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

## Syntax

**ARMAX_GOF**(**Y**, **X**, Order, **Beta**, **mean**, **sigma**, **phi**, **theta**, Type)

**Y**- is the response or the dependent variable time series data array (a 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)).
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). **Beta**- are the coefficients array of the exogenous factors.
**mean**- is the ARMA long-run mean (i.e., mu).
**sigma**- is the standard deviation of the model's residuals.
**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

- The underlying model is described here.
- The Log-Likelihood Function (LLF) is described here.
- Each column in the explanatory factors input matrix (i.e., X) corresponds to a separate variable.
- Each row in the explanatory factors input matrix (i.e., X) corresponds to an observation.
- Observations (i.e., rows) with missing values in X or Y are assumed missing.
- The number of rows of the explanatory variable (X) must be equal to the number of rows of the response variable (Y).
- The time series is homogeneous or equally spaced.
- The time series may include missing values (e.g., #N/A) at either end.
- The residuals/innovations standard deviation (i.e., $\sigma$) should be greater than zero.
- 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.

- The maximum likelihood estimation (MLE) is a statistical method for fitting a model to the data and provides estimates for the model's parameters.
- The long-run mean can take any value or be omitted, in which case a zero value is assumed.
- The residuals/innovations standard deviation (sigma) must be greater than zero.
- 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 ARMA).
- The order of the parameters defines how the exogenous factor input arguments are passed.
- One or more parameters may have missing values or error codes (i.e., #NUM!, #VALUE!, etc.).

- 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 may have missing values or error codes (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).

- 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).

- 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.

## Comments

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