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

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

Original Data 3

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

Original Data ADF Test Statistic % Critical Value* % Critical Value % Critical Value *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: :04 Included observations: 256 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. PETROP(-1) C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) The ADF test statistic confirms the notion of non- stationary data 6

Application of Step Function and Dummy Variables 7

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Dependent Variable: PETROP Method: Least Squares Date: 05/28/10 Time: 11:45 Sample: 1988: :04 Included observations: 257 VariableCoefficientStd. Errort-StatisticProb. C D D D D D STEP R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Regression of petroleum price against dummy variables and step function 9

Application of Step Function and Dummy Variables 10

Application of Step Function and Dummy Variables 11

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

Application of Step Function and Dummy Variables Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID Method: Least Squares Date: 05/28/10 Time: 11:49 VariableCoefficientStd. Errort-StatisticProb. C D D D D D STEP RESID(-1) RESID(-2) R-squared Mean dependent var5.73E-14 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) The F-statistic indicates that there is still serial correlation in this revised data Further steps must be taken 13

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: :04 Included observations: 244 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. C DD DD DD DD DD DSTEP R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) 14

Logarithm Transformation and First Differenced Data 15

Logarithm Transformation and First Differenced Data 16

Logarithm Transformation and First Differenced Data ADF Test Statistic % Critical Value* % Critical Value % Critical Value *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: :04 Included observations: 251 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. DLNPETROP(-1) D(DLNPETROP(-1)) D(DLNPETROP(-2)) D(DLNPETROP(-3)) D(DLNPETROP(-4)) C R-squared Mean dependent var-6.30E-05 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) The ADF test statistic indicates that the data is now stationary 17

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

Building The Model Dependent Variable: DLNPETROP Method: Least Squares Date: 05/29/10 Time: 23:26 Sample(adjusted): 1989: :04 Included observations: 242 after adjusting endpoints Convergence achieved after 6 iterations Backcast: 1988: :02 VariableCoefficientStd. Errort-StatisticProb. C DD DD DD DD DD DSTEP AR(1) AR(2) MA(11) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Inverted AR Roots i i Inverted MA Roots i i i i i i i i i i -.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

Building The Model Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID Method: Least Squares Date: 05/29/10 Time: 23:30 VariableCoefficientStd. Errort-StatisticProb. C-8.72E DD DD DD DD DD DSTEP AR(1) AR(2) MA(11) RESID(-1) RESID(-2) R-squared Mean dependent var-6.23E-05 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) The F-statistic indicates that there is no longer serial correlation in the data 20

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

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: :04 Included observations: 242 after adjusting endpoints Convergence achieved after 6 iterations Backcast: 1988: :02 VariableCoefficientStd. Errort-StatisticProb. C DD DD DD DD DD DSTEP AR(1) AR(2) MA(10) MA(11) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Inverted AR Roots i i Inverted MA Roots i i i i i i i i i i

Building The Model 23

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

Building The Model Once again, there is no serial correlation in this model Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID Method: Least Squares Date: 05/29/10 Time: 23:59 VariableCoefficientStd. Errort-StatisticProb. C-8.47E DD DD DD DD DD DSTEP AR(1) AR(2) MA(10) MA(11) RESID(-1) RESID(-2) R-squared Mean dependent var2.72E-06 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

Building The Model ARCH Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 05/30/10 Time: 00:00 Sample(adjusted): 1989: :04 Included observations: 241 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. C RESID^2(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) We ran the ARCH test to check for heteroskedasticity The F-statistic indicates the data is homoskedastic 26

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

Building The Model Dependent Variable: DLNPETROP Method: ML - ARCH Date: 05/30/10 Time: 00:03 Sample(adjusted): 1989: :04 Included observations: 242 after adjusting endpoints Convergence achieved after 97 iterations Backcast: 1988: :02 CoefficientStd. Errorz-StatisticProb. C DD DD DD DD DD DSTEP AR(1) AR(2) MA(10) MA(11) Variance Equation C ARCH(1) GARCH(1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Inverted AR Roots i i Inverted MA Roots i i i i i i i i i i -.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

Forecasting The Last 12 Months 29

Forecasting The Last 12 Months 30

Forecasting Through

Forecasting Through

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

The End 34