Calculates the out-of-sample simulated values.

## Syntax

**ARMAX_SIM**(**Y**, **X**, Order, **Beta**, **mean**, **sigma**, **phi**, **theta**, **T**, **Seed**)

**Y**- is the response or the dependent variable time series data array (a 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)).
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). **Beta**- are the coefficients array of the exogenous factors.
**mean**- is the ARMA long-run mean (i.e., mu).
**sigma**- is the standard deviation of the model's residuals.
**phi**- are the parameters of the AR(p) component model (starting with the lowest lag).
**theta**- are the parameters of the MA(q) component model (starting with the lowest lag).
**T**- is the simulation time/horizon (expressed in 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.
- ARMAX_SIM returns an array of one simulation path starting from the end of the input data.
- The response input data argument (i.e., latest observations) is optional. If omitted, an array of zeroes is assumed.
- The number of observations in the factors (exogenous variables) input data must be greater than or equal to the size of response input data plus horizon.
- The time series (response and factors) are homogeneous or equally spaced.
- The time series may include missing values (e.g., #N/A) at either end.
- The observation at any given time is examined using the response and factors value, so missing values (e.g., #N/A) in any input time series, deem the whole observation missing.
- The long-run 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:
- The input argument is optional and can be omitted, in which case no AR component is included.
- The order of the parameters starts with the lowest lag.
- One or more parameters may have a missing value or an error code (i.e., #NUM!, #VALUE!, etc.).
- The order of the 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:
- The input argument is optional and can be omitted, in which case no 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 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 function ARMAX_SIM is available starting with 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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