Econometrics Mini Assignment

8/10/2019 Econometrics Mini Assignment

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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


t-Statistic Prob.*



5% level -2.891550

10% level -2.582846



Dependent Variable: D(OIL,2)


Date: 02/03/14 Time: 11:48

Sample (adjusted): 4 100Included observations: 97 after adjustments


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






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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




Convergence achieved after 3 iterations


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




Inverted AR Roots -.03

1.6.2. ARMA(1,1) Model






MA Backcast: 2


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








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


Included observations: 98 after adjustmentsConvergence achieved after 9 iterations

MA Backcast: 1 2


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








1.6.4.ARMA(2,2)



Date: 02/03/14 Time: 13:32



MA Backcast: 2 3


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


Adjusted R-squared 0.107775 S.D. dependent var 1.091733S.E. of regression 1.031226 Akaike info criterion 2.929813



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


Date: 02/03/14 Time: 21:10



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



Log likelihood -154.9728 Hannan-Quinn criter. 3.565788F-statistic 1.721433 Durbin-Watson stat 1.976410



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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


t-Statistic Prob.*



5% level -2.891234

10% level -2.582678



Dependent Variable: D(CORN)Method: Least Squares

Date: 02/03/14 Time: 22:38

Sample: 3 100

Included observations: 98


CORN(-1) -0.139034 0.034922 -3.981207 0.0001

C 59.82638 15.18912 3.938764 0.0002







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


t-Statistic Prob.*



5% level -2.891871

10% level -2.583017



Dependent Variable: D(CORN,2)

Method: Least SquaresDate: 02/03/14 Time: 22:40


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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








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


Date: 02/03/14 Time: 22:51





C -0.605620 0.525980 -1.151412 0.2524

AR(1) -0.093248 0.101519 -0.918531 0.3606










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2.6.2. ARMA(1,1)Dependent Variable: DCORN


Date: 02/03/14 Time: 22:47



Convergence achieved after 16 iterationsMA Backcast: 2


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


Adjusted R-squared 0.025252 S.D. dependent var 5.687865S.E. of regression 5.615590 Akaike info criterion 6.319104


Log likelihood -306.6361 Hannan-Quinn criter. 6.351111F-statistic 2.256454 Durbin-Watson stat 2.081236Prob(F-statistic) 0.110306


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


Date: 02/03/14 Time: 22:57Sample (adjusted): 8 100



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


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



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


t-Statistic Prob.*



5% level -2.893589

10% level -2.583931



Dependent Variable: D(STATRESID,2)


Date: 02/03/14 Time: 23:11




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





Log likelihood -273.6713 Hannan-Quinn criter. 6.279632F-statistic 57.05371 Durbin-Watson stat 2.072421


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




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







Econometrics Mini Assignment

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