# SARIMA_GOF - Goodness of fit of a SARIMA Model

Computes the goodness of fit measure (e.g. log-likelihood function (LLF), AIC, etc.) of the estimated SARIMA model.

## Syntax

SARIMA_GOF(X, Order, mean, sigma, d, phi, theta, period, sd, sPhi, sTheta, Type)

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)).

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)

mean is the ARMA model mean (i.e. mu). If missing, mean is assumed to be 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).

Type is an integer switch to select the goodness of fitness measure: (1=LLF (default), 2=AIC, 3=BIC, 4=HQC).

Order Description
1 Log-Likelihood Function (LLF) (default)
2 Akaike Information Criterion (AIC)
3 Schwarz/Bayesian Information Criterion (SIC/BIC)
4 Hannan-Quinn information criterion (HQC)

## 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. The residuals/innovations standard deviation (i.e. $\sigma$) should be greater than zero.
6. The ARMA model has independent and normally distributed residuals with constant variance. The ARMA log-likelihood function becomes:

$$\ln L^* = -T\left(\ln 2\pi \hat \sigma^2+1\right)/2$$

Where:
• $\hat \sigma$ is the standard deviation of the residuals.
7. The value of the input argument - period - must be greater than one, or the function returns #VALUE!.
8. The value of the seasonal difference argument - sD - must be greater than one, or the function returns #VALUE!.
9. The maximum likelihood estimation (MLE) is a statistical method for fitting a model to the data and provides estimates for the model's parameters.
10. The long-run mean argument (mean) can take any value or be omitted, in which case a zero value is assumed.
11. The residuals/innovations standard deviation (sigma) must be greater than zero.
12. 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 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).
13. 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).
14. 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 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).
15. 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).
16. The non-seasonal integration order - d - is optional and can be omitted, in which case d is assumed to be zero.
17. The seasonal integration order - sD - is optional and can be omitted, in which case sD is assumed to be zero.
18. The season length - s - is optional and can be omitted, in which case s is assumed to be zero (i.e. plain ARIMA).
19. The function was added in version 1.63 SHAMROCK.

## References

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