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

## Syntax

**ARMA_GOF** (**[x]**, order, µ, **σ**, [φ], [θ], return)

**[X]**- Required. Is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
**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). **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.
**[φ]**- Optional. Are the parameters of the 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).
**Return**- Optional. Is an integer switch to select the goodness of fitness measure: (1 = LLF (default), 2 = AIC, 3 = BIC, 4 = HQC).
Value Return 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.
- The time series is homogeneous or equally spaced.
- The time series may include missing values (e.g., #N/A) at either end.
- The long-run mean can take any value or be omitted, in which a zero value is assumed.
- The residuals/innovations standard deviation (σ) must be greater than zero.
- For the input argument - ([φ]):
- The input argument is optional and can be omitted, so no 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 last value solely determines the order of the AR component model in the array with a numeric value (vs. missing or error).
- For the input argument - ([θ]):
- The input argument is optional and can be omitted, so 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 last value solely determines the order of the MA component model in the array with a numeric value (vs. missing or error).
- Missing parameter values reduce the model's actual number of overall parameters, thus improving the AIC, BIC, and HQC statistics.
- 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

- D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906.
- 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.
- Box, Jenkins and Reinsel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848.
- Walter Enders; Applied Econometric Time Series; Wiley; 4th edition(Nov 03, 2014), ISBN: 1118808568.

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