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, [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

1. The underlying model is described here.
2. The Log Likelihood Function (LLF) is described here.
3. The time series is homogeneous or equally spaced.
4. The time series may include missing values (e.g., #N/A) at either end.
5. Each column in the explanatory factors input matrix (i.e., X) corresponds to a separate variable.
6. Each row in the explanatory factors input matrix (i.e., X) corresponds to an observation.
7. Observations (i.e., rows) with missing values in X or Y are assumed to be missing.
8. The number of rows of the explanatory variable (X) must equal the number of rows of the response variable (Y).
9. The intercept or the regression constant term input argument is optional. If omitted, a zero value is assumed.
10. 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.).
11. The long-run mean argument (µ) can take any value or be omitted, in which case a zero value is assumed.
12. The residuals/innovations standard deviation - (σ) - must be greater than zero.
13. 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).
14. 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).
15. 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).
16. 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).
17. The non-seasonal integration order - (d) - is optional and can be omitted, in which case d is assumed to be zero.
18. The seasonal integration order - (sD) - is optional and can be omitted, in which case sD is assumed to be zero.
19. The season length - (s) - is optional and can be omitted, in which case s is assumed to be zero (i.e., plain ARIMA).
20. 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.