# SARIMAX_FORE - Forecasting for SARIMAX Model

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

1. The underlying model is described here.
2. The Log-Likelihood Function (LLF) is described here.
3. Each column in the explanatory factors input matrix (i.e. X) corresponds to a separate variable.
4. Each row in the explanatory factors input matrix (i.e. X) corresponds to an observation.
5. Observations (i.e. rows) with missing values in X or Y are assumed missing.
6. 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.
7. The time series is homogeneous or equally spaced.
8. The time series may include missing values (e.g. #N/A) at either end.
9. The intercept or the regression constant term input argument is optional. If omitted, a zero value is assumed.
10. 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 SARIMA).
• The order of the parameters defines how the exogenous factor input arguments are passed.
• One or more parameters may have missing value or an error code(i.e. #NUM!, #VALUE!, etc.).
11. The long-run mean argument (mean) of the differenced regression residuals can take any value. If omitted, a zero value is assumed.
12. The residuals/innovations standard deviation (sigma) must greater than zero.
13. 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 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).
14. 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).
15. 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 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).
16. 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).
17. The non-seasonal integration order - d - is optional and can be omitted, in which case d is assumed zero.
18. The seasonal integration order - sD - is optional and can be omitted, in which case sD is assumed zero.
19. The season length - s - is optional and can be omitted, in which case s is assumed zero (i.e. Plain ARIMA).
20. The function SARIMAX_FORE is available starting with version 1.63 SHAMROCK.