Modelling and forecasting using Time Series ARIMA Models

Modelling and forecasting using Time Series ARIMA Models

CHAPTER ONE: INTRODUCTION

1.0 INTRODUCTION 

Inflation is a persistent rise in the general price levels of goods and services in an economy over a period of time. Inflation rate has been regarded as one of the major economic indicators in any country. According to Olatunji et al. (2010), inflation is undeniably one of the leading and most dynamic macro-economic issues confronting almost all economies of the world. Its dynamism has made it an imperative issue to be considered.

The monetary authorities of a large number of countries including Nigeria have decided that price stability, that is, a low and stable inflation rate, is the main contribution that monetary policy can give to economic growth. Accordingly, predicting the future course of inflation in a precise manner is a crucial objective to maintain this goal, especially in inflation targeting environment. The Central Bank of Nigeria has the maintenance of monetary and price stability as one of its mandate. Therefore, the Bank is attempting to develop and improve models that will provide relatively precise and reliable forecast of the headline, food and core inflation so that it can react in time and neutralize inflationary or deflationary pressures that could appear in the future.

The concept of inflation has been intrinsically linked to money, as captured by the often heard maxim "inflation is too much money chasing too few goods". Inflation has been widely described as an economic situation when the increase in money supply is “faster” than the new production of goods and services in the same economy (Hamilton, 2001).   Usually, economists try to distinguish inflation from an economic phenomenon of a onetime increase in prices or when there are price increases in a narrow group of economic goods or services (Piana, 2001). Thus, the term inflation describes a general and persistent increase in the prices of goods and services in an economy (Ojo, 2000; Melberg, 1992).

Inflation (on all items less farm produce and energy) is a sustained increase in the general price level of goods and services (minus farm produce and energy) in an economy over a period of time. (Wyplosz & Burda 1997, Blanchard 2000, Barro 1997) When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation reflects a reduction in the purchasing power per unit of money a loss of real value in the medium of exchange and unit of account within the economy. 

 All items less farm produce and energy is all items minus farm produce and energy. One way of accomplishing this is by excluding items frequently subject to volatile prices, like food and energy. The concept was introduced in a 1975 paper by Robert J. Gordon. (Gordon, R. J. (1975))

1.1 HISTORICAL BACKGROUND OF INFLATION

Increases in the quantity of the money or in the overall money supply (or debasement of the means of exchange) have occurred in many different societies throughout history, changing with different forms of money used. For instance, when gold was used as currency, the government could collect gold coins, melt them down, mix them with other metals such as silver, copper or lead, and reissue them at the same nominal value. By diluting the gold with other metals, the government could issue more coins without also needing to increase the amount of gold used to make them. When the cost of each coin is lowered in this way, the government profit from an increase in seignior age. 

Historically, infusions of gold or silver into an economy also led to inflation. From the second half of the 15th century to the first half of the 17th, Western Europe experienced a major inflationary cycle referred to as the "price revolution", with prices on average rising perhaps six-fold over 150 years. This was largely caused by the sudden influx of gold and silver from the New World into Habsburg Spain.

 By the nineteenth century, economists categorized three separate factors that cause a rise or fall in the price of goods: a change in the value or production costs of the good, a change in the price of money which then was usually a fluctuation in the commodity price of the metallic content in the currency, and currency depreciation resulting from an increased supply of currency relative to the quantity of redeemable metal backing the currency. Following the proliferation of private banknote currency printed during the American Civil War, the term "inflation" started to appear as a direct reference to the currency depreciation that occurred as the quantity of redeemable banknotes outstripped the quantity of metal available for their redemption. At that time, the term inflation referred to the devaluation of the currency, and not to a rise in the price of goods. 

This relationship between the over-supply of banknotes and a resulting depreciation in their value was noted by earlier classical economists such as David Hume and David Ricardo, who would go on to examine and debate what effect a currency devaluation (later termed monetary inflation) has on the price of goods (later termed price inflation, and eventually just inflation). 



1.2 STATEMENT OF THE PROBLEM

The assumption of constant mean and variance over some period when a series moves or progresses through time is statistically inefficient and inconsistent (Campbell et al, 1997). In real life, financial data mean and variance changes with time, hence there is a need of studying models which accommodates this possible variation in the mean and variance. In considering the issue inflation rate (on all items less farm produce and energy) modelling and forecasting in Nigeria, this work consequently intends to use the Box-Jenkins methodology, i.e. Autoregressive Integrated Moving Average (ARIMA) models to model and accommodate the dynamics of the inflation rate data. Moreover by finding appropriate models to represent the data, the study intends to use them to predict future values based on the past observations.

1.2.1 Causes of Inflation

Historically, a great deal of economic literature was concerned with the question of what causes inflation and what effect it has. There were different schools of thought as to the causes of inflation. Most can be divided into two broad areas: quality theories of inflation and quantity theories of inflation. The quality theory of inflation rests on the expectation of a seller accepting currency to be able to exchange that currency at a later time for goods that are desirable as a buyer. The quantity theory of inflation rests on the quantity equation of money that relates the money supply, its velocity, and the nominal value of exchanges. Adam Smith and David Hume proposed a quantity theory of inflation for money, and a quality theory of inflation for production.



1.2.2 Controlling Inflation

A variety of methods and policies have been proposed and used to control inflation.

Monetary policy: Governments and central banks primarily use monetary policy to control inflation. Central banks increase the interest rate, slow or stop the growth of the money supply, and reduce the money supply. Some banks have a symmetrical inflation target while others only control inflation when it rises above a target, whether express or implied.

Fixed exchange rates: Under a fixed exchange rate currency regime, a country's currency is tied in value to another single currency or to a basket of other currencies (or sometimes to another measure of value, such as gold). A fixed exchange rate is usually used to stabilize the value of a currency, vis-a-vis the currency it is pegged to. It can also be used as a means to control inflation.

Gold standard: The gold standard is a monetary system in which a region's common media of exchange are paper notes that are normally freely convertible into pre-set, fixed quantities of gold. The standard specifies how the gold backing would be implemented, including the amount of specie per currency unit. To mention but few are: Wage and price controls, Stimulating economic growth, Cost-of-living allowance

1.3 AIM AND OBJECTIVES OF THE STUDY

The aim of this project work is to use Box-Jenkins methodology for the rates of Inflation on All items less farm produce and energy (by month) in Nigeria from January, 2003 to December, 2014. This will then be used to predict the values for the subsequent months (of the years). With a view to achieve the following objectives

To determine the plot of the series.

To identify the correlogram of the data series (rates of Inflation on All items less farm produce and energy) through the use of autocorrelation function (ACF) and partial autocorrelation function (PACF).

To detect whether or not there exist periodicity (non-stationarity or seasonality) in the data.

To identify the ARIMA models for monthly rates of Inflation on All items less farm produce and energy in Nigeria.

1.4 SIGNIFICANCE OF THE STUDY

The research project is of significant to several parties as follows:

This research study is expected to provide knowledge and basis for further studies in the same area or other related fields of study.

The study is intended to contribute to policy making through developing models which can be used to forecast inflation rate in Nigeria, and thus, guide policy makers in formulating macroeconomic policies.

1.5 SCOPE AND LIMITATION

This research work is conducted based on the Box and Jenkins (methodology 1976); due to its power and flexibility in model fitting. The application in the study is limited to the monthly rates of Inflation on All items less farm produce and energy in Nigeria from January 2003 to December, 2014.

 

1.6 DEFINITION OF SOME TERMS

Time Series:-is defined as a set of observations taken at a specified times, usually at equal intervals.

Forecasting:- is a process of estimating future values of numerical parameters on the basis of the past. To do this, a model is created. This model is an artificial equation that captures the important features of the data.

Stationary Time Series:- is a time series whose statistics do not change over time in the mean (no trend), variance and periodic (seasonal) variation of the series.

Non-stationary Time Series:-is a time series which doesn’t have a constant mean and variance. In other words, non-stationary time series is one that contains trend seasonal variation.

Trend:- is defined as long term change in the mean, i.e. it is a smooth upward and downward increment of a time series over a long period of time.

Differencing:- is defined as a mathematical technique used for removing trend in time series. i.e. the series of changes from one period to the next.

Variance Function:- this is the central moment of a stochastic process, the function measures the degree to which the process is spread out over the real line.

Autocovariance Function:-is a second mixed central moment of a stochastic process, the function measures the similarity between two points on the same series observed at different times.

Less farm produce and energy:- is the total rate of inflation on all items minus farm produce and energy, such as, agriculture, health, transport, e.t.c.


CHAPTER TWO: LITRATURE REVIEW

2.0 INTRODUCTION TO TIME SERIES

Time, in terms of years, months, days or hours is a device that enables one to relate phenomena to a set of common, stable reference points. In making conscious decisions under uncertainty, we all make forecasts. Almost all managerial decisions are based on some form of forecast. Essentially the concept of time series is based on the historical observations. It involves explaining past observations in order to try to predict those in the future (Ahiati, 2007). A time series is a collection of observations measured sequentially through time. These measurements may be made continuously through time or be taken at a discrete set of time points (Chatfield, 2000).

A time series is a sequence of measurements of some numerical quantity made at or during successive periods of time. It is a subclass of stochastic process, Nichola and Laverne (1989). A time series is a set of observations taken at specified times, usually at equal intervals (Schaum’s out lines Series). Some of examples of time series data are given below:-

Unit sales of a product over time.

Total Dollar sales for a company over time 

Production  of steel in the united states over a number of years, 

The daily closing price of a share on the stock exchange

The hourly temperatures announced by the weather bureau of a city.

Granger and New bold (1986) describe a time series as a sequence of observation or random variable ordered by time parameter i.e t=1,2,3,……, is called a time  series where t is the time. This time points may be annually, quarterly, monthly, weekly, daily, hourly, or even in minutes. The use of prediction intervals, and more recently prediction densities, has become much more common over the past 25 years as practitioners have come to understand the limitation of point forecasts. An important and through review of interval forecasts is given by Chatfield (1993) summarizing the literature of that time.

Mathematically a time series is define by the values y1, y2, … of a variable Y, (temperature, closing price of a share, e.t.c) at time t1, t2, … Thus, Y is a function of t, symbolized by Y=F(t). (Schaum’s out lines Series)

According to Nicholas and Laverne (1989), time series components are defined as follows:-

TREND (T):- Often called secular trend, refers to the long term (“secular”) tendency of a time series to rise or fall. This may be thought of as the underlying growth (“upward” or “positive” trend), or decline (“downward” or “negative” trend) component in the series.

SEASONAL VARIATION (S):- The seasonal component is concerned with the periodic fluctuation in the series within each year. Such fluctuations from relatively fixed pattern that tends to repeat year after year. Seasonal fluctuation or “measurement” are most often attributable to social customs, weather changes, or various institutional arrangements, such as school calendars, tax filing schedule and their likes.

CYCLE (C):- Cyclical behavior in a time series is similar to seasonality in that is evidenced by wave-like pattern of ups and downs. But it differs from seasonality in two important ways; first, cycles are viewed as broad contraction and expansions, they take place over a period of years not within each year. Second, the length of time between the successive peaks (or thoughts) of a cycle is not necessarily fixed, as it is with seasonal movements.

The familiar expansion–recession-expansion of the “business cycle” is a good example of cyclical movement.

IRREGULAR FLUCTUATION (I) OR (F):- Trend, seasonal and cyclical movements may explain much of the behavior of time series. The movement left over after accounting for these factors is called the irregular component. Borrowing from engineering terminology the irregular component is often called the noise in the series while the other three components are often called the noise in the series, while the other three components comprise the signal.

Time series application can be found in sciences, engineering and so on. Medical scientist may be studying the effect of drug over time; the engineer may be working on machines, the geographer on seasonal climate changes. 

An Ancient weather forecasting methods usually relied on observed patterns of events, also termed pattern recognition. For example, it might be observed that if the sunset was particularly red, the following day often brought fair weather. This experience accumulated over the generations to produce weather lore. However, not all of these predictions prove reliable, and many of them have since been found not to stand up to rigorous statistical testing, Jerry Wilson (2007). The clearest way to examine a regular time series manually is with a line chart such as the one shown for tuberculosis in the United States, made with a spreadsheet program. The number of cases was standardized to a rate per 100,000 and the percent change per year in this rate was calculated. The nearly steadily dropping line shows that the TB incidence was decreasing in most years, but the percent change in this rate varied by as much as +/- 10%, with 'surges' in 1975 and around the early 1990s. The use of both vertical axes allows the comparison of two time series in one graphic.

However, ideas of stationarity must be expanded to consider two important ideas: strict stationarity and second-order stationarity. Both models and applications can be developed under each of these conditions, although the models in the latter case might be considered as only partly specified. In addition, time-series analysis can be applied where the series are seasonally stationary or non-stationary. Situations where the amplitudes of frequency components change with time can be dealt with in time-frequency analysis which makes use of a time–frequency representation of a time-series or signal, Boashash, B. (2003). 

Boashash, B. (2003) highlighted the reason for using time series to analyze data as follows:-

It helps in describing the behavior of a particular series in terms of other variable and to developed structural model of behavior. Also a model is set up to account for the observations.

It helps in describing the behavior of variables under consideration and in determining the rate of growth as well as the direction of periodic fluctuation.

2.1 REVIEW OF TIME SERIES ANALYSIS

Box, G. E. P., and Jenkins, G. M. (1976) stated that time series analysis is used for the purpose of monitoring the performance of a company or national economy over time; data is often collected and published on a regular basis e.g. Weekly, Monthly, Quarterly or Yearly depending upon the particular item. It is a set of data in chronological order with initial purpose to gain an understanding of the changes that took place as revealed by the data over protected period of time. And also agreed that we undertake the analysis of time series in order to help in determination of historical data patterns and that this provides information about the way in which economic and social variables have been behaving in the recent past. According to summer-A major objective of time series is forecasting. Simply describing past history series can aids our understanding of the process at work. The attempt to anticipate the future is much more explicit when we make quantitative forecasts of future values. These factors are based on the assumption that certain element of systematic variation found in past data will persist in to the future.

2.1.1 The publications of Time series Analysis:

The ARIMA model is essentially an approach to economic forecasting based on time series data. However, the ARIMA model requires the use of stationary time-series data (Dickey and Fuller, 1981; Granger and Newbold, 1974; Tse; 1996). Under current practice, developing such data requires that observed data series should be tested for unit roots. The tests for unit roots are also known as Dickey-Fuller (DF) and augmented Dickey-Fuller (ADF) tests.

The ARIMA technique does not assume any particular pattern in the historical data of the series to be forecast. The application of an ARIMA model are well documented in Box and Jenkins (1976), e.t.c. for example, An ARIMA model uses an iterative approach of identifying a possible useful model from a general class of models. Another tool for identification of stationarity in an ARIMA models is the ordinary and partial autocorrelation functions. Non-stationarity may be present if the values plotted in the correlogram do not diminish at large lags. When the original series or correlogram exhibits non-stationarity, successive differencing is carried out the non-seasonal ARIMA model which classified as an ARIMA (p,d,q,) model. Where p is the order of autoregressive terms, d is the order of integrating, q is the order of moving average.  

Besides trend modeling, another method of fitting that is based only on the information in the Yi series itself and uses lagged values of Yt in regression models. Such model is of the form  and is called autoregressive models where p is the number of lagged terms in the order of the model. Autoregressive models are used when the current level of the series is thought to depend on the recent history of the series. For example, series that measure prediction and spending are often modeled as autoregressive processes since the amount produced or spent in one period may well affect the amounts produced or spent in successive periods. Three problems immediately arise when considering the uses of autoregressive models as follows:-

i If Yt has a trend, so will yt-1, yt-2… leading to the problems associated with regression trended series.

ii. Because the independence variables are actually previous values of dependent variable, there are problems with regression assumption.

iii. The order of the models p has to be selected.

However, the appropriate number of lags to use (i.e the order p of the model) can be determined by analyzing the partial autocorrelation co-efficient of the series. The partial autocorrelation co-efficient function (PACF) of lag K denoted by ρkk is a measure of the correlation between Yt and Yt-k after adjusting for the presence of all the Yt’s of shorter lag i.e. Yt-1, Yt-2, ……..Yt-k.

There are a variety of checks which should be performed before we can satisfy that a model is adequate. 

Plot the residuals and looks for unusual values or for increasing (decreasing) dispersion which may suggest the need to transform data.

Examine the approximate t-ratios to see whether any term should be dropped from the model.

Examine the correlogram derived from the residuals to determine whether additional term is required.

Check whether the selected model (after differencing) is stationary.

Check the overall fit of the model (although this is less often done than in regression analysis), Box and Jenkins (1976). 

Forecasting and control by Box and Jenkins (1976) integrated the existing knowledge.

Assis, Amran and Remali (2006) compared the forecasting performance of different time series methods for forecasting cocoa bean prices at Bagan Datoh cocoa bean. The monthly average data of Bagan Datoh cocoa bean prices graded for the period of January 1992 to December 2006 was used. Four different types of univariate time series models were compared namely the exponential smoothing, autoregressive integrated moving average (ARIMA), generalized autoregressive conditional heteroscedasticity (GARCH) and the mixed ARIMA/ GARCH models. Root mean squared error (RMSE), mean absolute percentage error (MAPE), mean absolute error (MAE) and Theil`s inequality coefficient (U-statistics) were used as the selection criteria to determine the best forecasting model. The study revealed that the time series data were influenced by a positive linear trend factor while a regression test results showed the non-existence of seasonal factors. Moreover, the autocorrelation function (ACF) and the Augmented Dickey-Fuller (ADF) test showed that the time series data was not stationary but became stationary after the first differentiating process was carried out. In their paper they pointed out that forecasting the future prices of cocoa bean through the most accurate time series model could help the Malaysia government as well as the buyers (exporters and millers) and sellers (farmers and dealers) in cocoa bean industry to perform better strategic planning and also to help them in maximizing revenue and minimizing the cost. Based on the results of the ex-post forecasting (starting from January until December 2006) GARCH model performed better compared to exponential smoothing, ARIMA and the mixed ARIMA/GARCH model for forecasting Bagan Datoh cocoa bean prices.

2.2 REVIEW OF THEORETICAL AND EMPIRICAL LITERATURE ON INFLATION 

An empirical analysis of causes of inflation in Nigeria by Asogu (1991) indicated that real output, money supply, domestic food prices, exchange rate and net exports were the major determinants of inflation in Nigeria. Moser (1995) and Fakiyesi (1996) studied Nigeria’s headline inflation using both the long-run and the dynamics error correction model and autoregressive distributed lag approaches, respectively. Their results confirmed that the basic findings of Asogu (1991) and agro-climatic conditions were the major factors influencing inflation in Nigeria. Also, using the framework of error correction mechanism, Olubusoye and Oyaromade (2008) found that the lagged CPI, expected inflation, petroleum prices and real exchange rate significantly propagate the dynamics of inflationary process in Nigeria. More recently, Adebiyi et al (2010) examined the different types of inflation forecasting models including ARIMA and showed that ARIMA models were modestly successful in explaining inflation dynamics in Nigeria. 

In recent years, inflation has become one of the major economic challenges facing most countries in the world especially those in Africa. Inflation is a major focus of economic policy worldwide as described by David (2001): Inflation dynamics and evolution can be studied using a stochastic modelling approach that captures the time dependent structure embedded in the time series inflation data. Inflation as described by Webster`s (2000), is the persistent increase in the level of consumer prices or persistent decline in the purchasing power of money. Inflation can also be expressed as a situation when the demand for goods and services exceeds their supply in the economy (Hall, 1982). In reality inflation means that your money can not buy as much as what it could have bought yesterday. People who are living with fixed income suffer most when prices of commodities rise, as they cannot buy as much as they could buy previously (Aidoo, 2010). Unanticipated inflation makes future prices unknown, as this situation discourages savings due to the fact that the money is worth more at present than in the future. It leads to inefficient allocation of economic resources. Therefore it reduces economic growth, because economy needs a certain level of savings to finance investments which boosts economic growth.

Jehovanes (2007) studied a time lag between a change in money supply and the inflation rate response. A modified generalized autoregressive conditional heteroscedasticity (GARCH) model was employed to monthly inflation data for the period 1994 to 2006: In the study he used the maximum likelihood estimation technique to estimate parameters of the model and to determine significance of the lagged value. GARCH model produced good results which indicated that a change in money supply would affect inflation rate considerably in seven months ahead.

A lot of empirical research has been conducted in the area of short-term forecasting using ARIMA models. Akdogan et.al (2012) produced short-term forecasts for inflation in Turkey, using a large number of econometric models such as the univariate ARIMA models, decomposition based models, a Phillips curve motivated time varying parameter model, a suit of VAR and Bayesian VAR models and dynamic factor models. Their result suggests that the models which incorporate more economic information outperformed the random walk model at least up to two quarters ahead. 

Mordi et al (2012) developed a short-term inflation forecasting framework to serve as a tool for analyzing inflation risks in Nigeria. Their framework follows mostly a structural time series model for each CPI component constructed at a certain level of disaggregation. Short term forecasts of the all items CPI is made as a weighted sum of the twelve CPI components forecasts. Thereafter, the all items CPI and the twelve CPI components are used to calculate short-term inflation forecasts. The framework is intended to serve as a tool for analyzing inflation risks with the aid of fan charts and given its disaggregated nature, appears informative and capable of improving the credibility of the policy maker. 

Pufnik and Kunovac (2006) provided a method of forecasting the Croatia’s CPI by using univariate seasonal ARIMA models and forecasting future values of the variables from past behavior of the series. Their paper attempts to examine whether separate modeling and aggregating of the sub-indices improves the final forecast of the all items index. The analysis suggests that given a somewhat longer time horizon (three to twelve months), the most precise forecasts of all items CPI developments are obtained by first forecasting the index’s components and then aggregating them to obtain the all items index. 

Alnaa and Ferdinand (2011) used ARIMA approach to estimate inflation in Ghana using monthly data from June 2000 to December 2010. They found that ARIMA (6,1,6) is best for forecasting inflation in Ghana. Also, Suleiman and Sarpong (2012) employed an empirical approach to modeling monthly CPI data in Ghana using the seasonal ARIMA model. Their result showed that ARIMA (3,1,3) and (2,1,1) model was appropriate for modeling Ghana’s inflation rate. Diagnostic test of the model residuals with the ARCH LM test and Durbin Watson test indicates the absence of autocorrelations and ARCH effect in the residuals. The forecast results inferred that Ghana was likely to experience single digit inflation values in 2012. 

Akhter (2013) forecasted the short-term inflation rate of Bangladesh using the monthly CPI from January 2000 to December 2012. The paper employs the seasonal ARIMA models proposed by Box et al (1994). Because of the presence of structural break in the CPI, the study truncates the series and using data from September 2009 to December 2012 fitted the seasonal ARIMA (1,1,1) (1,0,1)[12] model. The forecasted result suggests an increasing pattern and high rates of inflation over the forecasted period of 2013. 

Omane-Adjepong et al (2013) examined the most appropriate short-term forecasting method for Ghana’s inflation. The monthly dataset used was divided into two sets, with the first set used for modeling and forecasting, while the second set was used as test. Seasonal ARIMA and Holt-Winters approaches are used to obtain short-term out of sample forecast. From the results, they concluded that an out of sample forecast from an estimated seasonal ARIMA (2,1,2)(0,0,1)[12] model far supersedes any of the Holt-Winters’ approach with respect to forecast accuracy. 

Meyler et al (1998) outlined the practical steps which need to be undertaken to use ARIMA time series models for forecasting Irish inflation. They considered two alternative approaches to the issue of identifying ARIMA models – the Box Jenkins approach and the objective penalty function methods. The approach they adopted is ‘unashamedly’ one of model mining with the aim of optimizing forecast performance.

Gary (1995) analyzed the dominant factors influencing inflation in Nigeria using an error correction model based on money market equilibrium conditions. In his analysis, he found that monetary expansion driven mainly by expansionary fiscal policies, explains to a large extent the inflationary process in Nigeria. Some econometric models have been used to describe inflation rates, but they are restrictive in their theoretical formulations and often do not incorporate the dynamic structure of the data and have tendencies to inflict improper restrictions and specifications on the structural variables (Saz, 2011). 

Odusanya and Atanda (2010) determined the dynamic and simultaneous interrelationship between inflation and its determinants – growth rate of Gross Domestic Product (GDP), growth rate of money supply (M2), fiscal deficit, exchange rate (U.S dollar to Naira), importance and interest rates, using econometric time series model. Olatunji et al. (2010) examined the factors affecting inflation in Nigeria using cointegration and descriptive statistics. They observed that there were variations in the trend pattern of inflation rates and some variables considered were significant in determining inflation in Nigeria. These variables include annual total import, annual consumer price index for food, annual agricultural output, interest rate, annual government expenditure, exchange rate and annual crude oil export. 

Mordi et al. (2007), in their study of the best models to use in forecasting inflation rates in Nigeria identified areas of future research on inflation dynamics to include re-identifying ARIMA models, specifying and estimating VAR models and estimating a P-Star model, amongst others that can be used to forecast inflation with minimum mean square error.

Stockton and Glassman (1987) conducted a comparative study on three different inflation processes namely rational expectations model, monetarist model and the expectation augmented Philips curve that are based on economic theory relationships that explain and form inflation. These processes were compared to one another, utilizing their out-of-sample forecast performance on an eight-quarter horizon and in addition a simple Autoregressive Integrated Moving Average (ARIMA) model was used as a bench mark to substantiate the theoretical validity of the econometric models. Their findings showed that the ARIMA model outperformed both the rational expectations model and the monetarist model and was found to perform just as good as the Philips curve in all specifications. They concluded that despite all theoretical efforts to explain causes of inflation, a simple ARIMA model of inflation turned in such a respectable forecast performance relative to the theoretically based specifications. 

A reason why simple time series models tend to outperform their theoretical counterparts lies in the restrictive nature of econometric models with their improper restriction and specifications on structural variables. The absence of restriction in the ARIMA model gives it the necessary flexibility to capture dynamic properties and thus significant advantage in short-run forecasting (Saz, 2011). Encouraged by these empirical results on the superiority of ARIMA models, Saz (2011) applied Seasonal Autoregressive Integrated Moving Average (SARIMA) model to forecast the Turkish inflation. Longinus (2004) examined the influence of the major determinants of inflation with a particular focus on the role of exchange rate policy of Tanzania from 1986 to 2002. He discovered that the parallel exchange rate had a stronger influence on inflation. Other works on modeling inflation rates are seen in the works of Fatukasi (2003), Eugen et al. (2007) and Tidiane (2011).















CHAPTER THREE: METHODOLOGY

3.0 INTRODUCTION

Early attempts to study time series, particularly in the 19th century, were generally characterized by the idea of a deterministic world. It was the major contribution of Yule (1927) which launched the notion of stochasticity in time series by postulating that every time series can be regarded as the realization of a stochastic process. Based on this simple idea, a number of time series methods have been develop since then. Workers such as Slutskey, Walker, Yaglom and Yule first formulated the concept of autoregressive (AR) and moving average (MA) models. World’s decomposition theorem led to the formulation and solution of the linear forecasting problem of Kolmogorov (1941). Since then, a considerable body of literature has appeared in the area of time series, dealing with parameter estimation, identification, model checking, and forecasting see, Box and Jenkins (1978) for an early survey.

Models for time series data can have many forms and represent different stochastic processes. When modeling variations in the level of a process, three broad classes of practical importance are the autoregressive (AR) models, the integrated (I) models, and the moving average (MA) models. These three classes depend linearly on previous data points. (Gershenfeld, N. (1999). The Nature of Mathematical Modeling New York: Cambridge University Press. pp. 205–208. ISBN 0521570956) Combinations of these ideas produce autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) models. The autoregressive fractionally integrated moving average (ARFIMA) model generalizes the former three. 

It is importance to the economic growth of a country makes many researchers and economists to apply various time series and econometric models to forecast or model inflation rates of countries. These models include Autoregressive Integrated Moving Average (ARIMA) Models and Seasonal Autoregressive Integrated Moving Average (SARIMA) Models all in the time domain, error correction, VAR and other econometric models, but not much have been done using the Frequency domain models. (Modeling the Nigerian Inflation Rates Using Periodogram and Fourier series Analysis Omekara et al.)

3.1 ARIMA MODELS

In passing note that our emphasis here is on univariate ARMA models, that is, ARIMA models pertaining to a single time series. Therefore in the rest of this chapter we discuss the fundamentals of Box-Jenkins approach to economic forecasting.

3.1.1 An Autoregressive (AR) process

Let  represent the logged data at time . if we model  as

                                                                             (3.1.1)

Where  is an uncorrelated random error term with zero mean and constant variance  (i.e. it is white noise), then we say that  follows a first order autoregressive, or AR(1), stochastic process. Here the value of  at time  depends on its value in the previous time period and a random term. In other words, this model says that the forecast value of  at time  is simply some proportion  of its value at time  plus a random shock or disturbance  at time .

But if we consider this model 

                                                                 (3.1.2)

Then we say that  follows a second-order autoregressive, or AR(2), process. That is, the value the value of  at time  on its value in the previous two time period.

In general, we can have


                                                             (3.1.3)

In which case  is a pth-order autoregressive, or AR(P), process.

Notice that in all the preceding models only the current and previous  values are involved; there are no other regressors. In this sense, we say that the data speak for themselves. 

3.1.2 A Moving Average (MA) Process

The AR process just discussed is not the only mechanism that may have generated.

Suppose we model  as follows:

                                                                     (3.1.4)

Where  is a constant and , as before, is the white noise stochastic error term, and .

Here  at time  is equal to a constant plus a moving average of the current and past error terms. Thus, in the present case, we say that  follows a first-order moving average, or MA(1), process.

But if we  follows the expression

                                                         (3.1.5)

Then it is an MA(2) process. More generally,


                                                                 (3.1.6)     is an MA(q)process. In short, a moving average process is simply a linear combination of white noise error terms.

3.1.3 An Autoregressive Moving Average (ARMA) Process

Of course, it is quite likely that  has characteristics of both AR and MA and is therefore ARMA. Thus,  follows an ARMA(1,1) process if it can be written as:

                                                        (3.1.7)

Because there is one autoregressive and one moving average term.

In general, in an ARMA(p,q) process, there will be p autoregressive and q moving average term as 


                                                                      (3.1.8)

3.1.4 An Autoregressive Integrated Moving Average (ARIMA) Process

The time series models already discussed are based on the assumption that the time series involved are (weakly) stationary in the sense that; the mean and the variance for a weakly stationary time series are constant and its covariance is time invariant. But we know that many economic time series are non-stationary, that is, they are integrated, and if a time series is integrated of order 1 (i.e., it is I(1)), its first difference is I(0), that is stationary. Similarly, if a time series is I(2), its second difference is I(0). In general, if a time series is I(d), after differencing it d times, we obtain an I(0) series.

Therefore, if we have to difference a time series d times to make it stationary and then apply the ARMA(p,q) model to it, we say that the original time series is ARIMA(p,d,q), that is, it is an autoregressive integrated moving average time series, where p denotes the number of autoregressive terms, d the number of times the series has to be differenced before it becomes stationary, and q the number of moving average terms. Thus, an ARIMA(2,1,2) time series has to be differenced once (d=1) before it becomes stationary and the (first differenced) stationary time series can be modeled as an ARMA(2,2) process, that is, it has two AR and two MA terms. Of course, if d=0 (i.e., a series is stationary to begin with), ARIMA(p,d=0,q) = ARMA(p,q). Note that an ARIMA(p,0,0) process means purely AR(p) stationary process; an ARIMA(0,0,q) means purely MA(q) stationary process. Given the values of p, d, and q, one can tell what process is being modeled.

The important point to note is that to use the Box-Jenkins methodology, we must have stationary time series or a time series that is stationary after one or more differencing. The reason for assuming stationarity is; the objective of B-J (Box-Jenkins) is to identify and estimate a statistical model which can be interpreted as having generated the sample data. If this estimated model is then to be used for forecasting, we must assume that the features of this model are constant through time, and particularly over future time periods. Thus, the simple reason for requiring stationary data is that any model which is inferred from these data can itself be interpreted as stationary or stable, therefore providing a valid basis for forecasting, Michael Pokorny (1987)

3.2 THE BOX-JENKINS (BJ) METHODOLOGY

The million-dollar question obviously is: Looking at a time series, such as the Oil Price series in figure… , how does one know whether it follows a purely AR process (and if so, what is the value of p) or a purely MA process(and if so, what is the value of q) or an ARMA process (and if so, what are the values of p and q) or an ARIMA process in which case we must know the values of p, d, and q. the BJ methodology comes in handy in answering the preceding question. The method consists of four steps:

Step1. (IDENTIFICATION) (that is, find out the appropriate values of p, d, and q)

The chief tools in identification are the autocorrelation function (ACF), the partial autocorrelation function (PACF), and the resulting correlograms, which are simply the plots of ACFs and PACFs against the lag length.

Graphical Analysis (Time plot of a series):- as noted earlier, before one pursues format tests, it is always advisable to plot the time series under study. Such plots give initial clue about the likely nature of the time series.

Autocorrelation Function (ACF):- one simple test of stationarity is based on the so-called Autocorrelation Function.

The ACF at lag , denoted by , is defined as 

                                                                (3.2.1) 

Where covariance at lag  and variance of the process are defined as:

                                                   (3.2.2)

                                                      (3.2.3)

Where , the covariance (or Autocovariance) at lag , is the covariance between the values of  and , that is, between two  values  periods apart. If , we obtain , which is simply the variance of ; if ,  is the covariance between two adjacent values of .

Partial Autocorrelation Function (PACF):- the partial autocorrelation function, , measures correlation between (time series) observations that are  time periods apart after controlling for correlations at intermediate lags (i.e., lags less than ). In other words, partial autocorrelation is the correlation between  and  after removing the effect of the intermediate .

TABLE 3.2.1: Theoretical Patterns of ACF and PACF

Type of model

Typical pattern ACF

Typical pattern of PACF


AR(p)

Decay exponentially or with damped sine wave pattern or both

Significant spikes through lag p


MA(q)

Significant spikes through lag q

Declines exponentially


ARMA(p,q)

Exponential decay

Exponential decay



Step2. (ESTIMATION) having identified the appropriate p and q values, the next stage is to estimate the parameters of the autoregressive and moving average terms included in the model.

Step3. (DIAGONASTIC CHECKING) having choosing a particular ARIMA model, and having estimated its parameters, we next see whether the chosen model fits the data reasonably well, for it is possible that another ARIMA model might do the job as well. This is why considerable skill is required to choose the right ARIMA model. One simple test of the chosen model is to see if the residuals estimated from this model are white noise; if they are, we can accept the particular fit; if not, we must start over. Thus, the BJ methodology is an iterative process. (see the figure below)

Identification of the model (choosing tentative p, d, and q)



Parameter estimation of the chosen model



Diagnostic checking. Are the estimated residuals white noise?

If yes, go to (4), else go back to (1)



Forecasting


Figure 3.2.1 Box-Jenkins Methodology

Step4. (FORECASTING) one of the popularity of the ARIMA modelling is its success in forecasting. In many cases, the forecasts obtained by this method are more reliable than those obtained from the traditional econometric modelling, particularly for short-term forecasts. Of course, each case must be checked.

Since ARIMA forecasting (BJ methodology) usually refer to some set of techniques, this process was developed to help remove trends and uncover hidden patterns in non-stationary data because ARIMA process can only model stationary data.

Forecasting is the process of making statements about events whose actual outcomes (typically) have not yet been observed. A commonplace example might be estimation of some variable of interest at some specified future date. Prediction is a similar, but more general term. Both might refer to formal statistical methods employing time series, cross-sectional or longitudinal data, or alternatively to less formal judgmental methods. Usage can differ between areas of application: for example, in hydrology, the terms "forecast" and "forecasting" are sometimes reserved for estimates of values at certain specific future times, while the term "prediction" is used for more general estimates, such as the number of times floods will occur over a long period. Risk and uncertainty are central to forecasting and prediction; it is generally considered good practice to indicate the degree of uncertainty attaching to forecasts. In any case, the data must be up to date in order for the forecast to be as accurate as possible. (Scott Armstrong, Fred Collopy, Andreas Graefe and Kesten C. Green. "Answers to Frequently Asked Questions" Retrieved May 15, 2013.) 

Some forecasting methods try to identify the underlying factors that might influence the variable that is being forecast. For example, including information about climate patterns might improve the ability of a model to predict umbrella sales. Forecasting models often take account of regular seasonal variations. In addition to climate, such variations can also be due to holidays and customs: for example, one might predict that sales of college football apparel will be higher during the football season than during the off season. (Nahmias, Steven (2009). Production and Operations Analysis) 

3.2.1 Applications of forecasting

Climate change and increasing energy prices have led to the use of Egain Forecasting for buildings. These attempts to reduce the energy needed to heat the building, thus reducing the emission of greenhouse gases. Forecasting is used in Customer Demand Planning in everyday business for manufacturing and distribution companies.

Forecasting has also been used to predict the development of conflict situations. Forecasters perform research that uses empirical results to gauge the effectiveness of certain forecasting models, J. Scott Armstrong, Kesten C. Green and Andreas Graefe (2010). However research has shown that there is little difference between the accuracy of the forecasts of experts knowledgeable in the conflict situation and those by individuals who knew much less. (Kesten C. Greene and J. Scott Armstrong (2007). "The Ombudsman: Value of Expertise for Forecasting Decisions in Conflicts".

3.3 INFORMATION AND MODEL SELECTION CRITERIA

In statistical modeling, one of the main challenges is to select a suitable model from a candidate family to characterize the underlying data. Model selection criteria provide useful tools in this regard. Selection criteria assess whether a fitted model offers an optimal balance between goodness-of-ft and parsimony. Ideally, a criteria will identify candidate models that are either too simplistic to accommodate the data or unnecessarily complex. The most common model selection criteria are the AIC (Akaike (1974, 1976)), HQC (Hannan-Quinn (1979)), and SIC (Schwarz (1978)). 

Let  be the maximum likelihood of a model with k parameters based on a sample of size n. The information criteria for selecting the most parsimonious correct model are:

                       

Since the Schwarz information is derived using Bayesian arguments, this criterion is also known as the Bayesian Information Criterion (BIC).

These criteria take the general form

                                                  (3.3.2)

Where  in Akaike case,  in Hannan-Quinn case and  in the Schwarz case. Using these criteria, a model is selected that correspond to: 

                                                                       (3.3.3)

3.3.1 APPLICATIONS

3.3.1.1 ARMA Model Specification

In ARMA (p,q) case,

                            

These criteria become        where now  is the ML estimator of the error  and n is the number of observation used in the ML estimation.

It can be shown (see Hannan (1980)) that in the case of common roots in AR and MA polynomials the Hannan-Quinn and Schwarz criteria still select the correct orders p and q consisitently: given upper bound  and , where  and , are the orders of an ARMA (p,q) process, we have , where . 

3.4 STATISTICAL TESTS IN THE TIME SERIES ANALYSIS

3.4.1 UNIT ROOTS TESTS

3.4.1.1 The Augmented Dickey-Fuller (ADF) Test

This is a test pioneered by Dickey and Fuller (1979), consisting of model estimation; In statistics and econometrics, an augmented Dickey–Fuller test (ADF) is a test for a unit root in a time series sample. It is an augmented version of the Dickey–Fuller test for a larger and more complicated set of time series models. The augmented Dickey–Fuller (ADF) statistic, used in the test, is a negative number. The more negative it is, the stronger the rejection of the hypothesis: that there is a unit root at some level of confidence. (Elliott, G., Rothenberg, T. J. & J. H. Stock (1996) "Efficient Tests for an Autoregressive Unit Root", Econometrica, 64 (4), 813–836.) 

Testing Procedure

The testing procedure for the ADF test is the same as for the Dickey–Fuller test but it is applied to the model

Whereis a constant,the coefficient on a time trend, p the lag order of the autoregressive process,  a pure white error term and, , and so on. The number of lagged difference terms to include is often determined empirically, the idea being to include enough terms so that the error term  is serially uncorrelated, so that we can obtain an unbiased estimate of , the coefficient of lagged . In ADF, we test whether the hypothesis: Null hypothesis, H0:  (i.e., there is a unit root or the time series is non-stationary, or it has a stochastic trend).

Alternative hypothesis, H1:  (i.e., there is no unit root or the time series is stationary, or it has no stochastic trend, possibly around a deterministic trend). Dickey and Fuller rule out the possibility that, because in that case, in which case the underlying time series will be explosive.

The ADF test statistic is given by

           

where SE is the standard error of , and (^) denotes estimate. The null hypothesis of the unit root is rejected if the test statistic is less than the critical values. 

3.4.1.2 KPSS Test

Kwiatkowski–Phillips–Schmidt–Shin (KPSS) tests are used for testing a null hypothesis that an observable time series is stationary around a deterministic trend. Such models were proposed in 1982 by Alok Bhargava in his Ph.D. thesis where several John von Neumann or Durbin–Watson type finite sample tests for unit roots were developed (see Bhargava, 1986). Later, Denis Kwiatkowski, Peter C.B. Phillips, Peter Schmidt and Yongcheol Shin (1992) proposed a test of the null hypothesis that an observable series is trend stationary (stationary around a deterministic trend).The series is expressed as the sum of deterministic trend, random walk, and stationary error, and the test is the Lagrange multiplier test of the hypothesis that the random walk has zero variance; H0:  Vs H1: .

KPSS type tests are intended to complement unit root tests, such as the Dickey–Fuller tests. By testing both the unit root hypothesis and the stationarity hypothesis, one can distinguish series that appear to be stationary, series that appear to have a unit root, and series for which the data (or the tests) are not sufficiently informative to be sure whether they are stationary or integrated.



3.4.2 RESIDUAL TESTS

3.4.2.1 The Jarque - Bera Test

Jarque and Bera (1987)proposed a test for normality based on Skewness and Kurtosis of a distribution. The Jarque-Bera test is a two sided goodness of fit test suitablewhen a fully specified null distribution is unknown and its parameters must be estimated. The test statistic is given by

                                                                      (3.4.3)

Where n is the sample size, S is the sample Skewness, and K is the sample Kurtosis.

Hypothesis:

Null hypothesis, H0:  (Skewness) and  (kurtosis). That is, the distribution is symmetric and hence normal.

Alternative hypothesis, H1:  (Skewness) and  (kurtosis). That is, the distribution is asymmetric and hence non-normal.

The null hypothesis is rejected if the test statistic is greater than the critical values.  

3.4.2.2 Portmanteau Tests

These test whether any group of autocorrelations of a time series are different from zero. That is, it is used for investigating the presence of autocorrelation in time series. Among Portmanteau tests are both the Ljung-Box test and the (now obsolete) Box-pierce (Q-statistic) tests. The Portmanteau tests are used to test the stationarity of a time series.

To test a joint hypothesis that all the  up to certain lags are simultaneously equal to zero, instead of testing the statistical significance of any individual autocorrelation coefficients (of ACFs and PACFs), we use the Q statistic developed by Box and Pierce, which is defined as: 

                                                                                     (3.4.4)

Where n is the sample size and m the lag length. The Q statistic is often used as a test of whether a time series is a white noise. In large samples, it is approximately distributed as the chi-square distribution with m degrees of freedom. In application, if the computed Q exceeds the critical Q values from the chi-square distribution at the chosen level of significance, one can reject the null hypothesis that the entire (true)  are zero; at least some of them must be non-zero.

A variant to of the Box-Pierce Q-statistic is the Ljung-Box (LB) statistic, which is defined as: (G. M. Ljung and G. E. P. Box (1978))

                                                                       (3.4.5)

Although in large samples both Q and LB follows the chi-square distribution with m degrees of freedom, the LB statistic has been found to have better (more powerful, in statistical sense) small sample properties than Q statistic. (The Q and LB statistics may not be appropriate in every case. 

CHAPTER FOUR: DATA ANALYSIS

4.0 INTRODUCTION

This chapter focuses on the inflation data analysis using Box-Jenkins Methodology, a technique pioneered by Box and Jenkins (1976), to build a model for the monthly series on All items less farm produce and energy inflation in Nigeria from 2003-2014 as aimed, with a view to achieve the main objectives of the research work.

4.1 DATA USED IN THE STUDY

The data used in this research work is a monthly series of all items less farm produce and energy inflation in Nigeria from January, 2003 to February, 2014. A secondry data collected from the Central Bank of Nigeria (CBN). 

4.2 MODEL IDENTIFICATION

4.2.1 Visual Analysis


Figure 4.2.1 Time plot of the series before defferencing

The time plot of the series shown above, shows that the time plot have very little fluctuation patterns, with some wide swings; i.e. the variance of the series is not stabilized as such the plot indicates that the series is somehow not stationary. When a plot of a series shows such features, we take the first difference of the original series. The ACF and PACF below justifies that, this is really the case. 

4.2.2 ACF and PACF plots of the series before defferencing

The ACF declines very slowly up to about 11 lags and goes up slowly and spikes down (decline), other lags are not individually statistically significantly different from zero (i.e. they lie within the boundaries or confidence intervals) and the PACF drops after the first lag and most PACFs after lag 1 are statistically significant. This signifies that the series is not stationary.

Figure 4.2.2 ACF and PACF plots of the series before defferencing

The formal application of unit root tests (ADF and KPSS) will justify this as follows.

We reject the null hypothesis (H0) for the ADF Test since all the test statistics are less than the critical values and therefore conclude that the series does not contain unit root i.e. it is stationary. We accept the null hypothesis (H0) for the KPSS test since the test statistics are less than the p-values at some levels. But, since the plots reveals that the variance of the series is not stabilized, there is need to difference the data.

Table 4.2.1 ADF and KPSS Tests before differencing 

TEST

TEST STATISTIC

CRITICAL/P-VALUES


ADF Test without constant

-0.84656

0.3494


ADF Test with constant

-1.52841

0.5193


ADF Test with constant and Trend

-1.53645

0.8173


KPSS Without Trend

0.246115

1%           5%           10%

0.739     0.463        0.347


KPSS With Trend

0.125867

1%           5%

    0.146



The figure below plotted the first differences for the rate of inflation series. Unlike Figure 4.2.1, we do not observe any trend or unstabilized variance in the plot. We can also see this visually from the ACF and PACF correlogram given below the plot of the first difference of the original series. 




Figure 4.2.3 Time plot of the series after defferencing

4.2.3 ACF and PACF after differencing

The ACFs at some few lags seem statistically different from zero (at the 95% confidence limit, those lags are asymptotic and so can be considered approximate), but at all other lags, they are not statistically different from zero. We therefore conclude that the data series is now stationary. A formal application of the Augmented Dickey-Fuller and KPSS unit root tests below may show that this is indeed the case.


Figure 4.2.4 ACF and PACF plots of the series after defferencing

Table 4.2.2 ADF and KPSS Tests after differencing 

TEST

TEST STATISTIC

CRITICAL/P-VALUES


ADF Test without constant

-1.10433

0.2451


ADF Test with constant

-1.16868

0.6902


ADF Test with constant and Trend

-1.32296

0.8821


KPSS Without Trend

0.144839

1%           5%           10%

0.739     0.463        0.347


KPSS With Trend

0.145105

1%           5%

    0.146


Since the entire test statistics of the ADFs are less than the critical regions, we reject the null hypothesis and therefore conclude that there is no unit root or the time series is stationary. Likewise the KPSS tests, the test statistics are all less than the p-values; we therefore accept the null hypothesis and conclude that the data series is stationary around a deterministic trend.  

TABLE 4.2.3 ARIMA (P,D,Q) MODEL IDENTIFICATION AND SELECTION

MODEL

AIC

HQC

SIC


ARIMA (0,1,1)

684.7715

688.2950

693.4425


ARIMA (0,1,2)

685.3532

690.0513

696.9146


ARIMA (0,1,3)

686.1580

692.0307

700.6098


ARIMA (1,1,0)

684.9311

688.4547

693.6022


ARIMA (1,1,1)

686.3164

691.0145

697.8778


ARIMA (1,1,2)

682.3719

688.2445

696.8236


ARIMA (1,1,3)

687.9725

695.0197

705.3146


ARIMA (2,1,0)

685.1202

689.8183

696.6816


ARIMA (2,1,1)

682.6202

688.4929

697.0720


ARIMA (2,1,2)

684.2515

691.2986

701.5936


ARIMA (2,1,3)

676.7232

684.9448

696.9556


ARIMA (3,1,0)

686.8228

692.6954

701.2745


ARIMA (3,1,1)

684.0761

691.1233

701.4182


ARIMA (3,1,2)

690.3326

698.5543

710.5650


ARIMA (3,1,3)

673.6670

683.0632

696.7898



We tested 15 models with low AIC, HQC, and SIC which is common in ARIMA modelling and find the best models among the nine models which happened to be: ARMA (0,1,1), ARMA (1,1,2) and ARMA (3,1,3).

4.3 MODEL ESTIMATION:

After the best models have been chosen, the parameters of these models are estimated. The result of these estimates is shown in the table below:-

Table 4.3.1 Result of the estimated ARIMA (0,1,1) 

PARAMETERS

COEFFICIENT

STD. ERROR

t- RATIO

P-VALUE


Constant           

-0.0416791

0.241337

-0.1727

0.8629


  theta_1               

-0.0829207

0.0957161

-0.8663

0.3863



Table 4.3.2 Result of the estimated ARIMA (1,1,2) 

PARAMETERS

COEFFICIENT

STD. ERROR

t- RATIO

P-VALUE


Constant                    

-0.0430820

0.236348

-0.1823

0.8554


phi_1                    

-0.873252

0.0495822

-17.61

1.99e-069


theta_1                     

0.853814

0.101001

8.453

2.83e-017


theta_2                  

-0.146186

0.0984351

-1.485

0.1375



Table 4.3.3 Result of the estimated ARIMA (3,1,3) 

PARAMETERS

COEFFICIENT

STD. ERROR

t- RATIO

P-VALUE


Constant

-0.0379746               

0.253396

-0.1499

0.8809


phi_1

-0.168628              

0.0727300

-2.319

0.0204


phi_2

-0.170392              

0.0862703

-1.975

0.0483


phi_3

-0.714567              

0.0703197

-10.16

2.94e-024


theta_1

0.0696750              

0.0678720

1.027

0.3046


theta_2

0.0696746             

0.0735475

0.9473

0.3435


theta_3

1.00000                 

0.113710

8.794

1.44e-018



4.4 MODEL CHECKING

Here we are to look at some tests to check whether the models are specified correctly. The following tests are applied to the residuals;

Test for ACF and PACF of residuals

Portmanteau tests (Ljung and Box and Box-Pierce Q test)

Jaque-Bera test (for Normality)

AUTOCORRELATION AND PARTIAL AUTOCORRELATION OF RESIDUALS 

In the ACF and PACF, if the residuals are uncorrelated then the result is adequate and normal.

 Figure 4.4.1 ACF and PACF plots of residuals for ARIMA (0,1,1)

 

Figure 4.4.2 ACF and PACF plots of residuals for ARIMA (1,1,2)

Figure 4.4.3 ACF and PACF plots of residuals for ARIMA (3,1,3)

The figure above shows that we have much more difference pattern. The ACF and PACF seems statistically different from zero at some few lags and applying the 95% confidence limit for the correlogram,  the confidence limit of those lags are asymptotic and so can be considered approximate. But at some other lags they are not statistically different from zero. Therefore they give the impression that the residuals estimated from the above plot are purely random and hence, we conclude that the plots are stationary.

TABLE 4.4.1 Jarque-Bera diagnostic tests of ARIMA (0,1,1) model

TEST

TEST STATISTIC

P-VALUE


Jarque-Bera test (Normality)

44.1973

2.52746e-010



TABLE 4.4.2 Jarque-Bera diagnostic tests of ARIMA (1,1,2) model

TEST

TEST STATISTIC

P-VALUE


Jarque-Bera test (Normality)

40.8134

1.37243e-009



TABLE 4.4.3 Jarque-Bera diagnostic tests of ARIMA (3,1,3) model

TEST

TEST STATISTIC

P-VALUE


Jarque-Bera test (Normality)

46.3981

8.40985e-011



TABLE 4.4.4 PORTMANTEAU (Box-Pierce Q-Statistic) TEST of ARIMA (0,1,1)

LAG

TEST STATISTIC

P-VALUE


5

4.7907  

0.442


10

14.2750  

0.161



TABLE 4.4.5 PORTMANTEAU (Box-Pierce Q-Statistic) TEST of ARIMA (1,1,2)

LAG

TEST STATISTIC

P-VALUE


5

1.8509  

0.869


10

12.0402  

0.282



TABLE 4.4.6 PORTMANTEAU (Box-Pierce Q-Statistic) TEST of ARIMA (3,1,3)

LAG

TEST STATISTIC

P-VALUE


5

1.2617  

0.939


10

8.1602  

0.613



These results shows that the distributions are symmetric and hence normal, we therefore conclude that the residuals of the Models are normally distributed and there is no serial correlation.

The estimated residual ACFs and PACFs are shown in figures 4.4.1, 4.4.2 and 4.4.3 above. As these figures showed, none of the autocorrelations and partial autocorrelations is individually statistically significant, nor is the sums of the 40 squared autocorrelations as shown by the Box-Pierce (Q) statistic, are statistically significant. In other words, the correlograms of both autocorrelation and partial autocorrelation give the impression that the residuals estimated are purely random [i.e. the residual ACFs and PACFs seem statistically different from zero at some lags, but all other lags are not statistically different from zero]. 

4.5 FORECASTING

After the best ARIMA (p,d,q) models have been gotten, the next is to see the capability to further testify the validity of the models. Below is the forecasted graph and forecasted values of the best models for a period of 6 months i.e. from 2014:03 to 2014:08. Figure 4.5.1 Six month Forecasted graph for ARIMA (0,1,1) model

TABLE 4.5.1: Six months forecasted values of ARIMA (0,1,1) model (For 95% CI, z(.025) = 1.96)

 Observations

Prediction

std. error

95% confidence interval


2014:03

7.90880

3.10436

(1.82436, 13.9932)


2014:04

7.86712

4.21214

(-0.388528, 16.1228)


2014:05

7.82544

5.08402

(-2.13905, 17.7899)


2014:06

7.78376

5.82686

(-3.63668, 19.2042)


2014:07

7.74208

6.48517

(-4.96862, 20.4528)


2014:08

7.70041

7.08255

(-6.18114, 21.5820)



Figure 4.5.2 Six month Forecasted graph for ARIMA (1,1,2) model

TABLE 4.5.2 Six months forecasted values of ARIMA (1,1,2) model (For 95% CI, z(.025) = 1.96)

 Observations

Prediction

std. error

95% confidence interval


2014:03

7.73496

3.00463

(1.84599, 13.6239)


2014:04

7.83356

4.20809

(-0.414143, 16.0813)


2014:05

7.66675

4.71138

(-1.56739, 16.9009)


2014:06

7.73172

5.50185

(-3.05172, 18.5151)


2014:07

7.59428

5.92925

(-4.02683, 19.2154)


2014:08

7.63359

6.54073

(-5.18600, 20.4532)


Figure 4.5.3 Six month Forecasted graph for ARMA (3,1,3) model

TABLE 4.5.3 Six month Forecasted values of ARMA (3,1,3) model (For 95% CI, z(.025) = 1.96)

 Observations

Prediction

std. error

95% confidence interval


2014:03

7.18450

2.79264

(1.71102, 12.6580)


2014:04

7.79609

3.75907

(0.428441, 15.1637)


2014:05

8.68701

4.50136

(-0.135490, 17.5095)


2014:06

8.93731

9.98469

(-10.6323, 28.5069)


2014:07

8.22829

12.8140

(-16.8866, 33.3432)


2014:08

7.59060

14.6553

(-21.1332, 36.3144)



 The Figures 4.5.1-4.5.3 displayed the original inflation rate and the predicted values produced by the ARIMA (0,1,1), ARIMA (1,1,2) and ARIMA (3,1,3) models. The figures also displayed how the forecasted values behave. It can be confirmed that the models somehow fit the data well.


4.5.1 A Comparison of the Models selected Based on Forecasting Power

To compare the models selected and choose the best model, we compare the actual and the fitted values of the data series used in the study. If the magnitude of the difference between the forecasted and actual values is small then the models have good forecasting power. In our case ARIMA (3,1,3) has shown good results as evident from Table 4.5.4 (minimum MSE). One can observe from Figure 4.5.3 that the forecasted series are closer to the actual series. Therefore it can be concluded that the prediction power of this model is better and suitable for six periods forecasting, as such they fit the data well.

Table 4.5.4: Forecast evaluation statistics


ARMA (0,1,1)

ARMA (1,1,2)

ARMA (3,1,3)


Mean Squared Error

9.6372

9.2226

8.3337


Root Mean Squared Error

3.1044

3.0369

2.8868


Absolute Mean Error

2.1629

2.1373

2.0443



Given a series of T observations and associated forecasts we can construct several measures of the overall accuracy of the forecasts. Some commonly used measures are the Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Mean Percentage Error (MPE) and Mean Absolute Percentage Error (MAPE). A further relevant statistic is Theil’s U (Theil, 1966), defined as the positive square root of Theil’s U . The more accurate the forecasts, the lower the value of Theil’s U .



CHAPTER FIVE

SUMMARY, CONCLUSION AND RECOMMENDATION

5.1 SUMMARY

This project work consists of five chapters; chapter one contains the Introduction, Historical background of the study, statement of problem, causes of inflation, controlling inflation, Aim and objectives, significant of the study, scope and limitations and definition of some terms. Chapter two reviews the literature of the study. Chapter three designed the basic methodology of the research work, explaining the various methods applied for the purpose of achieving the objectives of the study. Chapter four analyze the data and discuss the results, following the methods explained in chapter three, and chapter five summarize, make relevant conclusion and give recommendations on the study.

To forecast the values of a time series, the basic Box-Jenkins strategy is as follows:

First examine the series for stationarity. This step can be done by computing the ACF and PACF and or by formal unit root analysis.

If the time series is not stationary, difference it one or more times to achieve stationarity.

Parsimonious models are then selected with minimum AIC, SIC and HQC which is common in modeling.

The estimates and residuals from these models are examined to find out if they are white noise. If they are, the models are probably good approximation to underlying stochastic process. If they are not, the process is started all over again. Therefore, the Box-Jenkins method is iterative.

A tentative model is finally selected by comparing the forecasted values and the actual one (small magnitude of the difference between the forecasted and actual values) given by the model with minimum Mean Square Error (MSE) or Root Mean Square Error (RMSE) which can be used for forecasting.

5.2 CONCLUSION

The study has presented us with an opportunity to have an extensive understanding of the theory of time series analysis in the area of linear models and its application to real life situation. The stages in the model building (that is the identification, estimation and checking) strategy has been explored and utilized. Based on minimum AIC, SIC and HQC values, the best fit ARIMA models tend to be ARIMA (0,1,1), ARIMA (1,1,2) and ARIMA (3,1,3) models respectively. After estimation of the parameters of selected models, a series of diagnostic and forecast accuracy tests were performed. Having satisfied all the model assumptions, ARIMA (3,1,3) model were judged to be the best model for forecasting. The ARIMA (3,1,3) model has smaller MSE and RMSE which is an indication that the model explained the data better than the other models. 

5.3 RECOMMENDATION

Although the application in this research study is based on inflation data, future research can also consider;

The ARIMA, where other macro-economic variables such as exchange rate and money supply will be involved in the model.

Other areas of application for instance, environmental and pollution data, health researches in the context of longitudinal data, agriculture and geo-statistics to mention but few.

There is a strong need for policy makers to focus on policies that will strengthen/stabilize the macroeconomic structure of the Nigerian economy with specific focus on; all items less farm produce and energy, alternative sources of government revenue, reduction in monetization of the items, and so on.


















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APPENDIX

Monthly INFLATION RATE (PERCENT) IN NIGERIA 01/2013-02/2014


Year

Month

All Items (Year on Change)

All Items (12 Months Avg. Change)

Food (Year on Change)/1

Food (12 Months Avg. Change)/1

All Items Less Farm Produce (Year on Change)/2

All Items Less Farm Produce (12 Months Avg. Change)/2

All Items Less Farm Produce and Energy (Year on Change)/3

All Items Less Farm Produce and Energy (12 Months Avg. Change)/3


2003

1

10.6

12.3

7.5

11.6

16.1

13.4

13.6

8.6


2003

2

7.3

11.4

3.5

9.7

14.2

14.3

12.1

8.9


2003

3

5.9

10.5

0.1

7.9

15.4

15

13.9

9.4


2003

4

8.3

10.1

3.3

6.9

16.9

15.6

17.6

10.4


2003

5

8.7

10

3.3

6.3

17.6

16.3

15.5

11.3


2003

6

14

10.1

6.2

6

22.2

16.6

17.7

12.3


2003

7

12.9

10

0.1

4.7

41.2

19

22

13.8


2003

8

12.4

10

0.9

3.9

30.9

20.5

19.9

14.9


2003

9

18.4

10.7

6.2

4

36.3

22.2

22.5

15.8


2003

10

23.6

12.3

13.2

5

40.7

24.6

27.8

17.4


2003

11

21.3

13

12.4

5.4

35.7

26

27.8

18.9


2003

12

23.8

14

15.4

6

34.8

27.2

25

19.8


2004

1

22.4

15

11.9

6.3

38

29

26.3

20.9


2004

2

24.8

16.5

14.5

7.2

41.1

31.2

26.4

22


2004

3

22.5

17.8

15.6

8.5

32.6

32.6

27.4

23.2


2004

4

17.5

18.5

14.4

9.4

22.5

32.9

15.4

22.9


2004

5

19.8

19.4

18.1

10.6

23.4

33.2

17.5

23


2004

6

14.1

19.4

14.5

11.3

14.7

32.3

14.5

22.6


2004

7

10.7

19.1

12.2

12.4

5.8

29

10.8

21.6


2004

8

13

19.1

16.3

13.7

9.1

26.9

11.4

20.8


2004

9

9.1

18.2

14.6

14.4

3.6

23.9

6.6

19.3


2004

10

10.7

17.1

15.4

14.6

3.4

20.7

6.4

17.5


2004

11

10

16.1

15.1

14.8

3.2

17.9

4.6

15.5


2004

12

10

15

12.1

14.5

5.9

15.5

-1.8

13.1


2005

1

9.8

14

15.1

14.8

2

12.6

1.2

11


2005

2

10.9

12.9

18.7

15.2

-0.4

9.4

3.2

9.2


2005

3

16.3

12.5

25

16

3.6

7.3

0.7

7.1


2005

4

17.9

12.6

20.3

16.5

13.8

6.8

14.2

7


2005

5

16.8

12.5

15.7

16.3

16.6

6.5

15.2

7


2005

6

18.6

12.9

18

16.6

16.2

6.7

17.8

7.3


2005

7

26.1

14.2

35.7

18.6

10.7

7.1

16.9

7.9


2005

8

28.2

15.5

38.5

20.5

12.1

7.4

13.1

8.1


2005

9

24.3

16.8

29.5

21.8

14.6

8.3

14.7

8.7


2005

10

18.6

17.4

24.6

22.5

9.4

8.8

13.3

9.3


2005

11

15.1

17.8

19.7

22.9

6.8

9.1

7.7

9.6


2005

12

11.6

17.9

15.5

23.1

2.4

8.8

9.1

10.5


2006

1

10.7

17.9

14.7

22.9

4.3

9

11.7

11.4


2006

2

10.8

17.8

10.5

22.1

11.1

10

11.9

12.1


2006

3

12

17.4

9.3

20.7

16.3

11.1

11.8

13.1


2006

4

12.6

16.9

9.6

19.8

16.6

11.3

11.5

12.8


2006

5

10.5

16.4

8.6

19.1

14.5

11.2

13.5

12.7


2006

6

8.5

15.5

6.2

18

13.6

11

14

12.4


2006

7

3

13.5

-3.7

14.5

13.6

11.3

15.2

12.3


2006

8

3.7

11.4

-2.3

11

14.5

11.5

11.9

12.2


2006

9

6.3

10

4.3

9

10.6

11.2

11

11.9


2006

10

6.1

9

4.7

7.5

9

11.1

13.1

11.9


2006

11

7.8

8.5

5.4

6.4

12.1

11.6

18.3

12.8


2006

12

8.5

8.2

3.9

5.6

17.3

12.8

22.6

13.9


2007

1

8

8

-0.1

4.4

19.3

14

13.2

14


2007

2

7.1

7.7

3.3

3.9

11

14

9

13.7


2007

3

5.2

7.2

1.7

3.3

8.9

13.4

10.2

13.6


2007

4

4.2

6.5

2.1

2.7

5.4

12.4

6.5

13.1


2007

5

4.6

6

2.4

2.2

4.6

11.5

2.7

12.1


2007

6

6.4

5.9

3.2

2

9.6

11.1

-0.3

10.8


2007

7

4.8

6

1.1

2.4

9.9

10.8

-0.4

9.4


2007

8

4.2

6.1

-1.2

2.5

11.2

10.6

5.6

8.9


2007

9

4.1

5.9

-0.8

2.1

10.5

10.6

3.7

8.3


2007

10

4.6

5.7

-0.1

1.7

10.7

10.7

-0.9

7.1


2007

11

5.2

5.5

3.2

1.5

7.4

10.3

-4.3

5.2


2007

12

6.6

5.4

8.2

1.9

3.6

9.2

-6.5

2.9


2008

1

8.6

5.5

12.6

2.9

2.5

7.9

1.4

2


2008

2

8

5.5

8.7

3.3

6.4

7.5

6.4

1.9


2008

3

7.8

5.8

12.4

4.2

0.5

6.8

-1.3

1


2008

4

8.2

6.1

13.1

5.1

1.2

6.4

1.5

0.6


2008

5

9.7

6.5

14.7

6.1

3.3

6.3

3.71

0.6


2008

6

12

7

18.11

7.41

3.61

5.8

7.96

1.34


2008

7

14

7.8

20.9

9

4.8

5.4

6.61

1.93


2008

8

12.4

8.5

18.8

10.7

3.9

4.8

6.6

2.03


2008

9

13

9.2

17.1

12.3

6.9

4.5

8.33

2.43


2008

10

14.7

10.1

19.2

14

7.9

4.3

11.4

3.44


2008

11

14.8

10.9

18.1

15.3

9.3

4.5

13.05

4.88


2008

12

15.1

11.6

18

16.1

10.4

5.1

15.29

6.72


2009

1

14

12

18.4

16.5

8

5.5

12.59

7.65


2009

2

14.6

12.6

20

17.5

7.2

5.6

9.08

7.87


2009

3

14.4

13.1

16.2

17.7

11.8

6.5

15.31

9.25


2009

4

13.3

13.5

15.3

17.9

10.9

7.3

13.28

10.24


2009

5

13.2

13.8

15.7

17.9

9.9

7.8

12.58

10.98


2009

6

11.2

13.7

13.1

17.5

8.5

8.3

9.31

11.08


2009

7

11.1

13.4

12.9

16.8

8.3

8.6

12.34

11.55


2009

8

11

13.3

12.7

16.3

8

8.9

9.26

11.76


2009

9

10.4

13.1

12.5

15.9

7.4

8.9

9.3

11.8


2009

10

11.6

12.8

13.5

15.4

8.9

9

9.4

11.7


2009

11

12.4

12.6

13.5

15

10.7

9.1

11

11.5


2009

12

13.9

12.5

15.5

14.8

11.2

9.2

11.12

11.15


2010

1

14.4

12.6

15.9

14.7

12.1

9.6

10.8

11.01


2010

2

15.6

12.7

16.2

14.4

14

10.1

13.39

11.37


2010

3

14.8

12.8

15.8

14.4

13.2

10.3

12.35

11.16


2010

4

15

12.9

16.3

14.5

12.8

10.4

11.69

11.04


2010

5

12.9

12.9

13

14.3

11.7

10.6

11.64

10.98


2010

6

14.1

13.1

15.1

14.4

12.7

10.9

12.42

11.24


2010

7

13

13.3

14

14.5

11.3

11.2

10.6

11.1


2010

8

13.7

13.5

15.1

14.7

12.4

11.5

13

11.4


2010

9

13.6

13.8

14.6

14.9

12.8

12

13.4

11.8


2010

10

13.4

13.9

14.1

14.9

13.2

12.3

13

12


2010

11

12.8

13.9

14.4

15

11.7

12.4

11.3

12.1


2010

12

11.8

13.7

12.7

14.7

10.9

12.4

10.4

12


2011

1

12.1

13.5

10.3

14.2

12.1

12.4

10.1

11.9


2011

2

11.1

13.2

12.2

13.9

10.6

12.1

10

11.6


2011

3

12.8

13

12.2

13.6

12.8

12.1

12.3

11.6


2011

4

11.3

12.7

11.6

13.2

12.9

12.1

12.2

11.7


2011

5

12.4

12.6

12.2

13.2

13

12.2

11.7

11.7


2011

6

10.2

12.3

9.2

12.7

11.5

12.1

10.3

11.5


2011

7

9.4

12

7.9

12.1

11.5

12.1

10.2

11.5


2011

8

9.3

11.6

8.7

11.6

10.9

12

9.7

11.2


2011

9

10.3

11.4

9.5

11.2

11.6

11.9

9.9

10.9


2011

10

10.5

11.1

9.7

10.8

11.5

11.7

11.7

11.7


2011

11

10.5

11

9.6

10.4

11.5

11.7

11

10.7


2011

12

10.3

10.8

11

10.3

10.8

11.7

9.9

10.7


2012

1

12.6

10.9

13.1

10.5

12.7

11.8

14.3

11.01


2012

2

11.9

11

9.7

10.3

11.9

11.9

12.03

11.18


2012

3

12.1

10.9

11.8

10.3

15

12.1

14.47

11.38


2012

4

12.9

11.1

11.2

10.3

14.7

12.2

14.3

11.6


2012

5

12.7

11.1

12.9

10.4

14.9

12.4

14.5

11.8


2012

6

12.9

11.3

12

10.6

15.2

12.7

13.7

12.1


2012

7

12.8

11.6

12.1

11

15

13

13.8

12.4


2012

8

11.7

11.8

9.9

11.1

14.7

13.3

13.2

12.7


2012

9

11.3

11.9

10.2

11.1

13.1

13.5

12.2

12.9


2012

10

11.7

11.9

11.1

11.2

12.4

13.5

11.4

12.9


2012

11

12.3

12.1

11.6

11.4

13.1

13.6

12.3

13


2012

12

12

12.2

10.2

11.3

13.7

13.9

12.9

13.3


2013

1

9

11.9

10.1

11.1

11.3

13.7

8.59

12.75


2013

2

9.5

11.7

11

11.2

11.2

13.7

8.58

12.44


2013

3

8.6

11.4

9.5

11

7.2

13

6.3

11.7


2013

4

9.1

11.1

10

10.8

6.9

12.3

5.6

11


2013

5

9

10.8

9.3

10.5

6.2

11.5

4.1

10.1


2013

6

8.4

10.4

9.6

10.4

5.5

10.7

4.4

9.3


2013

7

8.7

10

10

10.2

6.6

10

6.3

8.7


2013

8

8.2

9.8

9.7

10.2

7.2

9.4

6.7

8.2


2013

9

8

9.5

9.4

10.1

7.4

8.9

7.3

7.8


2013

10

7.8

9.2

9.2

10

7.6

8.6

7.6

7.5


2013

11

7.9

8.8

9.3

9.8

7.8

8.1

7.8

7.1


2013

12

8

8.5

9.3

9.7

7.9

7.7

8

6.8


2014

1

8

8.4

9.3

9.6

6.6

7.3

7.4

6.7


2014

2

7.7

8.3

9.2

9.5

7.2

7

8

6.6


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