed research models has been empirically tested on the USD/KWD foreign exchange market over an historical period from January 2007 to November 2009 with daily frequency data.

The results show that the linear model is more appropriate for this market. This confirms the exchange markets nature of these countries where the exchange rate regime is fixed and currencies indexed to the U.S. dollar or a basket of currencies as is the case of Kuwait.

In order to submit the Gulf countries future common currency to the demand and supply rules so that there will be some volatility in the movements, it is necessary to built a floating exchange rate system. Only within this context we can finally use real time forecasting models.

A fixed exchange rate system does not always optimally work for the GCC countries. The high oil prices and strong domestic demand in the region coupled with the low level dollar values have rocked inflation and led to negative real yields. Insofar as the anchor currency requires similar

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monetary policies to remain viable, the difference between business cycles in the GCC countries and the United States will remain a problem in the future.

In the short term, the Gulf region is not ready for a flexible exchange rate system for several reasons. The transition to such system requires the adoption of a monetary or inflation targeting. But such policies require technical and institutional resources - not existing yet - to conduct monetary transactions. Furthermore, a central bank can be credible only if it is independent and whether monetary policy measures taken are transparent. The central bank credibility may also be compromised by the persisting limited data access and that, despite other factors, it remains an obstacle to a transparent monetary policy.

Therefore we believe that a fixed exchange rate system will remain the best option for the monetary union launching. However, adopting a more flexible regime seems reasonable since the region is likely going to diversify its trade flows and capital. The anchor to a currencies basket heavily weighted with euro seems appropriate and could be the next step towards a more flexible exchange rate regime in the long term.

Moreover, the proposed common currency project of the Gulf countries in particular is indeed far from over. Two members of the Gulf Cooperation Council, Oman and the United Arab Emirates have already withdrawn in 2007 and 2009 respectively.

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