Retail Gasoline Prices Time Series Analysis

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Retail Gasoline PricesTime Series Analysis

• Joseph B. Rickert
• Statistics, CSUEB
• June 2 ,2006

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West Coast Regular Conventional Retail Gasoline
Prices

• Each point is the average of retail gas prices
  for one week over the West Coast of the US
• 687 weeks of data
• Monday May 11,1992 to Monday
  August 15, 2005
• Data brovided by Economagic.com
• http//www.economagic.com/em-cgi/data.exe/doewkly/
  day-mg_rco_p5

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Part 1

• Fit a good ARIMA model
• to the entire series

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Some Notation

• Backshift operator Bxt xt-1
• Bkxt xt-k
• Differencing ?xt xt xt-1 (1 B)xt
• ?dxt (1 B)dxt
• AR(p) (1-f1B-f2B2- . . .fpBp)xt wt
• MA(q) xt (1q1Bq2B2 . . .qqBq)wt
• ARMA(p,q) f(B)xt q(B)wt
• ARIMA(p,d,q) f(B)(1-B)dxt q(B)wt

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Entire Series

• Candidate Models
• Model A ARIMA(0,1,3)(0,1,1)52
• Model B ARIMA (0,1,3)(2,1,0)52
• Model C ARIMA (2,1,0)(0,1,1)52
• (1 f1B f2B2)(1 B52)(1 B)yt (1Q1B52)wt

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Model A

• ARIMA (0,1,3)(0,1,1)52
• Coefficients
• ma1 ma2 ma3 sma1
• 0.5689 0.3369 0.0938 -0.8981
• s.e. 0.0311 0.0273 0.0295 0.0563
• p-value 0 0 0 0
• sigma2 estimated as 0.0001045
• log likelihood 1964.83,
• aic -3919.66

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Model B

• ARIMA (0,1,3)(2,1,0)52
• Coefficients
• ma1 ma2 ma3 sar1 sar2
• 0.5663 0.3254 0.1081 -0.7179 -0.4601
• s.e. 0.0298 0.0269 0.0285 0.0368 0.0375
• p-value 0 0 0 0 0
• sigma2 estimated as 0.0001174
• log likelihood 1949.39, aic
  -3886.77

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Model C

• ARIMA (2,1,0)(0,1,1)52
• Coefficients
• ar1 ar2 sma1
• 0.7762 -0.0789 -0.8438
• s.e. 0.0396 0.0396 0.0011
• p-value 0 0 0
• sigma2 estimated as 0.0001027
• log likelihood 1979.86, aic
  -3951.72

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Part 2

• Fit an ARIMA model to the second part of the
  series and forecast

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Partial Series

• Candidate Models
• Model E ARIMA (0,1,3)(0,1,1)52
• Model F ARIMA (2,1,0)(0,1,1)52

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Model E

• ARIMA (0,1,3)(0,1,1)52
• Coefficients
• ma1 ma2 ma3 sma1
• 0.6477 0.2923 0.0597 -0.8580
• s.e.
• 0.0491 0.0474 0.0478 0.1611
• p-value 0 0 0 0
• sigma2 estimated as 0.0001697
• log likelihood 793.08, aic
  -1576.15

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Model F

• ARIMA(2,1,0)(0,1,1)52
• Coefficients
• ar1 ar2 sma1
• 0.9386 -0.2071 -0.8079
• s.e. 0.0615 0.0639 0.1220
• p-value 0 0 0
• sigma2 estimated as 0.0001564
• log likelihood 809.19, aic
  -1610.39

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Model F Forecast
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Summary and Conclusions

• Models based on second part of the series
  provided a superior fit as measured by the amount
  of unexplained structure in the residuals.
• We conclude that the model
• (1 f1B f2B2)(1 B52)(1 B)yt (1
  Q1B52)wt
• where yt log(Gas Price) and wt is a white
  noise process provides reasonably good fit to the
  series and is useful for forecasting.