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APPLICATION OF ARIMA AND GARCH MODELS IN FORECASTING CRUDE OIL PRICES LEE CHEE NIAN UNIVERSITI TEKNOLOGI MALAYSIA APPLICATION OF ARIMA AND GARCH MODELS IN FORECASTING CRUDE OIL PRICES LEE CHEE NIAN A dissertation submitted in partial fulfillment of the requirement for the award of the degree of Master of Science (Mathematics) Faculty of Science Universiti Teknologi Malaysia NOVEMBER 2009 iii Specially dedicated to my beloved parents, brother, sisters and those people who have guided and inspired me throughout my journey of education iv ACKNOWLEDGEMENTS I would like to take this opportunity to express my heartiest gratitude to everyone who involved directly or indirectly in contributing to this completed study. I am grateful to all the supports given to me. This particular research would never be able to accomplish without the support of my beloved supervisor, Associate Professor Dr. Maizah Hura Ahmad who is extremely knowledgeable about the time series. She has spent her valuable time giving advice, shared her experience with me and assisted me all the way long. I am truly grateful for having such a wonderful supervisor. Lastly, I would like to thank to my family who always supports me and also thank to all my friends for their assistance to enable the completion of this study. v ABSTRACT Crude oil is an important energy commodity to mankind. Several causes have made crude oil prices to be volatile. The fluctuation of crude oil prices has affected many related sectors and stock market indices. Hence, forecasting the crude oil prices is essential to avoid the future prices of the non-renewable natural resources to raise sky-rocket. In this study, daily WTI crude oil prices data is obtained from Energy Information Administration (EIA) from 2nd January 1986 to 30th September 2009. We use the Box-Jenkins methodology and Generalized Autoregressive Conditional Heteroscedasticity (GARCH) approach in forecasting the crude oil prices. An Autoregressive Integrated Moving Average (ARIMA) model is set as the benchmark model. We found ARIMA(1,2,1) and GARCH(1,1) are the appropriate models under model identification, parameter estimation, diagnostic checking and forecasting future prices. In this study, the analyses are done with the aid of EViews software where the potential of this software in forecasting daily crude oil prices time series data is explored. Finally, using several measures, comparison performances between ARIMA(1,2,1) and GARCH(1,1) models are made. GARCH(1,1) is found to be a better model than ARIMA(1,2,1) model. Based on the study, we conclude that ARIMA(1,2,1) model is able to produce accurate forecast based on a description of history patterns in crude oil prices. However, the GARCH(1,1) is the better model for daily crude oil prices due to its ability to capture the volatility by the non-constant of conditional variance. vi ABSTRAK Minyak mentah merupakan komoditi tenaga yang penting untuk umat manusia. Beberapa penyebab telah memjadikan harga minyak mentah akan berubahubah. Fluktuasi harga minyak mentah telah mempengaruhi pelbagai sektor berkaitan serta indeks pasaran saham. Oleh sebab itu, ramalan kepada harga minyak mentah adalah agak penting untuk mengelakkan harga masa depan sumber alam yang tidak diperbaharui daripada meningkatkan mendedak. Dalam kajian ini, harga minyak mentah harian WTI data yang diperolehi daripada Energy Information Administration (EIA) dari 2 Januari 1986 sampai ke 30 September 2009. Kami menggunakan metodologi Box-Jenkins dan pendekatan Generalized Autoregressive Conditional Heteroscedasticity (GARCH) dalam meramalkan harga minyak mentah. Sebuah model Autoregressive Integrated Moving Average (ARIMA) ditetapkan sebagai model patokan. Kami menemukan ARIMA(1,2,1) dan GARCH(1,1) adalah model yang sesuai di bawah pengenalan model, estimasi parameter, diagnostik pemeriksaan dan peramalan harga masa depan. Dalam kajian ini, analisis yang dilakukan dengan bantuan perisian EViews di mana potensi perisian ini akan dieksplorasi dalam memprediksi harga minyak mentah harian data siri masa. Akhirnya, dengan menggunakan beberapa ukuran, perbandingan prestasi di antara ARIMA(1,2,1) dan GARCH(1,1) model diuji. GARCH(1,1) ditemui untuk menjadi model yang lebih baik daripada model ARIMA(1,2,1). Mengikuti kajian ini, kami membuat kesimpulan bahawa model ARIMA(1,2,1) boleh menghasilkan perkiraan yang tepat berdasarkan keterangan pola-pola dalam sejarah harga minyak mentah. Namun, GARCH(1,1) adalah model yang lebih baik untuk harga minyak mentah harian kerana kemampuannya untuk menangkap volatilitas oleh pemalar bukan varians bersyarat. vii TABLE OF CONTENTS TITLE CHAPTER 1 2 PAGE DECLARATION ii DEDICATION iii ACKNOWLEDGEMENTS iv ABSTRACT v ABSTRAK vi TABLE OF CONTENTS vii LIST OF TABLES xii LIST OF FIGURES xiii LIST OF ABBREVIATIONS xv LIST OF SYMBOLS xvii LIST OF APPENDICES xix INTRODUCTION 1 1.1 Introduction 1 1.2 Background of the Problem 3 1.3 Statement of the Problem 4 1.4 Objectives of the Study 4 1.5 Scope of the Study 5 1.6 Significance of the Study 5 1.7 Summary and Outline of the Study 5 LITERATURE REVIEW 7 2.1 Introduction 7 2.2 Highlight of Volatile Crude Oil Prices 7 2.3 Factors Contributing to Crude Oil Prices Volatility 8 viii 3 2.4 Time Series and Forecasting 10 2.5 Relevant Research in Crude Oil 11 2.6 Concluding Remarks 17 METHODOLOGY 18 3.1 Introduction 18 3.2 Data Sources 19 3.3 EViews 5.0 19 3.3.1 20 3.4 Regression in EViews 21 3.4.1 Coefficient Results 21 3.4.1.1 Regression Coefficients 22 3.4.1.2 Standard Errors 3.4.1.3 -Statistics 22 3.4.1.4 Probability 23 Summary Statistics 24 3.4.2.1 R-squared 24 3.4.2.2 Adjusted R-squared 24 3.4.2.3 Standard Error of the Regression 25 3.4.2.4 Sum-of-Squared Residuals 25 3.4.2.5 Log Likelihood 26 3.4.2.6 Durbin-Watson Statistic 26 3.4.2.7 Mean and Standard Deviation 26 3.4.2.8 Akaike Information Criterion 27 3.4.2.9 Schwarz Information Criterion 27 3.4.2.10 F-Statistic 28 3.4.2 3.5 Overview of EViews 23 Residual Tests 28 3.5.1 Correlograms and Q-statistics 28 3.5.1.1 Autocorrelation 29 3.5.1.2 Partial Autocorrelation 30 3.5.1.3 Q-Statistics 31 3.5.2 Correlograms of Squared Residuals 32 3.5.3 Histogram and Normality Test 32 ix 3.6 3.7 3.8 3.9 3.5.3.1 Mean 32 3.5.3.2 Median 33 3.5.3.3 Max and Min 33 3.5.3.4 Standard Deviation 33 3.5.3.5 Skewness 33 3.5.3.6 Kurtosis 34 3.5.3.7 Jarque-Bera Test 34 3.5.4 Serial Correlation Lagrange Multiplier Test 35 3.5.5 The ARCH-LM Test 37 Unit Root Tests for Stationarity 37 3.6.1 The Augmented Dickey-Fuller Test 38 3.6.2 The Phillips-Perron Test 39 Forecast Performance Measures 39 3.7.1 Mean Absolute Error 40 3.7.2 Root Mean Squared Error 40 3.7.3 Mean Absolute Percentage Error 40 3.7.4 Theil Inequality Coefficient 41 3.7.5 Mean Squared Forecast Error 41 Box-Jenkins Methodology 43 3.8.1 ARIMA Model 43 3.8.2 Model Identification 44 3.8.3 Parameter Estimation 46 3.8.4 Diagnostic Checking 46 3.8.5 Forecasting 47 GARCH Process 48 3.9.1 GARCH(1,1) Model 49 3.9.2 Parameter Estimation 51 3.9.3 Diagnostic Checking 52 3.9.4 Forecast 53 3.10 Comparison of ARIMA and GARCH Processes 54 3.11 Concluding Remarks 55 x 4 RESULTS AND ANALYSIS 56 4.1 Introduction 56 4.2 Data Management 56 4.3 Crude Oil Prices Time Series 57 4.4 Stationary Series 58 4.5 ARIMA Model 64 4.5.1 ARIMA Model Identification 64 4.5.2 Parameter Estimation ARIMA(1,2,1) Model 69 4.5.3 Diagnostic Checking ARIMA(1,2,1) Model 71 4.5.4 Forecasting using ARIMA(1,2,1) Model 74 4.6 4.7 4.8 Heteroscedasticity Test 77 4.6.1 ARCH-LM Test 77 4.6.2 Diagnostic Checking for Residuals Squared 79 GARCH Model 80 4.7.1 Model Identification GARCH Model 80 4.7.2 Parameter Estimation GARCH(1,1) Model 80 4.7.3 Diagnostic Checking GARCH(1,1) Model 83 4.7.4 Forecasting using GARCH(1,1) Model 88 Evaluation of ARIMA(1,2,1) and GARCH(1,1) 91 Models Performances 4.8.1 Information Criterion for ARIMA(1,2,1) and 92 GARCH(1,1) Models 4.8.2 Forecasting Performances of ARIMA(1,2,1) 92 and GARCH(1,1) Models 4.9 5 Concluding Remarks CONCLUSIONS AND SUGGESTIONS FOR FUTURE 93 95 STUDY 5.1 Introduction 95 5.2 Conclusions 95 5.3 Suggestions for Future Works 96 xi REFERENCES 98 Appendix A 104 Appendix B 124 xii LIST OF TABLES TABLE NO. 3.1 TITLE PAGE The behaviour of ACF and PACF for each of the 45 general models 3.2 Comparison of ARIMA and GARCH models 55 4.1 ADF test for crude oil prices 59 4.2 PP test for crude oil prices 59 4.3 ADF test for first difference of oil prices 60 4.4 PP test for first difference for crude oil series 61 4.5 ADF test for second order difference series 66 4.6 Estimation equation of ARIMA(1,2,1) 70 4.7 Serial correlation Breusch-Godfrey LM test for 72 ARIMA(1,2,1) 4.8 Forecast evaluation for ARIMA(1,2,1) model 76 4.9 ARCH-LM test for ARIMA(1,2,1) model 78 4.10 Parameter estimation of GARCH(1,1) model 81 4.11 ARCH-LM test for GARCH(1,1) model 85 4.12 Forecast evaluation for GARCH(1,1) model 90 4.13 Information criterion for ARIMA(1,2,1) and 92 and 93 GARCH(1,1) models 4.14 Forecasting performances GARCH(1,1) models of ARIMA(1,2,1) xiii LIST OF FIGURES FIGURE NO. 3.1 TITLE PAGE An example of correlogram and -statistics from 21 4.1 The time series for WTI daily crude oil prices 57 4.2 Histogram and normality test on WTI daily crude oil 58 3.2 An example of equation output from EViews 29 EViews prices 4.3 First order difference crude oil prices series 62 4.4 Histogram and normality test of first order difference 63 series 4.5 Correlogram of the first order difference series 65 4.6 First order difference of crude oil prices series 67 4.7 Histogram and normality test of second order 68 difference series 4.8 Correlogram of the second order difference series 69 4.9 Correlogram of residuals for ARIMA(1,2,1) 71 4.10 Second order difference of residuals plot 73 4.11 Histogram and normality test for residuals 74 ARIMA(1,2,1) 4.12 Forecast crude oil prices by ARIMA(1,2,1) model 75 4.13 The plot of actual prices against forecast prices by 76 ARIMA(1,2,1) model 4.14 Correlogram of residuals squared by ARIMA(1,2,1) 79 4.15 Conditional standard deviation for GARCH(1,1) model 82 4.16 Conditional variance for GARCH(1,1) model 83 4.17 Correlogram of standardized residuals squared for 84 xiv GARCH(1,1) model 4.18 First order difference of residuals plot 86 4.19 Standardized residuals plot for GARCH(1,1) model 87 4.20 Histogram and normality test for standardized residuals 88 4.21 Forecast crude oil prices by GARCH(1,1) model 89 4.22 Conditional variance forecast by GARCH(1,1) model 90 4.23 The plot of actual prices against forecast prices by 91 GARCH(1,1) model xv LIST OF ABBREVIATIONS ACF - Autocorrelation functions ADF - Augmented Dickey-Fuller AIC - Akaike Information Criterion ANFIS - Adaptive Network-based Fuzzy Inference System ANN - Artificial Neural Networks API - American Petroleum Institute AR - Autoregression ARCH - Autoregressive Conditional Heteroscedasticity ARIMA - Autoregressive Integrated Moving Average ARMA - Autoregressive Moving Average CBP - Correlated Bivariate Poisson CGARCH - Component GARCH DW - Durbin-Watson EGARCH - Exponential GARCH EIA - Energy Information Administration EViews - Econometric Views EVT - Extreme Value Theory FIAPARCH - Fractional Integrated Asymmetric Power ARCH FIGARCH - Fractionally Integrated GARCH GARCH - Generalized Autoregressive Conditional Heteroscedasticity GED - Generalized Exponential distribution GUI - Graphical User Interface HSAF - Historical Simulation with ARMA Forecasts HT - Heavy-tailed IGARCH - Integrated GARCH ILS - Interval Least Square xvi IPE - International Petroleum Exchange IV - Implied Volatility JB - Jarque-Bera LM - Lagrange Multiplier MA - Moving Average MAE - Mean Absolute Error MAPE - Mean Absolute Percentage Error MRS - Markov Regime Switching MSFE - Mean Squared Forecast Error NYMEX - New York Mercantile Exchange OPEC - Organization of the Petroleum Exporting Countries PACF - Partial Autocorrelation Functions PP - Phillips-Perron QMS - Quantitative Micro Software RMSE - Root Mean Squared Error SIC - Schwarz Information Criterion SVM - Support Vector Machine TAR - Asymmetric Threshold Autoregressive Theil-U - Theil Inequality Coefficient US - United State VaR - Value at Risk VECM - Vector Error Correction Model WTI - West Texas Intermediate 2SLS - Two-stage Least Squares xvii LIST OF SYMBOLS - adjusted R-squared - standardized residuals - estimated residual - sum-of-squared residuals - residuals - residuals squared Ω - white noise process - measurable function of time − 1 information set - ̂ ̂ ̂ - - R-squared - estimator of the residual spectrum at frequency zero null hypothesis likelihood of - residual of time series - optional exogenous regressors - mean of the dependent variable - differenced of crude oil prices time series - time series of crude oil prices - coefficients for ARCH - consistent estimate of the error variance - autocorrelation - estimator for the standard deviation - unconditional variance - conditional variance - Chi-squared - partial autocorrelation - difference linear operator ∆ xviii " # - - #-statistic % - Q-statistic - & × ( matrix of independent variables - number of regressors - log likelihood - number of observations + - order of the autoregressive part - order of the moving average part - standard error of the regression - time , - (-vector of coefficients $ ) ( & * - - - backshift operator likelihood of the joint realizations amount of differencing &-dimensional vector of dependent variable &-vector of disturbances xix LIST OF APPENDICES APPENDIX TITLE PAGE A WTI Daily Crude Oil Prices Data 104 B Actual and Forecast Value 124 CHAPTER 1 INTRODUCTION 1.1 Introduction Crude oil or petroleum is a naturally occurring and flammable liquid found in rock formations in the earth. It has consisting of a complex mixture of hydrocarbons of various molecular weights plus other organic compounds. The main characteristics of crude oil are generally classifies according to its sulphur content and its density which the petroleum industry measured by its American Petroleum Institute (API) gravity. Obviously, crude oil may be considered light if it has low density with API gravity less than about 40. Typically, heavy crude has high density with API gravity 20 or less. In other words, the higher the API gravity, the lower in its density. Brent crude is important benchmark crude which has an API gravity of 38 to 39. Crude oil may be referred to as sweet if it contains less than 0.5% sulphur or sour if it contains substantial amounts of sulphur. Sweet crude is preferable to sour one because it is more suited to the production of the most valuable refined products. Moreover, the geographical location of crude oil production is another main count. In the crude oil market, the two current references or pricing markers are West Texas Intermediate (WTI) and Europe Brent. The former is the base grade traded, as ‘light sweet crude’, on the New York Mercantile Exchange (NYMEX) for delivery at Cushing, Oklahoma. While the latter is traded on London’s International Petroleum 2 Exchange (IPE) for delivery at Sullom Voe and is also one of the grades acceptable for delivery of the NYMEX contract (Lin and Tamvakis, 2001). The price of a barrel of oil is highly dependent on both its grade, determined by factors such as its specific API gravity, sulphur content and also location of production. The vast majority of oil is not traded on an exchange but on an over-thecounter basis. Some other important benchmarks include Dubai, Tapis (Malaysia), Minas (Indonesia) and Organization of the Petroleum Exporting Countries (OPEC) basket. The Energy Information Administration (EIA) uses the imported refiner acquisition cost where the weighted average cost of all oil imported into the United State (US) known as "world oil price". Look back into the past, the increasing oil prices has affected certain benchmark indices widely followed and traded. On the other hand, the scientific community is confused over the absolute quantities of oil reserves. In fact, crude oil is a limited and non-renewable natural reserve. The on going demand of crude oil and its refined products will consequently in oil supply scarcity. In the end, this energy commodity is most likely to keep an upward trend in the future if without any alternative replacements for crude oil. There are ample studies addressing the accuracy of crude oil volatility modelling and forecasting. These include Autoregressive Conditional Heteroscedasticity, ARCH-type models (Fong and See, 2002; Giot and Laurent, 2003), Asymmetric threshold autoregressive (TAR) model (Godby et al., 2000), and artificial based forecast methods (Fan et al., 2008a), Interval Least Square (ILS) (Xu et al., 2008), Support Vector Machine (SVM) (Xie et al., 2006), Artificial Neural Networks (ANN) (Kulkarni and Haidar, 2009), Adaptive Network-based Fuzzy Inference System (ANFIS) (Ghaffari and Zare, 2009), Fuzzy Neural Network (Liu et al., 2007), Autoregressive Moving Average (ARMA) (Cabedo and Moya, 2003), and etc. However, the complexity of the model specification does not guarantee high performance on out-performed out-of-sample forecasts. 3 One of the model that has gained enormous popularity in many areas and forecasting research practice is Box-Jenkins method. Thus, the purpose of this study is to forecast crude oil prices using Box-Jenkins method. However, despite the fact that the Box-Jenkins method is powerful and flexible, it is not able to handle the volatility that is present in the data series. To handle the volatility in the crude oil data, the current study proposes the use of the Generalized ARCH (GARCH) model. Using the forecasts obtained from the Box-Jenkins model as a benchmark, the forecasts obtained from the GARCH will be evaluated. 1.2 Background of the Study In statistics, a sequence of random variables is heteroscedastic if the random variables have different variances. The term means "differing variance" and comes from the Greek "hetero" ('different') and "skedasis" ('dispersion'). In contrast, a sequence of random variables is called homoscedastic if it has constant variance. In particular, we consider crude oil prices data as heteroscedastic time series models where the conditional variance given in the past is no longer constant(Palma, 2007). In a financial analysis, forecast of future volatility of a series under consideration are often of interest to assess the risk associated with certain assets. In that case, variance forecasts are of direct interest, of course (Lütkepohl, 2005). One of the most prominent stylized facts of returns on crude oil prices is that their volatility changes over time. In particular, periods of large movements in crude oil prices alternate with periods during which prices hardly change. This characteristic feature commonly is referred to as volatility clustering. It was first observed by Nobel Prize winner, Robert Engle (1982) that although many financial time series, such as, stock returns and exchange rates are unpredictable, there is apparent clustering in the variability or volatility. This is often referred to as conditional heteroscedasticity since it is assumed that overall the series 4 is stationary but the conditional expected value of the variance may be timedependent. Later, Bollerslev (1986) had modified Engle’s ARCH model into a more generalized model called GARCH model with is a simplified model to ARCH model but more powerful. Currently, this model has been widely used in many financial time series data. The simple GARCH model is able to detect the financial volatility in a time trend. 1.3 Statement of the Problem The price of the energy commodity is highly volatile throughout the time. Since crude oil prices variability does affect other sectors and stock market, the prediction of future crude oil prices becomes crucial. This study will explore the following question : “Which method between the Box-Jenkins and GARCH performs better in forecasting crude oil prices, which is of high volatility?” 1.4 Objective of the Study The objectives of this study are as follows: 1.4.1 To estimate suitable Box-Jenkins and GARCH models for forecasting crude oil prices. 5 1.4.2 To evaluate the performance of the GARCH and Box-Jenkins models in forecasting crude oil prices. 1.4.3 To forecast using EViews software. 1.5 Scope of the Study This study focuses on the Box-Jenkins and GARCH models to forecast crude oil prices. Since the oil price volatility is the main concern, the study uses only daily data. The data were obtained from EIA from 2nd January 1986 to 30th September 2009. 1.6 Significance of the Study Since crude oil market is highly volatile, the estimation of the time series model must be able to detect its volatility. We have to determine the precisely BoxJenkins and GARCH models when forecasting the volatility of crude oil prices. The process will be done with the aid of software. As a result of this study, a model and software that can be used to forecast volatile time series can be proposed. 1.7 Summary and Outline of the Study This dissertation is organized into 5 chapters. Chapter 1 discusses the research framework. It begins with the introduction to crude oil and the background of the study. The objectives, scope and the significance of this study are also presented. 6 Chapter 2 reviews crude oil prices in forecasting. First, crude oil prices will be reviewed. Then, the volatility in crude oil prices will be discussed. The discussions start on the past researchers’ work in Box-Jenkins methodology and GARCH-type models are also presented. Finally, conditional heteroscedasticity are explained. Chapter 3 begins methodology. In this chapter analysis of data sets using the Autoregressive Integrated Moving Average (ARIMA) and GARCH models are carried out. In chapter 4, a detail present on the analysis of the same data sets using the ARIMA and GARCH models. Also, comparison between the ARIMA and GARCH models are made. Chapter 5 summarizes and concludes the whole study and makes some suggestions for future investigation. CHAPTER 2 LITERATURE REVIEW 2.1 Introduction In this chapter, the volatility of the crude oil prices will be examined. Hence, some important causes affecting the significance changes of the crude oil prices will be discussed. A discussion on the ARCH model developed by Engle (1982) will also be presented. Engle (1982) was the first to introduce the concept of conditional heteroscedasticity. Based on the studies from other researchers’ works, we summarize the results related to the crude oil prices forecast. We highlight the BoxJenkins methodology and GARCH approach of forecasting crude oil market volatility since these are the focus of the current study. Finally, an extension some useful GARCH type models which brought an essential result of forecasting crude oil prices will be discussed. 2.2 Highlight of Volatile Crude Oil Prices Several world events have led to major oil disruptions in the past. Most of these disruptions were related to political or military upheavals, especially occurred in the Middle East. Since 1973, there were four crises which have caused oil prices to 8 be volatile. These include the 1973 Arab–Israeli war, the 1978–89 Iranian revolution, the 1980 Iran–Iraq war and the 1990–91 Gulf war which have resulted in initial shortfalls of between 4.0 and 5.6 million barrels per day (Marimoutou et al., 2009). In 1999, the increase in Iraq oil production coincided with the Asian financial crises which caused the oil price to drop due to a reduced in demand. In September 2007, WTI crude crossed $80 per barrel. There were several factors causing a raise in crude oil price. One of the main factors was when OPEC announced an output increase lower than expected (OPEC Press Release, 2007). US stocks fell lower than what the experts predicted (EIA, 2007), the changes in federal oil policies (Edmund L. Andrews, 2007), and six pipelines were attacked by a leftist group in Mexico (Imelda Medina, 2007). In October 2007, US light crude rose above $90 per barrel due to a combination of tensions in eastern Turkey and the reduced strength of the US dollar (BBC News, 2007). On July 11, 2008, oil prices hit a new highest record of $147.27 per barrel following concern over recent Iranian missile tests (BBC News, 2008). The extraordinary spike in prices represented to a large extent the consequences of a brief period where global oil demand outran supply. Commentators attributed these price increases to many factors, including reports from the United States Department of Energy and others showing a decline in petroleum reserves (Peter J. Cooper, 2006), worries over peak oil (Energy Bulletin, 2009), Middle East tension, and oil price speculation (Moira Herbst, 2008). However after all these events, oil prices started to decline. A strong contributor to this price decline was the drop in demand for oil in the US. Prices did not rebound even during the beginning of 2009. 2.3 Factors Contributing to Crude Oil Prices Volatility Crude oil prices change due to many reasons. These reasons include OPEC policy, war and political uncertainty in several places such as the Middle East, supply 9 disruptions due to natural or other disasters, and changing demand and imbalances between physical supply and demand (Marimoutou et al., 2009). Cheong (2009) also stated that apart from the supply and demand condition in oil markets, fluctuations of crude oil prices have also been profoundly influenced by political events, speculations, military conflicts and natural disasters. For example, when OPEC decided to curtail crude oil production by 4.2 million barrels daily in December 2000, the price rose to approximately $36 per barrel in the WTI crude oil spot markets. After the US invasion of Iraq, the price slumped to around $18 per barrel in December 2001. Recently, the so-called third crude oil crisis occurred when prices reached rocket-high levels of $145.31 per barrel middle of 2008. However, when the crude oil bubble burst mostly due to speculations, the price plunged to $30 per barrel at the end of 2008. Starting in 2006, geo-political events indirectly related to the global oil market had strong short-term effects on oil prices, such as North Korean missile tests, the conflict between Israel and Lebanon, and worries over Iranian nuclear plans. The risk of war in oil producing regions, notably the Middle East, will increase the price of oil. Supply disruptions as well as fear of supply disruptions have been a significant recent influence on the crude oil market. For example, the price of crude oil briefly spiked to more than $35 a barrel in response to the Iraqi invasion of Kuwait at the end of 1990. Moreover, the flow of oil to the market is restricted through collusion and the underdevelopment of the vast oil resources controlled by the OPEC (Marimoutou et al., 2009). The relationship between levels of inventory and volatility, however, is not straightforward and can run in the opposite direction, which volatility of oil prices can affect inventory levels. First, oil price volatility causes increases in volatility in consumption and production and as a result market participants will want to hold greater inventories to buffer these fluctuations. Second, oil price volatility increases the opportunity cost of producing now such that producers will not be willing to 10 extract oil unless the spot price is higher than the future price, except when the market is in strong backwardation. Pindyck (2001) provides evidence that market variables such as inventory levels cannot explain crude oil price volatility whereas volatility can influence some market variables such as production although the effect is empirically very small. Fluctuation of crude oil prices is also due to speculation. In September 2008, a study of the oil market by Masters Capital Management was released which claimed that speculation did significantly impact the market. The study stated that over $60 billion was invested in oil during the first 6 months of 2008, helping drive the price per barrel from $95 to $147 per barrel, and that by the beginning of September, $39 billion had been withdrawn by speculators, causing prices to fall (Josef J. Hebert, 2008). Congressman’s Ron Paul (2008) argued that loose monetary policy from the Federal Reserve and other central banks is a major contributor to the increase in oil prices, and the cause of both commodity speculation and dollar devaluation. The price of oil is closely tied to the value of the U.S. dollar because oil is traded in dollars. This has caused concern among some economists that the principal earned from the sale of oil may lose value in the long run if the U.S. dollar loses real value. Some other unexpected issues have also had some effect on oil prices. For example, the post-9/11 war on terror, labour strikes, hurricane threats to oil platforms, fires and terrorist threats at refineries, and other short-lived problems are responsible for the higher prices. Such problems do push prices into a higher temporarily, but have not historically been fundamental to long-term price increases. 2.4 Time Series and Forecasting The forecast of crude oil prices is essential because it affect other sectors and stock markets. As a time series data, various forecasting techniques can be employed. 11 The merits of ARIMA models are two-fold (Wang et al., 2005). First, ARIMA models are a class of typical linear models which are designed for linear time series and captured linear characteristics in time series. Second, the theoretical foundation of ARIMA models is perfect. Therefore, ARIMA models are widely used in many practical applications. However, the disadvantage of the ARIMA is that it cannot capture nonlinear patterns of complex time series if nonlinearity exists. Some researchers have done the studies in forecasting crude oil prices with Box-Jenkins methodology. Liu (1991) employed Box-Jenkins transfer function models to study the dynamic relationships between US gasoline prices, crude oil prices and the stock of gasoline. Kumar (1992) investigated time series models on comparison forecast accuracy of futures prices for crude oil. An ARMA(1,2) model has compared with forecast crude oil futures prices. Chinn et al. (2005) studied on the predictive content of energy futures. They examined the relationship between spot and futures prices for energy commodities. One of these energy commodities was WTI crude oil. An ARIMA(1,1,1) was used for crude oil prices forecast. Moshiri and Foroutan (2006) modelled and forecasted daily crude oil futures prices that are listed in MYMEX from 1983 to 2003, applying linear and nonlinear time series models. They used EViews 4 to estimate and forecast crude oil futures prices. They discovered that linear ARMA(1,3) and nonlinear GARCH(2,1) models were the most suitable. However, the GARCH model outperformed the ARMA model. 2.5 Relevant Research in Crude Oil In literature, there were quite a numbers of relevant researches regarding the Box-Jenkins method and GARCH-type models in forecasting crude oil prices. 12 Sadorsky (1999) found that oil price volatility shocks have asymmetric effects on the economy. The changes in oil prices affect economic activity but the changes in economic activity have little impact on oil prices. Therefore, oil price fluctuations have large macroeconomic impacts. Sadorsky (2006) also found that the out-of-sample forecasts of a single equation GARCH model are more superior to those of state space, vector autoregression and bivariate GARCH models in forecasting petroleum futures prices. Agnolucci (2009) stated that GARCH-type models are able to have a better performance than the implied volatility (IV) models in terms of predictive accuracy. Moreover, some conclusions are drawn, which included the mean returns from oil futures can be assumed to be constant across time, shocks to the conditional variance of the series have been found to be highly persistent, the parameters in the models are robust to the distribution assumed for the errors and no leverage effect can be observed in the oil future series. Cabedo and Moya (2003) compared the Value at Risk (VaR) from historical simulation with ARMA forecasts (HSAF) approach to those from a standard GARCH model. They found that the VaRs from the HSAF method provides most flexible and efficient risk qualification that outperform than those from the standard GARCH model. Sadeghi and Shavvalpour (2006) introduced the HSAF and comparison with GARCH model to estimate the VaR of OPEC oil price. In their findings HSAF was shown to be more efficient. However, Costello et al. (2008) assessed the performance of the HSAF model and the semi-parametric GARCH model proposed by Barone-Adesi et al. (1999). They argued that the VaRs from GARCH model with historical simulation are superior to the ARMA method. Their findings suggested that Cabedo and Moya’s (2003) conclusion is driven mainly by the normal distributional assumption imposed on the future risk structure in the standard GARCH framework. 13 Morana (2001) showed how the GARCH properties of oil price changes can be employed to forecast the oil price distribution over short-term horizons. He used a semi-parametric approach to oil price forecasting and it was based on bootstrap approach. According to Marimoutou et al. (2009), GARCH(1,1)- model may provide equally good results when compared to a combined GARCH and Extreme Value Theory (GARCH-EVT). Marzo and Zagalia (2007) studied the forecasting properties of linear GARCH models for closing-day futures prices on crude oil traded in the NYMEX. They compared volatility models based on the normal, Student’s and Generalized Exponential distribution (GED). Their main focus was on out-of-sample predictability. From the tests for predictive ability, the results showed that the GARCH-GED model fares best for short horizons from one to three days ahead. Fan et al. (2008b) carried out an estimation using the GARCH type model based on Generalized Error Distribution (GED-GARCH) for VaR of returns in crude oil spot market. They stressed that the historical simulation with ARMA forecasts method did not have an advantage over others to forecast the return in out-of-sample data. Results revealed that there is significant two-way risk spillover effect between crude oil markets. Hung et al. (2008) adopted the GARCH model with the heavy-tailed (HT) distribution to estimate one-day-ahead VaR for WTI and Brent spots and further compares the accuracy and efficiency with the GARCH-N and GARCH- models. First for each series considered, the out-of-sample VaR forecast of GARCH-HT model outperform alternative models in terms of failure rate in backtests at all confidence levels. Moreover, solving the analytical quantile-operator of HT distribution enables convenient out-of-sample VaR predictions. Because of the inability to characterize the tail behaviours of energy commodities, the GARCH models with normal and student- distributions tend to overestimate and 14 underestimate the tail risk at low confidence levels for some cases. Secondly, regarding efficiency, the GARCH-N model is superior to alternatives in the case of low confidence levels for most series. Unfortunately, the GARCH-N model cannot fully pass the backtests. On the contrary, the GARCH-HT model outperforms the competitive models at high confidence levels with approval of backtests. It suggests that the VaR forecasts obtained by the GARCH-HT model provide more satisfactory results in both accurate and efficient concerns as a whole. They concluded that the heavy-tailed distribution is more suitable for energy commodities, particularly VaR calculation. Kang et al. (2009) investigated the efficacy of a volatility model for three crude oil markets. They are Brent, Dubai, and WTI with regard to its ability to forecast and identify volatility stylized facts, in particular volatility persistence. They assessed persistence in the volatility of the three crude oil prices using conditional volatility models. One of the models is component-GARCH (CGARCH) model that was developed by Engle and Lee (1999). This model was able to distinguish between short-run and long-run persistence of volatility. Another model is the fractionally integrated GARCH (FIGARCH) model that was introduced by Baillie et al. (1996) that allows for a fractional integrated process in conditional variance. This model is better equipped to capture persistence than are the GARCH and Integrated GARCH (IGARCH) models. The CGARCH and FIGARCH models also provide superior performances in out-of-sample volatility forecasts. Kang et al. (2009) concluded that the CGARCH and FIGARCH models are useful for modelling and forecasting persistence in the volatility of crude oil prices. Cheong (2009) evaluated the volatility behaviour of the two major crude oil markets particularly in the empirical stylized facts such as the asymmetric news impact, long-persistence volatility and tail behaviour in the crude oil series. A very powerful and flexible ARCH model is used to evaluate the aforementioned stylized facts for both the Brent and WTI markets. Although both the estimation and diagnostic evaluations are in favour of the Fractional Integrated Asymmetric Power ARCH (FIAPARCH) model, from empirical out-of-sample forecast it appears that the simplest parsimonious GARCH fits the Brent crude oil data better than the other 15 models. On the other hand, the FIAPARCH out-of-sample WTI forecasts provide superior performance. These findings suggest that energy economists and financial analysts should consider not only the complexity, but also the parsimonious principle and actual performance of out-of-sample forecasts, in choosing a crude oil volatility model. Narayan and Narayan (2007) examine the volatility of crude oil price using daily data for the period 1991–2006 by Exponential GARCH (EGARCH) model. Their main findings summarised that across the various sub-samples, there is inconsistent evidence of asymmetry and persistence of shocks; and over the full sample period, evidence suggests that shocks have permanent effects and asymmetric effects, on volatility. These findings imply that the behaviour of oil prices tends to change over short periods of time. Fong and See (2002) studied a Markov switching model of the conditional volatility of crude oil futures prices, and show that the regimes identified by their model capture major oil-related events. Hence, the significant regime shifts in the conditional volatility of crude oil futures contracts, which tends to dominate the GARCH effects. Alizadeh et al. (2008) investigated the hedging effectiveness of the Markov Regime Switching (MRS) models for WTI crude oil futures contracts. They then introduced a MRS vector error correction model (VECM) with GARCH error structure. This specification links the concept of disequilibrium with that of high uncertainty across high and low volatility regimes. The results indicated that using MRS models, markets agents may be able to obtain superior gains, measured in terms of both variance reduction and increase in utility. The traditional continuous and smooth models, like the GARCH model, may fail to capture extreme returns volatility. Therefore, Cheng (2008) applied the correlated bivariate Poisson (CBP)-GARCH model to study jump dynamics in price volatility of crude oil and heating oil during the past 20 years. The empirical results indicated that the variance and covariance of the GARCH and CBP-GARCH models were found to be similar in low jump intensity periods and to diverge during jump events. Significant overestimations occur during high jump time periods in the 16 GARCH model because of assumptions of continuity, and easily leading to excessive hedging and overly measuring risk. Nevertheless, in the CBP-GARCH model, the specific shocks are assumed to be independent of normal volatility and to reduce the persistence of abnormal volatility. Therefore, the CBP-GARCH model is appropriate and necessary in high volatility markets. Nobel Prize winner Robert Engle’s (1982) original work on ARCH was concerned with the volatility inflation. However, it is the applications of the ARCH model to financial time series that established and consolidated the significance of his contribution. Financial time series have characteristics that are well represented by the models with dynamic variances (Hill et al., 2008). In order to motivate the underlying ideas of ARCH processes, first consider the problem of predicting the future level of the mean of a random variable, which is recorded from time series data. The relative success of forecasting from any dynamic econometric model essentially comes from the use of the conditional mean rather than the unconditional mean. As an illustration, consider the simple scalar first-order autoregression, -(1) model = 1 + , where 2( ) = 0 , 232 4 = 2 and 2( 5 ) = 0 for ≠ . One of the insights in the pioneering work of Engle (1982) is to analyse conditional, rather than unconditional, second moments and to allow the conditional 2 variance, 71 , to be time dependent. The simplest model involving ARCH is the Martingale Difference Sequence (Baillie, 2006) with ARCH(+) innovations, which can be expressed as = (2.1) ~IID(0,1) (2.2) 2 2 = 0 + = (2.3) > ?1 17 where parameter restriction 0 > 0 and ≥ 0 , for B = 1,2, … , + , are required to avoid negativity of the conditional variance. The variable 2 is a time-varying positive, and measurable function of time − 1 information set, Ω1 . 2.5 Concluding Remarks From the literature review, the analysis of crude oil prices has been a topic of extensive research. In this study, daily crude oil prices will be forecasted using ARIMA and GARCH methods. CHAPTER 3 METHODOLOGY 3.1 Introduction In this chapter, the main discussion is on the time series approaches to estimate and forecast crude oil prices using ARIMA and GARCH models. From the literature, the crude oil prices can be estimated and forecasted by several statistical methods. However, in this study, ARIMA and GARCH approach to estimate from the current data and forecast for the future prices. We introduce a class of models that can produce accurate forecasts based on a description of historical patterns in the data. Autoregressive integrated moving average (ARIMA) models are a class of linear models that are capable of representing stationary as well as non-stationary time series. Since crude oil prices are volatile over the time trend, a heteroscedasticity approach shall be tested for the entire data series. Hence, we use a GARCH model which is able to capture volatility clustering in crude oil prices time series. Its performance is then compared with ARIMA model. EViews is a statistical and econometric software that is gaining much popularity among researchers. Since the objective of this study is also to explore the potential of the EViews software in forecasting, a section discussing this software will be included. 19 3.2 Data Sources The data collected is daily spot WTI crude oil prices from 2nd January 1986 to 30th September 2009 of 5-day-per-week frequencies. Values are quoted in US dollars per barrel and the source data is obtained from EIA. However, there are some missing prices in the original series due to holiday and stock market closing day. Considering the non-linear characteristics of the oil prices, we use the value on the previous day of trading (Malik and Ewing, 2009) to replace the missing prices. As a result, 6195 samples are obtained. We divide the whole time-period into two parts. The first time-period which is called “in-sample period” from 2nd January 1986 to 30th June 2009, is used to estimate GARCH and ARIMA models. The second timeperiod, which is called “out-of-sample period” from 1st July 2009 to 30th September 2009, is used to construct the out-of-sample forecasts. The data tested were affected by several world events that have lead to oil disruption from 1986 to 2009. These include Gulf war in 1990-1991, slowdown of Asian economic growth in the period in 1997-1998, OPEC curtailed the daily production of crude oil in 2000-2001, 9/11 terrorists attack in 2001, US military action in Iraq in March 2003, and some recent economic crises. 3.3 EViews 5.0 In this section, EViews 5.0 software will be described. EViews (Econometric Views) is a statistical package for Windows, used mainly for time series oriented econometric analysis. It is developed by Quantitative Micro Software (QMS). EViews can be used for general statistical analysis and econometric analyses, such as cross-section and panel data analysis and time series estimation and forecasting. It combines spreadsheet and relational database technology with the traditional tasks found in statistical software, and uses a Windows Graphical User Interface (GUI). This is combined with a programming language which displays limited object orientation. 20 3.3.1 Overview of EViews EViews provides sophisticated data analysis, regression, and forecasting tools on Windows based computers. With EViews user can quickly develop a statistical relation from the data and then use the relation to forecast future values of the data. Areas where EViews can be useful include: scientific data analysis and evaluation, financial analysis, macroeconomic forecasting, simulation, sales forecasting, and cost analysis. EViews is a new version of a set of tools for manipulating time series data originally developed in the Time Series Processor software for large computers. The immediate predecessor of EViews was MicroTSP, first released in 1981. Although EViews was developed by economists and most of its uses are in economics, its design does not limit its usefulness to economic time series. Quite a large crosssection projects can be handled in EViews. EViews provides convenient visual ways to enter data series from the keyboard or from disk files, to create new series from existing ones, to display and print series, and to carry out statistical analysis of the relationships among series. EViews takes advantage of the visual features of modern Windows software. Users just need to move their mouse to guide the operation with standard Windows menus and dialogs. Results appear in windows and can be manipulated with standard Windows techniques. Alternatively, EViews also contains powerful command and batch processing language. Users can enter and edit commands in the command window. Besides, user can create and store the commands in programs that document his/her research project for later execution. The next section discusses some functions in EViews 21 3.4 Regression in EViews Using matrix notation, the standard regression may be written as: = %, + (3.1) where is a &-dimensional vector containing observations on the dependent variable, % is a & × ( matrix of independent variables, , is a (-vector of coefficients, and is a & -vector of disturbances. Here, & is the number of observations and ( is the number of right-hand side regressors. 3.4.1 Coefficient Results The coefficient results (QMS, 2004) will be shown in the example of equation output from EViews as in Figure 3.1. Figure 3.1 An example of equation output from EViews 22 3.4.1.1 Regression Coefficients The column labelled “Coefficient” in Figure 3.1 depicts the estimated coefficients. The least squares regression coefficients E are computed by the standard OLS formula: E = (%′%) %′ (3.2) If the equation is specified by list, the coefficients will be labelled in the “Variable” column with the name of the corresponding regressor like the example in Figure 3.1; if the equation is specified by formula, EViews lists the actual coefficients, C(1), C(2), etc. For the simple linear models considered here, the coefficient measures the marginal contribution of the independent variable to the dependent variable, holding all other variables fixed. If present, the coefficient of the C is the constant or intercept in the regression. It is the base level of the prediction when all of the other independent variables are zero. The other coefficients are interpreted as the slope of the relation between the corresponding independent variable and the dependent variable, assuming all other variables do not change. 3.4.1.2 Standard Errors The “Std. Error” column in Figure 3.1 reports the estimated standard errors of the coefficient estimates. The standard errors measure the statistical reliability of the coefficient estimates, where the larger the standard errors, the more statistical noise is present in the estimates. If the errors are normally distributed, there are about 2 chances in 3 that the true regression coefficient lies within one standard error of the reported coefficient, and 95 chances out of 100 that it lies within two standard errors. 23 The covariance matrix of the estimated coefficients is computed as: GH(E) = (%′%); = IJ K ; ̂ = − %E (3.3) where ̂ is the residual. The standard errors of the estimated coefficients are the square roots of the diagonal elements of the coefficient covariance matrix. 3.4.1.3 L-Statistics The -statistic, which is computed as the ratio of an estimated coefficient to its standard error, is used to test the hypothesis that a coefficient is equal to zero. To interpret the -statistic, examine the probability of observing the -statistic given that the coefficient is equal to zero. This probability computation is described below. In cases where normality can only hold asymptotically, EViews will report a -statistic instead of a -statistic. 3.4.1.4 Probability The last column of the output in Figure 3.1 shows the probability of drawing a -statistic or (a -statistic) as extreme as the one actually observed, under the assumption that the errors are normally distributed, or that the estimated coefficients are asymptotically normally distributed. This probability is also known as the *-value or the marginal significance level. Given a *-value, we can tell at a glance if we reject or accept the hypothesis that the true coefficient is zero against a two-sided alternative that it differs from zero. The *-values are computed from a -distribution with & − ( degrees of freedom. 24 3.4.2 Summary Statistics The following are statistics calculated by EViews. These include R-squared, adjusted R-squared, error of the regression, sum-of-squared residuals, log likelihood, Durbin-Watson statistic, Akaike Information Criterion, Schwarz Information Criterion, and #-statistic. All these statistics will be shown in the estimated equation for the models computed as the example in Figure 3.1. 3.4.2.1 R-squared The R-squared ( ) statistic measures the success of the regression in predicting the values of dependent variable within the sample. In standard setting, may be interpreted as the fraction of the variance of the dependent variable explained by the independent variables. The statistic will equal one if the regression fits perfectly, and zero if it fits no better than the simple mean of the dependent variable. It can be negative for a number of reasons. For example, if the regression does not have an intercept or constant, if the regression contains coefficient restrictions, or if the estimation method is two-stage least squares or ARCH. EViews computes the as ̂ = 1− ( − ) (3.4) where is the mean of the dependent variable. 3.4.2.2 Adjusted R-squared One problem with using as a measure of goodness-of-fit is that the will never decrease as more regressors have being added. In the extreme case, we obtain 25 = 1 if many independent regressors are included, as there are sample observations. The adjusted ( ), penalizes the for the addition of regressors which do not contribute to the explanatory power of the model. The is computed as: = 1 − (1 − ) &−1 &−( (3.5) The is never larger than and can be decreased as adding regressors. For poorly fitting models, it may be negative. 3.4.2.3 Standard Error of the Regression The standard error of the regression is a summary measure based on estimated variance of the residuals. The standard error of the regression is computed as: =M ̂ &−( (3.6) 3.4.2.4 Sum-of-Squared Residuals The sum-of-squared residuals can be used in a variety of statistical calculations, and is presented separately for convenience: K ̂ = =( − ′,) ? (3.7) 26 3.4.2.5 Log Likelihood EViews reports the value of the log likelihood function evaluated at the estimated values of the coefficients. Likelihood ratio tests may be conducted by looking at the difference between the log likelihood values of the restricted and unrestricted versions of an equation. The Log likelihood is computed as: & ̂ = − O1 + log(2S) + log ( )T 2 & (3.8) When comparing EViews output to the reported from other sources, note that EViews does not ignore constant terms. 3.4.2.6 Durbin-Watson Statistic The Durbin-Watson (DW) statistic measures the serial correlation in the residuals. The statistic is computed as: DW = ∑K?(̂ − ̂ ) ∑K? ̂ (3.9) As a rule of thumb, if the DW is less than 2, there is evidence of positive serial correlation. The DW statistic in the output is very close to one, indicating the presence of serial correlation in the residuals. 3.4.2.7 Mean and Standard Deviation The mean and standard deviation of the dependent variable are computed using the standard formulae: 27 = ∑K? & ∑K ( − ) W = M ? &−1 (3.10) (3.11) 3.4.2.8 Akaike Information Criterion The Akaike Information Criterion (AIC) is computed as: where is the log likelihood. AIC = − 2 2( + & & (3.12) The AIC is often used in the model selection for non-nested alternatives – smaller values of the AIC are preferred. 3.4.2.9 Schwarz Information Criterion The Schwarz Information Criterion (SIC) is an alternative to AIC that imposes a larger penalty for additional coefficients: SIC = − where is the log likelihood. 2 ( log & + & & (3.13) 28 3.4.2.10 F-Statistic The # -statistic reported in the regression output is from a test of the hypothesis that of all the slope coefficients in a regression is zero. For ordinary least squares models, #-statistic is computed as: /(( − 1) #= (1 − )/(& − () (3.14) Under null hypothesis with normally distributed errors, this statistic has an #- distribution with ( − 1 numerator degrees of freedom and & − ( denominator degrees of freedom. The *-value given is just below the #-statistic, denoted “Prob(#-statistic)”, is the marginal significance level of the # -test. If the * -value is less than the significance level we are testing, say 0.05, we reject the null hypothesis that all slope coefficients are equal to zero. Note that the #-test is a joint test so that even is all the -statistics are insignificant, the #-statistic can be highly significant. 3.5 Residual Tests Eviews provides tests for serial correlation, normality, heteroscedasticity, and ARCH in the residuals from the estimated equation. 3.5.1 Correlograms and Q-statistics This view displays the autocorrelation functions (ACF) and partial autocorrelation functions (PACF) up to the specified order of lags. These functions characterize the pattern of temporal dependence in the series and typically make 29 sense only for time series data. An example of correlograms and -statistics illustrates in Figure 3.2. Figure 3.2 An example of correlogram and -statistics from EViews 3.5.1.1 Autocorrelation The autocorrelation of a series at lag ( is estimated by = ∑K?7( − )( − ) ∑K?( − ) (3.15) where is the sample mean of . This is the correlation coefficient for values of the series ( periods apart. If is non-zero, it means that the series is first order serially correlated. If dies off more or less geometrically with increasing lag (, it is a sign that the series obeys a low-order AR process. If drops to zero after a small number of lags, it is a sign that the series obeys a low-order MA process. 30 Note that the autocorrelations estimated by EViews differ slightly from theoretical descriptions of estimator: where \ = ∑K?7(( − )( − ))/(& − () = ∑K?( − ) /& ∑ W]^_ K (3.16) . The difference arises since EViews employs the same overall sample mean as the mean of both and . While both formulations are consistent estimators, the EViews formulation biases the result toward zero in finite samples. Refer to the example in Figure 3.2, the dotted lines in the plots of autocorrelations are the approximate two standard error bounds computed as ± √K . If the autocorrelation is within these bound, it is not significantly different from zero at the 5% significance level. 3.5.1.2 Partial Autocorrelation The partial autocorrelation at lag ( is the regression coefficient on when is regressed on a constant, , … , . This is a partial correlation since it measures the correlation of values that are ( periods apart after removing the correlation from the intervening lags. If the pattern of the autocorrelation is one that can be captured by AR of order less than (, then the partial autocorrelation at lag ( will be close to zero. EViews estimates the partial autocorrelation at lag ( recursively by = b − ∑e? 1 − ∑ e? ,e e ,e e , for ( = 1 , for ( > 1 f (3.17) 31 where is estimated autocorrelation at lag ( and ,e = ,e − ,e . This is a consistent approximation of the partial autocorrelation. The algorithm is described in Box and Jenkins (1976). To obtain a more precise estimate of , simply run the regression: = , + , + ⋯ + , () + + (3.18) where is a residual. Refer to the example in Figure 3.2, the dotted lines in the plots of the partial autocorrelations are the approximate two standard error bounds computed as± √K . If the partial autocorrelation is within these bounds, it is not significantly different from zero at the 5% significance level. 3.5.1.3 Q-Statistics The Q-statistic at lag ( is a test statistic for the null hypothesis that is no autocorrelation up to order ( and is computed as: e = &(& + 2) = &−h e? (3.19) where e is the h-th autocorrelation and & is the number of these observations. If the series is not based upon results of ARIMA estimation, then under the null hypothesis, is asymptotically distributed as a with degrees of freedom equal to the number of autocorrelations. If the series represents the residuals from ARIMA estimation, the appropriate degrees of freedom should be adjusted to represent the number of autocorrelations less the number of AR and MA terms previously estimated. 32 3.5.2 Correlograms of Squared Residuals Refer to the example in Figure 3.2, the view displays ACF and PACF of the squared residuals up to any specified number of lags and computes the Ljung-Box Qstatistics for the corresponding lags. The correlograms of the squared residuals can be used to check ARCH in the residuals. If there is no ARCH in the residuals, the ACF and PACF should be zero at all lags and the Q-statistics should be not significant. 3.5.3 Histogram and Normality Test This view displays the frequency distribution of the series in a histogram. The histogram divides the series range into a number of equal length intervals and displays a count of the number of observations that fall into each interval. A complement of standard descriptive statistics is displayed along with the histogram. All of the statistics are calculated using the observations in the current sample. 3.5.3.1 Mean Mean refers to the average value of the series, obtained by adding up the series and dividing by number of observations. ∑K? = & (3.20) 33 3.5.3.2 Median Median refers to the middle value of the series when the values are ordered from the smallest to the largest. The median is a robust measure of the center of the distribution that is less sensitive to outliers than the mean. 3.5.3.3 Max and Min Max and min are representing the maximum and minimum values of the series in the current sample. 3.5.3.4 Standard Deviation Standard deviation is a measure of dispersion or spread in the series. The standard deviation is given by: =M ∑K?( − ) &−1 (3.21) where & is the number of observations in the current sample and is the mean of the series. 3.5.3.5 Skewness Skewness is a measure of asymmetry of the distribution of the series around its mean. Skewness is computed as: 34 Skewness = 2n( − o)p q p t 1 − p = =r s & ? (3.22) where is an estimator for the standard deviation that is based estimator for variance, = u(& − 1)/&. The skewness of a symmetry distribution, such as the normal distribution, is zero. Positive skewness means that the distribution has a long right tail and negative skewness implies that the distribution has a long left tail. 3.5.3.6 Kurtosis Kurtosis measures the peakedness or flatness of the distribution of the series. Kurtosis is computed as Kurtosis = 2n( − o)z q z K 1 − z = =r s & ? (3.23) where is again based on the biased estimator for the variance. The kurtosis of the normal distribution is 3. If the kurtosis exceeds 3, the distribution is leptokurtic relative to the normal. If the kurtosis is less than 3, the distribution is platykurtic relative to the normal. 3.5.3.7 Jarque-Bera Test Jarque-Bera (JB) is a test statistic for testing whether the series is normally distributed. The test statistic measures the difference of the skewness and kurtosis of the series with those from the normal distribution. The statistic is computed as 35 JB = & (Kurtosis − 3) (Skewness + ) 6 4 (3.24) Under null hypothesis of a normal distribution, the Jarque-Bera statistic is distributed as with 2 degrees of freedom. The reported Probability is the probability that a JB statistic exceeds the observed value under the null hypothesis. A small probability value leads to the rejection of the null hypothesis of a normal distribution. We reject the hypothesis of normal distribution at 5% significance level. 3.5.4 Serial Correlation Lagrange Multiplier Test This test is an alternative to Q-statistics for testing serial correlation. The test belongs to the class of asymptotic tests known as Lagrange Multiplier (LM) tests. Unlike the DW statistic for AR(1) errors, the LM test may be used to test for higher order ARMA errors and is applicable whether or not there are lagged dependent variables. Therefore, its use is recommended (in preference to the DW statistic) whenever we are concerned with the possibility that our errors exhibit autocorrelation. The null hypothesis of the LM test is that there is no serial correlation up to lag order *, where * is a pre-specified integer. The local alternative is ARMA(H,+) errors, where the number of lag terms * = max(H, +). Note that this alternative includes both AR(*) and MA(+) error processes, so that the test may have power against a variety of alternative autocorrelation structures. The test statistic is computed by an auxiliary regression as follows. First, suppose we have estimated the regression; = % , + (3.25) 36 where , are the estimated coefficients and are the errors. The test statistic for lag order * is based on the auxiliary regression for the residuals = − %,: = % + = 5 5 + 5? (3.26) Following the suggestion by Davidson and MacKinnon (1993), EViews sets any pre-sample values of the residuals to 0. This approach does not affect the asymptotic distribution of the statistic. Davidson and MacKinnon (1993) argue that doing so provides a test statistic which has better finite sample properties than an approach which drops the initial observations. This is a regression of the residuals on the original regressors % and lagged residuals up to order *. EViews reports two test statistics from this test regression. The #-statistic is an omitted variable test for the joint significance of all lagged residuals. Because the omitted variables are residuals and not independent variables, the exact finite sample distribution of the #-statistic under is still not known. The Obs*R-squared statistic is the Breusch-Godfrey LM test statistic. This LM statistic is computed as the product of the number of observations and the (uncentered) from the test regression. Under quite general conditions, the LM test statistic is asymptotically distributed as a (*). The serial correlation LM test is available for residuals from either least squares or two-stage least squares (2SLS) estimation. The original regression may include AR and MA terms, in which case the test regression will be modified to take account of the ARMA terms. Testing in 2SLS settings involves additional complications. 37 3.5.5 The ARCH-LM Test The ARCH test is a Lagrange multiplier test for ARCH in the residuals by Engle (1982). This particular heteroscedasticity specification was motivated by the observation that in many financial time series, the magnitude of the residuals appeared to be related to the magnitude of the recent residuals. ARCH in itself does not invalidate standard LS inference. However, ignoring ARCH effects may result in loss of efficiency. The ARCH-LM test statistic is computed from an auxiliary test regression. To test the null hypothesis that there is no ARCH up to order + in the residuals, we run the regression: > = , + = ,5 5 + 5? (3.27) where is the residual. This is a regression of the squared residuals on a constant and lagged squared up to order + . EViews reports two test statistics from this test regression. The #-statistic is an omitted variable test for the joint significance of all lagged squared residuals. The Obs*R-squared statistic is Engle’s LM statistic, computed as the product of the number of observations and the from the test regression. The exact finite sample distribution of the # -statistic under is not known, but the LM test statistic is asymptotically distributed as (+) under quite general conditions. 3.6 Unit Root Tests for Stationarity EViews provides a variety powerful tools for testing a series for the presence of a unit root such as Augmented Dickey-Fuller (1979) and Phillips-Perron (1988) tests. These tests are used to determine whether the series is stationary or it should undergo differencing to achieve stationarity. 38 3.6.1 The Augmented Dickey-Fuller Test Sometimes, time series data are not in a stationary form. To transform it into a stationary form, an easy way is to difference the time series data. One way is to use the Augmented Dickey-Fuller (ADF) -statistic. The ADF test constructs a parametric correction for higher-order correlation by assuming that the series follows an AR( * ) process and adding * lagged difference terms of the dependent variable to the right-hand side of the test regression as follow ∆ = + ′ + , ∆ + , ∆ + ⋯ + , ∆ + (3.28) where are optional exogenous regressors which may consist of constant, or a constant and trend. The null hypothesis of the ADF -test is : = 0 (3.29) which means that the data needs to be differenced to make it stationary. The alternative hypothesis of : < 0 (3.30) which means that the data is trend stationary and needs to be analyzed by means of using a time trend in the regression model instead of differencing the data. The test statistic is conventional -ratio for : = () where is the estimate of , and () is the coefficient standard error. (3.31) 39 3.6.2 The Phillips-Perron Test Phillips and Perron (1988) propose an alternative method of controlling for serial correlation when testing for a unit root called Phillips-Perron (PP) test. The PP method estimates the non-augmented Dickey-Fuller test equation: ∆ = + ′ + (3.32) It modifies the -ratio of the coefficient so that serial correlation does not affect asymptotic distribution of the test statistic. The PP test is based on the statistic: ( − )(()) ̃ = ( )J − 2J (3.33) where is the estimate of , and is the -ratio of , () is the coefficient standard error, and is the standard error of the test regression. In addition, is a consistent estimate of the error variance. The remaining term, is an estimator of the residual spectrum at frequency zero. 3.7 Forecast Performance Measures There are several ways to evaluate the performances of forecasting models. In this study, mean absolute error, root mean squared error, mean absolute percentage error, and Theil inequality coefficient will be calculated. Suppose the forecast sample is h = & + 1, & + 2, … , & + ℎ. We will denote the actual and forecasted value in period as and respectively. 40 3.7.1 Mean Absolute Error Mean absolute error (MAE) is calculated from MAE = ∑K7 − | ?K7| ℎ (3.34) The MAE reflects the typical error. It does not distinguish between variance and bias. It is appropriate when the cost function is linear. 3.7.2 Root Mean Squared Error Root mean squared error (RMSE) is calculated from RMSE = M ∑K7 − ) ?K7( ℎ (3.35) The RMSE is similar to MAE. The MAE and RMSE depend on the scale of the dependent variable. These should be used as relative measures to compare forecasts for the same series across different models. 3.7.3 Mean Absolute Percentage Error Mean absolute percentage error (MAPE) is calculated from K7 100 − MAPE = = ℎ ?K7 (3.36) 41 The MAPE is similar to MAE except that it is dimensionless. It will be helpful in making comparison among forecasts from different situations. For instance, to compare forecasting methods in two different situations with different units of measure, one can calculate the MAPEs and then average across situation. (Armstrong, 1985) When the cost of errors is more closely related to the percentage error than to the unit error, the MAPE is appropriate. 3.7.4 Theil Inequality Coefficient The measure of Theil inequality coefficient (Theil-U) is calculated as follows: Theil − U = ∑?K7( − ) ℎ K7 K7 M∑?K7 ℎ + K7 ∑?K7 (3.37) ℎ While MAPE and Theil-U are scale invariant, the Theil-U always lies between zero and one, where zero indicates a perfect fit. 3.7.5 Mean Squared Forecast Error The mean squared forecast error (MSFE) can be decomposed as: ∑( − ) ∑ = O T − + (W − W ) + 2(1 − H)W W ℎ ℎ (3.38) 42 where ∑ /ℎ, , W , W are the means and (biased) standard deviations of and , and H is the correlation between and . The proportions are defined as bias, variance and covariance. The bias proportion tells us how far the mean of the forecast is from the mean of the actual series. It is calculated as Bias Proportion = ∑ Or s − T ℎ ∑( − ) ℎ (3.39) The variance proportion tells us how far the variation of the forecast is from the variation of the actual series. It is calculated as (W − W ) Variance Proportion = ∑( − ) ℎ (3.40) The covariance proportion measures the remaining unsystematic forecasting errors. It is calculated as Covariance Proportion = 2(1 − H)W W ∑( − ) ℎ (3.41) Note that the bias, variance and covariance proportions add up to one. If the forecast is good, the bias and variance proportions should be small so that most of the bias should be concentrated on the covariance proportions. 43 3.8 Box-Jenkins Methodology The Box-Jenkins methodology of forecast is different from most methods because it does not assume any particular pattern in the historical data of the series to be forecast. It uses an iterative approach of identifying a possible model from a general class of models. The chosen model is then checked against the historical data to see whether it accurately describes the series. The models fits well if the residuals are generally small, randomly distributed, and contain no useful information. If the specified model is not satisfactory, the process is repeated using a new model designed to improve on the original one. This iterative procedure continues until a satisfactory model is found. At this point, the model can be used for forecasting. 3.8.1 ARIMA Model The acronym ARIMA stands for ‘‘Auto-Regressive Integrated Moving Average’’, whose model is a generalization of an autoregressive moving average (ARMA) model. Box and Jenkins (1976) introduced the ARIMA (*,),+) class of processes which have been applied to a wide variety of time series forecasting applications. They are applied in cases where data show evidence of non-stationarity, where an initial differencing step can be applied to remove the non-stationarity. The general methodology of the Box–Jenkins approach involves model identification, model estimation and diagnostic checking followed by forecasting. There is a model that mixes the AR(*) and MA(+ ) models, called non- seasonal mixed autoregressive-moving average (ARMA) of order (*, +), = + + + ⋯+ + − − − ⋯ − > > (3.42) 44 with , , ,…,> are random shocks, , ,…, and , ,…,> are the autoregressive (AR) parameters and moving average (MA) parameters respectively. It can be proven that for this model, = o(1 − − −⋯− ). A highly useful operator in time-series theory is the lag or backshift operator, " defined by " = . Model for non-seasonal series are denoted by ARIMA(*,),+). Here * indicates the order of the autoregressive part, ) indicates the amount of differencing, and + indicates the order of the moving average part. If the original series is stationary, ) = 0 and the ARIMA models reduce to the ARMA models. The difference linear operator (∆), is defined by ∆ = − = − " = (1 − ") (3.43) The stationary series is obtained as the )-th difference (∆¡ ) of , = ∆¡ = (1 − ")¡ (3.44) ARIMA (*,),+) has the general form: (")(1 − ")¡ = + > (") (") 3.8.2 = + > (") (3.45) (3.46) Model Identification For the Box-Jenkins approach, it cannot be directly applied if the series is non-stationary. It is important to know whether the data contain any trend and seasonal components. We analyzed occurrence of an upward or downward trend in oil price movement and we also check for seasonality from the ACF and PACF. Hence, the input series for ARIMA needs to be stationary, that is, it should have a constant mean, variance and autocorrelation through time. To determine the 45 stationarity of the data, we can check through ADF or PP test, or even look through pattern of correlogram of ACF and PACF. If a graph of ACF of the time series values either cuts off fairly quickly or dies down fairly quickly, then the time series values should be considered stationary. If a graph of ACF dies down extremely slowly, then the time series values should be considered non-stationary. If the series is not stationary, it can be transformed to a stationary series by differencing. Differencing is done until a plot of data indicates the series varies about a fixed level and the ACF dies down fairly rapidly. The number of differences required to achieve stationarity is denoted by ). The behaviour of the ACF and PACF can be used to help us identify which model describes the time series value. Table 1 summarizes the behaviour of the ACF and PACF for each of the general non-seasonal models that have been discussed. Table 3.1 : The behaviour of ACF and PACF for each of the general models Model ACF Moving average (MA) of order + Cuts off after lag Dies down = + − − − ⋯ − + = + + Dies down > > Autoregressive (AR) of order * Mixed + + ⋯+ autoregressive-moving (ARMA) of order (*,+) = + + + ⋯+ − − − ⋯ − > > average Dies down PACF Cuts off after lag * Dies down + To judge the significance of ACF and PACF, the values are compared with ±2/√&. These limits work well when & is large. 46 According to the principle of parsimony, simple models are preferred to complex models when all things being equal (Hanke et al., 2001). With a limited amount of data, it is relatively easy to find a model with large number of parameters that fits the data well. However, forecasts from such a model are likely to be poor because much of the variation in the data due to random error is being modelled. The goal is to develop the simplest model that provides an adequate description of the major features of the data. 3.8.3 Parameter Estimation Once a tentative model has been selected, the parameters for that model must be estimated. The parameters in ARIMA models are estimated by minimizing the sum of squares of the fitting errors. In general, these least squares estimates must be obtained using a nonlinear least squares procedure. A nonlinear least squares procedure is simply an algorithm that finds the minimum of the sum of squared errors function. Once the least squares estimates and their standard errors are determined, values can be constructed and interpreted in the usual way. Parameters that are judged significantly different from zero are retained in the fitted model; parameters that are not significant are dropped from the model. 3.8.4 Diagnostic Checking Before using the model for forecasting, it must be checked for adequacy. Basically, a model is adequate if the residuals cannot be used to improve the forecasts. That is the residuals should be random. An overall check of model adequacy is provided by a test based on the Ljung-Box -statistic. This test looks at the sizes of the residual autocorrelations as a group. If the *-value associated with the -statistic is small, the model is considered 47 inadequate. One should consider a new or modified model and continue the analysis until a satisfactory model has been determined. Judgment plays an important role in model building effort. When two simple competing models may adequately describe the data and a choice may be made on the basis of the nature of the forecasts. Also, few large residuals may be ignored if they can be ignored by unusual circumstances, and the model is adequate of the observations. 3.8.5 Forecasting Financial decisions often involve a long-term commitment of resources, the returns to which will depend upon what happens in the future (Brooks, 2008). In this context, the decisions made today will reflect forecasts of the future state of the world, and the more accurate those forecasts are, the more utility is likely to be gained from acting on them. Time series forecasting involves trying to forecast the future values of a series given its previous values of an error term. If the magnitudes of the most recent errors tend to be consistently larger than previous errors, it may be time to reevaluate the model. Although ARIMA models involve differences, forecasts for the original series can be always computed directly from the fitted model. 48 3.9 GARCH Process Most of the time, GARCH models can accommodate volatility clustering and leptokurtosis very easily. Indeed, they are tailor-made for volatility clustering and it produces returns with fatter than normal tails even if the innovations and the random shocks are normally distributed (Dowd, 2002). Whenever we estimate a GARCH process, the model of the mean is = 2( |Ω) + = ′, + (3.47) (3.48) where 2( |Ω ) = ′,. The mean equation is a function of exogenous variables with error term given the information set available at time − 1. Bollerslev (1986) extended Engle’s original work on ARCH by developing a technique that allows the conditional variance to be an ARMA process. Let the error process to be where ~BB) ¢(0,1). = (3.49) > ? ? = + = + = , (3.50) Since is a white noise process, the conditional and unconditional means of are equal to zero. Taking the expected value of , it can be verified that 2n q = 2n q = 0 (3.51) 49 Thus, the conditional variance of will be 2n |Ωq = (3.52) The GARCH( + , * ) allows both AR and MA components in the heteroscedastic variance. If we set * = 0 and + = 1, it is clear that the first order ARCH model is simply a GARCH(1,0) model. Hence, if all values of , equal zero, the GARCH(+,*) model is equivalent to an ARCH(+) model. The benefits of the GARCH model should be clear; a higher order ARCH model may be more parsimonious GARCH representation that is much easier to identify and estimate. This is particularly true since all coefficients, > 0, ≥ 0, and , ≥ 0. Moreover, all characteristic roots in the variance equation must lie inside the unit circle to ensure that the variance is finite. Clearly, the more parsimonious model will entail fewer coefficient restrictions. The key feature of GARCH models is that the conditional variance of the disturbances of the sequence constitutes an ARIMA process. Hence, it is to be expected that the residuals from a fitted ARIMA model should display this characteristic pattern. Suppose that the estimate of is an ARIMA process. If the model of is adequate, the ACF and PACF of the residuals should be indicative of a white noise process. However, the ACF of the squared residuals can help identify the order of the GARCH process. 3.9.1 GARCH(1,1) Model One popular GARCH model is GARCH(1,1) because this model is easy to apply. The model often fits the data fairly well when number of parameters is small. A high value of , means that volatility is persistent and it takes a long time to change. A high value of means volatility is spiky and quick to react to market movements. The common estimates of , are over 0.7, but is usually less than 0.25 (Alexander, 1998). The GARCH(1,1) with positive intercept also has the 50 attraction that it allows us to model the volatility as mean-reverting. If the volatility is high, it will tend to fall over time. If the volatility is low, it will tend to rise over time. (Dowd, 2002) The simplest GARCH(1,1) specification is: = o + = ~BB) ¢(0,1) = + + , > 0, ≥ 0 and , ≥ 0 (3.53) (3.54) (3.55) (3.56) (3.57) where represents the dependent variable over period and o is a constant mean. Since is the one-period ahead forecast variance based on past information, it is called conditional variance. The conditional variance equation specified contains a constant term, ; news about volatility from the previous period, measured as the ; last period’s forecast lag of the squared residual from the mean equation, variance, . The last condition is to ensure nonnegative conditional variances, while the condition 0 < ( + , ) < 1 ensures stationarity and finite variance of the unconditional returns. This specification is often interpreted in a financial context, where an agent or trader predicts this period’s variance by forming a weighted average of a long term average ( ), the forecasted variance from last period ( ), and information about volatility observed in the previous period ( ). If the asset return was unexpectedly large in either the upward or the downward direction, then the trader will increase the estimate of the variance for the next period. This model is also consistent with the volatility clustering often seen in financial returns data, where large changes in returns are likely to be followed by further large changes. 51 The GARCH(1,1) model is very popular specification because it fits many data series well. It shows that the volatility changes with lagged shocks, , but there is also momentum in the system working via . This model becomes popular because it can capture long lags in the shocks with only a few parameters. A GARCH(1,1) model with three parameters ( , ,,) can capture similar effects to an ARCH(+) model requiring the estimation of (+ + 1) parameters, where + is large (+ ≥ 6) (Hill et al., 2008). 3.9.2 Parameter Estimation The parameters in GARCH models can be estimated by maximum likelihood estimator. In order to estimate models from the GARCH family, another technique known as maximum likelihood is employed. Essentially, the method works by finding the most likely values of the parameters given the actual data. More specifically, a log-likelihood function is formed and the values of the parameters that maximise it are sought. Maximum likelihood estimation can be employed to find parameter values for both linear and non-linear models (Brooks, 2008). First, specify the appropriate equations for the mean and the variance. Then, specify the log-likelihood function to maximise under a normality assumption for the disturbances. Finally, the computer will maximise the function and generate parameter values that maximise the log-likelihood function and will construct their standard errors. Suppose the values of drawn from a normal distribution having a mean of zero and conditional variance . From a standard distribution theory, the likelihood of realization of is 1 = O u2S where is the likelihood of . IJ ]J T ¤] (3.58) 52 Since the realization of are independent, the likelihood of the joint realizations of , , … , K is the product in the individual likelihoods. Hence, if all have the same variance, the likelihood of the joint realizations is K 1 $ = ¥O u2S ? IJ ]J T ¤] (3.59) so that the log likelihood function is K K ? ? & 1 1 ln $ = − ln(2S) − = ln − = 2 2 2 (3.60) Now, suppose that = − o and the conditional variance is = + + , . Substituting for and yields K &−1 1 ln $ = ln(2S) − = ln + + , 2 2 K ? 1 ( − o) − = 2 + + , ? (3.61) Note that the initial observation is lost since is outside the sample. Once it , it is possible to maximize ln $ with respect to , substitutes ( − o) for and ,. There are no simple solutions to the first order conditions for a maximum. Fortunately, EViews are able to select the parameter values that maximize this log likelihood function. 3.93 Diagnostic Checking In addition to provide a good fit, an estimated GARCH model should capture all dynamic aspects of the model of the mean and the model of the variance. The estimated residuals should be serially uncorrelated and should not display any 53 remaining conditional volatility. To ensure that the model has captured the properties by standardising the residuals, ̂ = ̂ ⁄ . Thus, standardize each residual using its conditional standard deviation. The resulting series should have a mean and a variance of unity (Enders, 2004). If there is any serial correlation in the ̂ sequence, the model of the mean is not properly specified. To test the model of the mean, form the Ljung-Box statistics for the ̂ sequence. Do not reject the null hypothesis that the various statistics are equal to zero. To test the remaining GARCH effects, form Ljung-Box -statistics of the squared standardized residuals. If there is no remaining GARCH effects, then should not reject the null hypothesis that the sample values of the -statistics are equal to zero. Hence, the properties of the ̂ sequence should mimic those of a white noise process. 3.9.4 Forecasting The one-step-ahead forecast of the conditional variance is easy to obtain. If update by one period, we will get 7 = + + , (3.62) To obtain the h-step-ahead forecasts, begin from using the fact that = , so that 7e = 7e 7e . Then take the conditional expectation when update by h periods, such as 237e §Ω 4 = 23 7e 7e §Ω 4 . Since 7e is independent of 7e and 23 7e §Ω 4 = 1, it follows that 237e §Ω 4 = 237e §Ω 4. Then, use 237e §Ω 4 = 237e §Ω 4 to obtain the forecasts of the conditional variance of the GARCH(1,1) process. This will yield 54 7e = + 7e + , 7e (3.63) Taking the conditional expectation will give 237e §Ω 4 = + 2(7e |Ω ) + , 2(7e |Ω ) (3.64) By combining this relationship with 237e §Ω 4 = 237e §Ω 4 , it can be verified that 237e §Ω 4 = + ( + , )2(7e |Ω ) (3.65) Thus, given that 7 , the above equation used to forecast all subsequent values of the conditional variance is 237e §Ω 4 = ¨1 + ( + , ) + ( + , ) + ⋯ + ( + ,)e © + ( + , )e (3.66) When ( + , ) < 1, the conditional forecasts of 7e will converge to long- run value 2( ) = 3.10 1 − − , (3.67) Comparison of ARIMA and GARCH Processes In table 3.2, Gaussian white noise, ARIMA, GARCH, and ARMA/GARCH processes are compared according to various properties such as conditional means, conditional variances, conditional distributions, marginal variances, and distributions. (Ruppert, 2004) 55 Table 3.2 : Comparison of ARIMA and GARCH models Property Gaussian ARIMA GARCH White Noise Conditional ARMA/ GARCH constant non-constant 0 non-constant constant constant non-constant non-constant normal normal normal normal constant constant constant constant normal normal heavy-tail heavy-tail mean Conditional variance Conditional distribution Marginal mean and variance Marginal distribution 3.11 Concluding Remarks Using the methodology described in this chapter, daily crude oil prices will be analysed in the next chapter. CHAPTER 4 RESULTS AND ANALYSIS 4.1 Introduction Crude oil prices variability is important to mankind. Forecasting crude oil price is important because it affects many related sectors which heavily rely on the use of crude and its refineries. The objectives in this study are estimating and forecasting volatile crude oil prices with ARIMA and GARCH models using EViews software. The performances of these two models for forecasting daily crude oil prices data will also be compared. 4.2 Data Management The WTI daily crude oil prices data are obtained from EIA time-varying from 2nd January 1986 to 30th September 2009. The data are divided into two parts. One is for models’ estimation and another is for forecasting oil prices series purposes. The first part is in-sample period varying from 2nd January 1986 to 30th June 2009. It will be used to estimate the models. Meanwhile, the second part which is called out-ofsample period, varies from 1st July 2009 to 30th September 2009. 57 4.3 Crude Oil Prices Time Series The in-sample period of crude oil prices data from 2nd January 1986 to 30th June 2009 will be plotted with aid of Eviews. We have to determine the trend of the series of being constant, linear or non-linear and etc. The oil prices series is shown in Figure 4.1. 160 140 120 100 80 60 40 20 0 86 88 90 92 94 96 98 00 02 04 06 08 PRICE Figure 4.1 The time series for WTI daily crude oil prices In Figure 4.1 it can be seen that the oil prices have mainly fluctuated in the range of about $10 to $145. So, we consider its characteristics to be non-linear (Fan et al., 2008). It can observe that the oil price was stable around $20 during the 1990s. During the 1990-1991 periods, there occurs a spike when the Gulf war started. In 2000s, the oil prices fluctuations are much greater than the ones observed in the 1990s due to a great number of contributing causes. 58 Using EViews, a histogram and a normality test for the crude oil prices series is plotted in Figure 4.2. 2000 Series: PRICE Sample 1/02/1986 6/30/2009 Observations 6129 1600 1200 800 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis 31.57972 21.53000 145.3100 10.25000 22.64852 2.178150 7.995938 Jarque-Bera Probability 11220.35 0.000000 400 0 20 Figure 4.2 40 60 80 100 120 140 Histogram and normality test on WTI daily crude oil prices From the histogram, it can be seen that a great number of observations are located around $20. As summarized in Figure 4.1, the mean and standard deviation of the crude oil are $31.58 and $22.65 respectively. The value for skewness is 2.1782 and kurtosis is 7.9959 which imply that the graph is asymmetric and leptokurtosis. Jarque-Bera test indicates that we do not reject null hypothesis of being normal distribution at 5% significance level. However, the original series is not suitable to be used for estimating and forecasting using any models. 4.4 Stationary Series A stationary series must be obtained before it can be used to estimate and develop a model. The unit roots test will help us to determine the stationarity of a 59 series. For example, ADF and PP tests are used to check the stationarity of daily crude oil prices series. Initially, we check the stationarity of the original oil prices. This is tabulated in Table 4.1. Table 4.1 : ADF test for crude oil prices Null Hypothesis: PRICE has a unit root Exogenous: Constant Lag Length: 5 (Automatic based on SIC, MAXLAG=33) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -1.123670 0.7087 Test critical values: 1% level -3.431239 5% level -2.861817 10% level -2.566960 *MacKinnon (1996) one-sided p-values. According to the ADF test for daily crude oil prices time series in Table 4.1, the ADF test statistic is −1.1237 which is greater than test critical values of −3.4312, −2.8618 and −2.5670 at 1%, 5% and 10% significance levels. The *- value of 0.7087 strongly disagrees that the series is stationary. Thus, we do not reject the null hypothesis of being non-stationary. Table 4.2 : PP test for crude oil prices Null Hypothesis: PRICE has a unit root Exogenous: Constant Bandwidth: 14 (Newey-West using Bartlett kernel) Phillips-Perron test statistic Test critical values: 1% level 5% level 10% level *MacKinnon (1996) one-sided p-values. Adj. t-Stat Prob.* -1.181442 0.6848 -3.431238 -2.861817 -2.566960 60 We also test the crude oil prices series with PP test in Table 4.2. A similar result is obtained where the PP test statistic of −1.1814 is greater than its test critical values at1%, 5% and 10% significance levels. We do not reject the null hypothesis of being non-stationary since the one-sided *-value of 0.684 is large. Thus, the crude oil prices time series need to be differenced to obtain a stationary series. The process is continued until a stationary series to be found. We check the stationarity for the first order difference of crude oil prices series. Similarly, we use ADF and PP tests to determine the series’ stationarity. Table 4.3 : ADF test for first difference of oil prices Null Hypothesis: D(PRICE) has a unit root Exogenous: Constant Lag Length: 4 (Automatic based on SIC, MAXLAG=33) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -36.92943 0.0000 Test critical values: 1% level -3.431239 5% level -2.861817 10% level -2.566960 *MacKinnon (1996) one-sided p-values. The ADF test for first order difference from original crude oil prices series is shown in Table 4.3. From Table 4.3, the -statistic for the first lagged difference series is −36.9294 which is much smaller than the 1% significance level of test critical value. Hence, the zero in *-value indicates the ADF -statistic is significant. Thus, we reject the null hypothesis that states that the first order difference for daily crude oil prices series is stationary. 61 Table 4.4 : PP test for first difference for crude oil series Null Hypothesis: D(PRICE) has a unit root Exogenous: Constant Bandwidth: 13 (Newey-West using Bartlett kernel) Phillips-Perron test statistic Test critical values: Adj. t-Stat Prob.* -83.31242 0.0001 1% level -3.431238 5% level -2.861817 10% level -2.566960 *MacKinnon (1996) one-sided p-values. The PP test on the first order difference from the original crude oil prices is shown in Table 4.4. It gives a similar inference to ADF test for the first order difference of crude oil prices series. The adjusted -statistic for first order lagged difference series is −83.3124 , which is very small compared to the 1% level of test critical value of −3.4312. The small *-value of 0.0001 indicates that the PP test is significant. Thus, we make the same inference as in the ADF test where the first order difference series is stationary. First lagged difference from the original data time series, which is a stationary series is shown in Figure 4.3. 62 20 16 12 8 4 0 -4 -8 -12 -16 86 88 90 92 94 96 98 00 02 04 06 08 D(PRICE) Figure 4.3 First order difference crude oil prices series In Figure 4.3, it can be seen that the difference with respect to the first order for crude oil prices series is stationary because most of the price values are located around mean of zero. However, there are some spikes in the figure, representing high volatility periods. Figure 4.4 illustrates the histogram and normality distribution test statistics including mean, median, maximum and minimum values, standard deviation, skewness, kurtosis, and Jarque-Bera test of first order difference crude oil prices series. 63 3000 Series: D(PRICE) Sample 1/02/1986 6/30/2009 Observations 6128 2500 2000 1500 1000 500 0 -15 -10 Figure 4.4 -5 0 5 10 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis 0.007223 0.000000 18.56000 -14.76000 1.017233 -0.011305 41.42323 Jarque-Bera Probability 376960.2 0.000000 15 Histogram and normality test of first order difference series In Figure 4.4, the histogram is centred and peaked at zero. The mean value is 0.0072 and standard deviation 1.0172 which gives a standard normal distribution of ¢(0,1). The median is equal to 0 because most of the values after first lagged difference fall within the intervals of zero. The skewness and kurtosis values are −0.0113 and 41.4232 respectively which show that the distribution is slightly asymmetric and highly leptokurtosis. Jarque-Bera test indicates that the null hypothesis which claims that the first order difference series is normally distributed, is not rejected at a 5% significance level. With the information obtained above, we have the stationary series after one lagged difference from the original crude oil prices series. In the next step, we will use the first order difference for crude oil prices series to find our models using BoxJenkins and GARCH approaches. 64 4.5 ARIMA Model One of our objectives is to forecast the future crude oil prices with ARIMA model. For instance, we have the stationary series after differencing of one lagged. Now, the model that we are looking at is ARIMA(*,1,+). We have to identify the model, estimate suitable parameters, diagnostic checking for residuals and finally achieve our objective of forecasting the future crude oil prices. 4.5.1 ARIMA Model Identification Firstly, we compute the series correlogram which consists of ACF and PACF values as in Figure 4.5. We also calculate the Ljung-Box -statistics. We observe the patterns of the ACF and PACF, then determine the parameter values * and + for ARIMA model. 65 Figure 4.5 Correlogram of the first order difference series From the Figure 4.5, there are 20 lags of correlogram computed from EViews. The autocorrelation and partial autocorrelation charts are located at the left-hand side of the figure and values for ACF, PACF and Q-statistic are at the right-hand side. However, we cannot identify any model from the correlogram in Figure 4.5. The values of ACF and PACF are relatively small and lie within the confidence intervals. Therefore, no ARIMA model can be identified from the first order difference of crude oil prices series. The process is continued until another higher order of difference that is stationary is found. For this purpose, a second order lagged difference from the original series is obtained. ADF test is conducted on this series to check for stationarity. This is tabulated in Table 4.5. 66 Table 4.5 : ADF test for second order difference series Null Hypothesis: D(PRICE,2) has a unit root Exogenous: Constant Lag Length: 26 (Automatic based on SIC, MAXLAG=33) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -27.10960 0.0000 Test critical values: 1% level -3.431243 5% level -2.861819 10% level -2.566961 *MacKinnon (1996) one-sided p-values. The ADF test shown in Table 4.5 shows that the series is stationary. The - statistic of −27.1096 is smaller than 1% of test critical value. The *-value for ADF test is zero indicating that we have sufficient evidence to reject the null hypothesis of the series being non-stationary. Next, we plot the graph of second order difference for crude oil prices series using EViews software. This is plotted in Figure 4.6. 67 20 10 0 -10 -20 -30 -40 86 88 90 92 94 96 98 00 02 04 06 08 D(PRICE,2) Figure 4.6 First order difference of crude oil prices series Figure 4.6 shows the difference of second order for the crude oil prices series from the original data. The graph also shows the characteristic of series being stationary because most of the values lie around mean zero. There are some spikes in the graph representing some of the high volatility periods but it is relatively lesser than the first order lagged difference series. In Figure 4.7, using the EViews software, a histogram and normality test is plotted for the second order lagged series. 68 3000 Series: D(PRICE,2) Sample 1/02/1986 6/30/2009 Observations 6127 2500 2000 1500 1000 500 0 -30 Figure 4.7 -20 -10 0 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis -0.000341 0.000000 14.89000 -33.32000 1.480933 -1.643366 58.27611 Jarque-Bera Probability 782788.3 0.000000 10 Histogram and normality test of second order difference series In Figure 4.7, the histogram is centred and peaked at zero. The mean value is −0.0003 and the standard deviation is 1.4809 which also gives a standard normal distribution of ¢(0,1). Similarly, the median is equal to 0 where most of the values after second order difference fall within the intervals of zero. The skewness and kurtosis values are −1.6434 and 58.2761 respectively, which indicate that its distribution is positively skewed and highly leptokurtosis. Jarque-Bera test shows that the null hypothesis which claims that the series is normally distributed, is not rejected at 5% of significance level. The correlogram for ACF and PACF of the second order difference series is plotted in Figure 4.8. 69 Figure 4.8 Correlogram of the second order difference series In Figure 4.8, 20 lags of autocorrelation and partial autocorrelation are generated. The ACF dies out after lag 1 and PACF dies out slowly after lag 1. Thus, the * and + values for the ARIMA(*,2,+) model are set at 1 respectively. So, we temporarily set our ARIMA model to be ARIMA(1,2,1). 4.5.2 Parameter Estimation ARIMA(1,2,1) Model When we have identified the ARIMA model, the next step is to estimste the parameter coefficients. The parameter estimation of the model is conducted using the EViews software. Table 4.6 tabulates the results. 70 Table 4.6 : Estimation equation of ARIMA(1,2,1) Dependent Variable: D(PRICE,2) Method: Least Squares Date: 10/03/09 Time: 16:03 Sample (adjusted): 1/07/1986 6/30/2009 Included observations: 6126 after adjustments Convergence achieved after 15 iterations Backcast: 1/01/1986 Variable Coefficient Std. Error t-Statistic Prob. C 4.84E-06 3.34E-05 0.144822 0.8849 AR(1) -0.060846 0.012790 -4.757425 0.0000 MA(1) -0.997454 0.001233 -808.7012 0.0000 R-squared 0.529073 Mean dependent var -0.000356 Adjusted R-squared 0.528920 S.D. dependent var 1.481053 S.E. of regression 1.016525 Akaike info criterion 2.871147 Sum squared resid 6327.040 Schwarz criterion 2.874438 F-statistic 3439.512 Prob(F-statistic) 0.000000 Log likelihood -8791.323 Durbin-Watson stat 2.005690 Inverted AR Roots -.06 Inverted MA Roots 1.00 From the -statistics for the coefficient variables AR(*) and MA(+) in Figure 4.6, the null hypotheses that the coefficients are equal to zero are rejected. The estimated parameter coefficients by ARIMA(1,2,1) model gives = 4.84 × 10° , = −0.0608 and = −0.9975. The value for is = 0.5291, which implies that the dependency on the estimated value by the series is not strong. The DW statistic is approximately 2 due to the existence of a positive serial correlation in the residuals. Thus, the model equation can be formed as = 4.84 × 10° − 0.0608 + − 0.9975 (4.1) 71 4.5.3 Diagnostic Checking ARIMA(1,2,1) Model After we have estimated the parameters for ARIMA(1,2,1) model, the next step will be diagnostic checking of the model adequacy. Figure 4.9 illustrates the correlogram of residuals for ARIMA(1,2,1). Figure 4.9 Correlogram of residuals for ARIMA(1,2,1) From Figure 4.9, we found that the residuals of the ACF and the PACF are both relatively small or approximately equal to zero. The -statistic shows that the model is adequate. An alternative test to Q-statistics for testing serial correlation is BreuschGodfrey LM test. This test is on the null hypothesis of the LM test claiming that 72 there is no serial correlation up to lag order *. The result of the Breusch-Godfrey LM test is tabulated in Table 4.7. Table 4.7 : Serial correlation Breusch-Godfrey LM test for ARIMA(1,2,1) Breusch-Godfrey Serial Correlation LM Test: F-statistic 16.48843 Probability 0.000050 Obs*R-squared 16.45170 Probability 0.000050 From Table 4.7, the #-statistic and Breusch-Godfrey LM test statistic are 16.4884 and 16.4517 respectively. Both of the *-values of #-statistic and Obs*R- squared are approximately zero indicate that there are significantly rejected the null hypothesis of no serial correlation up to lag *. Once again, we justify the model is adequate. Next, in Figure 4.10, we plot the residuals plot for second order difference series data. Since the residuals are also changing with time, thus a volatile series is obtained. 73 20 15 10 5 0 -5 -10 -15 86 88 90 92 94 96 98 00 02 04 06 08 D(PRICE,2) Residuals Figure 4.10 Second order difference of residuals plot In Figure 4.10, we can see some spiky residuals in high volatile periods such as the Gulf war in 1990-91 and during global economic crisis in 2008. The residuals plots are quite similar to the one for difference series. However, the dependent variable axis range is narrower. 74 2800 Series: Residuals Sample 1/07/1986 6/30/2009 Observations 6126 2400 2000 1600 1200 800 400 0 -10 Figure 4.11 -5 0 5 10 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis -0.000734 0.009876 18.90316 -13.73422 1.016359 0.049838 41.49635 Jarque-Bera Probability 378275.1 0.000000 15 Histogram and normality test for residuals ARIMA(1,2,1) In Figure 4.11, the histogram and normality test are plotted. The mean value of the residuals is −0.0007 and the standard deviation is 1.0164 which is standard normal distributed ¢(0,1). The values of skewness and kurtosis are 0.0498 and 41.4964 respectively. This means that he residuals have excessive kurtosis and slightly skewed to the left. Jarque-Bera test shows that the residuals series do not reject the null hypothesis of normally distributed at 5% significance level. 4.5.4 Forecasting using ARIMA(1,2,1) Model In the next step, the forecast of crude oil prices using ARIMA(1,2,1) model is conducted. EViews software provides the one-step ahead static forecasts which are more accurate than the dynamic forecasts. Static forecasting extends the forward recursion through the end of the estimation sample, allowing for a series of one-step ahead forecasts of both the structural model and the innovations. When computing static forecasts, EViews uses the entire estimation sample to backcast the innovations. 75 The duration of forecasts is from 1st July 2009 to 30th September 2009. The forecasts are plotted in Figure 4.12. 76 72 68 64 60 56 2009M07 2009M08 2009M09 PRICE_ARIMA_F Figure 4.12 Forecast crude oil prices by ARIMA(1,2,1) model In Figure 4.12, the solid line represents the forecast value of crude oil prices from 1st July 2009 to 30th September 2009. Meanwhile, the dotted lines which are above or below the forecasted daily crude oil prices show the forecast prices with ±2 of standard errors. The forecast evaluation for ARIMA(1,2,1) model is shown in Table 4.8 76 Table 4.8 : Forecast evaluation for ARIMA(1,2,1) model Forecast: PRICE_ARIMA_F Actual: PRICE Forecast sample: 7/01/2009 9/30/2009 Included observations: 66 Root Mean Squared Error Mean Absolute Error Mean Absolute Percentage Error Theil Inequality Coefficient Bias Proportion Variance Proportion Covariance Proportion 1.704196 1.280483 1.884376 0.012492 0.000063 0.000068 0.999869 In Table 4.8, some quality forecasting measurements such as RMSE, MAE, MAPE, Theil-U and MSFE are shown. We will use them to compare with GARCH model’s forecast performances as stated in our objective. 74 72 70 68 66 64 62 60 58 2009M07 2009M08 PRICE Figure 4.13 2009M09 PRICE_ARIMA_F The plot of actual prices against forecast prices by ARIMA(1,2,1) model 77 In Figure 4.13, the graph of actual daily crude oil prices is plotted using a solid line and while dotted line represents the forecasted daily crude oil prices by ARIMA(1,2,1). The forecast series follow the actual series closely. 4.6 Heteroscedasticity Test We must examine the existence of heteroscedasticity in daily crude oil prices series before starting to estimate the GARCH model. Daily crude oil prices data that are used in this study contains volatility periods. Thus, it is suitable to apply heteroscedasticity where conditional variance is not constant throughout the time trend. 4.6.1 ARCH-LM Test There is a heteroscedastic test developed by Engle (1982) called ARCH Lagrange Multiplier (LM) test. This test is used to determine the occurrence of ARCH effect in the residuals. The results of ARCH-LM test for ARIMA(1,2,1) model is tabulated in Table 4.9. 78 Table 4.9: ARCH-LM test for ARIMA(1,2,1) model ARCH Test: F-statistic 1235.601 Probability 0.000000 Obs*R-squared 1028.464 Probability 0.000000 Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 10/03/09 Time: 16:07 Sample (adjusted): 1/08/1986 6/30/2009 Included observations: 6125 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 0.609786 0.077564 7.861680 0.0000 RESID^2(-1) 0.409772 0.011657 35.15112 0.0000 R-squared 0.167912 Mean dependent var 1.032923 Adjusted R-squared 0.167777 S.D. dependent var 6.573567 S.E. of regression 5.996821 Akaike info criterion 6.420662 Sum squared resid 220194.4 Schwarz criterion 6.422857 F-statistic 1235.601 Prob(F-statistic) 0.000000 Log likelihood Durbin-Watson stat -19661.28 1.963897 The top part in Table 4.9 is the test statistic for ARCH-LM distributed with . The # -statistic value of 1235.601 is taken from the test equation for residuals squared. The *-value indicates that the #-statistic is significantly ARCH effects in the models. The ARCH-LM test statistic of 1028.464 also gives the same result for #-statistic as the one under (1). 79 4.6.2 Diagnostic Checking for Residuals Squared Another important criterion to determine whether a series contains heteroscedastic is by checking the correlogram of the residual squared. At this point, we also need to observe the patterns in the ACF and PACF of residuals squared for ARIMA(1,2,1) model. To check the ARCH effects, the ACF and PACF of residuals squared for ARIMA(1,2,1) model are plotted in Figure 4.14. Figure 4.14 Correlogram of residuals squared by ARIMA(1,2,1) Figure 4.14 shows that there are spikes at the first lag for both ACF and PACF of residuals. This indicates that the ARCH effect does occur in the residuals for the ARIMA(1,2,1) model. 80 4.7 GARCH Model In section 4.6, we have determined that ARCH effect occurred in the data series for ARIMA(1,2,1) model. This is due to the presence of volatility in crude oil prices data. We will use the stationary first order difference series for testing our GARCH model. 4.7.1 Model Identification for GARCH Model We selected GARCH(1,1) model because crude oil prices data have the characteristics of volatility clustering and leptokurtosis. Sadorsky (2006) has suggested that GARCH(1,1) model is superior among prominent GARCH-type models for giving the best out-sample period forecasts. 4.7.2 Parameter Estimation GARCH(1,1) Model The method to estimate the parameters is done by EViews software. The maximum likelihood estimator will find the parameter coefficients for conditional mean and conditional variance equations. Using EViews, the parameter coefficients on the dependent variable of the first order difference for daily crude oil prices are obtained and tabulated in Table 4.10. 81 Table 4.10 : Parameter estimation of GARCH(1,1) model Dependent Variable: D(PRICE) Method: ML - ARCH Date: 10/03/09 Time: 16:15 Sample (adjusted): 1/03/1986 6/30/2009 Included observations: 6128 after adjustments Convergence achieved after 18 iterations Variance backcast: ON GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*GARCH(-1) C Coefficient Std. Error z-Statistic Prob. 0.001631 0.004673 0.349054 0.7270 5.934480 25.65466 272.7620 0.0000 0.0000 0.0000 Variance Equation C RESID(-1)^2 GARCH(-1) 0.001228 0.097757 0.909097 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood -0.000030 -0.000520 1.017498 6340.188 -5502.838 0.000207 0.003811 0.003333 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Durbin-Watson stat 0.007223 1.017233 1.797271 1.801658 2.119071 From Table 4.10, for the conditional mean equation, the parameter found is o = 0.0016. The standard normal distribution -test has rejected the parameter coefficients equal to zero, while the conditional variance equation gives = 0.0012, = 0.0978 and , = 0.9091. A high value of , means that volatility is persistent and it takes a long time to change. A high value of means that volatility is spiky and quick to react to market movements (Dowd, 2002). Somehow, gives a negative value in the estimation equation. In reality, the measure of in GARCH model is not important because it is only used to test the ARCH effect of residuals. The DW test in GARCH(1,1) model estimation is significant since it exceeds 2. 82 The GARCH(1,1) model can be written into conditional mean and conditional variance equations as = 0.0016 + = 0.0012 + 0.0978 + 0.9091 (4.2) (4.3) In Figure 4.15, a graphical plot for conditional standard deviation and conditional variance is presented. The values for conditional standard deviation are obtained by taking the square root from the conditional variance. 9 8 7 6 5 4 3 2 1 0 86 88 90 92 94 96 98 00 02 04 06 08 Conditional Standard Deviation Figure 4.15 Conditional standard deviation for GARCH(1,1) model In Figure 4.15, the extraordinary long spikes are the high volatile periods of the series. With GARCH(1,1) model, the volatility clustering will be detected. 83 In Figure 4.16, another conditional variance graph is plotted. 70 60 50 40 30 20 10 0 86 88 90 92 94 96 98 00 02 04 06 08 PRICE_GARCH Figure 4.16 Conditional variance for GARCH(1,1) model In Figure 4.16, there are lesser spikes compared to conditional standard deviation graph. But it does point out some of the high volatile clusters in the series. However, the extraordinary long spikes are the high volatile periods in the data series. 4.7.3 Diagnostic Checking GARCH(1,1) Model After we have estimated the parameters, the next step will be diagnostic checking on the adequacy for GARCH(1,1) model. It can be done by checking the correlogram of standardized residuals squared which consists of autocorrelation and 84 partial autocorrelation. The correlogram of standardized residuals squared is plotted in Figure 4.17. Figure 4.17 Correlogram of standardized residuals squared for GARCH(1,1) model In the Figure 4.17, ACF and PACF of residuals are approximately zero. The insignificant Ljung-Box -statistic also provides the same evidence with *-value that GARCH(1,1) model is adequate. Once again, we can conclude that the model is adequate. Last but not least, in diagnostic checking stage, a test for presenting of conditional heteroscedasticity in the data with ARCH-LM test on the residuals. The test is tabulated in Table 4.11. 85 Table 4.11 : ARCH-LM test for GARCH(1,1) model ARCH Test: F-statistic Obs*R-squared 0.051911 0.051928 Probability Probability 0.819778 0.819742 Test Equation: Dependent Variable: STD_RESID^2 Method: Least Squares Date: 10/03/09 Time: 16:17 Sample (adjusted): 1/06/1986 6/30/2009 Included observations: 6127 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C STD_RESID^2(-1) 0.997921 0.002911 0.032226 0.012777 30.96633 0.227840 0.0000 0.8198 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.000008 -0.000155 2.315408 32836.82 -13836.98 2.000104 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 1.000834 2.315229 4.517375 4.519569 0.051911 0.819778 In Table 4.11, there is computed one lag difference from the residuals squared in the ARCH-LM test. The ARCH-LM for one lag difference of residuals squared is 0.0519 under (1). But, the null hypothesis is not rejected since the *- value is 0.8198 where it has greater than 5% of significance level. On the other hand, # -statistic the test is 0.0519 also not rejected the null hypothesis at the same condition. The ARCH-LM test on the residuals of this model indicates that the conditional heteroscedasticity is no longer present in the data. First order difference of residuals plot is presented in Figure 4.18. 86 20 16 12 8 4 0 -4 -8 -12 -16 86 88 90 92 94 96 98 00 02 04 06 08 D(PRICE) Residuals Figure 4.18 First order difference of residuals plot From Figure 4.18, it can be concluded that the plot is quite similar to the previous lagged difference plots. There are some spikes found during the high volatile period. Next, we will plot the standardized residuals for GARCH(1,1) model. The standardized residuals graph for GARCH(1,1) model is plotted in Figure 4.19. This plot is different from the first order difference of residuals in Figure 4.18 due to a constant mean and a variance of unity in data series. 87 8 4 0 -4 -8 -12 86 88 90 92 94 96 98 00 02 04 06 08 Standardized Residuals Figure 4.19 Standardized residuals plot for GARCH(1,1) model In Figure 4.19, a band of lines are joined together around mean zero with little spikes throughout the time series. The plot can be observed to have a uniform mean and a unity variance. The distribution of the standardized residuals will be summarized in the histogram chart and normality test as in Figure 4.20. 88 1600 Series: Standardized Residuals Sample 1/03/1986 6/30/2009 Observations 6128 1200 800 400 0 -8 -6 Figure 4.20 -4 -2 0 2 4 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis 0.018478 -0.001774 5.978222 -8.246397 1.000305 -0.213420 6.368166 Jarque-Bera Probability 2943.159 0.000000 6 Histogram and normality test for standardized residuals In Figure 4.20, the histogram chart for standardized residuals on the left shows that the residuals are evenly distributed. The mean value is equal to 0.01848 and the standard deviation is 1.0003 which implies that the standardized residuals are normally distributed ¢(0,1). The skewness and kurtosis are -0.2134 and 6.3682 respectively. The distribution is a bit positively skewed and fat tailed. The JarqueBera test indicates that the standardized residuals is normally distributed. 4.7.4 Forecasting using GARCH(1,1) Model Apart from forecasting the conditional variance, we also forecast the conditional mean at the same time. Here, our daily forecast crude oil prices are the conditional mean from the original series. The forecast from GARCH(1,1) model, we use one-step ahead static forecast from EViews. In Figure 4.21, we plot the forecast value for crude oil prices using GARCH(1,1) model. 89 80 76 72 68 64 60 56 2009M07 2009M08 2009M09 PRICE_GARCH_F Figure 4.21 Forecast crude oil prices by GARCH(1,1) model In Figure 4.21, the solid line presents the forecasted prices whereas the dotted lines are forecast prices with ±2 standard errors. The forecast crude oil prices fluctuate between $59 and $73 in 3-month out-sample period. The forecast of conditional variance is plotted in Figure 4.22. 90 4.5 4.0 3.5 3.0 2.5 2.0 1.5 2009M07 2009M08 2009M09 Forecast of Variance Figure 4.22 Conditional variance forecast by GARCH(1,1) model As shown in Figure 4.22, the forecast of conditional variance is not constant. Since conditional heteroscedasticity searches for the non-constant variance that exists in time series data, then its trend is non-linear. The forecast evaluation for GARCH(1,1) model is tabulated in Table 4.12. Table 4.12 : Forecast evaluation for GARCH(1,1) model Forecast: PRICE_GARCH_F Actual: PRICE Forecast sample: 7/01/2009 9/30/2009 Included observations: 66 Root Mean Squared Error Mean Absolute Error Mean Absolute Percentage Error Theil Inequality Coefficient Bias Proportion Variance Proportion Covariance Proportion 1.683475 1.255553 1.848057 0.012340 0.000023 0.000010 0.999967 91 In Figure 4.23, the actual and forecast daily crude oil prices by GARCH(1,1) model are being plotted. 74 72 70 68 66 64 62 60 58 2009M07 PRICE Figure 4.23 2009M08 2009M09 PRICE_GARCH_F The plot of actual prices against forecast prices by GARCH(1,1) model From Figure 4.23, it can be concluded that the trend of forecast prices follows the actual crude oil prices for 3 months out-sample period, closely. 4.8 Evaluation of ARIMA(1,2,1) and GARCH(1,1) Models Performances One of the objectives of this study is to evaluate the forecast performances by two univariate time series models, namely ARIMA and GARCH models. We will 92 evaluate the ARIMA(1,2,1) and GARCH(1,1) models in terms of their AIC and SIC values in the estimation stage , and forecast performances in the forecasting stage. 4.8.1 Information Criterion for ARIMA(1,2,1) and GARCH(1,1) Models In the model estimation step, we calculate the AIC and SIC values from ARIMA(1,2,1) and GARCH(1,1) models. We aim to investigate which model is a better estimate model for daily crude oil prices. In this context, the model with smaller AIC and SIC values are concluded to be the better estimation model. The results are tabulated in Table 4.13. Table 4.13 : Information criterion for ARIMA(1,2,1) and GARCH(1,1) models Model AIC SIC ARIMA(1,2,1) 2.8711 2.8744 GARCH(1,1) 1.7973 1.8017 In Table 4.13, AIC and SIC values are obtained from equation estimation from both ARIMA(1,2,1) and GARCH(1,1) models using EViews. We found that both the AIC and SIC values from GARCH(1,1) model are smaller than that from ARIMA(1,2,1) model. Therefore, it shows that GARCH(1,1) is a better model than ARIMA(1,2,1) for estimating daily crude oil prices. 4.8.2 Forecasting Performances of ARIMA(1,2,1) and GARCH(1,1) Models In the forecasting stage, we calculate RMSE, MAE, MAPE, Theil-U and MSFE values from ARIMA(1,2,1) and GARCH(1,1) models. These are tabulated in 93 Table 4.14. If the actual values and forecast values are closer to each other, a small forecast error will be obtained. Thus, smaller RMSE, MAE, MAPE, Theil-U and MSFE values are preferred. Table 4.14 : Forecasting performances of ARIMA(1,2,1) and GARCH(1,1) models Forecast Performance RMSE MAE MAPE Theil-U ARIMA(1,2,1) GARCH(1,1) 1.7042 1.6835 1.2805 1.2556 1.8844 1.8481 0.0125 0.0123 0.000063 0.000023 0.000068 0.000010 0.999869 0.999967 MSFE Bias Proportion Variance Proportion Covariance Proportion From Table 4.14 it can be concluded that all forecast errors from GARCH(1,1) model are smaller than that from ARIMA(1,2,1) model. Therefore, we can conclude that GARCH(1,1) model performs better than ARIMA (1, 2, 1). In other words, GARCH(1, 1) is a better forecast model for daily crude oil prices than ARIMA(1,2,1) model. 4.9 Concluding Remarks In this chapter analyses on daily crude oil have been conducted using two models. The ARIMA(1,2,1) model is able to produce forecasts based on the history 94 patterns in the data. The GARCH(1,1) model on the other hand, gives a slightly better estimate when there are volatility clustering in the data series. This is due to the GARCH model’s ability to capture the volatility by the conditional variance of being non-constant throughout the time. CHAPTER 5 CONCLUSIONS AND SUGGESTIONS FOR FUTURE STUDY 5.1 Introduction This chapter presents the conclusion and summary of the study. Finally, suggestions for future work are made. 5.2 Conclusions This study was undertaken to obtain a suitable GARCH and ARIMA models for forecasting crude oil prices. ARIMA is a popular forecasting method. It is a general class of Box-Jenkins model for stationary time series. Selection of an appropriate model is made by comparing the distributions of autocorrelation coefficients of the time series being fitted with the theoretical distributions for various models. The Box-Jenkins approach involves four separate stages. These stages are model identification, model estimation, diagnostic checking and forecasting. In the current study, the model that has been selected for forecasting crude oil prices is ARIMA(1,2,1). This model gives reasonable and acceptable forecasts. However, despite the fact that this approach has been used extensively in various fields such as economics, agriculture and business, it does not perform very 96 well when there exists volatility in the data series. To handle volatility, the current study uses the GARCH model. Most of the time, GARCH models can accommodate volatility clustering and leptokurtosis very easily. Dowd (2002) stated that GARCH are tailor-made for volatility clustering and it produces returns with fatter than normal tails even if the innovations and the random shocks are normally distributed. GARCH approach involves model identification, model estimation, diagnostic checking and forecasting. In the current study, the model that has been selected for forecasting crude oil prices is GARCH(1,1). The model performs better than ARIMA(1,2,1) because of its ability to capture the volatility by the conditional variance of being non-constant throughout the time. In this study, GARCH(1,1) was concluded to be a better model than ARIMA (1, 2, 1) in forecasting crude oil prices because the values for RMSE, MAE, MAPE, Theil-U and MSFE calculated using this model were smaller than those calculated using ARIMA(1,2,1) model. In this study, an exploration on a software named EViews was made. It is concluded that EViews is a potential software for modelling and forecasting time series data. 5.3 Suggestions for Future Works The focus of this study is to discuss two approaches in forecasting crude oil prices. These approaches are ARIMA and GARCH. However, the current study only concentrated in forecasting crude oil prices with normal distribution. Future work in this area can include working with student’s distribution or generalized exponential distribution. Since the characteristics of crude oil prices data are generally leptokurtosis, volatility clustering and have some leverage effects. Future studies in this area can also use a hybrid method, specifically ARMA/GARCH model. This model combines the Box-Jenkins with GARCH. The hybrid model is an alternative to forecast crude oil prices because it contains both qualities of Box-Jenkins and GARCH methods. 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Journal of Systems and Complexity, 18(2), 145-166. 104 APPENDIX A WTI Daily Crude Oil Prices Data Cushing, OK WTI Spot Price FOB (Dollars per Barrel) Week Of 1985 Dec-30 to Jan- 3 1986 Jan- 6 to Jan-10 1986 Jan-13 to Jan-17 1986 Jan-20 to Jan-24 1986 Jan-27 to Jan-31 Mon Tue Wed 25.87 25.18 20.25 19.61 Thu 25.56 26.03 23.98 19.93 19.58 Fri 26.00 25.65 23.63 19.45 18.95 26.53 25.08 21.33 20.87 25.85 24.97 20.61 19.45 1986 Feb- 3 to Feb- 7 1986 Feb-10 to Feb-14 1986 Feb-17 to Feb-21 1986 Feb-24 to Feb-28 1986 Mar- 3 to Mar- 7 17.42 16.78 14.68 11.98 15.58 16.28 14.70 14.68 11.98 16.28 15.74 15.08 14.62 12.03 16.60 16.43 14.13 14.05 13.13 17.70 16.03 13.63 13.23 12.24 1986 Mar-10 to Mar-14 1986 Mar-17 to Mar-21 1986 Mar-24 to Mar-28 1986 Mar-31 to Apr- 4 1986 Apr- 7 to Apr-11 12.94 13.28 12.20 10.25 14.39 13.23 14.03 12.43 11.13 12.83 14.05 13.25 12.03 11.35 13.00 12.60 12.75 11.35 11.70 13.45 12.55 13.95 1986 Apr-14 to Apr-18 1986 Apr-21 to Apr-25 1986 Apr-28 to May- 2 1986 May- 5 to May- 9 1986 May-12 to May-16 12.94 12.48 13.34 14.32 15.75 12.72 13.13 13.63 14.43 15.65 11.50 13.70 13.38 15.13 15.53 11.75 13.65 13.80 15.70 15.68 11.88 14.23 14.65 15.83 16.08 1986 May-19 to May-23 1986 May-26 to May-30 1986 Jun- 2 to Jun- 6 1986 Jun- 9 to Jun-13 1986 Jun-16 to Jun-20 17.13 13.80 12.61 13.65 16.18 15.10 13.35 12.38 13.65 15.53 14.65 13.15 13.52 13.62 16.04 14.50 13.21 13.69 13.73 16.95 14.30 12.73 13.83 14.44 1986 Jun-23 to Jun-27 1986 Jun-30 to Jul- 4 1986 Jul- 7 to Jul-11 1986 Jul-14 to Jul-18 1986 Jul-21 to Jul-25 14.05 12.80 11.18 11.23 13.07 13.98 12.39 11.19 11.85 10.88 13.23 12.04 11.00 12.68 10.83 13.14 11.70 11.13 12.30 10.95 13.38 1986 Jul-28 to Aug- 1 1986 Aug- 4 to Aug- 8 1986 Aug-11 to Aug-15 1986 Aug-18 to Aug-22 1986 Aug-25 to Aug-29 11.09 14.00 14.92 15.58 15.48 11.63 14.35 15.50 14.98 15.78 11.73 14.80 15.28 15.23 15.83 11.23 15.18 15.43 15.23 15.83 11.56 14.83 15.83 15.48 15.93 1986 Sep- 1 to Sep- 5 1986 Sep- 8 to Sep-12 1986 Sep-15 to Sep-19 1986 Sep-22 to Sep-26 1986 Sep-29 to Oct- 3 15.63 14.31 13.94 14.93 16.43 15.00 13.80 14.30 14.70 16.03 14.90 14.03 14.55 15.23 16.18 15.05 14.55 14.28 15.38 15.63 15.06 14.47 14.43 14.86 15.60 14.55 15.22 15.35 14.83 14.85 15.05 14.53 14.88 14.98 14.85 14.93 1986 Oct- 6 to Oct-10 1986 Oct-13 to Oct-17 1986 Oct-20 to Oct-24 14.83 15.17 12.75 13.63 11.13 12.80 10.83 105 1986 Oct-27 to Oct-31 1986 Nov- 3 to Nov- 7 14.40 14.70 14.18 15.05 13.73 14.93 15.08 15.08 15.25 15.15 1986 Nov-10 to Nov-14 1986 Nov-17 to Nov-21 1986 Nov-24 to Nov-28 1986 Dec- 1 to Dec- 5 1986 Dec- 8 to Dec-12 15.30 15.62 14.98 15.29 15.01 15.39 15.65 15.05 15.22 14.93 15.33 15.52 15.00 15.13 15.12 15.55 15.10 15.20 15.49 15.68 15.13 15.00 15.14 16.13 1986 Dec-15 to Dec-19 1986 Dec-22 to Dec-26 1986 Dec-29 to Jan- 2 1987 Jan- 5 to Jan- 9 1987 Jan-12 to Jan-16 16.38 16.95 17.65 17.98 19.00 16.11 16.93 17.73 18.21 18.86 15.83 17.26 17.93 18.28 19.13 16.28 16.55 18.63 19.09 18.13 18.78 19.13 1987 Jan-19 to Jan-23 1987 Jan-26 to Jan-30 1987 Feb- 2 to Feb- 6 1987 Feb- 9 to Feb-13 1987 Feb-16 to Feb-20 18.70 18.63 18.59 18.37 18.73 18.48 18.38 18.43 17.78 18.60 18.56 18.26 18.03 17.44 18.76 18.68 18.56 18.05 17.48 18.59 18.73 18.44 17.83 17.83 1987 Feb-23 to Feb-27 1987 Mar- 2 to Mar- 6 1987 Mar- 9 to Mar-13 1987 Mar-16 to Mar-20 1987 Mar-23 to Mar-27 17.15 16.43 18.13 18.61 18.60 16.75 17.40 18.27 18.94 16.43 17.40 18.33 18.75 18.48 16.98 18.00 18.42 18.61 18.63 16.45 18.13 18.39 18.70 18.65 1987 Mar-30 to Apr- 3 1987 Apr- 6 to Apr-10 1987 Apr-13 to Apr-17 1987 Apr-20 to Apr-24 1987 Apr-27 to May- 1 18.67 18.71 18.07 18.66 18.83 18.82 18.69 18.09 18.97 18.73 18.78 18.68 18.46 19.03 18.66 18.90 18.64 18.58 19.03 18.76 18.68 18.26 1987 May- 4 to May- 8 1987 May-11 to May-15 1987 May-18 to May-22 1987 May-25 to May-29 1987 Jun- 1 to Jun- 5 18.93 19.43 19.91 19.55 19.18 19.37 19.97 19.35 19.70 19.22 19.39 19.75 19.38 19.87 19.08 19.56 19.95 19.28 19.75 19.28 19.84 19.68 19.36 19.79 1987 Jun- 8 to Jun-12 1987 Jun-15 to Jun-19 1987 Jun-22 to Jun-26 1987 Jun-29 to Jul- 3 1987 Jul- 6 to Jul-10 19.94 20.07 20.49 20.38 20.92 19.84 20.27 19.95 20.22 20.76 19.83 20.41 20.13 20.47 20.94 19.85 20.50 20.15 20.61 21.32 19.93 20.65 20.34 20.61 21.34 1987 Jul-13 to Jul-17 1987 Jul-20 to Jul-24 1987 Jul-27 to Jul-31 1987 Aug- 3 to Aug- 7 1987 Aug-10 to Aug-14 21.38 22.23 20.50 22.21 20.70 21.65 21.75 21.35 21.82 21.07 22.23 21.73 21.49 21.37 20.96 22.44 21.23 21.47 21.17 20.76 22.44 20.58 21.43 21.01 20.53 1987 Aug-17 to Aug-21 1987 Aug-24 to Aug-28 1987 Aug-31 to Sep- 4 1987 Sep- 7 to Sep-11 1987 Sep-14 to Sep-18 19.85 19.18 19.76 19.34 19.64 19.84 19.30 19.61 18.99 19.66 19.71 19.49 19.62 19.43 19.71 19.47 19.69 19.48 19.72 19.58 19.20 19.44 19.34 19.42 19.58 1987 Sep-21 to Sep-25 1987 Sep-28 to Oct- 2 1987 Oct- 5 to Oct- 9 1987 Oct-12 to Oct-16 1987 Oct-19 to Oct-23 19.77 19.48 19.81 19.67 19.79 19.35 19.58 19.33 19.66 19.79 19.66 19.62 19.68 19.79 19.93 19.61 19.62 19.77 19.77 20.19 19.47 19.88 19.67 20.23 20.18 1987 Oct-26 to Oct-30 1987 Nov- 2 to Nov- 6 1987 Nov- 9 to Nov-13 1987 Nov-16 to Nov-20 1987 Nov-23 to Nov-27 20.00 19.64 18.66 18.69 19.31 20.15 19.39 18.98 18.28 18.73 20.10 19.09 18.92 18.62 18.63 19.93 19.02 18.93 18.55 19.96 18.73 18.89 18.87 18.63 1987 Nov-30 to Dec- 4 1987 Dec- 7 to Dec-11 1987 Dec-14 to Dec-18 1987 Dec-21 to Dec-25 1987 Dec-28 to Jan- 1 18.52 18.30 17.47 15.12 16.46 18.44 18.06 16.75 16.60 16.95 18.59 18.53 15.97 16.64 16.97 18.87 18.51 15.97 16.54 16.74 18.68 18.31 15.57 1988 Jan- 4 to Jan- 8 1988 Jan-11 to Jan-15 17.77 16.63 17.89 16.76 17.73 16.56 17.30 17.10 17.33 16.92 19.01 18.84 106 1988 Jan-18 to Jan-22 1988 Jan-25 to Jan-29 1988 Feb- 1 to Feb- 5 17.28 17.11 16.83 17.30 16.96 16.92 17.20 16.64 17.12 17.21 16.94 17.17 16.99 16.97 17.34 1988 Feb- 8 to Feb-12 1988 Feb-15 to Feb-19 1988 Feb-22 to Feb-26 1988 Feb-29 to Mar- 4 1988 Mar- 7 to Mar-11 17.70 16.84 16.64 15.98 15.35 17.38 16.80 16.80 15.57 15.58 17.12 16.62 16.58 15.69 15.52 17.17 16.50 15.86 15.35 16.04 16.84 16.73 15.77 15.63 16.20 1988 Mar-14 to Mar-18 1988 Mar-21 to Mar-25 1988 Mar-28 to Apr- 1 1988 Apr- 4 to Apr- 8 1988 Apr-11 to Apr-15 15.57 16.52 17.05 16.99 17.87 15.81 16.02 17.08 16.72 18.06 16.03 16.58 17.06 16.80 18.12 16.39 16.73 17.09 17.05 18.40 16.61 17.07 1988 Apr-18 to Apr-22 1988 Apr-25 to Apr-29 1988 May- 2 to May- 6 1988 May- 9 to May-13 1988 May-16 to May-20 18.50 18.36 17.12 17.56 17.70 17.92 18.54 17.30 17.50 17.72 17.93 18.32 17.23 17.48 17.39 18.34 17.91 17.41 17.49 17.43 18.13 18.10 17.63 17.53 17.41 1988 May-23 to May-27 1988 May-30 to Jun- 3 1988 Jun- 6 to Jun-10 1988 Jun-13 to Jun-17 1988 Jun-20 to Jun-24 17.01 17.45 17.28 16.43 16.02 17.04 17.54 17.32 16.85 15.82 17.39 17.60 17.35 16.56 16.03 17.54 17.67 17.09 16.62 15.86 17.45 17.51 16.70 16.43 16.03 1988 Jun-27 to Jul- 1 1988 Jul- 4 to Jul- 8 1988 Jul-11 to Jul-15 1988 Jul-18 to Jul-22 1988 Jul-25 to Jul-29 15.86 14.56 15.84 16.09 16.01 15.11 14.61 15.16 15.99 15.37 15.44 14.35 15.76 16.18 15.20 15.83 14.84 16.28 16.08 14.92 15.42 14.85 16.27 16.37 1988 Aug- 1 to Aug- 5 1988 Aug- 8 to Aug-12 1988 Aug-15 to Aug-19 1988 Aug-22 to Aug-26 1988 Aug-29 to Sep- 2 16.07 15.84 15.59 15.75 15.24 15.57 15.56 15.51 15.71 15.39 15.20 15.65 15.46 15.55 15.19 15.12 15.75 15.59 15.33 15.05 15.31 15.54 15.77 15.35 14.79 1988 Sep- 5 to Sep- 9 1988 Sep-12 to Sep-16 1988 Sep-19 to Sep-23 1988 Sep-26 to Sep-30 1988 Oct- 3 to Oct- 7 14.79 14.48 14.72 14.16 13.03 14.25 14.55 15.08 14.22 13.03 14.29 15.38 15.19 14.07 12.58 14.51 14.86 15.25 13.91 12.62 14.14 14.50 14.26 13.33 12.99 1988 Oct-10 to Oct-14 1988 Oct-17 to Oct-21 1988 Oct-24 to Oct-28 1988 Oct-31 to Nov- 4 1988 Nov- 7 to Nov-11 13.59 15.16 12.94 13.54 14.08 13.63 14.63 13.36 13.52 13.70 14.02 15.33 13.45 13.78 13.88 14.26 14.44 13.67 13.89 13.99 14.90 14.22 13.79 14.04 13.99 1988 Nov-14 to Nov-18 1988 Nov-21 to Nov-25 1988 Nov-28 to Dec- 2 1988 Dec- 5 to Dec- 9 1988 Dec-12 to Dec-16 14.25 13.73 14.93 15.36 16.08 14.03 14.78 15.00 15.51 15.93 13.68 14.11 15.42 15.80 16.33 13.30 14.11 15.63 15.48 16.39 13.47 15.43 15.69 15.90 16.81 1988 Dec-19 to Dec-23 1988 Dec-26 to Dec-30 1989 Jan- 2 to Jan- 6 1989 Jan- 9 to Jan-13 1989 Jan-16 to Jan-20 16.24 17.74 18.88 17.68 16.98 17.38 17.80 19.03 17.27 17.03 16.99 18.16 19.20 17.36 16.81 17.45 18.11 19.28 16.63 17.12 17.56 18.49 18.85 1989 Jan-23 to Jan-27 1989 Jan-30 to Feb- 3 1989 Feb- 6 to Feb-10 1989 Feb-13 to Feb-17 1989 Feb-20 to Feb-24 17.66 17.32 17.38 17.61 18.60 17.96 17.00 17.55 17.60 18.64 18.23 17.50 17.49 18.23 18.50 17.68 17.72 17.42 18.35 18.49 17.74 17.51 17.11 18.60 18.06 1989 Feb-27 to Mar- 3 1989 Mar- 6 to Mar-10 1989 Mar-13 to Mar-17 1989 Mar-20 to Mar-24 1989 Mar-27 to Mar-31 18.16 18.73 19.05 19.53 20.55 18.21 18.12 19.47 20.08 19.93 18.30 18.57 19.84 20.21 20.20 18.68 18.53 19.86 20.16 21.03 18.67 18.53 20.34 1989 Apr- 3 to Apr- 7 20.03 20.59 20.07 19.85 20.03 16.87 18.32 20.27 107 1989 Apr-10 to Apr-14 1989 Apr-17 to Apr-21 1989 Apr-24 to Apr-28 1989 May- 1 to May- 5 20.65 21.23 20.64 20.66 20.56 21.74 21.32 19.73 20.66 22.66 21.20 20.13 20.26 24.62 20.83 20.57 20.68 23.38 20.38 20.08 1989 May- 8 to May-12 1989 May-15 to May-19 1989 May-22 to May-26 1989 May-29 to Jun- 2 1989 Jun- 5 to Jun- 9 19.41 20.53 20.92 19.55 20.50 19.54 20.58 21.77 19.96 20.38 19.56 20.15 19.68 19.93 19.70 20.13 20.25 19.46 19.83 19.94 20.12 20.58 19.55 20.18 19.87 1989 Jun-12 to Jun-16 1989 Jun-19 to Jun-23 1989 Jun-26 to Jun-30 1989 Jul- 3 to Jul- 7 1989 Jul-10 to Jul-14 19.30 20.88 20.28 20.55 20.38 19.50 19.88 20.38 20.29 20.75 20.37 19.74 20.04 20.98 20.20 20.56 19.53 20.27 20.35 20.42 19.98 19.71 20.29 20.78 20.36 1989 Jul-17 to Jul-21 1989 Jul-24 to Jul-28 1989 Jul-31 to Aug- 4 1989 Aug- 7 to Aug-11 1989 Aug-14 to Aug-18 20.49 18.76 18.33 17.91 18.60 20.21 18.52 17.91 18.06 18.78 19.83 18.32 18.28 18.23 18.99 19.92 18.15 18.18 18.58 18.67 19.86 17.96 18.05 18.46 18.82 1989 Aug-21 to Aug-25 1989 Aug-28 to Sep- 1 1989 Sep- 4 to Sep- 8 1989 Sep-11 to Sep-15 1989 Sep-18 to Sep-22 19.10 18.69 18.88 19.77 19.87 19.01 18.59 19.08 19.68 19.68 19.15 18.85 19.41 19.87 19.68 19.02 18.83 19.40 19.73 19.69 18.53 18.88 19.77 19.96 19.21 1989 Sep-25 to Sep-29 1989 Oct- 2 to Oct- 6 1989 Oct- 9 to Oct-13 1989 Oct-16 to Oct-20 1989 Oct-23 to Oct-27 19.49 19.99 20.04 20.64 19.67 19.62 20.15 20.21 20.68 19.70 19.61 20.17 20.18 20.58 19.60 19.99 19.95 20.49 20.37 19.48 20.15 19.89 20.91 20.04 19.79 1989 Oct-30 to Nov- 3 1989 Nov- 6 to Nov-10 1989 Nov-13 to Nov-17 1989 Nov-20 to Nov-24 1989 Nov-27 to Dec- 1 19.74 20.06 19.62 20.13 19.65 19.88 19.97 19.58 20.52 19.33 20.09 19.97 19.69 19.80 19.40 20.01 19.62 19.88 19.80 19.87 20.23 19.84 19.93 19.84 20.26 1989 Dec- 4 to Dec- 8 1989 Dec-11 to Dec-15 1989 Dec-18 to Dec-22 1989 Dec-25 to Dec-29 1990 Jan- 1 to Jan- 5 20.24 20.69 22.20 20.22 20.80 22.17 21.91 22.88 20.46 20.75 21.58 21.78 23.81 20.45 20.67 21.53 21.60 23.41 20.41 21.13 21.31 21.84 23.07 1990 Jan- 8 to Jan-12 1990 Jan-15 to Jan-19 1990 Jan-22 to Jan-26 1990 Jan-29 to Feb- 2 1990 Feb- 5 to Feb- 9 21.64 22.36 22.57 22.79 22.44 22.25 22.61 22.34 22.29 22.36 22.90 22.11 23.43 22.69 22.40 23.15 22.78 24.48 22.71 22.11 23.17 23.70 22.56 23.04 21.82 1990 Feb-12 to Feb-16 1990 Feb-19 to Feb-23 1990 Feb-26 to Mar- 2 1990 Mar- 5 to Mar- 9 1990 Mar-12 to Mar-16 22.02 22.46 21.81 21.62 20.26 21.88 22.20 21.62 21.30 20.15 22.12 21.87 21.55 20.93 20.07 22.86 21.40 21.19 20.80 20.38 22.46 21.13 21.36 20.42 20.09 1990 Mar-19 to Mar-23 1990 Mar-26 to Mar-30 1990 Apr- 2 to Apr- 6 1990 Apr- 9 to Apr-13 1990 Apr-16 to Apr-20 19.60 20.48 20.51 18.32 17.87 19.34 20.36 20.23 17.56 17.37 19.59 20.09 19.78 18.19 16.95 19.85 20.03 19.48 17.76 17.93 20.28 20.34 19.15 1990 Apr-23 to Apr-27 1990 Apr-30 to May- 4 1990 May- 7 to May-11 1990 May-14 to May-18 1990 May-21 to May-25 18.60 18.50 18.30 19.73 18.26 17.85 18.76 18.26 19.52 17.51 17.49 18.63 19.01 19.05 16.25 18.50 17.98 19.03 18.89 16.02 18.57 17.98 18.96 18.78 16.12 1990 May-28 to Jun- 1 1990 Jun- 4 to Jun- 8 1990 Jun-11 to Jun-15 1990 Jun-18 to Jun-22 1990 Jun-25 to Jun-29 17.09 16.82 15.92 16.15 18.00 16.41 17.39 15.55 17.12 17.88 16.91 17.60 15.43 16.70 17.47 16.65 17.11 16.09 17.18 17.51 16.78 16.64 16.50 17.05 17.90 108 1990 Jul- 2 to Jul- 6 1990 Jul- 9 to Jul-13 1990 Jul-16 to Jul-20 1990 Jul-23 to Jul-27 1990 Jul-30 to Aug- 3 16.94 16.63 18.67 19.88 20.24 16.73 17.05 18.23 19.84 20.57 16.73 17.45 18.57 19.33 21.59 16.50 18.69 19.07 20.33 23.71 16.49 18.37 19.61 20.07 23.79 1990 Aug- 6 to Aug-10 1990 Aug-13 to Aug-17 1990 Aug-20 to Aug-24 1990 Aug-27 to Aug-31 1990 Sep- 3 to Sep- 7 28.73 27.07 28.63 27.36 27.45 29.60 26.70 28.46 27.73 29.30 26.19 26.54 30.52 26.15 30.00 25.69 27.40 31.67 26.96 31.51 26.38 28.65 31.10 27.45 30.09 1990 Sep-10 to Sep-14 1990 Sep-17 to Sep-21 1990 Sep-24 to Sep-28 1990 Oct- 1 to Oct- 5 1990 Oct- 8 to Oct-12 30.83 33.73 39.05 37.08 38.88 30.29 33.48 38.33 34.43 40.73 30.85 33.18 39.12 37.04 39.30 31.20 34.44 39.77 36.76 41.07 31.79 36.21 39.53 37.87 39.42 1990 Oct-15 to Oct-19 1990 Oct-22 to Oct-26 1990 Oct-29 to Nov- 2 1990 Nov- 5 to Nov- 9 1990 Nov-12 to Nov-16 38.00 28.46 35.28 32.05 32.08 39.34 29.95 34.93 32.41 33.30 36.03 30.80 35.31 35.48 31.18 37.03 34.35 35.30 35.61 31.05 33.82 33.03 33.95 33.91 29.91 1990 Nov-19 to Nov-23 1990 Nov-26 to Nov-30 1990 Dec- 3 to Dec- 7 1990 Dec-10 to Dec-14 1990 Dec-17 to Dec-21 31.45 33.28 29.40 27.08 27.10 29.50 33.05 29.05 26.50 27.85 30.08 33.28 27.18 25.30 28.26 30.08 32.93 26.35 26.45 27.50 32.35 29.08 26.61 26.60 27.08 1990 Dec-24 to Dec-28 1990 Dec-31 to Jan- 4 1991 Jan- 7 to Jan-11 1991 Jan-14 to Jan-18 1991 Jan-21 to Jan-25 26.95 28.48 27.25 30.13 21.63 27.50 30.35 24.91 27.35 26.53 28.00 32.25 24.08 26.95 25.61 27.55 21.48 25.63 27.58 24.88 27.43 20.05 24.15 1991 Jan-28 to Feb- 1 1991 Feb- 4 to Feb- 8 1991 Feb-11 to Feb-15 1991 Feb-18 to Feb-22 1991 Feb-25 to Mar- 1 21.03 21.23 22.44 20.73 17.43 21.73 20.70 22.78 20.08 18.48 21.08 21.48 22.38 20.18 19.03 21.90 21.33 22.25 18.48 19.28 21.33 21.78 20.73 17.43 19.43 1991 Mar- 4 to Mar- 8 1991 Mar-11 to Mar-15 1991 Mar-18 to Mar-22 1991 Mar-25 to Mar-29 1991 Apr- 1 to Apr- 5 20.33 19.00 19.78 19.47 19.28 20.50 20.06 20.68 19.78 19.38 19.78 20.48 20.40 19.45 19.48 19.33 20.05 20.78 19.63 19.98 19.34 19.88 19.88 1991 Apr- 8 to Apr-12 1991 Apr-15 to Apr-19 1991 Apr-22 to Apr-26 1991 Apr-29 to May- 3 1991 May- 6 to May-10 20.28 21.85 21.68 21.30 21.73 20.58 21.60 21.03 20.99 21.72 21.08 21.73 21.00 21.28 21.83 20.98 21.08 20.95 21.17 21.92 21.54 21.16 21.28 21.43 21.25 1991 May-13 to May-17 1991 May-20 to May-24 1991 May-27 to May-31 1991 Jun- 3 to Jun- 7 1991 Jun-10 to Jun-14 20.89 21.39 21.16 19.84 20.73 20.95 21.34 20.90 19.85 21.04 20.85 21.04 20.45 20.08 20.91 20.95 21.30 20.38 19.71 21.19 21.04 21.16 20.28 19.70 1991 Jun-17 to Jun-21 1991 Jun-24 to Jun-28 1991 Jul- 1 to Jul- 5 1991 Jul- 8 to Jul-12 1991 Jul-15 to Jul-19 19.97 19.92 20.78 21.29 21.50 20.11 20.03 20.92 21.35 21.70 19.99 20.12 20.69 21.43 22.20 20.22 20.41 20.69 21.32 21.97 20.10 20.56 20.91 21.74 22.16 1991 Jul-22 to Jul-26 1991 Jul-29 to Aug- 2 1991 Aug- 5 to Aug- 9 1991 Aug-12 to Aug-16 1991 Aug-19 to Aug-23 21.74 21.38 21.50 21.71 22.50 21.25 21.55 21.39 21.52 22.30 21.41 21.70 21.38 21.30 21.44 21.11 21.33 21.60 21.39 21.93 21.48 21.35 21.64 21.29 21.78 1991 Aug-26 to Aug-30 1991 Sep- 2 to Sep- 6 1991 Sep- 9 to Sep-13 1991 Sep-16 to Sep-20 21.98 21.86 22.28 21.44 21.73 21.77 21.76 21.66 21.87 22.02 21.75 21.51 21.79 22.28 21.56 21.71 22.02 21.36 21.84 20.03 109 1991 Sep-23 to Sep-27 22.20 22.24 22.11 22.23 22.42 1991 Sep-30 to Oct- 4 1991 Oct- 7 to Oct-11 1991 Oct-14 to Oct-18 1991 Oct-21 to Oct-25 1991 Oct-28 to Nov- 1 22.25 22.99 23.48 24.03 23.24 22.10 23.02 23.89 23.47 23.09 22.34 23.18 23.67 23.14 23.11 22.70 23.01 23.96 23.47 23.29 22.64 23.19 24.12 23.18 23.85 1991 Nov- 4 to Nov- 8 1991 Nov-11 to Nov-15 1991 Nov-18 to Nov-22 1991 Nov-25 to Nov-29 1991 Dec- 2 to Dec- 6 23.82 22.66 22.38 22.15 21.10 23.78 22.58 21.88 21.04 20.61 23.44 22.28 22.17 21.38 20.75 23.29 22.55 21.93 21.38 20.42 23.00 22.79 21.85 21.48 19.87 1991 Dec- 9 to Dec-13 1991 Dec-16 to Dec-20 1991 Dec-23 to Dec-27 1991 Dec-30 to Jan- 3 1992 Jan- 6 to Jan-10 19.39 19.78 18.58 18.63 19.24 19.24 19.45 18.68 19.15 18.72 19.54 19.39 17.95 19.94 19.12 18.63 19.43 17.89 20.09 18.28 18.82 19.22 18.26 1992 Jan-13 to Jan-17 1992 Jan-20 to Jan-24 1992 Jan-27 to Jan-31 1992 Feb- 3 to Feb- 7 1992 Feb-10 to Feb-14 18.77 18.92 19.35 18.97 19.69 18.40 18.50 19.09 19.26 19.36 18.83 18.73 18.92 19.50 19.29 18.94 18.38 18.95 19.48 19.70 19.11 18.75 18.93 19.91 19.42 1992 Feb-17 to Feb-21 1992 Feb-24 to Feb-28 1992 Mar- 2 to Mar- 6 1992 Mar- 9 to Mar-13 1992 Mar-16 to Mar-20 19.42 18.37 18.36 18.65 19.15 18.14 18.30 18.67 18.62 19.27 18.42 18.44 18.62 18.50 19.05 18.52 18.77 18.58 18.86 19.27 18.60 18.69 18.53 19.16 18.78 1992 Mar-23 to Mar-27 1992 Mar-30 to Apr- 3 1992 Apr- 6 to Apr-10 1992 Apr-13 to Apr-17 1992 Apr-20 to Apr-24 18.91 19.25 20.44 20.24 20.26 19.06 19.49 20.29 19.81 20.17 19.00 19.85 20.64 19.93 19.92 19.31 19.79 20.33 20.25 19.91 19.19 20.25 20.45 1992 Apr-27 to May- 1 1992 May- 4 to May- 8 1992 May-11 to May-15 1992 May-18 to May-22 1992 May-25 to May-29 20.31 21.13 21.02 20.54 20.29 20.78 20.98 20.12 21.80 20.77 20.79 20.75 20.20 22.00 20.88 20.73 20.56 20.79 21.97 20.88 20.85 20.70 20.79 22.13 1992 Jun- 1 to Jun- 5 1992 Jun- 8 to Jun-12 1992 Jun-15 to Jun-19 1992 Jun-22 to Jun-26 1992 Jun-29 to Jul- 3 22.07 22.43 22.38 22.47 22.27 22.20 22.23 22.23 22.74 21.84 22.45 22.49 22.30 23.03 21.90 22.53 22.33 22.23 22.73 22.08 22.65 22.28 22.16 22.42 22.08 1992 Jul- 6 to Jul-10 1992 Jul-13 to Jul-17 1992 Jul-20 to Jul-24 1992 Jul-27 to Jul-31 1992 Aug- 3 to Aug- 7 21.87 21.38 21.78 22.09 21.60 21.51 21.42 21.91 22.03 21.23 21.38 21.73 21.99 21.98 21.18 21.38 21.80 22.11 21.82 21.41 21.25 21.56 21.95 21.83 21.27 1992 Aug-10 to Aug-14 1992 Aug-17 to Aug-21 1992 Aug-24 to Aug-28 1992 Aug-31 to Sep- 4 1992 Sep- 7 to Sep-11 21.12 21.47 21.85 21.46 21.77 21.00 21.54 21.60 21.72 21.98 21.13 21.33 21.24 21.65 22.00 21.35 21.40 21.14 21.69 21.93 21.31 21.19 21.30 21.77 22.00 1992 Sep-14 to Sep-18 1992 Sep-21 to Sep-25 1992 Sep-28 to Oct- 2 1992 Oct- 5 to Oct- 9 1992 Oct-12 to Oct-16 22.35 21.91 21.78 21.76 22.28 22.13 21.78 21.66 21.85 22.12 22.39 21.78 21.83 21.89 22.16 22.23 21.57 21.83 21.99 22.37 21.96 21.51 21.93 21.99 22.31 1992 Oct-19 to Oct-23 1992 Oct-26 to Oct-30 1992 Nov- 2 to Nov- 6 1992 Nov- 9 to Nov-13 1992 Nov-16 to Nov-20 22.18 21.14 20.77 20.63 20.37 21.88 21.03 20.60 20.54 20.32 21.49 21.17 20.33 20.46 20.21 21.22 20.71 20.62 20.19 20.50 21.11 20.68 20.26 20.04 20.37 1992 Nov-23 to Nov-27 1992 Nov-30 to Dec- 4 1992 Dec- 7 to Dec-11 20.01 19.91 19.17 20.06 19.56 18.69 20.29 19.38 18.86 19.08 19.26 20.29 18.95 19.09 20.05 110 1992 Dec-14 to Dec-18 1992 Dec-21 to Dec-25 19.06 19.92 18.94 19.79 19.42 19.97 19.68 19.97 1992 Dec-28 to Jan- 1 1993 Jan- 4 to Jan- 8 1993 Jan-11 to Jan-15 1993 Jan-18 to Jan-22 1993 Jan-25 to Jan-29 19.77 19.03 18.78 18.94 19.51 19.62 19.13 18.21 18.40 19.67 19.63 19.03 18.51 18.35 19.66 19.49 18.92 18.71 18.71 20.38 18.90 18.89 18.64 20.27 1993 Feb- 1 to Feb- 5 1993 Feb- 8 to Feb-12 1993 Feb-15 to Feb-19 1993 Feb-22 to Feb-26 1993 Mar- 1 to Mar- 5 20.32 20.07 20.08 20.62 20.00 20.12 19.59 20.49 20.48 20.20 19.30 20.28 20.46 20.29 20.29 19.44 20.63 21.05 20.27 19.96 19.69 20.53 20.85 1993 Mar- 8 to Mar-12 1993 Mar-15 to Mar-19 1993 Mar-22 to Mar-26 1993 Mar-29 to Apr- 2 1993 Apr- 5 to Apr- 9 20.63 20.18 19.52 20.27 20.59 20.71 20.04 19.98 20.27 20.33 20.42 20.14 19.97 20.44 20.35 20.12 20.29 20.12 20.54 20.22 20.38 20.05 20.42 20.67 1993 Apr-12 to Apr-16 1993 Apr-19 to Apr-23 1993 Apr-26 to Apr-30 1993 May- 3 to May- 7 1993 May-10 to May-14 20.43 19.99 19.83 20.58 20.42 20.28 20.05 20.21 20.40 20.38 20.38 19.99 20.46 20.19 20.21 19.79 20.63 20.49 19.80 20.09 19.93 20.54 20.43 19.51 1993 May-17 to May-21 1993 May-24 to May-28 1993 May-31 to Jun- 4 1993 Jun- 7 to Jun-11 1993 Jun-14 to Jun-18 19.51 19.43 19.32 19.75 20.20 19.65 18.60 19.15 19.92 20.05 19.66 18.87 19.56 20.05 19.77 19.27 18.70 19.60 20.04 19.79 18.99 18.71 1993 Jun-21 to Jun-25 1993 Jun-28 to Jul- 2 1993 Jul- 5 to Jul- 9 1993 Jul-12 to Jul-16 1993 Jul-19 to Jul-23 18.63 18.89 18.12 17.68 18.35 18.99 18.29 18.05 17.20 18.44 18.82 18.06 17.43 17.60 18.36 18.44 17.79 17.73 17.57 18.83 17.93 17.92 17.29 17.67 1993 Jul-26 to Jul-30 1993 Aug- 2 to Aug- 6 1993 Aug- 9 to Aug-13 1993 Aug-16 to Aug-20 1993 Aug-23 to Aug-27 18.11 17.90 17.58 17.84 18.42 18.50 17.81 17.58 17.95 18.38 18.25 17.82 17.87 17.68 18.45 18.12 17.54 18.16 17.64 18.40 17.94 17.30 18.14 18.09 18.82 1993 Aug-30 to Sep- 3 1993 Sep- 6 to Sep-10 1993 Sep-13 to Sep-17 1993 Sep-20 to Sep-24 1993 Sep-27 to Oct- 1 18.63 16.98 17.68 17.52 18.24 17.14 16.94 18.12 18.07 17.92 17.00 16.85 17.53 18.73 17.97 16.97 16.85 17.52 18.72 17.72 16.74 17.10 17.52 18.63 1993 Oct- 4 to Oct- 8 1993 Oct-11 to Oct-15 1993 Oct-18 to Oct-22 1993 Oct-25 to Oct-29 1993 Nov- 1 to Nov- 5 18.40 18.76 18.13 17.36 17.50 18.43 18.70 18.04 17.55 17.03 18.43 18.65 18.25 17.66 17.47 18.49 18.50 18.28 17.30 17.38 18.53 18.24 17.92 16.97 17.09 1993 Nov- 8 to Nov-12 1993 Nov-15 to Nov-19 1993 Nov-22 to Nov-26 1993 Nov-29 to Dec- 3 1993 Dec- 6 to Dec-10 16.69 16.78 16.67 15.30 14.57 16.71 16.69 16.39 15.36 14.53 16.54 17.08 15.73 15.45 14.61 16.91 16.67 16.72 16.55 15.53 15.03 15.04 1993 Dec-13 to Dec-17 1993 Dec-20 to Dec-24 1993 Dec-27 to Dec-31 1994 Jan- 3 to Jan- 7 1994 Jan-10 to Jan-14 14.50 14.20 14.09 14.52 14.65 14.47 14.32 14.11 14.66 14.95 14.39 14.61 14.45 15.30 14.36 14.21 14.48 14.19 15.36 14.56 13.98 1994 Jan-17 to Jan-21 1994 Jan-24 to Jan-28 1994 Jan-31 to Feb- 4 1994 Feb- 7 to Feb-11 1994 Feb-14 to Feb-18 15.15 15.09 15.24 15.31 14.15 14.92 15.26 15.91 15.12 14.13 15.25 15.50 16.06 14.64 13.89 14.95 15.48 15.97 14.59 14.26 14.85 15.37 15.63 14.71 14.21 1994 Feb-21 to Feb-25 1994 Feb-28 to Mar- 4 14.50 14.28 14.78 14.18 14.80 14.82 14.74 14.48 14.57 19.60 18.90 14.93 14.65 19.81 15.33 14.81 111 1994 Mar- 7 to Mar-11 1994 Mar-14 to Mar-18 1994 Mar-21 to Mar-25 14.14 14.52 15.37 14.17 14.82 15.04 14.23 15.03 14.94 14.19 14.83 15.21 14.47 14.88 15.19 1994 Mar-28 to Apr- 1 1994 Apr- 4 to Apr- 8 1994 Apr-11 to Apr-15 1994 Apr-18 to Apr-22 1994 Apr-25 to Apr-29 14.15 15.67 15.86 16.64 17.66 14.41 15.67 15.74 16.43 16.82 14.40 15.77 15.96 16.65 16.70 14.78 15.59 16.20 17.56 16.57 15.57 16.58 17.84 16.92 1994 May- 2 to May- 6 1994 May- 9 to May-13 1994 May-16 to May-20 1994 May-23 to May-27 1994 May-30 to Jun- 3 17.18 17.72 17.99 18.69 16.93 17.62 17.61 18.40 18.30 16.85 17.84 18.02 17.83 18.29 17.28 18.27 18.49 17.76 18.25 17.74 18.20 18.93 18.02 18.11 1994 Jun- 6 to Jun-10 1994 Jun-13 to Jun-17 1994 Jun-20 to Jun-24 1994 Jun-27 to Jul- 1 1994 Jul- 4 to Jul- 8 18.10 18.82 20.68 18.96 17.78 18.94 20.08 19.14 19.62 18.28 19.75 19.59 18.82 19.30 18.67 19.83 19.38 19.37 19.15 18.48 20.72 19.28 19.52 19.48 1994 Jul-11 to Jul-15 1994 Jul-18 to Jul-22 1994 Jul-25 to Jul-29 1994 Aug- 1 to Aug- 5 1994 Aug- 8 to Aug-12 20.17 19.38 19.42 20.65 19.41 20.40 19.49 19.26 20.22 19.29 20.14 19.15 19.46 20.06 18.96 20.18 19.32 19.87 20.14 18.67 19.89 19.59 20.30 19.37 18.05 1994 Aug-15 to Aug-19 1994 Aug-22 to Aug-26 1994 Aug-29 to Sep- 2 1994 Sep- 5 to Sep- 9 1994 Sep-12 to Sep-16 18.24 16.87 17.62 17.39 17.64 17.09 17.40 17.62 17.00 18.08 17.54 17.60 17.84 16.72 17.73 17.49 17.47 17.62 16.72 17.56 17.14 17.52 17.53 16.84 1994 Sep-19 to Sep-23 1994 Sep-26 to Sep-30 1994 Oct- 3 to Oct- 7 1994 Oct-10 to Oct-14 1994 Oct-17 to Oct-21 17.21 17.68 18.16 17.95 17.11 17.24 17.48 17.97 17.74 17.34 17.06 17.69 18.01 17.13 17.43 17.65 17.96 18.24 17.03 17.57 17.76 18.36 18.27 16.98 17.31 1994 Oct-24 to Oct-28 1994 Oct-31 to Nov- 4 1994 Nov- 7 to Nov-11 1994 Nov-14 to Nov-18 1994 Nov-21 to Nov-25 17.50 18.16 18.38 17.48 17.43 17.71 18.66 18.46 17.58 17.70 18.01 18.90 18.16 17.38 18.05 18.17 18.97 18.18 17.63 18.25 18.76 18.07 17.47 1994 Nov-28 to Dec- 2 1994 Dec- 5 to Dec- 9 1994 Dec-12 to Dec-16 1994 Dec-19 to Dec-23 1994 Dec-26 to Dec-30 18.10 16.88 16.92 16.91 17.97 16.93 16.98 16.98 17.63 18.06 16.86 16.96 17.07 17.74 17.77 17.14 16.60 17.15 17.69 17.00 17.12 16.84 17.41 17.77 17.45 17.42 18.47 18.53 18.48 17.56 17.70 18.72 18.82 18.48 17.76 17.72 18.70 18.23 18.54 17.69 17.52 18.46 17.95 18.78 18.36 18.46 18.84 18.31 18.30 18.28 18.63 18.59 18.40 17.98 18.47 18.94 18.70 18.63 17.92 1995 Jan- 2 to Jan- 6 1995 Jan- 9 to Jan-13 1995 Jan-16 to Jan-20 1995 Jan-23 to Jan-27 1995 Jan-30 to Feb- 3 17.40 17.89 18.16 18.20 1995 Feb- 6 to Feb-10 1995 Feb-13 to Feb-17 1995 Feb-20 to Feb-24 1995 Feb-27 to Mar- 3 1995 Mar- 6 to Mar-10 18.58 18.29 18.64 18.56 18.43 18.44 18.86 18.52 18.59 1995 Mar-13 to Mar-17 1995 Mar-20 to Mar-24 1995 Mar-27 to Mar-31 1995 Apr- 3 to Apr- 7 1995 Apr-10 to Apr-14 18.19 18.58 19.08 19.05 19.61 17.90 18.44 18.95 19.11 19.84 18.12 18.91 19.21 19.57 19.50 18.18 18.85 19.17 19.74 19.16 18.27 18.68 19.18 19.67 1995 Apr-17 to Apr-21 1995 Apr-24 to Apr-28 1995 May- 1 to May- 5 1995 May- 8 to May-12 1995 May-15 to May-19 19.80 20.13 20.53 20.32 19.91 20.14 20.31 20.18 19.67 20.01 20.40 20.18 19.88 19.76 19.94 20.52 20.48 20.30 19.40 19.99 20.46 20.36 20.33 19.55 20.06 1995 May-22 to May-26 19.86 19.81 19.34 19.12 18.70 112 1995 May-29 to Jun- 2 1995 Jun- 5 to Jun- 9 1995 Jun-12 to Jun-16 1995 Jun-19 to Jun-23 19.25 18.87 18.23 18.78 19.16 18.93 18.01 18.88 19.08 19.05 17.59 18.89 18.94 18.90 17.76 19.14 18.83 18.80 17.81 1995 Jun-26 to Jun-30 1995 Jul- 3 to Jul- 7 1995 Jul-10 to Jul-14 1995 Jul-17 to Jul-21 1995 Jul-24 to Jul-28 17.67 17.33 17.35 17.28 17.17 17.95 17.31 17.33 17.45 17.98 17.21 17.49 17.33 17.47 17.59 17.42 17.25 17.03 17.51 17.38 17.14 17.32 17.07 17.43 1995 Jul-31 to Aug- 4 1995 Aug- 7 to Aug-11 1995 Aug-14 to Aug-18 1995 Aug-21 to Aug-25 1995 Aug-28 to Sep- 1 17.62 17.65 17.48 18.26 17.84 17.59 17.86 17.40 18.54 17.87 17.95 17.77 17.59 18.79 17.74 17.72 17.91 17.65 19.60 17.89 17.74 17.86 17.87 19.91 18.08 1995 Sep- 4 to Sep- 8 1995 Sep-11 to Sep-15 1995 Sep-18 to Sep-22 1995 Sep-25 to Sep-29 1995 Oct- 2 to Oct- 6 18.49 18.92 17.38 17.67 18.49 18.76 19.01 17.47 17.56 18.27 18.53 18.53 17.64 17.32 18.27 18.87 17.89 17.72 16.86 18.44 18.94 17.27 17.54 17.03 1995 Oct- 9 to Oct-13 1995 Oct-16 to Oct-20 1995 Oct-23 to Oct-27 1995 Oct-30 to Nov- 3 1995 Nov- 6 to Nov-10 17.36 17.60 17.48 17.67 17.68 17.29 17.59 17.61 17.67 17.62 17.28 17.57 17.58 17.74 17.82 17.08 17.34 17.61 17.94 17.83 17.38 17.44 17.54 17.93 17.83 1995 Nov-13 to Nov-17 1995 Nov-20 to Nov-24 1995 Nov-27 to Dec- 1 1995 Dec- 4 to Dec- 8 1995 Dec-11 to Dec-15 17.78 18.14 18.37 18.61 18.62 17.89 17.89 18.28 18.66 18.80 17.92 17.93 18.27 18.76 19.01 18.16 18.56 18.27 18.69 19.14 18.43 18.97 19.51 1995 Dec-18 to Dec-22 1995 Dec-25 to Dec-29 1996 Jan- 1 to Jan- 5 1996 Jan- 8 to Jan-12 1996 Jan-15 to Jan-19 19.71 19.05 19.25 19.83 19.86 18.12 18.87 19.49 19.90 19.66 18.58 18.83 19.47 19.96 18.87 19.12 19.12 19.54 20.26 18.28 18.94 1996 Jan-22 to Jan-26 1996 Jan-29 to Feb- 2 1996 Feb- 5 to Feb- 9 1996 Feb-12 to Feb-16 1996 Feb-19 to Feb-23 18.49 17.33 17.53 18.01 18.58 17.65 17.74 18.96 21.07 18.95 17.76 17.71 18.86 21.63 18.06 17.63 17.79 19.02 22.14 17.68 17.83 17.83 19.16 20.97 1996 Feb-26 to Mar- 1 1996 Mar- 4 to Mar- 8 1996 Mar-11 to Mar-15 1996 Mar-18 to Mar-22 1996 Mar-25 to Mar-29 19.45 19.24 19.92 23.23 23.23 19.65 19.65 20.42 24.56 22.38 19.30 20.16 20.49 22.74 21.64 19.59 19.87 21.18 22.44 21.45 19.45 19.66 21.99 22.85 21.43 1996 Apr- 1 to Apr- 5 1996 Apr- 8 to Apr-12 1996 Apr-15 to Apr-19 1996 Apr-22 to Apr-26 1996 Apr-29 to May- 3 22.29 23.01 25.13 23.94 22.07 22.68 23.23 24.48 24.39 20.95 22.22 24.08 24.67 24.00 20.81 22.75 25.15 23.47 24.35 20.78 24.29 23.96 22.33 21.19 1996 May- 6 to May-10 1996 May-13 to May-17 1996 May-20 to May-24 1996 May-27 to May-31 1996 Jun- 3 to Jun- 7 21.06 21.23 22.28 19.86 21.07 21.35 21.95 21.10 20.28 21.13 21.28 23.02 20.65 19.73 20.65 20.79 22.35 19.95 20.06 21.01 20.64 21.58 19.77 20.28 1996 Jun-10 to Jun-14 1996 Jun-17 to Jun-21 1996 Jun-24 to Jun-28 1996 Jul- 1 to Jul- 5 1996 Jul- 8 to Jul-12 20.18 21.43 20.10 21.48 21.33 20.13 21.53 20.13 21.08 21.52 20.13 20.73 20.63 21.38 21.56 20.03 20.68 20.98 21.96 20.28 20.38 20.92 21.65 21.90 1996 Jul-15 to Jul-19 1996 Jul-22 to Jul-26 1996 Jul-29 to Aug- 2 1996 Aug- 5 to Aug- 9 1996 Aug-12 to Aug-16 22.43 21.05 20.28 21.25 22.25 22.28 21.58 20.28 21.08 22.35 21.65 20.97 20.46 21.35 22.10 21.68 21.05 20.95 21.45 21.95 20.95 20.13 21.35 21.60 22.60 20.50 18.42 113 1996 Aug-19 to Aug-23 1996 Aug-26 to Aug-30 1996 Sep- 2 to Sep- 6 1996 Sep- 9 to Sep-13 1996 Sep-16 to Sep-20 23.10 21.55 23.75 23.30 22.53 21.35 23.35 24.15 23.40 22.15 21.75 23.25 24.75 23.85 22.48 22.20 23.45 24.95 23.45 22.23 22.25 23.85 24.50 23.60 1996 Sep-23 to Sep-27 1996 Sep-30 to Oct- 4 1996 Oct- 7 to Oct-11 1996 Oct-14 to Oct-18 1996 Oct-21 to Oct-25 23.83 24.20 25.25 25.50 25.85 24.55 24.35 25.45 25.45 25.79 24.60 24.05 25.00 25.10 24.78 24.05 24.85 24.30 25.40 24.55 24.60 24.75 24.65 25.80 24.80 1996 Oct-28 to Nov- 1 1996 Nov- 4 to Nov- 8 1996 Nov-11 to Nov-15 1996 Nov-18 to Nov-22 1996 Nov-25 to Nov-29 24.75 22.80 23.35 23.85 25.75 24.15 22.65 23.40 24.50 23.70 24.40 22.75 24.25 23.75 23.70 23.25 22.80 24.35 23.90 23.00 23.60 24.15 24.15 23.70 1996 Dec- 2 to Dec- 6 1996 Dec- 9 to Dec-13 1996 Dec-16 to Dec-20 1996 Dec-23 to Dec-27 1996 Dec-30 to Jan- 3 24.70 25.15 25.70 26.40 25.35 24.80 24.25 25.70 25.05 25.90 25.00 23.40 26.05 25.60 23.75 26.55 25.10 25.55 25.60 24.45 26.10 25.20 25.55 1997 Jan- 6 to Jan-10 1997 Jan-13 to Jan-17 1997 Jan-20 to Jan-24 1997 Jan-27 to Jan-31 1997 Feb- 3 to Feb- 7 26.25 25.20 25.10 23.85 24.20 26.25 25.10 24.80 23.90 24.00 26.55 25.95 24.40 24.50 23.90 26.30 25.45 23.85 24.80 23.05 26.15 25.40 23.85 24.15 22.30 1997 Feb-10 to Feb-14 1997 Feb-17 to Feb-21 1997 Feb-24 to Feb-28 1997 Mar- 3 to Mar- 7 1997 Mar-10 to Mar-14 22.45 21.00 20.25 20.50 22.35 22.50 20.95 20.75 20.15 21.75 22.65 21.10 20.50 20.60 22.10 21.95 20.80 21.00 20.65 22.40 21.60 20.30 21.35 21.30 1997 Mar-17 to Mar-21 1997 Mar-24 to Mar-28 1997 Mar-31 to Apr- 4 1997 Apr- 7 to Apr-11 1997 Apr-14 to Apr-18 20.90 21.00 20.35 19.25 19.90 22.00 20.95 20.30 19.35 19.60 22.00 20.65 19.55 19.25 19.35 22.00 20.70 19.45 19.55 19.50 21.70 1997 Apr-21 to Apr-25 1997 Apr-28 to May- 2 1997 May- 5 to May- 9 1997 May-12 to May-16 1997 May-19 to May-23 20.35 19.85 19.60 21.40 21.50 19.60 20.40 19.70 21.10 21.20 19.65 20.20 19.60 21.35 21.60 19.85 19.90 20.35 21.30 21.60 19.80 19.60 20.45 22.10 21.25 1997 May-26 to May-30 1997 Jun- 2 to Jun- 6 1997 Jun- 9 to Jun-13 1997 Jun-16 to Jun-20 1997 Jun-23 to Jun-27 21.15 18.85 19.10 18.90 20.75 20.35 18.85 19.30 18.71 20.70 20.30 18.75 18.90 19.12 21.15 19.80 19.00 18.75 18.84 21.00 19.00 18.95 18.60 19.42 1997 Jun-30 to Jul- 4 1997 Jul- 7 to Jul-11 1997 Jul-14 to Jul-18 1997 Jul-21 to Jul-25 1997 Jul-28 to Aug- 1 19.82 19.53 18.92 19.25 19.67 20.11 19.67 19.81 19.25 20.08 20.39 19.40 19.57 19.68 20.43 19.48 19.22 19.92 19.71 20.19 19.39 19.22 19.71 20.27 1997 Aug- 4 to Aug- 8 1997 Aug-11 to Aug-15 1997 Aug-18 to Aug-22 1997 Aug-25 to Aug-29 1997 Sep- 1 to Sep- 5 20.81 19.73 19.94 19.29 20.79 19.91 20.04 19.31 19.73 20.46 20.19 20.07 19.65 19.65 20.09 20.02 19.83 19.59 19.46 19.58 20.08 19.70 19.66 19.61 1997 Sep- 8 to Sep-12 1997 Sep-15 to Sep-19 1997 Sep-22 to Sep-26 1997 Sep-29 to Oct- 3 1997 Oct- 6 to Oct-10 19.43 19.30 19.61 21.29 21.97 19.50 19.67 19.71 21.13 21.88 19.50 19.44 20.03 21.02 22.19 19.41 19.41 20.38 21.82 22.01 19.37 19.37 20.87 22.86 22.01 1997 Oct-13 to Oct-17 1997 Oct-20 to Oct-24 1997 Oct-27 to Oct-31 1997 Nov- 3 to Nov- 7 21.34 20.69 21.03 20.91 20.75 20.68 20.40 20.64 20.65 21.44 20.82 20.44 20.98 21.18 21.26 20.46 21.41 20.99 21.10 20.80 19.15 19.50 19.90 114 1997 Nov-10 to Nov-14 20.46 20.54 20.53 20.76 21.01 1997 Nov-17 to Nov-21 1997 Nov-24 to Nov-28 1997 Dec- 1 to Dec- 5 1997 Dec- 8 to Dec-12 1997 Dec-15 to Dec-19 20.29 19.41 18.76 18.86 18.18 20.01 19.35 18.67 18.79 18.19 19.82 19.13 18.81 18.15 18.22 19.17 19.76 18.49 18.18 18.51 18.72 18.21 18.46 1997 Dec-22 to Dec-26 1997 Dec-29 to Jan- 2 1998 Jan- 5 to Jan- 9 1998 Jan-12 to Jan-16 1998 Jan-19 to Jan-23 18.29 17.64 16.95 16.53 18.39 17.60 16.64 16.44 16.43 18.36 17.65 16.91 16.54 16.05 17.01 16.35 15.93 18.19 17.41 16.65 16.48 15.65 1998 Jan-26 to Jan-30 1998 Feb- 2 to Feb- 6 1998 Feb- 9 to Feb-13 1998 Feb-16 to Feb-20 1998 Feb-23 to Feb-27 16.96 17.07 16.59 15.23 17.06 16.43 16.48 15.64 15.14 17.35 16.42 16.17 16.24 15.28 17.93 16.63 15.97 16.12 15.41 17.21 16.72 16.03 16.13 15.44 1998 Mar- 2 to Mar- 6 1998 Mar- 9 to Mar-13 1998 Mar-16 to Mar-20 1998 Mar-23 to Mar-27 1998 Mar-30 to Apr- 3 15.32 14.58 13.41 16.14 16.32 15.28 14.54 13.67 15.52 15.75 15.33 14.30 14.36 16.69 15.69 15.34 14.40 14.43 16.92 16.04 14.90 14.19 14.31 16.84 16.08 1998 Apr- 6 to Apr-10 1998 Apr-13 to Apr-17 1998 Apr-20 to Apr-24 1998 Apr-27 to May- 1 1998 May- 4 to May- 8 15.48 15.35 15.52 15.43 15.98 15.39 15.18 15.57 15.90 15.48 15.71 15.64 15.07 15.43 15.45 15.61 15.97 13.09 15.56 15.26 15.49 13.23 16.25 15.21 1998 May-11 to May-15 1998 May-18 to May-22 1998 May-25 to May-29 1998 Jun- 1 to Jun- 5 1998 Jun- 8 to Jun-12 15.26 14.15 14.99 14.49 15.21 13.01 14.90 14.81 13.76 15.01 13.36 14.98 14.94 13.54 15.15 14.21 14.88 15.24 12.67 14.51 14.81 15.21 15.13 12.66 1998 Jun-15 to Jun-19 1998 Jun-22 to Jun-26 1998 Jun-29 to Jul- 3 1998 Jul- 6 to Jul-10 1998 Jul-13 to Jul-17 11.69 13.54 14.29 13.87 14.06 12.38 14.65 14.30 13.83 14.87 12.55 14.54 14.47 14.01 14.90 11.80 13.90 14.58 14.01 14.54 11.80 14.21 1998 Jul-20 to Jul-24 1998 Jul-27 to Jul-31 1998 Aug- 3 to Aug- 7 1998 Aug-10 to Aug-14 1998 Aug-17 to Aug-21 13.44 14.37 13.93 13.11 13.26 13.83 14.20 13.69 12.87 13.13 14.29 14.13 13.80 12.76 13.21 13.97 14.25 13.87 13.44 13.62 13.93 14.27 13.85 13.40 13.44 1998 Aug-24 to Aug-28 1998 Aug-31 to Sep- 4 1998 Sep- 7 to Sep-11 1998 Sep-14 to Sep-18 1998 Sep-21 to Sep-25 13.78 13.29 14.46 15.58 13.90 13.62 14.34 14.72 15.73 13.67 13.72 14.37 14.60 15.80 13.35 14.70 14.71 15.07 15.99 13.54 14.59 14.39 15.53 15.80 1998 Sep-28 to Oct- 2 1998 Oct- 5 to Oct- 9 1998 Oct-12 to Oct-16 1998 Oct-19 to Oct-23 1998 Oct-26 to Oct-30 15.74 15.24 14.47 13.39 14.39 16.01 15.60 14.27 13.45 14.19 16.19 15.03 14.09 14.14 14.35 15.52 14.57 14.08 14.00 14.31 15.71 14.64 14.16 14.06 14.48 1998 Nov- 2 to Nov- 6 1998 Nov- 9 to Nov-13 1998 Nov-16 to Nov-20 1998 Nov-23 to Nov-27 1998 Nov-30 to Dec- 4 14.45 13.39 12.81 12.51 11.37 14.24 13.54 12.46 11.48 11.27 14.09 13.63 12.33 10.86 11.31 13.93 14.04 12.21 13.93 13.57 12.20 11.25 11.20 1998 Dec- 7 to Dec-11 1998 Dec-14 to Dec-18 1998 Dec-21 to Dec-25 1998 Dec-28 to Jan- 1 1999 Jan- 4 to Jan- 8 11.61 11.30 10.86 11.59 12.42 11.33 11.64 10.99 11.82 12.04 11.20 12.55 11.12 11.75 12.84 10.82 11.05 11.03 12.14 12.99 10.86 10.95 1999 Jan-11 to Jan-15 1999 Jan-18 to Jan-22 1999 Jan-25 to Jan-29 13.43 12.91 12.13 12.16 12.33 11.82 12.40 12.23 12.45 12.52 12.21 12.62 12.81 12.41 13.96 14.02 13.06 115 1999 Feb- 1 to Feb- 5 1999 Feb- 8 to Feb-12 12.36 11.65 12.21 11.66 12.42 11.88 11.94 11.94 11.85 11.90 1999 Feb-15 to Feb-19 1999 Feb-22 to Feb-26 1999 Mar- 1 to Mar- 5 1999 Mar- 8 to Mar-12 1999 Mar-15 to Mar-19 11.97 12.28 13.63 14.48 11.38 12.44 12.66 13.78 14.60 11.49 12.49 12.92 14.74 15.11 11.97 12.61 13.32 14.30 15.11 11.79 12.31 13.33 14.51 15.26 1999 Mar-22 to Mar-26 1999 Mar-29 to Apr- 2 1999 Apr- 5 to Apr- 9 1999 Apr-12 to Apr-16 1999 Apr-19 to Apr-23 15.39 16.46 16.94 16.44 17.82 15.36 16.66 16.79 16.73 17.79 15.18 16.66 16.02 16.51 18.05 15.66 16.65 15.92 16.85 18.02 16.16 1999 Apr-26 to Apr-30 1999 May- 3 to May- 7 1999 May-10 to May-14 1999 May-17 to May-21 1999 May-24 to May-28 17.67 18.83 18.57 17.76 16.77 17.82 18.94 17.84 17.09 17.25 18.43 18.89 17.44 16.77 17.32 18.53 18.23 18.11 17.04 17.18 18.69 18.23 18.04 17.22 16.85 1999 May-31 to Jun- 4 1999 Jun- 7 to Jun-11 1999 Jun-14 to Jun-18 1999 Jun-21 to Jun-25 1999 Jun-28 to Jul- 2 17.91 18.24 17.79 18.22 16.31 17.66 18.61 17.62 18.56 16.61 17.98 17.92 18.42 19.33 16.81 17.82 18.25 18.26 19.39 17.36 18.45 18.01 18.17 19.70 1999 Jul- 5 to Jul- 9 1999 Jul-12 to Jul-16 1999 Jul-19 to Jul-23 1999 Jul-26 to Jul-30 1999 Aug- 2 to Aug- 6 19.92 20.28 20.49 20.46 19.78 20.33 19.38 20.40 20.21 19.98 19.89 19.54 20.61 20.49 19.76 20.26 19.84 21.09 20.67 19.94 20.66 20.41 20.52 20.89 16.59 17.34 17.96 1999 Aug- 9 to Aug-13 1999 Aug-16 to Aug-20 1999 Aug-23 to Aug-27 1999 Aug-30 to Sep- 3 1999 Sep- 6 to Sep-10 21.26 21.34 21.61 21.93 21.31 21.70 21.36 22.15 22.58 21.54 21.54 20.66 21.79 22.64 21.42 21.86 21.09 21.54 23.24 21.67 21.66 21.31 22.00 23.55 1999 Sep-13 to Sep-17 1999 Sep-20 to Sep-24 1999 Sep-27 to Oct- 1 1999 Oct- 4 to Oct- 8 1999 Oct-11 to Oct-15 24.18 24.32 24.63 23.71 21.58 23.86 24.48 24.52 23.37 22.93 24.10 24.26 24.69 23.24 22.93 24.52 24.76 24.54 22.46 22.46 24.72 24.81 24.51 20.81 22.81 1999 Oct-18 to Oct-22 1999 Oct-25 to Oct-29 1999 Nov- 1 to Nov- 5 1999 Nov- 8 to Nov-12 1999 Nov-15 to Nov-19 22.44 23.22 22.44 23.28 25.31 22.36 23.17 22.49 24.21 26.04 22.21 22.71 22.59 24.37 26.57 22.74 21.59 23.17 24.33 25.71 23.48 21.79 22.93 24.91 26.61 1999 Nov-22 to Nov-26 1999 Nov-29 to Dec- 3 1999 Dec- 6 to Dec-10 1999 Dec-13 to Dec-17 1999 Dec-20 to Dec-24 28.03 25.84 26.67 25.37 26.55 26.91 24.87 26.36 25.87 26.34 27.22 25.01 26.69 26.28 25.51 25.88 26.00 26.74 25.86 1999 Dec-27 to Dec-31 2000 Jan- 3 to Jan- 7 2000 Jan-10 to Jan-14 2000 Jan-17 to Jan-21 2000 Jan-24 to Jan-28 26.36 26.81 25.56 25.69 28.98 30.28 26.41 24.65 26.30 29.11 27.66 25.76 24.79 26.63 29.67 27.22 24.79 28.01 29.71 27.27 2000 Jan-31 to Feb- 4 2000 Feb- 7 to Feb-11 2000 Feb-14 to Feb-18 2000 Feb-21 to Feb-25 2000 Feb-28 to Mar- 3 27.65 28.40 30.30 30.11 28.28 28.05 30.17 29.63 30.57 27.52 28.71 30.01 30.19 31.71 28.26 29.49 29.37 30.23 31.51 28.67 29.51 29.51 30.34 31.46 2000 Mar- 6 to Mar-10 2000 Mar-13 to Mar-17 2000 Mar-20 to Mar-24 2000 Mar-27 to Mar-31 2000 Apr- 3 to Apr- 7 32.19 32.11 29.33 27.59 26.28 33.90 31.93 28.01 27.10 25.46 31.22 30.47 27.28 26.36 25.76 31.61 31.10 27.47 26.67 25.51 31.76 30.86 27.86 26.86 24.97 2000 Apr-10 to Apr-14 2000 Apr-17 to Apr-21 23.91 26.06 24.17 25.92 25.49 27.38 25.29 27.29 25.48 24.71 29.25 27.22 25.71 25.21 26.76 116 2000 Apr-24 to Apr-28 2000 May- 1 to May- 5 2000 May- 8 to May-12 27.41 25.84 28.05 26.41 26.86 28.46 24.69 26.60 28.10 25.53 27.08 29.25 25.71 27.37 29.64 2000 May-15 to May-19 2000 May-22 to May-26 2000 May-29 to Jun- 2 2000 Jun- 5 to Jun- 9 2000 Jun-12 to Jun-16 29.95 28.62 29.87 31.76 29.76 28.61 30.36 29.78 32.73 29.38 29.89 29.03 29.29 32.72 30.28 30.43 30.19 29.78 32.70 30.02 29.76 30.34 30.22 32.35 2000 Jun-19 to Jun-23 2000 Jun-26 to Jun-30 2000 Jul- 3 to Jul- 7 2000 Jul-10 to Jul-14 2000 Jul-17 to Jul-21 31.57 31.56 33.07 32.01 29.51 30.71 29.61 31.94 33.64 31.86 30.76 30.34 31.10 34.72 32.73 30.19 31.41 30.94 34.76 32.44 30.26 31.31 28.56 2000 Jul-24 to Jul-28 2000 Jul-31 to Aug- 4 2000 Aug- 7 to Aug-11 2000 Aug-14 to Aug-18 2000 Aug-21 to Aug-25 27.97 27.50 28.93 31.92 32.42 27.89 27.85 29.26 31.43 31.24 27.81 28.27 30.38 31.91 31.23 28.21 28.92 31.09 31.88 33.41 28.22 29.94 31.02 31.97 34.01 2000 Aug-28 to Sep- 1 2000 Sep- 4 to Sep- 8 2000 Sep-11 to Sep-15 2000 Sep-18 to Sep-22 2000 Sep-25 to Sep-29 32.92 35.14 36.75 31.50 32.73 33.92 34.25 36.96 31.78 33.25 34.97 33.87 37.22 31.23 33.09 35.18 34.37 33.84 30.26 33.42 33.62 35.87 32.66 30.87 2000 Oct- 2 to Oct- 6 2000 Oct- 9 to Oct-13 2000 Oct-16 to Oct-20 2000 Oct-23 to Oct-27 2000 Oct-30 to Nov- 3 32.05 31.98 33.18 34.90 32.86 31.86 33.29 33.36 34.53 32.70 30.91 33.20 33.51 33.62 33.14 30.66 36.06 33.06 33.75 32.59 30.86 34.96 34.31 32.78 32.62 2000 Nov- 6 to Nov-10 2000 Nov-13 to Nov-17 2000 Nov-20 to Nov-24 2000 Nov-27 to Dec- 1 2000 Dec- 4 to Dec- 8 33.02 34.30 35.98 36.24 31.28 33.44 34.72 35.54 34.02 29.25 32.92 35.54 36.06 34.58 30.24 33.87 34.84 33.61 29.36 34.05 35.62 36.06 32.06 28.31 2000 Dec-11 to Dec-15 2000 Dec-18 to Dec-22 2000 Dec-25 to Dec-29 2001 Jan- 1 to Jan- 5 2001 Jan- 8 to Jan-12 29.75 29.52 27.44 29.81 29.34 27.00 27.29 27.72 28.75 25.83 26.55 27.93 29.42 28.06 26.06 25.82 27.95 29.42 28.86 26.16 26.72 28.02 30.07 2001 Jan-15 to Jan-19 2001 Jan-22 to Jan-26 2001 Jan-29 to Feb- 2 2001 Feb- 5 to Feb- 9 2001 Feb-12 to Feb-16 32.21 29.07 30.55 30.52 30.19 31.66 29.12 30.27 30.12 29.77 31.46 28.62 31.27 29.55 30.42 31.61 29.88 31.57 28.96 32.12 29.79 31.27 30.93 29.22 2001 Feb-19 to Feb-23 2001 Feb-26 to Mar- 2 2001 Mar- 5 to Mar- 9 2001 Mar-12 to Mar-16 2001 Mar-19 to Mar-23 28.27 28.61 27.91 26.17 28.60 28.26 28.40 27.45 25.99 28.64 27.35 28.95 26.49 26.43 28.52 27.78 28.31 26.56 26.19 28.83 27.89 27.97 26.68 27.31 2001 Mar-26 to Mar-30 2001 Apr- 2 to Apr- 6 2001 Apr- 9 to Apr-13 2001 Apr-16 to Apr-20 2001 Apr-23 to Apr-27 27.40 25.70 27.45 28.81 26.97 27.62 26.65 28.47 27.89 25.24 26.43 27.16 28.42 27.83 25.93 26.47 27.24 28.75 27.92 28.47 26.37 27.07 2001 Apr-30 to May- 4 2001 May- 7 to May-11 2001 May-14 to May-18 2001 May-21 to May-25 2001 May-28 to Jun- 1 28.48 27.79 28.83 29.96 28.37 27.35 28.90 29.75 28.76 27.89 28.39 28.77 28.85 28.71 28.66 28.50 29.02 27.97 28.39 28.41 28.59 29.90 28.08 27.88 2001 Jun- 4 to Jun- 8 2001 Jun-11 to Jun-15 2001 Jun-18 to Jun-22 2001 Jun-25 to Jun-29 2001 Jul- 2 to Jul- 6 28.14 28.94 27.55 27.11 26.02 27.84 29.13 27.49 26.97 26.28 27.56 28.81 26.52 25.67 27.91 29.12 26.85 25.75 27.07 28.43 28.52 27.02 26.37 28.10 2001 Jul- 9 to Jul-13 27.63 27.22 27.07 26.85 26.57 27.00 28.35 117 2001 Jul-16 to Jul-20 2001 Jul-23 to Jul-27 2001 Jul-30 to Aug- 3 2001 Aug- 6 to Aug-10 26.02 25.88 26.60 27.70 25.33 26.18 26.70 28.17 24.65 26.71 26.83 27.62 24.65 26.74 27.86 27.72 25.67 26.98 27.51 28.12 2001 Aug-13 to Aug-17 2001 Aug-20 to Aug-24 2001 Aug-27 to Aug-31 2001 Sep- 3 to Sep- 7 2001 Sep-10 to Sep-14 27.85 27.20 26.69 27.66 28.10 27.93 27.16 26.94 27.65 27.51 27.17 27.07 27.03 27.64 27.47 25.63 26.65 27.54 28.58 26.65 28.34 26.65 27.99 29.59 2001 Sep-17 to Sep-21 2001 Sep-24 to Sep-28 2001 Oct- 1 to Oct- 5 2001 Oct- 8 to Oct-12 2001 Oct-15 to Oct-19 28.84 21.46 23.12 22.25 22.37 27.81 21.63 22.70 22.55 22.01 26.73 22.40 22.17 22.51 21.89 26.60 22.80 22.70 23.49 21.33 25.46 23.44 22.32 22.49 21.99 2001 Oct-22 to Oct-26 2001 Oct-29 to Nov- 2 2001 Nov- 5 to Nov- 9 2001 Nov-12 to Nov-16 2001 Nov-19 to Nov-23 21.78 22.10 20.00 21.26 17.74 21.28 21.83 19.93 21.56 18.71 22.00 21.20 20.11 19.63 18.38 21.75 20.47 21.21 17.50 22.07 20.24 22.23 18.09 2001 Nov-26 to Nov-30 2001 Dec- 3 to Dec- 7 2001 Dec-10 to Dec-14 2001 Dec-17 to Dec-21 2001 Dec-24 to Dec-28 18.69 20.27 18.32 19.28 19.57 19.71 18.04 19.38 19.37 19.62 18.38 19.37 21.32 18.55 18.69 18.20 18.67 21.07 19.46 19.08 19.31 19.30 20.42 2001 Dec-31 to Jan- 4 2002 Jan- 7 to Jan-11 2002 Jan-14 to Jan-18 2002 Jan-21 to Jan-25 2002 Jan-28 to Feb- 1 19.96 21.42 18.88 20.82 18.99 18.36 19.30 21.13 20.32 18.96 19.10 19.11 20.65 20.48 18.20 19.57 19.71 21.47 19.67 18.02 19.80 20.40 2002 Feb- 4 to Feb- 8 2002 Feb-11 to Feb-15 2002 Feb-18 to Feb-22 2002 Feb-25 to Mar- 1 2002 Mar- 4 to Mar- 8 20.02 21.29 20.24 22.55 20.06 20.76 20.77 21.37 23.18 19.77 21.19 20.31 21.40 23.32 19.75 21.19 20.81 21.78 23.62 20.25 21.47 20.92 22.37 23.87 2002 Mar-11 to Mar-15 2002 Mar-18 to Mar-22 2002 Mar-25 to Mar-29 2002 Apr- 1 to Apr- 5 2002 Apr- 8 to Apr-12 24.36 25.03 25.69 26.82 26.16 24.55 25.02 25.75 27.75 25.45 24.14 24.92 25.79 27.55 26.15 24.48 25.74 26.21 26.64 24.93 24.47 25.56 2002 Apr-15 to Apr-19 2002 Apr-22 to Apr-26 2002 Apr-29 to May- 3 2002 May- 6 to May-10 2002 May-13 to May-17 24.53 26.28 27.45 26.11 28.62 24.92 26.28 27.32 26.79 29.17 25.94 26.28 26.58 27.76 28.17 25.86 26.36 26.31 27.78 28.00 26.43 27.12 26.75 27.92 28.19 2002 May-20 to May-24 2002 May-27 to May-31 2002 Jun- 3 to Jun- 7 2002 Jun-10 to Jun-14 2002 Jun-17 to Jun-21 28.24 25.10 24.24 25.98 27.35 25.08 25.32 24.21 25.36 27.01 25.64 25.02 24.79 25.57 26.60 24.78 24.89 25.54 25.62 26.69 25.37 24.72 25.90 25.51 2002 Jun-24 to Jun-28 2002 Jul- 1 to Jul- 5 2002 Jul- 8 to Jul-12 2002 Jul-15 to Jul-19 2002 Jul-22 to Jul-26 26.31 26.79 26.14 27.23 26.61 26.06 26.83 26.16 27.68 26.61 26.67 26.82 26.73 27.88 26.78 26.77 26.79 27.01 27.50 26.67 27.48 27.83 26.55 2002 Jul-29 to Aug- 2 2002 Aug- 5 to Aug- 9 2002 Aug-12 to Aug-16 2002 Aug-19 to Aug-23 2002 Aug-26 to Aug-30 26.54 26.55 27.84 29.86 29.23 27.43 27.18 28.35 30.12 28.84 27.02 26.58 28.19 30.37 28.31 26.51 26.67 28.99 30.11 28.83 26.87 26.87 29.24 29.99 28.97 2002 Sep- 2 to Sep- 6 2002 Sep- 9 to Sep-13 2002 Sep-16 to Sep-20 2002 Sep-23 to Sep-27 2002 Sep-30 to Oct- 4 29.80 29.14 30.85 30.59 27.76 29.62 29.08 30.79 30.71 28.28 29.77 29.57 30.69 30.59 29.06 28.95 29.49 30.31 29.73 29.51 29.83 29.65 30.53 29.65 20.05 26.21 23.51 118 2002 Oct- 7 to Oct-11 2002 Oct-14 to Oct-18 2002 Oct-21 to Oct-25 2002 Oct-28 to Nov- 1 2002 Nov- 4 to Nov- 8 29.65 30.06 28.31 27.25 26.89 29.56 29.73 27.93 26.81 26.06 29.31 29.28 28.19 26.85 25.72 28.96 29.61 27.87 27.18 25.36 29.36 29.56 27.09 27.04 25.83 2002 Nov-11 to Nov-15 2002 Nov-18 to Nov-22 2002 Nov-25 to Nov-29 2002 Dec- 2 to Dec- 6 2002 Dec- 9 to Dec-13 26.02 26.71 27.01 27.27 27.29 26.19 26.41 26.60 27.34 27.73 25.28 27.00 26.87 26.80 27.49 25.40 27.07 25.50 27.73 27.27 28.20 27.03 28.39 2002 Dec-16 to Dec-20 2002 Dec-23 to Dec-27 2002 Dec-30 to Jan- 3 2003 Jan- 6 to Jan-10 2003 Jan-13 to Jan-17 30.15 32.09 31.41 32.29 32.08 30.04 32.13 31.21 31.20 32.42 30.41 30.66 33.23 30.57 32.61 31.97 31.95 33.58 30.57 32.68 33.26 31.59 33.88 2003 Jan-20 to Jan-24 2003 Jan-27 to Jan-31 2003 Feb- 3 to Feb- 7 2003 Feb-10 to Feb-14 2003 Feb-17 to Feb-21 32.43 32.84 34.46 34.62 32.70 33.61 35.43 36.88 34.32 33.54 33.91 35.83 37.02 33.90 33.78 34.36 36.63 36.45 34.98 33.51 35.05 36.61 36.76 2003 Feb-24 to Feb-28 2003 Mar- 3 to Mar- 7 2003 Mar-10 to Mar-14 2003 Mar-17 to Mar-21 2003 Mar-24 to Mar-28 37.29 36.10 37.18 34.92 29.51 36.06 36.95 36.81 31.55 33.42 37.96 36.86 37.87 30.01 28.71 36.83 37.21 36.05 28.62 30.31 36.76 37.76 35.41 27.18 30.21 2003 Mar-31 to Apr- 4 2003 Apr- 7 to Apr-11 2003 Apr-14 to Apr-18 2003 Apr-21 to Apr-25 2003 Apr-28 to May- 2 31.14 27.76 28.41 30.76 25.25 29.48 27.97 29.46 29.92 25.32 28.55 28.93 29.16 28.04 26.09 29.05 27.20 30.10 27.52 26.05 28.41 28.28 2003 May- 5 to May- 9 2003 May-12 to May-16 2003 May-19 to May-23 2003 May-26 to May-30 2003 Jun- 2 to Jun- 6 26.43 27.34 28.84 30.72 25.65 28.51 29.29 29.24 30.78 26.24 29.21 29.51 28.46 29.81 26.94 28.57 29.09 29.15 30.84 27.65 29.07 29.74 29.56 31.26 2003 Jun- 9 to Jun-13 2003 Jun-16 to Jun-20 2003 Jun-23 to Jun-27 2003 Jun-30 to Jul- 4 2003 Jul- 7 to Jul-11 31.36 31.14 30.22 30.15 30.08 31.72 31.08 30.05 30.41 30.32 32.17 30.28 31.65 30.29 30.87 31.41 29.86 28.97 30.39 31.04 30.63 30.63 29.18 2003 Jul-14 to Jul-18 2003 Jul-21 to Jul-25 2003 Jul-28 to Aug- 1 2003 Aug- 4 to Aug- 8 2003 Aug-11 to Aug-15 31.20 31.67 29.98 31.80 31.91 31.60 30.20 30.21 32.34 31.91 31.20 30.13 30.69 31.77 30.85 31.44 30.72 30.56 32.41 30.85 31.96 30.31 32.23 32.23 31.01 2003 Aug-18 to Aug-22 2003 Aug-25 to Aug-29 2003 Sep- 1 to Sep- 5 2003 Sep- 8 to Sep-12 2003 Sep-15 to Sep-19 30.81 31.43 28.85 28.15 30.76 32.01 29.57 29.22 27.60 30.96 31.18 29.43 29.41 27.00 31.78 31.41 28.87 28.86 27.26 31.64 31.76 28.93 28.26 26.93 2003 Sep-22 to Sep-26 2003 Sep-29 to Oct- 3 2003 Oct- 6 to Oct-10 2003 Oct-13 to Oct-17 2003 Oct-20 to Oct-24 26.97 28.35 30.40 31.91 30.37 27.00 29.19 30.48 31.68 30.19 28.19 29.43 29.60 31.74 30.00 28.29 29.83 30.97 31.51 30.31 28.21 30.37 32.01 30.61 29.99 2003 Oct-27 to Oct-31 2003 Nov- 3 to Nov- 7 2003 Nov-10 to Nov-14 2003 Nov-17 to Nov-21 2003 Nov-24 to Nov-28 29.95 28.81 31.01 31.75 29.99 29.57 28.86 31.21 33.16 30.02 28.95 30.29 31.37 32.84 30.33 28.67 30.25 31.89 32.87 29.24 30.73 32.31 32.26 2003 Dec- 1 to Dec- 5 2003 Dec- 8 to Dec-12 2003 Dec-15 to Dec-19 2003 Dec-22 to Dec-26 29.89 32.08 33.17 31.71 30.74 31.72 32.94 32.03 30.61 31.92 33.36 32.99 31.24 32.01 33.72 30.68 33.06 32.81 25.92 25.74 31.33 119 2003 Dec-29 to Jan- 2 32.51 33.01 32.51 2004 Jan- 5 to Jan- 9 2004 Jan-12 to Jan-16 2004 Jan-19 to Jan-23 2004 Jan-26 to Jan-30 2004 Feb- 2 to Feb- 6 33.71 34.92 33.54 34.26 36.21 33.99 34.20 33.57 34.62 35.53 33.63 33.06 34.27 33.61 35.12 32.86 33.26 34.38 35.16 34.94 33.16 32.49 2004 Feb- 9 to Feb-13 2004 Feb-16 to Feb-20 2004 Feb-23 to Feb-27 2004 Mar- 1 to Mar- 5 2004 Mar- 8 to Mar-12 32.91 35.75 36.85 36.53 34.03 35.13 35.85 36.60 36.29 33.93 35.42 37.28 35.80 36.21 34.03 35.81 35.45 36.81 36.95 34.51 35.80 36.08 37.31 36.21 2004 Mar-15 to Mar-19 2004 Mar-22 to Mar-26 2004 Mar-29 to Apr- 2 2004 Apr- 5 to Apr- 9 2004 Apr-12 to Apr-16 37.44 37.12 35.41 34.29 37.79 37.36 37.81 36.15 35.09 37.09 38.21 37.06 35.75 36.28 36.62 37.81 35.67 34.47 37.14 37.74 38.09 35.61 34.39 2004 Apr-19 to Apr-23 2004 Apr-26 to Apr-30 2004 May- 3 to May- 7 2004 May-10 to May-14 2004 May-17 to May-21 37.46 37.02 38.26 38.90 41.53 37.61 37.49 38.86 40.30 40.32 36.61 37.23 39.69 40.30 41.61 37.70 37.50 39.41 40.94 40.92 37.22 37.31 39.98 41.42 39.83 2004 May-24 to May-28 2004 May-31 to Jun- 4 2004 Jun- 7 to Jun-11 2004 Jun-14 to Jun-18 2004 Jun-21 to Jun-25 42.03 41.45 42.33 37.18 37.18 38.11 40.60 39.96 37.60 37.33 37.56 39.25 39.29 38.45 38.51 37.81 39.90 38.44 2004 Jun-28 to Jul- 2 2004 Jul- 5 to Jul- 9 2004 Jul-12 to Jul-16 2004 Jul-19 to Jul-23 2004 Jul-26 to Jul-30 36.25 39.30 41.55 41.45 35.60 39.56 39.55 40.86 41.83 36.92 39.18 40.98 40.63 42.81 38.56 40.27 40.70 41.51 42.69 38.37 39.90 41.10 41.82 43.72 2004 Aug- 2 to Aug- 6 2004 Aug- 9 to Aug-13 2004 Aug-16 to Aug-20 2004 Aug-23 to Aug-27 2004 Aug-30 to Sep- 3 43.83 44.86 46.02 46.00 42.32 44.13 44.51 46.75 45.68 42.23 42.73 44.72 47.36 43.83 43.89 44.39 45.52 48.66 43.06 44.04 43.95 46.61 47.60 43.11 43.94 2004 Sep- 6 to Sep-10 2004 Sep-13 to Sep-17 2004 Sep-20 to Sep-24 2004 Sep-27 to Oct- 1 2004 Oct- 4 to Oct- 8 43.86 46.33 49.56 49.85 43.18 44.62 47.11 49.76 51.08 42.77 43.83 48.41 49.53 51.98 44.53 44.03 48.37 49.56 52.56 42.84 45.63 48.86 50.16 53.40 2004 Oct-11 to Oct-15 2004 Oct-18 to Oct-22 2004 Oct-25 to Oct-29 2004 Nov- 1 to Nov- 5 2004 Nov- 8 to Nov-12 53.65 53.59 55.52 50.10 49.10 53.49 53.28 56.37 49.60 47.40 53.86 54.93 52.52 50.90 48.70 54.69 54.51 50.95 48.80 47.50 54.89 55.83 51.78 49.65 47.30 2004 Nov-15 to Nov-19 2004 Nov-22 to Nov-26 2004 Nov-29 to Dec- 3 2004 Dec- 6 to Dec-10 2004 Dec-13 to Dec-17 46.95 48.48 49.71 42.96 41.06 46.10 48.74 49.16 41.51 41.76 46.85 49.14 45.56 41.96 44.21 46.30 48.90 43.31 42.41 44.16 42.56 40.71 46.31 2004 Dec-20 to Dec-24 2004 Dec-27 to Dec-31 2005 Jan- 3 to Jan- 7 2005 Jan-10 to Jan-14 2005 Jan-17 to Jan-21 45.57 41.26 42.16 45.31 45.76 41.78 43.96 45.66 48.46 44.05 43.69 43.41 46.46 47.61 42.19 43.36 45.51 48.11 47.01 45.32 48.41 48.31 2005 Jan-24 to Jan-28 2005 Jan-31 to Feb- 4 2005 Feb- 7 to Feb-11 2005 Feb-14 to Feb-18 2005 Feb-21 to Feb-25 48.61 48.25 45.35 47.50 49.43 47.10 45.40 47.30 51.00 48.80 46.65 45.45 48.35 51.73 48.80 46.40 47.05 47.50 52.05 47.15 46.45 47.15 48.45 52.20 2005 Feb-28 to Mar- 4 2005 Mar- 7 to Mar-11 2005 Mar-14 to Mar-18 51.75 53.90 54.90 51.67 54.55 55.05 53.00 54.75 56.50 53.60 53.52 56.40 53.70 54.40 56.80 34.41 34.02 38.72 37.58 37.69 37.70 38.68 37.34 120 2005 Mar-21 to Mar-25 2005 Mar-28 to Apr- 1 56.70 54.06 55.95 54.26 49.43 53.96 49.70 55.31 57.26 2005 Apr- 4 to Apr- 8 2005 Apr-11 to Apr-15 2005 Apr-18 to Apr-22 2005 Apr-25 to Apr-29 2005 May- 2 to May- 6 56.86 53.71 50.52 53.16 50.94 55.83 51.54 52.33 54.33 49.60 55.88 50.21 52.45 51.37 50.22 54.16 51.11 52.49 51.92 51.12 53.46 50.61 54.16 49.20 51.30 2005 May- 9 to May-13 2005 May-16 to May-20 2005 May-23 to May-27 2005 May-30 to Jun- 3 2005 Jun- 6 to Jun-10 52.04 48.64 48.68 51.76 48.97 49.14 52.08 53.84 50.39 46.99 50.37 54.40 52.51 48.83 47.00 50.89 53.46 54.36 48.65 47.25 51.65 55.08 53.55 2005 Jun-13 to Jun-17 2005 Jun-20 to Jun-24 2005 Jun-27 to Jul- 1 2005 Jul- 4 to Jul- 8 2005 Jul-11 to Jul-15 55.47 59.19 59.78 59.23 55.03 58.90 58.32 59.71 60.49 55.53 58.27 57.23 61.24 60.00 56.48 59.23 56.63 60.76 57.83 58.40 59.63 59.11 59.71 58.36 2005 Jul-18 to Jul-22 2005 Jul-25 to Jul-29 2005 Aug- 1 to Aug- 5 2005 Aug- 8 to Aug-12 2005 Aug-15 to Aug-19 57.12 58.16 61.51 63.92 66.21 57.61 59.05 61.87 63.13 66.11 56.73 59.12 60.76 64.80 63.29 57.31 59.91 61.60 65.67 63.47 57.75 60.71 62.44 66.71 65.51 2005 Aug-22 to Aug-26 2005 Aug-29 to Sep- 2 2005 Sep- 5 to Sep- 9 2005 Sep-12 to Sep-16 2005 Sep-19 to Sep-23 65.46 67.41 63.29 67.21 65.81 69.91 65.83 63.18 66.24 67.10 68.63 64.38 65.20 66.96 67.29 69.50 64.80 64.64 67.07 66.05 66.91 64.21 62.91 64.67 2005 Sep-26 to Sep-30 2005 Oct- 3 to Oct- 7 2005 Oct-10 to Oct-14 2005 Oct-17 to Oct-21 2005 Oct-24 to Oct-28 65.98 65.36 60.71 64.26 60.63 64.94 63.74 63.84 62.94 62.83 66.36 62.56 64.13 62.11 60.85 66.83 61.81 63.05 61.04 61.03 66.21 61.81 62.61 61.05 61.30 2005 Oct-31 to Nov- 4 2005 Nov- 7 to Nov-11 2005 Nov-14 to Nov-18 2005 Nov-21 to Nov-25 2005 Nov-28 to Dec- 2 59.80 59.40 57.60 57.75 57.36 59.85 59.70 57.05 58.30 56.46 59.75 59.65 57.85 58.35 57.33 61.70 57.80 56.20 60.60 57.45 56.30 58.46 59.31 2005 Dec- 5 to Dec- 9 2005 Dec-12 to Dec-16 2005 Dec-19 to Dec-23 2005 Dec-26 to Dec-30 2006 Jan- 2 to Jan- 6 59.91 61.36 57.31 59.96 61.36 57.81 58.16 63.11 59.21 60.86 58.56 59.81 63.41 60.66 60.01 58.08 60.26 62.81 59.41 58.01 58.08 61.06 64.21 2006 Jan- 9 to Jan-13 2006 Jan-16 to Jan-20 2006 Jan-23 to Jan-27 2006 Jan-30 to Feb- 3 2006 Feb- 6 to Feb-10 63.56 63.41 66.36 66.83 67.86 63.01 63.91 65.76 65.60 66.61 62.51 63.96 66.86 65.80 64.71 62.66 63.86 68.16 67.81 65.41 62.01 2006 Feb-13 to Feb-17 2006 Feb-20 to Feb-24 2006 Feb-27 to Mar- 3 2006 Mar- 6 to Mar-10 2006 Mar-13 to Mar-17 61.26 61.01 62.46 61.81 59.61 61.21 61.37 61.51 63.01 57.61 59.03 62.01 60.06 62.11 58.61 58.03 63.36 60.51 63.46 59.76 61.46 63.61 59.91 62.81 2006 Mar-20 to Mar-24 2006 Mar-27 to Mar-31 2006 Apr- 3 to Apr- 7 2006 Apr-10 to Apr-14 2006 Apr-17 to Apr-21 60.31 63.75 66.07 68.29 70.30 60.41 65.65 65.75 69.03 71.28 60.03 66.00 66.76 68.53 72.07 62.13 66.70 67.22 69.53 71.96 63.90 66.25 67.02 2006 Apr-24 to Apr-28 2006 May- 1 to May- 5 2006 May- 8 to May-12 2006 May-15 to May-19 2006 May-22 to May-26 70.19 73.75 69.75 69.25 69.23 67.43 74.62 70.71 69.40 70.78 71.71 72.26 72.15 68.65 69.47 70.76 69.98 73.00 69.63 70.92 71.80 70.09 71.87 68.44 71.35 2006 May-29 to Jun- 2 2006 Jun- 5 to Jun- 9 72.50 71.85 72.43 71.42 70.90 70.11 70.25 72.73 71.62 54.46 68.06 68.36 65.11 73.73 121 2006 Jun-12 to Jun-16 2006 Jun-19 to Jun-23 2006 Jun-26 to Jun-30 70.28 69.21 71.63 68.48 69.30 72.05 69.12 70.07 72.15 69.78 70.62 73.50 69.75 70.50 73.94 2006 Jul- 3 to Jul- 7 2006 Jul-10 to Jul-14 2006 Jul-17 to Jul-21 2006 Jul-24 to Jul-28 2006 Jul-31 to Aug- 4 73.50 75.70 74.29 74.56 74.05 73.87 73.46 74.93 75.20 74.99 72.79 73.82 76.16 75.00 76.70 74.00 74.50 75.59 73.76 76.80 73.52 73.30 74.78 2006 Aug- 7 to Aug-11 2006 Aug-14 to Aug-18 2006 Aug-21 to Aug-25 2006 Aug-28 to Sep- 1 2006 Sep- 4 to Sep- 8 77.05 73.33 72.45 70.47 76.29 72.95 72.55 69.74 68.70 76.28 71.64 71.45 70.20 67.75 74.17 70.12 72.02 70.38 67.37 74.38 70.93 72.13 69.24 66.30 2006 Sep-11 to Sep-15 2006 Sep-18 to Sep-22 2006 Sep-25 to Sep-29 2006 Oct- 2 to Oct- 6 2006 Oct- 9 to Oct-13 65.42 63.84 60.74 60.96 59.93 63.81 61.77 60.63 58.64 58.50 64.09 60.00 62.96 59.53 57.56 63.27 61.62 62.46 60.02 58.23 63.30 59.79 62.90 59.68 58.69 2006 Oct-16 to Oct-20 2006 Oct-23 to Oct-27 2006 Oct-30 to Nov- 3 2006 Nov- 6 to Nov-10 2006 Nov-13 to Nov-17 59.91 56.74 58.41 60.11 58.59 58.91 57.55 58.72 58.94 58.28 57.66 59.09 58.64 59.93 58.79 58.55 60.27 57.87 61.18 56.23 57.35 60.75 59.13 59.66 55.90 2006 Nov-20 to Nov-24 2006 Nov-27 to Dec- 1 2006 Dec- 4 to Dec- 8 2006 Dec-11 to Dec-15 2006 Dec-18 to Dec-22 56.42 60.30 62.39 61.26 62.19 58.01 60.97 62.40 61.06 62.87 57.28 62.45 62.20 61.34 63.08 62.97 62.54 62.48 62.05 63.43 62.06 63.40 61.81 56.08 61.07 60.77 55.65 51.23 53.61 60.31 58.31 53.95 52.30 54.24 60.39 55.65 51.91 50.51 53.49 60.85 56.29 52.96 51.98 55.38 58.17 57.75 58.00 59.40 61.78 57.35 59.76 57.92 60.28 61.97 59.01 59.86 59.38 60.28 61.58 2006 Dec-25 to Dec-29 2007 Jan- 1 to Jan- 5 2007 Jan- 8 to Jan-12 2007 Jan-15 to Jan-19 2007 Jan-22 to Jan-26 51.11 2007 Jan-29 to Feb- 2 2007 Feb- 5 to Feb- 9 2007 Feb-12 to Feb-16 2007 Feb-19 to Feb-23 2007 Feb-26 to Mar- 2 54.01 58.69 57.76 61.41 57.03 58.91 58.98 58.32 61.46 2007 Mar- 5 to Mar- 9 2007 Mar-12 to Mar-16 2007 Mar-19 to Mar-23 2007 Mar-26 to Mar-30 2007 Apr- 2 to Apr- 6 60.05 58.94 56.65 61.77 66.03 60.66 58.03 56.41 62.98 64.59 61.85 58.15 56.98 64.11 64.40 61.63 57.52 60.21 66.10 64.26 60.06 57.06 61.07 65.94 2007 Apr- 9 to Apr-13 2007 Apr-16 to Apr-20 2007 Apr-23 to Apr-27 2007 Apr-30 to May- 4 2007 May- 7 to May-11 61.51 63.63 65.33 65.78 61.48 61.92 63.14 64.10 64.43 62.26 61.98 63.14 65.33 63.78 61.54 63.87 61.81 65.08 63.23 61.85 63.63 63.56 66.45 61.89 62.35 2007 May-14 to May-18 2007 May-21 to May-25 2007 May-28 to Jun- 1 2007 Jun- 4 to Jun- 8 2007 Jun-11 to Jun-15 62.55 66.25 66.17 65.93 63.16 64.91 63.19 65.63 65.36 62.57 65.10 63.47 65.97 66.17 64.83 63.62 64.02 66.93 67.62 64.93 64.59 65.09 64.78 68.04 2007 Jun-18 to Jun-22 2007 Jun-25 to Jun-29 2007 Jul- 2 to Jul- 6 2007 Jul- 9 to Jul-13 2007 Jul-16 to Jul-20 69.06 68.83 71.11 72.14 74.11 69.15 67.78 71.41 72.80 74.03 68.50 68.98 72.58 75.03 68.35 69.61 71.81 72.55 75.90 68.85 70.47 72.80 73.89 75.53 2007 Jul-23 to Jul-27 2007 Jul-30 to Aug- 3 2007 Aug- 6 to Aug-10 2007 Aug-13 to Aug-17 2007 Aug-20 to Aug-24 74.65 76.82 72.03 71.60 71.12 73.38 78.20 72.25 72.40 69.49 75.74 76.49 72.23 73.36 69.30 74.96 76.84 71.62 70.99 69.86 77.03 75.41 71.49 71.90 71.17 2007 Aug-27 to Aug-31 71.98 71.79 73.52 73.37 73.98 122 2007 Sep- 3 to Sep- 7 2007 Sep-10 to Sep-14 2007 Sep-17 to Sep-21 2007 Sep-24 to Sep-28 77.53 80.55 82.51 75.07 78.16 81.51 81.20 75.74 79.85 81.99 80.31 76.34 80.05 83.85 82.86 76.70 79.14 83.38 81.64 2007 Oct- 1 to Oct- 5 2007 Oct- 8 to Oct-12 2007 Oct-15 to Oct-19 2007 Oct-22 to Oct-26 2007 Oct-29 to Nov- 2 80.31 78.97 86.19 87.60 93.45 80.00 80.23 87.58 86.45 90.33 79.97 81.30 87.19 88.30 94.16 81.48 83.05 89.48 92.09 93.53 81.20 83.73 88.58 91.73 95.81 2007 Nov- 5 to Nov- 9 2007 Nov-12 to Nov-16 2007 Nov-19 to Nov-23 2007 Nov-26 to Nov-30 2007 Dec- 3 to Dec- 7 94.06 94.40 95.75 97.66 89.29 96.65 91.18 99.16 94.39 88.31 96.46 94.02 98.57 90.71 87.45 95.51 93.37 90.98 90.25 96.36 94.81 98.24 88.60 88.23 2007 Dec-10 to Dec-14 2007 Dec-17 to Dec-21 2007 Dec-24 to Dec-28 2007 Dec-31 to Jan- 4 2008 Jan- 7 to Jan-11 87.72 90.69 94.00 95.95 95.08 90.12 89.93 96.43 94.41 91.11 95.89 99.64 95.64 92.35 90.88 96.63 99.17 93.92 91.31 93.19 96.03 97.90 92.74 2008 Jan-14 to Jan-18 2008 Jan-21 to Jan-25 2008 Jan-28 to Feb- 1 2008 Feb- 4 to Feb- 8 2008 Feb-11 to Feb-15 94.23 90.99 90.07 93.56 91.87 89.64 91.66 88.32 92.82 90.80 87.65 92.34 87.16 93.28 90.11 89.98 91.67 88.07 95.42 90.55 90.37 89.03 91.77 95.57 2008 Feb-18 to Feb-22 2008 Feb-25 to Feb-29 2008 Mar- 3 to Mar- 7 2008 Mar-10 to Mar-14 2008 Mar-17 to Mar-21 99.40 102.42 107.90 105.74 99.99 100.83 99.72 108.73 109.57 100.86 99.59 104.45 109.86 103.25 98.57 102.60 105.51 110.21 102.57 99.03 101.78 105.12 110.03 2008 Mar-24 to Mar-28 2008 Mar-31 to Apr- 4 2008 Apr- 7 to Apr-11 2008 Apr-14 to Apr-18 2008 Apr-21 to Apr-25 101.70 101.54 108.91 111.71 117.48 101.78 100.92 108.54 113.77 119.17 105.83 104.83 110.89 114.80 119.28 107.56 103.92 110.07 114.80 117.10 105.59 106.09 110.14 116.56 119.64 2008 Apr-28 to May- 2 2008 May- 5 to May- 9 2008 May-12 to May-16 2008 May-19 to May-23 2008 May-26 to May-30 118.78 119.94 124.02 127.15 115.67 121.82 125.83 128.93 128.81 113.70 123.56 124.21 132.99 131.00 112.60 123.77 124.25 130.04 126.70 116.36 125.94 126.50 131.58 127.35 2008 Jun- 2 to Jun- 6 2008 Jun- 9 to Jun-13 2008 Jun-16 to Jun-20 2008 Jun-23 to Jun-27 2008 Jun-30 to Jul- 4 127.75 134.44 134.52 135.98 139.96 124.33 131.38 133.99 136.49 141.06 122.30 136.43 136.54 133.92 143.74 127.93 136.91 131.88 138.91 145.31 138.51 134.84 134.78 139.69 2008 Jul- 7 to Jul-11 2008 Jul-14 to Jul-18 2008 Jul-21 to Jul-25 2008 Jul-28 to Aug- 1 2008 Aug- 4 to Aug- 8 141.38 145.16 131.43 124.72 121.45 136.06 138.68 127.25 122.21 118.71 135.88 134.63 123.73 126.74 118.57 141.47 129.43 124.62 124.17 119.84 144.96 128.94 122.59 125.03 115.42 2008 Aug-11 to Aug-15 2008 Aug-18 to Aug-22 2008 Aug-25 to Aug-29 2008 Sep- 1 to Sep- 5 2008 Sep- 8 to Sep-12 114.44 112.92 114.85 106.35 113.10 114.39 116.31 109.63 103.23 115.96 115.48 118.17 109.38 102.66 115.05 121.23 115.58 107.99 100.95 113.46 114.48 115.55 106.47 101.19 2008 Sep-15 to Sep-19 2008 Sep-22 to Sep-26 2008 Sep-29 to Oct- 3 2008 Oct- 6 to Oct-10 2008 Oct-13 to Oct-17 95.52 122.61 96.29 88.15 81.17 91.49 107.85 100.70 90.18 78.69 97.39 106.84 98.23 88.94 74.38 97.50 111.54 93.84 86.50 69.81 104.05 106.77 93.91 77.44 71.90 2008 Oct-20 to Oct-24 2008 Oct-27 to Oct-31 2008 Nov- 3 to Nov- 7 2008 Nov-10 to Nov-14 2008 Nov-17 to Nov-21 74.08 61.92 63.93 62.19 55.14 71.29 62.80 70.41 59.38 54.42 66.92 67.45 65.41 55.95 53.64 67.17 65.79 60.72 58.31 48.86 63.34 68.10 61.06 57.18 49.22 123 2008 Nov-24 to Nov-28 2008 Dec- 1 to Dec- 5 2008 Dec- 8 to Dec-12 2008 Dec-15 to Dec-19 2008 Dec-22 to Dec-26 53.63 49.34 43.69 44.61 31.10 50.02 47.05 42.00 43.84 30.28 54.20 46.79 43.10 40.17 32.94 2008 Dec-29 to Jan- 2 2009 Jan- 5 to Jan- 9 2009 Jan-12 to Jan-16 2009 Jan-19 to Jan-23 2009 Jan-26 to Jan-30 39.89 48.61 37.65 38.95 48.56 37.77 38.57 41.67 44.60 42.75 37.43 42.56 42.04 41.68 35.41 42.33 41.58 46.17 40.69 35.38 45.12 41.73 2009 Feb- 2 to Feb- 6 2009 Feb- 9 to Feb-13 2009 Feb-16 to Feb-20 2009 Feb-23 to Feb-27 2009 Mar- 2 to Mar- 6 41.35 39.58 37.66 40.07 40.87 37.54 34.96 38.86 41.57 40.27 35.93 34.67 41.64 45.28 41.15 34.03 39.60 43.18 43.54 40.24 37.63 39.35 44.15 45.43 2009 Mar- 9 to Mar-13 2009 Mar-16 to Mar-20 2009 Mar-23 to Mar-27 2009 Mar-30 to Apr- 3 2009 Apr- 6 to Apr-10 47.01 47.33 53.05 48.49 51.10 45.68 48.97 53.36 49.64 49.13 42.46 48.12 52.24 48.46 49.37 46.91 51.46 53.87 52.61 52.24 46.22 51.55 52.41 52.52 2009 Apr-13 to Apr-17 2009 Apr-20 to Apr-24 2009 Apr-27 to May- 1 2009 May- 4 to May- 8 2009 May-11 to May-15 50.22 45.82 49.29 54.45 57.79 49.51 46.65 49.01 53.81 58.81 49.26 47.41 50.19 56.29 58.00 49.97 48.46 50.35 56.67 58.58 50.36 50.65 52.18 58.58 56.52 2009 May-18 to May-22 2009 May-25 to May-29 2009 Jun- 1 to Jun- 5 2009 Jun- 8 to Jun-12 2009 Jun-15 to Jun-19 58.99 68.59 68.05 70.54 59.52 62.48 68.58 70.02 70.47 61.45 63.41 66.14 71.38 71.07 60.49 65.09 68.80 72.69 71.42 61.15 66.31 68.43 72.13 69.60 2009 Jun-22 to Jun-26 2009 Jun-29 to Jul- 3 2009 Jul- 6 to Jul-10 2009 Jul-13 to Jul-17 2009 Jul-20 to Jul-24 67.09 71.47 64.06 59.69 63.93 68.81 69.82 62.88 59.62 64.81 68.14 69.32 60.15 61.49 64.58 69.70 66.68 60.36 62.07 66.10 69.16 2009 Jul-27 to Jul-31 2009 Aug- 3 to Aug- 7 2009 Aug-10 to Aug-14 2009 Aug-17 to Aug-21 2009 Aug-24 to Aug-28 68.34 71.59 70.59 66.72 73.68 67.24 71.40 69.46 69.22 71.60 63.42 71.97 70.08 72.54 71.38 66.90 71.96 70.57 72.40 72.49 69.26 70.97 67.51 73.12 72.72 2009 Aug-31 to Sep- 4 2009 Sep- 7 to Sep-11 2009 Sep-14 to Sep-18 2009 Sep-21 to Sep-25 2009 Sep-28 to Oct- 2 69.97 68.11 71.08 70.81 71.50 66.56 68.03 71.27 72.50 68.74 70.46 67.90 71.95 72.48 65.74 67.95 69.34 71.95 65.91 46.50 68.86 69.74 66.69 43.80 47.77 36.73 55.21 41.01 46.27 33.17 37.58 59.93 63.56 66.96 124 APPENDIX B Actual and One-step Ahead Forecasts Values Date Jul 01, 2009 Jul 02, 2009 Jul 03, 2009 Jul 06, 2009 Jul 07, 2009 Jul 08, 2009 Jul 09, 2009 Jul 10, 2009 Jul 13, 2009 Jul 14, 2009 Jul 15, 2009 Jul 16, 2009 Jul 17, 2009 Jul 20, 2009 Jul 21, 2009 Jul 22, 2009 Jul 23, 2009 Jul 24, 2009 Jul 27, 2009 Jul 28, 2009 Jul 29, 2009 Jul 30, 2009 Jul 31, 2009 Aug 03, 2009 Aug 04, 2009 Aug 05, 2009 Aug 06, 2009 Aug 07, 2009 Aug 10, 2009 Aug 11, 2009 Aug 12, 2009 Actual 69.32 66.68 66.68 64.06 62.88 60.15 60.36 59.93 59.69 59.62 61.49 62.07 63.56 63.93 64.81 64.58 66.1 66.96 68.34 67.24 63.42 66.9 69.26 71.59 71.4 71.97 71.96 70.97 70.59 69.46 70.08 ARIMA(1,2,1) 69.91574733 69.34426212 66.8276942 66.66668883 64.19947377 62.92850022 60.28574204 60.31704794 59.92500917 59.67285507 59.59238173 61.34917702 62.00950948 63.4480926 63.88747281 64.73879542 64.57593568 65.99334055 66.8959658 68.24800792 67.29634526 63.63198178 66.67613011 69.11086277 71.44900619 71.41221939 71.93740162 71.9627552 71.02986182 70.61163061 69.524338 GARCH(1,1) 69.82163121 69.32163121 66.68163121 66.68163121 64.06163121 62.88163121 60.15163121 60.36163121 59.93163121 59.69163121 59.62163121 61.49163121 62.07163121 63.56163121 63.93163121 64.81163121 64.58163121 66.10163121 66.96163121 68.34163121 67.24163121 63.42163121 66.90163121 69.26163121 71.59163121 71.40163121 71.97163121 71.96163121 70.97163121 70.59163121 69.46163121 125 Aug 13, 2009 Aug 14, 2009 Aug 17, 2009 Aug 18, 2009 Aug 19, 2009 Aug 20, 2009 Aug 21, 2009 Aug 24, 2009 Aug 25, 2009 Aug 26, 2009 Aug 27, 2009 Aug 28, 2009 Aug 31, 2009 Sep 01, 2009 Sep 02, 2009 Sep 03, 2009 Sep 04, 2009 Sep 07, 2009 Sep 08, 2009 Sep 09, 2009 Sep 10, 2009 Sep 11, 2009 Sep 14, 2009 Sep 15, 2009 Sep 16, 2009 Sep 17, 2009 Sep 18, 2009 Sep 21, 2009 Sep 22, 2009 Sep 23, 2009 Sep 24, 2009 Sep 25, 2009 Sep 28, 2009 Sep 29, 2009 Sep 30, 2009 70.57 67.51 66.72 69.22 72.54 72.4 73.12 73.68 71.6 71.38 72.49 72.72 69.97 68.11 68.03 67.9 67.95 67.95 71.08 71.27 71.95 69.34 68.86 70.81 72.5 72.48 71.95 69.74 71.5 68.74 65.74 65.91 66.69 66.56 70.46 70.03927693 70.53854354 67.68684139 66.75626329 69.06235754 72.34132415 72.41200714 73.08148724 73.65275185 71.72816427 71.39410859 72.42597864 72.71027728 70.13462674 68.21532309 68.02654978 67.89927499 67.93845694 67.94153379 70.8890816 71.25894504 71.91089516 69.49453832 68.88332489 70.69037945 72.40081267 72.48506674 71.98474103 69.87125205 71.39384455 68.90211752 65.90867375 65.88579927 66.63073594 66.55593115 70.08163121 70.57163121 67.51163121 66.72163121 69.22163121 72.54163121 72.40163121 73.12163121 73.68163121 71.60163121 71.38163121 72.49163121 72.72163121 69.97163121 68.11163121 68.03163121 67.90163121 67.95163121 67.95163121 71.08163121 71.27163121 71.95163121 69.34163121 68.86163121 70.81163121 72.50163121 72.48163121 71.95163121 69.74163121 71.50163121 68.74163121 65.74163121 65.91163121 66.69163121 66.56163121