APPLICATION OF ARIMA AND GARCH MODELS IN FORECASTING CRUDE OIL PRICES

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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 = − %,ƒ:
…
= % + „= 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 :
‹ =
Œ()
where is the estimate of , and Œ() 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:
 ( − )(Œ())
̃‹ = ‹ ( )J −

2J (3.33)
where is the estimate of , and ‹ is the -ratio of , Œ() 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 =
œ∑?K7( − )
ℎ
K7•
K7•
M∑?K7 ℎ
+
K7•
Ϸ?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 Œ ¤]
(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 Œ ¤]
(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. Other GARCH-type models that
97
should be investigated to forecast crude oil prices data are Integrated GARCH
(IGARCH) and Exponential GARCH (EGARCH).
Another suggestion for future work is to compare the model with other
methods that are able to capture volatility in the crude oil prices such as implied
volatility and stochastic volatility.
Finally, this study may be applicable in a wide range of situations. It is
therefore suggested that this study should be replicated in other types of forecasts and
in other types of industries so as to determine the potential of the GARCH models in
those situations and industries.
98
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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
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