Calculates the out-of-sample simulated values.

## Syntax

**SARIMA_SIM**(**X**, Order, Mean, **Sigma**, **d**, **Phi**, **Theta**, **Period**, **sd**, **sPhi**, **sTheta**, **T**, **Seed**)

**X**- is the univariate time series data (a 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)).
Value Order 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 ARMA model mean (i.e., mu). If missing, the 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 MA(q) (starting with the lowest lag).
**Period**- is 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 MA(q) (starting with the lowest lag).
**T**- is the simulation time/horizon (expressed in terms of steps beyond the end of the time series).
**Seed**- is an unsigned integer for setting up the random number generator(s).

## Remarks

- The underlying model is described here.
- The Log-Likelihood Function (LLF) is described here.
- SARIMA_SIM returns an array of one simulation path starting from the end of the input data.
- The input data argument (i.e., latest observations) is optional. If omitted, an array of zeroes is assumed.
- 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 residuals/innovations standard deviation (sigma) must be greater than zero.
- For the input argument - phi (parameters of the non-seasonal AR component):
- The input argument is optional and can be omitted, 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 values 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 omitted, 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 omitted, 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 values 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 omitted, 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 omitted, in which case d is assumed to be zero.
- The seasonal integration order - sD - is optional and can be omitted, in which case sD is assumed to be zero.
- The season length - s - is optional and can be omitted, in which case s is assumed to be zero (i.e. Plain ARIMA).
- The function was added in version 1.63 SHAMROCK.

## Files Examples

## Related Links

- Wikipedia - Likelihood function.
- Wikipedia - Likelihood principle.
- Wikipedia - Autoregressive moving average model.

## 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.

## Comments

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