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Financial Econometrics
BU7510
Course ID: 6898722
Financial Econometrics Mini Assignment
Prarthana Ravi Kumar 13315488
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Problem Statement:
I aim to analyse the relationship between Daily Prices of Oil and Daily prices of Corn to see if they are
related to each other and to what magnitude can oil prices predict the price of corn.
Hypothesis: Oil Prices have a significant influence on Corn prices.
1. Oil
1.1. Data
Data on Oil is represented by daily closing price of NYMEX sweet crude oil composite energy
future continuation. The oil futures price is the price of a contract to buy British Thermal
Units of crude oil deliverable at the end of the month. These contracts are rolled over at
expiry to get prices from September 2013.
1.2. Plots
Plotting the series at level and first difference to check for breakages or data errors. The Oil
data seems to be showing a steady downtrend in the last 100 trading days. First
differencing shows that there are significant spikes in prices.
88
92
96
100
104
108
112
OIL
4
3
2
1
0
1
2
3
Differenced OIL
1.3. Histogram
The data is plotted as a histogramto check the distribution. The data is not normally
distributed.
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0
2
4
6
8
10
92 94 96 98 100 102 104 106 108 110
Series: OILSample 1 120Observations 100
Mean 98.54970
Median 97.46000Maximum 110.5300Minimum 91.66000Std. Dev. 4.747433Skewness 0.657890Kurtosis 2.522336
Jarque-Bera 8.164343Probability 0.016871
1.4. AutocorrelationA Correlogram is plotted to graphically display the autocorrelation relationship. The autocorrelation
looks significant for a large number of lags, but this could be because of stickiness of the
autocorrelation at lag 1. The Partial ACF confirms this with insignificant values after lag 1
The autocorrelation seems insignificant when first differenced. Significance is tested by
checking if the AC value is greater than +0.196 or less than -0.196. In this case none of the
lags are greater than 0.196 but the p value is very high. This means that we can say with25% confidence that first lagged autocorrelation is not significant. This is not an acceptable
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level of confidence.
1.5. Unit Root
Checking for Unit roots at level using Dickey Fuller Test
Null Hypothesis: OIL has a unit rootExogenous: Constant
Lag Length: 0 (Automatic - based on SIC, maxlag=12)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -2.103992 0.2436
Test critical values: 1% level -3.497727
5% level -2.890926
10% level -2.582514
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(OIL)
Method: Least Squares
Date: 02/03/14 Time: 11:41
Sample (adjusted): 2 100
Included observations: 99 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
OIL(-1) -0.048545 0.023073 -2.103992 0.0380
C 4.673909 2.276701 2.052931 0.0428
R-squared 0.043645 Mean dependent var -0.110707Adjusted R-squared 0.033786 S.D. dependent var 1.108432
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S.E. of regression 1.089546 Akaike info criterion 3.029395Sum squared resid 115.1498 Schwarz criterion 3.081822
Log likelihood -147.9550 Hannan-Quinn criter. 3.050607
F-statistic 4.426782 Durbin-Watson stat 1.984063
Prob(F-statistic) 0.037965
We can see that the Test Statistic is not more negative than critical values, so the null hypothesisthat " There are no unit roots in the series" cannot be rejected.
Performing unit root on the first differenced level, we can see that we can reject the null hypothesis.
At the first difference level the series is stationary.
Null Hypothesis: D(OIL) has a unit root
Exogenous: Constant
Lag Length: 1 (Automatic - based on SIC, maxlag=12)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -6.351790 0.0000
Test critical values: 1% level -3.499167
5% level -2.891550
10% level -2.582846
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(OIL,2)
Method: Least Squares
Date: 02/03/14 Time: 11:48
Sample (adjusted): 4 100Included observations: 97 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
D(OIL(-1)) -0.934210 0.147078 -6.351790 0.0000
D(OIL(-1),2) -0.079942 0.101895 -0.784552 0.4347
C -0.117339 0.113473 -1.034067 0.3038
R-squared 0.505436 Mean dependent var 0.027835
Adjusted R-squared 0.494913 S.D. dependent var 1.547158
S.E. of regression 1.099557 Akaike info criterion 3.058132Sum squared resid 113.6484 Schwarz criterion 3.137762
Log likelihood -145.3194 Hannan-Quinn criter. 3.090330
F-statistic 48.03313 Durbin-Watson stat 1.938430
Prob(F-statistic) 0.000000
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1.6. Model
The correlogram profiles shows that the ACF function geometrically decays but the partial ACF hasmany non zero points. This means that this is a pure Autoregressive Order.
1.6.1. Pure AutoRegressive Model
Dependent Variable: DOIL
Method: Least Squares
Sample (adjusted): 3 100
Included observations: 98 after adjustments
Convergence achieved after 3 iterations
Variable Coefficient Std. Error t-Statistic Prob.
C -0.133727 0.107125 -1.248325 0.2149
AR(1) -0.032407 0.101162 -0.320342 0.7494
R-squared 0.001068 Mean dependent var -0.133878
Adjusted R-squared -0.009338 S.D. dependent var 1.089766
S.E. of regression 1.094842 Akaike info criterion 3.039294
Sum squared resid 115.0731 Schwarz criterion 3.092048
Log likelihood -146.9254 Hannan-Quinn criter. 3.060632
F-statistic 0.102619 Durbin-Watson stat 1.929113
Prob(F-statistic) 0.749405
Inverted AR Roots -.03
1.6.2. ARMA(1,1) Model
Dependent Variable: DOIL
Method: Least Squares
Sample (adjusted): 3 100
Included observations: 98 after adjustments
Convergence achieved after 13 iterations
MA Backcast: 2
Variable Coefficient Std. Error t-Statistic Prob.
C -0.130707 0.107516 -1.215706 0.2271
AR(1) -0.201207 0.459320 -0.438053 0.6623
MA(1) 0.160931 0.474035 0.339491 0.7350
R-squared 0.004054 Mean dependent var -0.133878
Adjusted R-squared -0.016913 S.D. dependent var 1.089766
S.E. of regression 1.098943 Akaike info criterion 3.056708
Sum squared resid 114.7291 Schwarz criterion 3.135840
Log likelihood -146.7787 Hannan-Quinn criter. 3.088715
F-statistic 0.193350 Durbin-Watson stat 1.927546
Prob(F-statistic) 0.824517
Inverted AR Roots -.20Inverted MA Roots -.16
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But the p values for AR(1) and MA(1) are high implying that confidence levels are only about 30-40%.
Here it shows that Inverted AR and MA roots are significantly lower than 1. This implies that both AR
and MA parts are stationary and invertible.
1.6.3.ARMA(1,2)
Dependent Variable: DOILMethod: Least Squares
Date: 02/03/14 Time: 21:00
Sample (adjusted): 3 100
Included observations: 98 after adjustmentsConvergence achieved after 9 iterations
MA Backcast: 1 2
Variable Coefficient Std. Error t-Statistic Prob.
C -0.135456 0.119048 -1.137828 0.2581
AR(1) -0.020839 0.101423 -0.205464 0.8376
MA(2) 0.101433 0.104203 0.973421 0.3328
R-squared 0.012783 Mean dependent var -0.133878
Adjusted R-squared -0.008001 S.D. dependent var 1.089766
S.E. of regression 1.094116 Akaike info criterion 3.047905Sum squared resid 113.7236 Schwarz criterion 3.127037
Log likelihood -146.3473 Hannan-Quinn criter. 3.079912
F-statistic 0.615051 Durbin-Watson stat 1.930155
Prob(F-statistic) 0.542752
Inverted AR Roots -.02
1.6.4.ARMA(2,2)
Dependent Variable: DOIL
Method: Least Squares
Date: 02/03/14 Time: 13:32
Sample (adjusted): 4 100Included observations: 97 after adjustments
Convergence achieved after 15 iterations
MA Backcast: 2 3
Variable Coefficient Std. Error t-Statistic Prob.
C -0.112930 0.113431 -0.995588 0.3220
AR(2) -0.683025 0.117444 -5.815770 0.0000MA(2) 0.826784 0.115257 7.173411 0.0000
R-squared 0.126363 Mean dependent var -0.124845
Adjusted R-squared 0.107775 S.D. dependent var 1.091733S.E. of regression 1.031226 Akaike info criterion 2.929813
Sum squared resid 99.96207 Schwarz criterion 3.009443
Log likelihood -139.0959 Hannan-Quinn criter. 2.962011
F-statistic 6.798096 Durbin-Watson stat 2.013442Prob(F-statistic) 0.001748
ARMA(2,2) has the lowest AIC and SBIC values and so we accept ARMA(2,2)
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1.7. Heteroskedasticity
Testing for Heteroskedasticity in ARMA(2,2) Model we can see that the F Statistic is notsignificant, suggesting that there is no serial autocorrelation among the error terms in first differencedoil series.
Heteroskedasticity Test: ARCH
F-statistic 1.721433 Prob. F(5,86) 0.1383Obs*R-squared 8.369972 Prob. Chi-Square(5) 0.1370
Test Equation:Dependent Variable: RESID^2
Method: Least Squares
Date: 02/03/14 Time: 21:10
Sample (adjusted): 9 100Included observations: 92 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 1.834662 0.335701 5.465162 0.0000
RESID^2(-1) -0.094398 0.111096 -0.849694 0.3979
RESID^2(-2) -0.213451 0.110193 -1.937063 0.0560
RESID^2(-3) -0.234038 0.109978 -2.128055 0.0362RESID^2(-4) -0.037404 0.109696 -0.340978 0.7340
RESID^2(-5) -0.166391 0.112684 -1.476617 0.1434
R-squared 0.090978 Mean dependent var 1.082818Adjusted R-squared 0.038128 S.D. dependent var 1.375347
S.E. of regression 1.348873 Akaike info criterion 3.499409
Sum squared resid 156.4734 Schwarz criterion 3.663874
Log likelihood -154.9728 Hannan-Quinn criter. 3.565788F-statistic 1.721433 Durbin-Watson stat 1.976410
Prob(F-statistic) 0.138260
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2. Corn
2.1. Data
Data on corn is represented by daily closing price of Corn on the Chicago Board of Trade
commodity futures continuation. The quotes represent the price in US Dollars of a contract
to buy 5000 bushels of corn at the end of the month.
2.2. Plot
The corn prices too show a steady downtrend like that of oil. The first differenced data
shows less spikes than oil. It is mostly confined between +/-$5.
10
20
30
40
50
60
70
80
90
CORN
-30
-20
-10
0
10
20
30
Differenced CORN
2.3. Histogram
The prices of corn appear normal with a left skew.
0
2
4
6
8
10
12
14
410 420 430 440 450 460 470 480
Series: CORNSample 3 100Observations 98
Mean 434.0612Median 430.3750Maximum 479.7500Minimum 412.0000Std. Dev. 14.26778Skewness 1.242890Kurtosis 4.582799
Jarque-Bera 35.46111Probability 0 .000000
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2.4. Autocorrelation
Autocorrelation at level is similar to oil series, where the first lag is significant and then the
significance decays.
At first differenced level, the significance is reduced.
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2.5. Unit Root
Testing using Dickey-Fuller test, we can see that at level , we can reject the null hypothesis
with a confidence of 99.77%
Null Hypothesis: CORN has a unit root
Exogenous: Constant
Lag Length: 0 (Automatic - based on SIC, maxlag=11)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -3.981207 0.0023
Test critical values: 1% level -3.498439
5% level -2.891234
10% level -2.582678
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(CORN)Method: Least Squares
Date: 02/03/14 Time: 22:38
Sample: 3 100
Included observations: 98
Variable Coefficient Std. Error t-Statistic Prob.
CORN(-1) -0.139034 0.034922 -3.981207 0.0001
C 59.82638 15.18912 3.938764 0.0002
R-squared 0.141708 Mean dependent var -0.607143
Adjusted R-squared 0.132767 S.D. dependent var 5.687865
S.E. of regression 5.296843 Akaike info criterion 6.192296Sum squared resid 2693.428 Schwarz criterion 6.245051
Log likelihood -301.4225 Hannan-Quinn criter. 6.213634
F-statistic 15.85001 Durbin-Watson stat 2.123500
Prob(F-statistic) 0.000133
At first difference level, the unit root test gives a better results with greater confidence.
Null Hypothesis: D(CORN) has a unit root
Exogenous: Constant
Lag Length: 2 (Automatic - based on SIC, maxlag=11)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -7.694709 0.0000
Test critical values: 1% level -3.499910
5% level -2.891871
10% level -2.583017
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(CORN,2)
Method: Least SquaresDate: 02/03/14 Time: 22:40
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Sample (adjusted): 5 100Included observations: 96 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
D(CORN(-1)) -1.485120 0.193005 -7.694709 0.0000
D(CORN(-1),2) 0.347286 0.147411 2.355898 0.0206D(CORN(-2),2) 0.183560 0.099252 1.849429 0.0676
C -0.732061 0.569458 -1.285540 0.2018
R-squared 0.574468 Mean dependent var 0.028646
Adjusted R-squared 0.560592 S.D. dependent var 8.250348
S.E. of regression 5.468977 Akaike info criterion 6.276834
Sum squared resid 2751.694 Schwarz criterion 6.383682
Log likelihood -297.2880 Hannan-Quinn criter. 6.320024
F-statistic 41.40004 Durbin-Watson stat 1.985070
Prob(F-statistic) 0.000000
2.6. Models
As per the ACF Functions, the first lag is persistent. The partial ACF is insignificant apart
from that at lag 1. This usually means that a simple Autoregressive model with lag 1 is
sufficient.
2.6.1. AR(1)
Dependent Variable: DCORN
Method: Least Squares
Date: 02/03/14 Time: 22:51
Sample (adjusted): 3 100
Included observations: 98 after adjustments
Convergence achieved after 3 iterations
Variable Coefficient Std. Error t-Statistic Prob.
C -0.605620 0.525980 -1.151412 0.2524
AR(1) -0.093248 0.101519 -0.918531 0.3606
R-squared 0.008712 Mean dependent var -0.607143
Adjusted R-squared -0.001614 S.D. dependent var 5.687865
S.E. of regression 5.692453 Akaike info criterion 6.336357
Sum squared resid 3110.786 Schwarz criterion 6.389111
Log likelihood -308.4815 Hannan-Quinn criter. 6.357695
F-statistic 0.843700 Durbin-Watson stat 1.946677
Prob(F-statistic) 0.360643
Inverted AR Roots -.09
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2.6.2. ARMA(1,1)Dependent Variable: DCORN
Method: Least Squares
Date: 02/03/14 Time: 22:47
Sample (adjusted): 3 100
Included observations: 98 after adjustments
Convergence achieved after 16 iterationsMA Backcast: 2
Variable Coefficient Std. Error t-Statistic Prob.
C -0.585482 0.563059 -1.039824 0.3011
AR(1) -0.793624 0.199388 -3.980308 0.0001MA(1) 0.782703 0.214655 3.646325 0.0004
R-squared 0.045350 Mean dependent var -0.607143
Adjusted R-squared 0.025252 S.D. dependent var 5.687865S.E. of regression 5.615590 Akaike info criterion 6.319104
Sum squared resid 2995.811 Schwarz criterion 6.398236
Log likelihood -306.6361 Hannan-Quinn criter. 6.351111F-statistic 2.256454 Durbin-Watson stat 2.081236Prob(F-statistic) 0.110306
Inverted AR Roots -.79
Inverted MA Roots -.78
ARMA(1,1) is preferred over AR(1) because of greater significance of the coefficients.
AIC and SBIC criteria also support selecting ARMA(1,1).
2.7. Heteroskedasticity
ARMA(1,1) does not have ARCH problem. The F statistic is not significant and null
hypothesis can be rejected.Heteroskedasticity Test: ARCH
F-statistic 0.146802 Prob. F(5,87) 0.9805
Obs*R-squared 0.778066 Prob. Chi-Square(5) 0.9784
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 02/03/14 Time: 22:57Sample (adjusted): 8 100
Included observations: 93 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 23.07306 7.043929 3.275595 0.0015RESID^2(-1) -0.009989 0.063628 -0.156994 0.8756
RESID^2(-2) -0.012009 0.063644 -0.188695 0.8508
RESID^2(-3) -0.029897 0.063597 -0.470098 0.6395
RESID^2(-4) -0.007626 0.063586 -0.119937 0.9048RESID^2(-5) 0.043036 0.063550 0.677196 0.5001
R-squared 0.008366 Mean dependent var 22.62247
Adjusted R-squared -0.048624 S.D. dependent var 51.61077S.E. of regression 52.85064 Akaike info criterion 10.83516
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Sum squared resid 243007.5 Schwarz criterion 10.99855Log likelihood -497.8348 Hannan-Quinn criter. 10.90113
F-statistic 0.146802 Durbin-Watson stat 2.051665
Prob(F-statistic) 0.980500
3. Relationship between Corn and Oil
3.1. Cointegration check using Engel-Granger MethodThe first differenced corn and oil series are linearly regressed to get the residual values. The residualvalues are checked for unit roots to confirm if the series are cointegrated.
Null Hypothesis: D(STATRESID) has a unit root
Exogenous: Constant
Lag Length: 6 (Automatic - based on SIC, maxlag=12)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -7.282815 0.0000
Test critical values: 1% level -3.503879
5% level -2.893589
10% level -2.583931
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(STATRESID,2)
Method: Least Squares
Date: 02/03/14 Time: 23:11
Sample (adjusted): 10 100
Included observations: 91 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
D(STATRESID(-1)) -5.444545 0.747588 -7.282815 0.0000D(STATRESID(-1),2) 3.532707 0.682267 5.177891 0.0000
D(STATRESID(-2),2) 2.713732 0.578382 4.691935 0.0000
D(STATRESID(-3),2) 1.865882 0.459733 4.058623 0.0001
D(STATRESID(-4),2) 1.127552 0.326451 3.453966 0.0009
D(STATRESID(-5),2) 0.524432 0.200891 2.610535 0.0107
D(STATRESID(-6),2) 0.165213 0.091039 1.814742 0.0732
C 0.266542 0.539695 0.493874 0.6227
R-squared 0.827935 Mean dependent var 0.057282
Adjusted R-squared 0.813424 S.D. dependent var 11.86871
S.E. of regression 5.126629 Akaike info criterion 6.190579
Sum squared resid 2181.433 Schwarz criterion 6.411314
Log likelihood -273.6713 Hannan-Quinn criter. 6.279632F-statistic 57.05371 Durbin-Watson stat 2.072421
Prob(F-statistic) 0.000000
The test stat is more negative than the critical values and hence we can reject the null hypothesis of the
Augmented Dickey Fuller Unit root test, i.e. , we can conclude that series are co-integrated.
Linear combination of Oil and Corn Prices are stationary.
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4. ConclusionThe Error Correction Model used at level is able to predict the relationship between the series with R
2=75%.
The coefficients are significant. So as per the relationship, a dollar price increase in oil can cause a $2.74increase in corn prices.
Dependent Variable: CORN
Method: Least SquaresDate: 02/03/14 Time: 23:37
Sample (adjusted): 3 100
Included observations: 98 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 163.9211 16.07726 10.19583 0.0000
OIL 2.747343 0.163338 16.81999 0.0000
STATRESID(-1) 0.221929 0.131561 1.686897 0.0949
R-squared 0.748692 Mean dependent var 434.0612
Adjusted R-squared 0.743401 S.D. dependent var 14.26778
S.E. of regression 7.227428 Akaike info criterion 6.823778Sum squared resid 4962.393 Schwarz criterion 6.902909
Log likelihood -331.3651 Hannan-Quinn criter. 6.855785
F-statistic 141.5111 Durbin-Watson stat 0.756067
Prob(F-statistic) 0.000000