University of Ghana http://ugspace.ug.edu.gh UNIVERSITY OF GHANA A METHODOLOGY FOR STOCHASTIC MONITORING OF MACRO-ECONOMIC VARIABLES IN GHANA BY AMA ASANEWA DICKSON (10386728) THIS THESIS IS SUBMITTED TO THE UNIVERSITY OF GHANA, LEGON IN PARTIAL FUFILLMENT OF THE REQUIREMENT FOR THE AWARD OF MPHIL. STATISTICS DEGREE JULY, 2018 University of Ghana http://ugspace.ug.edu.gh DECLARATION This is to certify that this thesis is the result of my own research work and that no part of it has been presented for another degree in this university or elsewhere. SIGNATURE:…………………………. DATE……………………….. AMA ASANEWA DICKSON (10386728) 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. KWABENA DOKU-AMPONSAH (CO-SUPERVISOR) i University of Ghana http://ugspace.ug.edu.gh DEDICATION This Thesis is firstly dedicated to God for the strength and wisdom he gave me to complete this research work. I also dedicate it to my parents, Mr. Kwamena Dickson and Madam Ruth Ekua Akomaning and my entire family for all their support throughout my graduate studies. ii University of Ghana http://ugspace.ug.edu.gh ACKNOWLEDGEMENT My gratitude goes to God for the strength and insight He gave me for the completion of this work. Also to my supervisors, Dr. Felix Okoe Mettle and Dr. Kwabena Doku- Amponsah for their corrections, encouragements and fatherly advice they gave me to see me through the end of this work. The Almighty God bless them abundantly. I also acknowledge my siblings, Kobina Darko Dickson and Kwamena Sekyi Dickson and also the entire Akomaning family for supporting me. My gratitude also goes to all the lecturers and staff of the Statistics department especially; Dr. Minkah and Mr. Benedict Mbeah-Baiden. I finally want to acknowledge all my friends who helped and encouraged me including Eugene Darku, Stephen, Nafisa Ahmed, Jonathan Agyeman, James Acquah-Mantey and Nii Nai Sowah. iii University of Ghana http://ugspace.ug.edu.gh ABSTRACT The study is based on the stochastic monitoring of Macroeconomic variables in Ghana. The macroeconomic variables that were considered in the study included Inflation rate and the Gross Domestic Product (GDP). The data covered the period from 2009 to 2017 for the monthly inflation rate and 1961 to 2017 for the GDP. All analyses were done using the R software. The Augmented Dickey-Fuller test was used to test the data and the results showed that all the data were stationary after the data was transformed by differencing the data once. The performances of the models were tested using the Akaike Information Criterion (AIC). The models with the least AIC values were selected and subjected to the Box Pierce and L-Jung diagnostic tests. Based on the diagnostics tests, the ARIMA (1,1,2) with the highest p-value of 0.9669 at 5% level of significance was selected as the best model for the Inflation rate time series analysis and the ARIMA (1,1,1) with the highest p- value of 0.9892 was chosen as the best model for the GDP time series data. The residuals were obtained from the appropriate models to obtain the quantile values to set upper and lower bounds around the forecasted values using the expected trend lines. Both the linear and curve-linear expected trend lines were employed in the study. In monitoring the achievement of the set targets, the forecast accuracies were estimated using the Mean Absolute Percentage Error. The results showed that both variables performed well when preceding values were around a linear expected trend line. It also showed that the methodology performs well for both high targeted and low targeted variables. It is recommended from the study that Policy makers and governments should employ the methodology in monitoring achievement of set targets. It is also recommended that researchers should consider further studies using other non-linear models and different periods for the forecasting. iv University of Ghana http://ugspace.ug.edu.gh TABLE OF CONTENTS DECLARATION .................................................................................................................. i DEDICATION ..................................................................................................................... ii ACKNOWLEDGEMENT ................................................................................................ iii ABSTRACT ........................................................................................................................ iv TABLE OF CONTENTS .................................................................................................... v LIST OF TABLES ............................................................................................................ vii LIST OF FIGURES ........................................................................................................... ix LIST OF ABBREVIATIONS ........................................................................................... xi CHAPTER ONE ................................................................................................................. 1 INTRODUCTION ............................................................................................................... 1 1.0 Background of study ................................................................................................... 1 1.1 Problem Statement ................................................................................................... 3 1.2 Objectives .................................................................................................................... 4 1.2.1 Main Objective ..................................................................................................... 4 1.3 Significance of the Study ............................................................................................ 4 1.4 Scope and Methodology .............................................................................................. 5 1.5 Organization of Study ................................................................................................. 5 CHAPTER TWO ................................................................................................................ 7 LITERATURE REVIEW ................................................................................................... 7 2.0 Introduction ................................................................................................................. 7 2.1 Inflation ....................................................................................................................... 7 2.1.1 World Inflation ..................................................................................................... 8 2.1.2 Inflation in Ghana ............................................................................................... 10 2.1.3 Factors affecting Inflation rates in Ghana........................................................... 11 2.2 Gross Domestic Product (GDP) ................................................................................ 13 2.2.1 Factors Influencing GDP .................................................................................... 15 2.3 Review of Related Works .......................................................................................... 17 2.3.1 ARIMA Model .................................................................................................... 17 2.3.2 Other Fields where time series models have been used...................................... 20 2.3.3 Extreme Value Theory ........................................................................................ 24 2.4 Conclusion ................................................................................................................. 26 CHAPTER THREE .......................................................................................................... 27 METHODOLOGY ............................................................................................................ 27 3.0 Introduction ............................................................................................................... 27 3.1 Source of Data and Collection of Data Procedure .................................................... 27 3.2 Variables Description. ............................................................................................... 27 3.3 Method of Data Analysis ........................................................................................... 28 v University of Ghana http://ugspace.ug.edu.gh 3.4 Time Series and its Basic Concepts .......................................................................... 30 3.5 Stationarity Process ................................................................................................. 31 3.6 Augmented Dickey Fuller Test ................................................................................. 33 3.7 Differencing Process ................................................................................................. 33 3.8 ARIMA Model .......................................................................................................... 34 3.9 Identification of Model, Estimation and Diagnostic Checking ................................. 37 3.9.1 Model Identification ........................................................................................... 37 3.9.2 Estimation ........................................................................................................... 38 3.9.3 Diagnostic Checking ........................................................................................... 38 3.9.4 Model Validation ................................................................................................ 38 3.10 Extreme Value Theory ............................................................................................ 39 3.10.1 The Generalised Pareto Distribution................................................................. 41 3.10.2 The Asymptotic Model Characterisation of the Generalized Pareto Distribution ..................................................................................................................................... 42 3.10.3 Extreme Quantiles............................................................................................. 44 3.10.4 Diagnostic Check .............................................................................................. 44 3.11 Expected Trend Line ............................................................................................... 45 CHAPTER FOUR ............................................................................................................. 47 ANALYSIS OF DATA AND RESULTS DISCUSSION ............................................... 47 4.0 Introduction ............................................................................................................... 47 4.1 Data description and Summary Statistics .................................................................. 47 4.2 Preliminary Analysis ................................................................................................. 48 4.3 Fitting a Time Series Model ...................................................................................... 52 4.4 Estimation of quantile values of Variables ................................................................ 62 4.5 Monitoring the achievement of set target of Inflation rate ........................................ 64 4.5.1 Analysis of Inflation given Curve-linear expected trend line ............................. 68 4.6 Monitoring the achievement of set target of GDP values ......................................... 72 4.6.1 Analysis of GDP given Curve-linear expected trend line ................................ 75 4.7 Discussion of Results ................................................................................................ 79 CHAPTER FIVE ............................................................................................................... 82 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS .................................. 82 5.0 Introduction ............................................................................................................... 82 5.1 Summary ................................................................................................................... 82 5.2 Conclusion ................................................................................................................. 84 5.3 Recommendations ..................................................................................................... 85 REFERENCES .................................................................................................................. 87 APPENDICES ................................................................................................................... 92 vi University of Ghana http://ugspace.ug.edu.gh LIST OF TABLES Table 2.1.2: Yearly average Inflation in Ghana from 1992 to 2012 ................................... 11 Table 4.1.1: Descriptive Statistics of Ghana’s monthly Inflation rates (2009-2017) and GDP values (Billion$; 1961 – 2017) .............................................................. 48 Table 4.2.1: Augmented Dickey-Fuller Test for Ghana’s monthly inflation rate ............... 50 Table 4.2.2: Augmented Dickey-Fuller Test for Ghana’s transformed monthly inflation rate (2009 to 2017) and GDP values (Billion$; 1961 – 2017) ....................... 52 Table 4.3.1: Suggested ARIMA(p,d,q) models and their AIC values for Inflation. ........... 54 Table 4.3.2: Suggested ARIMA (p,d,q) models and their AIC values for the annual GDP ........................................................................................................................ 55 Table 4.3.3: ARIMA outputs of Box-Pierce test for Ghana’s monthly inflation rate ......... 56 Table 4.3.4: ARIMA model output of Box-Pierce test for Ghana’s GDP values. .............. 58 Table 4.3.5: Model Output for ARIMA (1,1,2) Inflation rate ............................................ 61 Table 4.3.6: Model Output for ARIMA (1,1,1) GDP ......................................................... 61 Table 4.3.7: Six-months out sample forecast of the monthly inflation from the ARIMA (1,1,2) ............................................................................................................. 62 Table 4.3.8: Three years out sample forecast for the yearly GDP from ARIMA (1,1,1) ... 62 Table 4.4.1: Descriptive Statistics and Quantile values of residuals of ARIMA models ... 64 Table 4.5.1: Forecasted values of Inflation using the linear expected trend line and lower and upper bounds of quantile values .............................................................. 65 Table 4.5.2: Forecasted Inflation values using ARIMA (1,1,2) for a linear expected trend line given a 0.5 quantile bound ...................................................................... 66 Table 4.5.1.1: Forecasted values of Inflation using the curve-linear expected trend line and lower and upper bounds of quantile values ............................................. 68 vii University of Ghana http://ugspace.ug.edu.gh Table 4.5.1.2: Forecasted Inflation values using ARIMA (1,1,2) for a curve-linear expected trend line given a 0.5 bound ............................................................ 69 Table 4.5.1.3: Proportions of Forecasted values within 5%and 10% of targets given the expected trend lines and the bounds............................................................... 71 Table 4.5.1.4: Mean Absolute Percentage Error for Inflation rate given the expected trend line and the various quantile bound ............................................................... 72 Table 4.6.1: Forecasted values of Inflation using the linear expected trend line and lower and upper bounds of quantile values .............................................................. 73 Table 4.6.2: Forecasted GDP values using ARIMA (1,1,1) for a linear expected trend given a 0.5 bound ........................................................................................... 73 Table 4.6.1.1: Forecasted values of GDP using the curve-linear expected trend line and lower and upper bounds of quantile values .................................................... 76 Table 4.6.1.2: Forecasted GDP values using the ARIMA (1,1,1) for a curve-linear expected trend given a 0.5 bound ................................................................... 76 Table 4.6.1.3: Proportions of forecasted values that fall within 5% and 10% of set target given the expected trend lines and the bounds ............................................... 78 Table 4.6.1.4: Mean Absolute Percentage error for GDP value given the expected trend line and the various quantile bounds. ............................................................. 79 viii University of Ghana http://ugspace.ug.edu.gh LIST OF FIGURES Figure 4.2.1: A plot of Ghana’s monthly rate of inflation (2009 to 2017) ........................ 49 Figure 4.2.2: A plot of annual GDP values of Ghana (1961 to 2016) ................................ 49 Figure 4.2.3: A plot of Ghana’s transformed monthly rate of inflation (2009 to 2017) ..... 51 Figure 4.2.4: A plot of Ghana’s transformed annual GDP values (1961 to 2017) ............. 51 Figure 4.3.1: Autocorrelation Plot for the first difference of monthly rate of inflation (2009 to 2017) .............................................................................................. 53 Figure 4.3.2: Partial Autocorrelation plot for the first difference of the monthly rate of inflation (2009 to 2017) ................................................................................ 53 Figure 4.3.3: ARIMA (1, 1, 1) output of Ljung-Box test for Ghana’s inflation rate. ......... 56 Figure 4.3.4: ARIMA (2, 1, 1) output of Ljung-Box test for Ghana’s inflation rate. ......... 57 Figure 4.3.5: ARIMA (1, 1, 2) output of Ljung-Box test for Ghana’s inflation rate. ......... 57 Figure 4.3. 6: ARIMA (0, 1, 0) output of Ljung-Box test for Ghana’s GPD values .......... 59 Figure 4.3.7: ARIMA (1, 1, 0) output of Ljung-Box test for Ghana’s GDP values. .......... 59 Figure 4.3.8: ARIMA (1, 1, 1) output of Ljung-Box test for Ghana’s GDP values. .......... 60 Figure 4.4.1: Mean excess plot for ARIMA (1,1,2) residuals. ........................................... 63 Figure 4.5.1: Time Series indicating forecasts from the linear expected trend line for a 0.5 bound ............................................................................................................ 66 Figure 4.5.2: Time series indicating forecasts from the linear expected trend line for a 0.7 bound ............................................................................................................ 67 Figure 4.5.3: Time series indicating forecasts from the linear expected trend line for a 0.9 bound ............................................................................................................ 67 Figure 4.5.1.1: Time series indicating forecasts for the curve-linear expected trend line for a 0.5 bound ................................................................................................... 69 ix University of Ghana http://ugspace.ug.edu.gh Figure 4.5.1.2: Time series indicating forecasts for the curve-linear expected trend line for a 0.7 bound ................................................................................................... 70 Figure 4.5.1.3: Time series indicating forecasts for the curve-linear expected trend line for a 0.9 bound ................................................................................................... 70 Figure 4.6.1: Time series indicating forecasts from the linear expected trend line for a 0.5 bound ............................................................................................................ 74 Figure 4.6.2: Time series showing forecasts from the linear expected trend line with a 0.7 bound ............................................................................................................ 74 Figure 4.6.3: Time series showing forecasts from the linear expected trend line for a 0.9 bound ............................................................................................................ 75 Figure 4.6.1.1: Time series showing forecasted values from curve-linear expected trend line with a bound of 0.5 ................................................................................ 77 Figure 4.6.1.2: Time series showing forecasts from the curve-linear expected trend line for a 0.7 bound ................................................................................................... 77 Figure 4.6.1.3: Time series showing forecasted values from curve-linear expected trend line for 0.9 bound.......................................................................................... 78 x University of Ghana http://ugspace.ug.edu.gh LIST OF ABBREVIATIONS ACF Autocorrelation function ADF Augmented Dickey Fuller AIC Akaike Information Criterion ARCH Autoregressive Conditional Heteroscedastic ARFIMA Autoregressive Fractionally Integrated Moving Average ARIMA Autoregressive Integrated Moving Average BIC Bayesian Information Criterion CPI Consumer Price Index EGARCH Exponential Generalised Autoregressive Conditional Heteroscedastic EVT Extreme Value Theory FDI Foreign Direct Investment GARCH Generalised Autoregressive Conditional Heteroscedastic GDP Gross Domestic Product GEV Generalised Extreme Value MAE Mean Absolute Error MAPE Mean Absolute Percentage Error PACF Partial Autocorrelation function POT Peak Over Threshold RMSE Root Mean Square Error SARIMA Seasonal Autoregressive Integrated Moving Average SBC Schwartz Bayesian Criterion TGARCH Threshold Generalised Autoregressive Conditional Heteroscedastic VAR Vector Auto Regression VECM Vector Error Correction Model xi University of Ghana http://ugspace.ug.edu.gh CHAPTER ONE INTRODUCTION 1.0 Background of study The numerous needs of people in the society and the limited resources available pose a challenge on policy makers, firms, economists and individuals. Individuals, firms and economies have to therefore properly plan and order their needs in order to satisfy the most pressing ones. Satisfying the needs of people in the society and a country at large is one of the most important responsibilities of governments in every growing nation. It is the prime duty of government and policy makers to create an environment and make policies that will achieve the aim of meeting the unlimited needs of people in the country. In this global world in which we live, the success of an economy’s performance of all countries can be attributed to the performance of some macro-economic variables. Macroeconomics as a branch of economics is the study of the behaviour of the overall economy of a nation or a country. It looks at sections of the economy including the Gross Domestic Product, Inflation rate, Unemployment rate, growth rate, Consumer Price index, Interest rates and other factors. Macroeconomic variables play an important role in the economy of every country. One can say that they control the economy. Particularly, three important macroeconomic variables that play significant role in the control of the economy are Gross Domestic Product (GDP), Unemployment rate and Inflation rate. The gross domestic product (GDP) is one of the basic tools for measuring how an economy is performing over a period. It is determined by computing the market value of 1 University of Ghana http://ugspace.ug.edu.gh all final goods and services produced in a country for a period usually a quarter or year. The GDP of nations change from year to year and some countries have higher GDPs compared to other countries. According to the World Bank in 2016, The country with the highest GDP was the United States of America with a total GDP of $18,569,100,000,000 followed by the People’s Republic of China with a GDP of $11,199,145,000,000. Ghana places 85th on the World Bank ranking with a GDP of $42,690,000,000. Unemployment rate is the fraction of the labour force within an economy, which are not gainfully employed but are actively looking for employment. Ghana’s Unemployment rate has been fluctuating over time as population increases and as a lot more people graduate from tertiary institutions, the numbers keep increasing. Governments and political parties in opposition continue to suggest ways to create employment in order to stop the increasing rate of unemployment in Ghana. Unemployment rate in Ghana hit a high of 10.36% in 2000, 3.6% in 2006 and 5.766% in 2016 (World Bank, 2016). Inflation rate is the percentage increase in the prices of goods and services overtime or the decline of purchasing power of money. In Ghana, the rate at which the prices of goods increase is a major concern for governments and citizens and there has always been a struggle to reduce the inflation rate to a single digit. It is the priority of every state to manage the indicators of growth in the economy to create a stable and conducive economy for individuals, firms, economists and policy makers in the country. A higher rate of GDP, a lower unemployment rate and a lower rate of inflation are indicators of high economic performance in every country. A vice versa of the aforementioned; implies a decline or a fall in economic growth. A poor performance in these variables leads to unmerited redistribution of wealth in the case of high inflation rate. The unmerited redistribution causes money lenders to loose while their borrowers gain and 2 University of Ghana http://ugspace.ug.edu.gh the same fate is suffered by employers while their subordinates gain and this is because usually loans and salaries are in monetary terms. A high inflation rate implies a loss in the value of money. In terms of the unemployment rate, a high unemployment rate implies less disposal income to be invested into the economy and this in the long run will affect the economy of any nation. 1.1 Problem Statement There is no well-known scientific way of monitoring targets set by governments. Governments make promises as they campaign to win elections but there are no scientific ways to the best of our knowledge of monitoring to see the execution of the promises once they assume office. The citizens and electorates usually wait for the end of the term of the government to assess whether the promises have been fulfilled or executed. In the same way companies and organizations have some organizational structures that aid monitoring and evaluation once a project kick starts but usually there is no scientific way of modelling to monitor exact progress of targets set. Most researches done on macroeconomic variables are more related to forecasting the variables, determining factors affecting these variables and the impact they make on economic growth but little has been done in the area of monitoring the performance of these macro-economic variables within an economy (Yergin & Stainslaw, 1997; Sumaila & Laryea, 2001; Ocran, 2007; Aziz & Azmi, 2017). The little or lack of information available on performance monitoring of the variables creates a discontinuity in literature. This poses a challenge as to how to monitor variables such as GDP, Inflation rate and Unemployment rate within an economy. 3 University of Ghana http://ugspace.ug.edu.gh In view of this, the study seeks to develop a stochastic model to monitor macro-economic variables in an economy. 1.2 Objectives The objectives of this study is divided into two sections namely; main objective and specific objectives. Below are the main and specific objectives of the study. 1.2.1 Main Objective To develop a stochastic model to determine whether or not preceding values of macroeconomic variables around an “expected trend line” will lead to achieving target set at a future point in time 1.2.2 Specific Objectives: 1. To check the performance of the model when preceding values are around a “linear expected trend line”. 2. To check the performance of the model when preceding values are around a “curve-linear expected trend line”. 3. To compare the model’s performance between high targeted and low targeted macroeconomic variables. 1.3 Significance of the Study The results and conclusions from this study would be significant to policy makers such as the government, businesses and the general public as well as academics and researchers due to the following reasons: 4 University of Ghana http://ugspace.ug.edu.gh 1. Developing a stochastic model that will aid in monitoring of macro-economic variables will be very useful in the planning activities of the government, businesses and the public in general. 2. The results from this study will benefit academia and research by contributing to existing literature by closing or the elimination of the gap in literature or information. The results will also aid in further research in academia and help improve operations in industry. 1.4 Scope and Methodology The study employed secondary time series data on two macro-economic variables namely: Inflation rate and GDP over the period 1960 to 2017 in Ghana. This gave a total of fifty seven data points which is sufficiently large for the study. The data was sourced from the Bank of Ghana. The time series modelling using the Autoregressive Integrated Moving Average (ARIMA) and a concept in extreme value theory was used in analyzing the data. The statistical software package that was used in the data analysis is the R software. 1.5 Organization of Study The rest of the thesis is organized as follows: Chapter Two provides a brief overview and review of the macro-economic variables that are being investigated in the study, which are inflation and GDP. In the same chapter, the applications of time series model specifically the Autoregressive Integrated Moving Average (ARIMA) to several data will be reviewed as well as a review of the applications of the extreme value theorem to various occurrences or events in literature. 5 University of Ghana http://ugspace.ug.edu.gh Chapter Three provides a detail of the methodology employed for the analysis of the data in the study, including the time series model and the extreme value theorem. In Chapter Four, the time series data on GDP and inflation in Ghana are subjected to the methods discussed in chapter Three. Chapter Five is a summary of the findings and a conclusion of the study with some recommendations based on the findings that will be available for further research or policy considerations. 6 University of Ghana http://ugspace.ug.edu.gh CHAPTER TWO LITERATURE REVIEW 2.0 Introduction This chapter reviews some relevant theoretical models and concepts associated with the study and related research made on the topic area. The chapter is divided into Introduction, reviewed literature on Inflation and GDP and some models in literature including ARIMA and extreme value. 2.1 Inflation Inflation is said to be the consistent and significant rise in the general price levels of goods and services within a country or an economy. It can also be seen as the consistent decrease in the purchasing power of money within an economy. In defining Inflation, the continuous and rapid increase of goods and services in an economy is key and therefore inflation measures change in price levels of goods and services. The Merriam-Webster’s Collegiate Dictionary defines Inflation as a continuing rise in the general price levels within a period. Inflation all over the world has been erratic over the years and increases and falls depending on other factors. Inflation can be measured by various indexes such as consumer price index, producer price index, cost of living index, commodity price index and the Gross Domestic Product (GDP) deflator however the Consumer Price Index (CPI) is often used in determining inflation and particularly in Ghana, the Ghana Statistical Service uses the CPI to determine the rate of inflation for a period (Ocran, 2007). Inflation ultimately affects the livelihood of citizens in a country and it is very sensitive in developing economies such as Ghana and therefore political parties have always included 7 University of Ghana http://ugspace.ug.edu.gh the need to reduce the inflation rate in their campaign messages. Ghana governments have strived to maintain single digit inflations in order to be able to score political points. 2.1.1 World Inflation There have been studies all over the world on Inflation in different economies. A lot of studies on Inflation in other parts of the world have focussed on the relationship between Inflation and other macroeconomic indicators. In examining the relationship between inflation and economic growth, Gunasinghe (2007) used Sri Lanka as a case study to access the controversy between the two policy views. Using an observation and employing some econometric techniques on it, Gunasinghe pointed out that there exists a negative relation between inflation and economic growth but its contribution to growth in the economy is minimal. The work done by Benkovskis et al (2011) also focussed on the impact of inflation on the economy and they revealed that inflation shows a strong persistence with a negative impact. The impact of inflation, according to their study was still very dominant in the economy even three years after the shock. They further investigated the link between output and inflation and they studied how the shocks that hit the economy affected economic growth. Their analysis suggested that the relationship between inflation and activity has indeed not been static over time and the growth was very much dependent on the shocks in the economy at a point in time. The relationship between inflation and unemployment has also been widely investigated. Results from the various researches made on the relationship can be grouped into two schools of thoughts. One thought being that there is a direct relationship between inflation and unemployment and the other being that there is an inverse relationship between the two macroeconomic variables. Wajid and Kalim (2012) used Pakistan as a case study to study the extent to which inflation and growth in the economy affect unemployment. From their study, it was revealed that inflation affects unemployment positively in the long run 8 University of Ghana http://ugspace.ug.edu.gh whiles economic growth in the long run has a significant negative impact on unemployment. It also revealed that trade openness in the long run really has no significant impact on unemployment but there exists a significant impact in the short run. Yergin and Stanislaw (1997) on the other hand researched into the causes of Inflation in the United States of America within the period and their study revealed that U.S. inflationary periods were highly associated by the increase in domestic spending and the Vietnam war in the year 1971. In addition to this, the country was running a trade and balance of payment deficits with a spontaneous rise in money supply in the first six months of the year. Also the depreciation of the dollar with respect to the Deutsche Mark of Germany and other factors such as increase in labour cost, high price of domestic goods and services among others can be attributed to the inflationary situation at that time. In conclusion one can say that U.S’s past and present situations in relation to inflation in the country are highly attributed to expanding fiscal and monetary policy. In Africa, a research by Sumaila and Laryea(2001) in relation to the determinants of inflation in Tanzania revealed that output and monetary factors are the major causes of inflation in the country in the short run whereas in the long-run exchange rate in addition to the other causes mentioned in the short run also influences inflation. Assessing the main determinants of inflation in Ethiopia and Kenya, Dureval and Sjo (2012) developed single- equation error correction models for the CPI in each country. Their research considered sources of current surge in Inflation such as exchange rate, money supply, world energy prices, food and non-food world prices and shocks in domestic agricultural supply. The study revealed that monetary growth and shocks in agricultural supply had short to medium run effects whiles exchange rate and world food prices had long run impact. 9 University of Ghana http://ugspace.ug.edu.gh In West Africa, Nigeria’s economy has not being left out with the battle of the effect of inflation and other macroeconomic issues. Inflation in the economy can be associated with external shocks, money supply, growth in bank credit, budget deficits among other fiscal policies. Also several military interventions, exchange rate depreciation, increase in money supply to finance government deficits among others has caused the steady increase of inflation (Enu and Havi, 2014). 2.1.2 Inflation in Ghana Since Independence, Ghana has had to deal with high rates of inflation and every government is faced with the challenge of either reducing the rate or maintaining a single digit. Ocran (2007) revealed that during the period after independence, inflation in Ghana was in single digits and inflation rates were typically estimated at less than 1%. Factors such as deficits realized on government accounts as a result of frequent borrowing from the central bank, inadequate consumer goods and restrictions on imports accounted to inflationary issues faced in Ghana right after gaining independence (Lawson, 1966). During the period between 1960 and 1962, inflation in Ghana averaged 8% per year and then shot up to 23% per annum between 1964 and 1966. The brief period between 1966 and 1967 saw Ghana having a negative inflation. Between 1981 and 1983, inflation rates in Ghana shot up to over 100% for all the months in the year. The inflationary rates in most of the period 1972-1983 were mostly high and this was largely due to expansionary fiscal and loose monetary policies (Ocran, 2007). The high inflation in 1983 was also as a result of the devaluation of the cedi and the great famine that hit the country in that year. The late 80s saw a sharp decrease in inflation with an average of 40% and fluctuated between single digits and 20% before the end of the decade. According to Apaloo (2001), the average inflation rate for the period between 1986 and 1990 was between 20% and 40%. 10 University of Ghana http://ugspace.ug.edu.gh Table 2.1.2 shows the yearly average inflation in Ghana between 1992 and 2012. Inflation was averagely low hitting a high of 59.5 in 1995, and reducing steadily until single digit was attained in 2011. Table 2.1.2: Yearly average Inflation in Ghana from 1992 to 2012 Year Yearly Average Inflation 1992 10 1993 25 1994 24.9 1995 59.5 1996 46.6 1997 27.6 1998 19.2 2000 25.2 2001 32.9 2002 17.3 2003 29.8 2004 18 2005 15.4 2006 11.7 2007 10.7 2008 16.5 2009 19.3 2010 10.75 2011 8.73 2012 11.88 Source: Bank of Ghana (2013) The period between 2013 and 2016 when the New Democratic Congress governed the country was characterised by relatively low rates of inflation. Even though there were double digits of inflation rate, the government put in place measures to try to reduce it. 2.1.3 Factors affecting Inflation rates in Ghana There are numerous factors and other macroeconomic indicators that affect inflation rate in Ghana and some studies have been done to ascertain the extent to which these indicators or factors affect inflation. Oti-Boateng (1979) studied the problems and prospects of Inflation in Ghana and he asserted that factors such as increased money supply, increase in 11 University of Ghana http://ugspace.ug.edu.gh wages and salaries, a fall in productivity in the agricultural sector and finance deficits affect inflation in Ghana than global and imported inflation. Devaluation of the Cedi has been a major contributing factor in determining inflation and so Chhiber and Shaffik in 1990 studied the Inflationary consequence of Devaluation and Parallel markets with Ghana as a case study and it was revealed from their study that given the several contributing factors to Ghana’s Inflationary pressures, monetary factors still dominate. They further pointed out that the effect of policies by stakeholders on public revenue and expenditures contribute to the degree to which the depreciation of the local currency harms inflation Using the excess demand and the cost-push theories in econometric analysis to reveal the major causes of inflation in Ghana, Appiah and Bohene (2000) revealed in both short run and long run term the growth rate, GDP and that of money supply were the main causes of inflation in the country. Bawumia and Abradu-Otoo (2003) also in exploring the association between exchange rate, monetary growth and inflation in Ghana, concluded that there exist a long-run parity association between money supply, inflation, exchange rate and real income. Also their finding showed that there is an inverse relationship between real income and inflation in the country while in terms of the relationship between money supply, exchange rate and Ghana’s inflation is a positive one. Following the research of Bawumia and Abradu-Otoo (2003), Atta-Mensah and Bawumia (2003) in their study on Simple Vector Error Correction Forecasting model for Ghana asserted that the inflationary process in Ghana was solely a result of monetary experiences in the country. In an analysis of determinants contributing to dynamics of inflation in Ghana, Adu and Marbuah (2011) identified fiscal deficit, real output, exchange rate, interest rate and 12 University of Ghana http://ugspace.ug.edu.gh money supply as major factors playing a key role in the inflationary process in Ghana. They employed econometric techniques to establish the determinants contributing to the inflation, the findings concluded that a mixture of structural and monetary factors best explain the inflation in Ghana and this is consistent with findings of other researches made in relation to determinants of inflation. Lots of works have been done in relation to monetary factors influencing inflation. Ghana’s financial sector has improved and in recent years the sector has put in place transformational structures such as allowing citizens or residents of the country to operate “foreign currency denomination bank accounts” and the quick expansion of banking activities among others. This led to a research by Kovanen (2011) in relation to monetary factors influencing inflation in Ghana. His study disclosed that the influence money had on future inflation in the country was limited but, rather demand pressures and currency depreciation matters a lot in relation to future inflation in Ghana. Enu and Havi (2014) studied the influence of macroeconomic indicators such as foreign exchange and population growth, agricultural and services output as well as foreign direct investment on the inflationary situations experienced in Ghana and they concluded that in the long run, population growth, foreign aid, foreign direct investment and service’s output influence inflation in the country. They used co-integration analysis to achieve this. 2.2 Gross Domestic Product (GDP) The gross domestic product (GDP) is one of the measures of national income and output for an economy within a certain period. It is a way of appraising a country in terms of the performance of the economy and it is usually defined as the total market value of all final goods and services produced within a nation in a year. Landerfeld, Seskin and Fraumeni 13 University of Ghana http://ugspace.ug.edu.gh (2008) pointed out that in calculating the GDP of a country, intermediate goods are not computed, but only “new” products and services; this they revealed would avoid repeating already computed products. GDP has been studied by researchers widely because of its implications on the livelihood of the citizens of a country. Inconsistent GDP growth in an economy will lead to a higher occurrence of poverty and also hinder the improvement of other sectors such as health, education, and energy among others. The factors that contribute towards GDP growth are very necessary in ensuring that there is stability in the political climate as well as the social environment in a country (Aziz and Azmi 2017). GDP growth rates of countries have over the years been in the lime light in terms of policy making and economic management because GDP growth remains one of the most important main economic goals of any country. Poor economic growth is reflected in the figures that are put out by statisticians and when the growth figures are low then the citizens might see the ruling government as performing poorly in terms of economic management. Therefore, political leaders do their best to prevent this by putting in place measures to sustain high rates (Van den Bergh, 2008). The GDP is used to determine the economic performance of countries within a specific period. The GDPs of countries are used to classify countries into classes by the World Bank yearly. The classification includes; Low income countries, Lower middle income countries, Upper middle income countries and High income countries (World Bank, 2017). A notion of two-component that can be used to study GDP was suggested by Kitov in the year 2006. Those components are namely economic growth – a deviation or business cycle and an economic trend component. GDP being a function of economic growth, a lot of studies have been done on it to determine the growth rate of economies all over the world. 14 University of Ghana http://ugspace.ug.edu.gh 2.2.1 Factors Influencing GDP The Keynes model was adopted by Kira (2013) in Tanzania to find out factors that affect GDP in developing countries. It came to light in his studies that GDP exhibits continuous growth with acceleration and deceleration periods. He also revealed that consumption which can be categorised into household and government as well as export do affect GDP. He recommended that the investment sector must be attractive enough to encourage investors thereby creating a good impact on GDP. Usman (2016) also sought to find the influence of some major crops, livestock, and other crops to the growth of the Agriculture sector and how they affect the GDP growth rate of Pakistan. Applying a regression model to a 25 year secondary data from 1990 to 2014, his study pointed out that there is a significant impact of the above mentioned factors to both the growth in Agriculture sector and the growth rate of GDP as well. Ghura and Hadjmicheal (1996) interrogated the economic growth of countries within Sub- Saharan Africa. They used time series data that spanned from 1981 to 1992, their study pointed out that the significant growth was dependent on private as well as public investment. They implied that there is a direct relationship between economic growth, private and public investment. These findings were made possible by employing the generalised least square technique on a dataset of 29 Sub-Saharan African countries. Kawussi (1984) had earlier investigated a number of middle and low-income developing countries and their growth rates. His research revealed that high economic growth rate was a result of the correlation between high export growth rate. He also found out that this positive correlation that was exhibited between economic growth rate and the export growth rate was for both the middle and low income countries but the impact reduces depending on the country’s level in its development process. 15 University of Ghana http://ugspace.ug.edu.gh Gyimah-Brempong (1989) focussed on the effect of military spending on economic growth of some Sub-Saharan African countries, and his studies showed that there is an inverse relationship between expenditure by military and that of economic growth. Aziz and Azmi (2017) also explored an annual time series data on Inflation, Foreign Direct Investment (FDI) and Female Labour Force participation using the augmented Dickey Fuller and Ordinary Least Square method. Their findings concluded that there is a direct association between the growth in GDP and FDI as well as Female Labour Forces. Inflation on the hand is inversely related to GDP growth. Also among the three, Foreign Direct Investment was found to be the only variable that significantly contributes to GDP growth. In Ghana, some comprehensive studies have been done in the area of GDP and its relation with other factors or economic variables. Lloyd, Morrissey and Osei in 2001 critically examined aid and export and their relation to economic growth in Ghana. They pointed out that aid, export and public investment have a direct relationship with economic growth in the long run. Using data between the year 1970 and 2002 in studying the influence of FDI inflows and trade on economic growth in Ghana, Frimpong and Oteng-Abayie (2008) also identified that Ghana’s economic growth in the long term can be explained by factors such as trade, capital investment and labour . Antwi, Atta-Mills & Zhao (2013) also contributed to the study of the relation between GDP and other macroeconomic factors and their study brought to light the fact that changes in labour force in the short-run has no influence on economic growth but rather; inflation, foreign direct investment, foreign aid, government spending and physical capital in the long run do affect the economic growth in the country. 16 University of Ghana http://ugspace.ug.edu.gh 2.3 Review of Related Works Lots of works have been done in the application of ARIMA models and the extreme value theory in the analysis of various data. This section of the review explores various literature that have applied to these models both locally and internationally 2.3.1 ARIMA Model Autoregressive Integrated Moving Average (ARIMA) model is one among various non- stationary models mostly used by researchers in relation to building models and forecasting variables that are time series in nature. An important requirement in employing any time series technique is the stationarity of the data but most data encountered in practical situations turn to be non-stationary. This among various reasons is why most researchers use the ARIMA model. Others refer to the model as Box-Jenkins due to the popularisation of the model by George Box and Gwilym Jenkins in the year 1970 through the various publications they made. The ARIMA model is made up of three processes namely; the Autoregression (AR) process, Differencing (I) and the Moving Average process (MA). The AR (p) and the MA (q) are the stationary processes. AR(p) simply means the process is regressed on its past value p times and MA (q) means the process is regressed on its error terms q times. Therefore ARIMA (p,d,q) simply means that the process is differenced d times to make it stationary and it is regressed on p number of past values of the process itself and q number of error terms. An example is an ARIMA (2,1,1) which means that the process is differenced once (d=1) to become stationary and the series which is stationary is modelled as an ARMA (2,1) process that is it has two AR and one MA terms. The ARIMA model has been employed in several studies to forecast inflation worldwide. A study by Junttila (2001) compared models for forecasting time series and from the study 17 University of Ghana http://ugspace.ug.edu.gh it was realised that ARIMA models outperforms other models for forecasting time series data. The predicting ability of ARIMA models in the short-run period surpasses that of more sophisticated structural models (Aidan et al, 1999). Naveed and Iqbal (2016) compared the performance of various autoregressive integrated moving average (ARIMA) models, using time series data. According to their research, ARIMA (4,1,1) turned out to be better than the model that uses the forecasted errors as the bench mark. The Box-Jenkins approach to forecasting was employed in the study and the CPI was used as a proxy for inflation. The major problem of their studies was applying the Box-Jenkins approach to a few observations. Therefore the model must be updated from time to time as more data become available. Maity and Chatterjee (2012) also used ARIMA (1,2,2) to forecast the GDP growth rate of India, using secondary data of a 60 year time period. Their results revealed that forecasted values follow an increasing trend for the following years. Zhang (2013) also employed ARIMA, VAR and AR (1) models in the forecasting of per-capita GDP for five regions of Sweden for the period between 1993 and 2009. His findings revealed that the three models were useful in forecasting in the short run though AR(1) model was judged the most suitable for forecasting the per-capita GDP for the regions of Sweden. In the year 2015, Dritsaki employed the ARIMA model in forecasting the real GDP rate of Greece. His study made use of the Box-Jenkins technique and arrived at ARIMA (1,1,1) as an appropriate model for forecasting real GDP of Greece. Similar to the studies done on world macroeconomic indicators, in Africa some studies have been done in the forecasting and modelling of indicators such as GDP and Inflation. In modelling Liberia’s monthly inflation rates, Fannoh et al (2014) applied SARIMA (0,1,0)x (2,0,0). The results of the study revealed that the residuals showed no evidence of 18 University of Ghana http://ugspace.ug.edu.gh ARCH effect and serial correlation after the ARCH-LM and Ljung-Box test respectively. The Box-Jenkins methodology considering seasonality was used on a monthly inflation data from January to December 2006 to build the ARIMA model. Olajide et al. (2012) conducted a study to predict the inflation rate in Nigeria using Jenkins approach. They came out with the fact that the ARIMA (1,1,1) was the most suitable model for the inflation rate in Nigeria. A yearly data for 1961 to 2010 period was used and a year ahead was forecasted using the model built. Based on their findings they recommended that to avoid consequences in the economy, effective fiscal policies made should be aimed at monitoring Nigeria’s inflation trend. Subsequently, Okafor and Shaibu (2013) built an autoregressive moving average model for Nigerian inflation. They came to conclusion using a data between 1982 and 2010 that ARIMA (2,2,3) was an adequate model for tracking the actual inflation in the country. Both of these studies underscored the importance of the ARIMA models in forecasting the Inflation rate in Nigeria. Alnaa and Ahiakpor (2011) employed the use of ARIMA model in predicting inflation in Ghana using secondary data from the Ghana Statistical Service covering the period 2000 to 2010. They considered the Box-Jenkins approach to build the model. Their findings revealed that ARIMA (6,2,6) was the appropriate model for predicting inflation. The root mean squared error indicating the efficiency of predictability of the model was 0.115453. There are a plethora of time series models. Some researchesnn have been done in the area of comparing various time series models such as ARIMA, SARIMA, VAR and ARCH among others. Karlsson (2016) revealed that SARIMA model performed better than the ARIMA model when it comes to both out-of-sample and in-sample forecast performance based on the RMSE and MAE respectively. The VECM model performed worse than the ARIMA and SARIMA models under both the in-sample and out-of-sample performance 19 University of Ghana http://ugspace.ug.edu.gh since it had the highest RMSE and MAE. The study also revealed that for both in-sample and out-of-sample forecast, the SARIMA and ARIMA models produce similar or same accuracy for the forecast figures. Unwilingiymana et al. (2015) also used the ARIMA- GARCH models in forecasting inflation in Kenya. It was revealed that the combination of ARIMA (1, 1, 2) and GARCH (1, 2) performed better than ARIMA (1, 1, 2) and GARCH (1, 2) individually. Also they found out that ARIMA (1, 1, 2) gave more accurate estimates as compared to GARCH (1, 2) using the ordinary least squares estimation technique. Gikungu et al. (2015) in their article used the SARIMA model to forecast rate of inflation in Kenya. They employed the AIC in identifying the best model for the forecasting and used the maximum likelihood estimation method to estimate the parameters. The Jarque- Bera Normality test, ACF and PACF plots were used for diagnostic checks. Their results showed that SARIMA (0,1,0),x (0,0,1) was the best model to be used. Also the RMSE, MAPE and MAE were employed to check the predictive ability of the model. Abu et al. (2012) also used the vector autoregressive model to estimate the impulse response functions and variance decompositions for inflation and output in order to determine how inflation and output respond to changes in the exchange rate and what proportion of inflation and output variance can be explained by the exchange rate. 2.3.2 Other Fields where time series models have been used Apart from macroeconomic indicators such as GDP, unemployment and inflation which occur over a period and so can generate time series data, several aspects of our lives also generates data over a period that can be analysed by time series models. Some of these sectors or aspects of our lives include health, education and industrial and agricultural production. The intensity of incidence of hand, foot and mouth disease in infants in China led to a study by Liu et al. (2015) which focussed on predicting the incidence of the 20 University of Ghana http://ugspace.ug.edu.gh disease in a province in China using the ARIMA model. In accessing the goodness of fit of the model, they considered the coefficient of determination, the mean absolute percentage of error and the normalized Bayesian Information Criterion. The selected model used for the forecast of the incidence of the disease was ARIMA (1,0,1) x (0,1,0)12. Cristina et al (1998) also applied the ARIMA model to data from the health sector in Europe. In their study, they employed analysis of time series data in investigating the occurrence of nosocomial infection in Spain. The data was analysed by curve fitting, Autoregressive Integrated Moving Average (ARIMA) modelling and intervention and dynamic regression analysis. The results obtained showed that the imposed control and training of personnel by the surveillance system was associated with a 3.63% decrease in accumulated monthly incidence of 4.34% corresponded to a medical strike. Celik, Karadas and Eyduran (2017) employed the ARIMA model in forecasting the production of groundnut in Turkey. Their study revealed that out of the six studied ARIMA models experimented on, ARIMA (0,1,1) was revealed to be the appropriate model for forecasting. In selecting the most appropriate model, they considered the Akaike Information Criterion and the Schwarts Bayerian Information Criterion of the six studied models. In India, Singh, Darji and Singh (2016) also forecasted the area, production and productivity of wheat in Gujarat state using ARIMA model. In selecting the appropriate model for forecasting minimum value of the Akaike’s Information Criterion (AIC), and Schwartz-Bayesian Criterion (SBC) were considered. Also the Shapiro-Wilk test and Run test were used to test for normality and randomness of residuals distribution respectively. Their research revealed that ARIMA (0,1,1) was appropriate for forecasting the pattern of wheat area and production whiles ARIMA (1,1,0) was appropriate for forecasting wheat productivity trend of Gujarat state. 21 University of Ghana http://ugspace.ug.edu.gh Karthika, Krishnareni and Thirunavukkarasu (2017) forecasted meterological drought in India using the ARIMA model. In their studies they used secondary data from the Department of Statistics and Economics, Teynampet, Chennai, Tamilnadu between 1980 and 2014 for eight rain gauge stations. In selecting the appropriate model, they considered the minimum value of the AIC and SBC (Schwartz-Bayesian Criterion). Also the residual ACF function and Pormantateau lack of fit test were used to check the adequacy of the model selected. Their findings revealed that ARIMA (0, 1, 1), ARIMA (2, 1, 1), and ARIMA (1, 1, 2) were the best fit model for Mohannor, Elachipalayan, Namakkal, and Rasipuram stations respectively. The quest to help solve the general problem of loss of interest in investing in the capital market by investors led to a study by Jiban, Hoque and Rahman in 2013. The study’s focus was selecting the most appropriate ARIMA model to forecast average daily share price index. In testing for the stationarity condition, the ACF and PACF plots were observed in addition to testing the series using the Dickey-Fuller and Ljung-Box-Pierce Q- statistic. Out of the ten models built, ARIMA (2,1,2) was revealed to be the most appropriate model for forecasting the average daily share. The model was selected using the AIC, MAPE, RMSE, SIC and AME as a benchmark. Nwanko (2014) in his study of modelling the exchange rate from Naira to Dollar using the autoregressive integrated moving average (ARIMA) employed the B-Jenkins process. He revealed that ARIMA (1,0,0) which is equal to AR(1) had the most preferred AIC criterion though AR(2) had the same structural features. He recommended that governments should be able to control inflation and encourage savings to increase the availability of resources for future production since depreciation to exchange rate can lead to the expensiveness of importation which in the long run will seriously affect the balance payment and lead to depreciation of the domestic economy. 22 University of Ghana http://ugspace.ug.edu.gh Adebiyi, Adewumi and Ayo (2014) also in a quest to predict stock prices for a short-term period, employed the ARIMA model on secondary data from two Stock Exchange markets. The stock exchanges they considered were that of Nigeria and New York. Their results revealed that ARIMA model was potentially suitable for predicting short term stock values and could possibly replace some already existing techniques. They also found out that among the several ARIMA models, ARIMA (2,1,0) was the best using the BIC, Adjusted R2 and S.E of the regression as a check. In the same country, the performance of ARIMA and MA models were compared by applying them to Road accidents in Nigeria. From the study, Balogun and Oguntunde (2015) revealed that the model that best describes the road accident cases in Nigeria are the ARIMA (3,1,1) and MA( 0, 1, 2) using the mean square error (MSE) and Akaike Information Criteria (AIC) as a yardstick of selecting the best. National data on cases of accidents collected by Federal Road Safety Commission (FRSC), Nigeria for the year 2004 to 2011 was used. The Box-Jenkins process for model selection was also employed In Ghana, Agyemang (2012) employed the ARIMA in his study to estimate and assess the nature of the impact of the establishment and operations of the community policing unit of the Ghana Police Service. The study revealed that the pre-intervention period could be modelled best using the AR (1). Mensah (2013) in his study also applied the approach of the Box-Jenkins ARIMA process in predicting the water levels in the Akosombo dam. The study revealed that a seasonal model SARIMA (1, 1, 0) x (0, 1, 1)12 was a good forecast for the average monthly water levels. Suleman and Sarpong (2011) in their study employed time series modelling and forecast on hypertension cases in Navrongo, Ghana. They modelled using the Autoregressive Integrated Moving Average (ARIMA) stochastic model and employed the Box-Jenkins approach for modelling and forecasting. Their findings revealed that ARMA (3, 2) was 23 University of Ghana http://ugspace.ug.edu.gh best for modelling and forecasting hypertension in the catchment of Navrongo and their forecasts showed a decline in monthly hypertension cases for the year 2011. In relation to the Agricultural sector in Ghana, Harris, Abdul and Avuglah (2012) used secondary data from Ghana cocoa board from 1990 to 2010 to model the annual coffee production in Ghana employing ARIMA time series model. Out of the three models proposed after testing stationarity and differencing to achieve a stationary data and applying the three-stages in Box-Jenkins time series approach on the data, ARIMA (0,3,1) was revealed to be the best model. The selection was made on the basis of various selection, evaluation and diagnostics criterion. 2.3.3 Extreme Value Theory Extreme value theory is basically the stochastic behaviour of the extreme values in a process. Extreme Value Theory encompasses different applications concerning regular varying functions, sophisticated mathematical results on point processes and natural occurrences. Therefore, the theory has been used across different facets of life and Academia from engineering to health to statistics. The maximal stochastic behaviour can be obtained for a single process by the three extreme value distributions; Gumbel, Weibull and Frechet (Fisher and Tippett, 1928). The application of extreme value distributions was started in 1914 by Fuller and following after him several researchers the world over used the principle in analyzing particularly data on the climate. According to Kotz and Nadajarah (2000) extreme value distributions gained popularity or recognition due to the efforts of Bernoulli in 1709. In 1709, N. Bernoulli studied extreme value problems by discussing “the mean largest distance from the origin with a straight line of length t having n points on it at random” (Johnson, et al., 1995). 24 University of Ghana http://ugspace.ug.edu.gh In 1922, Von Bortkiewicz conducted a research on distribution in random samples from a normal distribution. In his study, the concept of largest value distribution was introduced for the first time (Kotz and Nadajarah, 2000). The German statistician Bortkiewicz contributed to the formulation of the theory by introducing the concept of the distribution of largest values. Another German, Von Mises, improved on his work by working on Gaussian distributions (Kotz and Nadajarah, 2000). Other studies by several researches on distributions such as a study in 1928 by Fisher and Tippet were conducted but the detailed basis for the theory was presented in 1943 by Gnedenko. His study brought to light the formative ideas for the concept that birthed the Extreme Value Condition which is the basic assumption underlying the Extreme Value Theory (Kotz & Nadajarah, 2000). The Extreme Value Condition provides a “Semi- parametric model” solely for the tails of the distribution function. Later studies on weak convergence of Sample Extremes such as the study by Haan in 1970 contributed immensely to the development of the Theory (Beirlant et al., 2004). According to Kotz and Nadajarah (2000), researchers applied the extreme value statistics to real life data in the mid twentieth century. Data on natural happenings such as floods, rainfall patterns, and distribution of human lifetimes among others were submitted to extreme value statistics. The investigation done by Katz (2010) on the application of the Extreme Value Theory to climate change and its implications on the analysis of the economic impact of extremes is evident of the wide use of the theory to natural occurrences. Embrechts, Kloppelberg and Mikosch (1997) also applied the theory to fields such as risk management and insurance Longin (2000) applied the Extreme Value Theory in understanding the consequences of events resulting from extreme market movements. His work on the 1997 financial risk of 25 University of Ghana http://ugspace.ug.edu.gh Asia applied the theory and because of his success researchers have followed in the use of Extreme Value Theory approach in analysing tail-oriented models of risk. Chavez-Demoulin and Embrechts (2004) conducted more investigations into the usefulness of the theory in analysis of various types of operational loss data. Resnick continued further studies of extreme events in 2007 and he focussed on sectors such as finance, insurance and data networks. The other researchers to further work with the extreme value theory were Bi and Giles in 2007 where they studied the appreciation of the extreme value theory to the daily return of crude oil prices in the Canadian spot market from the year 1998 to 2006 by focussing on the peak over threshold (POT). They revealed that given a range of selected thresholds, the estimates of the risk measures computed under different high quantile levels exhibited strong stability. Bi and Giles (2007) also used the peaks over threshold (POT) technique to analyse GDP. 2.4 Conclusion Most of the studies reviewed are geared towards examining factors affecting inflation and GDP, the relationship that exists between these variables and other macroeconomic variables, the forecasting ability or the predictive power of the ARIMA model and employing the extreme value theory in modelling events such as floods and rainfall patterns, and distribution of human lifetimes. This study however seeks to monitor the achievement of inflation and GDP targets by employing the ARIMA time series model and the extreme value theory. 26 University of Ghana http://ugspace.ug.edu.gh CHAPTER THREE METHODOLOGY 3.0 Introduction This chapter gives a detailed description of the method used in the study. The source of data and the collection of data begin the section. This is rightly followed by description and definition of variables and a detailed description of the theory and concept of Time Series and Extreme Value Theory. 3.1 Source of Data and Collection of Data Procedure The study employed secondary time series data on two macro-economic variables namely: Inflation rate and Gross Domestic Product values from Ghana. A monthly inflation rate was collected over the period 2009 to 2017 and this gave a total of one hundred and eight data points which is sufficiently large for the study. The GDP on the other hand was collected annually over the period 1961 to 2017, also giving fifty-seven data points. The data on inflation rate was sourced from the Ghana Statistical Services and that of the GDP from Bank of Ghana. 3.2 Variables Description. In this study the variables of interest were Inflation rate and GDP values. These two economic variables play very important role in accessing the growth of any country. This explains why a lot of studies have been conducted in relation to these variables. Below is a brief definition of these variables; Inflation: The Merriam-Webster’s Collegiate Dictionary defines Inflation as a continuing rise in the general price levels within a period. In defining Inflation, the continuous and 27 University of Ghana http://ugspace.ug.edu.gh rapid increase of goods and services in an economy is key and therefore inflation measures change in price levels of goods and services. GDP: The gross domestic product (GDP) is one of the measures of national income and output for an economy within a certain period. It is a way of appraising a country in terms of the performance of the economy and it is usually defined as the total market value of all final goods and services produced within a nation in a year. 3.3 Method of Data Analysis This section gives a brief description of how the sampled data is used to achieve the objective of the study. The first procedure is to build a time series model that best represents the data. In doing this, the data set is divided into the train set which is used to build the model and the test set which is used to validate the model after all due process is being taken to select the most appropriate model. The residuals of the most appropriate model are used to estimate quantile value using the extreme value theory or the usual procedure of calculating quantile value using the probability and the data set. In monitoring a set target given the target length we deduce an expected trend line based on the last data point of the series, the point and the length of period within which the target is expected to be achieved. The expected trend line serves as a forecast path for preceding values up to the targeted value to be monitored. Let g+1 be the target length and YT be the last value of the series. Now, let YT̂ j (j= 1,2,……,g) be the estimate from the expected trend line at time T+j ; 28 University of Ghana http://ugspace.ug.edu.gh where T is the length of the time series data and g+1 is the target length. For each j, a bound (blj , buj) is set at time T+j (j=1,2,…,g) where b ˆlj YT j e ˆT j ,q and buj YT j  eT j ,q represent the lower and upper bounds respectively and eT j ,q represents the qth quantile of the residuals at time T+j . Also for each j (j=1,2,…,g) a number YT  j is selected at random from (blj , buj). These numbers are then added to the data and the first g data points are dropped. The resultant data; Yg1,Yg2 ,...,YT ,YT 1,YT 2 ,...,YT g : are used by means of the appropriate model to forecast a period ahead. The process of the randomly generating figures within the lower and the upper bounds, updating the series and forecasting a period ahead is repeated 100 times. The bounds are then assessed by computing the proportion of the forecasted values that fall within a defined percentage () of the targeted value and the mean absolute percentage error given respectively as follows. 1 100 1 Yˆ  P where  tag  1 Ytag ,1 Ytag  fp   i   in i  i0 0 otherwise and n Y ˆ MAPE 1 tag Ytagi   , where n is the number of simulations, Ytag represents the n i1 Ytag set target to be achieved and Yˆtagi represents the forecasted value using the most appropriate model. The major statistical tools and concepts are detailed in the subsequent sections. 29 University of Ghana http://ugspace.ug.edu.gh 3.4 Time Series and its Basic Concepts Time series can be defined as a set of observations on values that a variable takes at different times with a commonly spaced equal time interval. According to Akgun (2003), time series is an ordered sequence of random variables over time. Time series analysis is widely used in fields such as engineering, economics, statistics, business, meteorology, geophysics among others. The nature of data encountered in these fields appears as time series. Time series data can be put into two categories namely; discrete and continuous. Observations under the discrete time series data considers regular space time intervals for instance; daily rainfall, hourly observations made on the yield of a chemical process, weekly prices of shares, yearly profit made of a company and so on, whereas in the case of continuous time series data, observations are recorded continuously over time interval such as electrocardiograms. Time series analysis encompasses procedures that try to comprehend the underlying generation process of the data points and put together a mathematical model to illustrate the generation process. The model which is put together is then used to forecast and predict future events based on known past events. Time series usually uses the natural one-way ordering of time so that values in a series for a given time period will be portrayed as being obtained from past values rather than future values. Time series data are usually characterised by four main patterns namely; trend component, seasonal component, periodic (cyclical) component, and the residual (error) component. These patterns are mostly seen in business and economic related series. The trend component is a long term relatively smooth pattern that usually persists for more than a year. The seasonal component is a pattern that appears in a regular interval where by the frequency of the occurrence is within a year or less. The periodic component is a repeated pattern in time series that goes beyond a frequency of a year and the residual component is the pattern that is obtained after the three patterns have been seen in the series. 30 University of Ghana http://ugspace.ug.edu.gh Time series data can be represented in various stochastic processes and in different forms of models. This can be categorised into two main groups namely; linear and non-linear models. Some of the models considered under the linear model category include autoregressive (AR) model of order (p), moving average (MA) model of order (q) as well as autoregressive moving average (ARMA) model of order (p,q). These models under the linear model category are considered in the situation where the time series data is stationary but in situations where the series is non-stationary the autoregressive integrated moving average (ARIMA) model of order (p,d,q) is used. Other linear models are the seasonal autoregressive integrated moving average (SARIMA) model, the autoregressive fractional integrated moving average (ARFIMA) among others. Also some models considered under the non-linear models including the autoregressive conditional heteroscedastic (ARCH) model of order (p), the Generalised ARCH (GARCH) model of order (p,q), Threshold GARCH (TGARCH), Power ARCH (PARCH) and Integrated GARCH model all of order (p,q). For the purpose of our study the autoregressive integrated moving average (ARIMA) model of the linear model category will be considered. An explanation of the theory of the model will be seen in later sections of this chapter. 3.5 Stationarity Process A necessary condition to be satisfied before applying time series analysis and technique is the stationary property of the series. This makes stationarity property a core foundation in time series analysis. Though the assumption of stationarity is usually in research, a test is always performed to confirm this property before carrying out the time series analysis. In situations where this property does not hold mathematical techniques are applied to the series to attain this condition. This section defines and describes the stationarity process. 31 University of Ghana http://ugspace.ug.edu.gh A series is described as stationary when its mean variance and auto covariance are time invariant. The stationarity of a series takes two forms namely; strictly stationary and weakly stationary. A strictly stationary time series is one for which the probability behaviour of every collection of values {yt , yt ,..., yt } is identical to that of the time shifted set1 2 k {yt1i , yt2i ,..., ytk i} . That is: P{yt  c1,..., yt  ck} P{yt i  c1,..., yt i  c k} (3.1) 1 k 1 k For all k 1,2,... all time points t1,t2 ,...,tk , all numbers c1,c2 ,...,ck and all time shifts i  0,1,2,... If a time series is strictly stationary, then all of the multi-variate distribution functions for subsets of variables must agree with their counterparts in the shifted sets of all values of the shift parameter i. A weakly stationary time series, yt is a finite variance process such that 1. The mean value function μ is constant and does not depend on time. (i.e; E(yt)= μ) 2. The auto covariance function , Cov(yt, ys)=γ depends on s and t only through their difference ⎸s – t ⎸ For the purpose of this study, Augmented Dickey-Fuller (ADF) test for stationarity was carried out and the weakly stationarity form was considered. 32 University of Ghana http://ugspace.ug.edu.gh 3.6 Augmented Dickey Fuller Test In testing for the stationarity property the Augmented Dickey Fuller test was employed. Below are the hypotheses of the test. Ho: there exist non-stationarity H1: there exist stationarity Decision Rule 1. Fail to reject the null hypothesis if the calculated t-statistic is greater than the ADF critical value or if the p-value calculated is greater than the significance level. 2. Reject the null hypothesis if the calculated t-statistic is less than the ADF critical value or if the p-value calculated is less than the significance level. In the case where the null hypothesis is rejected, one can conclude that based on the available data used and the level of significance chosen, the series is stationary otherwise the series is non-stationary when we fail to reject the null hypothesis. 3.7 Differencing Process Though most time series data encountered are usually non-stationary, differencing the series is one among the numerous ways of achieving stationarity which happens to be the bedrock of time series analysis. Non-stabilized variance of the residuals, seasonality and trend are some of the factors causing non-stationarity of any time series. Differencing is a consecutive change between observations in a series. It is also an exceptional type of filtering that helps in eliminating trend and/or seasonality in a non- stationary series (Luruli, 2011). Taking a logarithm transformation of the series which is 33 University of Ghana http://ugspace.ug.edu.gh not stationary helps to stabilize the variance and eliminate the non-constant variance of the residuals in the series. The differencing filter 1 Bd Yt Yt Ytd (3.2a) where d is a positive integer, eliminates seasonality of period d in the seriesYt . Thus if Yt 0 t t , (3.2b) then 1 BYt  Yt Yt1 0 t  t 0 t1  t1  t t1 (3.2c) Also, the differencing filter for eliminating the trend of the polynomial of degree d in the series Yt if present is given by d d  Yt  1 B Yt (3.2d) 3.8 ARIMA Model A generalised model which includes a wide range of non-stationary series that has been stationarised through transformation process such as differencing and estimating the log is the Autoregressive Integrated Moving Average model (ARIMA) of order (p, d, q). The ARIMA model includes Autoregressive model (AR) which include memory of past variables of the series, Moving Average model (MA) which accounts for the error terms, Mixed Autoregressive-moving average model (ARMA) which combines the Autoregressive and Moving Average models and the integrated (I) form which accounts for the stationarity of all three. 34 University of Ghana http://ugspace.ug.edu.gh An autoregressive process regresses a dependent variable on the past values of the dependent variable. A p-order process is of the form: Yt  1Yt1 2Yt2  ........PYtP t (3.3) Where Yt is the dependent variable which is stationary, (1 , 2 ,…, p ) are coefficients to be estimated, ( Yt1 , Yt2 ,… , Yt p ) are the response variable at the various time lags,  t is  the error term. Applying the backshift operator reduces the AR (p) process to the form: (B)Yt t (3.4) The Moving average process of order q expresses the dependent variable which is stationary as a function of the error term. It takes the form: Yt   t 1 t1 2 t2  ........q tq (3.5) where; 1 ,2 , ...,q are the parameters to be estimated and q is the number of lags . The Mixed Autoregressive-Moving Average process of order p and q is the combination of the AR process of order p and the MA process of order q. It is represented by the form: Yt  1Yt1 2Yt2  ........PYtP 1 t1 2 t2  ........q tq  t (3.6) where  1Yt1 2Yt2  ........PYtP is the AR (p) process and  t 1 t1 2 t2  ........q tq is the MA (q) process. ARIMA models are usually applied to stationary data whose mean, variance and autocorrelation function remain constant through time. 35 University of Ghana http://ugspace.ug.edu.gh Differencing a non-stationary series usually reduces the process to a Mixed Autoregressive- Moving Average process of order (p,q) which qualifies for the application of time series techniques. The condition for process Yt to be an ARIMA (p,d,q) process is when Y 1 dt    B Yt (3.7) is a causal ARMA of order (p,q) process and d is a non-negative integer. This simply means that Yt satisfies a difference equation of the form  d  BYt  B1 B Yt  (B) t (3.8) where  B and (B) are polynomials of degree p and q respectively whose zero lies in a unit circle and  2t ~ N(0, ) . The process Yt is considered as a stationary process only if d=0 and this reduces the process to an ARMA (p,q) process. As proposed by Box-Jenkins (1976), building an appropriate model is usually achieved using the three-stage repetitive procedure based on identification, estimation and diagnostic checking. In the identification stage a subclass of parsimonious models to be considered are proposed by using information on the data and the generating process of the series. The efficient use of the data in making inference about the parameters conditional on the competence of the model is investigated in the estimation stage. Lastly under the diagnostic checking stage, the fitted model in reference to the used data is scrutinized with the motive of revealing the incompetency of the model to achieve model improvement. 36 University of Ghana http://ugspace.ug.edu.gh 3.9 Identification of Model, Estimation and Diagnostic Checking This section briefly describes how the fitted model is obtained through the three-stage repetitive procedure. 3.9.1 Model Identification The main objective at this stage is to obtain an appropriate order of the ARMA model. The Partial Autocorrelation function (PACF) plot and the Autocorrelation function (ACF) plot are employed in determining the order of the AR and the MA respectively. The value of d is usually determined by number of times the non-stationary data were differenced to achieve stationarity. After determining the order of the model, various models with different orders can be proposed but the best model must be chosen from a group of competing models characterized by the ordering data. Several selection criteria have been suggested to aid in choosing the most suitable model. Among these criteria are the Bayesian Information Criterion (BIC) introduced by Schwartz (1978), the Akaike Information Criterion (AIC) proposed by Akaike in the year 1974 and many others. In choosing the suitable ARIMA model this study uses the AIC criteria mentioned above. The AIC basically deals with the trade-off between the complexity of the model and goodness of fit. It uses the principle of the maximum likelihood in obtaining the criteria value and is given by: AIC  2(log likelihood) 2k  2k 2In(L) , (3.9) where k is the number of estimated parameters in the model and L is the maximized value of likelihood function for the model. 37 University of Ghana http://ugspace.ug.edu.gh 3.9.2 Estimation The process of identification leads to a tentative formation of the model and this leads to acquiring efficient estimates for the unknown parameters. Two basic methods of estimation of parameters popularly used in academia are the maximum likelihood methods and the least squares method (Box-Jenkins, 1976). This study employed the maximum likelihood method of estimating unknown parameters in the fitted model due to its unique property of adapting to structural changes in the model. 3.9.3 Diagnostic Checking The final stage after the model identification and the parameter estimation is the diagnostic check. The purpose for this stage is to inspect or investigate the inadequacy of the model under study. The usual approach in diagnostic checking is investigating the residuals from the fitted model for any signs of non-randomness. The Ljung-Box and the Box-Pierce tests are among numerous tests used in checking the independency of residuals from the fitted model. Both tests are based on the null hypothesis that the residual values of the fitted model are independent. In the case where the p-value is greater than the given significance level of the test, the null hypothesis is not rejected and it can be concluded that the residual values are independent of each other and therefore the fitted model is adequate for the purpose of forecasting. On the other hand when the p-value is less than the given significance level of the test, the null hypothesis is rejected and it can be concluded that the residual values are dependent. 3.9.4 Model Validation To determine the validity of fitted model and its ability to forecast the series, the data was divided into two parts. The two parts included the training set and the test set in the 38 University of Ghana http://ugspace.ug.edu.gh proportions 95% and 5% respectively. The first part which was the training set was used to determine the model parameters whereas the second part which was the test set was used to assess the accuracy of the model in forecasting. A chosen model is said to be valid if it gives a good description of the data set. It implies that such a model is valid and adequate for forecasting the series. 3.10 Extreme Value Theory Extreme Value Theory (EVT) as a unique statistical discipline is focused on developing techniques for building models for the purpose of describing the behaviour of the occurrence of sample extremes and measuring unusual events whose occurrence are associated with very small probabilities. Modelling the behaviour of such occurrence can be achieved through the limiting distribution function of the order statistics or their exact distribution. Generally, EVT seeks to answer statistical and probabilistic questions associated with very low or high values in sequences of random variables and in stochastic processes. Consider Ym max{Y1,Y2,....,Ym} which denotes the sample maximum, where Y1,Y2 ,...Ym is a sequence of independent random variables with a common distribution function F. The distribution of the sample maximum Ym can be theoretically derived for exactly all the m values: m P{Ym  y} P{Y1  y,Y m 2  y,...,Ym  y}P{Yi  y}{F (y)} (3.10) i1 The two major setbacks associated with this approach is that the distribution of F may not be known and F(y) < 1, so F(y)m 0 as m (Coles,2001). 39 University of Ghana http://ugspace.ug.edu.gh The impetus of EVT is discovering the behaviour of the sample extreme for a sufficiently large m using the limiting distribution function. Modelling in EVT considers two paramount approaches given a data set. The first approach is the block maxima approach which considerably reduces the data by taking maxima of long blocks of the data, for example the monthly maxima. The other approach analyses excesses over a high threshold. Both approaches however can be specified in terms of a Poisson process which enables simultaneously fitting of parameters concerning both frequency and intensity of extreme events. The generalised extreme value (GEV) distribution function has been theoretically justified for fitting to block maxima of a data set. Similarly the Generalised Pareto (GP) distribution function also has justification backed by theory for fitting to excesses over a high threshold. The GEV density function (df) has three well known extreme value distributions namely; Gumble, Frechet and Weibull embedded in it. The distribution function of the GEV is given by;  1 G(z) exp 1  z           (3.11a)        The GEV df is defined on the set 1  z    0 , where the parameters satisfy   ,   0 and    . The model has three parameters: a location parameter, ν; a scale parameter, σ; and a shape parameter, κ. Depending on the sign of the shape parameter, the GEV df give rise to any of the three extreme value distribution mentioned above. The distribution function becomes the heavy-tailed Frechet when κ>0. When κ<0 40 University of Ghana http://ugspace.ug.edu.gh the distribution function turns to the upper bounded Weibull. The Gumbel is obtained by taking the limit as κ→0 giving   G(z)  exp exp   z     ,  z   (3.11b)      3.10.1 The Generalised Pareto Distribution A disadvantage of the block maxima approach of modelling extreme sample or events is when one encounters more than one extreme value in a particular block. This simply means that GEV method becomes very useful when only block maxima data are available but in a situation where other extreme values are available it becomes wasteful. Consider a situation where there are two or three extreme values recorded in a particular block, employing the GEV method will only allow the inclusion of only one of the extreme values in the estimation of the model. This may lead to underestimating the frequency of the occurrence of that particular event. This is the advantage the excesses over high threshold approach of modelling have over the block maxima since it considers extreme values beyond a selected threshold. Given Y1,Y2 ,.... as a sequence of independent and identically distributed random variables with a marginal distribution function F , Yi can naturally be considered as an extreme events that exceeds some high threshold υ. If Y represent an arbitrary term in Yi , then the probabilistic behaviour of the extreme events is given by the conditional probability (Coles,2001), 1F   xP Y   x / Y    , x  0 (3.12) 1F   41 University of Ghana http://ugspace.ug.edu.gh The distribution of the threshold exceedances in the equation 3.12 can be easily obtained given that the parent distribution F is known. However, this is usually unknown in practice therefore, approximations that are often applicable for high values of the threshold are used. 3.10.2 The Asymptotic Model Characterisation of the Generalized Pareto Distribution Obtaining the main result of the GP distribution is enveloped in the following theorem 3.10.2.1 Theorem Considering Y1,Y2,Y3,... as a sequence of identically and independent random variable having a common distribution function F , let Ym max{Y1,Y2,....,Ym} . Assuming that F satisfies all conditions of the GEV distribution function and let Y denote an arbitrary term in Yi sequence , so that for a sufficiently large m, P Ym  z   G(z) Then for a sufficiently large υ the distribution function of (Y ) condition on Y  approximated as;  1  H (x) 1   x  1  (3.13)    Equation (3.13) is defined on x : x  0and 1  x    0 . This type of distribution is called the Generalised Pareto distribution. As the block maxima has an approximation distribution, this theorem implies that the threshold excess also have an approximate distribution enveloped in the Generalised Pareto distribution. 42 University of Ghana http://ugspace.ug.edu.gh Just as the GEV distribution the Generalised Pareto distribution give rise to three unique distribution functions based on the shape parameter of the GP df and the parameters of the GP distribution are specially determined by the related GEV distribution of the block maxima. When the GP distribution has its shape parameter  0 the distribution is said to have an upper bound, if  0 it is said to have no upper limit and if   0 it is said to be unbounded which result in H (x)  x 1 exp  , x  0 when interpreted as  0 . This corresponds to exponential    distribution with mean . 3.10.2.2 Selection of Threshold Extreme events are identified by defining a high threshold υ, for which the exceedances areyi : yi  . Labelling these exceedances by y(1) ,..., y(k ) , the threshold excess is defined by xi  yi  for i 1,...,k . Selecting an appropriate threshold is an important task in modelling extreme events when considering the excess over high threshold approach. Similar to the choice of block size in the block maxima approach it is a balance between bias and variance. A too high threshold will eventually increase the variance and a too low threshold may lead to bias due to its likeliness of violating the asymptotic basis for the model. The normal practice done is to adopt as low a threshold as possible, subject to the limit model providing a reasonable approximation. 43 University of Ghana http://ugspace.ug.edu.gh 3.10.3 Extreme Quantiles Extreme quantiles for both the GEV distribution and the GP distribution are obtained by inverting their corresponding distribution. The extreme quantile associated with the GEV distribution is given by:     1 yz p  ,   0 p      (3.14)     ln yp ,   0 where yp   log(1 p) . This is usually interpreted as the value expected to be exceeded on average once every 1 p period and 1 p is the specific probability associated with the quantile. Also the extreme quantile associated with the GP distribution is given by:   ( p 1),   0qm   (3.15)    log p,   0 The quantile value of the GP distribution is also interpreted as an estimate of the probability of exceeding the threshold. 3.10.4 Diagnostic Check For the purpose of investigating the inadequacy of both the GEV model and GPD model, good graphical plots such as the quantile-quantile plot (QQ-plot) and the probability plot (PP-plot) among others are thoroughly examined (Coles,2011). The plots involve the comparison of the empirical data to the fitted distribution. A straight one-to-one line of points of the plots indicating linearity of the QQ- plot and the PP-plot is an indication that the model is valid and any deviation from this implies otherwise (Coles,2011). 44 University of Ghana http://ugspace.ug.edu.gh 3.11 Expected Trend Line The expected trend line is a forecast function for predicting future values for a given variable. For the purpose of this study, the expected trend line is the path for predicting the future values of our variables which are Inflation rate and GDP. For this study, two types of expected trend lines will be considered in the analysis. The lines are the “linear expected trend line” and the “curve-linear expected trend line”. To obtain a linear expected trend line; the formula for the equation of a straight line is employed. Given any two points with their respective x and y coordinates, the slope and the intercept of the straight line is calculated as; y  mxc (3.16) where m represents the slope and c the intercept. m is given by m y2  y 1 (3.17) x2  x1 The intercept c, can then be obtained with one of the points by substituting the coordinates and the slope into equation (3.16). Thus c  ymx (3.18) For a curve-linear expected trend line, the equation for a quadratic equation with three points is used; y  ax2 bxc (3.19) where a, b and c are the coefficients. 45 University of Ghana http://ugspace.ug.edu.gh The coefficients are obtained by substituting the coordinates of three points into the equation to obtain three simultaneous equations with three unknowns. Solving the equations simultaneously gives the values of the three coefficients. 46 University of Ghana http://ugspace.ug.edu.gh CHAPTER FOUR ANALYSIS OF DATA AND RESULTS DISCUSSION 4.0 Introduction The analysis and results obtained from the study are discussed and presented in this chapter. A description of the data with reference to some basic statistics is presented in section 4.1. This is followed by the preliminary analysis of the data in section 4.2. Following this section is the fitting of the time series model, estimating and performing a diagnostic check of the model, these will be discussed in section 4.3. Section 4.4 describes the procedure of estimating quantile values of variables. Monitoring the achievement of a set target for the monthly inflation rate is discussed in section 4.5. The last section which is 4.6 focuses on the discussion of monitoring the achievement of a target for the annual GDP value . All analyses were carried out using the R statistical software. 4.1 Data description and Summary Statistics A sample data of monthly inflation rates and annual Gross Domestic Product values of Ghana were obtained from the Ghana Statistical Service and the Central Bank of Ghana respectively. The monthly inflation rate was collected over the period 2009 to 2017 giving a total of one hundred and eight data points and that of the GDP collected annually over the period 1961 to 2017, also given fifty-seven data points. Table 4.1.1 revealed some descriptive statistics of the monthly inflation rates and the annual GDP values obtained. 47 University of Ghana http://ugspace.ug.edu.gh Table 4.1.1: Descriptive Statistics of Ghana’s monthly Inflation rates (2009-2017) and GDP values (Billion$; 1961 – 2017) Variable Sample Mean median St. deviation Max Min size Inflation 108 13.56 13.2 3.85 20.7 8.4 GDP 57 10.9 5.25 13.496 47.81 1.30 Source: Researcher’s computation based on sampled data The results obtained from Table 4.1.1 reveal 13.56 as the average value of Ghana’s monthly inflation rates with a standard deviation of 3.85 and a median of 13.2. It also shows a 20.7 maximum value of inflation rate with a corresponding 8.4 minimum value. The table also reveals an average GDP value of 10.9 billion (US dollars) with a standard deviation of 13.496 and a median of 5.25 billion (US dollars). Table 4.1.1 further depicts that the GDP value ranges between 47.81 billion (US dollars) and 1.3 billion (US dollars). 4.2 Preliminary Analysis Figure 4.2.1 presents a plot of monthly rate of inflation in Ghana for the period of 2009 to 2017 and Figure 4.2.2 displays a plot of the annual GDP value ( in dollars) of Ghana for the period; 1961 to 2016. From the two figures, it can be noticed that the mean and variance for both plots vary over time. The characterisation of the unstable mean and variance is a clear indication that the two sets of data (both the monthly inflation rate and the annual GDP values) are non-stationary. This was further confirmed by performing the Augmented Dickey- Fuller test for the monthly rate of inflation and the GDP values. The result of the tests are shown in table 4.2.1. 48 University of Ghana http://ugspace.ug.edu.gh Figure 4.2.1: A plot of Ghana’s monthly rate of inflation (2009 to 2017) Figure 4.2.2: A plot of annual GDP values of Ghana (1961 to 2016) 49 University of Ghana http://ugspace.ug.edu.gh Table 4.2.1: Augmented Dickey-Fuller Test for Ghana’s monthly inflation rate (2009 to 2017) and GDP values (Billion$; 1961 – 2017) Variable Dickey-Fuller p-value Inflation rate -1.8814 0.6258 GDP -0.39705 0.9833 Source: Researcher’s computation based on sampled data From table 4.2.1, the test at a significance level of 5% fails to reject the null hypothesis as explained in section 3.5 in the previous chapter. The reason being that a p-value of 0.6258 and 0.9833 for the Augmented Dickey-Fuller test for the monthly inflation rate and the annual GDP are all greater than 5% significance level resulting in a failure to reject the null hypothesis and therefore concluding that there exists non-stationarity in the two data sets which confirmed the unstable or non-constant mean and variance statement made about the monthly inflation rate and the annual GDP values for the period of 2009 to 2017 and 1961 to 2016 respectively. Since stationarity of a data set ought to be achieved in order for time series modelling technique to be applied a transformation was carried out on the two sets of data to bring them to stationarity. The ordinary differencing form of transformation was employed on the monthly inflation rate data for stationarity to be attained, the data was differenced once. Also in attaining stationarity for the annual GDP data, the natural logarithm was taken and the data was differenced once. The Augmented Dickey-Fuller test was once again carried out to confirm the stationarity property of the two data sets after the data transformation. Figures 4.2.3 and 4.2.4 show the plots of the transformed monthly inflation rate and the transformed annual GDP values. Table 4.2.2 shows the outputs of the Augmented Dickey-Fuller test after the transformation of the data sets 50 University of Ghana http://ugspace.ug.edu.gh Figure 4.2.3: A plot of Ghana’s transformed monthly rate of inflation (2009 to 2017) Figure 4.2.4: A plot of Ghana’s transformed annual GDP values (1961 to 2017) 51 University of Ghana http://ugspace.ug.edu.gh Table 4.2.2: Augmented Dickey-Fuller Test for Ghana’s transformed monthly inflation rate (2009 to 2017) and GDP values (Billion$; 1961 – 2017) Variable Dickey-Fuller p-value Inflation -7.7005 0.01 GDP -6.0458 0.01 Source: Researcher’s computation based on sampled data Figures 4.2.3 and 4.2.4 reveal a more constant mean and variance indicating stationary data sets as compared to Figures 4.2.1 and 4.2.2. This was confirmed in the output of the augmented dickey-fuller tests in Table 4.2.2. The p-value of 0.01 for both tests which is lesser than 5% level of significance allows the test to reject the null hypothesis as stated in section 3.5 of the previous chapter and therefore a conclusion of the two transformed data sets being stationary can be made. 4.3 Fitting a Time Series Model Since the stationarity property which is a core feature in modelling in time series analysis has been achieved, the next step to take in relation to the Box-Jenkin’s stage of model building in time series which has been explained in detailed in section 3.8 of chapter three is the identification stage. Clearly since the original data were differenced once to attain stationarity, the ARIMA model with d=1 was considered. In identifying the orders of the AR (p) and the MA (q) terms of the model, the ACF plot and the PACF plot of the transformed data were observed. Figures 4.3.1 and 4.3.2 shows the ACF and the PACF plots of the first difference of the monthly rate of inflation. 52 University of Ghana http://ugspace.ug.edu.gh Figure 4.3.1: Autocorrelation Plot for the first difference of monthly rate of inflation (2009 to 2017) Figure 4.3.2: Partial Autocorrelation plot for the first difference of the monthly rate of inflation (2009 to 2017) Observing the PACF and the ACF plots of the first difference data of the monthly inflation rate, it can be seen that the PACF plot has three significant spikes in lags 1, 2 and 3 53 University of Ghana http://ugspace.ug.edu.gh respectively after which the PACF tails off. Now due to the spikes as seen in figure 4.3.2, an AR of order 3 was considered and an MA of order 1 was also considered by observing the ACF plot which tails off after the first lag. Since the monthly inflation rate data was differenced once to attain a stationary data and from the ACF and the PACF plots an ARIMA (3, 1, 1) was obtained. Estimating the unknown parameters of the suggested model becomes a peculiar interest once the order of the model has been obtained. To estimate these parameters the maximum likelihood method was employed. Usually the orders of the MA and the AR derived from the ACF and the PACF plots give a suggestion of the order of the appropriate model to be considered. In our case however, it is the ARIMA (3, 1, 1). Different models of order close to the appropriate model were fitted and the AIC values of these fitted models were used as a bench mark to obtain the most appropriate model. The most appropriate model was selected based on a smaller AIC value. Table 4.3.1 shows the suggested ARIMA models and their equivalent AIC values. Table 4.3.1: Suggested ARIMA(p,d,q) models and their AIC values for Inflation. Model AIC ARIMA (1, 1, 1) 165.26 ARIMA (0, 1, 2) 182.22 ARIMA (1, 1, 2) 165.60 ARIMA (1, 1, 3) 167.32 ARIMA (2, 1, 0) 175.03 ARIMA (2, 1, 1) 165.87 ARIMA (3, 1, 2) 169.00 ARIMA (3, 1, 1) 167.10 Source: Researcher’s computation based on sampled data 54 University of Ghana http://ugspace.ug.edu.gh From table 4.3.1, it can be seen that the most appropriate model based on the smallest AIC value is ARIMA (1, 1, 1) and it was therefore selected as the appropriate model for Ghana’s monthly inflation rate for the period of 2009 to 2017. To obtain the most appropriate model for Ghana’s annual GDP values a similar observation of the ACF and the PACF plots was employed in selecting the suggested order for the MA and the AR term of the ARIMA model. The suggested ARIMA model based on the order of the MA and AR terms from the ACF and the PACF plots respectively was compared with other ARIMA models whose MA and AR terms lie close to the suggested one and the smallest AIC value was used as a bench mark for selecting the most appropriate model as it was done for the monthly inflation rate. The ACF and the PACF plots can be found in the appendix A; figures 1A and 2A respectively. Table 4.3.2 also gives the suggested ARIMA models and their equivalent AIC values. From Table 4.3.2, ARIMA (0, 1, 0) appears to be the most appropriate model using the smallest AIC value as a bench mark. Table 4.3.2: Suggested ARIMA (p,d,q) models and their AIC values for the annual GDP Model AIC ARIMA (0, 1, 0) -51.89 ARIMA (0, 2, 0) -24.86 ARIMA (1, 1, 0) -51.76 ARIMA (1, 1, 1) -50.04 ARIMA (1, 1, 2) -46.86 ARIMA (2, 1, 1) -48.43 ARIMA (1, 2, 1) -46.12 ARIMA (2, 1, 2) -48.26 Source: Researcher’s computation based on sampled data Having the most appropriate model based on the minimum AIC selection criterion does not imply that the model is a good fit, therefore a diagnostic check was carried out on the residuals as explained in section 3.8.3 in the previous chapter. 55 University of Ghana http://ugspace.ug.edu.gh In performing the diagnostic check, three of the suggested ARIMA models with the very minimum AIC values were considered for both the monthly inflation rate and the annual GDP values. The Box-Pierce and the Ljung-Box tests were performed on the residuals of these models and the model that best fit the data was expected to have residuals which are random, independent and identically distributed. Table 4.3.3 gives the output of Box- Pierce test for ARIMA models of the monthly inflation rate. Table 4.3.3: ARIMA outputs of Box-Pierce test for Ghana’s monthly inflation rate Model X-squared Df p-value ARIMA (1,1,1) 1.029 1 0.3104 ARIMA (2,1,1) 0.036899 1 0.8477 ARIMA (1,1,2) 0.0017196 1 0.9669 Source: Researcher’s computation based on sampled data Figure 4.3.3, figure 4.3.4 and figure 4.3.5 also show the standardized residual plots, ACF plot of the residuals and the p-values for the Ljung-Box statistics of the models of the monthly inflation rate. Figure 4.3.3: ARIMA (1, 1, 1) output of Ljung-Box test for Ghana’s inflation rate. 56 University of Ghana http://ugspace.ug.edu.gh Figure 4.3.4: ARIMA (2, 1, 1) output of Ljung-Box test for Ghana’s inflation rate. Figure 4.3.5: ARIMA (1, 1, 2) output of Ljung-Box test for Ghana’s inflation rate. 57 University of Ghana http://ugspace.ug.edu.gh Observing figures 4.3.3, 4.3.4, and 4.3.5 and considering table 4.3.3 for the diagnotics tests for the ARIMA (2, 1, 1), ARIMA (1, 1, 2), and ARIMA (1,1,1), it can be deduced that, the residuals are independent and their respective standardized residual plots appear to be random. The ACF plots of the residuals show a non-existence of auto-correlation. The p-values for the Ljung-Box statistic for the first ten lags for the models are all greater than a 0.05 significance level. It can therefore be said that the three models which are ARIMA (1,1,1), ARIMA(2,1,1) and ARIMA (1,1,2) are all appropriate models for the representation of the data nevertheless using the Box-Pierce test as a yard stick, ARIMA (1,1,2) is considered as the most appropriate model that best represents the data because it has the highest probability value for confirming the independence of the residual value. In the same vein, three of the suggested ARIMA models for Ghana’s GDP values with the minium AIC values were considered and the most appropriate model that well represented the data was selected as the model with the best fit based on the findings obtained from the diagnostic check made on the residuals of the each of the three models. Table 4.3.4 shows the output of the Box-Pierce test for ARIMA (0, 1, 0), ARIMA(1, 1, 0) and ARIMA(1, 1, 1) respectively . Figures 4.3.6, 4.3.7 and 4.3.8 show their equivalent Ljung-Box tests. Table 4.3.4: ARIMA model output of Box-Pierce test for Ghana’s GDP values. Models X-squared Df p-value ARIMA (0,1,0) 1.8658 1 0.172 ARIMA (1,1,0) 0.026313 1 0.8711 ARIMA (1,1,1) 0.00018433 1 0.9892 Source: Researcher’s computation based on sampled data 58 University of Ghana http://ugspace.ug.edu.gh Figure 4.3. 6: ARIMA (0, 1, 0) output of Ljung-Box test for Ghana’s GPD values Figure 4.3.7: ARIMA (1, 1, 0) output of Ljung-Box test for Ghana’s GDP values. 59 University of Ghana http://ugspace.ug.edu.gh Figure 4.3. 8: ARIMA (1, 1, 1) output of Ljung-Box test for Ghana’s GDP values. Observing figures 4.3.6, 4.3.7 and 4.3.8 and considering table 4.3.4 for the diagnotics tests for the ARIMA (1,1,0), ARIMA (1,1,1), and ARIMA (0,1,0) it can be deduced that, the residuals are independent and their respective standardized residual plots appears to be random. The ACF plots of the residuals shows a non-existence of auto-correlation. The p- values for the Ljung-Box statistic for the first ten lags for both models are all greater than a 0.05 significance level. It can therefore be said that the three models which are ARIMA (0,1,0), ARIMA (1,1,0) and ARIMA (1,1,1) are all appropriate models for the representation of the data but nevertheless using the p-value of the output of the Box- Pierce test as a yard stick, ARIMA (1, 1, 1) is considered as the most appropriate model that best represent the data because it has the highest p-value for making the decision that the residual values are independent. 60 University of Ghana http://ugspace.ug.edu.gh Table 4.3.5 and Table 4.3.6 below show the model outputs for the ARIMA models. It presents the coefficients and standard errors of the models for the macroeconomic variables. Table 4.3.5: Model Output for ARIMA (1,1,2) Inflation rate Variable Coefficient Std Error Intercept -0.0612 0.1404 AR(1) 0.8747 0.0710 MA(1) -0.7909 0.1189 MA (2) 0.1723 0.1034 Source: Researcher’s computation based on sampled data Table 4.3.6: Model Output for ARIMA (1,1,1) GDP Variable Coefficient Std. Error Intercept 0.0634 0.0227 AR (1) -0.0637 0.4976 MA (1) 0.2595 0.4736 Source: Researcher’s computation based on sampled data From tables 4.3.5 and 4.3.6, the estimated models can be mathematically written as; Yt  0.060.87Yt1 0.79t1 0.17t2 (4.1a) Yt  0.060.06Yt1 0.26t1 (4.1b) The out sample forecast of the monthly inflation for six months from ARIMA (1,1,2) is shown in table 4.3.7. Table 4.3.8 also depicts the out sample forecasts for three years for the GDP values from the ARIMA (1,1,1) model. From Tables 4.3.7 and 4.3.8, it can be deduced that all the forecasted values fall within the lower and upper limits given a 95% 61 University of Ghana http://ugspace.ug.edu.gh confidence interval and the forecasted values are close to the actual values hence both models are appropriate for forecasting future values. Table 4.3.7: Six-months out sample forecast of the monthly inflation from the ARIMA (1,1,2) Month(2017) Actual Forecast Forecast error 95% Confidence Interval Lower limit Upper limit July 11.9 11.90 0.00 10.90 13.05 August 12.3 12.00 0.30 10.28 13.55 September 12.2 12.20 0.00 9.79 14.14 October 11.6 11.65 0.05 9.42 14.60 November 11.7 11.80 0.10 9.12 14.99 December 11.8 12.00 0.20 8.87 15.34 Source: Researcher’s computation based on sampled data Table 4.3.8: Three years out sample forecast for the yearly GDP from ARIMA (1,1,1) Year Actual Forecast Forecast error 95% Confidence Interval Lower limit Upper limit 2015 37.54 38.10 0.56 33.41 43.37 2017 42.69 43.00 0.31 40.53 45.88 2018 45.5 45.60 0.10 41.99 47.13 Source: Researcher’s computation based on sampled data 4.4 Estimation of quantile values of Variables The residuals of the most ideal models of both the Inflation and the GDP values were used to estimate the quantile values. In estimating the quantile values, the GP distribution model is fitted to the residuals as explained in the previous chapter. The mean excess plot aids in the selection of the threshold since it is an important task to consider when fitting the GP distribution . The selection is done by observing the point where there is a form of linearity on the mean excess plot. Observing the mean excess plot 62 University of Ghana http://ugspace.ug.edu.gh for ARIMA (1,1,2) residuals of the inflation data in figure 4.4.1, it can be seen that there is no ab line on the plot. This means that it is difficult to locate a point on the plot where a form of linearity exits. It is therefore difficult to select a particular threshold for fitting the GP model. Also the number of data points of the GDP do not allow for the fitting of a GP model therefore posing a challenge on estimating quantile values using the extreme value approach for both data sets. We therefore resort to usual method of calculating quantile values given the probabilities and the data sets. Figure 4.4.1: Mean excess plot for ARIMA (1,1,2) residuals. Table 4.4.1 gives a descriptive statistics and the quantile value of 0.5,0.7 and 0.9 respectively of the residuals of ARIMA (1,1,2) and ARIMA (1,1,1)which are the most appropriate models that best represent the Inflation and the GDP data respectively. 63 University of Ghana http://ugspace.ug.edu.gh Table 4.4.1: Descriptive Statistics and Quantile values of residuals of ARIMA models Variable Model Statistics Quantile Mean Median Max Min 0.5 0.7 0.9 Inflation ARIMA(1,1,2) 0.03136 0.03917 1.383 -1.709 0.0282 0.1925 0.5117 GDP ARIMA(1,1,1) 0.00006 0.01923 0.5576 -0.4938 0.0192 0.0634 0.1283 Source: Researcher’s computation based on sampled data It can be revealed from table 4.4.1 that the residuals of ARIMA (1,1,2) are centred around 0.03917, they have an average of 0.03136 and ranges between -1.709 and 1.383. That of ARIMA (1,1,1) ranges between -0.4938 and 0.5576 with an average of 0.00006 and centred around 0.01923. 4.5 Monitoring the achievement of set target of Inflation rate In monitoring the achievement of set targets, we consider both types of the expected trend lines and the quantile values for both variables as discussed in the previous chapter. To achieve a 9% inflation target in the next six months, two coordinate points; (108, 11.8) and (114, 9) which represent the last point of the inflation rate which is December 2017 with an inflation rate of 11.8% and June 2018 with a 9% inflation rate representing the target to be achieved in the next six months were used to obtain a linear expected trend line. The deduced linear expected trend line is given by y  0.47x62.2 In monitoring the achievement of a 9% inflation rate, a lower and an upper bound using the various quantile estimation obtained are set on the linear expected trend line. Assuming a uniform interval, five inflation rates are each randomly generated within each of the bounds. 64 University of Ghana http://ugspace.ug.edu.gh These generated inflation rates are then added to the series with the first five inflation rates taken out of the series simultaneously. With the updated series, ARIMA (1,1,2) model is used to forecast one period ahead. For the purpose of this study, this process is repeated 100 times. Table 4.5.1 shows the forecasted values for inflation using the linear expected trend line and the lower and upper bounds of 0.5, 0.7 and 0.9 quantile values respectively. Table 4.5.1: Forecasted values of Inflation using the linear expected trend line and lower and upper bounds of quantile values Months Inflation Quantile bounds (2018) 0.5 0.7 0.9 Lower Upper Lower Upper Lower Upper January 11.33 11.3018 11.3582 11.1375 11.5225 10.8188 11.8417 February 10.87 10.8418 10.8982 10.6775 11.0625 10.3583 11.3817 March 10.40 10.3718 10.4282 10.2075 10.5925 9.888 10.9117 April 9.93 9.9018 9.9582 9.7375 10.1225 9.4183 10.4417 May 9.47 9.4418 9.4982 9.2775 9.6625 8.9583 9.9817 Source: Researcher’s computation based on sampled data Table 4.5.2 shows the forecasted values using ARIMA (1,1,2) for a linear expected trend line given a 0.5 quantile bound. Other values forecasted using the ARIMA (1,1,2) given the 0.7 and 0.9 bounds are presented in the appendix A; Tables 1A and 2A. 65 University of Ghana http://ugspace.ug.edu.gh Table 4.5.2: Forecasted Inflation values using ARIMA (1,1,2) for a linear expected trend line given a 0.5 quantile bound 9.3889 9.3791 9.3877 9.3836 9.3844 9.3870 9.3828 9.3813 9.3896 9.3881 9.3854 9.3871 9.3876 9.3893 9.3909 9.3851 9.3860 9.3852 9.3877 9.3881 9.3871 9.3862 9.3903 9.3793 9.3883 9.3894 9.3886 9.3868 9.3850 9.3853 9.3896 9.3847 9.3829 9.3901 9.3921 9.3851 9.3933 9.3860 9.3883 9.3839 9.3874 9.3862 9.3871 9.3863 9.3845 9.3882 9.3856 9.3855 9.3837 9.3856 9.3845 9.3827 9.3868 9.3920 9.3936 9.3864 9.3861 9.3856 9.3851 9.3841 9.3853 9.3847 9.3834 9.3901 9.3925 9.3831 9.3829 9.3858 9.3807 9.3910 9.3862 9.3807 9.3801 9.3869 9.3918 9.3847 9.3852 9.3850 9.3871 9.3840 9.3865 9.3923 9.3919 9.3883 9.3894 9.3926 9.3929 9.3937 9.3853 9.3837 9.3813 9.3894 9.3884 9.3877 9.3855 9.3902 9.3893 9.3793 9.3839 9.3889 Source: Researcher’s computation based on sampled data The Time series graphical plots showing the forecasts from the linear expected trend line with quantile values 0.5, 0.7 and 0.9 are presented in Figures 4.5.1, 4.5.2 and 4.5.3 Figure 4.5.1: Time Series indicating forecasts from the linear expected trend line for a 0.5 bound 66 University of Ghana http://ugspace.ug.edu.gh . Figure 4.5.2: Time series indicating forecasts from the linear expected trend line for a 0.7 bound Figure 4.5.3: Time series indicating forecasts from the linear expected trend line for a 0.9 bound 67 University of Ghana http://ugspace.ug.edu.gh 4.5.1 Analysis of Inflation given Curve-linear expected trend line For a curve-linear expected trend line and a 9% inflation rate target to be achieved in the next six months, the procedure explained in the previous chapter is considered. Three coordinate points; (108, 11.8), (111, 13) and (114, 9) representing December 2017, March 2018 and June 2018 with their corresponding inflation rates were used to deduce a curve- linear expected trend line of the form y  0.29x2 63.67x3494.6 . The same procedure used to obtain the forecasted values using the ARIMA (1,1,2) as illustrated in the case of the linear expected trend line and their respective bounds was followed. Table 4.5.1.1 shows the forecasted values of inflation using the curve-linear expected trend line and the lower and upper bounds of 0.5, 0.7 and 0.9 respectively. Table 4.5.1.1: Forecasted values of Inflation using the curve-linear expected trend line and lower and upper bounds of quantile values Months Inflation Quantile bounds (2018) 0.5 0.7 0.9 Lower Upper Lower Upper Lower Upper January 12.78 12.7518 12.8082 12.5875 12.9725 12.2683 13.2917 February 13.18 13.1518 13.2080 12.9875 13.3725 12.6683 13.6917 March 13.00 12.9718 13.0282 12.8075 13.1925 12.4883 13.5117 April 12.24 12.2119 12.2682 12.0475 12.4325 11.7283 12.7517 May 10.91 10.8818 10.9382 10.7175 11.1025 10.3983 11.4217 Source: Researcher’s computation based on sampled data Table 4.5.1.2 shows the forecasted values using ARIMA (1,1,2) for a curve-linear expected trend line given a 0.5 quantile bound. Other values forecasted using the ARIMA (1,1,2) given the 0.7 and 0.9 bounds are presented in the appendix A; Tables 3A and 4A. 68 University of Ghana http://ugspace.ug.edu.gh Table 4.5.1.2: Forecasted Inflation values using ARIMA (1,1,2) for a curve-linear expected trend line given a 0.5 bound 9.8322 9.8196 9.8219 9.8186 9.8302 9.8223 9.8196 9.8274 9.8353 9.8280 9.8391 9.8301 9.8310 9.8244 9.8296 9.8351 9.8281 9.8309 9.8289 9.8195 9.8278 9.8341 9.8272 9.8359 9.8207 9.8296 9.8328 9.8310 9.8367 9.8380 9.8405 9.8382 9.8315 9.8236 9.8336 9.8364 9.8355 9.8420 9.8272 9.8413 9.8350 9.8358 9.8287 9.8305 9.8350 9.8316 9.8373 9.8418 9.8321 9.8308 9.8359 9.8316 9.8266 9.8316 9.8373 9.8418 9.8419 9.8344 9.8340 9.8175 9.8328 9.8262 9.8380 9.8189 9.8310 9.8422 9.8300 9.8315 9.8356 9.8260 9.8304 9.8341 9.8247 9.8253 9.8192 9.8335 9.8261 9.8345 9.8205 9.8325 9.8359 9.8353 9.8347 9.8334 9.8352 9.8276 9.8424 9.8424 9.8431 9.8384 9.8230 9.8284 9.8258 9.8359 9.8243 9.8295 9.8181 9.8241 9.8358 9.8188 Source: Researcher’s computation based on sampled data The Time series graphical plots showing the forecasts from the curve-linear expected trend line with quantile values 0.5, 0.7 and 0.9 are presented in Figures 4.5.1.1, 4.5.1.2 and 4.5.1.3 respectively Figure 4.5.1.1: Time series indicating forecasts for the curve-linear expected trend line for a 0.5 bound 69 University of Ghana http://ugspace.ug.edu.gh Figure 4.5.1.2: Time series indicating forecasts for the curve-linear expected trend line for a 0.7 bound Figure 4.5.1.3: Time series indicating forecasts for the curve-linear expected trend line for a 0.9 bound Table 4.5.1.3 represent the proportion of the forecasted value using ARIMA(1,1,2) that falls within a 5% and 10% range of 9 target of inflation rate given the expected trend lines and their respective quantile bounds. 70 University of Ghana http://ugspace.ug.edu.gh Table 4.5.1.3: Proportions of Forecasted values within 5%and 10% of targets given the expected trend lines and the bounds Expected trend line Quantile bound 5% (8.55, 9.45) 10% (8.1, 9.9) Linear Expected trend 0.5 100 100 line 0.7 100 100 0.9 94 100 Curve-linear Expected 0.5 None 100 trend line 0.7 None 100 0.9 None 94 Source: Researcher’s computation based on sampled data From table 4.5.1.3, it can be observed that for a 5% range of an inflation rate of 9 given a linear expected trend line, all the values within a 0.5 and 0.7 quantile bound fall within the range whereas for a quantile bound of 0.9, 94% of the values fall within the range. The table also depicts that for a 10% range of an Inflation rate of 9 given a linear expected line, all the values within 0.5, 0.7 and 0.9 quantile bounds fall within the range. It can further be deduced from the table that for a 5% range of inflation rate of 9 given a curve-linear expected trend line, none of the values within the 0.5, 0.7 and 0.9 quantile bounds fall within the range. Also for a 10% range of an inflation rate of 9 given a curve- linear expected, the values within 0.5, and 0.7 quantile bounds fall within the range whiles the values within a quantile bound of 0.9, 94% of them fall within the range. Table 4.5.1.4 depicts the mean absolute percentage error estimations for the Inflation rates given the expected trend lines and the various quantile bounds. 71 University of Ghana http://ugspace.ug.edu.gh Table 4.5.1.4: Mean Absolute Percentage Error for Inflation rate given the expected trend line and the various quantile bound Expected trend line Quantile bound MAPE (%) Linear Expected trend line 0.5 4.32 0.7 4.17 0.9 3.01 Curve-linear Expected trend line 0.5 9.10 0.7 8.42 0.9 7.16 Source: Researcher’s computation based on sampled data The Mean Absolute Percentage Error values from table 4.5.1.4 shows that the percentage errors for the inflation rate given the linear expected trend line are lower than that of the errors given the curve-linear expected trend line. This depicts that for Inflation rate, the forecast from ARIMA (1,1,2) after preceding values have been generated from the linear expected trend line performs well compared to that of the curve-linear expected trend line. 4.6 Monitoring the achievement of set target of GDP values Similar procedure is followed as illustrated in monitoring the achievement of a set target of a 9% inflation rate. With a target of 49 billion US dollars to be achieved in the next six years, the coordinate points; (57, 45.5) and (63, 49) which represents the last point of the series and the target to be achieved in the next six years is used to deduce a linear expected trend line. The deduced linear expected trend line is given by y  0.583x12.25 In monitoring the achievement of 49 billion dollars in the next six years, the updated series is used to forecast one period ahead using the ARIMA (1,1,1) model and the forecasted value recorded. Table 4.6.1 shows the forecasted values for GDP using the linear expected trend line and the lower and upper bounds of 0.5, 0.7 and 0.9 respectively. 72 University of Ghana http://ugspace.ug.edu.gh Table 4.6.1: Forecasted values of Inflation using the linear expected trend line and lower and upper bounds of quantile values Years GDP Quantile bounds 0.5 0.7 0.9 Lower Upper Lower Upper Lower Upper 2018 46.08 45.9888 46.1712 46.0166 46.1434 45.9517 46.2083 2019 46.67 46.5788 46.7612 46.6068 46.7334 46.5417 46.7983 2020 47.25 47.1588 47.7342 47.1866 47.3134 47.1217 47.3783 2021 47.83 47.7388 47.9212 47.7666 47.8934 47.7017 47.9583 2022 48.42 48.3288 48.5112 48.3566 48.4834 48.2917 48.5483 Source: Researcher’s computation based on sampled data Table 4.6.2 shows the forecasted values of the GDP using ARIMA (1,1,1) for a linear expected trend line given a 0.5 quantile bound. Other values forecasted using the ARIMA (1,1,1) given the 0.7 and 0.9 bounds are presented in the appendix B; Tables 1B and 2B. Table 4.6.2: Forecasted GDP values using ARIMA (1,1,1) for a linear expected trend given a 0.5 bound 49.9622 49.9617 49.9621 49.9620 49.9619 49.9622 49.9618 49.9619 49.9624 49.96214 49.9619 49.9620 49.9620 49.9620 49.9621 49.9618 49.9619 49.9619 49.9623 49.9619 49.9621 49.9619 49.9619 49.9618 49.9622 49.9621 49.9617 49.9618 49.9619 49.9620 49.9619 49.9621 49.9623 49.9619 49.9620 49.9621 49.9617 49.9621 49.9621 49.9622 49.9620 49.9621 49.9618 49.920 49.9619 49.9619 49.9620 49.9620 49.9621 49.9619 49.9621 49.9622 49.9618 49.9620 49.9618 49.9622 49.9620 49.9621 49.9622 49.9620 49.9618 49.9619 49.9618 49.9623 49.9619 49.9617 49.9618 49.9619 49.9624 49.9620 49.9619 49.9618 49.9617 49.9617 49.9621 49.9623 49.9624 49.9620 49.9619 49.9621 49.9623 49.9621 49.9618 49.9617 49.9620 49.9619 49.9621 49.9625 49.9618 49.9619 Source: Researcher’s computation based on sampled data The Time series graphical plots showing the forecasts from the linear expected trend line with quantile values 0.5, 0.7 and 0.9 are presented below in Figures 4.6.1, 4.6.2 and 4.6.3 respectively 73 University of Ghana http://ugspace.ug.edu.gh Figure 4.6.1: Time series indicating forecasts from the linear expected trend line for a 0.5 bound Figure 4.6.2: Time series showing forecasts from the linear expected trend line with a 0.7 bound 74 University of Ghana http://ugspace.ug.edu.gh Figure 4.6.3: Time series showing forecasts from the linear expected trend line for a 0.9 bound 4.6.1 Analysis of GDP given Curve-linear expected trend line For a curve-linear expected trend line and a 49 billion US dollars target to be achieved in the next period, three coordinate points; (57, 45.5), (60, 47) and (63, 49) representing the years 2017, 2020 and 2023 with their corresponding GDP value was employed to obtain the curve-linear trend line. The deduced curve-linear expected trend line is given by y  0.0278x2 2.75x112 Table 4.6.1.1 shows the forecasted value of GDP using the curve-linear expected trend line and the lower and the upper bound of 0.5, 0.7 and 0.9 respectively. 75 University of Ghana http://ugspace.ug.edu.gh Table 4.6.1.1: Forecasted values of GDP using the curve-linear expected trend line and lower and upper bounds of quantile values Years GDP Quantile bounds 0.5 0.7 0.9 Lower Upper Lower Upper Lower Upper 2018 45.94 45.9208 45.9592 45.8766 46.0034 45.8117 46.0683 2019 46.44 46.4208 46.4592 46.3766 46.5034 46.3117 46.5683 2020 47.00 46.9808 47.0192 46.9366 47.0634 46.8717 47.1283 2021 47.61 47.5908 47.6292 47.5466 47.6734 47.4817 47.7383 2022 48.28 48.2608 48.2992 48.2166 48.3434 48.1517 48.4083 Source: Researcher’s computation based on sampled data Table 4.6.1.2 shows the forecasted values of the GDP using ARIMA (1,1,1) for a curve- linear expected trend line given a 0.5 quantile bound. Other values forecasted using the ARIMA (1,1,1) given the 0.7 and 0.9 bounds are presented in the appendix B: Tables 3B and 4B. Table 4.6.1.2: Forecasted GDP values using the ARIMA (1,1,1) for a curve-linear expected trend given a 0.5 bound 48.9476 48.9475 48.9474 48.9475 48.9476 48.9474 48.9475 48.9476 48.9475 48.9475 48.9475 48.9476 48.9476 48.9475 48.9474 48.9475 48.9476 48.9475 48.9476 48.9474 48.9475 48.9476 48.9475 48.9475 48.9475 48.9476 48.9476 48.9475 48.9475 48.9476 48.9476 48.9475 48.9476 48.9475 48.9476 48.9476 48.9476 48.9477 48.9475 48.9476 48.9476 48.9476 48.9476 48.9476 48.9476 48.9475 48.9477 48.9476 48.9474 48.9473 48.9476 48.9475 48.9476 48.9475 48.9475 48.9476 48.9476 48.9476 48.9476 48.9476 48.9475 48.9476 48.9476 48.9476 48.9474 48.9475 48.9475 48.9476 48.9475 48.9476 48.9476 48.9476 48.9475 48.9475 48.9475 48.9474 48.9476 48.9476 48.9475 48.9476 48.9475 48.9476 48.9476 48.9475 48.9474 48.9475 48.9476 48.9475 48.9476 48.9475 48.9476 48.9476 48.9475 48.9474 48.9475 48.9475 48.9474 48.9475 48.9476 48.9475 Source: Researcher’s computation based on sampled data The Time series graphical plots showing the forecasts from the curve-linear expected trend line with quantile values 0.5, 0.7 and 0.9 are presented below in Figures 4.6.1.1, 4.6.1.2 and 4.6.1.3 respectively 76 University of Ghana http://ugspace.ug.edu.gh Figure 4.6.1.1: Time series showing forecasted values from curve-linear expected trend line with a bound of 0.5 Figure 4.6.1.2: Time series showing forecasts from the curve-linear expected trend line for a 0.7 bound 77 University of Ghana http://ugspace.ug.edu.gh Figure 4.6.1.3: Time series showing forecasted values from curve-linear expected trend line for 0.9 bound Table 4.6.1.3 represents the proportion of the forecasted value using ARIMA (1,1,1) that falls within a 5% and 10% range of 49 target of GDP given the expected trend lines and their respective quantile bounds Table 4.6.1.3: Proportions of forecasted values that fall within 5% and 10% of set target given the expected trend lines and the bounds Expected trend line Quantile bound 5% (46.55, 51.45) 10% (44.1, 53.9) Linear Expected trend 0.5 100 100 line 0.7 100 100 0.9 100 100 Curve-linear Expected 0.5 100 100 trend line 0.7 100 100 0.9 100 100 Source: Researcher’s computation based on sampled data 78 University of Ghana http://ugspace.ug.edu.gh From table 4.6.1.3, it can be observed that for a 5% range of a GDP of 49 given a linear expected trend line, all the values within a 0.5, 0.7 and 0.9 quantile bound fall within the range. The table also shows that for a 10% range of a GDP of 49 given a linear expected line, all the values within 0.5, 0.7 and 0.9 quantile bounds fall within the range. It can further be deduced from the table that for a 5% range of a GDP of 49 given a curve- linear expected trend line, all the values within the 0.5, 0.7 and 0.9 quantile bounds fall within the range. Similar deductions can be made for 10% of a GDP of 49 given a curve- linear expected trend line. Table 4.6.1.4 is a tabular representation of the mean absolute percentage error values for the GDP values given their expected trend lines and quantile bounds. Table 4.6.1.4: Mean Absolute Percentage error for GDP value given the expected trend line and the various quantile bounds. Expected trend line Quantile bound MAPE (%) Linear Expected trend line 0.5 1.96 0.7 1.96 0.9 1.96 Curve-linear Expected trend line 0.5 0.11 0.7 0.11 0.9 0.10 Source: Researcher’s computation based on sampled data From the table, it can be observed that the mean absolute percentage error values for the GDP values given a curve-linear expected trend line and the quantile bounds are lower than the error values for the linear expected trend line. 4.7 Discussion of Results The findings and results realised from the various stages of the analyses are discussed in this section. 79 University of Ghana http://ugspace.ug.edu.gh With the main objective of the study being stochastically monitoring the achievement of set targets in relation to variables, two macroeconomic variables were considered. The GDP values and Inflation rate values over a period were used for the analyses. In the initial stages of the analyses, the graphical plots of the data sets were generated and the results showed that the mean and variance for both plots vary over time. This indicated non-stationarity in the data sets. The Dickey-Fuller test was performed to confirm the non- stationarity observed and it was confirmed that the non-stationarity existed. Hence, the Inflation data with a p-value of 0.6258 at a 5% level of significance was differenced once to attain stationarity and was once again confirmed by the Augmented Dickey-Fuller test. The GDP data on the other hand was made stationary by taking the natural log of the series and differenced once. The ACF and PACF plots were done to find the order of the MA(q) and AR(p) and this was used to determine the possible models to be used. In the case of the Inflation data, AR(1) and MA(3) were suggested through observation of the ACF and the PACF plots. Likewise AR(0) and MA(0) were suggested by the plots for the GDP data. Since both data sets were differenced once to attain stationarity, an ARIMA model was built with d= 1. Eight different possible models were built for each of the data sets. Among the eight, three had the minimum AIC values for both of the data sets. The Box Pierce and Ljung diagnostics tests were used to confirm the best models for both data sets from the three that were selected. Based on the diagnostics tests, the ARIMA (1,1,2) with the highest p- value of 0.9669 at 5% level of significance was selected as the best model for the Inflation rate time series analysis and the ARIMA (1,1,1) with the highest p-value of 0.9892 was chosen as the best model for the GDP time series analysis. The expected trend lines for forecasting the future values were then deduced and used to forecast the future inflation rate values and GDP values. The linear expected trend line and 80 University of Ghana http://ugspace.ug.edu.gh the curve-linear expected trend line were considered for both data sets. The equation of a straight line and the quadratic equation were employed in deducing the lines. The residuals of the most appropriate models of the two data sets were used to obtain quantile values for 50%, 70% and 90% respectively. These quantile values were then used to set upper and lower bounds around the forecasted values using the expected trend lines. The proportions of the forecasted values that fall within 5% and 10% range of the targets were estimated to determine whether preceding values around the expected trend lines given the bounds are likely to meet the set target for a length of period. This was further confirmed by computing the mean absolute percentage error for the variables given the bounds and the expected trend lines. It was realised that for Inflation rate, the forecasts from ARIMA (1,1,2) after preceding values have been generated from the linear expected trend line performs well compared to that of the curve-linear expected trend line. The computation for the GDP showed that the forecasts from ARIMA (1,1,1) after preceding values have been generated from the curve-linear expected trend line performs better than forecasts from ARIMA (1,1,1) after preceding values have been generated from the linear expected trend line. 81 University of Ghana http://ugspace.ug.edu.gh CHAPTER FIVE SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 5.0 Introduction This chapter summarizes the findings from the entire study as well as conclusions on the study. Recommendations from the study are also presented for literature and policy makers. 5.1 Summary In most Statistical research done on Macro-Economic variables, the focus has been on forecasting variables, determining factors affecting these variables and the impact they make on economic growth but little has been done in the area of monitoring the performance of these macro-economic variables within an economy. The lag in literature on the performance monitoring of macro-economic variables necessitated the study. The main objective of this study was to develop a stochastic model to determine whether or not preceding values of Macro-economic variables around an “expected trend line” will lead to achieving target set at a future point in time. Specifically, the study sought to check the performance of the model when preceding values are around a “linear expected trend line” and a “curve-linear expected trend line”. The Macro-economic variables which were analysed under the study were Inflation rate and the GDP of Ghana and the data covered a period of 2009 to 2017 and 1961 to 2017 respectively. The graphical plots of the data sets showed a non-stationarity in both of the data sets as confirmed by the Augmented Dickey-Fuller test. 82 University of Ghana http://ugspace.ug.edu.gh The data was transformed to obtain the stationarity. The ACF and PACF plots were used to find the order of the MA and AR and this was used to determine the possible models to be used. The minimum AIC values of the models were used to select the best models and were confirmed by the Box-Pierce and Ljung-Box Diagnostic tests. Based on the diagnostics tests, the ARIMA (1,1,2) was employed as the best model for the Inflation rate time series analysis and the ARIMA (1,1,1) was chosen as the best model for the GDP time series analysis. The quantile values of the ARIMA model residuals were calculated for 50%, 70% and 90% quantiles. The expected trend lines were then deduced and used to forecast the future inflation rate values and GDP values. For the performance monitoring of the Macro- economic variables, Upper and lower bounds were set for the expected trend lines. The absolute differences between the values forecasted using the ARIMA models and the set target were then calculated. The focus for the inflation rate was monitoring the achievement of a 9% inflation rate in the next six months given the linear expected trend line and a particular quantile bound. A large proportion of the forecasted values fell within 5% and 10% of the targets set and the MAPE results confirmed the forecast accuracy. This implies that future inflation rate values that lie within 50%, 70% and 90% quantile bounds of the residuals of ARIMA (1,1,2) are certain to achieve the 9% inflation rate by the sixth month. Likewise for the monitoring of the achievement of a GDP of 49 in the next six years given the expected trend lines and a particular bound. A large proportion of the forecasted values fell within 5% and 10% of a target of 49 billion US dollars and the MAPE results confirmed the forecast accuracy implying that future GDP values that lie within the set bounds of 50%, 70% and 90% quantile bounds of the residuals of ARIMA (1,1,1) are certain to achieve a GDP of 49 billion USD by the sixth month. 83 University of Ghana http://ugspace.ug.edu.gh 5.2 Conclusion Based on the analyses, the findings and the summary, it can be concluded that: i. For both of the macroeconomic variables that were considered; Inflation rate and GDP values, the models that were performed well when preceding values were around the linear expected trend lines. It can also be concluded that looking at the forecasted values and the targets set and the forecast accuracies of the models, future forecasted values using the ARIMA models for a linear expected trend line within a given bound are likely to be achieved within six months for inflation and six years for the GDP. ii. The models performed well when the preceding values were around the curve- linear expected trend line. It can also be concluded from the forecasted values and the targets looking at the forecast errors that, future forecasted values using the ARIMA models for curve-linear expected trend lines considering different bounds are certain to be achieved. iii. Considering the performance of the models between high targeted and low targeted macro-economic variables, the macroeconomic variables that were studied were Inflation rate for the low targeted variable and Gross Domestic Product (GDP) for the high targeted macro-economic variable. Looking at the results of the MAPE calculated for the expected trend lines given the bounds, it can be concluded that for monitoring a low targeted macroeconomic variable, preceding values around a linear expected trend line performs well compared to the preceding values around a curve-linear expected trend line. On the other hand, for monitoring high targeted macroeconomic variable, preceding values around a curve-linear expected trend line given their respective bounds performed better compare to those around a linear trend line. It can be 84 University of Ghana http://ugspace.ug.edu.gh concluded from the study that the methodology performs well for both low targeted macro-economic variables and high targeted macro-economic variables. iv. Finally, looking at the proportions of the forecasted values that fell within 5% and 10% of the targets given both linear and curve-linear expected trend lines, it can be concluded that subsequent time series values that fall within 0.5, 0.7 and 0.9 quantile bounds have a high level of confidence in meeting the set targets. 5.3 Recommendations Based on the findings from the study and the conclusions reached, the following recommendations have been made for further study, industry and to influence future policy decisions by policy makers. 1. Other studies in this field have looked more at forecasting macro-economic variables and not monitoring their achievement within a time frame. This study focussed on monitoring the attainment of set targets within a period hence it is recommended that Policy makers such as governments and organisations should consider this methodology in the monitoring of their set targets instead of just using the traditional models to forecast. This methodology will also aid in monitoring and assessing the attainment of set targets during the implementation period of policies. 2. Secondly, it is recommended that in research, further studies should be conducted on the methodology to ascertain the efficiency of the model in monitoring set targets for different periods. The study is limited by the length of the period chosen to monitor the attainment of the set targets and hence other research should be 85 University of Ghana http://ugspace.ug.edu.gh undertaken by increasing the length of the period to assess the performance of the model for longer periods. 3. It is finally recommended that further or similar research in the area or on the topic should focus on other non-linear models such as; Autoregressive Conditional Heteroscedastic model (ARCH), the Generalised Autoregressive Conditional Heteroscedastic model (GARCH) among others. 86 University of Ghana http://ugspace.ug.edu.gh REFERENCES Abu B. 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Unwilingiymana, C., Munga’tu, J. & Harerimana, J.D.D. (2015). Forecasting Inflation. Journal of Management and Commerce Innovation 3, 15-27. Usman, M. (2016). Contribution of Agriculture Sector in GDP Growth Rate of Pakistan. Journal of Global Economics, ISSN: 2375-4389. Van den Bergh, & Jeroen C.J.M. (2008), "The GDP Paradox", Journal of Economic Psychology, no. 30, pp. 117-135 Wajid, A. & Kalim, R. (2012). The Impact of Inflation and Economic Growth. www.worldbank.org/opendata/new-country-classifications-income-level-2017-2018. Assessed on 15/02/2018 Yergin, D. & Stanislaw (1997). Nixon Tries Price Controls. www.en.wikipedia.org/wiki/Nixonshock Zhang, H. (2013). Modelling and forecasting regional GDP in Sweden using autoregressive models. Working Paper, Hoyskolan Dalama University, Sweden. data.worldbank.org/indicator. Assessed 10/01/2018 91 University of Ghana http://ugspace.ug.edu.gh APPENDICES APPENDIX A Figure 1A: Autocorrelation plot of Ghana’s transformed annual GDP values Figure 2A: Partial Autocorrelation plot of Ghana’s transformed annual GDP values. 92 University of Ghana http://ugspace.ug.edu.gh Table 1A: Forecasted Inflation values using ARIMA (1,1,2) for a linear expected trend line given a 0.7 quantile bound 9.4021 9.3341 9.3937 9.3656 9.3704 9.3891 9.3588 9.3485 9.3989 9.4064 9.3953 9.3776 9.3919 9.3932 9.4159 9.3752 9.3813 9.3762 9.4051 9.3960 9.3973 9.3894 9.3831 9.4120 9.3349 9.3975 9.4050 9.3998 9.3871 9.3745 9.3761 9.3822 9.3717 9.3602 9.3874 9.4243 9.3752 9.4328 9.3818 9.3971 9.3651 9.4070 9.3829 9.3894 9.4095 9.3705 9.3967 9.3772 9.3780 9.3658 9.3793 9.3907 9.3586 9.3871 9.3830 9.4353 9.3823 9.3821 9.3792 9.3451 9.3678 9.3714 9.3701 9.3645 9.4236 9.4279 9.3609 9.3588 9.3798 9.3890 9.4170 9.3769 9.3454 9.3410 9.4105 9.4227 9.3729 9.3757 9.3749 9.3773 9.3669 9.3843 9.4225 9.4221 9.3976 9.4042 4.4281 9.4305 9.4350 9.3666 9.3656 9.3491 9.4052 9.3992 9.3932 9.3801 9.4115 9.4048 9.3362 9.3752 Source: Researcher’s Computation based on sampled data Table 2A: Forecasted Inflation values using ARIMA (1,1,2) for a linear expected trend line given 0.9 quantile bound 9.4266 9.2551 9.4135 9.3319 9.3437 9.3984 9.2857 9.3154 9.4169 9.4343 9.4007 9.3627 9.3985 9.4114 9.4419 9.4628 9.3542 9.4516 9.3803 9.3929 9.4181 9.4500 9.3589 9.3716 9.2521 9.4392 9.3418 9.3478 9.3819 9.4196 9.4325 9.4154 9.3419 9.4066 9.3201 9.4413 9.4847 9.3540 9.5031 9.3797 9.3620 9.5154 9.4886 9.3876 9.3103 9.3647 9.3302 9.3526 9.3388 9.3189 9.3108 9.3329 9.3578 9.3636 9.4486 9.4704 9.3713 9.2770 9.3604 9.2971 9.3344 9.4961 9.2766 9.3510 9.3624 9.3912 9.3512 9.4818 9.3995 9.2710 9.4322 9.4745 9.4139 9.4894 9.4300 9.3729 9.4957 9.5085 9.3842 9.4040 9.3678 9.4312 9.2896 9.3291 9.3659 9.3346 9.2636 9.4310 9.4115 9.4237 9.3260 9.3548 9.4016 9.3319 9.3204 9.2713 9.5432 9.2334 9.1423 9.7710 Source: Researcher’s Computation based on sampled data Table 3A: Forecasted Inflation values using ARIMA (1,1,2) for curve-linear expected trend line given a 0.7 quantile bound 9.8386 9.7480 9.7683 9.7524 9.8270 9.7756 9.7546 9.8031 9.8598 9.8088 9.8863 9.8246 9.8288 9.7830 9.8209 9.8604 9.8105 9.8298 9.7518 9.8045 9.8539 9.8050 9.8622 9.7604 9.8146 9.8395 9.8300 9.8626 9.8689 9.8851 9.8329 9.7803 9.8478 9.8640 9.8570 9.9031 9.8040 9.9039 9.8558 9.8617 9.8154 9.8270 9.860 9.8530 9.8866 9.8880 9.8393 9.8250 9.8618 9.8326 9.7972 9.8317 9.8745 9.9044 9.8514 9.8486 9.7454 9.8419 9.7962 9.8172 9.8419 9.7962 9.8737 9.7509 9.8269 9.9105 9.8174 9.8262 9.8584 9.7898 9.8246 9.8484 9.7835 9.7782 9.7532 9.8405 9.7994 9.8493 9.7623 9.8413 9.8547 9.8556 9.8545 9.8425 9.86063 9.8074 9.9118 9.9120 9.8775 9.7757 9.7757 9.7915 9.8591 9.7829 9.8168 9.7491 9.7719 9.8617 9.7389 9.758 Source: Researcher’s Computation based on sampled data 93 University of Ghana http://ugspace.ug.edu.gh Table 4A: Forecasted Inflation values using ARIMA (1,1,2) for curve-linear expected trend line given a 0.9 quantile bound 9.8375 9.5788 9.6614 9.6432 9.8170 9.6952 9.6285 9.7295 9.8908 9.7536 9.9606 9.8046 9.8061 9.6871 9.7964 9.9083 9.7664 9.8134 9.7890 9.64080 9.7348 9.8919 9.7552 9.8976 9.6303 9.7556 9.826 9.8134 9.8631 9.8613 9.9004 9.9704 9.8161 9.6853 9.8607 9.8915 9.8641 9.9894 9.7470 10.0282 9.8650 9.8891 9.7827 9.8068 9.9056 9.8858 9.9580 9.9787 9.8461 9.7847 9.8856 9.8114 9.7114 9.7992 9.9483 10.0232 10.0000 9.8570 9.8340 9.6285 9.8397 9.7225 9.8929 9.6210 9.7927 10.0527 9.7480 9.7578 9.8663 9.6720 9.7967 9.8475 9.6658 9.6208 9.6422 9.8133 9.7490 9.8340 9.6540 9.8488 9.8409 9.8552 9.8814 9.8312 9.9040 9.7626 10.0523 10.0221 9.9238 9.6778 9.7564 9.6980 9.8737 9.6985 9.7698 9.6391 9.6347 9.8911 9.5409 9.6447 Source: Researcher’s Computation based on sampled data 94 University of Ghana http://ugspace.ug.edu.gh APPENDIX B Table 1B: Forecasted GDP values using ARIMA (1,1,1) for a linear expected trend line given a 0.7 bound 49.9626 49.9609 49.9622 49.9621 49.9618 49.9625 49.9614 49.9616 49.9622 49.9627 49.9627 49.9617 49.9624 49.9617 49.9619 49.9621 49.9621 49.9620 49.9618 49.9614 49.9629 49.9616 49.9613 49.9624 49.9619 49.9614 49.9625 49.9625 49.9609 49.9610 49.9613 49.9616 49.9619 49.9615 49.9623 49.9630 49.9614 49.9622 49.9615 49.9623 49.9611 49.9623 49.9625 49.9627 49.9627 49.9622 49.9621 49.9620 49.9618 49.9620 49.9615 49.9619 49.9622 49.9623 49.9619 49.9624 49.9622 49.9624 49.9613 49.9619 49.9614 49.9625 49.9621 49.9621 49.9625 49.9620 49.9614 49.9615 49.9612 49.9613 49.9629 49.9617 49.9610 49.9613 49.9618 49.9632 49.9617 49.9615 49.9628 49.9624 49.9610 49.9609 49.9623 49.9631 49.9618 49.9619 49.9626 49.9638 49.9630 49.9625 49.9624 49.9611 49.9612 49.9619 49.9618 49.9625 49.9635 49.9616 49.9610 49.9616 Source: Researcher’s Computation based on sampled data Table 2B: Forecasted GDP values using ARIMA (1,1,1) for a linear expected trend line given a 0.9 quantile bound 49.9632 49.9599 49.9624 49.9621 49.9617 49.9632 49.9607 49.9611 49.9623 49.9635 49.9631 49.9614 49.9619 49.9622 49.9621 49.9624 49.9617 49.9606 49.9615 49.9614 49.9639 49.9612 49.9629 49.9618 49.9617 49.9607 49.9629 49.9632 49.9598 49.9600 49.9606 49.9613 49.9617 49.9609 49.9628 49.9639 49.9609 49.9625 49.9610 49.9625 49.9601 49.9626 49.9629 49.9634 49.9624 49.9623 49.9606 49.9620 49.9615 49.9612 49.9609 49.9619 49.9624 49.9627 49.9618 49.9627 49.9624 49.9624 49.9628 49.9620 49.9609 49.9629 49.9622 49.9622 49.9632 49.9620 49.9608 49.9610 49.9604 49.9605 49.9638 49.9614 49.9601 49.9606 49.9615 49.9644 49.9618 49.9609 49.9637 49.9628 49.9600 49.9598 49.9626 49.9643 49.9615 49.9619 49.9631 49.9639 49.9641 49.9630 49.9628 49.9602 49.9604 49.9621 49.9617 49.9630 49.9651 49.9611 49.9600 49.9611 Source: Researcher’s Computation based on sampled data Table 3B: Forecasted GDP values using ARIMA (1,1,1) for a curve-linear trend line given a 0.7 quantile bound 48.9475 48.9476 48.9472 48.9474 48.9476 48.9473 48.9474 48.9477 48.9476 48.9474 48.9478 48.9476 48.9475 48.9473 48.9475 48.9476 48.9475 48.9476 48.9476 48.9471 48.9474 48.9476 48.9474 48.9475 48.9476 48.9474 48.9475 48.9475 48.9477 48.9479 48.9479 48.9476 48.9477 48.9475 48.9475 48.9476 48.9478 48.9476 48.9475 48.9479 48.9478 48.9477 48.9475 48.9476 48.9477 48.9477 48.9479 48.9479 48.9477 48.9477 48.9478 48.9477 48.9477 48.9476 48.9475 48.9475 48.9479 48.9478 48.9478 48.9473 48.9477 48.9474 48.9480 48.9473 48.9476 48.9476 48.9478 48.9477 48.9478 48.9478 48.9474 48.9476 48.9477 48.9478 48.9472 48.9475 48.9479 48.9473 48.9474 48.9477 48.9478 48.9477 48.9475 48.9476 48.9473 48.9478 48.9477 48.9476 48.9474 48.9474 48.9477 48.9476 48.9472 48.9475 48.9472 48.9474 48.9477 48.9472 48.9476 48.9473 Source: Researcher’s Computation based on sampled data 95 University of Ghana http://ugspace.ug.edu.gh Table 4B: forecasted GDP values using ARIMA (1,1,1) for a curve-linear trend line given a 0.9 quantile bound 48.9475 48.9476 48.9468 48.9472 48.9476 48.9471 48.9473 48.9472 48.9478 48.9471 48.9481 48.9477 48.9473 48.9469 48.9474 48.9476 48.9475 48.9477 48.9475 48.9467 48.9472 48.9476 48.9473 48.9477 48.9472 48.9473 48.9474 48.9479 48.9483 48.9474 48.9484 48.9477 48.9477 48.9474 48.9475 48.9476 48.9481 48.9481 48.9476 48.9482 48.9479 48.9478 48.9475 48.9476 48.9478 48.9479 48.9474 48.9482 48.9478 48.9478 48.9481 48.9479 48.9478 48.9477 48.9476 48.9475 48.9482 48.9480 48.9480 48.9469 48.9479 48.9473 48.9484 48.9471 48.9472 48.9477 48.9478 48.9481 48.9481 48.9480 48.9472 48.9477 48.9477 48.9481 48.9468 48.9473 48.9474 48.9482 48.9472 48.9478 48.9481 48.9478 48.9474 48.9475 48.9478 48.9471 48.9478 48.9478 48.9475 48.9471 48.9478 48.9477 48.9474 48.9481 48.9478 48.9474 48.9468 48.9476 48.9476 48.9471 Source: Researcher’s Computation based on sampled data . 96