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Comparison of the models

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Concentration data

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Its ACF

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Its PACF. AR(2)?

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AR(1)? m = 17.06, p-value a 1 = , p-value Portmanteau test 47.44, p-value 0.003

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AR(2)? m = 17.06, p-value a 1 = , p-value a 2 = , p-value Portmanteau test 26.97, p-value

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AR(3)? m = 17.06, p-value a 1 = , p-value a 2 = , p-value a 3 = , p-value Portmanteau test 26.14, p-value

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Increments of the data

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ACF of the increments

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PACF of the increments

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MA(1)? b 1 = 0.701, p-value Portmanteau test 29.14, p-value

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MA(2)? b 1 = , p-value b 2 = , p-value Portmanteau test 24.77, p-value

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AR(4)? a 1 = , p-value a 2 = , p-value a 3 = , p-value a 4 = , p-value Portmanteau test 25.5, p-value

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AR(5)? a 1 = , p-value a 2 = , p-value a 3 = , p-value a 4 = , p-value a 5 = , p-value Portmanteau test 25.4, p-value

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AR(6)? a 1 = , p-value a 2 = , p-value a 3 = , p-value a 4 = , p-value a 5 = , p-value a 6 = , p-value Portmanteau test 14.81, p-value

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AR(7)? a 1 = , p-value a 2 = , p-value a 3 = , p-value a 4 = , p-value a 5 = , p-value a 6 = , p-value a 7 = , p-value Portmanteau test 14.83, p-value

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Coal Production data

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Its ACF

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Its PACF. AR(2)?

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AR(1)? m = 3.772, p-value a 1 = , p-value Portmanteau test 21.44, p-value

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AR(2)? m = 3.802, p-value a 1 = , p-value a 2 = , p-value Portmanteau test 12.03, p-value 0.97

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AR(3)? m = 3.809, p-value a 1 = , p-value a 2 = , p-value a 3 = , p-value Portmanteau test 10.61, p-value

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Profit Margin data

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Its ACF

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Its PACF. AR(1)?

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AR(1)? m = 4.699, p-value a 1 = 0.876, p-value Portmanteau test 22.34, p-value Still, ρ(4) is out of range

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AR(2)? m = 4.716, p-value a 1 = 1.026, p-value a 2 = , p-value Portmanteau test 21.98, p-value

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ARMA(1,1)? m = 4.714, p-value a 1 = , p-value b 1 = , p-value Portmanteau test 19.06, p-value

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Parts Availability data

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Its increments

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ACF for the increments. MA(1)?

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PACF for the increments. AR(2)?

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AR(1)? a 1 = , p-value Portmanteau test 22.4, p-value ρ(2) is way out of range though

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AR(2)? a 1 = , p-value a 2 = , p-value Portmanteau test 14.5, p-value

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AR(3)? a 1 = , p-value a 2 = , p-value a 3 = , p-value Portmanteau test 12.24, p-value

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MA(1)? b 1 = , p-value Portmanteau test 12.23, p-value

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Treasury Bonds Yield data

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Its increments

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ACF for the increments. MA(1)??

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PACF for the increments. AR(1)?

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AR(1)? a 1 = , p-value Portmanteau test 29.02, p-value

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AR(2)? a 1 = , p-value a 2 = , p-value Portmanteau test 28.96, p-value

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MA(1)? b 1 = , p-value Portmanteau test 36.57, p-value

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MA(2)? b 1 = , p-value b 2 = , p-value Portmanteau test 30.89, p-value

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ARMA(1,1)? a 1 = , p-value b 1 = , p-value Portmanteau test 28.94, p-value

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Crops Prices data

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Take the Logarithm

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Take the difference

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Its ACF

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MA(3) might work?

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MA(3)? b 1 = , p-value b 2 = , p-value b 3 = , p-value Portmanteau test 27.11, p-value

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Residual ACF for MA(3) model. Looks good, but the estimate for b(3) has a big p-value 0.157

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MA(2)? b 1 = , p-value b 2 = , p-value Portmanteau test 54.11, p-value

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Residual ACF for MA(2) model, Portmanteau test has p-value

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Switch to AR models. Here is PACF of the data.

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AR(8)? Or AR(4)? 8 th value is almost 4 standard deviations

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AR(4)? a 1 = , p-value a 2 = , p-value a 3 = , p-value a 4 = , p-value Portmanteau test 39.62, p-value

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Residual ACF for AR(4). Portmanteau test is a No (p-value 0.008)

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AR(8)? a 1 = , p-value a 2 = , p-value a 3 = , p-value a 4 = , p-value a 5 = , p-value a 6 = , p-value a 7 = , p-value a 8 = , p-value Portmanteau test 21.27, p-value

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Residual ACF for AR(8). Portmanteau test is a Yes (p-value 0.215). Finally?

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Since 8 is sort of too many, let’s try mixed models

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ARMA(2,1)? a 1 = , p-value a 2 = , p-value b 1 = , p-value Portmanteau test 28.44, p-value

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Residuals for ARMA(2,1). Portmanteau test is a Yes (p-value 0.161)

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Best AIC score = ARMA(8,1) a 1 = , p-value a 2 = , p-value a 3 = , p-value a 4 = , p-value a 5 = , p-value a 6 = , p-value a 7 = , p-value a 8 = , p-value b 1 = , p-value Portmanteau test 12.1, p-value

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Residual ACF for ARMA(8,1). Note that the first 5 values are practically zeroes, one of the symptoms of over- parametrization

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