SARIMA_SIM - Simulated values of an SARIMA Model

Calculates the out-of-sample simulated values.

Syntax

SARIMA_SIM(X, Order, Mean, Sigma, d, Phi, Theta, Period, sd, sPhi, sTheta, T, Seed)

X

is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).

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).

Mean

is the ARMA model mean (i.e., mu). If missing, the 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 MA(q) (starting with the lowest lag).

Period

is 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 MA(q) (starting with the lowest lag).

T

is the simulation time/horizon (expressed in terms of steps beyond the end of the time series).

Seed

is an unsigned integer for setting up the random number generator(s).

Remarks

1. The underlying model is described here.
2. The Log-Likelihood Function (LLF) is described here.
3. SARIMA_SIM returns an array of one simulation path starting from the end of the input data.
4. The input data argument (i.e., latest observations) is optional. If omitted, an array of zeroes is assumed.
5. The time series is homogeneous or equally spaced.
6. The time series may include missing values (e.g., #N/A) at either end.
7. The long-run mean argument (mean) can take any value or be omitted, in which case a zero value is assumed.
8. The residuals/innovations standard deviation (sigma) must be greater than zero.
9. 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 values 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).
10. 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).
11. 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 values 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).
12. 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).
13. The non-seasonal integration order - d - is optional and can be omitted, in which case d is assumed to be zero.
14. The seasonal integration order - sD - is optional and can be omitted, in which case sD is assumed to be zero.
15. The season length - s - is optional and can be omitted, in which case s is assumed to be zero (i.e. Plain ARIMA).
16. 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.