Calculates the out-of-sample conditional forecast (i.e. mean, error and confidence interval)

## Syntax

**SARIMAX_FORE**(

**Y**,

**X**,

**Order**,

**Beta**,

**mean**,

**sigma**,

**d**,

**phi**,

**theta**,

**period**,

**sd**,

**sPhi**,

**sTheta**,

**T**,

**Type**,

**alpha**)

**Y** is the response or the dependent variable time series data array (one dimensional array of cells (e.g. rows or columns)).

**X** is the independent variables (exogenous factors) time series data matrix, such that each column represents one variable.

**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) |

**Beta** are the coefficients array of the exogenous factors.

**mean** is the SARIMA model mean (i.e. long-run of the differenced time series). If missing, mean is assumed zero.

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

**d** is the non-seasonal difference order.

**phi** are the parameters of the non-seasonal AR model component AR(p) (starting with the lowest lag).

**theta** are the parameters of the non-seasonal MA model component (i.e. MA(q)) (starting with the lowest lag).

**period** is the the number of observations per one period (e.g. 12=Annual, 4=Quarter).

**sd** is the seasonal difference order.

**sPhi** are the parameters of the seasonal AR model component AR(p) (starting with the lowest lag).

**sTheta** are the parameters of the seasonal MA model component (i.e. MA(q)) (starting with the lowest lag).

**T** is the forecast time/horizon (expressed in terms of steps beyond 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 value of 5% is assumed.

## Remarks

- The underlying model is described here.
- The Log-Likelihood Function (LLF) is described here.
- Each column in the explanatory factors input matrix (i.e. X) corresponds to a separate variable.
- Each row in the explanatory factors input matrix (i.e. X) corresponds to an observation.
- Observations (i.e. rows) with missing values in X or Y are assumed missing.
- The number of rows of the explanatory variable (X) must be greater or equal to the number of rows of the response variable (Y) plus forecast horizon.
- The time series is homogeneous or equally spaced.
- The time series may include missing values (e.g. #N/A) at either end.
- The intercept or the regression constant term input argument is optional. If omitted, a zero value is assumed.
- For the input argument - Beta:
- The input argument is optional and can be ommitted, in which case no regression component is included (i.e. plain SARIMA).
- The order of the parameters defines how the exogneous factor input arguments are passed.
- One or more parameters may have missing value or an error code(i.e. #NUM!, #VALUE!, etc.).

- The long-run mean argumen (mean) of the differenced regression residuals can take any value. If ommitted, a zero value is assumed.
- The residuals/innovations standard deviation (sigma) must greater than zero.
- For the input argument - phi (parameters of the non-seasonal AR component):
- The input argument is optional and can be ommitted, in which case no non-seasonal AR component is included.
- The order of the parameters starts with the lowest lag
- One or more parameters may have missing value or an error code(i.e. #NUM!, #VALUE!, etc.).
- The order of the non-seasonal AR component model is solely determined by the order of the last value in the array with a numeric value (vs. missing, or error).

- For the input argument - theta (parameters of the non-seasonal MA component):
- The input argument is optional and can be ommitted, in which case no non-seasonal MA component is included.
- The order of the parameters starts with the lowest lag
- One or more values in the input argument can be missing or an error code(i.e. #NUM!, #VALUE!, etc.).
- The order of the non-seasonal MA component model is solely determined by the order of the last value in the array with a numeric value (vs. missing, or error).

- For the input argument - sPhi (parameters of the seasonal AR component):
- The input argument is optional and can be ommitted, in which case no seasonal AR component is included.
- The order of the parameters starts with the lowest lag
- One or more parameters may have missing value or an error code(i.e. #NUM!, #VALUE!, etc.).
- The order of the seasonal AR component model is solely determined by the order of the last value in the array with a numeric value (vs. missing, or error).

- For the input argument - sTheta (parameters of the seasonal MA component):
- The input argument is optional and can be ommitted, in which case no seasonal MA component is included.
- The order of the parameters starts with the lowest lag
- One or more values in the input argument can be missing or an error code(i.e. #NUM!, #VALUE!, etc.).
- The order of the seasonal MA component model is solely determined by the order of the last value in the array with a numeric value (vs. missing, or error).

- The non-seasonal integration order - d - is optional and can be ommitted, in which case d is assumed zero.
- The seasonal integration order - sD - is optional and can be ommitted, in which case sD is assumed zero.
- The season length - s - is optional and can be ommitted, in which case s is assumed zero (i.e. Plain ARIMA).
- The function SARIMAX_FORE is available starting with version 1.63 SHAMROCK.

## 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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