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ABSTRACT
This study is aimed at investigating the long memory properties and to obtain the estimation technique that will be the best in estimation of long memory variables, using the Naira to Dollar exchange rates series of Nigeria from January 2000 to Dec. 2016. The auto-correlation functions associated with the series and Hurst exponent coefficient indicated that long memory variables are present. Three parametric and the three semi-parametric methods each utilized during the estimation of the long memory variables d. The estimates gotten from the six estimators were then deployed to fitting an ARFIMA model with the aid of Box-Jenkins methodology. The outcome of the forecast evaluation criteria indicated that the best model was found to be the model derived with the method credited to Geweke& Porter-Hudak.
TABLE OF CONTENTS
Title page i
Declaration iii
Certification iv
Dedication v
Acknowledgements vi
Abstract vii
Table of contents viii
List of Figures xii
CHAPTER ONE: INTRODUCTION
1.1 Background of the Study 1
1.2 Statement of the Problem 3
1.3 Aim and Objectives of the Study 4
1.4 Significance of the Study 4
1.5 Scope of the Study 4
CHAPTER TWO: LITERATURE REVIEW
2.1 Test for presence of long memory 5
2.2 Estimating the fractional parameter d 7
2.3 Performance of estimators of the difference parameter d. 10
2.4 Modeling and forecasting using exchange rate 12
2.5 Modeling and forecasting using ARFIMA 15
CHAPTER THREE: MATERIALS AND METHODS
3.1 Time Series Plot 18
3.2 Long Memory Process 18
3.3 Autocorrelation Function (ACF) w.r.t Long Memory 18
3.4 Testing for Long Memory Using the Hurst effect 20
3.5 Estimation of the Fractional Parameter d 21
3.5.1 Parameter Methods 21
3.5.1.1 Exact Maximum Likelihood Estimator (EML) 21
3.5.1.2 Modified Profile Likelihood Estimator (MPL) 22
3.5.1.3 Non Linear Least Squares (NLS) Estimator 23
3.5.2 Semi-Parametric Methods 24
3.5.2.1 Estimation of GPH 24
3.5.2.2 Smoothed Periodogram 25
3.5.2.3 Wavelet Method 27
3.6 Differencing 27
3.8 Box and Jenkins Estimation Procedure 32
3.8.1 Model Identification 32
3.8.2 Model Estimation 32
3.8.3 Model Diagnostic 33
3.8.3.1 Akalike Information Criteria 33
3.8.3.2 Ljung and Box Test 34
3.9 Unit Root for Stationarity 34
3.9.1 Kwiatkowshi Philips Schmidt and Shin (KPSS) 35
3.9.2 Augmented Dickey Fuller Test 35
3.10 Forecasting 36
CHAPTER FOUR: RESULTS AND DISCUSSIONS
4.1 Data Presentation 37
4.2 Time Plots of the Data 37
4.3 Unit Root Test 38
4.3.1 Augmented Dickey-fuller Test (ADF) 39
4.3.2 KPSS Test 39
4.4 Test for long Memory 40
4.5 Test for Long Memory Parameter 40
4.5.1 Parametric Method of Estimating d 40
4.5.2 Semi Parametric Method of the Long Memory Estimating d 41
4.6 ACF and PACF of the Fractionally Differenced Series 41
4.7 ARFIMA Model Identification 45
4.8 Model Fitting 46
4.9 Diagnostic Checking 46
4.9.1 Ljung-Box Test 50
4.10 Forecast Evaluation values 51
4.11 Discussion of Findings 51
CHAPTER FIVE: SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
5.1 Summary 54
5.2 Conclusion 56
5.3 Recommendation 56
5.4 Contribution to Knowledge 57
Reference 58
Appendix A 62
Appendix B 63
Appendix C 65
CHAPTER ONE
INTRODUCTION
1.1 Background of the Study
Time series data represents sets of data points collected in sequential manner in a specific equal time interval. As a time series presents its values sequentially over time, it is expected to present a serial correlation in time that is characteristic of dependence between the present and previous values (Ribeiro 2003).
Modeling using time series concept intends the model that is to be fitted to have correlations developed theoretically closer to the samples correlations that is calculated using data. Correlation from models designed for time series are usually expected to diminish when the perspective of the observer system is far apart with respect to time but the decay speed may be different. Time series can have a long-range-dependency or long memory in situation where the correlation decay occurs at slower rate in a hyperbolic manner.
Benoit B. Mandelbrot was the initiator of long memory concept developed popularly as an instrument for the description of time series in economic concepts. Long memory process is known for having high order correlation structure indicting there is non-negligible dependency between the previous and present points.
Robinson (1995) defines time series long memory concept as a process where the autocorrelation decay at hyperbolic rate (slowly vanishing) or unboundedness of the processes density function. Robinson (1994) and Baillie (1996) have all reviewed long memory that are present in most data in economics while Beran (1994) have reviewed long memory modeling that exist in other disciplines
Reisen (2007) stated that long memory can only be indicated by non-zero existence and the shift from zero to non-zero is the measure of the long memory strength Cheung&Diebold (1994), Chow et al (1995), Cheung & Lai (1993), Booth et al (1982) all tested for long memory availability and presence in their different studies.
In time series modeling, specifically in cases of long memory’s presence, it is pertinent to look for the non-integer variable d by differentiating in order to incorporate the long memory. Hosking (1981) and Granger (1978) laid the foundation for a different class modeling long memory systems called ARFIMA autoregressive fractional integrated moving averages, the most useful process insolving problems of time series having long memory features. ARFIMA model allow the integration order of a series to take on fractional values which presents a very important instrument that could be deployed in time series forecasting and modeling of systems having long memory features.
Reisen (2007) defines ARFIMA as a special case of ARIMA in the level of difference variable d which takes a non-integral value and involves a fractional differentiation. The ARFIMA process has widely been utilized in many fields such as astronomy, hydro-logy, mathematics e.t.c to represent long memory time series system (see Beran 1994)
Many researchers in various literatures have stated that the major difficulty encountered in using ARFIMA time series process is the fractional variable d estimation. See Olatayo&Adedotun (2014), Ekonomi&Butka (2011), Reisen (1994), Smith et al (1997). Recently a range of estimators for the fractional variable d have appeared in time series literatures for instance see Hipel&Mcleod (1978), Hassler (1993), Reisen (1994), Chen et al (1993).
The estimators of d can generally be categorized into three group namely parametric, non-parametric and semi-parametric method. Parametric method, all the parameters i.e. the autoregressive (p), fractional parameter (d) and moving averages (q) can be simultaneously estimated, examples of parametric methods are Exact maximum, likelihood (EML), modified profile likelihood (MPL) whittle Estimator etc. while semi-parametric method, the fractional parametric d is estimated before the autoregressive parameter (p) and the moving average parameter q are estimated, examples of semi-parametric methods are Geweke and Porter -Hudek GPH method, Rescaled Range Analysis (R/S), modified Rescaled Range Analysis (MRS), Bootstrap method, Jackknife & Bootstrap method, truncated geometric bootstrap method.
ARFIMA model is employed in modeling financial times series data like stock prices, exchange rates crude oil prices etc. Exchange rates in the rate of which one currency is exchanged for another. It represents price of one countries currency relative to another currency (Jhingan, 2005). Exchange rate is the price of a unit of foreign currency relative to a domestic currency. Exchange rate dynamics are very essential in estimation of flow of international trade tradable goods prices, prices of foreign-exchange futures and international access portfolios. Numerous effects are created in attempt to understand exchange rate dynamics since the inception of the floating rates regime in 1973. A core understanding of the time series properties of exchange rate has significant economic implications like determination of the economic growth of a nation.
Exchange rates have become unfavorable to Nigeria owing to usage of floating foreign exchange determination system hence it has become very pertinent to construct model for exchange rate determination which could evaluate the exchange rate performance in time domain.
1.2 Statement of the Problem
One of the main issues associated with fitting an ARFIMA model is variable d estimation. Literatures reviewed indicated that many methods have been employed to estimate d. Oftentimes these methods outperform each other relative to the properties and data set used.
This situation might result in loss of information, misspecification, and misclassification of the data. In Nigeria, the exchange rate which obviously has long memories features has been model using ARFIMA. In this research work, we set to get the best estimation method of the variable d when modeling exchange rate of Naira to Dollar.
1.3 Aim and Objectives of the Study
The aim of this study is to examine the long memory features of the Naira to Dollar exchange rate series of Nigeria and get the most appropriate method of estimation for long memory variable d. To accomplish this aim, the objectives of this research work are:
- To investigate the presence or availability of long memory in the series
- To obtain long memory variable estimates using three parametric namely Exact maximum likelihood method, modified profile likelihood method and Non-linear least square method and three semi-parametric methods namely Geweke and Porter-Hudak, Smooth periodogram and Wavelet
iii. To fit ARFIMA models using the estimates generated using the six methods of estimating the variable d that is the fractionally differenced parameter.
- To test or investigate the model adequacy
- To compare their forecasting performance of the six estimators of the variable d.
1.4 Significance of the Study
The justifications of this research work lie on believe that the outcome from this work will be helpful in estimation of the fact that misuse of methodology lead to inappropriate results and conclusion. It will be helpful in affirming the best method of estimating the d parameter in modeling an ARFIMA model.
1.5 Scope of the Study
This work is focused on estimation of fractionally different variable d based on parametric and semi-parametric method using the Nigeria exchange rate series from January 2000 to December 2016 gotten from the central bank of Nigeria (2016).