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

## Syntax

**SARIMAX_FORE** (**[y]**, **[x]**, order, [β], µ, **σ**, **d**, [φ], [θ], s, sd, [sφ], [sθ], **t**, return, α)

**[Y]**- Required. Is the response or the dependent variable time series data array (a one-dimensional array of cells (e.g., rows or columns)).
**[X]**- Required. Is the independent variables (exogenous factors) time series data matrix, so each column represents one variable.
**Order**- Optional. 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). **[β]**- Optional. Is the coefficients array of the exogenous factors.
**µ**- Optional. Is the ARMA model long-run mean (i.e., mu). If missing, the process mean is assumed to be zero.
**σ**- Required. Is the standard deviation value of the model's residuals/innovations.
**D**- Required. Is the non-seasonal integration order.
**[φ]**- Optional. Are the parameters of the non-seasonal AR(p) component model: [φ1, φ2 … φp] (starting with the lowest lag).
**[θ]**- Optional. Are the parameters of the MA(q) component model: [θ1, θ2 … θq] (starting with the lowest lag).
**S**- Optional. Is the number of observations per period (e.g., 12 = Annual, 4 = Quarter).
**sD**- Optional. Is the seasonal integration order.
**[sφ]**- Optional. Are the parameters of the seasonal AR(P) component model: [sφ1, sφ2 … sφpp] (starting with the lowest lag).
**[sθ]**- Optional. Are the parameters of the seasonal MA(Q) component model: [sθ1, sθ2 … sθqq] (starting with the lowest lag).
**T**- Required. Is the forecast time/horizon (expressed in steps beyond the end of the time series).
**Return**- Optional. Is an integer switch to select the forecast output type: (1 = mean (default), 2 = Std. Error, 3 = Term Struct, 4 = LL, 5 = UL).
Value Return 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. **α**- Optional. 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.
- 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 to be missing.
- The number of rows of the explanatory variable (X) must equal the number of rows of the response variable (Y).
- The intercept or the regression constant term input argument is optional. If omitted, a zero value is assumed.
- For the input argument - ([β]):
- The input argument is optional and can be omitted, in which case no regression component is included (i.e., plain SARIMA).
- The order of the parameters defines how the exogenous factor input arguments are passed.
- One or more parameters may have a missing value or an error code (i.e., #NUM!, #VALUE!, etc.).
- The long-run mean argument (µ) can take any value or be omitted, in which case a zero value is assumed.
- The residuals/innovations standard deviation - (σ) - must be greater than zero.
- For the input argument - ([φ]) (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 error codes (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 - ([θ]) (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 - ([sφ]) (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 error codes (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 - ([sθ]) (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 SARIMAX_FORE is available starting with version 1.63 SHAMROCK.

## Files Examples

## Related Links

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

## References

- James Douglas Hamilton; Time Series Analysis, Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896.
- Tsay, Ruey S.; Analysis of Financial Time Series, John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740.

## Comments

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