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Zhen Tian
Jeff Lee
Visut Hemithi
Huan Zhang
Diana Aguilar
Yuli Yan
A Deep Analysis of A Random WalkA Deep Analysis of A Random Walk
Identification
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94 96 98 00 02 04 06 08
PRICE
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Series: PRICESample 4/05/1993 5/25/2009Observations 821
Mean 169.6928Median 144.0000Maximum 416.5000Minimum 94.90000Std. Dev. 72.61291Skewness 1.377958Kurtosis 4.121595
Jarque-Bera 302.8479Probability 0.000000
Unit Root
Null Hypothesis: PRICE has a unit root
Exogenous: Constant
Lag Length: 3 (Fixed)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -2.431974 0.1332
Test critical values: 1% level -3.438129
5% level -2.864863
10% level -2.568594
Pre-Whitening
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DLNPRICE
Log Transformation
Trend in Var.
Difference
Trend in Mean
Pre-Whitening
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-0.10 -0.05 -0.00 0.05 0.10 0.15
Series: DLNPRICESample 4/05/1993 5/25/2009Observations 820
Mean 0.000574Median -0.000909Maximum 0.161180Minimum -0.098922Std. Dev. 0.019665Skewness 0.259126Kurtosis 11.59559
Jarque-Bera 2533.555Probability 0.000000
Unit Root
Null Hypothesis: DLNPRICE has a unit root
Exogenous: Constant
Lag Length: 4 (Fixed)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -9.819079 0.0000
Test critical values: 1% level -3.438149
5% level -2.864872
10% level -2.568599
Dependent Variable: DLNPRICEMethod: Least SquaresSample (adjusted): 5/17/1993 12/22/2008Included observations: 815 after adjustmentsConvergence achieved after 6 iterationsBackcast: 1/11/1993 5/10/1993
Variable Coefficient Std. Error t-Statistic Prob.
C 0.000421 0.001769 0.238003 0.8119AR(1) 0.503803 0.034771 14.48920 0.0000AR(2) 0.104044 0.039420 2.639331 0.0085AR(3) 0.143444 0.036466 3.933679 0.0001AR(5) -0.124364 0.032993 -3.769439 0.0002MA(8) 0.103332 0.037023 2.791007 0.0054
MA(18) 0.118155 0.036499 3.237194 0.0013
R-squared 0.389743 Mean dependent var 0.000545Adjusted R-squared 0.385211 S.D. dependent var 0.019718S.E. of regression 0.015461 Akaike info criterion -5.492468Sum squared resid 0.193141 Schwarz criterion -5.452073Log likelihood 2245.181 F-statistic 86.00538Durbin-Watson stat 2.000720 Prob(F-statistic) 0.000000
Model Validation-1
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Residual Actual Fitted
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Series: ResidualsSample 5/17/1993 12/22/2008Observations 815
Mean 4.29e-06Median -0.000768Maximum 0.145377Minimum -0.113527Std. Dev. 0.015404Skewness 0.656675Kurtosis 17.17994
Jarque-Bera 6886.600Probability 0.000000
Model Validation-2
Breusch-Godfrey Serial Correlation LM Test:
F-statistic 0.105569 Prob. F(2,806) 0.899825
Obs*R-squared 0.213376 Prob. Chi-Square(2) 0.898806.000
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RESIDSQ
ARCH GARCH (2)Dependent Variable: DLNPRICEMethod: ML - ARCH (Marquardt) - Normal distribution
MA backcast: 1/11/1993 5/10/1993, Variance backcast: ONGARCH = C(8) + C(9)*RESID(-1)^2 + C(10)*GARCH(-1)
Coefficient Std. Error z-Statistic Prob.
C 0.000121 0.001162 0.104385 0.9169
AR(1) 0.543373 0.037777 14.38374 0.0000
AR(2) 0.188020 0.038719 4.855993 0.0000
AR(4) -0.095035 0.036572 -2.598555 0.0094
AR(5) 0.006074 0.030310 0.200393 0.8412
MA(9) -0.034453 0.033775 -1.020079 0.3077
MA(18) 0.105656 0.028083 3.762291 0.0002
Variance Equation
C 7.30E-06 1.18E-06 6.189402 0.0000
RESID(-1)^2 0.288911 0.040133 7.198888 0.0000
GARCH(-1) 0.724386 0.031554 22.95716 0.0000
ARCH GARCH (3)Dependent Variable: DLNPRICEMethod: ML - ARCH (Marquardt) - Normal distribution
MA backcast: 1/11/1993 5/10/1993, Variance backcast: ONGARCH = C(8) + C(9)*RESID(-1)^2 + C(10)*GARCH(-1)
Coefficient Std. Error z-Statistic Prob.
C 0.000359 0.001198 0.299488 0.7646
AR(1) 0.539575 0.037382 14.43418 0.0000
AR(2) 0.198400 0.038456 5.159169 0.0000
AR(4) -0.094842 0.030904 -3.068937 0.0021
MA(18) 0.106256 0.027612 3.848250 0.0001
Variance Equation
C 7.55E-06 1.21E-06 6.252662 0.0000
RESID(-1)^2 0.294422 0.040314 7.303273 0.0000
GARCH(-1) 0.718210 0.031285 22.95731 0.0000
Histogram
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Series: Standardized ResidualsSample 5/10/1993 12/22/2008Observations 816
Mean 0.048932Median -0.086097Maximum 5.942007Minimum -2.775767Std. Dev. 0.999207Skewness 1.024413Kurtosis 6.494198
Jarque-Bera 557.8417Probability 0.000000
ARCH Test
ARCH Test:
F-statistic 0.190541 Prob. F(1,812) 0.662583
Obs*R-squared 0.190965 Prob. Chi-Square(1) 0.662115
Forecast
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DLNPRICEF1DLNPRICEF1+2*SEFDLNPRICEF1-2*SEFDLNPRICE
Comparison
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08M01 08M04 08M07 08M10 09M01 09M04
PRICEALOWER
UPPERPRICEF
A Little Bit Further
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DLNPRICEF1+2*SEFDLNPRICEF1-2*SEFDLNPRICEF1DLNPRICE
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PRICEFPRICE
UPPERLOWER
Story Behind the Scene
To Investigate the Sources of Shock Geopolitical Events (War & Disasters)
GDP / Mean Personal Income
Vehicle Sales (SUV Sales)
China Petro Consumption
Speculation (Future Contract Price)
Key Bibliography “Causes and Consequences of the Oil Shock of 2007-08”
James D. Hamilton, UCSD (2009)