# ARIMA_GOF - Goodness of fit of an ARIMA Model

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

## Syntax

ARIMA_GOF(X, Order, d, mean, sigma, phi, theta, 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)

d is the degree of the differencing (i.e. d).

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

sigma is the standard deviation value of the model's residuals/innovations.

phi are the parameters of the AR(p) component model (starting with the lowest lag).

theta are the parameters of the MA(q) component model (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 maximum likelihood estimation (MLE) is a statistical method for fitting a model to the data and providing estimates for the model's parameters.
8. The integration order argument (d) must be a positive integer.
9. The long-run mean can take any value or may be omitted, in which case a zero value is assumed.
10. The residuals/innovations standard deviation (sigma) must be greater than zero.
11. For the input argument (phi):
• The input argument is optional and can be omitted, in which case no AR component is included.
• The order of the parameters starts with the lowest lag.
• One or more parameters can be missing or an error code (i.e. #NUM!, #VALUE!, etc.).
• The order of the 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 (theta):
• The input argument is optional and can be omitted, in which case no 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 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 function was added in version 1.63 SHAMROCK.