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UNIVERSITY OF GHANA
A COMPARATIVE ANALYSIS OF FORECAST
PERFORMANCE BETWEEN SARIMA AND SETAR MODELS
USING MACROECONOMIC VARIABLES IN GHANA
BY
NAFISAH BADAWIISE AHMED
(10598809)
A THESIS SUBMITTED TO THE DEPARTMENT OF STATISTICS AND
ACTUARIAL SCIENCE, UNIVERSITY OF GHANA IN PARTIAL FUFILLMENT
OF THE REQUIREMENT FOR THE DEGREE OF MPHIL. STATISTICS
July, 2018
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DECLARATION
CANDIDATE’S DECLARATION
This is to certify that this thesis is the result of my own extensive research and reading,
under the guidance of my supervisors and that no part of it has been presented for
another degree in this university or elsewhere.
SIGNATURE:............................. DATE:..........................
NAFISAH BADAWIISE AHMED (10598809)
SUPERVISORS’ DECLARATION
We hereby certify that this thesis was prepared from the candidate’s own work and
supervised in accordance with guidelines on supervision of thesis laid down by the
University of Ghana.
SIGNATURE:.................................. DATE:..................................
DR. FELIX OKOE METTLE
(Principal Supervisor)
SIGNATURE:.................................. DATE:..................................
DR. RICHARD MINKAH
(Co-Supervisor)
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DEDICATION
This work is dedicated to my beloved parents Hajia and Mr. Issah Bawourun Ahmed
and also to my dear cousin Mr. Mohammed Nazif Senje.
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ACKNOWLEDGMENT
This research has had a long development period, to which I am indebted to many
en route. My sincerest gratitude first and fore mostly goes to my supervisors, Dr.
Felix O. Mettle and Dr. Richard Minkah, thanks so much for the fatherly advices and
corrections given me throughout my research. This acknowledgment would not be
complete without the mention of my friends Mr. Osman Sherif, Dr. Nasiru Suleman,
Mr. Abdul Ghanniyu Abubakari, Miss Ama Dickson, Mr. John Essel, Mr. Badi Abdul
Sayibu and Miss Christophelia Adjei-Danso. I am forever grateful for all the help
rendered me in studies and words of encouragement given me each time I needed to
share my frustrations with someone. I pray God grant you all your heart desires. Also,
I am grateful to all the lecturers and staff of the university.
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ABSTRACT
Most macroeconomic variables such as; inflation, GDP and others have been described
by most financial and economics time series analysts to exhibit nonlinear behaviour.
Therefore, to cater for this behaviour, the nonlinear class of models have been largely
adopted to model and forecast such time series. In this study, the Keenan and
Tsay tests for linearity showed inflation and CIC rates follow threshold nonlinear
processes. Hence, the two-regime SETAR model was adopted to accommodate these
nonlinearities in the datasets. Using the linear SARIMA model as a benchmark
for comparative analysis. Results from both in-sample and out- of- sample forecast
performance using MAE and RMSE measures revealed that, the nonlinear SETAR
model outperformed the linear SARIMA model for inflation. This was however
different for CIC rates, since the Linear SARIMA model turned to outperform the
nonlinear SETAR model. Further analysis of forecast accuracy using the Diebold-
Mariano test showed there was no significant difference between the two models for
inflation but, there was significant difference between both models for CIC rates.
Nevertheless, it is recommended that, continuous monitoring of these models, review
market conditions and necessary adjustments are vital to make realistic use of these
models.
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CONTENTS
DECLARATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
ABBREVIATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background of study . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Objectives of the study . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 General objective . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Specific Objectives . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Significance of study . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6 Organisation of Study . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Brief introduction to SARIMA model . . . . . . . . . . . . . . . . . 7
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2.3 Related studies on forecast performance of SARIMA model . . . . . . 8
2.4 Brief introduction to SETAR model . . . . . . . . . . . . . . . . . . 12
2.5 Related studies on forecast performance of SETAR models . . . . . . 13
2.6 Related studies on comparative studies of other linear and nonlinear
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.7 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Research Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Source of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Trend Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Unit Root Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.5.1 Phillips-Perron (PP) Test . . . . . . . . . . . . . . . . . . . . 24
3.5.2 Kwiatkowski-Philips-Schmidt-Shin (KPSS) Test . . . . . . . 25
3.6 Autoregressive AR(p)Model . . . . . . . . . . . . . . . . . . . . . . 27
3.7 Moving Averages (MA(q)) Model . . . . . . . . . . . . . . . . . . . 28
3.8 Autoregressive Moving Averages (ARMA (p,q)) . . . . . . . . . . . . 29
3.9 Autoregressive Integrated Moving Averages (ARIMA) . . . . . . . . 29
3.10 Seasonal Autoregressive Integrated Moving Averages (SARIMA) model 30
3.11 Self Exciting Threshold Autoregressive (SETAR) model . . . . . . . 32
3.12 Linearity Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.13 Ramsey Regression Equation Specification Error Test (RESET) . . . . 34
3.14 Keenan-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.15 Tsay’s F -test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.16 Criterion for Model Selection . . . . . . . . . . . . . . . . . . . . . . 37
3.17 Model Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.17.1 Ljung-Box Test . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.17.2 ARCH-LM Test . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.18 Diebold-Mariano Test . . . . . . . . . . . . . . . . . . . . . . . . . . 40
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3.19 Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.20 Jarque-Bera Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.21 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 DATA ANALYSIS AND DISCUSSIONS . . . . . . . . . . . . . . . . . . 43
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.1 Preliminary Analysis of Inflation rate . . . . . . . . . . . . . 44
4.2.2 Preliminary Analysis for CIC rate . . . . . . . . . . . . . . . 46
4.3 Further Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.1 SARIMA modelling . . . . . . . . . . . . . . . . . . . . . . 49
4.3.2 SETAR Modelling . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.3 Comparative Analysis of forecast between SARIMA and
SETAR models . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 CONCLUSION AND RECOMMENDATIONS . . . . . . . . . . . . . . 70
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
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LIST OF ABBREVIATION
AIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Akaike Information Criterion
AICc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Akaike Information Criterion Corrected
ACF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Autocorrelation Function
ADF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Augmented Dickey-Fuller
ANN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Artificial Neural Network
AR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Autoregressive
ARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . .Autoregressive Conditional Heteroscedasticity
ARCH-LM . . . . Autoregressive Conditional Heteroscedasticity Lagrange Multiplier
ARIMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Autoregressive Integrated Moving Average
ARMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Autoregressive Moving Average
ASEAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Association of Southeast Asian Nations
BIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bayesian Information Criterion
BIVARMA . . . . . . . . . . . . . . . . . . . . . . Bilinear Vector Autoregressive Moving Average
BoG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bank of Ghana
CIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Currency in Circulation
CNB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Czech National Bank
DF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dickey-Fuller
DGP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Generating Process
ECB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .European Central Bank
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EGARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exponential Generalised ARCH
GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalised ARCH
GDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Gross Domestic Product
GNP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gross Net Product
KPSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kwiatkowski-Phillips-Schmidt-Shin
LCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lower Confidence Limit
LSTAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Linear Smooth Transition Autoregressive
MA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Moving Average
MAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean Absolute Error
MAPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean Absolute Percent Error
Max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Maximum
Min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum
MPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Mean Percent Error
MSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean Square Error
OLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ordinary Least Square
PACF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Autocorrelation Function
PP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phillips -Perron
RESET . . . . . . . . . . . . . . . . . . . . Ramsey Regression Equation Specification Error Test
RMSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Mean Square Error
RMSPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Root Mean Square Percentage Error
SARIMA . . . . . . . . . . . . . . . . . . . Seasonal Autoregressive Integrated Moving Average
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SETAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-Excited Threshold Autoregressive
SSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sum of Squared Error
STAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Smooth Transition Autoregressive
Std . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Deviation
TAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Threshold Autoregressive
UCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Upper Confidence Limit
UK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . United Kingdom
VAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vector Autoregressive
VARMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vector Autoregressive Moving Average
ZA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zivot Andrews
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LIST OF TABLES
4.1 Descriptive Statistics of Inflation Rates . . . . . . . . . . . . . . . . . 44
4.2 Descriptive Statistics of CIC Rates . . . . . . . . . . . . . . . . . . . 47
4.3 Unit root test for inflation . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4 AIC, RMSE, MAE and MAPE of candidate seasonal ARIMA Models
for Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 Estimated Parameters of ARIMA(3,1,3)(1,0,1)12 for inflation . . . . 51
4.6 Estimate Parameters of ARIMA(1,1,2)(1,0,1)12 for inflation . . . . . 52
4.7 Estimate Parameters of ARIMA(3,1,3)(2,0,1)12 for inflation . . . . . 52
4.8 Residual diagnostic tests for Seasonal ARIMA models for inflation . . 53
4.9 Unit Root test for CIC rate . . . . . . . . . . . . . . . . . . . . . . . 54
4.10 AIC, RMSE, MAE and MAPE of candidate seasonal ARIMA Models
for CIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.11 Estimated parameters of ARIMA (0,1,0) (3,0,3)12 for CIC . . . . . . . 55
4.12 Estimated parameters of ARIMA(0,1,0)(1,0,3)12 for CIC . . . . . . 56
4.13 Estimated parameters of ARIMA(0,1,0)(1,0,1)12 for CIC . . . . . . 56
4.14 Residual diagnostic tests for Seasonal ARIMA models for CIC . . . . 56
4.15 Linearity test for inflation rate . . . . . . . . . . . . . . . . . . . . . 58
4.16 AIC and MAPE values of tentative SETAR models . . . . . . . . . . 58
4.17 Estimate Parameters of SETAR (2;14,4) for inflation . . . . . . . . . . 59
4.18 Estimate Parameters of SETAR (2;14,5) for inflation . . . . . . . . . . 60
4.19 Estimate Parameters of SETAR (2;14,13) for inflation . . . . . . . . . 61
4.20 Residual diagnostics of tentative SETAR models for inflation . . . . . 61
4.21 Linearity test for CIC rate . . . . . . . . . . . . . . . . . . . . . . . . 62
4.22 AIC and MAPE candidate models for CIC . . . . . . . . . . . . . . . 62
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4.23 Estimate Parameters of SETAR(2;13,9) for CIC . . . . . . . . . . . . 63
4.24 Residual Diagnostics of suggested SETAR models for CIC . . . . . . 63
4.25 Forecast results of competing SARIMA models for inflation . . . . . 64
4.26 Forecast results of competing SETAR models for inflation . . . . . . . 65
4.27 Forecast accuracy test for inflation . . . . . . . . . . . . . . . . . . . 65
4.28 Forecast results of competing SARIMA models for CIC . . . . . . . . 66
4.29 Forecast result of SETAR model for CIC . . . . . . . . . . . . . . . . 66
4.30 Forecast accuracy test for CIC . . . . . . . . . . . . . . . . . . . . . 67
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LIST OF FIGURES
4.1 Time series plot of inflation . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 ACF and PACF plots of inflation rates . . . . . . . . . . . . . . . . . 45
4.3 ACF and PACF plots of differenced inflation . . . . . . . . . . . . . . 46
4.4 Time Series plot of CIC . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5 ACF and PACF plots of CIC . . . . . . . . . . . . . . . . . . . . . . 48
4.6 ACF and PACF plots of differenced CIC . . . . . . . . . . . . . . . . 48
4.7 ACF plots of residuals for inflation . . . . . . . . . . . . . . . . . . . 54
4.8 ACF plots of residuals for CIC . . . . . . . . . . . . . . . . . . . . . 57
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CHAPTER 1
INTRODUCTION
1.1 Background of study
The simple linear model gained popularity in modeling and forecasting most
macroeconomic variables until the 1980’s. Tong (1978) and Tong and Lim (1980)
first proposed the threshold concept of nonlinear class of models after a challenge was
thrown to a group of time series analysts in 1977 during an ordinary Royal Statistical
Society meeting.
For the purpose of this study, macroeconomics is defined as a branch of economics
which deals with the study of the whole economy characterized by variables. Inflation,
Gross Domestic Product (GDP) and unemployment rates are major examples of
macroeconomic variables that have received great attention in most economic and
statistical researches worldwide with Ghana not being an exception. This is so because
macroeconomic variables are the major indicators of an economy. For this reason,
governments, policy makers and investors have developed keen interests in researches
on macroeconomic variables. It is therefore not surprising to hear governments
boasting of attaining single digit inflation, creating jobs especially for the youth
and stabilizing an economy during its regime. Also, it is no lame reason that per
Ghanaian setting, governments appoint a statistician whose sole responsibility is to
constructively advise government on strategies to keep the economy in check and
balance.
According to Marcellino (2007), the World War II crisis socially, economically and
politically had an influence on modelling of macroeconomic variables worldwide. In
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this, many literatures have suggested time varying and non-linear class of models to
be better off in dealing with such macroeconomic variables as compared to linear
specifications in modelling and forecasting.
In a typical Ghanaian market, the dynamics in the business cycle associated with
events such as economic expansion and recession results in discontinuities in a time
series either as a result of fiscal or monetary policies by governments. Researchers
such as Williams and Miller (1990) and Carreno and Madinaveita (1990) proposed
some modifications to deal with these discontinuities. Other researchers such as
Rosas and Guerrero (1994) attempted to cater for discontinuities in a time series data
by examining an exponential smoothing forecast subject to a number of constraints
taking into consideration seasonal variations.
Tsay and Tiao (1984), Campbell and Shiller (2001), Clements and Jeremy (2001) and
Aidoo (2011) have also described most macroeconomic variables as nonlinear and
hence propose the nonlinear class of models as best for forecasting such variables.
However, other researchers such as Gooijer and Kumar (1992) are of divergent view
that there is no clear evidence either in favour of linear or nonlinear models in terms
of their forecast performance.
The fact that time series generated from macroeconomic variables appear irregular by
nature, it is imperative to study, model and detect this asymmetric behaviour. Again,
because of this irregular behaviour of these variables, the simple linear models cannot
be used in their modelling and forecasting to produce accurate results.
Recently, quite a number of nonlinear class of models such as the Artificial Neural
Networks (ANN) and the Threshold Autoregressive (TAR) models, have been
proposed and extensively used for modelling and forecasting time series that appear
not to be linear. These nonlinear models are classified as regime switching models
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but are further divided into “Threshold models” and “Markov Switching models”
depending on the state process evolution between the two models.
The nonlinear models, “threshold models” introduced by Tong (1978) and later
reviewed by Potter (1999) triggered by regime shifts in relation to an unknown
threshold are considered to capture irregularities in the data generation process of a
time series to be used for modelling and forecasting. The Self- Exciting Threshold
Autoregressive (SETAR) models, a special case of the TAR model is accommodating
in terms of structural changes in regimes or discontinuities in the data generation
process. For this reason, the SETAR model is considered in this study to model
some macroeconomic variables in Ghana. Again, according to Marcellino (2007) the
nonlinear ANN model is a very powerful tool for any kind of nonlinear behavior but
they are quite difficult to interpret from an economic perspective. Likewise, seasonal
ARIMA (SARIMA) model is proposed as a benchmark for the study since it is
comparatively easier to model and use for forecasting trended time series at regular
intervals. Researchers such as Swanson and White (1995) have described the in-
sample forecast of the SETAR models to outperform their linear counterpart but cannot
vouch for their out of sample forecast. It is therefore imperative to consider both the in
sample and the out of sample performance of the two models.
1.2 Problem Statement
The Linear class of time series models have received much attention over the years
in both theory and applied research. Although, the linear class of models has widely
been used in several areas of study, they usually do not capture vital features of most
macroeconomic and financial data. Since the business cycle for most economies is
bound to go through either an expansion or recession, data from these variables such
as inflation, unemployment rate and GDP go through a system of structural breaks and
behavioural changes. It is therefore reasonable to assume that each kind of data set
may require a different type of time series model to explain and model it at different
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times.
In detecting and modelling the nonlinear behaviour of a time series, it seems a natural
concept is to consider the nonlinear class of models as proposed by researchers
like Watier and Richardson (1995), Camphell and Shiller (2001), Hsu et al. (2010)
and Gharleghi et al. (2014) to effectively explain irregular behaviourial changes in
macroeconomic and financial time series.
Models that assume the dynamic changes or discontinuities in time series are usually
termed as Autoregressive models. Examples of such models include threshold
autoregressive model, self-exciting threshold autoregressive (SETAR) and smooth
transition autoregressive (STAR) models. The simple autoregressive models are
arguably most frequently used time series models that can be easily estimated and
interpreted. An extension of an AR process to cater for the nonlinear behaviour of
time series makes the entire process nonlinear thereby making the nonlinear models
easy to understand and explain.
Recently, Gharleghi et al. (2014) found that the nonlinear SETAR model outperformed
their linear counterpart, the ARIMA model. According to Aidoo (2011), little of
research has been done in comparing the forecast performance between SARIMA and
SETAR models in an application to Ghana’s inflation rate. It is in this similar spirit that
this research will consider a case study of macroeconomic variables in Ghana. This
research will produce multiple models which will be more accurate in choosing the best
performing model since macroeconomic variables operate under different conditions
though interrelated.
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1.3 Objectives of the study
1.3.1 General objective
To identify macroeconomic variables that satisfy the conditions of SARIMA and
SETAR models and compare their in-sample and out-of-sample forecast performance.
1.3.2 Specific Objectives
i. Identify appropriate linear and nonlinear models for forecasting macroeconomic
variables.
ii. To examine the in-sample forecast performance of both models.
iii. To examine the out- of- sample forecast performance of both models.
1.4 Significance of study
This study will contribute to existing literature for researchers who wish to work on
similar studies in the near future. The study will also identify appropriate models that
will be useful to policy makers at all relevant levels, such as investors, governmental
and non-governmental organisations, in the effective design of economic strategies to
combat expected and unexpected changes in business cycles and also influence their
financial planning.
1.5 Research Questions
i. What is the behavior of these macroeconomic variables?
ii. Which linear SARIMA model is appropriate for modelling each macroeconomic
variable?
ii. Which nonlinear SETAR model is appropriate for modelling each
macroeconomic variable?
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ii. Which of the two models performs better in forecasting?
1.6 Organisation of Study
The structure of the thesis is as follows: Chapter one comprises the introduction
and it caters for the following: background, problem statement, research questions,
objectives of study and organisation of study. Chapter two reviews literature, which
comprises: introduction, brief introductions of SARIMA and SETAR models and
examines existing literature on the performance of both models and other linear and
nonlinear models. It goes further to show what this study seeks to achieve. Chapter
three focuses on the methodology of the study, whereas chapter four deals with data
analysis and results and chapter five deals with conclusions and recommendations of
the study.
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CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
This chapter deals with brief introduction to SARIMA and SETAR models, related
studies on forecast performance of SARIMA model, related studies on forecast
performance of SETAR model and lastly other related studies on comparative forecast
performance between linear and nonlinear models.
2.2 Brief introduction to SARIMA model
The simple linear model Autoregressive Integrated Moving Averages (ARIMA)
has received much attention in modelling and forecasting time series data ranging
from economics, health, agriculture and finance just to mention a few. The ARIMA
model is considered to be a special kind of a regression model which comprises an
Autoregressive (AR) and a Moving Average (MA) component that are dependent on
past values and past errors respectively. A mixture of the AR and the MA models
produces a mixed model called the ARMA model.
The ARIMA model was popularized by Box and Jenkins (1976), the “I” represents a
differencing or an integration of a time series to become stationary. ARIMA models
are comparatively good in modelling and forecasting as compared to other linear time
series models. Researchers such as Meese and Geweke (1984), Marcellino (2007),
Stock and Watson (2003) have confirmed this in their works.
However, these models are not capable of capturing both seasonal and non-seasonal
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patterns in a time series. An extension of these models to capture seasonality is a
Seasonal Autoregressive Integrated moving Average (SARIMA) model. A time series
is described as exhibiting seasonality if there exist a regular pattern in changes that
repeats over a number of time intervals until the patterns repeat again. For monthly
data, even changes usually occurs in specific months in the time series data. SARIMA
models are well known for their statistical modelling and forecasting but are however
unable to extract nonlinear relationships within the time series data.
2.3 Related studies on forecast performance of
SARIMA model
This section reviews existing literature from various fields where the SARIMA model
has been used.
An economics related thesis by Aidoo (2010), revealed that SARIMA(1,1,1)(0,0,1)12
best represent the behaviour of Ghana’s inflation rate considering monthly data
from July 1991 to December 2009 obtained from the Ghana statistical service. In
the study, the best model based on various selection criteria was used to forecast
Ghana’s inflation rate for the next seven months. The forecasted values demonstrated
a decrease in Ghana’s inflation rate for the forecast period with a turning point in the
month of July.
In a related study by Nasiru and Sarpong (2012), SARIMA(3,1,3)(2,1,1)12 was
found to be the most appropriate modelling approach for monthly inflation in Ghana
using Box-Jenkins methodology for the period between January 1990 to January
2012. Fannoh et al. (2014) further revealed SARIMA(0,1,0)(2,0,0)12 to best examine
short term data on inflation from January 2006 to December 2006 for Liberia while
considering seasonality.
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Moreover, Otu et al. (2014) developed a model for forecasting Nigeria’s inflation rate.
In their study, Seasonal ARIMA model was applied to short term monthly inflation
rate collected from November 2003 to October 2013. SARIMA(1,1,1)(0,0,1)12 was
adjudged the best model to forecast inflation rates for Nigeria after meeting all model
assumptions. Forecasted inflationary rates for November 2013 to November 2014
displayed a downwards trend for the first quarter of 2014 but made a U-turn at the
beginning of the second quarter in the same year. Additionally, there was an upwards
trend in inflation values till the month of September. They posited that volatility in
Nigeria’s inflation rates is attributed to many economic factors.
Again, Nasiru et al. (2013) compared the efficiency of ARIMAX and SARIMA
models in predicting monthly currency in circulation for Ghana. Their predicted
results indicated that, both models were good for prediction. Further diagnostic checks
revealed both models were free from conditional heteroscedasticity and high-level
autocorrelations. However, they recommended continuous evaluation of these models
to improve forecasting performance.
Luguterah et al. (2013) proposed SARIMA(0,1,1)(0,1,1)12 as the best model for
predicting currency in circulation for Ghana. They employed Seasonal ARIMA in
their model building and forecasting for Monthly data obtained from the Bank of
Ghana from January 2000 to December 2011. From their results, the model with
minimum values of AIC, BIC and AICc was chosen. Diagnostic checks of the model
residuals with Ljung-Box and ARCH-LM tests showed the model was free from
higher level autocorrelation and conditional heteroscedasticity.
In Agriculture, Akter and Rahman (2010) generated forecasts for milk supply of
dairy cooperative in UK applying seasonal Holt winter’s model and seasonal ARIMA
model. They revealed that the forecasts produced less than three-percent errors and
hence proposed both techniques to be effective in forecasting.
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A health-related study by Martinez and da Silva (2011), resulted in developing
the most appropriate prediction model for incidence of dengue cases in Campinas,
southeast Brazil using SARIMA model in application to reported monthly dengue
data. From their results, SARIMA(2,1,3)(1,1,1)12 was proposed to be the best
model fit in forecasting dengue cases in Brazil after satisfying all model developing
assumptions. By their study, researchers were advised to take into account climate
changes when dealing with data on dengue cases for Brazil in future.
Yet again, a health monitored study by Zhang (2013) predicted the incidence of
typhoid fever in China using SARIMA and three different inspired neural network
models. They applied the various models to data obtained from the Chinese center
for disease control and prevention from 2005 to 2010. Their results showed that the
SARIMA model outperformed the three other models.
An investigative research in the energy sector by Oduro-Gyimah et al. (2012) on
microwave transmission of the Yeji- Salaga link, collated receive signal level data
between January 2006 to December 2010. They found SARIMA(1,1,1)(0,2,2)12
as the best model to predict microwave transmission pattern for 24-months. They
also reported that the SARIMA model was the best model to use for time series with
seasonal complexity.
Another energy sector study by Abledu (2013), pointed SARIMA(1,1,1)(0,1,2)12 as
the best fit model to explain the pattern of short-term energy data collected within the
time frame 2010–2011. The developed model was then used to make predictions for
the year 2013.
In addition, Asamoah-Boaheng (2014) determined an efficient model using seasonal
ARIMA model to forecast average temperature for the Ashanti Region of Ghana with
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data between 1980 to 2013 from the meteorology and climate department. Out of
the three models built in their study, the SARIMA(2,0,2)(2,1,1)12 model had the
least values of AICs and BICs and so was chosen as most adequate for predicting
average temperatures. There was neither a decreasing nor increasing pattern in the
average temperatures but there existed some amount of seasonality in the data after
decomposition of the time series.
While examining the rainfall pattern by Abdul-Aziz et al. (2013) in the Ashanti
region of Ghana, the study revealed rainfall patterns changes over time using data
from 1974 to 2010. Their results further showed a slight decrease in rainfall figures
from August to December months and also that, there was likely to be an increase
in rainfall figures from February to march months. Afrifa-Yamoah et al. (2016)
too, examined the rainfall pattern for Brong Ahafo Region of Ghana and proposed
SARIMA(0,0,0)(1,1,1)12 as the best model to forecast rainfall patterns for the region.
Their results revealed increases in rainfall figures for the months of September and
October and minimum figures for January, February and December months for the
region using data from 1975-2009.
Recently, Dritsaki (2016) examined the precision of forecasting of Greece’s
unemployment rate following procedures of Box-Jenkins methodology and
demonstrated that SARIMA(0,2,1)(1,2,1)12 had the best predictive power and hence
was used to forecast unemployment rates for Greece. Correspondingly, Mahipan et al.
(2013) compared SARIMA and ANN models and found SARIMA(0,1,1)12 to be the
best predictive model for unemployment rates in Thailand after satisfying all model
assumptions. They suggested that the ANN model did not fit the unemployment data
adequately hence the choice of the SARIMA model for them.
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2.4 Brief introduction to SETAR model
Tong in an editorial report, recounts an ordinary royal statistical society meeting in
1977 presided over by the then David Cox. He explained the need to revisit all the
lynx data models widely researched by Mr. Campbell, Dr. Tong and Professor Walker
since they are time reversible. In addition, Dr. Granville Tunnicliffe asked, “would we
not prefer a model which would exhibit stable periodic deterministic behaviour a limit
cycle?” such limit cycles cannot arise from linear models. Long before this meeting,
there hardly existed the concept of nonlinearity in time series since time series at that
time operated on the assumption of linearity.
As events unfolded during the meeting, a challenge was thrown to all time series
analysts to go find a practical explanation to the chaotic nature of time series like data
for lynx which appeared not to be linear. This task was indeed a daunting one, since
the concept was new and therefore needed much experimentation.
Tong determined on raveling the mystery of this “non-linearity”, focused on data of
animal population and river flow. He observed two advantages for these data sets.
Firstly, a better insight to the dynamics underlying the data. Which indeed was a
revelation to Dr. Granville’s comments on “limit cycles”. Secondly, specific sets
of data were needed to constantly check the methodology if it headed in the right
direction. Keen on his quest to finding a better explanation to nonlinearity, between
the years 1977 -78, asked his research student, Lim to carry out a task using simulation
but she misunderstood and showed the results obtained by recursion of a SETAR
model. This actually was a blessing in disguise as it offered Tong the opportunity to
have a glimpse of the SETAR model for the first time. By the later part of 1980, after
reading widely and huge experimentation Tong wrote a paper entitled “Threshold auto
regression, limit cycles and cyclical data” to a UK journal but unfortunately for him it
was rejected. Though saddened, he tried the platform of the Royal Statistical Society
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and fortunately for him the paper was accepted. Although accepted, there was quite a
lot to be done to smoothen the rough edges like “how to choose the threshold variable”
in a time series. As a matter of fact, this threshold concept was a very broad one and
therefore needed further research to explain adequately the behaviour of a time series.
This therefore gave other researchers like Watier and Richardson (1995) and Clements
and Smith (1999) to research further on nonlinear models.
Narrowing the concept of this threshold Autoregression to internal changes within a
time series brought about “SETAR”. The SETAR model actually is a typical example
of the threshold Autoregressive (TAR) model. The threshold model deals with general
discontinuities in a time series unlike the SETAR model, internal discontinuities in
a time series derives the term “self- exciting”. A set of different linear AR process
characterized by regime switches according the value of their threshold makes the
entire process a nonlinear process.
2.5 Related studies on forecast performance of SETAR
models
Quiet a number of literature exists in various areas where SETAR models have been
applied. This section makes a casual review of such literature ranging from health,
agriculture to economics just to mention a few.
Investigating several approaches to obtain multi-step forecast ahead for SETAR
models, Clements and Smith (1997) based their study on existing literature of h-step
ahead forecast values. In their study, they compared the forecast performance of
SETAR models to AR models using various methods of simulation for US Gross
National Product (GNP). Their results expressed some doubt on the robustness of
SETAR models over two sample periods and on the size of the forecast accuracy
there was no doubt SETAR models could deliver. They therefore concluded that
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improvements for forecast performance of linear AR model depended largely on
historical events. They also concluded that, the type of nonlinearity in a time series
contributes to the out of sample forecast performance.
Further investigations by Feng and Liu (2002) revealed features of nonlinearity in
Canadian GDP. In their study, they compared the out of sample forecast performance
of non-linear SETAR model and a standard linear ARIMA model. Employing
techniques for forecasting like the1- step forecast ahead and multi-step forecast ahead,
they revealed that both models appears satisfactory in terms of their MAPEs and
RSMPEs. As a result, they proposed both models for predicting the Canadian GDP.
They however are of the view that, it is more practical to use the multi-step technique
in real life situations.
For more than forty years of existence and application of nonlinear models, Clements
et al. (2004) reviewed the current state -of- art of these models ranging from
estimation, evaluation and selection of best forecasting models among others for
some economic and financial time series. They argued that, evidence of non-linearity
in these time series are patchy and hence proposed a variety of future topics for
upcoming researchers.
In an economics related study, Christos (2005) evaluated the performance of
competing models for UK’s unemployment rate from January 1971 to December
2002. Firstly, it was found that, MA(4)− ARCH(1) provided better forecasts for
unemployment rate. On the other hand, two forecasting samples pointed that MA(4)
model does well, whereas both MA(1) and AR(4) proved to be the best forecasting
models for the other two forecasting periods. Hence, an empirical evidence suggests
that there exist a close relationship between forecasting theory and labour market
conditions. Further findings bring to the conclusion that, forecasting methods are
nearer to the realities of UK labour market.
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Another economics related study by Marcellino (2007), evaluated the relative
performance of a bootstrap algorithm to standard linear models in an application to
bootstrap series of GDP and inflation. Considering a variety of evaluation criteria, the
standard linear models was suggested to be hardly beaten in prediction if only they are
carefully specified.
Similar economics study by Sjoberg (2010), compared the forecast performance of
ARIMA, LSTAR and SETAR models using monthly Swedish Industrial production
from 28 branches over the period January1990 to December 2008. In terms of
their mean square error of prediction, the ARIMA model was adjudged the best
model for majority of the industrial branches. For some industrial branches and in
a few branches, the LSTAR and SETAR models also outperformed the ARIMA model.
Equally, Gharleghi et al. (2014) evaluated the prediction performance of SETAR-
EGARCH, ARIMA-EGARCH and AR-EGARCH models by employing data from
three ASEAN currencies, namely; Indonesian Rupiah, Malaysian Ringgit and Thai
Baht. They revealed that, the non-linear SETAR–EGARCH model fits better than the
other two models considering factors like the data generation process and therefore
concluded that, the data generation process of the variables is nonlinear.
van Ruth (2014), further applied several linear and non-linear models to monthly
inflation rates for Netherlands over the period 1970-2009. In an empirical approach,
the nonlinear STAR model was revealed to outperform all other linear and nonlinear
approaches to explain the dynamics of inflation in the Netherlands.
A comparative study by Gibson and Nur (2011) investigated the performance of a
two regime AR and STAR models. From their results they revealed that, the in-
sample forecast of the STAR remains problematic since it properly does not replicate
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the observed behaviour of the time series like the SETAR model does. They further
revealed that, the SETAR model outperforms the STAR model in terms of their out- of
-sample forecast.
Another comparative study by Akeyede et al. (2015), compared different forecasting
methods proposed in existing literature to obtain h-step ahead forecast. They further
compared the multi- step forecast performance of linear and nonlinear time series
via method of simulation. In conclusion, they proposed AR model to be best for
forecasting AR functions and that LSTAR model supersedes all other non-linear
forecasting methods except for polynomials. SETAR models beats all other models in
terms of their prediction ability.
Likewise, Badawi (2016) in an unpublished master’s thesis, compared the forecast
performance between SARIMA and SETAR models using data on pneumonia cases
in the northern region of Ghana. The data was obtained from the Tamale Teaching
Hospital over the period January, 2000 to October, 2015 based on the forecast values
from the two models, the SETAR model was adjudged the best in terms of their MSE,
RMSE and also other forms of criterion selection processes.
Again, Karlsson and Karlsson (2016) modelled Sweden’s unemployment rate by
exploiting model efficiency of SARIMA, SETAR and VAR models. As a measure of
goodness of fit test, they concluded that, the SARIMA and SETAR models performs
better than the VAR model in terms of their out of sample forecast using quarterly
unemployment data within the life span 1983 to 2010 from OECD. They further
revealed that, short term forecasts are better off than longer term forecasts.
Analysing structural changes in the Taiwan stock market, Hsu et al. (2010) found
the SETAR model to better examine the out of sample forecast of the non-linear
time series. Employing monthly data from January 2005 to December 2009, the best
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model was chosen by performing a unit root test and comparing the out of sample
forecast between the standard linear ARIMA model and the nonlinear SETAR model.
A graphical view of their results showed that the structural change in the time series
occurred in the month of June 2008. Thereby they built a 2–regime SETAR model. In
conclusion the nonlinear SETAR model was found to be superior in forecasting to the
linear ARIMA model in the Taiwan stock market.
Contrary to the norm of comparing the performance of models and prediction, Öznur
and GÜNERݧ (2015) developed a hybrid models whose components are either
parametric or non- parametric models. They used the technique of back fitting
algorithm based on smoothing spline to producer price index in Turkey. Their results
showed that, the hybrid model AAR and SETAR performs best amongst all other
hybrid models.
Recently, Zak (2017) analysed the forecast performance of nonlinear models in
application to Czech Republic’s exchange rate against EUR. Secondary data from
three sources namely; Czech National Bank (CNB) Board decisions minutes,
Statistical Data Warehouse of European Central Bank (ECB) website and from the
press releases of Governing Council of ECB was obtained within the periods 1999 to
2016 for the study. Employing SETAR models and simple random walk they showed
that the simple random walk model out performs the complex SETAR model. This
further supports the assertion exchange rates are generally difficult to forecast.
Again, Fırat (2017) applied SETAR modelling process to explain the non-linear pattern
in modelling currencies in EUR/USD, EUR/ TRY and USD/TRY parities. The study
compared the forecast performance of the observed values for both linear and nonlinear
models. It was concluded that, the non-linear SETAR model, outperformed the other
forms of nonlinear and linear models.
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2.6 Related studies on comparative studies of other
linear and nonlinear models
In the field of engineering, Rodas et al. (2017) compared forecast performance
between linear and nonlinear models in application to a DC motor. In their study,
they adopted two criteria for measuring the performance of both models namely;
tracking error and the control effort. Their results indicated that, the linear controller
presents a significant advantage over their nonlinear counterpart when experimenting
at operation speeds very close to the operation argument.
A medical evaluation by Murphy and Dietrich (2006), assessed two classes of control
loop algorithms namely; the linear adaptive filter and the adaptive nonlinear neural
networks for highly irregular patient behaviours. From their discussion, they explained
the nonlinear neural networks demonstrated robust adaptability to all of the observed
breathing patterns of patients but their linear counterpart failed.
Motivated by the need to establish a Vector Autoregressive Moving averages
(VARMA) model, Iwok and Etuk (2009) developed a general bivariate Vector
Autoregressive Moving Averages (BIVARMA) model. In their model development,
they graphically compared the two competing models based on their estimates and
residual variances. Furtherly, they established the fact that, the BIVARMA model
performed better than the VARMA model.
Other aspects of human life are not left out in the application of linear and nonlinear
models as demonstrated by Isaac et al. (2016) in modelling road accidents in
Ghana. They used count regression models in their study as they compared several
models like Poisson, Negative Binomial and Conway- Maxwell-Poisson to fit count
data expected in the field of transportation. The best models were then chosen
using Akaike Information Criterion and Bayesian Information Criterion. From their
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results, the Negative Binomial model appeared to perform better than the other models.
A number of peer review researches showed the existence of rich literature on
macroeconomic variables. For instance, van Gysen et al. (2013) comprehensively
compared the predictive accuracy of linear and non-linear models in forecasting
financial returns. In their study, they considered varieties of macroeconomic variables
from Johannesburg’s stock exchange and their results showed that markov switching
models provided the best in- sample fit. Nevertheless, they are of the view that, the
out- of-sample forecast of the mixed models, EGARCH linear models and 2-state
dynamic regression models outperform their alternative models.
As well, Cuestas et al. (2011) considered inflation for a study by applying non-linear
and fractional integration as a hypothesis of empirical inflation for a pool of African
countries. Their discussions revealed that, most African countries do not support the
hypothesis. They therefore proposed that, the LSTAR models are more stable.
Furthermore, Hadrat et al. (2015) compared performance between Artificial Neural
Networks (ANN) and the traditional Autoregressive and Vector Autoregressive in
application to Ghana’s inflation rate. Their results indicated that, ANNs had less errors
than their traditional counter parts. In conclusion, the ANNs models were suggested
to accurately forecast Ghana’s inflation.
Several inflationary studies in Ghana has found nonlinear models to be the best model
but Antwi (2017) is of a divergent view. Firstly, the existence of nonlinear pattern
in monthly inflation for Ghana using nonlinear models was applied. Secondly, the
threshold models; SETAR and LSTAR were applied on the monthly inflation data
obtained from the Ghana statistical service within the period January 1981 to August
2016.
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Finally, the performance of the threshold models was compared to AR(1) and AR(2)
models using minimum values of AIC and BIC. In conclusion, both SETAR and
LSTAR models fitted the data best but in terms of their forecasting ability the simple
AR models outperformed their non-linear alternatives. Also, it was revealed that
Ghana’s inflation was likely to experience double digit in the year 2017.
Lastly, investigations by Nchor et al. (2016) established a linear and nonlinear
relationship between macroeconomic variables and oil prices in Ghana. They
considered six variables namely; real industry value added, real government
expenditure, the real effective exchange rate, real oil price, real imports and inflation.
Vector Auto regression (VAR) and Vector Error Correction Model (VECM) used in the
study pointed that, irrespective of a positive or a negative change in oil prices, there
definitely would be an impact on macroeconomic variables.
2.7 Chapter Conclusion
This chapter has comprehensively dealt with review of literature from other works
relevant to the study. Review of these literatures offered the opportunity to encounter
diverse methodology employed by researchers to forecast and predict various forms
of macroeconomic variables, health related cases, climate related cases and more.
However, in most of the studies, various forms of linear and nonlinear models such
as ARIMA and SETAR, SARIMA and ANN’s have extensively been used. In the
case of Ghana, little has been done in relation to applying macroeconomic variables to
compare their forecast performance. Based on this research gap identified, this study
employed the Seasonal Autoregressive Integrated Moving Average (SARIMA) models
and Self Excited Threshold Autoregressive (SETAR) models in predicting inflation and
Currency in circulation (CIC) of Ghana
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CHAPTER 3
METHODOLOGY
3.1 Introduction
This chapter presents the methods and techniques used to achieve the objectives of
the study. It contains the statistical techniques, analytical procedures and theories that
were employed to achieve the stated objectives of the study. Similarly, the chapter will
be sectioned into source of data, unit root test, trend analysis, AR(p) model, MA(q),
mixed ARMA(p,q) model, ARIMA(p,d,q) model, SARIMA(p,d,q)(P,D,Q)s model,
SETAR(2; p,d) model, linearity test, model estimation, model diagnosis and Diebold-
Mariano test.
3.2 Research Design
This study takes the form of an experimental approach since this concept is imperative
to forecasting phenomenon and research design to explain better the possible trends in
the datasets to be considered.
3.3 Source of Data
The study obtained secondary datasets of two macroeconomic variables; namely
inflation and currency in circulation rates spanning from January 1990 to August 2016,
from the website of the Bank of Ghana (BoG). An analytical method was employed
for the description of these macroeconomic variables and also for comparative
performance of the nonlinear SETAR and linear SARIMA models using R statistical
software.
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3.4 Trend Analysis
Generally, the long- term change in time series levels over time longer than a year
tends to reflect a trend in the time series. Linear, exponential, nonlinear and quadratic
trends are some types of trends exhibited by the time series. This study will evaluate
the performance of the aforementioned trend models in forecasting inflation and
currency in circulation. These models are however, presented either as additive or
multiplicative. The linear trend in the additive model does not depend on the trend
unlike the multiplicative model which intensifies in its variation as the trend increases.
If the trend in the time series is a linear function of time then,
Z(t) = β0 +β1t + εt , t = 1,2, . . . ,n. (3.1)
The logarithmic representation of the linear trend is
logZ(t) = β0 +β1t + εt . (3.2)
And also, for an exponential representation of the series
Z(t) = eβ0+β1t+εt . (3.3)
where Z(t) represents observations of the time series, time (t), (t = 1,2, · · · ,n− 1,n)
and εt is the error term, whereas β0 and β1 are constants.
If the series shows a parabola shape then it exhibits a quadratic trend represented as
Z(t) = β0 +β1t +β2t2 + εt . (3.4)
If β1 < 0, then a plot of the series eventually turns to negative. But if β2 > 0, then a
plot of the series sky-rockets up.
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For a polynomial of order k,
Z(t) = β +β t +β t2 + · · ·+β tk0 1 2 k εt . (3.5)
Equations (3.1) to (3.5) were obtained by the application of the concept of ordinary
least squares (OLS).
3.5 Unit Root Test
As described by Aidoo (2010) and Nasiru et al. (2013), most time series of
macroeconomic variables exhibit trending behaviour in their mean, autocorrelation
and variance through graphical plots of their series. By this, the time series turns
to be non-stationary if the coefficients of the estimated model are in absolute value
less than one. This means that, if the roots of the characteristic equation lie outside
the unit circle, then the series is considered stationary. Principal examples of such
macroeconomic variables include; inflation, exchange rates, stock prices and currency
in circulation.
An important task to dealing with such macroeconomic variables is to first check for
stationarity in the dataset, this idea is what is called a unit root. The unit root test is
usually performed to check for the existence of a stochastic or a deterministic trend in
the series. In practice the ARMA model is non-stationarity, but attains stationarity if
transformed and differenced. This process brings about the integration factor (I) in the
term ARIMA model, this in reality makes the model stationary and effective for any
forecast to produce accurate results.
A good number of time series related research have employed graphical and statistical
tests in checking for the presence of unit root in a series. Dickey-Fuller and Augmented
Dickey-Fuller (DF-ADF), Kwiatkowski, Philips, Schmidt and Shin (KPSS), Zivot
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Andrews (ZA), Phillips-Perron tests are typically used examples for statistical tests. In
the graphical approach an Autocorrelation Function (ACF) plot is observed and if there
exists a strong and slow decaying ACF, then it suggests a deviation from stationarity.
Hence, the series must be transformed and differenced to attain stationarity. However,
one needs to be cautious when differencing a non-stationary series since over
differencing may exclude relevant information to building an appropriate model for
any meaningful predictions. To determine seasonal and non-seasonal behaviours in a
series, various forms of tests must be conducted under both seasonal and non-seasonal
parts each using their ACF and PACF plots. Relevant tests to this study will be
discussed below.
3.5.1 Phillips-Perron (PP) Test
The Phillips- Perron test has received much attention in analysis of most financial time
series. This unit root test is quite different from the Augmented Dickey Fuller (ADF)
test in terms of their approach in dealing with errors arising from serial correlation and
heteroskedasticity. Also, this test unlike the ADF test, makes use of non-parametric
procedures in their test regression. By so doing, they correct any form of serial
correlation in the test regression by modifying the Dickey-Fuller test statistic. The
test regression for the PP tests can be written as follows:
∆yt = βDt +πyt−1 +µt (3.6)
where ∆ is a difference operator, β is a constant, D(t) is a time trend, π is the dickey
fuller test statistic and µt is an error term.
Under the null hypothesis π = 0, this means non-stationarity.
By modifying the Dickey-Fuller test statistic tπ=0 and Tπ , the statistics zt and zπ are
formed. Just like the ADF test statistic, these modified statistics are asymptotically
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distributed and normalized bias with a limiting distribution.
( ) 1 ( ) ( )
σ̂2 2 1 λ̂ 2− σ̂2 2
Zt = ·
T SE(λ̂ )
tπ=0− · 2 (3.7)λ̂ 2 2 λ̂ 2 σ̂
− 1 T
2SE(π̂)
Zπ = T π̂ (λ̂ 22 − σ̂
2) (3.8)
2 σ̂
with parameters λ̂ 2 and σ̂2 as consistent estimates of sample and long-run variances
respectively.
T
σ2 = lim T−1 E[µ2] (3.9)
T−→∞ ∑ tt=1
T
λ 2 = lim ∑ E[T−1S2T ] (3.10)T−→∞ t=1
(3.11)
where ST = ∑Tt=1 µt . The estimators σ̂2 and λ̂ 2, are consistent residual estimates (µt)
of the sample and the long-run variances respectively.
Although, the ADF has some advantages over the PP test, the PP tests are relatively
good in dealing with large samples because of their asymptotic assumption, their
robustness to any form of heteroskedasticity in error terms makes it more advantageous
than the ADF test. Furthermore, there are no mandatory specification for lagged values
as in the case of the ADF.
3.5.2 Kwiatkowski-Philips-Schmidt-Shin (KPSS) Test
The KPSS test is a complementary unit root test for testing a null hypothesis whose
observable time series is trend-stationary against an alternative hypothesis of a unit
root, this however is the opposite hypothesis for most unit root tests. In addition, the
existence of a unit root does not guarantee stationarity but, by design, trend stationary.
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Thus, it has become imperative to distinguish between some behaviour of time series
which may be non-stationary, non-unit root yet trend- stationary. If the data generating
process assumes no linear trend, then
Yt = zt + εt (3.12)
where zt is a random walk i.e. zt = zt−1 +µ 2t , and µt is a random error term ∼ (0,σv )
assumed to be i.i.d and ε is also a stationary error.
The hypothesis to tested is as follows;
H 20 : σv = 0
H1 : σ2v > 0.
If H0 is true, then it means that, Yt becomes a stationary process since it has components
of stationary εt and constant terms. The test statistics proposed by Kwiatkowski et al.
(1992) is given below:
1 T s2
KPSS = ∑ tT 2 σ̂2t=1 ∞
where T is the number of observations, st = ∑ti=1Ŵi with Ŵi = Yi− Ŷ and σ̂2i ∞ is an
estimator of σ2∞ = T−1Var(∑Tt=1 εt) is a long-run variance estimator of the εt process.
The observable time series yt attains stationary with an integration of order one (I(1))
and the denominator quantity in KPSS statistic is an estimator of its variance, which
has a deterministic limit. The term in the denominator ensures that overall; the limiting
distribution is free of unknown nuisance parameters. If, however, it is integrated of
order one (I(1)), the numerator will grow without bounds, causing the statistic to
become large for large sample sizes. The null hypothesis of stationarity is rejected
for large values of KPSS.
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3.6 Autoregressive AR(p)Model
A time series yt is said to be an autoregressive model if future values of the series
depend on past values. Where there exists some amount of correlation between
preceded and succeeded values of the time series, then the model becomes more
effective for prediction. The word “auto” from the term “autoregressive” meaning
“self” only considers past values of a time series to model its behaviour. The AR
model is measured by a simple linear regression model which usually comprises a
time series regression of past values plus an error term. The order of AR model is
paramount to model building since it measures the immediate preceding values in the
time series to be used to predict the succeeding value. In that case, if the preceding
model is of first-order or order one, then it is written as AR(1) and represented as:
yt = φ0 +φ1yt−1 + εt . (3.13)
For second order, AR(2)
yt = φ0 +φ1yt−1 +φ2yt−2 + εt . (3.14)
More generally, for order p, AR(p)
yt = φ0 +φ1yt−1 +φ2yt−2 + · · ·+φpyt−p + εt . (3.15)
where observations of the time series are denoted by the y’s, εt is the random error
at time period t, the terms t − j and φ j are the AR parameters to be estimated with
j = 1,2, · · · , p as AR orders. For simplicity, the backward shift operator B, such that
By 2t = yt−1, B yt = yt−2 and so on is used to simply the above general model of order
p into:
φ(B)yt = εt
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where φ(B) = 1−φ1B−φ B22 −·· ·−φpBp is a polynomial of order p.
For a stationary AR(p) model,
y = φ−1t (B)εt
converges to zero.
3.7 Moving Averages (MA(q)) Model
A time series yt is said to be a moving average process if present values of the series
depends on past errors or white noise values. Usually this model is classified as a
“naive” model due to its ability to stabilise datasets by creating another dataset with
less variations and also it does not show traces of seasonal trend in the dataset. For
this reason, they are designed to provide smoothing effect within a time series which
may be relevant or not to future estimations of a variable. But more importantly, it
provides a “tidier” dataset for estimation.
The general form of the MA model is in the form:
yt = θ1εt−1 +θ2εt−2 + · · ·+θqεt−q + εt . (3.16)
where θ j are the model parameters to be estimated, with j = (1,2, · · · ,q), q is the
order of the model and ε is the error term assumed to be i.i.d with mean zero and
constant variance σ2. No matter the independent average values of the error terms, the
MA process always assumes stationarity.
Equivalently, using the backward shift operator (B), this can be written as:
yt = θ(B)εt
where θ(B) = 1+ θ1B+ θ2B2 + · · ·+ θ Bqq is a polynomial in B of order q, and as
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usual εt is the normally distributed error term with mean zero and a constant variance
σ2. If the characteristic roots of the polynomial mq + θ mq−1 + θ q−21 2m + · · ·+ θq is
an invertible MA(q) process equals to zero, then the absolute terms are less than one.
3.8 Autoregressive Moving Averages (ARMA (p,q))
The mixed model ARMA(p,q) is a linear combination of the Autoregressive (AR) and
Moving Average (MA) model time series models, where p and q are orders of AR and
MA processes respectively. Mostly the ARMA(p,q) model is given as:
yt = φ1yt−1 +φ2yt−2 + · · ·+φpyt−p + εt +θ1εt−1 +θ2εt−2 + · · ·+θqεt−q
where φi and θ j are the respective AR and MA parameters to be estimated with
i = 1,2, · · · , p and j = 1,2, · · · ,q, εt’s are the i.i.d random errors with mean zero and a
constant variance σ2.
For simplicity, the backward shift operator (B) is used to express the ARMA(p,q)
model as:
φ(B)yt = θ(B)εt
with φ(B) and θ(B) as polynomials in B of orders p and q. The ARMA(p,q) process
is said to be stationary if the roots of the AR component of the characteristic function
converges to zero and also, if the polynomial of the MA component is less than one in
absolute terms, then the process is described to be invertible.
3.9 Autoregressive Integrated Moving Averages
(ARIMA)
The ARIMA model has been extensively used in theory and practice to study and
forecast the behaviour of most time series variables with macroeconomic variables not
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being an exception. In general, they are assumed to be the stationary form of ARMA
models by transforming and differencing the time series. However, when differencing
the model to become stationary, care should be taken so as not to over difference
the series otherwise the data might not fit well into the model and hence results in
inaccurate predictions. What the ARIMA model actually does is to remove any form
of “noise” from both the AR and MA components of the model to make it stationary
and then predicts future values with the transformed model. The non-seasonal ARIMA
model is basically denoted as ARIMA(p,d,q) model, where the parameters p,d,q are
the AR, differenced and MA terms greater than or equal to zero respectively.
The general form of the ARIMA model using the backward shift operator is:
φ(B)(1−B)dyt = θ(B)εt .
3.10 Seasonal Autoregressive Integrated Moving
Averages (SARIMA) model
A time series model is said to be a seasonal ARIMA model if it has the structure of an
ARIMA model with two components to capture seasonal and non-seasonal parts of the
model. When a time series is found to exhibit seasonal patterns, it is usually advisable
to carefully identify models that are relevant to capture this behaviour. The Seasonal
ARIMA model is usually denoted in the form ARIMA(p,d,q)∗ (P,D,Q)s where;
p is the number of non-seasonal AR component
q is the number of non-seasonal MA component
d is the order of non-seasonal differencing
P is the number of seasonal AR component
Q is the number of seasonal MA component
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D is order of seasonal differencing
s can either be represented monthly as, s = 12, or on quarterly basis, s = 4 and
so on.
Using the backward shift operator (B), the Seasonal ARIMA is given as:
φ(B)Φ(B)s.
When a time series data exhibit seasonal behaviour, the ARIMA model is usually not
able to capture the behaviour along the seasonal part of the series, hence, the tendency
for wrong order selection for the non-seasonal component. Identification of relevant
models and inclusion of suitable seasonal variables are therefore necessary when a time
series data exhibit periodic patterns. The SARIMA model therefore has the advantage
of capturing both seasonal and non seasonal components. The general SARIMA model
can be expressed using the backward shift operator (B) as:
φ(B)Φ(B)s(1−B)d(1−B)sy = θ(B)Θ(Bst )εt .
where
φ(B) = 1−φ B−φ B2−·· ·−φ Bp1 2 p ,
Φ(Bs) = 1−Φ s1B −Φ 2s2B −·· ·−φPBPs,
θ(B) = 1+θ B+θ B2 q1 2 + · · ·+θqB ,
θ(Bs) = 1+θ1Bs +θ2B2s + · · ·+θ QsQB ,
and
yt is the time series observation at time period (t),
B is the backward shift operator,
εt is a sequence of error term with mean zero and constant variance (σ2),
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φ(i) and Φ j are the non- seasonal and seasonal AR components respectively,
θi and Θ j are the non- seasonal and seasonal MA components respectively.
3.11 Self Exciting Threshold Autoregressive (SETAR)
model
An extension of an AR model results in a special case of the general Threshold
Autoregressive (TAR) model called the SETAR model. This model was first introduced
by Howell Tong in 1977 through a seminal paper and later by Tong and Lim (1980).
Also, this model explains better the data generation process of a time series especially
where changes in behaviour occur from one regime to another. Typically, these models
are employed to understand and predict regime switching behaviours for parameters
of higher orders with flexibility. The switch from one regime to another depends on
the past values of the time series (hence the Self-Exciting portion of the name). This
model is usually presented in the form (k, p) where k indicates the different regime
changes, p also indicates the order of AR. Furthermore, this model can be thought of
as a set of different AR models changing according to their lagged values or threshold
variables triggered by changes in a delay parameter d. The two regime SETAR model
of order p as given by Boero and Marrocu (2004) can be presented as follows:
 (1) p(1) (i) (1)
yt =
φ0 +∑i=1 φ1 yt−i + εt if yt−d ≤ τ
(3.17)
φ (2) ∑p(2) (i) (2)0 + i=1 φ1 yt−i + εt if yt−d > τ
where φ (1) and φ (2)i i are coefficients in lower and higher regimes respectively needed
to be estimated, τ is the threshold value, d is the delay parameter, yt−d is the threshold
variable which describes the switch from one regime to another regime, p(1) and p(2)
are orders of AR in lower and higher regimes respectively and εt’s are random error
terms which are i.i.d with mean zero and constant variance σ2. One major advantage
of the nonlinear SETAR model is its ability to capture the irregular behaviour of a time
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series which cannot be captured by the “naïve” linear models. In this study, the two-
regime SETAR model of the simplest form (2; p;d) will be considered and also the
procedure discussed by Franses and van Dijk (2000) will be used for effective model
selection, furthermore the basic approach and tests for modelling SETAR model will
be discussed below.
3.12 Linearity Test
For an observable time series yt , the SETAR model can only be applied if the series
under consideration is found to be nonlinear or irregular in nature under the hypothesis:
H0 : linearity exists
H1 : nonlinearity exists.
Sometimes, in testing for a certain nonlinear feature in the Data Generating Process
(DGP) of a series, the H1 is specifically pre-chosen but in some cases, the H1 is
relatively general.
In order to apply the SETAR model, one must first check for the existence of nonlinear
behaviour in the series, if there exists any, then an appropriate linear AR(p) model is
specified to test for nonlinearity in the series. A discussion paper by Franses and van
Djik revealed that, the maximum lag order is based on the choice of the minimum
AIC value in an AR(p) model. After careful determination of the linear AR(p), then
various forms of linearity tests such as; Tsay F-test and Keenan test, a special case of
the Ramsey Regression Equation Specification Error Test (RESET) test will be used to
test for linearity. Below are brief discussions of the tests to be used in this study.
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3.13 Ramsey Regression Equation Specification Error
Test (RESET)
Ramsey (1969) proposed linear least square regression analysis for specific tests based
on the view that, nonlinearity occurs in the diagnostics of a fitted model only if the
linear model residuals are correlated in some power terms. Thus, this test focuses on
specific errors in the linear regression including those from unmodeled nonlinearity
and its readily applicable to linear AR(p) models.
Now consider a linear AR(p) model:
yt = φ0 +φ1yt−1 +φ2yt−2 + · · ·+φpyt−p + εt .
In the first stage in a RESET test, the least square estimate φ̂ , the error term which
is denoted by εt = yt − ŷt and sum of square residual given by SSR0 = ∑n 2t=1+p ε̂t are
obtained.
Also, in the Second stage, we consider the linear regression
εt = X ′t−1b+Y
′
t−1d +Vt
where Xt−1 = (1,X 2 s+1t−1, · · · ,Xt−p) and Yt−1 = (Xt , · · · ,Xt ) for s≥ 1.
Then the least square residuals are computed as follows;
V̂t = ε̂t−X ′t−1b̂−Y ′t−1d̂.
Thirdly, SSR1 is computed as:
n
SSR1 = ∑ V̂ 2t
t=1+p
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For an adequate AR(p), b and d parameters must be zero. This can be tested in the
final stage by the usual F statistic given by:
(SSR0−SSR1)/gF = with g = s+ p+1
(SSR1)/(n− p−g)
which under normality and linearity has Fg,n−p−g.
3.14 Keenan-test
Considering an irregularly behaved observation, Keenan (1985) proposed nonlinearity
test for such time series. This test can be well- thought as a special case of
the Ramsey Regression Equation Specification Error Test (RESET) as discussed
above. Precisely, a modification of the second stage of the RESET test using X2t avoids
any form of multicollinearity between X2t and Xt−1. Particularly, Keenan assumes that
the series can be approximated by the Volterra expansion as follows:
∞ ∞ ∞ ∞
Xt = µ + ∑ ∑ θuεt−u + ∑ ∑ θuvεt−uεt−v
u=−∞ v=−∞ u=−∞ v=−∞
where µ is the mean level of the nonlinear observation Xt , with the error terms εt−u and
εt−v. The idea of the Keenan’s test is of a similar view as the F-test and so clearly from
linear approximation ∑∞ ∞u=−∞ ∑v=−∞ θuεt−u = 0. Similarly, the Keenan test follows the
same procedure used in the RESET test above to obtain fitted estimate (X̂t), error terms
(ε̂t) , the residual set (ξt)and SSR. And lastly, we calculate
∑n
η t=p+1
ε̂t ξ̂t
= .
∑n ξ̂ 2i=p+1 t
The test statistic of the Keenan test is:
2 (n−2p−2)F̂ = η
SSR−η2
where;
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P = lag order of the autoregressive process
n = Sample size considered
SSR = Sum of Square Residual of the AR(p) process
and the null hypothesis;
∞ ∞
H0 : ∑ ∑ θuεt−u = 0
u=−∞ v=−∞
with assumption that εt’s has zero mean and constant variance.
When the null hypothesis is satisfied, F̂ is approximately F-distributed with 1 and
n−2p−2 degrees of freedom. The H0 of linearity is rejected if the p – value associated
with F̂ is small (p-value <α) or when the value of F̂ is greater than the selected critical
value of the F- distribution with 1 and n−2p−2 degrees of freedom.
3.15 Tsay’s F -test
Tsay (1989) introduced the Tsay test for detecting nonlinearity in an observable time
series after improving on the power of the Keenan (1985) test in 1986. He generalized
the Keenan test by explicitly checking for quadratic serial dependence in the series,
by doing so, he catered for the disaggregated nonlinear variables. The first stage of
this test is the same as the first stage of the Keenan test. Unlike the second stage
in the Keenan test, the second stage of this test involves regression of the products
Xt−iXt− j, i = j = 1, . . . , p on (1,Xt−1,Xt−2, . . . ,Xt−p). Hence, its corresponding test
statistic F is asymptotically distributed as Fm,n−m−p−1, where m =
p(p−1)
2 , with the
hypothesis as;
H0 : linear AR(p) model
H1 : non-linear threshold model
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3.16 Criterion for Model Selection
In order to select tentative models that fits the datasets well, it is imperative to
consider various model selection criteria. The Akaike Information Criterion (AIC),
the Akaike Information Criterion corrected (AICc) and the Bayesian Information
Criterion (BIC) are examples of model selection criteria frequently used in most
time series research. In this study too, the AIC and other model selection criteria
are used. To select appropriate models from competing models, models with
minimum or least values of AIC are considered for further validation. This is because
these least values have been described as closest to truth. Again, this study considers
models with more parameters as tentative models since it improves goodness of fit test.
The AIC model can be given as:
AIC = 2k−2log(L)
where
K denotes the number of parameters in the model,
L is the maximized value of the likelihood function.
In the case of the SETAR model, the AIC model is given as follows:
AIC (p1, p2) = n1 ln σ̂2 21 +n2 ln σ̂2 +2(p1 +1)+2(p2 +1)
where
n1 and n2 are the observations in regimes1 and 2.
σ̂2 and σ̂21 2 represents the variance of residuals in regimes 1 and 2.
p1and p2 denotes the minimized information criterion selected lag orders in
regimes 1 and 2.
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3.17 Model Diagnostics
After carefully selecting tentative models to be used for forecasting, we check the
residuals of the models to ensure that, their time series satisfy all model assumptions.
This step is paramount to making any meaningful inferences with the models. So,
we employ the Ljung-Box and ARCH-LM tests on the model residuals as discussed
below:
3.17.1 Ljung-Box Test
Ljung and Box (1978), described this test as a diagnostic tool used to check for the
presence or absence of serial correlations in the residuals of a fitted model. Thus, a
time series with any specified lag orders, say order m examines autocorrelations in
the residuals. Instead of testing randomness at each distinct lag, usually it tests the
"overall" randomness based on a number of specified lags, and so for this reason it is
referred as a portmanteau test. The test procedure is given as follows;
The hypothesis to be tested is;
H0 : Residuals are uncorrelated up to order m
H1 : Residuals are uncorrelated up to order m
The test statistic is
m ρ̂2
Qm = n(n+2) ∑ k
k=1 n− k
where
ρ̂2k represents residual autocorrelations of the series at lag m
m is the number lags being tested,
n is the number of residuals.
We reject the null hypothesis if the value of Qm is greater than the chi-square table
value. Hence, residuals are correlated. However, if the value of Qm is less than the chi-
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square table value, we do not reject the null hypothesis. So, we conclude that the model
residuals are uncorrelated and therefore the model fits the dataset well. Otherwise the
whole estimation process has to be repeated again in order to get the most adequate
model.
3.17.2 ARCH-LM Test
For residuals of a fitted model to become adequate enough to produce meaningful
forecast values, the model must satisfy all model assumptions. Thus, the residuals
should have constant variance. Hence, the reason why Engle (1982) proposed
the ARCH-LM test to cater for issues of conditional heteroscedasticity in squared
residuals. The test procedure is as follows;
The hypothesis is;
H0 : There is no heteroscedasticity in the model residuals
H1 : There is heteroscedasticity in the model residuals.
and the test statistic is
LM = nR2 (3.18)
where n is the number of observations and R2 is the coefficient of determination of
auxiliary residual regression.
e2t = β
2 2
0 +β1et−1 +β2et−2 + · · ·+β e2q t−q + εt
where et is the residual and q is the length of ARCH lags.
We reject the null hypothesis if P-values are less than 5% level of significance, and
hence we conclude the existence of heteroscedasticity in the model residuals. Also,
p-values greater than 5% significance level indicates there is no heteroscedasticity.
Hence, we conclude the model has constant variance.
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3.18 Diebold-Mariano Test
Forecasting has been described as the last step to modelling procedures following the
approach of Box-Jenkins. To ensure competing models are statistically significant,
Diebold and Mariano (1995) proposed a test popularly known as the Diebold-Mariano
test. The purpose of this test is to verify forecast accuracy between competing models.
In this test, we assess forecasts between two competing models by considering their
mean squared errors for statistical significance. Nevertheless, lower values of mean
square errors of one forecast in comparison to the other do not necessarily translate
into superiority of that forecast.
The hypothesis is given as;
H0 : There is no significant difference between the two models
H1 : There is significant difference between the two models.
Following a standard normal distribution, the test statistic is given as;
[ ]− 1
DM = V̂ (d̂) 2 d̂ (3.19)
where;
d 2 2t = et1−et2 is the difference between the sets of squared error forecasts from the two
competing models.
And d̂= 1 ∑TT t=1 dt is the mean, T= no of residuals.
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3.19 Markov Chain
The Markov chain is the smallest form of a Markov model. This Chain models the state
of a system with changing random variable throughout time. This model operates on
the Markov property which suggests that, the distribution of a current variable depends
on previous distribution states. The Markov Chain Monte Carlo, an application of
the Markov Chain uses the Markov property to prove that a particular method for
performing a random walk will sample from its joint distribution. The model has the
ability to model the behaviour of a system by constructing consistent block diagrams,
fault trees and more. This model too can handle errors and recovery at detailed levels
in modelling. Despite its advantages, it is cumbersome to handle since it requires large
number of states for analysis. It is also quite difficult to construct and interpret though
it models types of complexity solution techniques that are feasible but for only small
models. In the study of a particular system, it does not offer the required speed and
accuracy for the study.
3.20 Jarque-Bera Test
The Jarque-Bera test, a type of Lagrange multiplier named after Carlos Jarque and Anil
K. Bera is used for checking normality in a data. Normality is one of the assumptions
used in most statistical tests, just like the t or F tests. In order to check normality in
a data, it is prudent to consider values of Skewness, (i.e. perfect symmetry around
the mean) and Kurtosis (i.e. how much tailed a data is and how peaked the data is).
However, it is needless to know the mean or standard deviation for the data before
performing the test.
The test statistic is defined as;
[ ]
b1 (b2−3)
2
JB = n +
6 24
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where;
n is the sample size
b1 is the sample skewness
b2 is the sample Kurtosis
with the hypothesis is given as;
H0 : The data is normally distributed
H1 : The data does not come from a normal distribution.
Generally, the null hypothesis is rejected for smaller p-values less than 5% level of
significance. Hence, the data under consideration is said not to be normally distributed.
3.21 Chapter Conclusion
The chapter has dealt with statistical techniques and methods which include; the unit
root tests, linearity tests and Box-Jenkins methodology for both linear and nonlinear
models. All these techniques are appropriate for the study due to their advantages over
other time series.
Due to the cumbersome nature of large data, the unit root test is preferred for checking
stationarity as compared to the graphical approach. The Box-Jenkins methodology is
also most appropriate in terms of forecasting for shorter periods of time as compared to
other time series like Markov chain which is difficult to interpret. Again, the normality
test is an appropriate approach to understand the distribution of most data rather than
assuming all datasets follow normal distribution.
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CHAPTER 4
DATA ANALYSIS AND DISCUSSIONS
4.1 Introduction
This chapter deals extensively with the results and discussions obtained from the study.
It includes preliminary and further analyses of datasets, the results and discussions.
In the preliminary analysis, the source from which data was obtained and general
characteristics of the time series datasets is explained. Also, in the further analysis,
the various models and techniques employed in derivation of the best model(s) is
explained. Furthermore, in the results and discussion, results obtained from analysis
using the R statistical software are discussed. The chapter ends with the conclusion,
which summarises the entire chapter.
4.2 Preliminary Analysis
Monthly data of Ghana’s inflation and CIC rates were obtained from the website of
the Bank of Ghana ranging from January 1990 to August 2016. The datasets each of
which consists of 320 observations, were divided into two parts namely; the train and
test datasets. A casual view of time series plots of these datasets appeared seasonally
adjusted and asymmetric in nature. Following the Box and Jenkins (1976) procedure
for forecasting, the first 296 points which starts from January 1990 to August 2015
represents the train dataset used to build the model whiles the remaining points of 12
months from September 2015 to August 2016 represented the test dataset retained for
out of sample performance assessment for each variable.
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4.2.1 Preliminary Analysis of Inflation rate
In this section, descriptive statistics of monthly inflation rates is discussed.
By inspection, Table 4.1 recorded a minimum value of 7.3 and a maximum value of
70.8 for inflation rate in March 1992 and December 1995 respectively. The sample
moments of inflation for the time period under consideration appeared positively
skewed and leptokurtic in nature about their mean values. The leptokurtic nature of
the data means that, the inflation values were more peaked about the mean value of the
inflation rate. The Jarque-Bera test for normality was used to confirm the asymmetric
nature of the inflation rate. At 5% significance level, the test confirmed that the test was
not normally distributed as its p-value, 2.2e−16, was less that the significant level.
Table 4.1: Descriptive Statistics of Inflation Rates
Mean Median Std Min Max Skewness Kurtosis Jarque-Bera
(P-value)
21.16 17.30 12.88 7.30 70.80 1.86 3.83 2.2e-16
Figure 4.1, shows both increasing and decreasing pattern in inflation rate overtime.
These fluctuations in the rate could be as a result of unstable market conditions.
Somewhere around 1991, the figure shows volatility in the time series plot. Volatility
in this plot could be as a result of some monetary factors such as poor agricultural
production and world economic crisis as described by Aidoo (2011) and Ocran (2007).
The years 2004 to 2008 saw consistent decline and pattern in inflation rate. This may
have resulted due to the implementation of policies like reduced taxes leading to the
reduction in price of goods and services. Yet again, inflation rates appeared stable
from somewhere around the year 2011 to 2013. A trendy nature in the plot shows
periodic pattern over time in regular intervals.
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Figure 4.1: Time series plot of inflation
Figure 4.2 confirms Ghana’s inflation is non-stationary, since there exists slow decay
in the series for all lag values in the ACF plot. Likewise, the PACF plot shows a
significant spike in lag 1 with marginal spikes in some few lags of the series.
Figure 4.2: ACF and PACF plots of inflation rates
After Figure 4.2 confirmed the series was nonstationary, the data was logarithmically
transformed and differenced to attain stationarity in the series as shown in Figure 4.3.
Significant spikes for both ACF and PACF plots appear to revolve about fixed points,
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this affirms stationarity. Moreover, these plots demonstrate a sine wave-like pattern.
This confirms Ghana’s inflation has both seasonal and non-seasonal components. In
this figure too, the PACF plot shows significant spikes in lags 12, 24 and 36 and at lag
12 for the ACF plot. This affirms the trendy pattern in the series at regular intervals.
For further analysis, the transformed and differenced series is used.
Figure 4.3: ACF and PACF plots of differenced inflation
4.2.2 Preliminary Analysis for CIC rate
An examination of descriptive statistics for the CIC data in Table 4.2, revealed the
value 53.1 as the minimum CIC rate recorded in March 1996 and the value 9536.5 as
the maximum CIC rate recorded in December 2015. The sample moments for the CIC
rate under study appeared positively skewed and platykurtic in nature about the mean
value. The platykurtic nature of the data indicates the CIC values are broadly spread
about the mean value of the CIC rate. Just like the inflation, the test for normality in
the CIC rates confirms the data is far from a normal distribution. So, we conclude the
CIC rate is asymmetric in nature.
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Table 4.2: Descriptive Statistics of CIC Rates
Mean Median Std Min Max Skewness Kurtosis Jarque-Bera
(P-value)
1729.6 730.5 2273.82 53.1 9536.5 1.68 1.75 2.2e-16
Although, the years 1996 and 1997 experienced steep decline in CIC rates, the years
1990 to 1993 recorded constant values for the CIC rate. In later years, CIC rates
picked up until somewhere around the year 2000. From the year 2000 onwards, a
gradual increase occurred in CIC rate at regular intervals. This indicates that as years
go by the total supply of money in Ghana increases. Volatility explains better the
consistent peaks in the time series plot of the CIC rates as shown in Figure 4.4. This
volatility, may be due to instability in prices and market interest rates.
Figure 4.4: Time Series plot of CIC
A graphical view of the CIC rate presented in Figure 4.5 revealed that, Ghana’s CIC
rate is non- stationary. Since, the graphs do not revolve around a zero mean and a
constant variance. In addition, the plot of ACF shows a slow decay in its series. But
then, the PACF plot displays a single significant spike at lag 1 with several marginal
spikes across all other lags.
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Figure 4.5: ACF and PACF plots of CIC
In Figure 4.6, the spikes in plots of ACF and PACF of CIC rate decay rapidly, this
indicates the series is stationary. Stationarity of the series is again confirmed as the
plots are perceived to revolve around fixed points.
Figure 4.6: ACF and PACF plots of differenced CIC
4.3 Further Analysis
To explain the procedures used for model building, evaluating and forecasting in this
study, we derived appropriate model(s) by employing SARIMA and SETAR modelling
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procedures for both inflation and CIC rates. This section too represents and explains
actual statements and observations including tables and figures. Sufficient details of
the study as well are discussed to serve as reference to other related studies.
4.3.1 SARIMA modelling
In this section, the seasonal ARIMA approach is followed to fit appropriate models to
forecast Ghana’s inflation and CIC rates. In this modelling cycle, stationarity is first
checked in the original datasets for both rates using Phillips-Perron and KPSS unit root
tests. Later, we follow the procedures described by Box and Jenkins in model building
and forecasting.
SARIMA Modelling for inflation
To model inflation, we first checked for stationarity in the data using the KPSS and PP
tests for unit root testing. For the PP-test, the null hypothesis of non-stationarity was
not rejected because the resulting P-value was greater than 5% level of significance.
So, we conclude the raw data is not stationary. Hence, the data was logarithmically
transformed and differenced resulting in a P-value smaller than the 5% level of
significance. This indicates the data is now stationary.
On the other hand, the alternative unit root test, the KPSS test was employed to cater
for shocks within the series under the null hypothesis of stationarity. The P-Value that
resulted from this test, suggests the raw data is not stationary. Just as in the PP-test, the
raw data is transformed to attain stationarity. This concept of stationarity is important
to most time series analysts. Since, it enables them to make meaningful predictions.
Table 4.3 summarises the results from the unit root tests performed for inflation.
Table 4.3: Unit root test for inflation
Unit root test Raw Data Transformed Data
Phillips-Perron (p-value) 0.29 0.01
KPSS (p-value) 0.01 0.1
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Now that our data is stationary, our next task is to build appropriate models that fit
the data well. We do so, by further examination of Figure 4.3. Suggested AR and
MA orders for the models under construction are chosen from Figure 4.3. While we
selected AR orders from significant spikes of lag orders in the differenced PACF plot,
MA orders were selected from the differenced ACF plot with significant spike of lag
orders to make up tentative models. In this approach of selecting model components,
we considered all spikes above zero as non-seasonal components and below zero as
the seasonal components. By inspection of the differenced PACF plot in Figure 4.3,
lag orders 1 and 3 were chosen as non-seasonal components and lag orders 1, 2 and 3
as seasonal components of the AR order to be added to the model. In the differenced
ACF plot too, lag orders 1,2 and 3 were chosen as non-seasonal components with lag
order 1 as the seasonal MA component to be added to the model.
The various model combinations are evaluated by considering models with least
values of AIC, RMSE, MAE and MAPE. These are shown in Table 4.4.
Table 4.4: AIC, RMSE, MAE and MAPE of candidate seasonal ARIMA Models for
Inflation
Model AIC RMSE MAE MAPE
ARIMA(1,1,1)(1,0,1)12 -696.55 0.07550 0.05060 1.7985
ARIMA(3,1,1)(1,0,1)12 -693.47 0.07540 0.05043 1.78758
ARIMA(1,1,1)(2,0,1)12 -695.00 0.07543 0.05032 1.78517
ARIMA(3,1,1)(2,0,1)12 -691.82 0.07534 0.05019 1.78018
ARIMA(1,1,2)(1,0,1)12 -694.68 0.07549 0.05050 1.78991
ARIMA(1,1,2)(2,0,1)12 -693.69 0.07542 0.05022 1.78136
ARIMA(3,1,2)(1,0,1)12 -692.64 0.07524 0.05046 1.78775
ARIMA(3,1,2)(2,0,1)12 -691.02 0.07518 0.05021 1.77997
ARIMA(1,1,3)(1,0,1)12 -693.75 0.07537 0.05038 1.78537
ARIMA(1,1,3)(2,0,1)12 -692.07 0.07535 0.05016 1.77862
ARIMA(3,1,3)(1,0,1)12 -698.18 0.07377 0.04940 1.74989***
ARIMA(3,1,3)(2,0,1)12 -694.36 0.07411 0.04974 1.76357
*** means best based on the selection criteria.
Carefully scrutinizing the twelve tentative models in Table 4.4 for inflation,
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ARIMA(3,1,3)(1,0,1)12 could be adjudged the best model in terms of its minimum
values of AIC, RMSE, MAE and MAPE for inflation rate.
In line with a said objective of the study, we sought to make an out of sample forecast
from the best model. Hence, tentative models with the most minimum values were
selected and assessed using the forecast accuracy test. In this test, the method of
maximum likelihood was used to evaluate the values of parameter estimates of the
models with their corresponding standard errors as shown in Table 4.5. Model
variables with either corresponding positive or negative values for parameter estimates
and standard errors are considered significant to the model. However, the greater
the number of variables resulting from either a positive standard error and a positive
estimate and vice-versa, the more accurate the model.
By the inspection of Table 4.5, seven parameter variables demonstrated to be more
significant to the model ARIMA(3,1,3)(1,0,1)12. This means that the variables
AR(1), AR(2), AR(3), MA(1), MA(3), SAR(1) and SMA(1) were added to the model,
since they proved to be significant.
Table 4.5: Estimated Parameters of ARIMA(3,1,3)(1,0,1)12 for inflation
Variable Estimate Standard Error 95% confidence Interval
Lower limit Upper Limit
∗AR(1) -0.8740 0.0651 -1.0016 -0.7464
∗AR (2) 0.5570 0.1067 0.3478 0.7662
∗AR(3) 0.8286 0.0599 0.7112 0.9459
∗MA(1) 1.1159 0.1238 0.8732 1.3585
MA(2) -0.2394 0.1679 -0.5684 0.0897
∗MA(3) -0.6872 0.0886 -0.8610 -0.5135
∗SAR(1) 0.1830 0.0821 0.0221 0.3439
∗SMA (1) -0.8027 0.0592 -0.9186 -0.6867
σ̂2 = 0.00546
∗ means significant at 5% significant level.
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Table 4.6 revealed just SAR (1) model variable as the significant parameter to the
model ARIMA(1,1,2)(1,0,1)12 and hence, this model was considered in the model.
Table 4.6: Estimate Parameters of ARIMA(1,1,2)(1,0,1)12 for inflation
Variable Estimate Standard Error 95% confidence Interval
Lower limit Upper Limit
AR(1) -0.8708 0.0620 0.7493 0.9923
∗MA(1) -0.6669 0.0844 -0.8323 -0.5016
MA(2) -0.0231 0.0642 -0.1490 0.1028
SAR(1) 0.1562 0.0872 -0.0147 0.3270
∗SMA (1) -0.7591 0.0650 -0.8864 -0.6317
σ̂2 = 005717
∗ means significant at 5% significant level.
In terms of the number of significant variables to the model, the model
ARIMA(3,1,3)(2,0,1)12 displayed six significant parameter variables as shown
in Table 4.7. This indicates the model is significant. Hence, we consider the model for
estimation.
Table 4.7: Estimate Parameters of ARIMA(3,1,3)(2,0,1)12 for inflation
Variable Estimate Standard Error 95% confidence Interval
Lower limit Upper Limit
AR(1) 0.0647 0.0644 -0.0616 0.1910
∗AR(2) -0.3129 0.0508 -0.4124 -0.2135
∗AR(3) 0.8572 0.0631 0.7335 0.9810
MA(1) 0.1283 0.0962 -0.0602 0.3169
∗MA(2) 0.4864 0.0784 0.3328 0.6401
∗MA(3) -0.6552 0.0916 -0.8347 -0.4756
SAR(1) 0.1664 0.0976 -0.0250 0.3577
SAR(2) 0.0795 0.0795 -0.0763 0.2353
∗SMA(1) -0.7934 0.0791 -0.9484 -0.6385
σ̂2 = 0.005511
∗ means significant at 5% significant level.
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Obtaining parameter estimates of tentative models is not enough to conclude that,
a particular model best fits a data. An important task therefore is to check if all
assumptions of the SARIMA model are satisfied. This was done by performing
diagnostic checks on model residuals as shown in Table 4.8. Thus, we checked
if residuals have zero mean, constant variance and are also uncorrelated, that is if
residuals follow a white noise process.
The Ljung- Box test, a portmanteau test, was performed on the residuals of the model
under the null hypothesis of residuals being random, P-values resulting from this test
revealed the residuals were random. This was so, because P-values obtained from
the test were greater than 5% level of significance. Equally, the ARCH-LM test was
performed on the model residuals under the null hypothesis of there is no ARCH
effect. This test was included in the study due to its ability to test for variations in
model variances. The resulting P-values greater than the 5% level of significance
indicated, there were no ARCH effect in the model residuals. This therefore, enables
us conclude the presence of constant variance with zero mean in all the tentative
models.
Table 4.8: Residual diagnostic tests for Seasonal ARIMA models for inflation
Model P-values
ARCH-LM test Ljung-test
ARIMA(3,1,3)(1,0,1)12 0.9111 0.8702
ARIMA(1,1,2)(1,0,1)12 0.7321 0.4823
ARIMA(3,1,3)(2,0,1)12 0.6512 0.7015
The ACF plots of the residual models was observed to be zero, depicting residuals are
uncorrelated. Also, it was observed that all spikes of the plots in Figure 4.7 are within
the confidence boundaries. This means the models fits best the data under study.
Clearly, all these tentative models appear to be satisfactory of all SARIMA model
assumptions, thereby we consider all these models for further valuation.
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Figure 4.7: ACF plots of residuals for inflation
SARIMA Modelling for CIC
As shown in Table 4.9, the resulting P-values from the raw data for CIC was
not stationary for both PP and KPSS tests under their respective null hypotheses.
Statistically, if a P-value is greater than 5% significance level, the null hypothesis is not
rejected and vice- versa. Contrarily to the PP test, the KPSS test produced a P-value
of 0.01, which is less than a 5% significance level. This affirms non-stationarity in
the raw data considering both unit root tests. As usual, the non-stationary data was
transformed to attain stationarity in the data.
Table 4.9: Unit Root test for CIC rate
Unit root test Raw Data Transformed Data
Phillips-Perron (p-value) 0.99 0.01
KPSS (p-value) 0.01 0.1
After attaining stationarity in the data, the next task was to select model constituents
to enable us derive appropriate model(s) for further analysis. Just as it was done
earlier for inflation, the same procedure was followed to select our seasonal and
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non-seasonal components of the model(s) to be derived by examining Figure 4.6. On
successfully deriving our models with their respective values of AIC, RMSE, MAE
and MAPE, models with minimum values were considered as the selection criterion
for the tentative models for further assessment. Table 4.10 below provides models
with their respective measure of values.
Table 4.10: AIC, RMSE, MAE and MAPE of candidate seasonal ARIMA Models for
CIC
Model AIC RMSE MAE MAPE
ARIMA(0,1,0)(1,0,1)12 -16.49 0.23235 0.07934 1.32993
ARIMA(0,1,0)(1,0,3)12 -27.72 0.22494 0.08272 1.36114
ARIMA(0,1,0)(3,0,1)12 -26.01 0.22675 0.08745 1.44632
ARIMA(0,1,0)(3,0,3)12 -52.20 0.20165 0.07955 1.30287***
∗∗∗Means minimum values of the selection criterion.
Later, the estimated parameters were of the tentative models were checked for their
significance with the least values of selection criterion. By this, the forecast accuracy
was employed as discussed earlier. Results from Tables 4.11, 4.12 and 4.13 indicates
all estimates of the model components are relevant. Here too, variable parameters
with resulting positive values for both estimates and standard errors or vice-versa were
considered as relevant to the model under construction.
Table 4.11: Estimated parameters of ARIMA (0,1,0) (3,0,3)12 for CIC
Variable Estimate Standard Error 95% confidence Interval
Lower limit Upper Limit
∗SAR(1) -1.0917 0.1203 -1.3274 -0.8559
∗SAR(2) -0.7941 0.1551 -1.0981 -0.4790
∗SAR(3) -0.2543 0.1076 -0.4652 -0.0435
∗SMA(1) 1.0499 0.1509 0.7542 1.3456
∗SMA(2) 0.8201 0.2140 0.4007 1.2394
SMA(3) -0.1499 0.1316 -0.4079 0.1081
σ̂2 = 0.04079
∗ means significant at 5% significant level.
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Table 4.12: Estimated parameters of ARIMA(0,1,0)(1,0,3)12 for CIC
Variable Estimate Standard Error 95% confidence Interval
Lower limit Upper Limit
∗SAR(1) -0.5825 0.0798 -0.7390 -0.4261
∗SMA(1) 0.5118 0.0781 0.3587 0.6650
SMA(2) 0.0455 0.0618 -0.0757 0.1667
∗SMA(3) -0.2883 0.0815 -0.4481 -0.1285
σ̂2 = 0.05076
∗ means significant at 5% significant level.
Table 4.13: Estimated parameters of ARIMA(0,1,0)(1,0,1)12 for CIC
Variable Estimate Standard Error 95% confidence Interval
Lower limit Upper Limit
SAR(1) -0.7978 0.0821 -0.9587 -0.6369
SMA(1) 0.6319 0.1059 0.4243 0.8394
σ̂2 = 0.05416
∗ means significant at 5% significant level.
To certify the models are appropriate SARIMA models, all the model assumptions of
constant variance, zero mean and autocorrelation were checked for residuals of the
models by employing the Ljung- Box and ARCH- LM tests under their respective null
hypothesis. Resulting P-values from Table 4.14 indicates, all models have zero means
and constant variances.
Table 4.14: Residual diagnostic tests for Seasonal ARIMA models for CIC
Model P-values
ARCH-LM test Ljung-test
ARIMA(0,1,0)(3,0,1)12 0.9997 0.9636
ARIMA(0,1,0)(1,0,3)12 0.9094 0.8556
ARIMA(0,1,0)(3,0,3)12 0.9996 0.9006
Lastly, the assumption of autocorrelation were checked. Figure 4.8 revealed the ACF
plots of residuals to be zero, describing residuals are uncorrelated. A supplementary
observation showed except for lag zero that all spikes in all other lags are within the
confidence boundaries. This means that, all the three models fit CIC data well. Hence,
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we consider all the models for further analysis.
Figure 4.8: ACF plots of residuals for CIC
4.3.2 SETAR Modelling
In this section, the SETAR modelling procedure proposed by Franses and Van Dijk
(2000) is followed to model and forecast Ghana’s inflation and CIC rates. The section
includes the two- regime SETAR model and forecast approach for the two rates under
study.
SETAR modelling for inflation rate
In order to model inflation rate with the SETAR model, its time series must first
satisfy the condition of nonlinearity. So, the nonlinearity was checked in this series
by specifying the order of the linear AR(p) model. The order of the linear AR(p)
is chosen as the AR(p) model based on the maximum lag order with the least value
of AIC. Using the Keenan and Tsay tests of linearity, AR(14) was found to be the
linear AR order with the least AIC value. In addition, P-values from these tests were
both less than the 5% significant level, suggesting that Ghana’s inflation rate follow
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threshold nonlinear. While considering the null hypothesis of linearity for Keenan-1-
degree test and the alternative hypothesis of threshold-type nonlinearity in the Tsay
F-test. Employing regime switching models can best account for these discontinuities
and switching of inflation as compared to the simple linear model. Table 4.15 displays
a summary of the linearity tests performed on inflation rate.
Table 4.15: Linearity test for inflation rate
Test Test statistic P- value Order Decision
Keenan- 4.2232 0.0041 14 Reject Linearity
1-degree
Tsay 3.034 1.936e-11 14 No threshold nonlinearity
rejected
Following that, the time series of inflation is confirmed to be nonlinear, the appropriate
SETAR model identified best fit the dataset. In doing so, the lag order p in each regime
and the threshold variable yt−d , here d represents the delay parameter were determined.
After performing grid search of tentative model combinations for SETAR models that
best fit the dataset, lag orders with minimum values of AIC and MAPE were chosen as
the tentative model(s). Tables 4.16 below provides a summary of competing models to
be used in forecasting the two rates. The table suggest that SETAR (2;14,4) with
threshold variable yt−4, SETAR (2;14,5) with threshold variable yt−5 and SETAR
(2;14,13) with threshold yt−13 was the appropriate models in explaining nonlinearity in
Ghana’s inflation rate. Hence, the three models were considered for further assessment.
Table 4.16: AIC and MAPE values of tentative SETAR models
Model AIC MAPE
SETAR (2;14,4) -1541.411 1.5680
SETAR (2;14,5) -1549.031 1.7785
SETAR (2;14,13) -1538.501 1.7000
In accordance with Franses and Dijk’s (2000) approach, the method of conditional
least squares is used to estimate parameters of the candidate models as performed in
Tables 4.17, 4.18 and 4.19 with their corresponding threshold values.
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Resulting proportions from Table 4.17 with delay parameter of four and threshold value
of 0.03944, can attribute structural breaks in inflation to data emanating from the low
regime rather than the high regime as displayed.
Table 4.17: Estimate Parameters of SETAR (2;14,4) for inflation
Low Regime High Regime
Coefficient
Estimate Std Error t- value Estimate Std Error t-value
Constant -0.0055 0.0063 -0.8821 0.0232 0.0171 1.3566
φ 1 0.2311 0.0629 3.6766 -0.1208 0.1617 -0.7467
φ 2 -0.0584 0.0648 -0.9018 0.7412 0.1738 4.2647
φ 3 0.0559 0.0709 0.7890 -0.0279 0.1055 -0.2640
φ 4 0.0544 0.0664 0.8197 0.0228 0.1055 0.2157
φ 5 0.1563 0.0908 1.7218 -0.0773 0.1097 -0.7050
φ 6 -0.0669 0.0789 -0.8479 0.0437 0.0880 0.4965
φ 7 0.0549 0.0688 0.7978 -0.0933 0.1081 -0.8625
φ 8 0.0568 0.0683 0.8306 0.0364 0.0963 0.3776
φ 9 -0.0840 0.0681 -1.2336 0.1066 0.0933 1.1422
φ 10 0.1396 0.0644 2.1678 -0.0387 0.1044 -0.3704
φ 11 -0.0563 0.0617 -0.9115 -0.0807 0.1222 -0.6606
φ 12 -0.3874 0.0628 -6.1712 -0.3996 0.1161 -3.441
φ 13 0.1143 0.0801 1.4268 0.0248 0.1104 0.2249
φ 14 -0.0341 0.0666 -0.5115 0.1700 0.1435 1.1851
Threshold value = 0.03944
Proportion 75.09% 24.91%
In Table 4.18 too, more than half the data recorded switches and discontinues in
the low regime. This indicates the earlier years of the inflation data demonstrates
discontinuities in the dataset given a delay parameter of five with threshold value of
0.01342.
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Table 4.18: Estimate Parameters of SETAR (2;14,5) for inflation
Low Regime High Regime
Coefficient
Estimate Std Error t- value Estimate Std Error t-value
Constant -0.0137 0.0078 -1.7433 0.0225 0.0111 2.0377
φ 1 0.3381 0.0789 4.2820 0.1130 0.0856 1.3205
φ 2 -0.0597 0.0686 -0.8706 0.3225 0.1202 2.6834
φ 3 0.0873 0.0737 1.1841 0.0315 0.0864 0.3648
φ 4 0.1221 0.0776 1.5737 -0.0060 0.0772 -0.0775
φ 5 -0.0176 0.0699 -0.2516 0.0979 0.0944 1.0371
φ 6 -0.0927 0.0989 -0.9375 -0.0280 0.0918 -0.3054
φ 7 0.0680 0.0808 0.8408 -0.0519 0.0769 -0.6746
φ 8 0.0155 0.0712 0.2182 0.1419 0.0912 1.5565
φ 9 -0.0293 0.0755 -0.3884 0.0038 0.0779 0.0490
φ 10 0.2872 0.0820 3.5007 -0.0726 0.0723 -1.0046
φ 11 -0.1464 0.0678 -2.1583 -0.0382 0.0892 -0.4283
φ 12 -0.3134 0.0739 -4.2402 -0.5414 0.0792 -6.8377
φ 13 0.0797 0.0780 1.0217 0.2289 0.0926 2.4729
φ 14 -0.0064 0.0792 -0.0808 0.0311 0.0914 0.3403
Threshold value = 0.01342
Proportion 61.77% 38.23%
Unlike Tables 4.17 and 4.18, Table 4.19 explains that, more than 80% of the inflation
data experienced switching regimes and discontinuities in the higher regimes as
compared to the low regime.
Similarly, in SETAR modelling we check models to confirm if they satisfy all
model assumptions as mentioned earlier. Thus, we check the model residuals
for no auto correlation, zero mean and constant variance of residuals. Also, we
perform the ARCH-LM and Ljung -Box tests on the residuals under their specific
hypotheses as discussed earlier. The results from Tables 4.20 confirms non-existence of
autocorrelation and random residuals in all tentative models under the null hypothesis
of residuals being random and residuals being non-correlated. A critical look at
the table again, reveals SETAR(2;14,4) failed its ARCH-test for lag order 12, this
indicates the possibility of autocorrelation, nevertheless we will still consider this
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model for further assessment.
Table 4.19: Estimate Parameters of SETAR (2;14,13) for inflation
Low Regime High Regime
Coefficient
Estimate Std Error t- value Estimate Std Error t-value
Constant -0.0412 0.0326 -1.2643 -0.0032 0.0053 -0.5957
φ 1 -0.0105 0.1322 -0.0792 0.3050 0.0673 4.5348
φ 2 0.1790 0.0937 1.9099 0.0405 0.0825 0.4916
φ 3 0.2177 0.0862 2.5269 0.0254 0.0740 0.3428
φ 4 -0.0780 0.1049 -0.7434 0.1051 0.0715 1.4697
φ 5 -0.1625 0.1468 -1.1070 0.0750 0.0635 1.1805
φ 6 0.2319 0.1053 2.2024 0.0294 0.0644 0.4567
φ 7 -0.1882 0.0957 -1.9670 0.0611 0.0687 0.8895
φ 8 -0.1880 0.1425 -1.3193 0.0794 0.0612 1.2972
φ 9 -0.0910 0.1105 -0.8232 0.0033 0.0617 0.0532
φ 10 0.0822 0.0967 0.8496 0.1159 0.0682 1.6990
φ 11 -0.1485 0.1287 -1.1543 -0.0950 0.0638 -1.4897
φ 12 -0.3674 0.1453 -2.5280 -0.3932 0.0632 -6.2224
φ 13 0.1225 0.1223 1.0013 0.0935 0.0761 1.2296
φ 14 -0.1817 0.1972 -0.9214 0.0223 0.0804 0.2779
Threshold value = -0.07091
Proportion 17.41% 82.59%
Table 4.20: Residual diagnostics of tentative SETAR models for inflation
Model Ljung-Box test ARCH-LM test
SETAR (2;14,4) 0.8371 0.0030
SETAR (2;14,5) 0.8667 0.4480
SETAR (2;14,13) 0.9776 0.5915
SETAR modelling for CIC rate
We follow the same procedure for testing linearity as we did earlier. In this section,
we found AR (13) to be our AR(p) model order with the least AIC while employing the
Keenan and Tsay tests for linearity. The P-values from Table 4.21 indicates Ghana’s
CIC rate follow Threshold nonlinear under the null hypotheses of Keenan and Tsay
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tests as mentioned earlier. Again, P-values from the Keenan test indicates nonlinearity
in the data as well as the presence of threshold from the Tsay test. Equally, regime
switching models best caters for these discontinuities and switching of inflation as
compared to the simple linear model.
Table 4.21: Linearity test for CIC rate
Test Test statistic P- value Order Decision
Keenan- 29.405 1.236e-07 13 Reject Linearity
1-degree
Tsay 3.905 5.066e-16 13 No threshold nonlinearity
rejected
After confirming the CIC rate is nonlinear, we identified an appropriate SETAR model
that best fits the data as shown in Table 4.22. Interestingly, a grid search of tentative
model combination for SETAR model revealed only one model as the tentative model
with close AIC and MAPE values for the CIC rate. Hence, we consider this model for
further assessment.
Table 4.22: AIC and MAPE candidate models for CIC
Model AIC MAPE
SETAR (2;13,9) 3148.395 0.1662
Looking at Table 4.23, it can be inferred early years of CIC values resulted in more
than 80% of structural breaks in the data. The standard errors from the table are
relatively low in the low regime as compared to the high standard error in the high
regime. The t- values in the table was obtained by dividing the estimate of each
constant by its corresponding standard error.
Again, we check the model if it satisfies the model assumptions of autocorrelation,
zero mean and constant variance. The ARCH -LM and Ljung-Box tests on the model
residual confirms the constructed model satisfied model assumptions after lag 6. Also,
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the ACF plot of the model residual is not correlated. Hence, we conclude the model fit
the data well.
Table 4.23: Estimate Parameters of SETAR(2;13,9) for CIC
Low Regime High Regime
Coefficient
Estimate Std Error t- value Estimate Std Error t-value
Constant 13.6062 15.8325 0.8594 318.4621 94.3598 3.3750
φ 1 0.6990 0.0640 10.9183 0.8922 0.1096 8.1432
φ 2 0.2298 0.0788 2.9180 -0.0661 0.1198 -0.5520
φ 3 -0.0254 0.0801 -0.3164 -0.0229 0.1190 -0.1924
φ 4 0.0106 0.0799 0.1327 -0.0808 0.1187 -0.6807
φ 5 0.0337 0.0800 0.4209 0.1599 0.1177 1.3590
φ 6 0.0317 0.0800 0.3956 -0.2129 0.1198 -1.7772
φ 7 0.0214 0.0804 0.2665 0.1279 0.1269 1.0074
φ 8 -0.0501 0.0832 -0.6020 0.0769 0.1198 0.6416
φ 9 0.0580 0.1015 0.5708 -0.2244 0.0971 -2.3109
φ 10 0.0297 0.1076 0.2761 0.0083 0.0939 0.0884
φ 11 0.0166 0.1072 0.1547 0.2542 0.0919 2.7654
φ 12 0.0367 0.1072 0.3422 0.5410 0.0915 5.9121
φ 13 -0.0922 0.0829 -1.1126 -0.4658 0.0967 -4.8164
Threshold value = 2159
Proportion 82.71% 17.29%
Table 4.24: Residual Diagnostics of suggested SETAR models for CIC
Model Ljung-Box test Arch-test
SETAR (2;13,9) 0.9163 0.4476
4.3.3 Comparative Analysis of forecast between SARIMA and
SETAR models
As described by most researchers, forecasting is an important phenomenon to effective
decision making and policy planning. It is therefore necessary to obtain good models
that provide best forecast values. The main task in this research is to compare the
forecasts performance between the linear SARIMA model and the non-linear SETAR
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Model for both inflation and CIC rates. This is core to the policy maker and the planner
for effective economic growth.
Inflation rate
The tentative models for inflation have shown to adequately fit the inflation data
well and satisfied all model assumptions. Confidently, we use these models to
forecast inflation rate over 12 months period from 2015(8) to 2016(9). Thus, we
compare forecast performance between the tentative models in terms of their RMSE
and MAE. As shown in Table 4.25, none of these models recorded consistent least
values from both in sample and out of sample forecasts. Inconsistently, the model
ARIMA(3,1,3)(1,0,1)12 recorded the most least values of RMSE and MAE for the
in-sample forecasts but failed to record same for the out of sample forecasts. The
ARIMA(1,1,2)(1,0,1)12 model also recorded the most least values for the out of
sample forecasts but recorded otherwise for the in-sample forecasts. This makes
it quite difficult to select any of the models to represent inflation in terms of the
most least error term measures. Hence, the maximum P-values resulting from the
Ljung-Box and ARCH tests in Table 4.20 were relied on to make a choice on the most
appropriate linear SARIMA model to consider for comparison with the nonlinear
SETAR model. Finally, we consider ARIMA(3,1,3)(1,0,1)12 as most appropriate for
comparison because it produced the most maximum P-values from table 4.8. This is
because the greater the P-value from 5%, the better the model. The chosen model as
well satisfies all SARIMA modelling assumptions.
Table 4.25: Forecast results of competing SARIMA models for inflation
Out of sample In-sample
Model
RMSE MAE RMSE MAE
ARIMA(3,1,3)(1,0,1)12 2.7711 2.7709 0.0738 0.0494
ARIMA(1,1,2)(1,0,1)12 2.7686 2.7683 0.0755 0.0506
ARIMA(1,1,1)(2,0,1)12 2.7834 2.7832 0.0754 0.0503
As explained earlier for Table 4.25, Table 4.26 demonstrates similar results. Yet again,
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no particular model shows consistent values for both in sample and out of sample
forecasts. Also, the P-values from Table 4.20 with the largest values from the Ljung-
Box and ARCH -LM tests were relied on to make a choice on the most appropriate
model to consider for valuation. The model SETAR (2;14,13) is chosen to be the
most appropriate model. This is because, it displayed to have the largest P- values.
Table 4.26: Forecast results of competing SETAR models for inflation
Out of sample In-sample
Model
RMSE MAE RMSE MAE
SETAR (2;14,4) 0.0015 0.0350 0.0058 0.0528
SETAR (2;14,5) 0.0021 0.0394 0.0055 0.0514
SETAR (2;14,13) 0.0017 0.0353 0.0057 0.0514
The selected models from the two modelling approaches have shown to fit the inflation
data well and satisfied all model assumptions. Although, the nonlinear SETAR model
outperformed the linear SARIMA model for inflation rates by their least error terms,
the two models were compared to see which of the models is more significant. This
comparison is done using the Diebold - Mariano test under the null hypothesis of
the existence of significant difference between the two models. Results from Table
4.27 indicates all the P-values from the five different forecast horizons was rejected.
Hence, it was concluded that the linear SARIMA and nonlinear SETAR models are
statistically significant to forecast inflation.
Table 4.27: Forecast accuracy test for inflation
Forecast Horizon DM statistic P-value
1 -0.2488 0.8037
2 -0.2397 0.8107
3 -0.2443 0.8071
4 -0.2767 0.7822
5 -0.2958 0.7676
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Currency in Circulation
To choose the most appropriate linear SARIMA model for comparison as we did for
the inflation rate, the three competing SARIMA models shown to satisfy all model
assumptions are examined. In Table 4.28 below, no model seems to have consistent
least values of RMSE and MAE for their out of sample and in sample forecasts over
our sample forecast period from 2015(8) to 2016(9). So, we inferred from the results
of Table 4.14 and adjudged ARIMA(0,1,0)(3,0,1)12 as our best model. We did this,
by choosing the model with the most maximum P-values resulting from the Ljung-
Box and ARCH -LM tests.
Table 4.28: Forecast results of competing SARIMA models for CIC
Out of sample In-sample
Model
RMSE MAE RMSE MAE
ARIMA(0,1,0)(3,0,1)12 8.8428 8.8428 0.2268 0.0874
ARIMA(0,1,0)(1,0,3)12 8.8700 8.8699 0.2249 0.0827
ARIMA(0,1,0)(3,0,3)12 8.9813 8.9812 0.2016 0.0795
Interestingly, the SETAR model for the CIC rate is just one and once it satisfies all
SETAR modelling assumptions, therefore the model was chosen from Table 4.29 for
further investigation.
Table 4.29: Forecast result of SETAR model for CIC
Out of sample In-sample
Model
RMSE MAE RMSE MAE
SETAR (2;13,9) 191086.54 280.35587 23697.61 74.85154
The Diebold - Mariano test was employed to test the forecast accuracy of the two
selected models for SARIMA and SETAR models. The test was performed under the
null hypothesis of the two models being significant. From Table 4.30, five forecasts
were considered to test the statistical significance of the models. The results from
the test indicates only one model is significant to forecast inflation, indicating that the
null hypothesis can be rejected. However, by comparing the error terms from both
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models, out rightly it can be concluded that the linear SARIMA model outperforms
the nonlinear SETAR model.
Table 4.30: Forecast accuracy test for CIC
Forecast Horizon DM statistic P-value
1 -3.0383 0.0026
2 -2.6836 0.0077
3 -2.6397 0.0087
4 -2.6492 0.0085
5 -2.6525 0.0084
4.4 Results and Discussion
An observation of datasets for inflation and CIC rates did not appear to follow a
normal distribution, this was confirmed using the Jarque -Bera test. While inflation
rate had a broader surface due to its positively skewed value greater than three,
CIC rate showed a positively skewed value less than three indicating CIC values
are peaked about the mean value as compared to a normal distribution. The series
for both rates were observed to be nonstationary through their time series plots.
These series were therefore logarithmically transformed and differenced to attain
stationarity. Additionally, the differenced series showed a sine- wave like structure in
the ACF’s of the datasets, indicating the presence of both seasonal and non-seasonal
behaviour in the series. However, after the first non-seasonal differencing the series
became stationary, hence there was no need seasonally differencing the series. These
transformed and differenced series was then used to build the appropriate models
following procedures of Box-Jenkins. All chosen tentative models for both inflation
and CIC performed well in terms of their ACF plots of residuals indicating that any
of these models can serve as a good model for forecasting. This was confirmed
through diagnostic checks on residuals of the tentative models. Further results from
the ARCH-LM and Ljung-Box tests depict all chosen models are good for forecasting
since they were above 5% level of significance and hence, satisfy all assumptions of
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the SARIMA model.
Before fitting the SETAR model, the datasets for both inflation and CIC rates must
be nonlinear. So, a linearity test was performed on both rates using the Keenan and
Tsay tests for linearity under their respective null hypothesis. The resulting P-values
were less than 5%, implying the datasets follow threshold nonlinear. In like manner
as in the SARIMA modelling, a diagnostic check was performed on the residuals of
the fitted model(s) after performing a grid search of tentative models. Again, results
from the ARCH and Ljung -Box tests showed P-values satisfying the assumptions of
SETAR model.
The fear of the “unknown” makes the policy maker and the planner disorganized, since
it becomes difficult to plan into the future. However, as discussed by Box and Jenkins
(1976), forecasting provides the basis for effective economic planning, optimization
of industrial activities and improved inventory and production control. Therefore,
obtaining good models to provide accurate forecasts is core to the researcher, policy
makers, investors and governments. Inflation and CIC rates are some examples of
the major drivers of an economy of which Ghana is not left out. Hence, forecasting
of these variables will serve as a guiding tool to financial and economic analysts
to effectively make decisions and advise key stakeholders to formulate important
policies to reduce inflation rates and regulate CIC rates. As a result, the study adjudges
ARIMA(3,1,3)(1,0,1)12 and SETAR (2;14,13) as best models in predicting inflation
rates. Concurrently, ARIMA(0,1,0)(3,0,1)12 and SETAR(2;13,9)are best models to
predict CIC rates in Ghana.
Further checks on the chosen models in terms of their forecast accuracy using the
Diebold-Mariano test revealed that both SARIMA and SETAR models can be used to
forecast inflation. Since, the test proved there was no significant difference between the
two models. Divergently, there was significant difference between the SARIMA and
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SETAR models used in predicting the CIC rate. This could have resulted due to vast
difference in the criterion used to measure forecast accuracy. Therefore, we predict
inflation using both models for a twelve months period. However, for CIC rates we
prefer to use the SARIMA model for predictions.
4.5 Chapter Conclusion
This chapter has comprehensively dealt with analyses and discussions from the study.
It also represented concise and precise major findings of the study. The chapter
also showed that the two rates considered increased in regular intervals, implying the
datasets are seasonal by nature. Further test using the Jarque-Bera test revealed the
datasets do not follow a normal distribution. In affirmation, the linearity test proposed
by Keenan and Tsay proved the datasets are threshold nonlinear. Additionally, forecast
performance indicates that the behaviour of a dataset does not necessarily inform the
best model fit. This was confirmed in the analysis for the CIC rate. Even though, the
dataset appeared to be nonlinear, the linear model fitted and forecasted the rate better.
Hence, it can be concluded that the SARIMA model is best for forecasting CIC rates.
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CHAPTER 5
CONCLUSION AND RECOMMENDATIONS
5.1 Introduction
This chapter represents conclusion and recommendations of the study.
5.2 Conclusion
In the study, SARIMA and SETAR models were fitted to time series datasets on
monthly inflation and currency in circulation (CIC) rates obtained from the Bank of
Ghana. Observation of the ACF and PACF of the datasets showed that the data were
not stationary. This was confirmed by the KPSS and PP unit root tests on the datasets.
The datasets were logarithmically transformed and differenced non-seasonally to
obtain a stationary time series. In the linear SARIMA modelling, the Box-Jenkins
approach was used and identified twelve and four tentative models for inflation and
CIC rates respectively. Amongst the tentative models, ARIMA(3,1,3)(1,0,1)12 and
ARIMA(0,1,0)(3,0,1)12 were respectively identified as best models for inflation and
CIC rates in the sense that, it gave least values of AIC, MAPE, RMSE and MAE.
Further diagnosis proved that all SARIMA model assumptions have been satisfied.
Thus, the models are adequate.
In the nonlinear SETAR modelling, Keenan and Tsay-F tests showed the datasets
were threshold nonlinear, so a two regime SETAR model was applied. A grid search
of tentative models revealed three and one SETAR models for inflation and CIC,
respectively. SETAR (2;14,13) and SETAR (2;13,9) were identified amongst the
tentative models as the best models. Again, further diagnosis showed the datasets
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satisfied all model assumptions. After modelling both rates, the performance between
the SARIMA and SETAR models were compared for inflation and CIC rates by
employing forecast measures, Root Mean Square Error (RMSE) and Mean Absolute
Error. Based on forecast results for the period of twelve months, the nonlinear SETAR
model outperformed the linear SARIMA model for inflation. However, the case
was different for CIC rates, the Linear SARIMA model outperformed the nonlinear
SETAR model. This confirms the stand of Gooijer and Kumar (1992) that there is no
clear evidence either in favour of linear or nonlinear models in terms of their forecast
performance.
Although, the results of forecast performance between the nonlinear SETAR model
and the linear SARIMA model showed the SETAR model has superiority over the
SARIMA model, the Diebold-Mariano test suggests both models are significant for
forecasting Inflation rates.
Finally, the results for forecast performance between models showed the linear
SARIMA model is superior to the nonlinear SETAR model. Hence, we conclude that
the SARIMA model is best to model and forecast inflation. This is confirmed from the
Diebold-Mariano test for different forecast horizons.
5.3 Recommendations
The research compared the linear SARIMA and nonlinear SETAR models. Since
models approximate reality, it is recommended that more research should be done
considering other linear and nonlinear models.
The results from the research revealed that the SARIMA model is better off in
predicting CIC rates as compared to the SETAR model; it is therefore recommended
close monitoring of financial markets should be done on regular basis in order to
regulate the total amount of money supply in the economy.
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