Calculates the out-of-sample conditional mean and error forecast

## Syntax

**AIRLINE_FORE**(

**X**,

**Order**,

**mean**,

**sigma**,

**s**,

**theta**,

**theta2**,

**T**,

**Type**,

**alpha**)

**X** is the univariate time series data (one dimensional array of cells (e.g. rows or columns)).

**Order** is the time order in the data series (i.e. the first data point's corresponding date (earliest date=1 (default), latest date=0)).

Order | Description |
---|---|

1 | ascending (the first data point corresponds to the earliest date) (default) |

0 | descending (the first data point corresponds to the latest date) |

**mean** is the model mean (i.e. mu).

**sigma** is the standard deviation of the model's residuals/innovations.

**s** is the length of seasonality (expressed in terms of lags, where s > 1).

**theta** is the coefficient of non-seasonal MA component (see model description).

**theta2** is the coefficient of seasonal MA component (see model description).

**T** is the forecast time/horizon (expressed in terms of steps beyond the end of the time series).

**Type** is an integer switch to select the forecast output type: (1=mean (default), 2=Std. Error, 3=Term Struct, 4=LL, 5=UL)

Order | Description |
---|---|

1 | Mean forecast value (default) |

2 | Forecast standard error (aka local volatility) |

3 | Volatility term structure |

4 | Lower limit of the forecast confidence interval. |

5 | Upper limit of the forecast confidence interval. |

**alpha** is the statistical significance level. If missing, a default of 5% is assumed.

## Remarks

- The underlying model is described here.
- The time series is homogeneous or equally spaced.
- The time series may include missing values (e.g. #N/A) at either end.
- The long-run mean argument (mean) can take any value or be omitted, in which case a zero value is assumed.
- The value of the residuals/innovations standard deviation (sigma) must be positive.
- The season length must be greater than one.
- The input argument for the non-seasonal MA parameter - theta - is optional and can be omitted, in which case no non-seasonal MA component is included.
- The input argument for the seasonal MA parameter - theta2 - is optional and can be omitted, in which case no seasonal MA component is included.

## Files Examples

## References

- Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
- Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740

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