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TAKE HOME PROJECT 2 Group C: Robert Matarazzo, Michael Stromberg, Yuxing Zhang, Yin Chu, Leslie Wei, and Kurtis Hollar.

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Presentation on theme: "TAKE HOME PROJECT 2 Group C: Robert Matarazzo, Michael Stromberg, Yuxing Zhang, Yin Chu, Leslie Wei, and Kurtis Hollar."— Presentation transcript:

1 TAKE HOME PROJECT 2 Group C: Robert Matarazzo, Michael Stromberg, Yuxing Zhang, Yin Chu, Leslie Wei, and Kurtis Hollar

2 Introduction We chose to forecast the imported petroleum price index. Petroleum has many uses but is mainly used to produce fuels. The price of petroleum heavily influences the price of gasoline. 2

3 Original Data 3

4 4

5 Decay in the autocorrelation Large spike at lag 1 in the partial correlation This structure indicates non- stationary data 5

6 Original Data ADF Test Statistic-0.443169 1% Critical Value*-3.4575 5% Critical Value-2.8729 10% Critical Value-2.5728 *MacKinnon critical values for rejection of hypothesis of a unit root. Augmented Dickey-Fuller Test Equation Dependent Variable: D(PETROP) Method: Least Squares Date: 05/28/10 Time: 11:42 Sample(adjusted): 1989:01 2010:04 Included observations: 256 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. PETROP(-1)-0.0042200.009521-0.4431690.6580 C1.4509601.3706141.0586210.2908 R-squared0.000773 Mean dependent var0.960156 Adjusted R-squared-0.003161 S.D. dependent var12.89963 S.E. of regression12.92000 Akaike info criterion7.963212 Sum squared resid42399.31 Schwarz criterion7.990909 Log likelihood-1017.291 F-statistic0.196399 Durbin-Watson stat0.728269 Prob(F-statistic)0.658021 The ADF test statistic confirms the notion of non- stationary data 6

7 Application of Step Function and Dummy Variables 7

8 8

9 Dependent Variable: PETROP Method: Least Squares Date: 05/28/10 Time: 11:45 Sample: 1988:12 2010:04 Included observations: 257 VariableCoefficientStd. Errort-StatisticProb. C105.35474.90660021.472030.0000 D1314.145375.536144.1588740.0000 D2266.145375.536143.5234170.0005 D3183.545375.536142.4299010.0158 D496.2453475.536141.2741630.2038 D545.4453475.536140.6016370.5480 STEP130.576619.472466.7057040.0000 R-squared0.241580 Mean dependent var117.0074 Adjusted R-squared0.223378 S.D. dependent var85.53265 S.E. of regression75.37662 Akaike info criterion11.50973 Sum squared resid1420409. Schwarz criterion11.60640 Log likelihood-1472.000 F-statistic13.27215 Durbin-Watson stat0.111977 Prob(F-statistic)0.000000 Regression of petroleum price against dummy variables and step function 9

10 Application of Step Function and Dummy Variables 10

11 Application of Step Function and Dummy Variables 11

12 Application of Step Function and Dummy Variables Decay in the autocorrelation Large spike at lag 1 in the partial correlation This structure indicates non- stationary data 12

13 Application of Step Function and Dummy Variables Breusch-Godfrey Serial Correlation LM Test: F-statistic5887.195 Probability0.000000 Obs*R-squared251.6986 Probability0.000000 Test Equation: Dependent Variable: RESID Method: Least Squares Date: 05/28/10 Time: 11:49 VariableCoefficientStd. Errort-StatisticProb. C1.3062290.7150461.8267760.0689 D1-373.860211.44297-32.671590.0000 D262.9775926.148382.4084700.0167 D3-1.30622910.89304-0.1199140.9046 D4-1.30622910.89304-0.1199140.9046 D5-1.30622910.89304-0.1199140.9046 STEP1.9043662.8103810.6776180.4986 RESID(-1)1.2073290.06257719.293630.0000 RESID(-2)-0.1787420.065813-2.7159080.0071 R-squared0.979372 Mean dependent var5.73E-14 Adjusted R-squared0.978706 S.D. dependent var74.48806 S.E. of regression10.86954 Akaike info criterion7.644198 Sum squared resid29300.44 Schwarz criterion7.768484 Log likelihood-973.2794 F-statistic1471.799 Durbin-Watson stat1.900201 Prob(F-statistic)0.000000 The F-statistic indicates that there is still serial correlation in this revised data Further steps must be taken 13

14 Logarithm Transformation and First Differenced Data Regression of the first difference of the logarithm of petroleum price against dummy variables and step function Dependent Variable: DLNPETROP Method: Least Squares Date: 05/29/10 Time: 23:20 Sample(adjusted): 1989:01 2009:04 Included observations: 244 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. C0.0105590.0044792.3574060.0192 DD1-0.1135330.069245-1.6395840.1024 DD2-0.2456070.098132-2.5028240.0130 DD3-0.5076340.120437-4.2149500.0000 DD4-0.8779880.139356-6.3003090.0000 DD5-1.1788780.156127-7.5507780.0000 DSTEP-1.2369680.171380-7.2177020.0000 R-squared0.222473 Mean dependent var Adjusted R-squared0.202789 S.D. dependent var S.E. of regression0.069100 Akaike info criterion Sum squared resid1.131627 Schwarz criterion Log likelihood309.3474 F-statistic Durbin-Watson stat1.102828 Prob(F-statistic) 14

15 Logarithm Transformation and First Differenced Data 15

16 Logarithm Transformation and First Differenced Data 16

17 Logarithm Transformation and First Differenced Data ADF Test Statistic-7.814147 1% Critical Value*-3.4580 5% Critical Value-2.8731 10% Critical Value-2.5729 *MacKinnon critical values for rejection of hypothesis of a unit root. Augmented Dickey-Fuller Test Equation Dependent Variable: D(DLNPETROP) Method: Least Squares Date: 05/30/10 Time: 15:57 Sample(adjusted): 1989:06 2010:04 Included observations: 251 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. DLNPETROP(-1)-0.7001270.089597-7.8141470.0000 D(DLNPETROP(-1))0.3159750.0822443.8419210.0002 D(DLNPETROP(-2))0.0987280.0772061.2787680.2022 D(DLNPETROP(-3))0.1439670.0673442.1378000.0335 D(DLNPETROP(-4))0.0694450.0633191.0967410.2738 C0.0042220.0040911.0320670.3031 R-squared0.286618 Mean dependent var-6.30E-05 Adjusted R-squared0.272059 S.D. dependent var0.075215 S.E. of regression0.064173 Akaike info criterion-2.630867 Sum squared resid1.008940 Schwarz criterion-2.546593 Log likelihood336.1738 F-statistic19.68690 Durbin-Watson stat2.007980 Prob(F-statistic)0.000000 The ADF test statistic indicates that the data is now stationary 17

18 Logarithm Transformation and First Differenced Data Notice the significant spikes at lags 1, 2, and 11 The spike at lag 10 may also be significant 18

19 Building The Model Dependent Variable: DLNPETROP Method: Least Squares Date: 05/29/10 Time: 23:26 Sample(adjusted): 1989:03 2009:04 Included observations: 242 after adjusting endpoints Convergence achieved after 6 iterations Backcast: 1988:04 1989:02 VariableCoefficientStd. Errort-StatisticProb. C0.0105500.0066951.5756590.1165 DD1-0.1185520.060062-1.9738140.0496 DD2-0.2531430.109485-2.3121240.0216 DD3-0.5305450.146073-3.6320450.0003 DD4-0.8948820.172724-5.1809990.0000 DD5-1.1948210.194876-6.1311790.0000 DSTEP-1.2457890.212066-5.8745410.0000 AR(1)0.5190820.0650707.9772730.0000 AR(2)-0.1967110.064619-3.0441670.0026 MA(11)0.1760800.0664172.6511090.0086 R-squared0.428059 Mean dependent var0.004986 Adjusted R-squared0.405871 S.D. dependent var0.077414 S.E. of regression0.059671 Akaike info criterion-2.759510 Sum squared resid0.826055 Schwarz criterion-2.615339 Log likelihood343.9007 F-statistic19.29289 Durbin-Watson stat1.980779 Prob(F-statistic)0.000000 Inverted AR Roots.26+.36i.26 -.36i Inverted MA Roots.82+.24i.82 -.24i.56 -.65i.56+.65i.12 -.85i.12+.85i -.35+.78i -.35 -.78i -.72+.46i -.72 -.46i -.85 Based off of the correlogram, we tried modeling with an AR(1) AR(2) MA(11) All of the coefficients are significant at a 5% level 19

20 Building The Model Breusch-Godfrey Serial Correlation LM Test: F-statistic0.776204 Probability0.461351 Obs*R-squared1.622180 Probability0.444374 Test Equation: Dependent Variable: RESID Method: Least Squares Date: 05/29/10 Time: 23:30 VariableCoefficientStd. Errort-StatisticProb. C-8.72E-050.006703-0.0130050.9896 DD10.0011030.0602350.0183110.9854 DD20.0027160.1100760.0246760.9803 DD30.0064930.1471300.0441300.9648 DD40.0097900.1739970.0562650.9552 DD50.0109480.1960740.0558360.9555 DSTEP0.0110140.2131720.0516660.9588 AR(1)0.5830910.7725790.7547330.4512 AR(2)0.0285830.1589900.1797810.8575 MA(11)0.0012980.0665380.0195080.9845 RESID(-1)-0.5727960.773672-0.7403590.4598 RESID(-2)-0.3622300.330054-1.0974880.2736 R-squared0.006703 Mean dependent var-6.23E-05 Adjusted R-squared-0.040802 S.D. dependent var0.058546 S.E. of regression0.059728 Akaike info criterion-2.749708 Sum squared resid0.820517 Schwarz criterion-2.576703 Log likelihood344.7147 F-statistic0.141104 Durbin-Watson stat1.998401 Prob(F-statistic)0.999515 The F-statistic indicates that there is no longer serial correlation in the data 20

21 Building The Model The resulting correlogram indicates that the spike at lag 10 may still be significant We will try incorporating an MA(10) into a new model 21

22 Building The Model Adding an MA(10) term made the coefficients more significant in general All of the coefficients are significant at a 5% level Dependent Variable: DLNPETROP Method: Least Squares Date: 05/29/10 Time: 23:56 Sample(adjusted): 1989:03 2009:04 Included observations: 242 after adjusting endpoints Convergence achieved after 6 iterations Backcast: 1988:04 1989:02 VariableCoefficientStd. Errort-StatisticProb. C0.0106510.0075661.4076880.1606 DD1-0.1222850.059446-2.0570940.0408 DD2-0.2743870.108398-2.5313040.0120 DD3-0.5533640.144543-3.8283590.0002 DD4-0.9230810.171140-5.3937150.0000 DD5-1.2167430.192998-6.3044290.0000 DSTEP-1.2748600.210151-6.0664060.0000 AR(1)0.5154310.0644877.9928470.0000 AR(2)-0.1915170.064415-2.9731520.0033 MA(10)0.1578070.0646702.4401970.0154 MA(11)0.1850340.0647152.8592260.0046 R-squared0.441873 Mean dependent var0.004986 Adjusted R-squared0.417712 S.D. dependent var0.077414 S.E. of regression0.059073 Akaike info criterion-2.775696 Sum squared resid0.806103 Schwarz criterion-2.617107 Log likelihood346.8592 F-statistic18.28842 Durbin-Watson stat1.977425 Prob(F-statistic)0.000000 Inverted AR Roots.26+.35i.26 -.35i Inverted MA Roots.86+.26i.86 -.26i.56+.70i.56 -.70i.08 -.88i.08+.88i -.41 -.75i -.41+.75i -.71 -.39i -.71+.39i -.78 22

23 Building The Model 23

24 Building The Model The correlogram has no significant spikes All remaining lags are even less significant than without the MA(10) term 24

25 Building The Model Once again, there is no serial correlation in this model Breusch-Godfrey Serial Correlation LM Test: F-statistic0.617507 Probability0.540183 Obs*R-squared1.298123 Probability0.522536 Test Equation: Dependent Variable: RESID Method: Least Squares Date: 05/29/10 Time: 23:59 VariableCoefficientStd. Errort-StatisticProb. C-8.47E-050.007580-0.0111800.9911 DD10.0003630.0596340.0060850.9952 DD20.0001840.1089730.0016880.9987 DD30.0018320.1455440.0125870.9900 DD40.0040070.1723130.0232530.9815 DD50.0044650.1940910.0230060.9817 DSTEP0.0048250.2111810.0228460.9818 AR(1)0.3438240.8019850.4287160.6685 AR(2)0.0642310.1625700.3950970.6931 MA(10)-0.0056560.064992-0.0870200.9307 MA(11)0.0014450.0648210.0222950.9822 RESID(-1)-0.3311810.802074-0.4129050.6801 RESID(-2)-0.2734380.337831-0.8093930.4191 R-squared0.005364 Mean dependent var2.72E-06 Adjusted R-squared-0.046757 S.D. dependent var0.057834 S.E. of regression0.059171 Akaike info criterion-2.764545 Sum squared resid0.801779 Schwarz criterion-2.577123 Log likelihood347.5100 F-statistic0.102918 Durbin-Watson stat1.998171 Prob(F-statistic)0.999950 25

26 Building The Model ARCH Test: F-statistic0.213734 Probability0.644277 Obs*R-squared0.215330 Probability0.642622 Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 05/30/10 Time: 00:00 Sample(adjusted): 1989:04 2009:04 Included observations: 241 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. C0.0032380.0004447.2997470.0000 RESID^2(-1)0.0298970.0646680.4623140.6443 R-squared0.000893 Mean dependent var0.003338 Adjusted R-squared-0.003287 S.D. dependent var0.006005 S.E. of regression0.006015 Akaike info criterion-7.380936 Sum squared resid0.008646 Schwarz criterion-7.352016 Log likelihood891.4027 F-statistic0.213734 Durbin-Watson stat2.001542 Prob(F-statistic)0.644277 We ran the ARCH test to check for heteroskedasticity The F-statistic indicates the data is homoskedastic 26

27 Building The Model The residual^2 correlogram shows slight structure in the earlier lags 27

28 Building The Model Dependent Variable: DLNPETROP Method: ML - ARCH Date: 05/30/10 Time: 00:03 Sample(adjusted): 1989:03 2009:04 Included observations: 242 after adjusting endpoints Convergence achieved after 97 iterations Backcast: 1988:04 1989:02 CoefficientStd. Errorz-StatisticProb. C0.0101610.0071761.4159260.1568 DD1-0.1211033.887236-0.0311540.9751 DD2-0.2733656.337982-0.0431310.9656 DD3-0.55006915.65666-0.0351330.9720 DD4-0.92031821.19041-0.0434310.9654 DD5-1.21250621.40283-0.0566520.9548 DSTEP-1.27124121.42383-0.0593380.9527 AR(1)0.5080520.0733226.9290410.0000 AR(2)-0.1962080.075139-2.6112540.0090 MA(10)0.1635580.0668762.4456910.0145 MA(11)0.1643190.0661872.4826570.0130 Variance Equation C0.0004320.0005550.7771280.4371 ARCH(1)0.0676950.0653801.0354130.3005 GARCH(1)0.8031140.2141453.7503370.0002 R-squared0.441553 Mean dependent var0.004986 Adjusted R-squared0.409711 S.D. dependent var0.077414 S.E. of regression0.059477 Akaike info criterion-2.771904 Sum squared resid0.806566 Schwarz criterion-2.570065 Log likelihood349.4004 F-statistic13.86729 Durbin-Watson stat1.961365 Prob(F-statistic)0.000000 Inverted AR Roots.25+.36i.25 -.36i Inverted MA Roots.86 -.26i.86+.26i.56 -.69i.56+.69i.07 -.87i.07+.87i -.41 -.74i -.41+.74i -.70 -.37i -.70+.37i -.75 To fix this slight structure, we tried adding ARCH/GARCH to the model However, most variable coefficients became insignificant We returned to an AR(1) AR(2) MA(10) MA(11) model 28

29 Forecasting The Last 12 Months 29

30 Forecasting The Last 12 Months 30

31 Forecasting Through 2010 31

32 Forecasting Through 2010 32

33 Conclusion There is an upward trend in the forecast suggesting an increase in future petroleum price Because of this, companies that heavily rely on oil may want to hedge against this Ex: Southwest Airlines (2007) 33

34 The End 34

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