Modelling and forecasting using time series GARCH models - NOMA

MODELLING AND FORECASTING USING TIMESERIES GARCH MODELS: AN APPLICATION OFTANZANIA INFLATION RATE DATAByNgailo EdwardA Dissertation Submitted in (Partial) Fulfillment of the Requirements for theDegree of Master of Science(Mathematical Modelling) of the University ofDaresSalaamUniversity of Dar es SalaamJuly 2011

iCERTIFICATIONThe undersigned certify that they have read and hereby recommend for acceptanceby the University of Dar es Salaam a dissertation entitled:Modellingand Forecasting Using Time Series GARCH Models: An Applicationof Tanzania Inflation Rate Data, in fulfilment of the requirements for thedegree of Master of Science (Mathematical Modelling) of the University of Dares Salaam.—————————————Dr. E.S.Massawe(Supervisor).Date:. . . . . . . . . . . . . . . . . . . . . . . .—————————————Dr. E.Luvanda(Supervisor)Date:. . . . . . . . . . . . . . . . . . . . . . . .

viTABLE OF CONTENTSCertification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iDeclaration and Copyright . . . . . . . . . . . . . . . . . . . . . . . . .iiAcknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ivAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ixList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . .xii1 CHAPTER ONEINTRODUCTION 11.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Inflation performance in Tanzania . . . . . . . . . . . . . . 31.2 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . 41.3 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.1 General objective . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Specific objectives . . . . . . . . . . . . . . . . . . . . . . . 51.4 Significance of the study . . . . . . . . . . . . . . . . . . . . . . . 6

vii2 CHAPTER TWOLITERATURE REVIEW 72.1 Overview of Related Literature . . . . . . . . . . . . . . . . . . . 73 CHAPTER THREECONDITIONAL HETEROSCEDASTICITY: ARCH-GARCH MOD-ELS 143.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 ARCH(q) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.1 ARCH(1) Model . . . . . . . . . . . . . . . . . . . . . . . 163.2.2 Estimation of the ARCH(1) and the ARCH(q) Models . . 173.2.3 Forecasting with the ARCH model . . . . . . . . . . . . . 203.3 The GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.1 Estimation of the GARCH(1, 1) . . . . . . . . . . . . . . . 243.3.2 The GARCH(p, q) Model . . . . . . . . . . . . . . . . . . . 273.3.3 Estimation of GARCH(p, q) Model . . . . . . . . . . . . . 313.3.4 Model Checking . . . . . . . . . . . . . . . . . . . . . . . . 313.3.5 Forecasting with GARCH(p, q) models . . . . . . . . . . . 323.4 Model selection criteria . . . . . . . . . . . . . . . . . . . . . . . . 333.4.1 The Akaike Information Criterion . . . . . . . . . . . . . . 33

viii3.4.2 The Bayesian information criterion . . . . . . . . . . . . . 343.5 Forecasting performance . . . . . . . . . . . . . . . . . . . . . . . 354 CHAPTER FOURDATA ANALYSIS AND METHODOLOGY 374.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Data description and basic statistics . . . . . . . . . . . . . . . . . 374.3 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4 Testing for ARCH effects and Serial correlation in the return series 414.5 Model estimation and evaluation . . . . . . . . . . . . . . . . . . 444.5.1 Model selection and analysis . . . . . . . . . . . . . . . . . 444.5.2 A Comparison of the fitted GARCH models . . . . . . . . 484.6 Diagnostic checking of the GARCH(1, 1) model . . . . . . . . . . 494.7 Forecasting with the GARCH(1, 1) model . . . . . . . . . . . . . . 524.7.1 Forecast Evaluation and Accuracy Criteria . . . . . . . . . 535 CHAPTER FIVEDISCUSSION, CONCLUSION AND FUTURE WORK 575.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.3 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . 58

ix6 REFERENCES 597 APPENDIX 64

xLIST OF FIGURES4.1 Time plot of monthly inflation in Tanzania . . . . . . . . . . . . . 394.2 First Difference of Log of CPI . . . . . . . . . . . . . . . . . . . . 404.3 ACF with Bounds for Raw Return Series . . . . . . . . . . . . . . 434.4 PACF with Bounds for Raw Return Series . . . . . . . . . . . . . 434.5 ACF of the squared Returns . . . . . . . . . . . . . . . . . . . . . 444.6 Plot of Return, Estimated Volatility and Innovations (residuals) . 494.7 Time plot of residuals from GARCH(1,1) . . . . . . . . . . . . . . 504.8 ACF of the squared standardized residuals . . . . . . . . . . . . . 514.9 (a)Histogram of residuals(top) and (b)Normal probability(bottom)plot from GARCH(1,1) . . . . . . . . . . . . . . . . . . . . . . . 554.10 Time plot of Inflation rate and one year forecasts from GARCH(1,1) 56

xiLIST OF TABLES4.1 Descriptive Statistics for the Inflation rate series. . . . . . . . . . 384.2 Ljung-Box-Pierce Q-test for autocorrelation: (at 95% confidence). 414.3 Ljung-Box Q-test for squared returns: (at 95% confidence). . . . . 424.4 Engle ARCH test for heteroscedasticity: (at 95% confidence). . . . 424.5 Comparison of suggested GARCH models. . . . . . . . . . . . . . 454.6 Parameter estimates for GARCH(1,1). . . . . . . . . . . . . . . . 464.7 Parameter estimates for GARCH(1,2). . . . . . . . . . . . . . . . 474.8 Parameter estimates for GARCH(2,2). . . . . . . . . . . . . . . . 484.9 Ljung-Box-Pierce Q-test on standardized innovations. . . . . . . . 524.10 Engle ARCH test on standardized innovations. . . . . . . . . . . . 524.11 Forecast Accuracy Test on the most likely suggested GARCH models. 534.12 One year in-sample forecasts of Inflation obtained from the GARCH(1,1). 54

xiiABBREVIATIONSACFADFAICARARCHARIMABICCPIEGARCHGARCHIGARCHMAERMSETNBSPACFQTMSARIMAAucorrelation FunctionAugumented Dickey FullerAkaike Information CriteriaAutoRegressiveAutoRegressive Conditional HeteroscedasticAutoRegressive Integrated Moving AverageBayesian Information CriteriaConsumer Price IndexExponential Generalized AutoRegressive Conditional HeteroscedasticGeneralized AutoRegressive Conditional HeteroscedasticIntegated Generalized AutoRegressive Conditional HeteroscedasticMean Absolute ErrorRoot Mean Square ErrorTanzania National Bureau of StatisticsPartial Autocorrelation FunctionQuantum Theory of MoneySeasonal AutoRegressive Integrated Moving Average

CHAPTER ONEINTRODUCTION1.1 General IntroductionTime, in terms of years, months, days or hours is a device that enables one torelate phenomena to a set of common, stable reference points. In making consciousdecisions under uncertainty, we all make forecasts. Almost all managerialdecisions are based on some form of forecast.Essentially the concept of timeseries is based on the historical observations. It involves explaining past observationsin order to try to predict those in the future (Ahiati, 2007 ). A timeseries is a collection of observations measured sequentially through time. Thesemeasurements may be made continuously through time or be taken at a discreteset of time points (Chatfield, 2000 ).In recent years, inflation has become one of the major economic challenges facingmost countries in the world especially those in Africa including Tanzania.Inflation is a major focus of economic policy worldwide as described by David(2001). Inflation dynamics and evolution can be studied using a stochastic modellingapproach that captures the time dependent structure embedded in the timeseries inflation data. The autoregressive conditional heteroscedasticity (ARCH)models, with its extension to generalized autoregressive conditional heteroscedasticity(GARCH) models as introduced by Engle (1982) and Bollerslev (1986)respectively accommodates the dynamics of conditional heteroscedasticity (thechanging variance nature of the data). Heteroscedasticity affects the accuracy offorecast confidence limits and thus has to be handled properly by constructing1

2appropriate non-constant variance models (Amos, 2009).Inflation as described by Webster‘s (2000), is the persistent increase in the level ofconsumer prices or persistent decline in the purchasing power of money. Inflationcan also be expressed as a situation when the demand for goods and servicesexceeds their supply in the economy (Hall, 1982). In reality inflation means thatyour money can not buy as much as what it could have bought yesterday.The most common way of measuring inflation is the Consumer P rice Index(CP I) over monthly, quarterly or yearly. The formula for calculating the inflationdata can be defined as follows; Let P t be the current average price level of aneconomic basket of goods and services and P t−1 be the average price level of thebasket a year ago, then the inflation rate I t at time t is calculated as ;I t = P t − P t−1P t−1× 100%.An alternative measure is the gross domestic product (GDP ) which is usuallytaken annually. In Tanzania the CP I is calculated by the National Bureau ofStatistics under the auspices of the Ministry of Finance.The national CP Icovers prices collected in twenty towns in Tanzania mainland. All prices are theprevailing market prices and are gathered for two hundred and seven items. TheCP I is a dynamic figure obtained by measuring the price variations of numerousgoods and services consumed by typical Tanzanian households on a monthly basis.Specific weights are assigned to categories of expenditure based on consumerspending statistics. The different types of expenditure are grouped together in abasket and the high volume spending items carry the most weight and thereforehave the most material impact on the calculated index. Once the CP I is known,the rate of inflation is the rate of change in the CP I over a period for example

3month-on-month inflation rate and usually its units is in percentages. Inflationcan be caused by either too much money in circulation in the country or toofew goods offered for sale. People who are living with fixed income suffer mostwhen prices of commodities rise, as they cannot buy as much as they could buypreviously (Aidoo, 2010). Unanticipated inflation makes future prices unknown,as this situation discourages savings due to the fact that the money is worthmore at present than in the future. It leads to inefficient allocation of economicresources. Therefore it reduces economic growth, because economy needs a certainlevel of savings to finance investments which boosts economic growth.1.1.1 Inflation performance in TanzaniaThe historical trend of inflation in Tanzania shows that inflation has always beena two-digit figure starting in 1966. In the 1970s, it has been limited to fluctuatebetween 10% and 20%. At the end of the 1970s and the beginning of 1980, aradical increase was recorded. Inflation rose to the level of 30%. It stabilized atthis level, only dropping to 20% at the end of 1980s. The government of Tanzania‘sstrategy for reducing inflation has since 1986 focused on tight monetarypolicy and increased output production. Inflation remained high during theseperiods, although at a slightly lower level than the pre-reform level of 22.3% in1985 (Rutayisire, 1986) The significant decline in the inflation rate since 1994reflects the impact of tight monetary and fiscal policies pursued by the centralbank and government. Inflation declined from 15.5% in December 1996 to 5.5%in December 2000; nonetheless, it rose to 6.7% in December 2006. The periodbetween December 2000 and December 2005, the inflation went down to 6.2%and 4.4% respectively. From 6.7% of 2006 inflation has been rising and it reachedto 11.6% in 2010 (Tanzania National Bureau of Statistics, 2010)

4Within time series modelling, there are two approaches available for forecasting;the univariate and multivariate. In particular this research study has forecastedin-sample future values of inflation time series data using the univariate forecastingapproach; in which forecasts depend only on present and past values of a singleseries being forecasted. An alternative approach available is a multivariate forecasting;in which the forecasts of a given variable depend, on more additional timeseries data. The autoregressive conditional heteroscedasticity models are specificsubset of univariate modelling in financial time series and statistics, in which acurrent conditional variance depends not only on the previous observations, butalso on the previous conditional variance.The stochastic models; the autoregressiveintegrated moving average (ARIMA) and the seasonal autoregressiveintegrated moving average (SARIMA) models are also other approaches underunivariate modelling. The main differences from GARCH models come about dueto the assumptions and characteristics of the models, particularly the assumptionof constant variance.This research study is therefore intended to model long term behaviour of monthlyinflation rate data of Tanzania from 1997 to 2010 and predict future values. Thebest way to study and understand the underlying phenomenon of the inflationprocess is to construct practical mathematical stochastic model that accounts forvariations in the process of inflation.1.2 Statement of the problemThe assumption of constant variance over some period when the resulting seriesmoves or progress through time, statistically is inefficient and inconsistent(Campbell et al., 1997). In real life financial data for instance inflation data,

5variance changes with time (a phenomenon defined as heteroscedasticity), hencethere is a need of studying models which accommodates this possible variationin variance. In considering the issue inflation modelling and forecasting in Tanzania,it has been found that there are many researches done, but no previousmathematical study has used Tanzania monthly inflation rate data to model theconditional variance and forecast future values. Consequently, this work intendsto use the autoregressive conditional heteroscedasticity (ARCH) models with itsextension to generalized ARCH (GARCH) models to model and accommodatethe dynamics of conditional heteroscedasticity in inflation data.Moreover byfinding an appropriate model to represent the data, the study intends to use itto predict future values based on the past observations.1.3 Research objectives1.3.1 General objectiveThe general objective of the study is to establish the generalized autoregressiveconditional heteroscedasticity time series model to analyse inflation data basedon Tanzania.1.3.2 Specific objectivesThe specific objectives of this study are:• To develop time series model (GARCH model) for inflation data in Tanzania.

6• To determine the accuracy of the model.• To determine future inflation rate.1.4 Significance of the studyThe research study is of significant to several parties as follows:• To Academicians and Researchers;This research study is expected to provide knowledge and basis for furtherstudies in the same area or other related fields of study.• To policy makers;The study is intended to contribute to policy making through developing amodel which can be used to forecast inflation thus guide policy makers informulating macroeconomic policies.

CHAPTER TWOLITERATURE REVIEW2.1 Overview of Related LiteratureA brief survey on previous works provides the context of the proposed study.Amos (2009) studied financial time series modelling using inflation data spanningfrom January 1994 to December 2008 for South Africa. In the study two timeseries models which are the seasonal autoregressive integrated moving average(SARIMA) model and the generalized autoregressive conditional heteroscedasticity(GARCH) model were fitted to the data for encountering trend and seasonalterms and accomodating time varying variance respectively. A best fitting modelfor each family of models offering an optimal balance between goodness of fit andparsimony was selected. The SARIMA model of order (1,1,0)(0,1,1) and GARCHmodel of order (1,1) were chosen to be the best fitting models for determiningthe two years forecasts of inflation rate of South Africa. However GARCH modelof order (1,1) was observed to be superior in producing future forecasts becauseof its ability to capture variations in the data.Perniagaan, Ekonomi and Nor (2004) explored the varying volatility dynamicsof inflation rate data in Malaysia for the period from August 1980 to December2004. The Generalized autoregressive conditional heteroscedasticity (GARCH)models and the exponential generalized autoregressive conditional heteroscedasticity(EGARCH) models were used to capture the stochastic variations andasymmetries in the data. An in-sample evaluation of the sub-periods volatility7

8was done. Based from the results using both models the GARCH model of order(1, 1) and EGARCH model of order (1, 1) produced good estimates of sub-periodsvolatility. But due to nature of the data of having highly irregular fluctuationEGARCH model was selected to be the best.Dijk and Franses (1996) studied the performance of the GARCH model and two ofits non-linear modifications to forecast weekly stock market volatility. The modelswere the Quadratic GARCH and the Glosten, Jagannathan and Runkle (1992)or (GJR) models which had been proposed to describe the negative skewness instock market indices. Their forecasting results for five weekly stock market indicesshowed that the QGARCH model could significantly be improved on the linearGARCH model so that it could be good at calibrating data including extremeevents.Based on the results they concluded that the forecasting of GARCHtype models appeared sensitive to extreme within sample observations. The GJRmodel on the other hand was not recommended for forecasting.Garcia et al., (2003) pointed out that, the generalized autoregressive conditionalheteroscedasticity (GARCH) models consider, the price series as not invariant(does not have zero mean and constant variance) as it happens in an autoregressiveintegrated moving average (ARIMA) process. In their study GARCHprocess measured the implied volatility of a series due to price spikes. Their paperfocused on day-ahead forecast of daily electricity markets with high volatilityperiods using a GARCH methodology. The GARCH model in their study was toprovide the twenty four (24) forecasts of the clearing prices for the next day basedon historical data. They observed a good performance of the prediction method.The daily mean errors were around 4%, where the lowest mean error was 2.60%and the highest one was 7.60%. In general, they concluded that the results obtainedby the model were quite reasonable, as the errors obtained were not largerthan 10%. Nevertheless to verify the predictive accuracy of the GARCH model,

9different statistical measures were utilized such as mean week error (MW E) andforecast mean square error (F MSE).Assis, Amran and Remali (2006) compared the forecasting performance of differenttime series methods for forecasting cocoa bean prices at Bagan Datoh cocoabean. The monthly average data of Bagan Datoh cocoa bean prices graded forthe period of January 1992 to December 2006 was used. Four different types ofunivariate time series models were compared namely the exponential smoothing,autoregressive integrated moving average (ARIMA), generalized autoregressiveconditional 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) wereused as the selection criteria to determine the best forecasting model. The studyrevealed that the time series data were influenced by a positive linear trend factorwhile 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 stationaryafter the first differentiating process was carried out. In their paper theypointed out that forecasting the future prices of cocoa bean through the mostaccurate time series model could help the Malaysia government as well as thebuyers (exporters and millers) and sellers (farmers and dealers) in cocoa bean industryto perform better strategic planning and also to help them in maximizingrevenue and minimizing the cost. Based on the results of the ex-post forecasting(starting from January until December 2006) GARCH model performed bettercompared to exponential smoothing, ARIMA and the mixed ARIMA/GARCHmodel for forecasting Bagan Datoh cocoa bean prices.Kontonikas (2004) analyzed the relationship between inflation and inflation uncertaintyin the United Kingdom from 1973 to 2003 with monthly and quarterly

10data. Different types of GARCH Mean (M)-Level (L) models that allow for simultaneousfeedback between the conditional mean and variance of inflation wereused to test the relationship. They found that there was a positive relationshipbetween past inflation and uncertainty about future inflation, in line with Friedmanand Ball (2006) to which in their study of testing for rate of dependence andasymmetric in inflation uncertainty they concluded that there was a link betweeninflation rate and inflation uncertainty.Jehovanes (2007) studied a time lag between a change in money supply andthe inflation rate response.A modified generalized autoregressive conditionalheteroscedasticity (GARCH) model was employed to monthly inflation data forthe period 1994 to 2006. In the study he used the maximum likelihood estimationtechnique to estimate parameters of the model and to determine significance ofthe lagged value. GARCH model produced good results which indicated that achange in money supply would affect inflation rate considerably in seven monthsahead.Chong, Loo and Muhammad (2002) used seven years of daily observed Sterlingexchange rate. The GARCH model including the family of GARCH Mean modelwere used to explain the commonly observed characteristics of the unconditionaldistribution of daily rate of returns series. The results indicated that the hypothesesof constant variance model could be rejected, at least within sample,since almost all the parameter estimates of the ARCH and GARCH models weresignificant at 5 percent level. The Q− statistics and the Lagrange Multiplier testrevealed that the use of the long memory GARCH model was preferable to theshort memory and high order ARCH model. The results from various goodnessof fit statistics were not consistent for Sterling exchange rates. It appeared thatthe BIC and AIC test proposed GARCH models to be the best for within samplemodelling while the mean square error (MSE) test suggested the GARCH mean

11model to be the best for modelling the heteroscedasticity of daily exchange rates.Engle (1982) studied on ARCH and GARCH models, and revealed that, thesemodels were designed to deal with the assumption of non-stationarity found inreal life financial data. He further pointed out that these models have becomewidespread tools for dealing with time series heteroscedastic. The ARCH andGARCH models treat heteroscedasticity as a variance to be modelled. The goalof such models is to provide a volatility measure like a standard deviation thatcan be used in financial decisions concerning risk analysis, portfolio selection andderivative pricing.Hamilton (1994) carried out a study on the importance of forecasting the conditionalvariance and pointed out that sometimes we might be interested in forecastingnot only the level of the series, r t but also its changing variance. He furtherdescribed that changes in the variance are quite important for understanding financialmarkets, since investors require higher expected returns as compensationfor holding riskier assets. A variance that changes over time also has implicationsfor the validity and efficiency of statistical inference about the parameters thatdescribe dynamics of the level of the series, r t .Chatfield (2000), explored in his book that, the idea behind a GARCH modelis similar to that behind ARMA model in the sense that a higher order AR orMA model may often be approximated by a mixed ARMA model, with fewerparameters, using a rational polynomial approximation. Thus a GARCH modelcan be thought of as an approximation to a higher-order ARCH model.TheGARCH(1, 1) model has become the standard model for describing changing variancefor no obvious reason other than relative simplicity. In practice, if such amodel is fitted to data, it is often found that (α + β) < 1 so that the stationaritycondition may be satisfied. If α + β = 1,, then the process does not have finite

12variance, although it can be shown that the squared observations are stationaryafter taking first differences leading what is called an integrated GARCH orIGARCH model.Brooks (2008) studied stochastic volatility models and found that most time seriesmodels such as GARCH, will have forecasts that tend towards the unconditionalvariance of the series as the prediction horizon increases. This is a good propertyfor a volatility forecasting model to have, since it is well known that volatilityseries are mean-reverting. This implies that if they are at a low level relative totheir historic average they will have a tendency to rise back towards the average.This feature is accounted for in GARCH volatility forecasting models.Mugume and Kasende (2009) investigated inflation and inflation forecasting inUganda from 1993 to 2009. They employed various inflation forecasting modelslike Philips curve, P-star model based on Quantum Theory of Money (QTM),and the price equation and ARIMA model. In addition, M3 was employed andthe results of both short-run dynamics and long-run equilibrium indicated thatinflation had not been a result of money growth. The long-run inflation equationseemed to indicate that exchange rate depreciation could have had a strongerimpact in driving inflation upwards than money supply, although it had no shortrunimpact.Igogo (2010) did a study on the effect of real exchange rate volatility on tradeflows in Tanzania for period of 1968 to 2007. The study employed recent ARCHfamly models to measure volatility. Firstly the GARCH(1, 1) model was employedand found to violate the non-negativity condition. However the study employedEGARCH(1, 1) model proposed by Nelson(1991) to resolve the problem.Theadequacy of the EGARCH(1, 1) model to measure the real exchange rate volatilitywas confirmed by testing for ARCH effect after running the model. Furthermore,

13the study reveals that it is the real exchange rate rather than its volatility thatis found to have a significant effect on trade flows although the effect is larger onexports than on imports. Therefore in the short-run imports are mainly affectedby the domestic income while exports are mainly affected by the real exchangerate.

CHAPTER THREECONDITIONAL HETEROSCEDASTICITY:ARCH-GARCH MODELS3.1 IntroductionThe analysis of financial data has received considerable attention in the literatureover the last 20 years. Several models have been suggested for capturing specialfeatures of financial data, and most of these models have the property that theconditional variance (or the conditional scaling) depends on the past. One of thebest known and most often used is the autoregressive conditionally heteroscedastic(ARCH) process introduced by Engle, (1982). The ARCH model has beeninvestigated and generalized by several authors, including Bollerslev (1986) andGourieroux (1997). The theoretical results on ARCH and related properties haveplayed a special role in empirical work in the analysis of data on exchange rates,stock prices and inflation rate data to mention but a few.In real life financial data, variance may change with time, therefore there is a needof studying models which accommodate this possible variation in variance. Theautoregressive conditional heteroscedasticity (ARCH) models, with its extensionto generalized ARCH, (GARCH) models as introduced by Engle and Bollerslevrespectively, accommodates the dynamics of conditional heteroscedasticity. Thisis done by relating the current variance errors to the previous errors in the caseof ARCH where as in GARCH, the previous conditional variances are included.Heteroscedasticity affects the accuracy of forecast confidence limits and thus has14

15to be handled properly by constructing appropriate non-constant variance models.The ARCH-GARCH modelling considers the conditional error variance as a functionof the past realization of the series. Campbell et al., (1997) argued that “itis both logically inconsistent and statistically inefficient to use and model volatilitymeasures that are based on the assumption of constant variance over someperiod when the resulting series moves or progress through time.” In the case offinancial data for example, large and small errors occur in clusters, which implythat large returns are followed by more large returns and small returns are furtherfollowed by small returns. In this context of the current study this is equivalentto saying that periods of high inflation are usually followed by further periods ofhigh inflation, while low inflation is likely to be followed by much low inflation.The theory of the ARCH and GARCH models are detailed in sections 3.2 and3.3 respectively.3.2 ARCH(q) ModelLet {r t } be the mean-corrected return or rate of inflation, ε t be the Gaussianwhite noise with zero mean and unit variance and H t be the information set attime t given by H t = {r 1 , r 2 , . . . , r t−1 }. Then the process {r t } is ARCH(q) (Engle,1982) ifr t = σ t ε t , (3.1)whereE(r t|Ht ) = 0, (3.2)V ar(r t|Ht ) = σ 2 t = α 0 +q∑α i rt−i, 2 (3.3)i=1

16and the error term ε t is such thatE(ε t|Ht ) = 0, (3.4)V ar(ε t|Ht ) = 1. (3.5)Equations (3.4) and (3.5) show that the error term ε t is a conditionally standardizedmartingale difference defined as follows: A stochastic series {r t } is amartingale difference if its expectation with respect to past values of anotherstochastic series X i is zero, that is,E(r t+i|Xi ,X i−1 ,...,) = 0for i = 1, 2, . . . . In this type of the impact of the past on the present volatilityis assumed to be a quadratic function of lagged innovations. The coefficient(α 0 , α 1 , . . . , α q ) can consistently be estimated by regressing {rt 2 } on rt−1, 2 rt−2, 2 . . . , rt−q.2To ensure non-negative volatility we require α 0 ≥ 0, α i ≥ 0 for all i = 1, 2, . . . , q.3.2.1 ARCH(1) ModelThe ARCH(1) model is a particular case of the general ARCH(q) model and isdefined as follows: Let {r t } be the mean-corrected return and ε t be a Gaussianwhite noise with mean zero and unit variance. If H t is the information set availableat time t then the process {r t } is ARCH(1) if q = 1 given byr t = σ t ε t , (3.6)and the conditional variance σ 2 tis given byσ 2 t = α 0 + α 1 r 2 t−1, (3.7)

17where α 0 and α 1 are unknown parameters. Thus under the normality assumptionof ε t the process can be stated conditionally in terms of H t similar to thevariance σ 2 tas given above. To ensure non-negativity condition for the conditionalvariance, the constraints α 0 ≥ 0 and α 1 ≥ 0 must be satisfied. Equations(3.6) and (3.7) suggest that a large past squared mean-corrected return impliesa large conditional variance σ 2 tresulting in r t being large in absolute value. TheARCH(1) is a special case of ARCH(q) and therefore what applies for ARCH(q)also applies for ARCH(1) .3.2.2 Estimation of the ARCH(1) and the ARCH(q) ModelsBased on the assumption of the normality made on the ε t the method of maximumlikelihood estimation is adopted. Let r 1 , r 2 , . . . , r t , be a realization from anARCH(1) process, then the likelihood of the data can be written as a product ofthe conditionals asf(r 1 , r 2 , . . . , r t |θ) = f(r t |r t−1 )f(r t−1 |r t−2 ) . . . f(r 2 |r 1 )f(r 1 |θ), (3.8)where θ = (α 0 , α 1 ) ′ . It is more practical to set condition on r 1 since the formf(r 1 |θ) is difficult to obtain. Usually r 1 is assumed to be known and equal to itsobserved value. This allows us to use the conditional likelihood given byf(r 1 , r 2 , . . . , r t |θ; r 1 ) = f(r t |r t−1 )f(r t−1 |r t−2 ) . . . f(r 2 |r 1 )f(r 1 |θ; r 1 ). (3.9)Since r t |H t ∼ N(0, σ 2 t ) it follows that

18f(r t |H t ) ={ }1√ exp − r2 t2πσ2t2σt2(3.10)where σ 2 t = α 0 + α 1 r 2 t−1. The conditional log-likelihood is expressed asl = l c (α 0 , α 1 |r 1 ),= − 1 2= lnf(r 2 , . . . , r 1 ; θ),t∑ln(2πσi 2 ) − 1 t∑{ r2t2i=2i=2σ 2 t}. (3.11)The maximum likelihood estimates are obtained by maximizing this function withrespect to α 0 and α 1 , (Tsay, 2002). Note that the function is non linear in theseparameters and thus its maximization must be done using appropriate non linearoptimization routine. Let a process [r t ] T t=1be a series generated by an ARCH(1)process, where T is the sample size. Conditioning on the initial observation, thejoint density function can be written asf(r) =T∏f(r t |H t ). (3.12)t=2To find the conditional maximum likelihood estimates of α 0 and α 1 , first oneneeds the derivatives of the conditional log-likelihood with respect to α 0 and α 1given by∂l t= 1 ( r2t∂α 0 2σt2 σt2) ∂lt− 1 ,∂α 0= 1 ( r2t2σt2 σt2) ∂lt− 1∂σt2× ∂σ2 t∂α 0, (3.13)and∂l t= 1 ( r2t∂α 1 2σt2 σt2− 1) ∂lt∂α 1,

19= 1 ( r2t2σt2 σt2) ∂lt− 1∂σt2× ∂σ2 t∂α 1. (3.14)More generally the partial derivative of l is= − 1 2∂lT∑∂θ =T∑t=2( 1σ 2 tt=2∂l t ∂σt2∂σt2 ∂θ ,⎛)1⎜⎝− r2 tσ 4 tr 2 t−1⎞⎟⎠ ,= 1 2T∑( r2tσ 2 t=2 t) 1− 1σt2⎛⎜⎝1r 2 t−1⎞⎟⎠ . (3.15)recalling that σ 2 t = α 0 + α 1 r 2 t−1. Since ∂2 σ 2 t∂θ∂θ ′= 0, the Hessian is given by= − 1 2T∑t=2∂ 2 l∂θ∂θ ′ =T∑t=2( r2t(σ 2 t ) 3 + ( r2tσ 2 t∂ 2 l t∂σ 4 t∂σ 2 t∂θ∂σ 2 t∂θ ′ ,⎛) ) 1 − 1 ⎜σt 4 ⎝1 r 2 tr 2 tr 4 t⎞⎟⎠ .The Fisher information matrix denoted by g is defined to be the negative of theexpected value of the Hessian, that is,( ) ∂ 2 lg = −E ,∂θ∂θ ′sinceE rt|H t( r2tσ 2 t) 1− 1σt4⎛⎜⎝1 r 2 t−1r 2 t−1r 4 t−1⎞⎟⎠ = 0,

20andIt then follows that,as in Engle, (1982).{ } { }r2E t Ert|Hrt|Ht =trt2 .(σt 2 ) 3 (σt 2 ) 3g = 1 2T∑t=2( 1σ 4 t⎛)⎜⎝1 r 2 t−1r 2 t−1r 4 t−1⎞⎟⎠ (3.16)Non-linear optimization routines are iterative, thus if θ idenotes the parameter estimates after the i th iterations, then θ i+1 has the form{ } ∂lθ i+1 = θ i + λM −1 ∂θwhere λ is a step-length chosen to maximize the likelihood function in the direction∂l . For Newton Raphson based routines λ = 1 and M =∂2 ∂θl∂θ∂θ ′Fisher scoring method λ = 1 and M = g (Mills, 1994 and Engle, 1982).and for the3.2.3 Forecasting with the ARCH modelForecasting is one of the main aims of developing a time series model.TheARCH models also provide good estimates of the series before it is realized. Wenow provide the theory of forecasting with the ARCH models in detail.Letr 1 , r 2 , r 3 , . . . , r t , be an observed time series , then the l− step ahead forecast, forl = 1, 2, . . . , at the origin t, denoted as r t (l), is taken to be the minimum meansquared error predictor, that is, r t (l) minimizesE(r t+l − f(r)) 2where f(r) is a function of the observations, thenr r (l) = E[r t+l |r 1 , . . . , r t ],

21(Tsay, 2002)However for the ARCH(1) modelr t (l) = E[r t+l |r 1 , . . . , r t ] = 0The forecasts for the r t series provide no much helpful information. It is thereforeimportant to consider the squared returns r 2 t given asr 2 t = E[r 2 t+l|r 2 1, . . . , r 2 t ],(Shephard, 1996)Hence the l− step ahead forecast for r 2 t is given byr 2 t (1) = ˆα 0 + ˆα 1 r 2 twhich is equivalent toσ 2 t (1) = E[σ 2 t+1|r t ]= ˆα 0 + ˆα 1 r 2 twhere ˆα 0 and ˆα 1 are the conditional maximum likelihood estimates of α 0 and α 1 .Similarly a 2− step ahead forecast for r 2 t is given byrt 2 (2) = E[rt+2|r 2 t ]= E[σ t+2 |r t ]= ˆα 0 + ˆα 1 E[rt+1|r 2 t ]= ˆα 0 + ˆα 1 ( ˆα 0 + ˆα 1 rt 2 )

22= ˆα 0 (1 + ˆα 1 ) + ( ˆα 1 2 r 2 t )= σ 2 t (2)In general the l− step ahead forecast for the r 2 t is given by,r 2 t (l) = E[r 2 t+l|r t ]= ˆα 0 (1 + ˆα 1 + ˆα 1 2 + . . . + ˆα 1 l−1 ) + ˆα 1 l r 2 t ,= σ 2 t (l).The obvious possible problem in using the ARCH formulation is that the approachcan lead to a high parametric model if the lag q is large. This necessitated theintroduction of the GARCH model. Section 3.3 below gives an account of thetheory of the GARCH modelling.3.3 The GARCH ModelThe Generalized ARCH (GARCH), as developed by Bollerslev (1986), is an extensionof the ARCH model similar to the extension of an AR to ARMA process.When modelling using ARCH, there might be a need for a large value of thelag q , hence a large number of parameters. This may result in a model witha large number of parameters, violating the principle of parsimony and this canpresent difficulties when using the model to adequately describe the data. AnARMA model may have fewer parameters than AR model, similarly a GARCHmodel may contain fewer parameters as compared to an ARCH model, and thusa GARCH model may be preferred to an ARCH model. There are a variety ofextensions of the ARCH family of models that include the Exponential GARCH(EGARCH), the Integrated GARCH (IGARCH) (Amos, 2009). These are not

23discussed in this research study. For interested reader, thesis by Talke (2003)is good source of such information. The GARCH(p, q) model employs the sameequation (3.1) for the log-returns r t but the equation for the volatility, includesq new terms, that isr t = σ t ε t , ε t ∼ N(0, 1), (3.17)σt 2 = α 0 + α 1 rt−1 2 + . . . + α q rt−q 2 + β 1 σt−1 2 + . . . + β p σt−p. 2 (3.18)where now t >max(p, q) and the remaining components are as in the ARCHmodel. The parameters of the model are α 0 , α 1 , . . . , α q , β 1 , . . . , β q , for some positiveintegers p, q .We see that if p = 0 the above model is reduced to theARCH(q) . Thus the GARCH model generalizes the ARCH by introducing valuesof σ 2 t−1, σ 2 t−2, . . . in equation(3.17). Let {r t } be the mean corrected return, ε tbe a Gaussian white noise with mean zero and unit variance. Let also H t be theinformation set or history at time t given by H t = { r 1 , r 2 , . . . , r t−1}as in theARCH model. Then the process {r t } is GARCH(1, 1) ifandr t = σ t ε t , ε t ∼ N(0, 1), (3.19)σ 2 t = α 0 + α 1 r 2 t−1 + β 1 σ 2 t−1. (3.20)The restrictions α 0 ≥ 0, α 1 ≥ 0, and β 1 ≥ 0 are imposed in order for the varianceσ 2 t to be positive. Clearly equations (3.18) and (3.19) show that large past meancorrected squared returns r 2 t−1 or past conditional variances σ 2 t−1 give rise to largevalues of σ 2 t (Tsay, 2002). The conditional mean E(r t |H t ) = 0 implies that { r t}is a martingale difference, thus E(r t ) = 0 and {r t } is an uncorrelated series. Atthis point it is important to point out that the information set is now strictlygiven by[r 1 ; σ 2 1, . . . , r t−1 ; σ 2 t−1].

24Furthermore ARCH(1) can also be defined as GARCH(0, 1) . Taking v t = rt 2 −σt 2 ,we writert 2 = σt 2 + v t ,= α 0 + α 1 rt−1 2 + β 1 (rt−1 2 − v t−1 ) + v t ,= α 0 + (α 1 + β 1 )rt−1 2 + v t − β 1 v t−1 . (3.21)As in ARCH(1) model, E(v t |H t ) = 0 and v t is another martingale difference,meaning that E(v t ) = 0 and cov(v t , v t−k ) = 0 for k ≥ 1 and v t is series uncorrelatedfrom equation (3.18). SoE(r 2 t ) = σ 2 t = α 0 + (α 1 + β 1 )E(r 2 t−1),σt 2 = α 0 + (α 1 + β 1 )E(σt 2 ε 2 t ),= α 0 + (α 1 + β 1 )σt 2 E(ε 2 t ),= α 0 + (α 1 + β 1 )σt 2 .1,(1 − α 1 B − β 1 B)σt 2 = α 0 ,provided |α 1 + β 1 | < 1=α 01 − (α 1 + β 1 ) .3.3.1 Estimation of the GARCH(1, 1)Estimation of the parameters of the GARCH(1, 1) model is performed in the sameapproach as in the ARCH(1) model. However, since the conditional variance ofthe GARCH(1, 1) model depends also on the past conditional variance, an initialvalue of the past conditional variance σ1 2 is needed. Suppose as before that wehave a sample of log-returns r 1 , . . . , r n and we wish to find estimates ˆα 0 , ˆα 1 and

26∂l(α 0 , α 1 , β 1 ; r n )∂α 1= − 1 2n∑t=1r 2 t−1σ 2 t+ 1 2n∑t=1rt 2 rt−12 , (3.26)σt4∂l(α 0 , α 1 , β 1 ; r n )∂β 1= − 1 2n∑t=1σ 2 t−1σ 2 t+ 1 2n∑t=1rt 2 σt−12 , (3.27)σt4Recalling that σ 2 t = α 0 + α 1 r 2 t−1 + β 1 σ 2 t−1 and equating equations (3.25) , (3.26)and (3.27) to zero, we obtain the system of three (3) equations with three (3)unknowns.n∑(t=1)1ˆα 0 + ˆα 1 rt−1 2 + ˆβ=1 σt−12n∑()rt2( ˆα 0 + ˆα 1 rt−1 2 + ˆβ, (3.28)1 σt−1) 2 2t=1n∑(t=1r 2 t−1ˆα 0 + ˆα 1 r 2 t−1 + ˆβ 1 σ 2 t−1)=n∑()rt 2 rt−12( ˆα 0 + ˆα 1 rt−1 2 + ˆβ, (3.29)1 σt−1) 2 2t=1n∑( ˆα0 + ˆα 1 rt−2 2 + ˆβ )1 σt−22ˆα 0 + ˆα 1 rt−1 2 + ˆβ=1 σt−12t=1n∑t=1( r2t ( ˆα 0 + ˆα 1 rt−2 2 + ˆβ )1 σt−2)2( ˆα 0 + ˆα 1 rt−1 2 + ˆβ. (3.30)1 σt−1) 2 2where σt−1 2 and σt−2 2 can be expressed in terms of the log-returns only, given thatsome initial values for σ1 2 and σ0 2 are given.The maximum likelihood estimator of∑σt 2 is σt 2 = 1 nn i=1 r2 i . Alternatively, the above sums can start from t = 2 or t = 3. The above system can be solved for ˆα 0 , ˆα 1 , ˆβ 1 using numerical methods and itcan be verified that the Hessian matrix evaluated at α 0 = ˆα 0 , α 1 = ˆα 1 , β 1 = ˆβ 1defined by⎛⎞H =⎜⎝∂ 2 l∂α 2 0∂ 2 l∂α 1 α 0∂ 2 l∂β 1 α 0∂ 2 l∂α 0 α 1∂ 2 l∂α 2 1∂ 2 l∂β 1 α 1∂ 2 l∂α 0 β 1∂ 2 l∂α 1 β 1∂ 2 l∂β 2 1α 0 = ˆα 0 , α 1 = ˆα 1 , β 1 = ˆβ 1⎟⎠

27is a negative definite matrix and so ˆα 0 , ˆα 1 , ˆβ 1 are the maximum likelihood estimatesof α 0 , α 1 , β 1 . However, since the conditional variance of the GARCH(1, 1)model depends also on the past conditional variance, an initial value of the pastconditional variance, σ 2 1 is needed. The unconditional variance of r t can be takenas an initial value for this variance, that is, σ 2 1can be taken to beα 01−α 1 −β 1.Sometimes the sample variance of the return series can be taken to be the initialvalue.3.3.2 The GARCH(p, q) ModelThe GARCH(p, q) is a generalization of GARCH(1, 1) with p as the autoregressivelag p and lag q is the moving average lag. Formally a process { }r t is GARCH(p, q)ifr t = σ t ε t ,q∑p∑σt 2 = α 0 + α i rt−i 2 + β j σt−j,2i=1j=1= α 0 + α(B)rt 2 + β(B)σt 2 . (3.31)where ε t is a Gaussian white noise, while α(B) and β(B) are polynomials in thebackshift operator given byα(B) = α 1 B + . . . + α q B q ,andβ(B) = β 1 B + . . . + β p B p ,with the restrictions α 0 > 0, α i ≥ 0 and β j ≥ 0 for i = 1, 2, . . . , q and j =1, 2, . . . , p, being imposed in order to have the conditional variance remainingpositive. Equation (3.31) can be expressed as(1 − β(B))σ 2 t = α 0 + α(B)r 2 t .

28Note that GARCH(0, q) is the same as an ARCH(q) model and that for p =q = 0 we have a GARCH(0, 0) model, which is a simple white noise. Similarto the ARCH(q) model, the conditional mean of {r t } is zero, that is E(r t |H t )implies the series {r t } is a martingale difference and observing {r t } is uncorrelatedGourieroux, et al., (1997). Assuming the GARCH(p, q) process is second orderstationary, that isV ar(r t ) = E(rt 2 ),= E(σt 2 ε 2 t ),= E(σt 2 E(ε 2 t |r t−1 )),= E(α 0 + α(B)rt 2 + β(B)σt 2 ),= α 0 + α(B)E(rt 2 ) + β(B)E(σt 2 ),= α 0 + α(B)σt 2 + β(B)σt 2 ,=α 01 − ∑ qi=1 α i − ∑ pj=1 β j. (3.32)The autocovariance of a GARCH(p, q) model for k ≥ 1 where k is the lag,E(r t r t−k ) = 0,since r t is a martingale difference Gourieroux, et al., (1997). Thus the GARCH(p, q)model does not show autocorrelation in the return series {r t } .However thesquared returns show autocorrelation even though the returns are not correlated.Considering writing r 2 t in terms of v t = r 2 t − σ 2 t , yieldsrt 2 = σt 2 + v t ,q∑p∑= α 0 + α i rt−i 2 + β j (rt−j 2 − v t−j ) + v ti=1j=1q∑p∑p∑= α 0 + α i rt−i 2 + β j rt−j 2 − β j v t−j + v t . (3.33)i=1j=1j=1

29Now let m =max(p, q) thenr 2 t = α 0 +q∑p∑(α i + β i )rt−j 2 − β j v t−j + v t , (3.34)i=1j=1where α i = 0 for i > q and β j = 0 for i > p . Thus the equation of r 2 t hasARMA(m, p) representation. In order to find the GARCH(p, q) process, we considersolving for α 0 in equation (3.34) and let the variance of r t be σ 2 εwhichyieldsα 0 = σ 2 ε(1 −q∑ p∑α i − β j ), (3.35)i=1 j=1and substituting the value of α 0 of equation (3.35) into equation (3.34) givesm∑m∑p∑rt 2 = σε(1 2 − (α i + β j )) + ( (α i + β j ))rt−j 2 − β j v t−j + v t ,i,j=1i,j=1j=1= σε 2 +Therefore,m∑p∑(α i + β j )(rt−j 2 − σε) 2 − β j v t−j + v t . (3.36)i,j=1m∑p∑rt−k 2 − σε 2 = (α i + β j )(rt−j 2 − σε) 2 − β j v t−j + v t . (3.37)j=1i,j=1j=1Multiplying both sides by (r 2 t−k − σ2 ε), yields(rt−k 2 − σ2 ε)(rt 2 − σε) 2 = ∑ mi,j=1 (α i + β j )(rt−i 2 − σε)(r 2 t−k 2 − σ2 ε)−∑ pj=1 β jv t−j ((rt−k 2 − σ2 ε) + v t (rt−k 2 − σ2 ε),(3.38)and taking expectationsE[(r 2 t−k − σ2 ε)(r 2 t − σ 2 ε)] = E[ ∑ mi,j=1 (α i + β j )(r 2 t−i − σ 2 ε)(r 2 t−k − σ2 ε)]−E[ ∑ pj=1 β jv t−j ((r 2 t−k − σ2 ε)] + E[v t (r 2 t−k − σ2 ε)].(3.39)

30ButE[v t (r 2 t−k − σ 2 ε)] = E[(r 2 t−k − σ 2 ε)E(v t |r t )] = 0,since v t is a martingale difference and alsoE[β j v t−j ((r 2 t−k − σ 2 ε)] = E[(r 2 t−k − σ 2 ε)E(v t−j |r t−k )] = 0. (3.40)for k < j. . Thus the autocovariance of the squared returns for the GARCH(p, q)model is given bym∑cov(rt 2 , rt−k) 2 = E[ (α i + β j )(rt−i 2 − σε)(r 2 t−k 2 − σε)],2i,j=1m∑= (α i + β j )cov(rt 2 , rt−k+i), 2 (3.41)i,j=1Dividing both sides by var(r 2 t ) gives the autocorrelation function at lagk asρ k =m∑(α i + β i )ρ k−i . (3.42)i=1for k ≥ (p + 1). This result is analogous to the Yule-Walker equations for an ARprocess. Thus the autocorrelation function (ACF) and the partial ACF (PACF)of the squared returns in a GARCH process has the same pattern as those of anARMA(m, p) process. As in ARMA modelling the ACF and PACF are useful inidentifying the orders p and q of the GARCH(p, q) process. The ACF are alsoimportant for checking model accuracy, in which case, the ACF‘s of residualsshould be indicative of a white noise if the model is adequate. Thus the firstautocorrelations depend on the parameters α 0 , α 1 , . . . , α q ; β 1 , β 2 , . . . , β q , but givenρ p , . . . , ρ p+1−m , , the expression in (3.39) determines uniquely the autocorrelationsat higher lags, (Bollerslev, 1986 ).Thus letting φ mm denotes the m th partialautocorrelation for r 2 t , thenρ k =m∑φ mm ρ k−i , k = 1, . . . , m.i=1

31By equation (3.39) φ mm cuts off after lag q for an ARCH(q) process such thatφ mm ≠ 0 for k ≤ q and φ mm = 0 for k > q . This is identical to the PACF for anAR(q) process and decays exponentially (Bollerslev, 1986). After identifying theorders p and q , we can now estimate the parameters (α 0 , α 1 , . . . , α q , β 1 , . . . , β q ),of the GARCH(p, q) model as explained in section 3.3.3 .3.3.3 Estimation of GARCH(p, q) ModelThe maximum likelihood estimation is also applicable in estimating the parametersof the model. As in the GARCH(1, 1) model estimation, initial values of boththe squared returns and past conditional variances are needed in estimating theparameters of the model. As suggested by Bollerslev, (1986) and Tsay, (2002),the unconditional variance given in equation (3.31) or the past sample varianceof the returns for the past variances may be used as initial values. Therefore assumingr 1 , . . . , r q and σ1, 2 . . . , σp 2 are known, the conditional maximum likelihoodestimates can be obtained by maximizing the conditional log-likelihood given byl = logf(r q+1 , . . . , r t , σ 2 p+1, . . . , σ 2 t |θ, r 1 , . . . , r q , σ 2 1, . . . , σ 2 p),= − 1 2T∑t=m+1log(2πσ 2 t ) − 1 2T∑t=m+1{ r2twith, θ = (α 0 , α 1 , . . . , α p ; β 1 , β 2 , . . . , β q ) and m =max(p, q).σ 2 t}.3.3.4 Model CheckingGoodness of fit of the ARCH-GARCH model is based on residuals and is morespecifically on the standardized residuals (Takle, 2003).The residuals are assumedto be independently and identically distributed following either a normal

32or standardized t−distribution (Tsay, 2002) and (Gourieroux, 2001). Plots of theresiduals such as the histogram, the normal probability plot and the time plotof the residuals can be used. If the model fits the data well the histogram ofthe residuals should be approximately symmetric. The normal probability plotshould be a straight line while the time plot should exhibit random variation.The ACF and the PACF of the standardized residuals are used for checking theadequacy of the conditional variance model. The Engle’s test and the Ljung BoxQ−test are used to check the validity of the ARCH effects in the data. Havingestablished that our model fits the data well, we can now use the fitted model tocompute forecasts.3.3.5 Forecasting with GARCH(p, q) modelsAs we have outlined, the GARCH and the ARMA models are similar, such thatforecasting using the GARCH model is the same as using the ARMA model.Thus the conditional variance of {r t } is obtained simply by taking the conditionalexpectation of the squared mean corrected returns. Assuming a forecasting originof T, then the l−step ahead volatility forecast is given byr 2 t (1) = E[r 2 t+1|r t ],= α 0 +m∑(α i + β i )E(rt+l−i|r 2 t ) −i=1p∑β i (v t+l−i |r t ).i=1where rt 2 , . . . , rt+l−m 2 , σ2 t , . . . , σt+1−p 2 are assumed known at time t and the trueparameter values α i and β i for i = 1, . . . , m are replaced by their estimates. Furthermore,the l−step ahead forecast of the conditional variance in a GARCH(p, q)model is given byσt 2 (l) = E[rt+1|r 2 t ],

33= α 0 +m∑(α i + β i )E(rt+l−i|r 2 t ) −i=1p∑β i (v t+l−i |r t ).i=1where E[rt+1|r 2 t ] for i < l can be given recursively as for i ≥ l, E(v t+l−i |r t ) = 0for i < l, E(v t+l−i |r t ) = v t+l−i for i ≥ l We now consider the techniques that areused for selecting the best fitting model in line of two or more competing modelsbased on the likelihood ratios.3.4 Model selection criteriaIn statistical modelling, one of the main challenges is to select a suitable modelfrom a candidate family to characterize the underlying data.Model selectioncriteria provide useful tools in this regard. Selection criteria assess whether afitted model offers an optimal balance between goodness-of-fit and parsimony.Ideally, a criteria will identify candidate models that are either too simplisticto accommodate the data or unnecessarily complex. The most common modelselection criteria are the AIC and the BIC.3.4.1 The Akaike Information CriterionThe Akaike information criterion (AIC) was the first model selection criterionto gain widespread acceptance. AIC was introduced in 1973 by Hirotogu Akaikeas an extension to the maximum likelihood principle. Conventionally, maximumlikelihood is applied to estimate the parameters of a model once the structureof the model has been specified. Akaikes seminal idea was to combine estimationand structural determination into a single procedure. The minimum AIC

34procedure is employed as follows. Given a family of candidate models of variousstructures, each model is fit to the data through maximum likelihood. An AIC iscomputed based on each model fit. The fitted candidate model corresponding tothe minimum value of AIC is then selected. AIC serves as an estimator of Kullback’sdirected divergence between the generating, or true, model (that is, themodel that presumably gave rise to the data) and a fitted candidate model. Thedirected divergence assesses the disparity or separation between two statisticalmodels. Thus, when entertaining a family of fitted candidate models, by selectingthe fitted model corresponding to the minimum value of AIC, one is hoping toidentify the fitted model that is closest to the generating model (Salkind, 2007).The AIC is defined byAIC = −2(loglikelihood) + 2(N).where N denotes the number of observations3.4.2 The Bayesian information criterionThe Bayesian information criterion (BIC) is related to the Bayes factor andis useful for model comparison in its own right. The BIC of a model is defined asBIC = −2ln(likelihood) + (k + klnN).where k denotes the number of parameters and N denotes the number of observationsor equivalently, the sample size. BIC penalizes more complex models(those with many parameters) relative to simpler models. This definition permitsmultiple models to be compared at once; the model with the highest posteriorprobability is the one that minimizes BIC.A desirable model is one that minimizes the AIC or the BIC. The other criteria arethe R 2 associated with the model which is the proportion of variability in a data

35set that is accounted for by the statistical model (Salkind, 2007 ). However, asHarvey (1991) indicated that the coefficient of determination (R 2 ) has a limitationin that, a model which can pick out the trend reasonably well, will have R 2 closeto unit. In general a model selected by two different criteria mentioned above maydiffer and thus it should be emphasized that the selection of an ARCH-GARCHmodel depends on the selection criteria used (Talke, 2003 ).3.5 Forecasting performanceAs pointed out in section 3.2.3, forecasting is one of the most important objectivesof time series modelling, thus one of the criteria for selecting the best model canbe based or centred on the best forecasting model.Forecast in this contextmeans the estimates of the conditional variances obtained from models. Thereare several measures for assessing the predictive accuracy of an ARCH-GARCHmodel. Among these is the mean square error (MSE). The MSE is defined asthe average of the squared difference between the actual variance and volatilityforecast σt 2 . In the absence of the observed true variance the squared time seriesobservations rt 2 is used. The MSE is thus given byMSE = 1 T∑ ( )r2 2,T t − ˆσ t2 (3.43)t=1where ˆσ t 2 , for t = 1, . . . , T is the estimated conditional variance obtained fromfitting ARCH-GARCH model. The MSE is critized, as in Tsay (2002), in that,although the r 2 tis a consistent estimator of σ 2 t , it is nevertheless noisy, thereforeunstable. Among the alternative measures are the mean absolute error (MAE)by Lopez (1999) defined byMAE = 1 TT∑ ∣∣rt 2 − ˆσ t2 ∣ 2 , (3.44)t=1

36and the MSE of the log of the squared error defined byMSLE = 1 TT∑ (ln(r2t ) − ln(ˆσ t 2 ) ) 2. (3.45)t=1The advantage of the MSE of the log of the squared error is that it penalizesinaccurate variance forecasts more heavily when the squared innovations, r 2 tarelow. Another criterion which is popular is Theil‘s U−statistic. The Theil‘s Utest which is used to test accuracy of the future forecasts/predictions is definedas (Diebold and Lopez, 1995)whereU =is forecasted relative change, andis actual relative change.( ∑T) 1−1t=1 (F P E t+1 − AP E t+1 ) 2 2∑ T −1t=1 (AP E . (3.46)t+1) 2F P E t+1 = ( ˆX t+1 − X t )X tAP E t+1 =( )Xt+1 − X tX tIf the forecasts are good then U should be close to zero. A U−statistic of oneimplies that the model under consideration and the benchmark model are equally(in)accurate, while a value of less than one implies that the model is superior tothe benchmark, and vice versa for U > 1.

CHAPTER FOURDATA ANALYSIS AND METHODOLOGY4.1 IntroductionHaving explored the general theory of ARCH-GARCH models in the precedingchapter, this chapter is dedicated to fitting the GARCH family of models tothe Tanzania inflation rate data which we obtained courtesy of the TanzaniaNational Bureau of Statistics. A description of the data is given in section 4.2and application of GARCH in real life data is given in section 4.3 through 4.7. Insection 4.2 we investigate the general statistical features of the given time series:Inflation rate data based on Tanzania.4.2 Data description and basic statisticsThe data employed in this study comprise of 168 monthly observations of theTanzania inflation rates spanning from January 1, 1997 to December 31, 2010. InTanzania the Consumer Price Index (CPI) is calculated by the National Bureauof Statistics under the auspices of the Ministry of Finance. The national CPIcovers prices collected from twenty towns in Tanzania mainland. All prices arethe prevailng market prices and are gathered for two hundred and seven items.Table 4.1 gives the descriptive statistics for monthly consumer prices and returns,where Std. Dev. stands for Standard Deviation. The inflation series which isthe basic data in this work, has a constant general mean of 8.1, the standard37

38deviation of 3.7, positive skewness of 0.8138 implying that the distribution has along right tail and kurtosis of 2.8019. The minimum value in this data is 0.2 andthe maximum value is 17.2 , giving a data range of 17.0. On the other hand, thereturn series has a negative skewness of −0.3811 implying that the distributionhas a long left tail. Skewness is a measure of symmetry. For univariate data setsX 1 , X 2 , . . . , X N , the formula for Skewness is: Skewness =∑ Ni=1 (X i− ¯X) 3(N−1)s 3, where ¯Xis the mean, s is the standard deviation, and N is the number of data points.Table 4.1: Descriptive Statistics for the Inflation rate series.PriceReturnMean 8.0500 1.9773Median 6.9000 1.9314Std. Dev. 3.6475 0.5023Kurtosis 2.8019 6.9162Skewness 0.8138 -0.3811Maximum 17.2 2.6391Minimum 0.2 -1.6094Jarque-Bera 55.8432 53.8218ADF(Level) -0.576881ADF(1st diff) -11.642264The value for kurtosis is high (close to three), so, the distribution are peakedrelative to normal.Kurtosis is a measure of whether the data are peaked orflat relative to a normal distribution. That is, data sets with high kurtosis tendto have a distinct peak near the mean, decline rather rapidly, and have heavy

39tails. Data sets with low kurtosis tend to have a flat top near the mean rathera sharp peak. For univariate data X 1 , X 2 , . . . , X N , the formula for Kurtosis is:Kurtosis =∑ Ni=1 (X i− ¯X) 4(N−1)s 4where ¯X is the mean, s is the standard deviation andN is the number of data points. The Kurtosis for standard normal distributionis three. For this reason, ExcessKurtosis =∑ Ni=1 (X i− ¯X) 4(N−1)s 4− 3. The Jarque-Beratest rejects the null hypothesis at 5% level that the distribution is normal. So,the sample has all financial characteristics, that is, volatility clustering and leptokurtosis.Furthermore, in terms of stationary the results from the AugmentedDickey Fuller(ADF) test in Table 4.1 indicates that the series has unit root andtherefore time series models can be used to examine the behavioour of volatilityover time.4.3 TransformationsAs stated earlier most applied time series are non stationary, their variance ischanging with time.Consider, for example, a plot of the CPI series for theFigure 4.1:Time plot of monthly inflation in Tanzania

40period January 1997 through December 2010 in Figure 4.1 , it is evident thatvariances are changing with time. The inflation series we have in this study isnon-stationary. Therefore we convert the prices to returns by logarithmic transformations.The logarithmic returns is based upon the following mathematicaldefinition, that is,wherer t = ln P tP t−1• r t is the return for any time, t,• P t is the Consumer Price Index value at time , t,• P t−1 is the Consumer Price Index value at time , t − 1.Figure 4.2: First Difference of Log of CPIFigure 4.2 illustrates the plot of return series converted from original data. Thereturns appear to be quite stable overtime and the transformation has produceda stationary process.This behaviour of inflation series returns is in line withmost financial theories and models which usually assume the prices to returns,to be stationary process. This is also noticed in Figure 4.9(b) where the normal

41probability plot shows a nearly straight line suggesting that the residuals followan approximately normal distribution.4.4 Testing for ARCH effects and Serial correlationin the return seriesThis section uses a formal statistical test to establish the presence of ARCH effectsin the data. The test for heteroscedasticity has been done using Ljung-Box-PierceQ − test and Engle’s ARCH test. Table 4.2 shows the Ljung-Box-Pierce Q − testin the form of binary number that is 0 and 1. H = 0 implies that no significantcorrelation exists. H = 1 means that significant correlation exists. From Table4.2 it can be verified that there is no significant correlation in the raw returnswhen tested for up to 10, 15 and 20 lags of the ACF at the 5% level of significance.However, there is significant serial correlation in the squared returns in Table 4.3and Table 4.4, all the p − values show that the Q − test and the ARCH test atlag 10, lag 15 and lag 20 are significant showing presence of ARCH effects.Table 4.2: Ljung-Box-Pierce Q-test for autocorrelation: (at 95% confidence).Lag H p − value Stat CriticalV alue10 0.0000 0.0721 17.1019 18.307015 0.0000 0.2359 18.529 24.995820 0.0000 0.3888 21.1416 31.4104The dependence in the data x 1 , . . . , x n are ascertained by computing correlationsfor data values at varying time lags. This is done by plotting the sample auto-

42Table 4.3: Ljung-Box Q-test for squared returns: (at 95% confidence).Lag H p − value Stat CriticalV alue10 1.0000 0.0000 57.3782 18.307015 1.0000 0.0000 76.7057 24.995820 1.0000 0.0000 82.7525 31.4104Table 4.4: Engle ARCH test for heteroscedasticity: (at 95% confidence).Lag H p − value Stat CriticalV alue10 1.0000 0.0000 68.6467 18.307015 1.0000 0.0000 67.9727 24.995820 1.0000 0.0000 66.2913 31.4104correlation function (ACF):ACF (h) = ρ(h) = γ(h)γ(0) ,against the time lags h = 0, 1, . . . , n − 1 and where ρ(h) is the sample autocovariancefunction (ACVF) given by:ACV F (h) = ρ(h) = 1 ∑n−h(x t+h − ¯x)(x t − ¯x),nand ¯x is the sample mean. If the time series is an outcome of a ‘completely’random phenomenon, the autocorrelation should be near zero for all time-lagseparations. Otherwise, one or more of the autocorrelations will be significantlynon-zero.t=1Another useful method to examine serial dependencies is to examine the partial

43Figure 4.3: ACF with Bounds for Raw Return SeriesFigure 4.4: PACF with Bounds for Raw Return Seriesautocorrelation function (PACF) ( an extension of autocorrelation),where thedependence on the intermediate elements (those within the lag) is removed. Thepartial autocorrelation is similar to autocorrelation, except that when calculatingit, the autocorrelation with all the elements within the lag are eliminated.In Figure 4.3 and Figure 4.4 the ACF and PACF provide no indication of thecorrelation characteristics of the returns. The ACF of squared returns in Figure4.5 show significant correlation and die out slowly. This results indicates thatthe variance of returns series is conditional on its past history and may change

44Figure 4.5: ACF of the squared Returnsover time. Each of these tests extracts the sample mean from the actual inflationseries. This is consistent with the definition of the conditional mean equation ofthe default model, in which the innovations process is ε t = r t − C and C is themean of r t .4.5 Model estimation and evaluation4.5.1 Model selection and analysisThe strategy used in selecting the appropriate model from competing modelsis based on the Akaike information criterion (AIC), the Bayesian informationcriterion (BIC), standard error (SE) and on the significance tests. MAT LABsoftware is used to perform trial and error to determine the best fitting model. Table4.5 gives the suggested models with their respective fit statistics. The idea isto have a parsimonious model that captures as much variation in the data as possible.Usually the simple GARCH model captures most of the variability in moststabilized series. Small lags for p and q are common in applications. Typically

45GARCH(1, 1), GARCH(2, 1) or GARCH(1, 2) models are adequate for modellingvolatilities even over long sample periods (Bollerslev, Chou and Kroner, 1992).However in the Table 4.5 below we have included GARCH(1, 0), GARCH(0, 2) andGARCH(2, 2) in order to check if they are appropriate for modelling time varyingvariances of our data. In the Table 4.5 below the smaller the AIC and BIC thebetter. Larger AIC ′ s, BIC ′ s and standard error makes the model unfavourable.The model given in bold is judged to be the most appropriate according to thecriteria above.Table 4.5: Comparison of suggested GARCH models.Model AIC BIC SE Log − LikelihoodGARCH(0, 1) 489.0200 498.3560 0.1026 241.5100GARCH(1, 1) 474.8236 487.2715 0.0544 233.4118GARCH(0, 2) 478.5589 491.0068 0.0860 235.2794GARCH(1, 2) 475.5820 491.1419 0.0863 232.7910GARCH(2, 1) 476.8236 492.3835 0.0735 233.4118GARCH(2, 2) 477.1047 495.7766 0.0654 232.5524The Table 4.5 shows the competing models to the data with their respectiveAIC, BIC and SE. From our derived models, using the method of maximumlikelihood, the estimated parameters of the models with their corresponding standarderror and other statistical tests are shown in the Table 4.6, 4.7 and 4.8. Westarted in Table 4.6 by fitting a GARCH(1, 1) to the data.

46Table 4.6: Parameter estimates for GARCH(1,1).Parameter C K GARCH(1) ARCH(1)Estimates 0.0272 0.0753 0.5427 0.4573Standard Error 0.0283 0.0254 0.0681 0.0958t − value 0.9627 2.9656 6.7136 5.6619Pr(> |t|) 0.8315 < 0.0001 < 0.0001 < 0.0001r t = 0.0272 + ε t , (4.1)σ 2 t = 0.0753 + 0.45730r 2 t−1 + 0.54266σ 2 t−1. (4.2)The standard errors are used to assess the accuracy of the estimates, the smallerthe better. The model fit statistics used to assess how well the model fit thedata are the AIC and BIC. The corresponding values are: AIC = 474.8 andBIC = 487.3 with the log likelihood value of 233.4. The standard errors arequiet small suggesting precise estimates. Based on 95% confidence level, the coefficientsof the GARCH(1, 1) model are significantly different from zero and theestimated values satisfy the stability condition, that is, α 1 +β 1 < 1, with β 1 > α 1 .The second model to be considered is the GARCH(1, 2). This model is chosenbecause it seemed to fit the data well, as it has small log likelihood function,small standard error and both their AIC and BIC are very close to that ofGARCH(1, 1). However the model was found to be violating α i +β j < 1, i, j ∈ Z +as α 1 = 0.34705, α 2 = 0.01764, β 1 = 0.47655, that is α 1 + β 1 + β 2 = 1, whereα i + β j < 1 is a necessary condition for the model to be stationary. The correspondingfit statistics are: AIC = 475.6 and BIC = 491.1 with log likelihood

47Table 4.7: Parameter estimates for GARCH(1,2).Parameter C K GARCH(1) ARCH(1) ARCH(2)Estimates 0.0198 0.0892 0.4766 0.3471 0.1764Standard Error 0.0422 0.0290 0.0915 0.1219 0.1468t − value 0.4706 3.0819 3.7920 3.9109 1.2013function of 232.8. Thus the GARCH(1, 2) has bigger AIC and BIC values thanthe stochastic volatility GARCH(1, 1) model, indicating that, GARCH(1, 1) is superiorin extracting statistical information from the data hence accommodatingconditional variances. The parameter estimates given in Table 4.7 giving rise tothe model,r t = 0.0198 + ε t , (4.3)σt 2 = 0.0892 + 0.3471rt−1 2 + 0.04766σt−1 2 + 0.1764σt−2. 2 (4.4)A corresponding stochastic GARCH(2, 1) model was also seen not to fit the datawell, as the α 2 coefficient of the GARCH(2, 1) was not significant, also , the stationaritycondition was violated.Another candidate model fitted to data and tested was the GARCH(2, 2) . Theanalysis showed a non significant α 1 coefficient, this imply that the ARCH(1) parameteradds little explanatory power to the model. The fit statistics are: AIC =477.1 and BIC = 495.8 which are also greater than the GARCH(1, 1) model, suggestingthat the GARCH(2, 2) is less appropriate than the GARCH(1, 1) model.Note also that the mean standard error under GARCH(2, 2) is bigger as comparedto the stochastic volatility GARCH(1, 1) model.

48Table 4.8: Parameter estimates for GARCH(2,2).Parameter Estimates Standard Error t − valueC 0.0205 0.0418 0.4864K 0.1055 0.0527 3.2315GARCH(1) 0.4543 1.5e-17 0.0015GARCH(2) 0.3661 0.0572 3.1373ARCH(1) 2.204e-20 0.1281 0.3.5456ARCH(2) 0.1796 0.1124 3.2573At this point in time series, it can be established that, amongst all the identifiedmodels, the GARCH(1, 1) is proven to be the best fit model. Section 4.5.2 belowprovides a comprehensive comparison of the identified models.4.5.2 A Comparison of the fitted GARCH modelsTable 4.5 shows the competing GARCH models together with their correspondingestimates, standard errors, significant statistical tests, AIC and BIC for eachparameter. The GARCH(1, 1) has the smallest AIC = 474.8 and BIC = 487.3and all parameters are significant. The standard error of the GARCH(1, 1) isthe smallest of the all models, suggesting that the model parameters are precise.Therefore this model is selected to be the best fit of all the models fitted. Theother models have greater AIC, BIC and standard errors, therefore they wereprovided only for comparison purposes. Having constructed our model, the nexttask is to assess how well the model fits the data. This is done through residual

49analysis or model checking as explained in section 4.64.6 Diagnostic checking of the GARCH(1, 1) modelAs was explained in section 3.3.4 , model checking is done through analyzingthe residuals from the fitted model. In time series modelling, the selection ofthe best model fit to the data is directly related to whether residual analysis isperformed well. One of the assumptions of GARCH model is that, for a goodmodel, the residuals must follow a white noise process, that is if the model fits thedata well, the residuals are expected to be random, independent and identicallydistributed following the normal distribution. Figure 4.6 inspects the relationshipbetween the innovations (residuals) derived from the fitted model, the correspondingconditional standard deviations and the observed returns. We can see thatboth innovations (top plot) and returns (bottom plot) in the Figure 4.6 exhibitvolatility clustering.Figure 4.6: Plot of Return, Estimated Volatility and Innovations (residuals)However if we plot the, standardized innovations (the innovations divided by their

50conditional standard deviation), they appear generally stable with little clusteringas in Figure 4.7.The time plot of the residuals given in Figure 4.7 is used to checkwhether the residuals are random. The normality check is also done by analyzingthe histogram of residuals and normal probability plot. Figure 4.9(a) gives thehistogram of the residuals from the GARCH(1, 1) model. The histogram showsalmost a symmetric bell-shaped distribution which is indicative of the residualsfollowing a normal distribution. The slight negative skewness is expected sincethe residuals may also come from student‘s t distribution. The negative skewnesstendency is also supported by negative large residuals in Figure 4.7. The normalitycheck of residuals can also be done by the normal probability plot. As we saw insection 3.3.4 , if the residuals come from the normal distribution the plot shouldbe a straight line. Figure 4.9(b) gives the probability plot of the residuals. Theresiduals are indeed normally distributed the graph Fig 4.9(b) shows a nearlystraight line were almost all the points has fallen along the dashed line.Figure 4.7: Time plot of residuals from GARCH(1,1)Moreover if the model is successful at modelling the serial correlation structurein the conditional mean and conditional variance, then there should be no autocorrelationleft in the standardized residuals and squared standardized residuals.

51Figure 4.8: ACF of the squared standardized residualsFigure 4.8 gives the plot of the ACF of the squared standardized innovations;the plot shows no correlation left. Table 4.9 and 4.10 compared the results of theQ−test and the ARCH test with the results of the same tests in the pre-estimationanalysis in Table 4.2 and 4.3 respectively. In the pre-estimation analysis, both theQ−test and the ARCH test indicated rejection (H = 0 with p value = 0) of theirrespective null hypothesis showing significant evidence in support of GARCH effects.In the post estimation analysis using standardized innovations based onthe estimated model, these same tests indicate acceptance (H = 0 with highlysignificant p−values) of their respective null hypothesis and confirm the explanatorypower of GARCH(1, 1). The tests showed that no any ARCH effects left (noheteroscedasticity). Also from t − test result Table 4.9 , since the p − values of0.9964, 0.9997, 1.0000 are greater than 5% alpha level, we fail to reject the nullhypothesis that, there is no autocorrelation left in the residuals. Therefore weproceed to use the model to forecast future values of the inflation series.

52Table 4.9: Ljung-Box-Pierce Q-test on standardized innovations.Lag H p − value Stat CriticalV alue10 0.0000 0.99412 2.2420 18.307015 0.0000 0.9992 3.3378 24.995820 0.0000 1.0000 4.0144 31.4104Table 4.10: Engle ARCH test on standardized innovations.Lag H p − value Stat CriticalV alue10 0.0000 0.9964 1.9937 18.307015 0.0000 0.9997 2.8363 24.995820 0.0000 1.0000 3.5752 31.41044.7 Forecasting with the GARCH(1, 1) modelOne of the main objectives of this study and time series analysis in general isto use the constructed model to compute forecasts. In time series, forecastingis a mathematical way of estimating future values using present and historicalvalues of the series (Aidoo, 2010). Forecasting as described by Box and Jenkins(1976) provide basis for economic and business planning, inventory and productioncontrol and optimization of industrial process. From previous studies, mostresearch work has found that the selected model is not necessary the model thatprovides best forecasting. In this sense further forecasting accuracy test such asMAE, RMSE and MSE must be performed.

534.7.1 Forecast Evaluation and Accuracy CriteriaEmpirically taking, we have examined that Table 4.12 reports the various measuresof forecasting errors, namely the mean absolute error (MAE), the rootmean squared error (RMSE), and Thieles U for two models seemed to be adequatelyfit the data. The first two forecast error statistics depend on the scale ofthe dependent variable. These are used as relative measure to compare forecastsfor the same series across different models, the smaller the error the better theforecasting ability of that model accordingly. The remaining two statistics arescale invariant. The Theil inequality coefficient always lies between zero and one,where zero indicates a perfect fit. From the results, it can be seen that, mostTable 4.11: Forecast Accuracy Test on the most likely suggested GARCH models.Model MSE MAE RMSE Thiel‘s U − testGARCH(1, 1) 0.5848 0.7483 0.8651 0.8528GARCH(1, 2) 0.6034 0.9812 0.7768 0.9821of the accuracy test favour GARCH(1, 1) model. Also the Thieles statistics isless than one (0.8528 as from Table 4.11 ) which indicates that, the forecastsare fairly accurate. To measure the forecasting ability we have estimated withinsample forecasts. The purpose of forecasting within the sample is to test for thepredictability power of the model. If the magnitude of the difference between theforecasted and actual values is small then the model has good forecasting power.In our case GARCH(1, 1) has shown good results as evident from Table 4.12 .One can observe from the Figure 4.9 that the forecasted series are closer to theaccuracy series. Therefore it can be concluded that the prediction power of themodel is better and suitable for twelve periods forecasting.

54Table 4.12:One year in-sample forecasts of Inflation obtained from theGARCH(1,1).Month Forecast(%) Observed Value(%) Forecast ErrorJanuary 11.32 10.9 0.42February 10.43 9.6 0.83March 9.66 9.0 0.66April 10.08 9.4 0.68May 7.90 7.2 0.70June 8.74 7.9 0.84July 7.29 6.3 0.99August 7.48 6.6 0.88September 5.03 4.5 0.53October 4.81 3.9 0.91November 5.14 4.3 0.84December 6.30 5.6 0.70The Figure 4.10 displays the original inflation rate and the predicted values producedby the GARCH(1, 1) model. The figure also displays how the forecastedvalues behave. It can be confirmed that the model somehow fit the data well.

55Figure 4.9: (a)Histogram of residuals(top) and (b)Normal probability(bottom)plot from GARCH(1,1)

56Figure 4.10: Time plot of Inflation rate and one year forecasts from GARCH(1,1)

CHAPTER FIVEDISCUSSION, CONCLUSION AND FUTURE WORK5.1 DiscussionThe GARCH(1, 1) model has been fitted. From this model we get α 0 = 0.0753, α 1 =0.45730 and β 1 = 0.54266 with small standard error 0.05441 . Ljung Box-PierceQ−test is applied to test the uncorrelatedness of the data, Engles test is appliedto test for the presence of heteroscedasticity. A forecasting plot has been drawn,and both AIC = 474.8236 and BIC = 487.2715 has been obtained. According tothe GARCH(1, 1) model fitted α 1 reflects the influence of random deviations inthe previous period on σ t , where as β 1 measures the part of the realized variancein the previous period that is carried over into the current period.The sizesof α 1 and β 1 determine the short-run dynamics of the resulting volatility timeseries. Large GARCH coefficient, α 1 mean that volatility reacts intensely to theinflation rate movements. Large GARCH coefficient β 1 , indicates that, shocksto the conditional variance take long time to die out. Our results also show thatthere exist some persistent in volatility as measured by the sum of α 1 and β 1 inthe GARCH(1, 1) model.5.2 ConclusionThe study has presented us with an opportunity to have an extensive understandingof the theory of time series analysis in the area of non linear models57

58and its application to real life situation. The stages in the model building (thatis the identification, estimation and checking) strategy has been explored andutilized. Based on minimum AIC and BIC values, the best fit GARCH modelstend to be GARCH(1, 1) and GARCH(1, 2) . The GARCH(1, 1) model hassmaller AIC and BIC which is an indicative that it explains the data better thanGARCH(1, 2) model. After estimation of the parameters of selected models, aseries of diagnostic and forecast accuracy test were performed. Having satisfiedall the model assumptions, GARCH(1, 1) model was judged to be the best modelfor forecasting.5.3 Suggestions for future workAlthough the application in this research study is based on inflation data, futureresearch can also consider;• the multivariate GARCH, where other macro-economic variables such asexchange 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 geostatisticsjust to mention a few.

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APPENDIXMatlab Codes for the CPI data% reading data for CPIload CPIdata;returns=diff(log(CPIdata));plot([0:166],CPIdata,’LineWidth’,2);set(gca,’XTick’,[1 14 28 42 56 70 84 98 112 126 140 154 168 ]);set(gca,’XTickLabel’,’1997’ ’1998’ ’1999’ ’2000’ ’2001’ .. ’2010’);xlabel(’Time’);ylabel(’Rate(%)’);title(’Tanzania Montly Inflation Rate 1997:1-2010:12’);legend(’CPI data’);figure;plot(returns, ’LineWidth’,2);ylabel(’Return’);title(’Tanzania‘s Monthly Difference of Log of CPI-Inflation ’);axis tight;figure;plot(returns, ’LineWidth’,2);ylabel(’Return’);title(’Tanzania‘s Monthly Difference of Log of CPI-Inflation ’);axis tight;figure;% Pre-estimation Analysisautocorr(returns);title(’ACF with Bounds for Raw Return Series’);64

65figure;parcorr(returns);title(’PACF with Bounds for Raw Return Series’);figure;autocorr(returns.square);title(’ACF of the Squared Returns’);figure;% Ljung-Box-Pierce Q-test for ACF(H,pValue,Stat,CriticalValue) =lbqtest(returns-mean(returns),(101520)’,0.05);(H pValue Stat CriticalValue)% Ljung-Box-Pierce Q-test for squared ACF(H,pValue,Stat,CriticalValue) =lbqtest(returns-mean(returns).square,(10 15 20)’,0.05); (H pValue Stat Critical-Value)Comment:Heteroscedasticity Engle ARCH test(H,pValue,Stat,CriticalValue) =archtest(returns-mean(returns),(10 15 20)’,0.05); (H pValue Stat CriticalValue)% Estimating Parameters GARCH(1,1)(coeff,errors,LLF,innovations,sigmas,summary) = garchfit(returns);garchdisp(coeff,errors);spec11 = garchset(’P’,1,’Q’,1,’Display’,’off’);spec21 = garchset(’P’,2,’Q’,1,’Display’,’off’);(coeff21,errors21,LLF21) = garchfit(spec21,returns);garchdisp(coeff21,errors21);% Estimating garch(0,1)spec01 = garchset(’P’,0,’Q’,1,’Display’,’off’);(coeff01,errors01,LLF01) = garchfit(spec01,returns);

66garchdisp(coeff01,errors01);% Estimating garch(0,2)spec02 = garchset(’P’,0,’Q’,2,’Display’,’off’);(coeff02,errors02,LLF02) = garchfit(spec02,returns);garchdisp(coeff02,errors02);% Estimating garch(2,2)spec22 = garchset(’P’,2,’Q’,2,’Display’,’off’);(coeff22,errors22,LLF22) = garchfit(spec22,returns);garchdisp(coeff22,errors22);% Estimating garch(1,2)spec12 = garchset(’P’,1,’Q’,2,’Display’,’off’);(coeff12,errors12,LLF12) = garchfit(spec12,returns);garchdisp(coeff12,errors12);% specifying number of parameters for garch(2,1) and garch(1,1) modelsn21 = garchcount(coeff21);n11 = garchcount(coeff11);% the AIC and BIC for garch(1,1) modelformat long(AIC,BIC) = aicbic(LLF21,n21,166);(AIC BIC)% the AIC and BIC for garch(0,1) modelformat long(AIC,BIC) = aicbic(LLF11,n11,166);(AIC BIC)% the AIC and BIC for garch(0,1) modelformat long(AIC,BIC) = aicbic(LLF01,3,166);(AIC BIC)

67% the AIC and BIC for garch(0,2) modelformat long(AIC,BIC) = aicbic(LLF02,4,166);(AIC BIC)% the AIC and BIC for garch(1,2) modelformat long(AIC,BIC) = aicbic(LLF12,5,166);(AIC BIC)% the AIC and BIC for garch(2,2) modelformat long(AIC,BIC) = aicbic(LLF22,6,166);(AIC BIC)% Post estimation Analysisgarchplot(innovations,sigmas,returns);figure;% standardised innovationsstdresiduals=innovations./sigmas;mean(stdresiduals);plot(stdresidiuals, ’LineWidth’,2);ylabel(’Innovation’);title(’Standardised Innovations’);figure;histfit(stdresiduals);figure;normplot(stdresiduals);figure;% ACF of standardized innovationsautocorr((innovations./sigmas).square);

68title(’ACF of the Squared Standardized Innovations’);figure;% Ljung-Box-Pierce Q-test and Engle ARCH test on standardized innovations(H, pValue,Stat,CriticalValue) =lbqtest((innovations./sigmas).square,(10 15 20)’,0.05);(H pValue Stat CriticalValue)(H, pValue, Stat, CriticalValue) =archtest(innovations./sigmas,(10 15 20)’,0.05);(H pValue Stat CriticalValue)% loading data: non stationary time series with 168 data transform into a stationaryTS with 166 data specifying the modelspec=garchset(’VarianceModel’,’GARCH’,’P’,1,’Q’,2);spec=garchset(spec,’Display’,’off’,’Distribution’,’T’);%taking the first 154 values to fit the model andforecasting the next 12 values of datafor i=1:12;(coeff,errors,LLF,eFit,sFit)=garchfit(spec,data(i:153+i));(sigmaForecast,meanForecast,sigmaTotal,meanRMSE)=garchpred(coeff,data(i:153+i),1);garchdisp(coeff,errors)(sigmaForecast,meanForecast,sigmaTotal,meanRMSE)=garchpred(coeff,data(i:153+i),1);sForecast(i,1)=sigmaForecast;mForecast(i,1)=meanForecast;sTotal(i,1)=sigmaTotal;mRMSE(i,1)=meanRMSE;i=i+1;(sigmaForecast,meanForecast,sigmaTotal,meanRMSE)

69end% time plot and predicted plot of inflation rate.plot((0:167),CPIdata,’m’,(0:167),CPIdata,’k’, ’LineWidth’,2);title(’Actual and Forecasted Values from GARCH(1,1)’);xlabel(’Time’);ylabel(’Rate(%)’);title(’Actual and Forecasted values from GARCH(1,1)’);legend(’Forecasted values’,’Actual values’);