Modeling and Forecasting in Energy and
Financial Market
Submitted by:
Aravind Joni : NMP 08
Nikhil Padalia : EM 04
Prithwish Sinha : EM05
Rajesh Kumar : EM06
Table of Contents
1. Regression Analysis ............................................................................................................................... 4
1.1. Introduction ...................................................................................................................................... 4
1.2. Data description (data frequency, data span, source of data) ......................................................... 4
1.3. Model (brief theory, assumptions etc) ............................................................................................. 5
1.4. Empirical analysis .............................................................................................................................. 6
1.5. Conclusion ....................................................................................................................................... 11
2. DOW Effect and MOY Effect ............................................................................................................... 12
2.1. Introduction .................................................................................................................................... 12
2.2. DOW (Day of Week) effect .............................................................................................................. 13
2.2.1. Dummy Variables ........................................................................................................................ 14
2.2.2. Empirical Analysis ........................................................................................................................ 14
2.2.3. Regression Output ...................................................................................................................... 15
2.2.4. Conclusion ................................................................................................................................... 15
2.3. MOY (Month of Year) Effect ........................................................................................................... 16
2.3.1. Data ............................................................................................................................................. 16
2.3.2. Empirical Analysis ........................................................................................................................ 17
2.3.3. Regression Output ...................................................................................................................... 17
2.3.4. Conclusion ................................................................................................................................... 18
3. MSARIMA EGARCH MODEL ................................................................................................................. 19
3.1. Introduction .................................................................................................................................... 19
3.2. Data of Crude oil: ............................................................................................................................ 20
3.3. Analysis ........................................................................................................................................... 21
3.3.1. Graphical Representation of data: .............................................................................................. 21
3.3.2. Check Correlogram: .................................................................................................................... 22
3.3.3. Unit Root Test: ............................................................................................................................ 23
3.3.4. Detrend the series: ..................................................................................................................... 25
3.3.5. Deseasonalize the data: .............................................................................................................. 26
3.3.6. Estimation Stage ......................................................................................................................... 27
3.3.7. Residual Analysis stage(Diagnostic Checking): ........................................................................... 28
3.3.8. Forecasting Stage: ....................................................................................................................... 29
3.3.9. GARCH MODELLING: ................................................................................................................... 29
3.3.10. Static and Dynamic forecast ....................................................................................................... 32
3.4. Conclusion: ...................................................................................................................................... 33
4. VAR-Cointergration ............................................................................................................................. 34
4.1. Introduction .................................................................................................................................... 34
4.2. Objective ......................................................................................................................................... 34
4.3. Data Desciption ............................................................................................................................... 34
4.4. Empirical Analysis ............................................................................................................................ 35
4.5. Conclusion ....................................................................................................................................... 40
4.5.1. Long term relations ..................................................................................................................... 40
4.5.2. Short term relations .................................................................................................................... 40
1. Regression Analysis
1.1. Introduction
We have taken into account certain economic factors such as GDP (Gross Domestic
Product), Export, Installed Generation Capacity (MW), RBI credit to Government, WPI
(Wholesale Price Index).
As we all know that GDP is a function of Export, Installed Generation capacity, RBI credit to
Government and WPI. Therefore GDP is the dependent variable here. We have tried to find
out with the help of Regression model how the above mentioned economic factors affect
GDP.
1.2. Data description (data frequency, data span, source of data)
The data has been taken for our country. Data spans from June2004 to June2014, financial quarter wise.
Data Sample
GDP Export
Installed Gen Capacity
(MW)
RBI Credit to
Govt WPI
Jun-04 68,15,520.00 7,85,265.00 1,09,852.90 75,16,790.00 97.9
Sep-04 69,26,820.00 8,48,027.90 1,11,560.50 74,04,880.00 100.1
Dec-04 78,77,230.00 9,17,523.30 1,14,739.80 73,28,240.00 100.9
Mar-05 80,95,070.00 11,40,289.50 1,18,419.10 75,24,360.00 101.2
Jun-05 77,63,780.00 10,28,953.20 1,21,175.00 75,92,100.00 102.7
Sep-05 78,26,590.00 10,70,639.80 1,23,014.80 75,45,890.00 104.3
Dec-05 90,21,210.00 11,35,718.70 1,23,667.80 74,90,340.00 105.3
Mar-06 92,93,440.00 13,22,705.30 1,24,287.20 75,94,160.00 105.6
Jun-06 89,92,100.00 13,21,345.40 1,26,089.00 78,36,370.00 108.8
Sep-06 91,48,490.00 14,80,695.30 1,27,423.00 79,74,930.00 111.5
Dec-06 105,16,360.00 13,59,003.10 1,27,752.50 78,42,430.00 112.5
Mar-07 108,75,810.00 15,14,685.40 1,32,329.20 82,76,260.00 112.6
Jun-07 105,55,130.00 14,43,577.10 1,34,716.60 85,52,710.00 114.7
Sep-07 105,48,580.00 15,20,652.30 1,35,781.60 87,13,660.00 115.9
Dec-07 121,24,140.00 15,85,730.40 1,40,301.80 83,54,290.00 116.6
Mar-08 125,93,000.00 18,76,387.40 1,43,061.00 89,95,180.00 119.3
Jun-08 126,70,290.00 23,47,333.10 1,44,913.00 93,56,420.00 125
Sep-08 127,02,820.00 22,97,170.70 1,45,555.00 96,72,960.00 128.7
Dec-08 139,68,400.00 18,84,692.30 1,47,402.80 109,66,310.00 126.7
Mar-09 136,94,170.00 18,78,355.10 1,47,965.40 127,73,330.00 123.7
Jun-09 137,37,440.00 18,72,260.00 1,50,323.40 139,81,490.00 125.9
1.3. Model (brief theory, assumptions etc)
With the help of Regression model, we have tried to analyze how the dependent
variable, GDP is related to the independent variables, Export, Installed Generation
capacity, RBI credit to Government and WPI.
Regression analysis explains the relationship between two variables. The purpose of
this model is to test a theory or hypothesis.
From Regression equation, Y= 0 + 1X1 + 2X2 + 3X3 + 4X4 + e
X1 is Export
X2 is Installed Generation capacity
X3 is RBI credit to Government and
X4 is WPI.
Apart from X, there might be other variables which are unknown, but might have an
influence on Y. We capture this influence by error,
Actual value = Systematic part + random error
is randomly distributed
Econometric model = Mathematical model + Error term ()
Our objective is to create a best fitted line among the scattered plot. The OLS (Ordinary
Least Squares Method) is to create the best fitted line
Assumptions:-
We then check for the validity of the four important assumptions of a multiple
regression model.
The errors are normally distributed Normality
The mean of the errors is zero
Errors have a constant variance No heteroskedasticity
1.4. Empirical analysis
Step 1: First, we have done t
statistic of all the independent variables (which should be ideally greater than 2), we can
observe that only WPI is significant.
Only Probability value of WPI is
insignificant.
The Durbin Watson stat value is 1.8(tending to 2) which means that we can say there is no
auto correlation. Same cannot be confirmed now.
Probability value of F statistic is 0 which is
Empirical analysis
First, we have done the LS (Least Squares) test. We check the mod value of t
statistic of all the independent variables (which should be ideally greater than 2), we can
observe that only WPI is significant.
Only Probability value of WPI is significant (
Step 2: Next we have done the pair wise correlation test between independent variables. If pair
wise correlation is
Step 4: Now we have removed the Credit to Government variable in the LS test. We can
observe that except WPI, remaining variables are insignificant (
Step 5: Now we have removed the Credit to Government variable and Exchange rate variable in
the LS test. We can observe that except WPI, remaining variables are insignificant (
Step 6: Now we have done the LM test which is the confirmatory test.
Here the hypothesis is:-
Ho: There is no auto correlation
H1: There is auto correlation
From the above table we can observe that the Probability value is 0.58. Therefore we can say
that there is 58% chance of no auto correlation.
Now we have done the LM test which is the confirmatory test.
From the above table we can observe that the Probability value is 0.58. Therefore we can say
that there is 58% chance of no auto correlation.
From the above table we can observe that the Probability value is 0.58. Therefore we can say
1.5. Conclusion
From the above results, we can conclude that only the independent variable WPI (Wholesale
Price Index) has an impact on the GDP in the data period which we have taken.
2. DOW Effect and MOY Effect
2.1. Introduction
The efficient market hypothesis (EMH) postulates that stock prices must efficiently reflect all
available information about their intrinsic value. According to the EMH, stocks always trade at
their fair value on stock exchanges, making it impossible for investors to either purchase
undervalued stocks or sell stocks for inflated prices.
As such, it should be impossible to outperform the overall market through expert stock
selection or market timing, and that the only way an investor can possibly obtain higher returns
is by purchasing riskier investments. The opponents of efficient market theory asserts that
stock prices are largely determined based on investor expectation, and that price movements
will follow any patterns or trends and that past price movements can be used to predict future
price movements.
Besides, the efficient market hypothesis was contradicted by anomalies such as calendar
anomalies, fundamental anomalies and technical anomalies. Calendar anomalies refer to the
tendency of securities to behave differently on a particular day-of-the-week, or month-of-the-
year. Among such anomalies, the day-of-the-week effect has been seen as one of the most
important patterns and it has been found in several emerging stock markets (French, 1980;
Jaffe and Westerfield, 1985; Balaban, 1995; Lian and Chen,2004).
The day-of-the-week effect indicates that returns are abnormally higher on some days of the
week than on other days. Specifically, results derived from many empirical studies have
documented that the average return on Friday is abnormally high, and the average return on
Monday is abnormally low.
Besides, the rational investor should consider the risk or volatility of returns while making of
investment decisions. It is expected that there exist significant differences in volatility across
day of the week in stock markets. The day-of-the-week effects have been significantly
documented in the financial literature in the context of both developed and emerging stock-
markets.
Examination of day-of-the week effects is immense helpful for rational decision-makers to be
sentient of variation in the volatility of stock returns dependent on the day-of-the week and
whether high or low returns are associated with a correspondingly high or low volatility for a
given day.
If investors can identify a certain pattern of volatility, it is easier to make investment decisions
based on both the projected returns and the risks associated with the particular security.
Besides, the investigation of anomalous patterns may reveal evidence about the extent of
market efficiency.
2.2. DOW (Day of Week) effect
The data is Return on Sensex for 5 days viz Monday, Tuesday, Wednesday, Thursday and Friday.
The data has 1259 observations from Wednesday, September 10, 2014 to Friday, August 21,
2009.
A sample of that data is shown below
Date Sensex Return Monday Tuesday Wednesday Thursday Friday Wednesday,September10,2014 27057.41 -0.762543774 0 0 1 0 0
Tuesday, September 09, 2014 27265.32 -0.19959846 0 1 0 0 0
Monday, September 08, 2014 27319.85 1.084668124 1 0 0 0 0
Friday, September 05, 2014 27026.7 -0.218674419 0 0 0 0 1
Thursday, September 04, 2014 27085.93 -0.199005598 0 0 0 1 0
Wednesday,September03,2014 27139.94 0.446161072 0 0 1 0 0
Tuesday, September 02, 2014 27019.39 0.565142709 0 1 0 0 0
Monday, September 01, 2014 26867.55 0.861322369 1 0 0 0 0
Thursday, August 28, 2014 26638.11 0.293522439 0 0 0 1 0
Wednesday, August 27, 2014 26560.15 0.443750116 0 0 1 0 0
Tuesday, August 26, 2014 26442.81 0.021901107 0 1 0 0 0
Monday, August 25, 2014 26437.02 0.066125275 1 0 0 0 0
Friday, August 22, 2014 26419.55 0.22549223 0 0 0 0 1
Thursday, August 21, 2014 26360.11 0.174125922 0 0 0 1 0
Wednesday, August 20, 2014 26314.29 -0.402639297 0 0 1 0 0
Tuesday, August 19, 2014 26420.67 0.112576428 0 1 0 0 0
Monday, August 18, 2014 26390.96 1.102277381 1 0 0 0 0
Thursday, August 14, 2014 26103.23 0.710985592 0 0 0 1 0
2.2.1. Dummy Variables
In statistics and econometrics, particularly in regression analysis, a dummy variable (also known
as an indicator variable, design variable, Boolean indicator, categorical variable, binary variable,
or qualitative variable is one that takes the value 0 or 1 to indicate the absence or presence of
some categorical effect that may be expected to shift the outcome.
Dummy variables are "proxy" variables or numeric stand-ins for qualitative facts in a regression
model. In regression analysis, the dependent variables may be influenced not only by
quantitative variables (income, output, prices, etc.), but also by qualitative variables (gender,
religion, geographic region, etc.).
A dummy independent variable (also called a dummy explanatory variable) which for some
observation has a value of 0 will cause that variable's coefficient to have no role in influencing
the dependent variable, while when the dummy takes on a value 1 its coefficient acts to alter
the intercept.
For example, suppose Gender is one of the qualitative variables relevant to a regression. Then,
female and male would be the categories included under the Gender variable. If female is
arbitrarily assigned the value of 1, then male would get the value 0. Then the intercept (the
value of the dependent variable if all other explanatory variables hypothetically took on the
value zero) would be the constant term for males but would be the constant term plus the
coefficient of the gender dummy in the case of females.
Dummy variables are used frequently in time series analysis with regime switching, seasonal
analysis and qualitative data applications.
2.2.2. Empirical Analysis
OLS Technique is used for Empirical Analysis of Time Series data. Regression is performed and
then significance of variables is checked. If Monday comes out to be significant then it is
removed and again OLS is performed to check any other day significant or not.
Based on that inference is evaluated. The output of Regression is shown below
2.2.3. Regression Output
Here Return on Sensex is dependent variable and rest are independent variable.
The confidence interval is 95% so as we can see all variables are in significant with p value >0.05
and variability of model is also poor at 2.3%.
Dependent Variable: RETURN
Method: Least Squares
Date: 10/21/14 Time: 03:36
Sample (adjusted): 8/24/2009 9/10/2014
Included observations: 1259 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.114682 0.066743 1.718263 0.086
MON -0.007397 0.094953 -0.0779 0.9379
TUE -0.061726 0.094481 -0.65332 0.5137
THR -0.129488 0.094858 -1.36508 0.1725
FRI -0.120317 0.095245 -1.26324 0.2067
R-squared 0.002588 Mean dependent var 0.051291
Adjusted R-squared -0.000594 S.D. dependent var 1.067568
S.E. of regression 1.067885 Akaike info criterion 2.973201
Sum squared resid 1430.035 Schwarz criterion 2.993607
Log likelihood -1866.63 F-statistic 0.813382
Durbin-Watson stat 1.861889 Prob(F-statistic) 0.516589
2.2.4. Conclusion
Now we remove independent variable Wednesday and use C and calculate OLS. We find that all
variables are insignificant and variability explained by model is also low at 2.5% . Also overall
explanatory power of the model explained by F statistic is also insignificant so we can conclude
that there is no DOW week in Sensex return.
2.3. MOY (Month of Year) Effect
In the context of financial markets, calendar effects, that contradict the EMH, have been
documented over several years. These calendar effects are trends seen in stock returns, where
the returns tend to rise or fall on a particular day or month as compared to the mean.
They are called anomalies because they cannot be explained by traditional asset pricing models
and they violate the weak-form of market efficiency (i.e. asset prices fully reflect all past
information).
Examples of such patterns include the Month-of-the-year effect, Day-of-the-week effect, Intra-
month effect, Turn-of-the-month effect, Holiday effect, Halloween effect, and Daylight savings
effect.
As the name suggests, the month-of-the-year effectis a seasonal phenomenon where exchange
traded equities tend to produce abnormal returns during particular months of the year. This
effect is sometimes identified as the 'January effect' since most developed countries tend to
produce abnormal returns in January.
2.3.1. Data
The observations are from October 1, 2014 to August 12, 2002.
Return is the dependent variable and Months from January to December are independent
variable. A sample of data is shown below
Date return JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
October 1, 2014 -2.51 0 0 0 0 0 0 0 0 0 1 0 0
September 1, 2014 -10.54 0 0 0 0 0 0 0 0 1 0 0 0
August 1, 2014 -7.27 0 0 0 0 0 0 0 1 0 0 0 0
July 1, 2014 4.73 0 0 0 0 0 0 1 0 0 0 0 0
June 2, 2014 11.22 0 0 0 0 0 1 0 0 0 0 0 0
May 1, 2014 18.58 0 0 0 0 1 0 0 0 0 0 0 0
April 1, 2014 1.70 0 0 0 1 0 0 0 0 0 0 0 0
March 3, 2014 14.59 0 0 1 0 0 0 0 0 0 0 0 0
February 3, 2014 -3.44 0 1 0 0 0 0 0 0 0 0 0 0
January 1, 2014 -16.07 1 0 0 0 0 0 0 0 0 0 0 0
2.3.2. Empirical Analysis
OLS method is used on E Views software to establish significance of the Variables.
2.3.3. Regression Output
Dependent Variable: RETURN
Method: Least Squares
Date: 10/21/14 Time: 03:47
Sample (adjusted): 2002M09 2014M10
Included observations: 146 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
JAN -3.494242 4.428802
-
0.788981 0.4315
FEB -0.030156 4.428802
-
0.006809 0.9946
MAR 1.99687 4.428802 0.450883 0.6528
APR 3.783424 4.428802 0.854277 0.3945
MAY 5.687706 4.428802 1.284254 0.2013
JUN -1.092606 4.428802
-
0.246705 0.8055
JUL 4.406515 4.428802 0.994968 0.3215
AUG -2.713964 4.428802
-
0.612799 0.541
SEP 4.605324 4.255055 1.082318 0.2811
OCT -0.917103 4.255055
-
0.215533 0.8297
NOV -0.017103 4.428802
-
0.003862 0.9969
DEC 12.33089 4.428802 2.784251 0.0061
R-squared 0.075945 Mean dependent var 2.042705
Adjusted R-squared 0.000089 S.D. dependent var 15.3425
S.E. of regression 15.34182 Akaike info criterion 8.377658
Sum squared resid 31539.77 Schwarz criterion 8.622886
Log likelihood -599.5691 Durbin-Watson stat 1.866882
According to regression output only DEC is significant and variability explained by model is 7.5%
Next Regression output
Dependent Variable: RETURN
Method: Least Squares
Date: 10/21/14 Time: 03:48
Sample (adjusted): 2002M09 2014M10
Included observations: 146 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 12.33089 4.428802 2.784251 0.0061
JAN -15.82514 6.263271
-
2.526657 0.0127
FEB -12.36105 6.263271
-
1.973577 0.0505
MAR -10.33402 6.263271 -1.64994 0.1013
APR -8.547469 6.263271
-
1.364697 0.1746
MAY -6.643188 6.263271
-
1.060658 0.2908
JUN -13.4235 6.263271
-
2.143209 0.0339
JUL -7.924378 6.263271
-
1.265214 0.208
AUG -15.04486 6.263271
-
2.402077 0.0177
SEP -7.72557 6.141643 -1.2579 0.2106
OCT -13.248 6.141643
-
2.157077 0.0328
NOV -12.348 6.263271
-
1.971493 0.0507
R-squared 0.075945 Mean dependent var 2.042705
Adjusted R-squared 0.000089 S.D. dependent var 15.3425
S.E. of regression 15.34182 Akaike info criterion 8.377658
Sum squared resid 31539.77 Schwarz criterion 8.622886
Log likelihood -599.5691 F-statistic 1.001176
Durbin-Watson stat 1.866882 Prob(F-statistic) 0.448805
2.3.4. Conclusion Now DEC is removed and C is added and now Jan is significant .Overall model is insignificant
because F statistic is insignificant. So we can conclude there is no MOY effect and DOW effect.
3. MSARIMA EGARCH MODEL
3.1. Introduction
The world oil market is a capital-intensive environment characterized by complex
interactions deriving from the wide variety of products, transportation/ storage issues and
stringent environmental regulation.
Worldwide consumption of oil exceeds $500 billion, roughly 10% of the US GDP. Crude oil is
also the worlds most actively traded commodity, accounting for about 10% of total world
trade.
The economic importance of oil derives not only from the sheer size of the market, but also
from the crucial, almost strategic, role it plays in the economies of oil-exporting and oil-
consuming countries. Oil prices drive revenues to oil-exporting countries in a large number
of which, oil exports comprise over 20% of the GDP. On the other hand, costs of oil imports
(typically over 20% of the total import bill) have a substantial impact on growth initiatives in
developing countries. Energy price shocks have often been cited as causing adverse
macroeconomic impacts on aggregate output and employment, in countries across the
world.
Oil price generally shows Non-stationarity, Seasonality,Time-varying volatility, Regime
switching properties among others,Hence Forecasting volatility is fundamental to the risk
management process in order to price derivatives, devise hedging strategies and estimate
the financial risk of a firms portfolio of positions. In recent years, Autoregressive
Conditional Heteroscedasticity (ARCH) type models have become popular as a means of
capturing observed characteristics of financial returns like thick tails and volatility clustering.
This study forecasts day-ahead Crude Oil price using of multiplicative seasonal
autoregressive integrated moving average (MSARIMA) model and compares MSARIMA
forecasting performance with that of MSARIMA-exponential generalized autoregressive
conditional heteroskedasticity (EGARCH) model with an additional objective of modeling
time-varying volatility present in the time series data.
3.2. Data of Crude oil:
No of data points :501
Data span: 2-Jan-2013 to 5-Dec-2014
Data Frequency: Daily(5 working days / week)
Date Oil price Date Oil price Date Oil price Date Oil price
5-Dec-14 65.84 27-Oct-14 81 11-Sep-14 91.86 29-Jul-14 100.97
4-Dec-14 66.81 24-Oct-14 81.01 10-Sep-14 90.84 28-Jul-14 101.67
3-Dec-14 67.38 23-Oct-14 82.09 9-Sep-14 91.89 25-Jul-14 102.09
2-Dec-14 66.88 22-Oct-14 80.52 8-Sep-14 92.05 24-Jul-14 102.07
1-Dec-14 69 21-Oct-14 82.49 5-Sep-14 93.29 23-Jul-14 103.12
30-Nov-14 64.31 20-Oct-14 81.91 4-Sep-14 94.45 22-Jul-14 102.39
28-Nov-14 66.15 17-Oct-14 82.06 3-Sep-14 95.54 21-Jul-14 102.86
27-Nov-14 68.55 16-Oct-14 81.95 2-Sep-14 92.88 18-Jul-14 101.95
26-Nov-14 73.69 15-Oct-14 80.94 1-Sep-14 95.83 17-Jul-14 102.2
25-Nov-14 74.09 14-Oct-14 81.2 29-Aug-14 95.96 16-Jul-14 100.6
24-Nov-14 75.78 13-Oct-14 84.98 28-Aug-14 94.55 15-Jul-14 99.53
23-Nov-14 76.56 10-Oct-14 85.82 27-Aug-14 93.88 14-Jul-14 100.48
21-Nov-14 76.51 9-Oct-14 85.77 26-Aug-14 93.86 11-Jul-14 100.83
20-Nov-14 75.85 8-Oct-14 87.31 25-Aug-14 93.35 10-Jul-14 102.93
19-Nov-14 74.5 7-Oct-14 88.85 22-Aug-14 93.65 9-Jul-14 102.29
18-Nov-14 74.64 6-Oct-14 90.34 21-Aug-14 93.96 8-Jul-14 103.4
17-Nov-14 75.66 3-Oct-14 89.74 20-Aug-14 93.45 7-Jul-14 103.53
16-Nov-14 75.61 2-Oct-14 91.01 19-Aug-14 92.86 4-Jul-14 103.76
14-Nov-14 75.82 1-Oct-14 90.73 18-Aug-14 93.75 3-Jul-14 104.06
13-Nov-14 74.16 30-Sep-14 91.16 15-Aug-14 95.32 2-Jul-14 104.48
12-Nov-14 77.15 29-Sep-14 94.57 14-Aug-14 94.08 1-Jul-14 105.34
11-Nov-14 77.87 26-Sep-14 93.54 13-Aug-14 96.74 30-Jun-14 105.37
10-Nov-14 77.38 25-Sep-14 92.53 12-Aug-14 96.48 27-Jun-14 105.74
7-Nov-14 78.65 24-Sep-14 92.8 11-Aug-14 97.21 26-Jun-14 105.84
6-Nov-14 77.91 23-Sep-14 91.56 8-Aug-14 97.65 25-Jun-14 106.5
5-Nov-14 78.68 22-Sep-14 90.87 7-Aug-14 97.34 24-Jun-14 106.03
4-Nov-14 77.19 19-Sep-14 91.65 6-Aug-14 96.92 23-Jun-14 106.17
3-Nov-14 78.78 18-Sep-14 91.98 5-Aug-14 97.38 20-Jun-14 106.83
31-Oct-14 80.54 17-Sep-14 93.2 4-Aug-14 98.29 19-Jun-14 106.05
30-Oct-14 81.12 16-Sep-14 93.81 1-Aug-14 97.88
29-Oct-14 82.2 15-Sep-14 91.99 31-Jul-14 98.17
28-Oct-14 81.42 12-Sep-14 91.37 30-Jul-14 100.27
3.3. Analysis
3.3.1. Graphical Representation of data:
Note that from above graph we are not sure about the trend and seasonality present in the
data.
3.3.2. Check Correlogram:
ACF graph indicates that there is a Trend in the data with decay and a pattern of AR(1) as per
PACF, hence both tend and seasonality present. Also, it looks like that the data is Stationary.
3.3.3. Unit Root Test:
Null Hypothesis: The data has unit root and the data is not stationary.
Alternate hypothesis: data has no root and the data is stationary.
Check the data at Level Difference & observed that the Null Hypothesis accepted at 5% level of
significance, hence there is unit root and the data series is not stationary.
Hence we have to check the data at 1st difference.
Null Hypothesis Rejected at 5% level of signifiance, There is no unit root.
Thus, the data series is Stationary. Hence we need differencing to make the series stationary
3.3.4. Detrend the series:
Let us detrend the series with the equation series dd=d(price,1,0) and check correlogram
Inference: Dead end is reached. From correlogram it is difficult to identify AR, MA patterns.
Let us de-seasonalize the series & check correlogram
3.3.5. Deseasonalize the data:
Deseasonalize the series with the equation series dd=d(price,0,5)
From the correlogram, AR(1) SAR(5) signatures observed.
3.3.6. Estimation Stage
All the variables are statistically significant and < 1 Hence Stationarity & invertibility
conditions satisfied
Next, check if the residual series reach White Noise
3.3.7. Residual Analysis stage(Diagnostic Checking):
Inference: - White noise has been reached as alll prob value more than 0.05 as per the
correlogram.
3.3.8. Forecasting Stage:
Inference: - MAE error is 1.75 using static forecast processes and so we can go ahead with the
dynamic forecast process.
3.3.9. GARCH MODELLING:
Residuals graph after achieving white noise through ARIMA modelling
Results Of Arch Test@Lag1 For Conforming Volatility
Inferences Null Hypothesis for ARCH TEST is ARCH effect does not exist.
Here, the Null Hypothesis is rejected at approximately 95% confidence interval
Hence we can say that the model has an ARCH effect @ Lag 1
Variance Equation Will Be Generated By Using Garch Model At Lag 1
Inference: Coefficients of Constant value, size effect (coeff C5) and perseverance (coeff C7) are
significant and sign effect(Coeff C6) is insignificant at 95% confidence interval.
Result Of Arch Lm Test@ Lag 1 To Confirm Absence Of Arch Effect Now
Inference The variance equation is generated after the arch effect at lag 1 is taken into
consideration and further the arch effect is not present after that.
3.3.10. Static and Dynamic forecast
3.4. Conclusion:
By using ARIMA time series modeling we forecasted crude oil prices using Eviews software and
further to improve the accuracy of forecast by including the variance effect, we used ARIMA-
GARCH model. From the result it is observed that all the coefficients in the mean equation are
statistically significant and size effect, , is negative and statistically significant 5% level with
coefficient estimate of -0.153 indicating that a shock to crude oil prices has the impact on
volatility with respective of the direction of shock. The sign effect, , is statistically insignificant
indicating the presence of symmetric effects. i.e. negative shocks and positive shocks give rise
to similar volatility of prices. The volatility persistence term, , is statistically significant at 5 %
level, however, the coefficient is large indicating that the shocks effect will stay for a longer
time.
4. VAR-Cointergration
4.1. Introduction Co-integration is a statistical property of time series variables. Two or more time series are
co-integrated if they share a common stochastic drift. Vector auto regression (VAR) is an
econometric model used to capture the linear interdependencies among multiple time
series.
VAR models generalize the univariate auto regression (AR) models by allowing for more
than one evolving variable. A VAR model describes the evolution of a set of k variables
(called endogenous variables) over the same sample period (t = 1... T) as a linear function of
only their past values.
4.2. Objective We have taken major stock exchange index: BSE SENSEX -BSE, NYSE COMPOSITE -NYSE,
FTSE 100 -FTSE.
Through this analysis we intend to understand the relationship between these series in
terms of how they impact each other and how the return on each stock exchange is related
to each other.
4.3. Data Desciption BSE SENSEX -BSE, NYSE COMPOSITE -NYSE, FTSE 100 FTSE rates have been taken for 5
years (from 19/Dec/2014 to 4/Jan/2010) from Yahoo Finance.
Sample Data
Date BSE NYSE FTSE
12/19/2014 27371.84 4765 6545.3
12/18/2014 27126.57 4748 6466
12/17/2014 26710.13 4644 6336.5
12/16/2014 26781.44 4548 6331.8
12/15/2014 27319.56 4605 6182.7
12/12/2014 27350.68 4654 6300.6
12/11/2014 27602.01 4708 6461.7
12/10/2014 27831.1 4684 6500
12/9/2014 27797.01 4766 6529.5
12/8/2014 28119.4 4741 6672.2
12/5/2014 28458.1 4781 6742.8
4.4. Empirical Analysis We conducted tests to determine the nature of each of the series.
We found Each series is non stationary as per ADF unit root test. Each of the series is I (1)
in nature.
Figure 1- Return on stock exchange
Figure
Figure 2- Estimation of lag of VAR
Figure
Figure 3- Cointegration Testing
Figure
Figure 4- Error Correction Estimate
Figure 5
5-Granger Causlity Test (Short Run)
4.5. Conclusion
4.5.1. Long term relations
FTSE has long term relation with lag of order 1 and 2 with NYSE, NYSE has long term relation of
lag order 1 and 2 with FTSE, and BSE has long term relation of lag order 1 with NYSE.
4.5.2. Short term relations
BSE has short term relation with NYSE. NYSE & FTSE has mutual short term relationship.