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, µ, σ, d, [φ], [θ], s, sd, [sφ], [sθ], 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.

D

Required. Is the non-seasonal integration order.

[φ]

Optional. Are the parameters of the non-seasonal 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).

S

Optional. Is the number of observations per period (e.g., 12 = Annual, 4 = Quarter).

sD

Optional. Is the seasonal integration order.

[sφ]

Optional. Are the parameters of the seasonal AR(P) component model: [sφ1, sφ2 … sφpp] (starting with the lowest lag).

[sθ]

Optional. Are the parameters of the seasonal MA(Q) component model: [sθ1, sθ2 … sθqq] (starting with the lowest lag).

Return

Optional. Is an integer switch to select the output array: (1 = Quick Guess (default), 2 = Calibrated, 3 = Std. Errors).

Value	Return
1	Quick guess (non-optimal) of parameters' values (default).
2	Calibrated (optimal) values for the model's parameters.
3	Standard error of the parameters' values.

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 SARIMA model has independent and normally distributed residuals with constant variance. The SARIMA 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 provides estimates for the model's parameters.
7. The long-run mean argument (µ) can take any value or be omitted, in which case a zero value is assumed.
8. The residuals/innovations standard deviation - (σ) - must be greater than zero.
9. For the input argument - ([φ]) (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).
10. For the input argument - ([θ]) (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).
11. For the input argument - ([sφ]) (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).
12. For the input argument - ([sθ]) (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).
13. The non-seasonal integration order - (d) - is optional and can be omitted, in which case d is assumed to be zero.
14. The seasonal integration order - (sD) - is optional and can be omitted, in which case sD is assumed to be zero.
15. The season length - (s) - is optional and can be omitted, in which case s is assumed to be zero (i.e., plain ARIMA).
16. 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

• 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.