Jory Namrata Devi 2012 (1)

8/20/2019 Jory Namrata Devi 2012 (1)

1/50

A Comparison of Neural Network, ARIMA andGARCH Forecasting of Exchange Rate Data

Namrata Devi Jory

UNIVERSITY OF MAURITIUS

Dissertation submitted to the Department of Mathematics, Faculty of Science,

University of Mauritius, as a partial fulfilment of the requirement for the degree

of BSc (Hons) Mathematics with Computer Science.

April 2012


2/50

Contents

List of Figures iii

List of Tables iv

Acknowledgements v

Abstract vi

1 Introduction 1

1.1 Forecasting Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Aims and Organization of the Project . . . . . . . . . . . . . . . . . . . . . . . 3

2 Artificial Neural Networks 4

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Feedforward Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Backpropagation Learning Algorithm . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Training set and Testing set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Linear and Non-linear Time Series 10

3.1 Stationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.1 Autoregressive process . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.2 Moving Average Process . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.3 Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 ARIMA ( p, d, q ) Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.1 Autocorrelation and Partial Autocorrelation Functions . . . . . . . . 13

i


3/50

CONTENTS

3.3 GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.1 ARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.2 GARCH model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Analysis of Exchange Rate Data 16

4.1 Analysis of Period I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1.1 Analysis of Period II data . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Forecasting Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Fitting Neural Network Model to the Exchange Rates Data . . . . . . . . . . 23

4.4 Fitting ARIMA Model to the Foreign Exchange Rates Data . . . . . . . . . . 244.5 Fitting the GARCH model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.6 Empirical findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.6.1 Forecasts performance of Period I . . . . . . . . . . . . . . . . . . . . 26

4.6.2 Results and Discussions for Period I . . . . . . . . . . . . . . . . . . . 32

4.6.3 Forecasts performance of Period II . . . . . . . . . . . . . . . . . . . . 34

4.6.4 Results and Discussions for Period II . . . . . . . . . . . . . . . . . . 39

5 Conclusion 41

Bibliography 42

ii


4/50

List of Figures

2.1 Three-Layer feedforward Network . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The neuron weight adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1 Daily MUR/USD data and its corresponding first differences of the logs . . 17

4.2 Daily MUR/EU data and its corresponding first differences of the logs . . . 17

4.3 Daily MUR/GBP data and its corresponding first differences of the logs . . 18

4.4 Histogram and Kernel Density estimate of daily log return of MUR/USD . 19

4.5 Histogram and Kernel Density estimate of daily log return of MUR/GBP . 19

4.6 Histogram and Kernel Density estimate of daily log return of MUR/EU . . 19

4.7 Scatter plot of yt against yt−1 for the MUR/USD log return data. . . . . . . . 20

4.8 Scatter plot of yt against yt−1 for the MUR/EU log return data. . . . . . . . . 20

4.9 Scatter plot of yt against yt−1 for the MUR/GBP log return data. . . . . . . . 20

4.10 Daily MUR/USD Jan 03 - Dec 11 . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.11 Daily MUR/EU Jan 03 - Dec 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.12 Daily MUR/GBP Jan 02 - Dec 11 . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.13 In-sample and out-of-sample forecast for MUR/USD . . . . . . . . . . . . . 33

4.14 In-sample and out-of-sample forecast for MUR/EU . . . . . . . . . . . . . . 33

4.15 In-sample and out-of-sample forecast for MUR/GBP . . . . . . . . . . . . . 34

iii


5/50

List of Tables

3.1 Behaviour of ACF and PACF . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1 Summary statistics for the daily exchange rates: log first difference . . . . . 18

4.2 Summary statistics of log first difference daily exchange rate . . . . . . . . . 21

4.3 In-sample performance of MUR/USD for data Jan 2003-Dec 2008 . . . . . . 26

4.4 In-sample performance of MUR/EU for data Jan 2003-Dec 2008 . . . . . . . 27

4.5 In-sample performance of MUR/GBP for data Jan 2002-Dec 2008 . . . . . . 28

4.6 Out-of-sample performance of MUR/USD for data Jan 2003-Dec 2008 . . . . 29

4.7 Out-of-sample performance of MUR/EU for data Jan 2003-Dec 2008 . . . . 30

4.8 Out-of-sample performance of MUR/GBP for data Jan 2002-Dec 2008 . . . . 31

4.9 First 10 forecasts values of MUR/GBP for ANN, ARIMA and GARCH models 31

4.10 In-sample performance MUR/USD data . . . . . . . . . . . . . . . . . . . . . 34

4.11 In-sample performance MUR/EU data . . . . . . . . . . . . . . . . . . . . . . 35

4.12 In-sample performance MUR/GBP data . . . . . . . . . . . . . . . . . . . . . 36

4.13 Out-of-sample performance MUR/USD data . . . . . . . . . . . . . . . . . . 37

4.14 Out-of-sample performance of MUR/EU data . . . . . . . . . . . . . . . . . 38

4.15 Out-of-sample performance of MUR/GBP data . . . . . . . . . . . . . . . . . 39

4.16 In-sample and Out-of-sample forecasts of MUR/USD . . . . . . . . . . . . . 40

4.17 In-sample and Out-of-sample forecasts of MUR/EU . . . . . . . . . . . . . . 40

4.18 In-sample and Out-of-sample forecasts of MUR/GBP . . . . . . . . . . . . . 40

iv


6/50

Acknowledgements

I am deeply indebted to my supervisor, Professor Muddun Bhuruth, for his excep-

tional guidance and encouragement which helped me to carry out this research and write

this dissertation. It was a wonderful experience to work with such a mentor and I am very

grateful for all the criticisms to make my work better each time. I am also thankful to my

co-supervisor Associate Professor Ravindra Boojhawon who has helped and motivated me

throughout the project.

I am very grateful to my parents for their love and for always being by my side in goodand bad times and special thanks to my dearest sister for her understanding and help. I

would like to thank my friends khush, Prish and Bho for always being there for me.

I also extend my gratitude to all those who helped me directly or indirectly to make this

work a success.

Finally, I thank GOD for his blessings from the bottom of my heart.

v


7/50

Abstract

We consider linear and non linear models for forecasting exchange rates of the Mauri-

tian Rupee against US Dollar, Euro and the British Pound. The linear models considered

are the ARIMA processes and the non linear models considered are the Artificial Neural

Networks and the GARCH model. Since no guidelines were available to choose the pa-

rameters of the Neural Network, they were chosen through extensive experimentation.

Two periods of analysis were carried out first from January 2002 to December 2008 and

for January 2002 to December 2011 and the in-sample and out-of-sample forecasts were

produced. The reason for this choice is that we wanted to test the ability of our forecastingprocedures during the financial crisis.

Using three forecast evaluation criteria RMSE, MAE and MAPE, we found that the ARMA-

GARCH model performs slightly better than the ARIMA model. The two mentioned mod-

els give better performance when compared with the ANN model for the in-sample fore-

casts of the first period data. However the ANN was found to outperform the ARIMA and

ARMA-GARCH models in the out-sample forecasting for both periods data.

vi


8/50

Chapter 1

Introduction

Foreign exchange rates are among the most important determinants of the economic health

of a country. They describe the price of one currency in terms of another one and they

play a vital role in the trading relationship between countries, which in turn affect the

world economy. For this reason, they are the most watched, analysed and governmen-

tally manipulated economic measures. Understanding the evolution of exchange rates is

important for many essential issues in international economics and finance, such as inter-

national trade, capital flows, international portfolio management, currency option pricing,

and foreign exchange risk management. The foreign exchange market has experienced

many unexpected growth and downfall over the last few decades. The dynamics of the

exchange market depend entirely on the exchange rates. Thus the appropriate prediction

of exchange rates is a very crucial factor for the success of many businesses on the global

market. The exchange market is in itself well known for being extremely unpredictable

and volatile. A volatile exchange market makes international trade and investment deci-

sions more difficult because volatility increases exchange rate risk which may result into a

potential loss due to a change in the rates.Forecasting any time series accurately is very difficult and to add up to it, exchange

rates prediction is one of the most challenging applications of modern time series forecast-

ing. The exchange rates are generally noisy, non-stationary and deterministically chaotic

(Yaser & Atiya 1996) which suggest that there is no past behaviour information which can

produce a relation between the past and the future behaviour. However, though all the

constraints mentioned, numerous techniques have been devised by researchers to forecast

1


9/50

1.1 Forecasting Models

the exchange rates. And the search for a reliable model to predict exchange rates is still

ongoing.

1.1 Forecasting Models

Forecasting foreign exchange rates is a very important issue in the economic world. During

the past years, different models were developed using the linear and nonlinear framework.

A linear model is one which can predict the future values of exchange rates by identifying

and magnifying the existing linear structure in the data. And the most commonly used

linear models for exchange rates forecasting are the Box and Jenkins’ Autoregressive Inte-

grated Moving Average (ARIMA) models. One of the attractive features of Box and Jenkins

approach to forecasting is that there is a very rich class of possible models and it is usually

possible to find a process which provides an adequate description of the data. The ARIMA

model is also a very powerful instrument for construction of accurate forecasts with small

distance of forecasting. Since it is extremely popular, this model is used as a benchmark

to evaluate new modelling approaches. ARIMA models are very effective techniques for

forecasting when the dynamics of the time series is linear and stationary (Cao & Tay 2001).

However, nonlinearities in exchange rates are supported by many evidences. Thus approx-

imation by ARIMA models may not be adequate since it cannot capture nonlinear patterns

in the exchange rates data.

Ever since the inadequacy of linear models (Racine 2001) was observed, there has

been considerable development in modeling nonlinearities. Thus nonlinear models such

as the Autoregressive Conditional Heteroscedasticity (ARCH) model and the Generalized

Autoregressive Conditional Heteroscedasticity (GARCH) model were developed. They

have been found to be useful in capturing certain nonlinear features of financial time series

such as clusters of outliers. (Bollerslev & Ghysels 1996) had shown successful applicationof GARCH model in describing the dynamic behavior of the exchange rates. However

there exists no distinguished theory that can be applied to all such nonlinear models as

they require specific assumptions concerning the precise form of nonlinearity. Moreover

there exist too many possible nonlinear patterns in a particular data set, thus a specific

nonlinear model may not be sufficient in capturing all of them.

In response to all the constraints of the linear and nonlinear models, artificial neural

2


10/50

1.2 Aims and Organization of the Project

networks (ANNs) have been used to forecast the exchange rates. ANNs resemble and op-

erate in the same way as our biological neural system. Due to their unique non-parametric,

non-assumable, noise-tolerant and adaptive properties (Haoffi & Han 2007), ANNs can

deal better with non-stationary and volatile data. ANNs have flexible nonlinear function

mapping which can approximate any continuous measurable function with arbitrary de-

sired accuracy. They are found to have an upper hand on various traditional linear and

nonlinear models in the exchange rates forecasting. Many researchers such as (Wang

& Leu 1996), (Tang & Fishwich 1993), (T. Hill & Remus 1996) have shown that ANNs

perform better than ARIMA models (linear models), especially for more irregular series

and for multiple-period-ahead forecasting. (Gencay 1999) and (R. K. Bissoondeeal &

Mootanah 2008) find that forecasts generated by neural network are superior to those of

ARIMA and GARCH models. (Panda & Narasimhan 2007) have shown that the perfor-

mance of ANNs is better than the linear autoregressive model and random walk model

for the one-step-ahead prediction, thus suggesting that there always exists a possibility of

forecasting exchange rate. However other studies have reported inconsistent results for

example (W. R. Foster & Ungar 1992) have shown the ANNs are inferior to linear regres-

sion and (Meade 2002) find no evidence that the foreign exchange rate behaviour is better

represented by the ANNs than the linear model.

1.2 Aims and Organization of the Project

This project studies neural network methods for modeling the Mauritian Rupee (MUR)

against the three most important currencies which are the US Dollar, the Euro and the

British Pound. We focus on various neural network models and assess their performance

for in-sample and out-of-sample forecasts and the results are compared against forecasts

produced by ARIMA and GARCH models.The organization of the project is as follows: In Chapter 2, we discuss how the Artificial

Neural Network works and learning algorithm used. Chapter 3 describes linear and non

linear time series models. In Chapter 4, we analyse our data and compare the forecasting

accuracies of the different models.

3


11/50

Chapter 2

Artificial Neural Networks

2.1 Introduction

Artificial neural networks, which imitate the human brain’s ability to classify patterns and

make prediction based on past experience, have found applications in different areas such

as; financial forecasting, medical diagnostics, flight control and product inspection. The

artificial neural networks have been widely used in applied forecasting due to their ability

to model complex relationships between input and output variables and also because of

the presence of nonlinearities in many time series.

2.2 Feedforward Network

The feedforward neural network is the most commonly used network in applied work

due to its capability of resolving a large number of problems. It consists of a considerable

number of simple processing units known as neurons which are organised in layers. A

feedforward neural network begins with an input layer which is connected to a hidden

layer. The hidden layer can then be connected to another hidden layer or directly to the

output layer. In this architecture data enters at the input layer and passes through the net-

work layer by layer until it arrives at the output layer. The hidden layer is known to be

a very important layer in the neural network since it is responsible to approximate a con-

tinuous function and achieve the desired accuracy. And since a single hidden layer is less

4


12/50


13/50

2.2 Feedforward Network

The hidden Layer Vector is Z =(z1, z2, · · · , zk)T

Each hidden neuron has a bias α which is being added to the received weighted sum of all

the inputs to form a net input. The bias may be viewed as simply being added to shift the

function to the left by an amount α. It is much like a weight, except that it has a constant

input of 1.

net input =n

i=1

wi,jxi (2.2)

The net input is the argument of the transfer function f which is applied to construct the

output of a specific neuron.

z j = f

α +

ni=1

wi,jxi

where j = 1, 2, · · · , k (2.3)

In the output layer, the output neuron receives the weighted sum of the processed signals

obtained from the hidden layer neurons. Another function ϕ is applied to produce the

final output.

y = ϕ

β + k

j=1

µ jz j

(2.4)

Where ϕ is the transfer function, β is the bias unit, µ j is the weight from the hidden neuron

j to the output unit.

Replacing z j in the equation 2.4, we get

y = ϕ

β + k

j=1

µ j f

α +

ni=1

wi,jxi

(2.5)

In our feedforward network, a hyperbolic tangent sigmoid transfer function is used in

the hidden layer and a linear transfer function is used in the output layer. This is so be-

6


14/50

2.3 Backpropagation Learning Algorithm

cause these two transfer function had proved to be successful earlier.

The hyperbolic tangent sigmoid is given by f (x) = tanh(x) and the linear function transfer

function is f (x) = x. A node’s transfer function serves the purpose of controlling the out-

put signal strength for the node. These functions set the output signal strength between 1.0

and −1.0. It was also noticed that the hyperbolic tangent sigmoid function can accelerate

learning for some models and also have an impact on predictive accuracy. The learning

rule commonly used in this type of network is the backpropagation algorithm.

2.3 Backpropagation Learning Algorithm

The process of learning is implemented by modifying the weights iteratively until the de-

sired response is achieved at the output node. Backpropagation is known to be the most

popular supervised learning algorithm. A supervised learning algorithm is one which ac-

cepts input values, computes the output values and compares it with the desired output

values, then adjusts the weights to minimize the deviance. This process is carried out until

the network cannot further reduce the error.

Figure 2.2: The neuron weight adjustment

Backpropagation learning updates the network weights and biases in the direction in

which the performance function decreases the most. The gradient is computed and the

weights are updated after each input in the increment mode. Backpropagagtion starts at

7


15/50

2.4 Training set and Testing set

the output layer with the following equations:

wij = w

ij + l.e j .xi (2.6)

Where ith input of the j th neuron, wij is the weight, w

ij is the previous weight value, l is

the learning rate, e j is the error term and xi is the ith input.

The backpropagation algorithm looks for the global error from the error function in the

weight space using the method of gradient descent. In gradient descent, weights are

changed in proportion to the negative of an error derivative with respect to each weight. In

this specific algorithm, the network follows the curvature of the error surface with weight

update moving in the direction of the steepest descent. However there is a high probabil-

ity that the network does not reach the global minimum since it may be stuck at the local

minima which does not represent optimal solution.

Momentum is a technique that can help the network out of the local minima. It is an

extension of the backpropagation algorithm which can be helpful in speeding the con-

vergence and avoiding local minima. With momentum, if the weights are moving in a

particular direction, they tend to continue in the same direction. It is also noticed that

momentum smoothes the weight changes. The momentum factor determines the effect of

past changes on current changes of weights and also increases the speed of the learning

rate. The momentum factor which is constantly used has a value very close to 1. The ratio

which influences the speed and quality of learning is called the learning rate. The learning

rate plays a very important role in the learning process of a network, as it controls the size

of the changes in weight at each iteration. Thus the right choice of the learning rate is ex-

tremely important since a too small or a too big change in the size of the weight will affect

the result of the network. A learning rate between 0.05 and 0.5 was found to provide good

results in many practical cases.


In neural network forecasting, we divide our data into a training set and a test set. The

training set consists of the input data and the target data which need to be presented to the

8


16/50


network. However some transfer functions need the input and target data to be scaled so

as they fall within a specified range. Thus in order to meet this requirement we pre-process

our data by normalising the inputs and targets so that they fall in the interval of [−1, 1].

When these input data are presented to the network, the latter makes a guess of the correct

answer and compares it with the target data. The network goes through the data again

and again depending on the number of epochs used, adjust the weight value so as to reach

a value close to the target value. The training set is used to build up the model whereas

the test set which is independent of the training set is used to measure the performance of

the model. More precisely it is used to evaluate the out-of-sample performance. After the

forecasts are obtained from the network we need to convert the data back to the original

scale by the process called post-processing. Moreover, we find from previous studies that

there is no precise rule on the optimum size of the two data set.

9


17/50

Chapter 3

Linear and Non-linear Time Series

In this chapter we consider the widely used time series models for forecasting. We first

briefly describe the Autoregressive Integrated Moving Average (ARIMA) model which

is a linear model and we then consider the Generalized Autoregressive Conditional Het-

eroscedasticity (GARCH) model as the non-linear model.

3.1 Stationary Processes

A stochastic process X tZ is a family of random variables defined on a probability space.

The joint cumulative distribution function of X t is defined as

F t1,t2,...,tn(x1, x2, . . . , xn) = P (X t1 ≤ x1, X t2 ≤ x2, . . . , X tn ≤ xn) (3.1)

and the process X t is said to be strictly stationary if

F t1+s,t2+s,...,tn+s(x1, x2, . . . , xn) = F t1,t2,...,tn(x1, x2, . . . , xn) (3.2)

for all n-tuples (x1, x2, . . . , xn), (t1, t2, . . . , tn) and for any s The mean function of X t is

given by

µt = E (xt) =

xdF t(x) (3.3)

10


18/50

3.1 Stationary Processes

and the autocovariance function is given by

γ (t, ) = E [(xt − µt)(xt− − µt−)] (3.4)

The process is said to be weakly stationary if µt = µ for all t and γ (t, ) = γ . In this case we

have γ − = γ .

A white noise process εt is a process such that µ = 0 and γ = σ2 and γ = 0 for = 0. We

denote such a process by εt ∼ WN (0, σ2).

3.1.1 Autoregressive process

The autoregressive process of order p is denoted by AR( p) and it is defined as follows:

X t =

pr=1

φrX t−r + εt (3.5)

where εt ∼ WN (0, σ2) or φ(B)X t = εt, where B is the backshift operator, such that

φ(B) = 1 − φ1B − φ2B

2

− · · · · · · − φ pB

p

(3.6)

The process is invertible since p

r=1 |φr| < ∞ and for the process to be stationary the roots

of φ(B) = 0 must lie outside the unit circle.

3.1.2 Moving Average Process

The moving average process of order q is denoted by MA(q ) and is defined by

X t =q

s=0

θsεt−s (3.7)

or the process can also be written as X t = θ(B)εt where,

θ(B) = 1 − θ1B − θ2B2 − · · · · · · − θqB

q (3.8)

11


19/50

3.2 ARIMA ( p, d, q ) Processes

Because MA processes consist of a finite sum of stationary white noise terms, they are sta-

tionary hence they have mean zero. The process is invertible when the roots of θ(B) all

exceed unity in absolute value.

3.1.3 Random Walk

In a random walk model at each point of time, the series moves randomly away from its

current position. The model can then be written as

X t = X

t−1 + ε

t (3.9)

We see that the random walk model has the same form as an AR(1) process, but, since

φ = 1, it is not stationary.

Repeatedly substituting for past values gives

X t = X 0 +t−1 j=0

εt− j (3.10)

We find that the first difference of the random walk is stationary, as it is just white noise:

X t −X t−1 = εt (3.11)


An ARIMA model is a combination of the Autoregressive (AR), differencing and Moving

Average (MA) processes.

If the original process X

t is not stationary, we can look at the first order difference process

Y t = X t −X t−1 (3.12)

or the second order differences and so on.

A general ARIMA model of order ( p, d, q ) can be represented as follows

12


20/50


φ(B)∇dX t = θ(B)εt (3.13)

where φ(B) is the AR operators of order p and θ(B) is the MA operators of order q .

X t is the observed value at time t, εt ∼ WN (0, σ2) and d is the number of time the data

series must be differenced to produce a stationary time series.

Fitting an ARIMA model to the raw data involves the following four steps iterative cycles:

model identification, estimation of parameters p, d, q , diagnostic checking and the fore-

casting process.

3.2.1 Autocorrelation and Partial Autocorrelation Functions

The autocorrelation function (ACF) can be used to detect non-randomness in data and also

to identify an appropriate time series model if the data are not random.

Given the data, x1, x2, . . . , xN at time t1, t2, . . . , tN , the lag k autocorrelation function is

defined as:

γ k =

N −ki=1 (xi − x̄)(xi+k − x̄)

ki=1(xi − x̄)

2(3.14)

It is assumed that the observation is equi-spaced thus the time variable t, is not used in the

formula for autocorrelation. The correlation is between two values of the same variable

at times ti and ti+k. When the autocorrelation is used to identify an ARIMA model, the

autocorrelations are plotted for many lags.

The use of the partial autocorrelation function (PACF) was introduced to time series mod-

elling, where one could easily determine the appropriate lags p in AR( p) model or the

extended ARIMA ( p, d, q ) model by just plotting the PACF

The PACF is the conditional correlation,

Correlation(xt, xt+k|xt+1, .......,xt+k−1) (3.15)

Given a time series , the partial autocorrelation of lag k, denoted, is the autocorrelation

between xt and xt+k with the linear dependence of xt+1 through to xt+k−1 removed.

13


21/50

3.3 GARCH Model

Process ACF PACF

AR( p) Dies down exponentially or sinusoidally cuts off after lag p

MA(q ) Cuts off after lag q Dies down exponentially of sinusoidally

ARMA( p, q ) Dies down exponentially or sinusoidally Dies down exponentially or sinusoidally

Table 3.1: Behaviour of ACF and PACF

3.3 GARCH Model

The GARCH model which is a generalisation of the autoregressive conditional heteroske-

daticity (ARCH) model was introduced by (Bollerslev 1986) and has been used by manyresearchers in modeling financial time series. It has been found that a wide range of fi-

nancial data exhibit time varying volatility clustering which is the property that there are

periods of high and low variance. In response to these, (Engle 1982) suggested the ARCH

model as an alternative to the usual time series process.

3.3.1 ARCH Model

The autoregressive conditional heteroskedasticity (ARCH) model is the very first model of

conditional heteroskedasticity. It is a forecasting model which forecasts the error variance

at time t on the basis of information known at time t − 1 and it is expressed as a moving

average of the past error terms. The following conditional variance defines an ARCH

model of order q :

σ2t = σ0 +

qi=1

αiε2t−i (3.16)

α0 ≥ 0, αi ≥ 0 where αi must be estimated from the data.

The error term will have the form:

εt = σtzt (3.17)

where zt is a sequence of identically and independent distributed random variables with

zero mean and unit variance.

14


22/50

3.3 GARCH Model

3.3.2 GARCH model

The GARCH model is an extended version of the ARCH model. In this GARCH( p, q )model the variance is a linear function of its own lags and has the form

σ2t = α0 + α1ε2t−1 + · · · + αqε

2t−q + β 1σ

2t−1 + · · · + β pσ

2t− p (3.18)

= α0 +

qi=1

αiε2t−i +

pi=1

β iσ2t−i (3.19)

The rate of decay of the ARCH model is considered to be to be too fast when compared

with the usual financial series, unless the value of q in 3.16, is large. Thus the GARCH

model is prefered since it enables very complicated heteroscedasticity patterns to be mod-

eled at low orders of p and q . The most popular GARCH model in applications has been

the GARCH(1, 1) model, that is, p = q = 1 in 3.18.

15


23/50

Chapter 4

Analysis of Exchange Rate Data

We study the daily rates of MUR against the US dollar (MUR/USD), the Euro (MUR/EU)

and the British Pound (MUR/GBP). The data sets were obtained from the Bank of Mau-

ritius for the period of January 2003 to December 2011 for MUR/USD and MUR/EU and

for the period of January 2002 to December 2011 for MUR/GBP.

For our analysis, we choose two periods: the first period January 2003 to December 2008

for MUR/USD and MUR/EU and January 2002 to December 2008 for MUR/GBP which

represents the data till the financial crisis and for our second period data we take the orig-

inal data obtained.

The daily returns are calculated as the log differences of the levels. Let xt be a given ex-

change rate time series, then the exchange rate return series yt is given by

yt = ln

xt

xt−1

(4.1)

4.1 Analysis of Period I

Figure 4.1, 4.2 and 4.3 the daily exchange rates data and their corresponding return of the

MUR against USD, EU and GBP:

16


24/50


MUR/USD

Time

U S D

2003 2004 2005 2006 2007 2008 2009

2 6

2 8

3 0

3 2

Log differenced of MUR/USD

Time

r e t u r n s

2003 2004 2005 2006 2007 2008 2009

− 0

. 0 1 0

− 0

. 0 0 5

0 . 0

0 0

0 . 0

0 5

0 . 0

1 0

Figure 4.1: Daily MUR/USD data and its corresponding first differences of the logs

MUR/EURO

Time

E U R O

2003 2004 2005 2006 2007 2008 2009

3 0

3 5

4 0

4 5

Log differenced of MUR/EURO

Time

r e t u r n s

2003 2004 2005 2006 2007 2008 2009 −

0 . 0

3

− 0

. 0 2

− 0

. 0 1

0 . 0

0

0 . 0

1

0 . 0

2

0 . 0

3

Figure 4.2: Daily MUR/EU data and its corresponding first differences of the logs

17


25/50


MUR/GBP

Time

G B P

2002 2003 2004 2005 2006 2007 2008 2009

4 5

5 0

5 5

6 0

6 5

Log differenced of MUR/GBP

Time

r e t u r n s

2002 2003 2004 2005 2006 2007 2008 2009

− 0

. 0 6

− 0

. 0 4

− 0

. 0 2

0 . 0

0

0 . 0

2

0 . 0

4

Figure 4.3: Daily MUR/GBP data and its corresponding first differences of the logs

In each of the three daily log returns, we observe that there is large random fluctua-

tions. However all the three series appear to be stationary which mean thats the random

variation is constant over time. There is volatility clustering in each of the log return series

since we can see periods of high and low variation.

Mean Median Max Min S.D Skewness Kurtosis

MUR/USD 6.46 e-005 0.0000 0.0119 -0.0101 0.0018 1.0889 11.9486

MUR/EU 2.70 e-004 1.32 e-004 0.0311 -0.279 0.0061 0.0698 5.1469

MUR/GBP 3.80 e-005 0.0000 0.0462 -0.416 0.0058 -0.1038 9.8907

Table 4.1: Summary statistics for the daily exchange rates: log first difference

The standard deviations indicate that the MUR/EU return data is more volatile than

that of the MUR/USD and MUR/GBP return series. The skewness coefficient for MUR/USD

and MUR/EU are positive which indicate that the tail on the right is longer than the left

side. For MUR/GBP the bulk of values lie to the right of the mean. Kurtosis is a measure

of whether the data peaks or flattens relative to a normal distribution. For all the three

18


26/50


return series the kurtosis is larger than that of the normal distribution (which is equal to

3), which in turn indicates leptokurtosis. The leptokurtosis indicates that the series are

clustered during certain periods and the volatility changes at a relatively low rate thats is

large changes tend to be followed by large changes and small changes by small changes.

Histogram with Normal Curve

return

F r e q u e n c y

−0.010 −0.005 0.000 0.005 0.010

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

−0.010 −0.005 0.000 0.005 0.010

0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

kernel density estimation

N = 1500 Bandwidth = 0.0001054

D e n s i t y

Figure 4.4: Histogram and Kernel Density estimate of daily log return of MUR/USD


return

F r e q u e n c y

−0.06 −0.04 −0.02 0.00 0.02 0.04

0

2 0 0

4 0 0

6 0 0

8 0 0

−0.06 −0.04 −0.02 0.00 0.02 0.04

0

2 0

4 0

6 0

8 0


N = 1750 Bandwidth = 0.00095

D e n s i t y

Figure 4.5: Histogram and Kernel Density estimate of daily log return of MUR/GBP


return

F r e q u e n c y

−0.03 −0.02 −0.01 0.00 0.01 0.02 0.03

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

−0.03 −0.02 −0.01 0.00 0.01 0.02 0.03

0

2 0

4 0

6 0

8 0


N = 1500 Bandwidth = 0.001075

D e n s i t y

Figure 4.6: Histogram and Kernel Density estimate of daily log return of MUR/EU

19


27/50


The plots indicate that the normality assumption is questionable for all the three daily

log return series.

Figure 4.7: Scatter plot of yt against yt−1 for the MUR/USD log return data.

Figure 4.8: Scatter plot of yt against yt−1 for the MUR/EU log return data.

Figure 4.9: Scatter plot of yt against yt−1 for the MUR/GBP log return data.

In the scatter plots shown above the data at time t is plotted against the value at time

t− 1. It is one way of showing the degree of correlation in the data

20


28/50


4.1.1 Analysis of Period II data

Mean Median Max Min S.D Skewness Kurtosis

MUR/USD 8.91 e-006 0.0000 0.0122 -0.0101 0.0021 0.7574 7.871

MUR/EU 1.07 e-004 1.04 e-004 0.0311 -0.0279 0.0061 0.0843 5.044

MUR/GBP 2.02 e-005 0.0000 0.0462 -0.0416 0.0060 -0.2152 8.379

Table 4.2: Summary statistics of log first difference daily exchange rate

We find that the data sets for the period of January 2003 to December 2011 for MUR/USD

and MUR/EU and for the period of January 2002 to December 2011 produce the samecharacteristics as described above for the sample data sets. We observe that the minimum

and the maximum value remain the same for the two periods, showing that in the sample

data sets the three series had already attain their maximum and minimum value.

MUR/USD

Time

U S D

2004 2006 2008 2010 2012

2 6

2 8

3 0

3 2

3 4

Figure 4.10: Daily MUR/USD Jan 03 - Dec 11

21


29/50

4.2 Forecasting Accuracy

MUR/EURO

Time

E U R O

2004 2006 2008 2010 2012

3 0

3 5

4 0

4 5

Figure 4.11: Daily MUR/EU Jan 03 - Dec 11

MUR/GBP

Time

G B P

2002 2004 2006 2008 2010 2012

4 5

5 0

5 5

6 0

6 5

Figure 4.12: Daily MUR/GBP Jan 02 - Dec 11

4.2 Forecasting Accuracy

For comparing forecasts produced by different models, the root mean square error, the

mean absolute error and the mean absolute percentage error are calculated.

et = Y t − Ŷ t

denote the forecast error where Y t is the forecasted value of the observed value Ŷ t.

1. Root Mean Square Error

RMSE =

1n

nt=1

e2t

22


30/50

4.3 Fitting Neural Network Model to the Exchange Rates Data

2. Mean Absolute Error

MAE = 1n

nt=1

|et|

3. Mean Absolute Percentage Error

MAPE =

1

n

nt=1

|et|

Y t

× 100

The MSE which is the most common measure of forecasting acccuracy indicates the de-gree of spread, however large errors are given additional weight. RMSE is considered,

since the forecast error is then denoted in the same dimensions as the actual and forecast

values themselves. It is found to be most informative for the errors with near normal dis-

tribution. The mean absolute error is a very popular measure of forecast error since it

compares forecast with their eventual outcomes. However it was emphasized by previous

research that the MAPE is the most useful measure to compare the accuracy of forecasts

since it measures relative performance.

4.3 Fitting Neural Network Model to the Exchange Rates Data

The choice of an appropriate architecture is in itself a very difficult task and to add up to it

there is a large number of parameters which must be estimated.

For our Neural Network model we used the feedforward neural network with a single

hidden layer. The number of neurons in the hidden layer was varied between 1 and 5.

The activation function in the hidden layer neuron is Tansigmoid (tansig) and that in the

output layer in the linear function (purelin). To set up the learning rate we ran the net-

work with a large number of different learning rates between 0.05 and 0.5 before settling

on 0.25 which gave us the best results. The momentum value was analyzed and varied

between 0.1 and 0.9 and we found that 0.4 and 0.8 gave us better results. However we

use 0.8 during the experiments since it is common to choose the momentum value close to

1. The training algorithm ’Traingdm’ was used since it provides faster convergence in our

23


31/50

4.4 Fitting ARIMA Model to the Foreign Exchange Rates Data

feedforward network. The number of epochs used while training was between 30000 and

50000 depending on the performance graph which shows us the point where the network

was sufficiently trained.

After having estimated these parameters, we focus our case study on the following issues

which are firstly the number of input variables and secondly the number of hidden neu-

rons in the hidden layer. All the computations regarding the neural network were carried

out in Matlab.

The number of input variables are based on the number of lagged past observation. The

first input consists of training the data using only the first previous value Lag (1) that is Y t

is the target value thus we used Y t−1 as the input. When we have 2 input variables we use

Lag (1, 2) that is we use Y t−1 and Y t−2 as input and Y t as target. The experiment is carried

out using Lag (1) to Lag (1-12).

We use the daily data of MUR/USD, MUR/EU and MUR/GBP we divide the data into

the training set (In-sample) and testing set (Out-of-sample). The test set for all the data

consists of the values of the whole year of 2008 which we shall try to forecast. For each

input variable set we only experiment with 1-5 hidden nodes.

4.4 Fitting ARIMA Model to the Foreign Exchange Rates Data

ARIMA modeling consists of three stages which are the identification stage, the estimation

and diagnostic checking stage and the forecasting stage.

During the identification stage, we take our data and convert it into time series and find

ACF and PACF. The stationary tests can be performed to determine if differencing is

needed. The analysis of the AFC and PACF graphs usually suggest one or more tenta-tive models that can be fitted in our data.

In the estimation and diagnostic checking stage, the diagnostic statistics help us to judge

the adequacy of the models found in the identification stage. The goodness-of-fit statistics

aid in compairing the model to others. The model with the least Akaike Information Cri-

terion (AIC) and Bayesian information criterion (BIC) is retained.

After inspecting the ACF and PACF for identifying the best model for ARIMA forecasting

24


32/50

4.5 Fitting the GARCH model

different values of p, d and q are fitted and compared, if any of these model proved to

be better then we fit model for the forecasting stage. Moreover there exists an auto.arima

function in R which can help us to find a fitted model for the data set.

4.5 Fitting the GARCH model

The first step while fitting a GARCH model is to select the parameters of the specific model

however we have check for the ARCH effect in the exchange rate data. From the log return

plots of the MUR/USD, MUR/EU and MUR/GBP we find that there are heavy fluctua-

tions in the data along with the presence of volatility clustering which gives us a hint that

the data may not be identically and independently distributed (iid). The ACF and PACF

for the log return, the absolute log return and the square of the log return is taken and if

they produce significant autocorrelation then the data is not iid and thus we can say that

there exists the ARCH effect.

The ARMA-GARCH model is fitted to the data and the parameters of the equations of the

model is estimated by the ACF, PACF and EACF plot however the parameters values are

varied and different models are fitted to the in-sample data. Both the in-sample and out-

of-sample forecast performance are produced.

25


33/50

4.6 Empirical findings


4.6.1 Forecasts performance of Period I

Lags No of epochs No of neurons RMSE MAE MAPE

ARIMA(3,1,0) 0.0273 0.0172 0.0586

ARIMA(2,1,3) 0.0270 0.0172 0.0587

AR(1)-GARCH(1,1) 0.0271 0.0172 0.0587

1 50000 2 0.0384 0.0249 0.1000

1-2 50000 2 0.0321 0.0219 0.0744

1-3 50000 2 0.0308 0.0194 0.0658

1-4 50000 4 0.0303 0.0199 0.0673

1-5 50000 5 0.0322 0.0207 0.0706

1-6 50000 3 0.0327 0.0216 0.0735

1-7 50000 4 0.0322 0.0210 0.0712

1-8 50000 2 0.0338 0.0224 0.0760

1-9 50000 3 0.0310 0.0197 0.0672

1-10 50000 4 0.0306 0.0197 0.0670

1-11 50000 2 0.0310 0.0203 0.0689

1-12 50000 1 0.0354 0.0264 0.0894

Table 4.3: In-sample performance of MUR/USD for data Jan 2003-Dec 2008

26


34/50



ARIMA(1,0,0) 0.1993 0.1553 0.4219

ARIMA(0,1,0) 0.1993 0.1552 0.4214

AR(1)-GARCH(1,2) 0.1990 0.1551 0.4213

1 40000 1 0.2026 0.1595 0.4336

1-2 50000 2 0.2007 0.1569 0.4260

1-3 50000 4 0.2008 0.1579 0.4282

1-4 50000 3 0.2007 0.1573 0.4273

1-5 50000 2 0.1996 0.1558 0.4230

1-6 50000 4 0.1995 0.1559 0.42331-7 50000 4 0.2010 0.1574 0.4269

1-8 40000 1 0.2029 0.1599 0.4343

1-9 50000 4 0.2020 0.1578 0.4278

1-10 50000 5 0.1996 0.1563 0.4235

1-11 50000 4 0.2006 0.1563 0.4320

1-12 50000 5 0.2001 0.1564 0.4243

Table 4.4: In-sample performance of MUR/EU for data Jan 2003-Dec 2008

27


35/50



ARIMA(1,0,0) 0.2710 0.2032 0.3828

ARIMA(0,1,0) 0.2709 0.2030 0.3823

AR(1)-GARCH(1,1) 0.2700 0.2026 0.3943

1 40000 1 0.2776 0.2097 0.3960

1-2 50000 2 0.2733 0.2050 0.3868

1-3 40000 1 0.2732 0.2052 0.3872

1-4 50000 2 0.2739 0.2052 0.3872

1-5 40000 1 0.2773 0.2096 0.3957

1-6 50000 3 0.2750 0.2067 0.38881-7 50000 2 0.2716 0.2057 0.3879

1-8 50000 2 0.2710 0.2041 0.3848

1-9 50000 5 0.2752 0.2095 0.3945

1-10 50000 3 0.2736 0.2058 0.3865

1-11 50000 4 0.2725 0.2056 0.3865

1-12 50000 2 0.2718 0.2057 0.3874

Table 4.5: In-sample performance of MUR/GBP for data Jan 2002-Dec 2008

28


36/50



ARIMA(3,1,0) 2.011 1.588 5.331

ARIMA(2,1,3) 2.331 1.615 5.250

AR(1)-GARCH(1,1) 2.161 1.562 5.131

1 50000 2 0.0979 0.0682 0.2338

1-2 50000 2 0.0704 0.0493 0.1687

1-3 30000 1 0.0754 0.0548 0.1871

1-4 50000 2 0.0760 0.0537 0.1835

1-5 30000 1 0.0746 0.0554 0.1896

1-6 30000 1 0.0864 0.0639 0.21881-7 30000 1 0.0843 0.0621 0.2121

1-8 50000 5 0.0786 0.0576 0.1970

1-9 30000 1 0.0791 0.0589 0.2014

1-10 50000 2 0.0815 0.0570 0.1942

1-11 50000 3 0.0864 0.0638 0.2178

1-12 50000 2 0.0784 0.0574 0.1964

Table 4.6: Out-of-sample performance of MUR/USD for data Jan 2003-Dec 2008

29


37/50



ARIMA(1,0,0) 1.271 0.8503 1.955

ARIMA(0,1,0) 1.119 0.8037 1.868

AR(1)-GARCH(1,2) 1.265 0.8439 1.940

1 30000 1 0.3646 0.2642 0.6196

1-2 50000 2 0.3640 0.2654 0.6229

1-3 30000 1 0.3647 0.2643 0.6200

1-4 30000 1 0.3637 0.2636 0.6182

1-5 50000 2 0.3634 0.2632 0.6174

1-6 30000 1 0.3642 0.2642 0.61971-7 30000 1 0.3650 0.2649 0.6219

1-8 50000 3 0.3631 0.2646 0.6213

1-9 30000 1 0.3650 0.2653 0.6224

1-10 50000 2 0.3638 0.2658 0.6237

1-11 30000 1 0.3638 0.2657 0.6236

1-12 30000 1 0.3631 0.2655 0.6230

Table 4.7: Out-of-sample performance of MUR/EU for data Jan 2003-Dec 2008

30


38/50



ARIMA(1,0,0) 4.429 3.755 7.244

ARIMA(0,1,0) 4.537 3.853 7.431

ARMA(1,1)-GARCH(1,1) 4.137 3.477 6.708

1 30000 1 0.4796 0.3431 0.6497

1-2 30000 1 0.4772 0.3410 0.6456

1-3 50000 2 0.4772 0.3405 0.6447

1-4 30000 1 0.4794 0.3402 0.6443

1-5 50000 2 0.4776 0.3438 0.6498

1-6 30000 1 0.4845 0.3433 0.65011-7 50000 2 0.4838 0.3477 0.6581

1-8 50000 2 0.4825 0.3450 0.6530

1-9 30000 1 0.4841 0.3431 0.6498

1-10 30000 1 0.4848 0.3435 0.6506

1-11 50000 5 0.4807 0.3430 0.6495

1-12 30000 1 0.4850 0.3439 0.6512

Table 4.8: Out-of-sample performance of MUR/GBP for data Jan 2002-Dec 2008

ANN ARIMA GARCH

Year/daily Actual values Forecasts Errors Forecasts Errors Forecast Errors

2008/1 57.1351 57.4654 -0.3303 57.2970 -0.1619 57.3049 -0.1698

2008/2 57.1221 57.2047 -0.0826 57.2961 -0.174 57.3041 -0.182

2008/3 57.1823 57.1134 0.0689 57.2953 -0.113 57.3032 -0.1209

2008/4 57.3500 57.1434 0.2066 57.2944 0.0556 57.3024 0.0476

2008/5 56.9235 57.253 -0.3295 57.2936 -0.3701 57.3106 -0.3871

2008/6 57.2858 56.8477 0.4381 57.2928 -0.007 57.3007 -0.0149

2008/7 57.2978 57.2938 0.004 57.2919 0.0059 57.2999 -0.0021

2008/8 57.2383 57.1821 0.0562 57.2911 -0.0528 57.2991 -0.0608

2008/9 57.3006 57.2182 0.0824 57.2903 0.0103 57.2982 0.0024

2008/10 57.4102 57.235 0.1752 57.2894 0.1208 57.2974 0.1128

Table 4.9: First 10 forecasts values of MUR/GBP for ANN, ARIMA and GARCH models

31


39/50


4.6.2 Results and Discussions for Period I

The tables 4.3 to 4.8 show the minimum error obtained when we vary the number of neu-rons (between 1 − 5) in the hidden layer for each Lag (1) to Lag (1 − 12). We observe from

our data that as the number of neurons in the hidden layer increases the amount of epochs

to train the network must also increase so as to reach a point where the forecast value does

not change anymore. Thus increasing the number of epochs when the network has already

reached the global minima is useless and very time consuming. We reach the conclusion

that the number of epochs used is dependent to the number of neurons in the hidden layer.

Moreover we also notice that as the number of neurons increases the performance of the

network changes. We tend to achieve the best forecasts when we have 2 or 3 neurons inthe hidden layer and as the number of neurons increases from 5 the forecast values tend

to move away from the target values since great increase in the RMSE, MAE and MAPE

values are observed.

In the tables above the in-sample performance are presented followed by the out-of-sample

performance. Here the In-sample data set for MUR/USD and MUR/EU is taken from Jan-

uary 2003-December 2007 and for MUR/GBP is taken from January 2002-December 2007.

The out-of-sample data set in all the three cases is taken for January 2008-December 2008.

As for the number of inputs presented to the network, we cannot find any trend about how

it affects the performance. In our experiment for both the in-sample and out-of-sample

forecast we observe that each data set has a different number of inputs where the neural

network works best.

Considering the in-sample forecast we find that for MUR/USD the ARIMA(2, 1, 3) model

outperforms the ANN models however the AR(1)-GARCH(1, 1) model gives almost the

same forecast accuracy. As for the MUR/EU and MUR/GBP model the GARCH models

perform much better than the other models. Thus we can conclude that for in-sample fore-

casts the GARCH model produces better forecasts than the ARIMA and the ANN models.

For out-of-sample forecasts the GARCH models used for both MUR/USD and MUR/GBP

gives a superior results than the ARIMA models when the RMSE, MAE and MAPE are

considered as evalution criteria. However when compared with the ANN model the fore-

cast performance is found to be far behind.

The table 4.9 show that first 10 out-of-sample forecasts of MUR/GBP for ANN, ARIMA

and the GARCH models. We notice that the three models perform very accurately for the

32


40/50


first 10 forecasts and since the MAPE values for the forecasts of year 2008 of the ARIMA

and GARCH models are quite inferior to the ANN model value we can say that as the

ANN continue to predict more values it out-perform the other two models.

The diagrams 4.13, 4.14 and 4.15 shows the In-sample and Out-of-sample forecasts

produced by the ANN models:

Figure 4.13: In-sample and out-of-sample forecast for MUR/USD

Figure 4.14: In-sample and out-of-sample forecast for MUR/EU

33


41/50


Figure 4.15: In-sample and out-of-sample forecast for MUR/GBP

4.6.3 Forecasts performance of Period II

Lags No of Epochs No of Neurons RMSE MAE MAPE

ARIMA(5,1,0) 0.0489 0.0313 0.1025

ARIMA(2,1,3) 0.0489 0.0314 0.1026

AR(1)-GARCH(1,1) 0.0049 0.0314 0.1031

1 50000 1 0.0671 0.0454 0.15051-2 50000 2 0.0547 0.0368 0.1214

1-3 50000 2 0.0551 0.0364 0.1200

1-4 50000 2 0.0537 0.0363 0.1198

1-5 50000 3 0.0544 0.0374 0.1236

1-6 40000 1 0.0562 0.0396 0.1312

1-7 50000 2 0.0536 0.0363 0.1200

1-8 50000 2 0.0534 0.0359 0.1186

1-9 40000 1 0.0561 0.0392 0.12991-10 50000 2 0.0544 0.0362 0.1193

1-11 50000 3 0.0581 0.0389 0.1284

1-12 50000 5 0.0536 0.0357 0.1177

Table 4.10: In-sample performance MUR/USD data

34


42/50



ARIMA(1,0,0) 0.2445 0.1806 0.4571

ARIMA(0,1,0) 0.2445 0.1805 0.4568

AR(1)-GARCH(1,1) 0.2245 0.1806 0.4571

1 50000 2 0.2442 0.1817 0.4587

1-2 50000 3 0.2445 0.1818 0.4591

1-3 50000 2 0.2453 0.1833 0.4629

1-4 50000 3 0.2461 0.1841 0.4646

1-5 50000 2 0.2436 0.1807 0.4560

1-6 50000 2 0.2448 0.1815 0.45811-7 50000 5 0.2437 0.1813 0.4575

1-8 50000 4 0.2431 0.1805 0.4553

1-9 50000 3 0.2445 0.1820 0.4589

1-10 50000 2 0.2439 0.1811 0.4565

1-11 50000 5 0.2442 0.1809 0.4560

1-12 50000 2 0.2452 0.1826 0.4611

Table 4.11: In-sample performance MUR/EU data

35


43/50



ARIMA(1,0,0) 0.3181 0.2318 0.4443

ARIMA(0,1,0) 0.3181 0.2318 0.4441

AR(1)-GARCH(1,1) 0.2704 0.2026 0.3942

1 50000 2 0.3120 0.2270 0.4383

1-2 50000 2 0.3118 0.2273 0.4358

1-3 30000 1 0.3098 0.2262 0.4263

1-4 50000 3 0.3113 0.2284 0.4405

1-5 50000 4 0.3088 0.2260 0.4363

1-6 50000 2 0.3115 0.2287 0.44101-7 50000 4 0.3110 0.2275 0.4391

1-8 50000 2 0.3092 0.2251 0.4392

1-9 50000 2 0.3143 0.2316 0.4477

1-10 50000 3 0.3113 0.2278 0.4396

1-11 50000 4 0.3105 0.2278 0.4398

1-12 50000 2 0.3098 0.2271 0.4381

Table 4.12: In-sample performance MUR/GBP data

36


44/50



ARIMA(5,1,0) 1.234 1.220 4.082

ARIMA(2,1,3) 1.233 1.220 4.080

AR(1)-GARCH(1,2) 1.233 1.221 4.082

1 30000 1 0.0501 0.0356 0.1189

1-2 30000 2 0.0539 0.0422 0.1441

1-3 30000 1 0.0502 0.0404 0.1349

1-4 30000 1 0.0494 0.0394 0.1317

1-5 50000 2 0.0495 0.0386 0.1291

1-6 30000 1 0.0501 0.0387 0.12951-7 50000 2 0.0512 0.0410 0.1371

1-8 30000 1 0.0505 0.0396 0.1324

1-9 50000 2 0.0512 0.0406 0.1357

1-10 30000 1 0.0472 0.0368 0.1229

1-11 30000 1 0.0518 0.0405 0.1352

1-12 30000 1 0.0222 0.0166 0.0553

Table 4.13: Out-of-sample performance MUR/USD data

37


45/50



ARIMA(1,0,0) 1.772 1.735 4.408

ARIMA(0,1,0) 1.780 1.742 4.426

AR(1)-GARCH(1,1) 1.776 1.739 4.419

1 30000 1 0.1427 0.1099 0.2783

1-2 30000 2 0.1424 0.1103 0.2791

1-3 50000 4 0.1381 0.1047 0.2645

1-4 30000 1 0.1424 0.1102 0.2790

1-5 30000 1 0.1423 0.1107 0.2802

1-6 30000 1 0.1437 0.1122 0.28411-7 50000 2 0.1433 0.1120 0.2835

1-8 30000 1 0.1452 0.1147 0.2904

1-9 50000 2 0.1590 0.1237 0.3133

1-10 50000 2 0.1470 0.1105 0.2798

1-11 30000 1 0.1410 0.1107 0.2803

1-11 50000 2 0.1394 0.1170 0.2961

1-12 30000 1 0.1389 0.1079 0.2730

Table 4.14: Out-of-sample performance of MUR/EU data

38


46/50



ARIMA(1,0,0) 1.993 1.981 4.249

ARIMA(0,1,0) 1.934 1.921 4.119

AR(1)GARCH(1,2) 1.996 1.985 4.257

1 30000 1 0.2396 0.1807 0.3694

1-2 30000 1 0.2440 0.1831 0.3745

1-3 50000 3 0.1985 0.1624 0.3332

1-4 30000 1 0.2048 0.1648 0.3381

1-5 50000 5 0.1990 0.1463 0.3003

1-6 30000 1 0.1998 0.1470 0.30071-7 50000 3 0.1994 0.1506 0.3089

1-8 50000 3 0.2025 0.1593 0.3270

1-9 50000 4 0.1955 0.1535 0.3149

1-10 30000 1 0.2141 0.1689 0.3466

1-11 50000 2 0.2142 0.1711 0.3511

1-12 50000 2 0.2029 0.1506 0.3087

Table 4.15: Out-of-sample performance of MUR/GBP data

4.6.4 Results and Discussions for Period II

In this experiment the in-sample set for MUR/USD and MUR/EU consists the values from

January 2003 to November 2011 whereas the in-sample set for MUR/GBP consists the val-

ues from January 2002 to November 2011. The out-of-sample set for each of the three series

consists the value of December 2011.

Considering the in-sample forecasts of the MUR/USD data we find that the ARIMA model

outperform all the other models. As for the MUR/EU data the ANN model using the

Lag(1 − 8) produces a superior results since the RMSE, MAE and MAPE values are small

compare to the random walk, ARIMA and GARCH model. In the MUR/GBP forecasts the

GARCH model used produces better forecasts.

For the out-of-sample forecast we find that the ANN models outperform the ARIMA,

GARCH and random walk model for all the three series.

39


47/50


Period I Period II

ANN ARIMA GARCH ANN ARIMA GARCH

RMSE 0.0308 0.0273 0.0271 0.0536 0.0489 0.0499

In-sample MAE 0.0194 0.0172 0.0172 0.0357 0.0313 0.0314

MAPE 0.0658 0 .0586 0.0587 0.1177 0 .1025 0.1031

RMSE 0.0704 2.011 2.161 0.0222 1.233 1.233

Out-of-sample MAE 0.0493 1.588 1.562 0.0166 1.220 1.221

MAPE 0.1687 2.331 5.131 0.0553 4.080 4.082

Table 4.16: In-sample and Out-of-sample forecasts of MUR/USD

Period I Period II


RMSE 0.1996 0.1993 0.1990 0.2436 0.2445 0.2245

In-sample MAE 0.1563 0.1553 0.1551 0.1807 0.1806 0.1806

MAPE 0.4235 0 .4219 0.4213 0.4560 0 .4571 0.4571

RMSE 0.3634 1.271 1.265 0.1389 1.772 1.776

Out-of-sample MAE 0.2632 0.8503 0.8439 0.1079 1.735 1.739

MAPE 0.6174 1.955 1.940 0.2730 4.408 4.419

Table 4.17: In-sample and Out-of-sample forecasts of MUR/EU

Period I Period II


RMSE 0.2710 0.2710 0.2700 0.3098 0.3181 0.2704

In-sample MAE 0.2041 0.2032 0.2026 0.2262 0.2318 0.2026

MAPE 0.3848 0 .3828 0.3843 0.4263 0 .4443 0.3942

RMSE 0.4794 4.429 4.137 0.1990 1.993 1.996

Out-of-sample MAE 0.3402 3.755 3.477 0.1463 1.981 1.985

MAPE 0.6443 7.244 6.708 0.3003 4.249 4.257

Table 4.18: In-sample and Out-of-sample forecasts of MUR/GBP

40


48/50

Chapter 5

Conclusion

In this project, linear and nonlinear models were used to forecast the exchange rates of the

Mauritius Rupee against three foreign currencies: the US Dollar, the Euro and the British

Pound. Accuracy of forecasting models for two periods of data first from January 2002

to December 2008 and the second from January 2002 to December 2011 were taken into

consideration. The reason for this choice is that we wanted to test the ability of our fore-

casting procedures to provide accurate out-of-sample forecasts during the financial crisis.

The artificial neural network (ANN) which is a nonlinear model was used as an alternative

model to the linear ARIMA processes and the nonlinear GARCH models. The empirical

results show that the ANN models has a superior out-of-sample forecasting performance

for both periods data when compared to the other models. For the in-sample forecasts we

observed that ARIMA and ARMA-GARCH models provided better goodness-of-fit than

the ANN models.

One of the reasons behind this is that there is still no well defined guidelines to build up an

ANN model to solve a specific problem. Thus to obtain the best possible ANN forecasting

model rigorous experiments were carried out so as to determine the different parametersto build up the model. Considering the out-of-sample performance of the ANN models

we can say that they proved to be successful when fitted to the foreign exchange rates data

provided extreme care is taken in designing the network. Thus we can conclude that the

ANN model can be used as a complementary tool to different time-series models and the

forecast accuracies can be further improved.

41


49/50

Bibliography

Bollerslev, T. (1986), ‘Generalized autoregressive conditional heteroscedasticity’, Journal of

economics 31, 307–327.

Bollerslev, T. & Ghysels, E. (1996), ‘Periodic autoregressive conditional heteroscedasticity’,

J.Bus.Eco.Stat 14, 139–151.

Cao, L. & Tay, F. (2001), ‘Financial forecasting using support vector machines’, Neural Com-

puter and Application 10, 184–192.

Engle, R. F. (1982), ‘Autoregressive conditional heteroscedasticity with estimates of the

variance of united kingdom inflation’, Economica 50, 909–927.

Gencay, R. (1999), ‘Linear, non-linear and essential foreign exchange rate prediction withsimple technical trading rules’, Journal of International Economics 47, 91–107.

Haoffi, Z., X. G. Y. F. & Han, Y. (2007), ‘A neural network model based on the multistage

optimization approach for short term food price forecasting in china’, Expert Syst. Appl.

33, 347–356.

Meade, N. (2002), ‘A comparison of the accuracy of short term foreign exchange forecasting

methods’, International Journal of Forecasting 18, 67–83.

Panda, C. & Narasimhan, V. (2007), ‘Forecasting exchange rate better with artificial neuralnetwork’, Journal of Policy Modeling 29, 227–236.

R. K. Bissoondeeal, J. M. Binner, M. B. A. G. & Mootanah, V. P. (2008), ‘Forecasting exchange

rates with linear and nonlinear models’, Global Business and Economics Review 10, 414–

429.

Racine, J. (2001), ‘On the nonlinear predictability of stock returns using financial and eco-

nomic variables’, Business Econ.Stat. 19, 380–382.

42


50/50

BIBLIOGRAPHY

T. Hill, M. O. & Remus, W. (1996), ‘Neural network models for time series forecasts’, Man-

agement Science 42, 1082–1092.

Tang, Z. & Fishwich, P. A. (1993), ‘Backpropagation neural nets as models for time series

forecasting’, ORSA Journal on Computing 5, 374–385.

W. R. Foster, F. C. & Ungar, L. H. (1992), ‘Neural network forecasting of short, noisy time

series’, Computers and Chemical Engineering 16, 293–297.

Wang, J. H. & Leu, J. Y. (1996), ‘Stock market trend prediction using arima-based neural

networks’, Proc.of IEEE Int.Conf.on Neural Networks 4, 2160–2165.

Yaser, S. & Atiya, A. (1996), ‘Introduction to financial forecasting’, Applied Intelligence6, 205–213.

Documents

Jory Namrata Devi 2012 (1)