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, 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).

D

Required. Is the integration order.

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

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 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.
6. The maximum likelihood estimation (MLE) is a statistical method for fitting a model to the data and providing estimates for the model's parameters.
7. The integration order argument (d) must be a positive integer.
8. The long-run mean can take any value or may be omitted, in which case a zero value is assumed.
9. The residuals/innovations standard deviation (σ) must be greater than zero.
10. For the input argument ([φ]):
• 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).
11. For the input argument ([θ]):
• 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).
12. The function was added in version 1.63 SHAMROCK.

Files Examples

Related Links

• Wikipedia - Likelihood function.

References

• 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.