ETW3410 Assignment 2 final draft ex5

1
Table of Contents
Overview .....................................................................................................................................2
Methodology............................................................................................................................ 2
Part 1: A Dynamic Model for Crude Oil and Vegetable Oil Price Change Series..................................3
A. Analysis on the Lags on the Model of Crude Oil and Palm Oil Price Changes ............................. 3
Amount of lagged values and what problem might occur......................................................... 4
Testing for second order serial correlation.............................................................................. 5
Linear Multi-Co-Linearity Test(LM)......................................................................................... 5
B. Proving forindividual and multiple variable significance.......................................................... 6
Checkingfor improvements afterincreasing the number of explanatory variables.................... 6
Testingthe presence of normal,long-runandshort-runrelationshipbetweencrude oil and
palm oil price changes...........................................................................................................7
C. Proving the theoretical covariance......................................................................................... 8
D. Application of the Almonlag scheme to help in preventing multi-co-linearity........................... 9
Part 2: Relationship between Crude Oil and Vegetable Oil Price Changes....................................... 10
A. Application of OLS method onexplanatory variables............................................................. 10
Problems with Estimated Results.......................................................................................... 10
B. Testing the validity of OLS method on Simultaneous Equations.............................................. 11
Contemporaneous correlation between equation anderror terms......................................... 11
Recommendation of Econometric Technique to Solve the Issue............................................. 11
C. Application of VAR (4) for all variables and testing for Granger causality ................................ 12
Conclusion ................................................................................................................................. 15
Summary................................................................................................................................ 15
Problems Encountered and Suggestions................................................................................... 16

2
Overview
The data obtained to conduct this report comes from the time frame of January 1982 up until December
2012.
The sequence of the report are as follows:
Methodology
Part 1: A Dynamic Model for Crude Oil and Vegetable Oil Price Change Series
1. Analyse the lags from the model of crude oil and palm oil price changes
a. Taking into account the amount of lagged values and what problem might occur
b. Testing for second order serial correlation
2. Proving for individual and multiple variable significance
a. Checking for improvements after increasing the number of explanatory variables
b. Testing the presence of normal, long-run and short-run relationship between crude oil and palm oil
price changes
3. Proving the theoretical covariance
4. Application of the Almon lag scheme to help in preventing multi-co-linearity
Part 2: Relationship between Crude Oil and Vegetable Oil Price Changes
1. Application of OLS method on explanatory variables
2. Testing the validity of OLS method
3. Application of VAR for all variables and testing for Granger causality

3
A. Analysis on the Lags on the Model of Crude Oil and Palm Oil Price Changes
Palm Oil versus Crude Oil Price Changes
𝑃𝑎𝑙𝑚𝑅𝑒𝑡 𝑡 = 𝛾0 + ∑ 𝛾𝑖 𝑃𝑎𝑙𝑚𝑅𝑒𝑡 𝑥1𝑡−𝑖 + 𝜖 𝑡
8
𝑖=0

4
Amount of lagged values and what problem might occur
Multi-co-linearity is a situation when too many lagged values are in a model. When there are too many
lagged values in a model, the true results obtained from tests would be unreliable because the errors in
the model are not behaving well.
Residual Time plot
Observation:
From the residual plot above of all the exogenous variables, there seem to be no pattern present.
Therefore, there is no presence of Hetero-Skedasticity visually. In order to attain a stronger
confirmation, we have to perform a formal second order correlation LM test.

5
Testing for second order serial correlation
Linear Multi-Co-Linearity Test (LM)
Regressors Condition P-value (Chi Squared&
𝛼 (𝐴𝑙𝑝ℎ𝑎)
Decision
𝛽 11(𝑅𝑒𝑠𝑖𝑑(−1))
𝛽 12( 𝑅𝑒𝑠𝑖𝑑(−2))
𝐻 0: 𝛽 11 = 𝛽 12 =
0
(𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜
𝑆𝑒𝑐𝑜𝑛𝑑 𝑂𝑟𝑑𝑒𝑟
𝑆𝑒𝑟𝑖𝑎𝑙
𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛)
𝐻 1: 𝐴𝑡𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝛽𝑗 ≠
0
(There is a Second
Order Serial
Correlation)
P-value: 0.0000
𝛼 : 0.05
Since 0.0000 (p-value) <
0.05 (level of significance),
we reject the null
hypothesis at a 5% level of
significance.
Conclusion:
There is enough statistical
evidence to conclude that
there is a second order
correlation. Hence, there a
second order correlation in
the model.

6
B. Proving for individual and multiple variable significance
Palm Oil versus Crude Oil, Soybean Oil, Sun Flower Oil, and Rape Seed Oil Price Changes
Checking for improvements after increasing the number of explanatory variables
R-Squared: Increased from 16.24% to 16.71%
Log likelihood: Increased from -1,196.079 to -1,214.679
This shows that there are improvements in the model after adding in other exogenous variables.

7
Testing the presence of normal, long-run and short-run relationship between crude oil and
palm oil price changes
Type Condition Test Criteria Decision
Short Run
Method: Using
the test
statistics of the
constant based
on the output
above.
𝐻 0: 𝛽 1 = 0
(𝑃𝑎𝑙𝑚 𝑜𝑖𝑙 𝑝𝑟𝑖𝑐𝑒 𝑐ℎ𝑎𝑛𝑔𝑒𝑠 𝑑𝑜
𝑛𝑜𝑡 𝑟𝑒𝑠𝑝𝑜𝑛𝑑 𝑡𝑜 𝑐𝑟𝑢𝑑𝑒 𝑜𝑖𝑙
𝑝𝑟𝑖𝑐𝑒 𝑐ℎ𝑎𝑛𝑔𝑒𝑠)
𝐻 1: 𝛽1 ≠ 0
(Palm oil price changes
responds to crude oil price
changes)
Test Statistics:
2.107
𝑇𝑒𝑠𝑡 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 :
1.96
Since 2.107 (t-stat) > 1.96
(t-crit), we reject the null
significance.
palm oil price changes do
respond to crude oil price
changes.
Long Run
Method: Using
the Wald’s test
where c(2) +
c(6) + c(7) +
c(8) + c(9) +
c(10) = 0
𝐻 0: 𝛽 1 = 𝛽 5 = 𝛽 6 = 𝛽 7 =
𝛽 8 = 𝛽 9 = 0
𝑝𝑟𝑖𝑐𝑒 𝑐ℎ𝑎𝑛𝑔𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑙𝑜𝑛𝑔
𝑟𝑢𝑛)
𝐻 1: 𝐴𝑡𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝛽𝑗 ≠ 0
changes in the long run)
P-value (Chi
Squared):
0.4199
𝛼 : 0.05
Since 0.0.4199 (p-value) >
we do not reject the null
significance.
There is not enough
statistical evidence to
conclude that there is a
long run effect between
crude oil price changes and
palm oil price changes.
Normal
Method: Using
the Wald’s test
where c(2) =
c(5) = c(6) =
c(7) = c(8) =
c(9)
𝐻 0: 𝛽 1 = 𝛽 5 = 𝛽 6 = 𝛽 7 =
𝛽 8 = 𝛽 9 = 0
𝑝𝑟𝑖𝑐𝑒 𝑐ℎ𝑎𝑛𝑔𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠ℎ𝑜𝑟𝑡
𝑟𝑢𝑛)
𝐻 1: 𝐴𝑡𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝛽1 ≠ 0
changes in the short run)
P-value(Chi
Squared):
0.0268
𝛼 : 0.05
we reject the null
significance.
palm oil price changes do
respond to crude oil price
changes.
(Models will be put into the Appendix Table 1 and Table 2)

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C. Proving the theoretical covariance
Considering a Koyk Model:
𝑦𝑡 = 𝜙0 (1 − 𝜆) + 𝜙1 𝑥 𝑡 + 𝜆𝑦𝑡−1 + 𝜉𝑡 − 𝜆𝜉𝑡−1
Disturbance term 𝜉𝑡 is assumed to satisfy the following assumptions:
𝐸( 𝜉𝑡) = 0,𝐸( 𝜉𝑡
2) = 𝜎𝜉
2
,𝐶𝑜𝑣(𝜉𝑡, 𝜉𝑡+𝑠) ≠ 0, for 𝑠 ≠ 0
𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 𝐶𝑜𝑣 ( ξ 𝑡 − 𝜆𝜉𝑡−1 , 𝜉𝑡−𝑠 − 𝜆 𝜉𝑡−𝑠−1) ≠ 0
1. 𝐸[ξ 𝑡 − 𝜆𝜉𝑡−1 − 𝐸(ξ 𝑡 − 𝜆𝜉𝑡−1)]
= 𝐸[ξ 𝑡 − 𝜆𝜉𝑡−1 − 𝐸(ξ 𝑡) − 𝜆𝐸(𝜉𝑡−1)]
= 𝐸[ξ 𝑡 − 𝜆𝜉𝑡−1 − 0 − 0]
= 𝐸[ξ 𝑡 − 𝜆𝜉𝑡−1]
2. 𝐸[ 𝜉𝑡−𝑠 − 𝜆 𝜉𝑡−𝑠−1 − 𝐸( 𝜉𝑡−𝑠 − 𝜆 𝜉𝑡−𝑠−1)]
= 𝐸[ 𝜉𝑡−𝑠 − 𝜆 𝜉𝑡−𝑠−1 − 𝐸( 𝜉𝑡−𝑠)− 𝜆𝐸( 𝜉𝑡−𝑠−1)]
= 𝐸[ 𝜉𝑡−𝑠 − 𝜆 𝜉𝑡−𝑠−1 − 0 − 0]
= 𝐸[ 𝜉𝑡−𝑠 − 𝜆 𝜉𝑡−𝑠−1]
3. 𝐶𝑜𝑣 ( ξ 𝑡 − 𝜆𝜉𝑡−1 , 𝜉𝑡−𝑠 − 𝜆 𝜉𝑡−𝑠−1) = 𝐸{ (ξ 𝑡 − 𝜆𝜉𝑡−1)( 𝜉𝑡−𝑠 − 𝜆 𝜉𝑡−𝑠−1) }
𝐸[ ξ 𝑡 𝜉𝑡−𝑠 − ξ 𝑡 𝜆 𝜉𝑡−𝑠−1 − 𝜆𝜉𝑡−1 𝜉𝑡−𝑠 + 𝜆𝜉𝑡−1 𝜆 𝜉𝑡−𝑠−1 ]
= 0 − 0 + 𝜆𝜉𝑡−1 𝜆 𝜉𝑡−𝑠−1 − 𝜆𝜉𝑡−1 𝜉𝑡−𝑠
= 𝜆2 𝜉2
𝑡−1,𝑡−𝑠−1 − 𝜆𝜉2
𝑡−1,𝑡−𝑠
= 𝜆2 𝜎𝑡−1,𝑡−𝑠−1
2
− 𝜆𝜎𝑡−1,𝑡−𝑠
2
𝐻𝑒𝑛𝑐𝑒, 𝐶𝑜𝑣 ( ξ 𝑡 − 𝜆𝜉𝑡−1 , 𝜉𝑡−𝑠 − 𝜆 𝜉𝑡−𝑠−1) ≠ 0

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D. Application of the Almon lag scheme to help in preventing multi-co-linearity
Criteria:
Criteria Almon Lag Scheme
(PDL(Crude, 4,4))
Almon Lag Scheme
(PDL(Crude, 4,3))
Almon Lag Scheme
(PDL(Crude, 4,2))
Akaike Info Criterion 7.041729 7.044515 7.039952
Schwarz Criterion 7.107173 7.099052 7.083582
𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝑅2
0.020780 0.015322 0.017075
Concluding section: It is observed that the Almon lag scheme with two degree of polynomial
performed better than the three degree and four degree polynomial.

10
A. Application of OLS method on explanatory variables
(Full output in Appendix Table 3)
Problems with Estimated Results
When Var (3) is applied on the simultaneous equations, the OLS method is able to help in getting
consistent and asymptotically efficient estimators. However, when the structural equations are over
identified, the OLS estimations of the reduced equations will be inefficient. In this case, the OLS
estimations are not over identified and therefore there would be no problems with the estimated results.
It is seen that the 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝑅2 for the parameters (soybean oil, sunflower oil, rapeseed oil and palm
oil) are quite significant and most of the F-Statistics for the parameters are above 1.96. This indicates
that the endogenous variables estimated are all significant.

11
B. Testing the validity of OLS method on Simultaneous Equations
Contemporaneous correlation between equation and error terms
No OLS does not take into account the contemporaneous correlation between the errors of different
equations. Therefore,the results is inefficient if the model is not consistent and asymptotically normally
distributed.
Recommendation of Econometric Technique to Solve the Issue
One of an econometric technique that would be able to solve the contemporaneous correlation
problem is through applying the Seemingly Unrelated Regression Equations Model (SURE). This
model would have separate OLS estimations of each equation which makes it as efficient as joint
estimations.
This inefficiency of estimates issue also can be solved by making sure the VAR model is stationary.
This can be achieved by differencing techniques. When the VAR model is consistent and
asymptotically normally distributed, the OLS estimations would no longer be inefficient.

12
C. Application of VAR (4) for all variables and testing for Granger causality
CrudeRet
CRUDERET = C(1,1)*CRUDERET(-1) + C(1,2)*CRUDERET(-2) + C(1,3)*CRUDERET(-3) +
C(1,4)*CRUDERET(-4) + C(1,5)*PALMRET(-1) + C(1,6)*PALMRET(-2) + C(1,7)*PALMRET(-3) +
C(1,8)*PALMRET(-4) + C(1,9)*RAPESEEDRET(-1) + C(1,10)*RAPESEEDRET(-2) +
C(1,11)*RAPESEEDRET(-3) + C(1,12)*RAPESEEDRET(-4) + C(1,13)*SOYBEANRET(-1) +
C(1,14)*SOYBEANRET(-2) + C(1,15)*SOYBEANRET(-3) + C(1,16)*SOYBEANRET(-4) +
C(1,17)*SUNFLOWERRET(-1) + C(1,18)*SUNFLOWERRET(-2) + C(1,19)*SUNFLOWERRET(-3) +
C(1,20)*SUNFLOWERRET(-4) + C(1,21)
PalmRet
PALMRET = C(2,1)*CRUDERET(-1) + C(2,2)*CRUDERET(-2) + C(2,3)*CRUDERET(-3) +
RapeseedRet
RAPESEEDRET = C(3,1)*CRUDERET(-1) + C(3,2)*CRUDERET(-2) + C(3,3)*CRUDERET(-3) +
SoybeanRet
SOYBEANRET = C(4,1)*CRUDERET(-1) + C(4,2)*CRUDERET(-2) + C(4,3)*CRUDERET(-3) +
SunflowerRet
SUNFLOWERRET = C(5,1)*CRUDERET(-1) + C(5,2)*CRUDERET(-2) + C(5,3)*CRUDERET(-3) +

13
Estimated Equation:
𝑦0𝑡 = 𝑥1𝑡 + 𝑥2𝑡 + 𝑥3𝑡 + 𝑥4𝑡 + 𝜀𝑡
𝑦0𝑡 = 𝛽01 + ∑ 𝛿0𝑖 𝑦0𝑡 −𝑖
4
𝑖=1 + ∑ 𝛾0𝑖 𝑥1𝑡−𝑖
4
𝑖=1 + ∑ 𝜑0𝑖 𝑥2𝑡−𝑖
4
𝑖=1 + ∑ 𝜓0𝑖 𝑥3𝑡−𝑖
4
𝑖=1 + ∑ 𝜃0𝑖 𝑥4𝑡−𝑖
4
𝑖=1 + 𝑢0𝑡
𝑥1𝑡 = 𝛽11 + ∑ 𝛿1𝑖 𝑦0𝑡 −𝑖
4
𝑖=1 + ∑ 𝛾1𝑖 𝑥1𝑡−𝑖
4
𝑖=1 + ∑ 𝜑1𝑖 𝑥2𝑡−𝑖
4
𝑖=1 + ∑ 𝜓1𝑖 𝑥3𝑡−𝑖
4
𝑖=1 + ∑ 𝜃1𝑖 𝑥4𝑡−𝑖
4
𝑖=1 + 𝑢1𝑡
𝑥2𝑡 = 𝛽21 + ∑ 𝛿2𝑖 𝑦0𝑡 −𝑖
4
𝑖=1 + ∑ 𝛾2𝑖 𝑥1𝑡−𝑖
4
𝑖=1 + ∑ 𝜑2𝑖 𝑥2𝑡−𝑖
4
𝑖=1 + ∑ 𝜓2𝑖 𝑥3𝑡−𝑖
4
𝑖=1 + ∑ 𝜃2𝑖 𝑥4𝑡−𝑖
4
𝑖=1 + 𝑢2𝑡
𝑥3𝑡 = 𝛽31 + ∑ 𝛿3𝑖 𝑦0𝑡 −𝑖
4
𝑖=1 + ∑ 𝛾3𝑖 𝑥1𝑡−𝑖
4
𝑖=1 + ∑ 𝜑3𝑖 𝑥2𝑡−𝑖
4
𝑖=1 + ∑ 𝜓3𝑖 𝑥3𝑡−𝑖
4
𝑖=1 + ∑ 𝜃3𝑖 𝑥4𝑡−𝑖
4
𝑖=1 + 𝑢3𝑡
𝑥4𝑡 = 𝛽41 + ∑ 𝛿4𝑖 𝑦0𝑡−𝑖
4
𝑖=1 + ∑ 𝛾4𝑖 𝑥1𝑡−𝑖
4
𝑖=1 + ∑ 𝜑4𝑖 𝑥2𝑡−𝑖
4
𝑖=1 + ∑ 𝜓4𝑖 𝑥3𝑡−𝑖
4
𝑖=1 + ∑ 𝜃4𝑖 𝑥4𝑡−𝑖
4
𝑖=1 + 𝑢5𝑡
(1)
Where
𝑦0𝑡 - Palm oil price changes;
𝑥1𝑡 - Crude oil price changes;
𝑥2𝑡 - Soybean oil price changes;
𝑥3𝑡 - Sunflower oil price changes;
𝑥4𝑡 - Rapeseed oil price changes;
The Granger Causality test helps in identifying whether one variable in the VAR model affects
other variables in the VAR model or vice versa. It is concerned with whether one variable is
useful in predicting future values of another variable, but not whether one variable actually
causes the other.
Type Regressors Condition Test Criteria Decision
Palm oil 𝑦0𝑡(𝐶𝑟𝑢𝑑𝑒𝑅𝑒𝑡) 𝐻 0: 𝛽 0 = 0
(𝐶𝑟𝑢𝑑𝑒 𝑜𝑖𝑙 𝑝𝑟𝑖𝑐𝑒 𝑐ℎ𝑎𝑛𝑔𝑒𝑠 𝑑𝑜
𝑛𝑜𝑡 𝐺𝑟𝑎𝑛𝑔𝑒𝑟 𝑐𝑎𝑢𝑠𝑒 𝑝𝑎𝑙𝑚 𝑜𝑖𝑙
(Crude oil Granger cause palm
oil price changes)
P-value: 0.0681
𝛼 : 0.05
Since 0.0681 (p-value) >
significance.
There is not enough
conclude that crude oil
price changes Granger
cause palm oil price
changes.
Soybean Oil 𝑥 2𝑡(𝐶𝑟𝑢𝑑𝑒𝑅𝑒𝑡) 𝐻 0: 𝛽 0 = 0
(Crude oil Granger cause
Soybean oil price changes)
P-value: 0.0778
𝛼 : 0.05
significance.
There is not enough
conclude that Crude oil
Granger cause Soybean oil
price changes.

14
Type Regressors Condition Test Criteria Decision
Sunflower Oil 𝑥 3𝑡(𝐶𝑟𝑢𝑑𝑒𝑅𝑒𝑡) 𝐻 0: 𝛽 0 = 0
Sunflower oil price changes)
P-value: 0.0332
𝛼 : 0.05
we reject the null
significance.
Crude oil Granger cause
Sunflower oil price
changes.
Rapeseed Oil 𝑥 4𝑡(𝐶𝑟𝑢𝑑𝑒𝑅𝑒𝑡) 𝐻 0: 𝛽 0 = 0
Rapeseed oil price changes)
P-value: 0.1267
𝛼 : 0.05
significance.
There is not enough
conclude that Crude oil
Granger cause Rapeseed
oil price changes.
(Model will be put into the Appendix Table 6)
Concluding section: The results from the Granger Causality test implies that crude oil price changes
do not Granger cause palm oil, soybean oil and rapeseed oil prices. However it Granger causes the
sunflower oil prices.

15
Conclusion
Summary
In theory, multi-co-linearity will be present when too many lagged values are in a model. This will
cause the true results obtained from tests would be unreliable because the errors in the model are not
behaving well.
By looking at the scatter plot, there were different variations in spikes but there were no obvious
patterns. However,through analysing the errors from the model estimated with eight lags of crude oil,
the results showed that there was serial correlation present.
Knowing there is serial correlation presentin the previous model, a newmodel will be formed to achieve
some improvements. The results show that there were minor improvements but not huge.
Then, the model is used to test for short-run, long-run and normal relationship between the crude oil
and palm oil prices. It is observed that there was not long-run between the palm oil price changes and
the crude oil price changes but there was evidence that the normal and short run crude oil price changes
is a function of palm oil price changes.
Multicolinearity is an issue that causes results to be unreliable. The covariance for the estimated
equation is used to prove the presence of this issue in theory. The result shows that there would be
bound to be serial correlation.
Knowing this, the Almon lag scheme was then applied to help prevent multi-co-linearity. Three types
of Almon degree polynomial was used to improve the model. The best model was Almon Lag Schme
of degree polynomial 2.

16
Firstly, theory wasapplied to understand whether using OLS method to estimate would cause problems.
Theory states that OLS’s estimates would only be unreliable if it does not follow the assumption of
asymptotic distribution. The output from estimating the parameters using the OLS method shows that
the results were not so significant and the residual covariance was large.
Then the issue of contemporaneous error terms from different equations by applying OLS method was
discussed. Theory says that OLS does not take into account of the correlation between the error terms
in different equations.
Therefore,recommendations was made to solve this problem. This issue can be solved by applying the
Seemingly Unrelated Residual Equations Model and by stabilizing the VAR model.
After that, a VAR(4) model was generated to estimate the parameters to test the Granger cause
relationship between the Crude oil price changes with the other exogenous variables. The results show
that Crude oil price changes only have a granger cause relationship with Sunflower oil price changes.
Problems Encountered and Suggestions
Problem Suggestions
Accuracy of estimation
methods
Making sure the models follow an asymptotic distribution
Making sure the variance is not serially correlated
Heteroskedasticity Making sure the errors of the exogenous variables are not correlated to
each other
Finding models that takes into account of serial correlation
Such as GARCH and ARCH models

17
Appendix
Long Run Effect Test
Table 1
Normal Effect Test
Table 2

20
Granger Causality Test
Table 5

21
Pair Granger Causality Test
Table 6

ETW3410 Assignment 2 final draft ex5

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