ABSTRACT
This project presents an empirical study of time series modeling and forecasting of the official daily Exchange rate of Nigeria Naira for US Dollar in terms of buying rate, central rate and selling rate, from the period of 1st January 2016 to 19th of May 2017 (recession period). In this view, Box Jenkins approach was applied for the modelling of naira/dollar daily exchange rate using ARIMA model. The results of the analysis show that the series became stationary after first differencing. Based on AIC and BIC selection criteria, the best model that explains the series was found to be ARIMA (0, 1, 1). The diagnosis checking on such model was confirmed, the error was white noise, and the presence of no serial correlation. The performance of the three ARIMA (0, 1, 1) models for buying rate, central rate and selling rate shows that the selling rate model had the Minimum ME, MSE, RMSE and MAPE.
TABLE OF CONTENT
Abstract vi
Table of Content vii
LIST OF TABLES xi
LIST OF FIGURES xiii
CHAPTER ONE: INTRODUCTION 1
1.1 Background of the Study 1
1.2 Statement of the Problem 3
1.3 Aim and Objectives of the Study 5
1.4 Justification for the Study. 6
1.5 The Scope of the Study 7
1.6 Limitations: 7
1.7 Definition of Key Concept Terms 8
CHAPTER TWO: LITERATURE REVIEW 10
CHAPTER THREE: METHODOLOGY 14
3.1 Data and Source 14
3.2 Time Series Analysis 14
3.3 Component Analysis 15
3.4 Box and Jenkins Time Series Methodology 16
3.5 Test for Stationarity: 16
3.5.1. The graphical approach includes 19
3.5.2. Quantitative methods includes: 19
3.5.3 Augmented Dickey-Fuller (ADF) Test 19
3.5.4 Phillips – Perron test: 20
3.6 Differencing to achieve the stationarity: 21
3.6.1 Differencing operations 22
3.7 Box and Jenkins Modelling Approach 23
3.7.1 Autoregressive (AR) models 23
3.7.2 Moving – Average MA(q) Model 24
3.7.3 ARMA (p, q) Model 25
3.8 Model Identification 25
3.8.1 Autocorrelation Function: 25
3.8.2 Partial Autocorrelation Function 27
3.8.3 Summarizes the behavior of the theoretical 28
3.9 Best Model Selection Criteria 29
3.10 Estimation of Model parameters: 30
3.11 Model Diagnostics check 30
3.11.1 Test for Heteroscedasticity 30
3.11.2Ljung-Box test 31
3.12 Forecasting 33
3.12.1 ARIMA Model Forecast Performance 34
CHAPTER FOUR : ANALYSIS OF DATA 36
4.1 Graphical presentation of the exchange rate time series data 36
4.2 Component of the series 38
4.3 Test for Stationarity of Exchange Rate of Naira/USD 38
4.3.1 Unit Root Test (Buying Rate Series) of Naira/USD 38
4.3.3 Unit Root Test (Selling Rate Series) of Naira/USD 40
4.4 ACF and PACF for the Non-Stationary Exchange Rate Series are Reported in the following Autocorrelation and Partial-autocorrelation Correlogram. 41
4.5 Next we take the first differencing of each of the series and repeat
the test for Stationarity. 44
4.5.1 Unit Root Test for (difference buying rate series) of Naira/USD 44
4.5.2 Unit Root Test (Difference Central Rate Series) of Naira/USD 46
4.5.3 Unit Root Test (Difference Selling Rate Series) 47
4.6 Model identification 48
4.6.1 ACF and PACF for the Stationary Exchange Rate Series are Reported
in the following Autocorrelation and Partial-autocorrelation Correlogram. 48
4.6.2 Best Model Selection 52
4.7 ARIMA Model Estimation 54
4.8 Model Diagnostic Check 55
4.8.1 Test for Heteroscedasticity 55
4.9 Forecasting 60
CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMENDATION 68
5.1 Summary 68
5.2 Conclusion 69
5.3 Recommendation 70
REFERENCE 71
ABBREVIATION WORDS 75
APPENDIX 76
LIST OF TABLES
Table 4.1: Dickey-Fuller test (ADF(stationary) / k: 6 / Buying Rate): 38
Table 4.2 Philips –perron test result (Buying rate) 38
Table 4.3 Dickey- fuller test result (Central Rate) 39
Table 4.4 philips perron test result (Central Rate) 39
Table 4.5 Dickey- fuller test result (selling Rate) 40
Table 4.6 philips perron test result (Selling Rate) 40
Table 4.7 Dickey- fuller test result (Stationary) (Buying Rate) 44
Table 4.8 philips perron test result (Stationary) (Buying Rate) 44
Table 4.9 Dickey- fuller test result (Stationary) (Central Rate) 46
Table 4.10 philips perron test result (Stationary) (Central Rate) 46
Table 4.11 Dickey- fuller test result (Stationary) (Selling Rate) 47
Table 4.12 philips perron test result (Stationary) (Selling Rate) 47
Table 4.13: Results of ARIMA modeling of the Buying Rate series: 52
Table 4.14: Results of ARIMA modeling of the Central Rate series: 53
Table 4.15: Results of ARIMA modeling of the Selling Rate series: 53
Table 4.16: Final Estimates of Parameters for Buying Rate model 54
Table 4.17: FinalEstimates of Parameters for Central Rate model 54
Table 4.18: Final Estimates of Parameters for Selling Rate model 55
Table 4.19 ARCH-LM test result for buying rate 56
Table 4.20 ARCH-LM test result for Central rate 57
Table 4.21 ARCH-LM test result for Selling rate 58
Table 4.22: Ljung-Box Q-test Residual autocorrelation 59
test results for Buying rate series
Table 4.23: Ljung-Box Q-test Residual autocorrelation
test results for central rate series 59
Table 4.24: Ljung-Box Q-test Residual autocorrelation
test results for selling rate series 60
Table 4.25 Daily Exchange Rate Forecast on Selling
Rate of Naira/ US dollar 61
Table 4.26: Daily Exchange Rate Forecast on Central
Rate of Naira/ US dollar 63
Table 4.27: Daily Exchange Rate Forecast on Selling
Rate of Naira/US-Dollar 64
Table 4.28: In-sample forecast error performance of
ARIMA (0, 1, 1) model. 66
LIST OF FIGURES
Figure 3.1: Diagrammatic representation of Box -Jenkins process. 18
Figure 4.1: Time plot of (Buying Rate) of Daily exchange
rate of naira to us dollar from January 2016 to May 2017. 36
Figure 4.2: Time plot of (Central Rate) of Daily exchange
rate of naira to us dollar from January 2016 to May 2017. 37
Figure 4.3: Time plot of (Selling Rate) of Daily exchange rate
of naira to US dollar (US$) from January 2016 to May 2017. 37
Figure 4.4: ACF plot of (Buying Rate) of Daily exchange
rate of naira to US dollar from January 2016 to May 2017 41
Figure 4.5: PACF plot of (Buying Rate) of Daily exchange rate
of naira to US dollar from January 2016 to May 2017 41
Figure 4.6: ACF plot of (Central Rate) of Daily exchange
rate of naira to US dollar from January 2016 to May 2017. 42
Figure 4.7: PACF plot of (Central Rate) of Daily exchange
rate of naira to US dollar from January 2016 to May 2017. 42
Figure 4.8: ACF plot of (Selling Rate) of Daily exchange
rate of naira to US dollar from January 2016 to May 2017. 43
Figure 4.9: PACF plot of (Selling Rate) of Daily exchange
rate of naira to US dollar from January 2016 to May 2017. 44
Figure 4.10: Difference plot of (Buying Rate) of Daily exchange rate
of naira to US dollar from January 2016 to May 2017 44
Figure 4.11: Differencing plot of (Central Rate) of Daily exchange
rate of naira to US dollar from January 2016 to May 2017 45
Figure 4.12: Differencing plot of (Selling Rate) of Daily exchange
rate of naira to US dollar from January 2016 to May 2017 47
Figure 4.13: ACF of (Buying Rate) of Daily exchange
rate of naira to US dollar from January 2016 to May 2017. 48
Figure 4.14: PACF plot of (Buying Rate) of Daily exchange
rate of naira to US dollar from January 2016 to May 2017. 49
Figure 4.15: ACF plot of (Central Rate) of Daily exchange
rate of naira to US dollar from January 2016 to May 2017. 49
Figure 4.16: PACF plot of (Central Rate) of Daily exchange
rate of naira to US dollar from January 2016 to May 2017. 50
Figure 4.17: PACF plot of (Selling Rate) of Daily exchange
rate of naira to US dollar from January 2016 to May 2017. 50
Figure 4.18: PACF plot of (Selling Rate) of Daily exchange
rate of naira to US dollar from January 2016 to May 2017. 51
Figure 4.19: Residuals QQ-Plot of (Buying Rate) of Daily exchange
rate of naira to US dollar from January 2016 to May 2017. 56
Figure 4.20: QQ-Plot of the (Central Rate) of Daily exchange rate of naira to
US dollar from January 2016 to May 2017. 57
