Comparison of the models

Concentration data

Its ACF

Its PACF. AR(2)?

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

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

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

Increments of the data

ACF of the increments

PACF of the increments

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

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

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

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

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

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

Coal Production data

Its ACF

Its PACF. AR(2)?

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

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

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

Profit Margin data

Its ACF

Its PACF. AR(1)?

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

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

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

Parts Availability data

Its increments

ACF for the increments. MA(1)?

PACF for the increments. AR(2)?

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

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

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

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

Treasury Bonds Yield data

Its increments

ACF for the increments. MA(1)??

PACF for the increments. AR(1)?

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

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

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

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

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

Crops Prices data

Take the Logarithm

Take the difference

Its ACF

MA(3) might work?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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