Select the method or formula of your choice.

The coefficients are estimated using an iterative algorithm that calculates least squares estimates. At each iteration, the back forecasts are computed and SSE is calculated. For more details, see Box and Jenkins^{1}.

The ARIMA algorithm is based on the fitting routine in the TSERIES package written by Professor William Q. Meeker, Jr., of Iowa State University^{2}. We are grateful to Professor Meeker for his help in the adaptation of his routine to Minitab.

Back forecasts are calculated using the specified model and the current iteration's parameter estimates. For more details, see Cryer^{3}.

Term | Description |
---|---|

n | total number of observations |

residuals using that iteration's parameter estimates, including back forecasts |

Term | Description |
---|---|

n | total number of observations |

a _{t} | residuals using the final parameter estimates, excluding back forecasts |

For a model with a constant term:

(*n* – *d*) – *p* – *q* – 1

For a model without a constant term:

(*n* – *d*) – *p* – *q*

Term | Description |
---|---|

n | total number of observations |

d | number of differences |

p | number of autoregressive parameters included in the model |

q | number of moving average parameters included in the model |

SS / DF

Term | Description |
---|---|

n | total number of observations |

d | number of differences |

K | 12, 24, 36, 48 |

k | lag |

autocorrelation of the residuals for the k ^{th} lag |

For a model with a constant term:

*K* – *p* – *q* – 1

For a model without a constant term:

*K* – *p* – *q*

Term | Description |
---|---|

K | 12, 24, 36, 48 |

p | number of autoregressive parameters included in the model |

q | number of moving average parameters included in the model |

P(*X* < *χ* ^{2})

Term | Description |
---|---|

X | distributed as χ ^{2} _{(DF)} |

Forecasts are calculated recursively, based on the model and the parameter estimates. For example, if an ARIMA model is fit with 1 autoregressive term (AR(1)) and one seasonal differencing term with a seasonal period of 12, this model is fit:

*Y _{t} * –

To estimate , the first forecast, where *k* is the origin, find:

Then, you find , in the same manner, and so on.

Term | Description |
---|---|

Y_{t} | actual value at time t |

Φ | autoregressive term |

estimated autoregressive term | |

γ | constant term |

estimated constant term |