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

## Syntax

**ARMAX_FORE**(**Y**, **X**, Order, **Beta**, **mean**, **sigma**, **phi**, **theta**, **T**, Type, alpha)

**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).
**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 The lower limit of the forecast confidence interval. 5 The upper limit of the forecast confidence interval. **alpha**- is the statistical significance level (i.e., alpha). If missing or omitted, an alpha value of 5% is assumed.

## Remarks

- The underlying model is described here.
- The Log-Likelihood Function (LLF) 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 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 (beta):
- The input argument is optional and can be omitted, in which case no regression component is included (i.e., plain ARMA).
- The order of the parameters defines how the exogenous factor input arguments are passed.
- One or more parameters may have missing values or error codes (i.e., #NUM!, #VALUE!, etc.).

- 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 missing values or error codes (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 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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