SARIMAX_GOF - Goodness of fit of a SARIMAX Model

1. The underlying model is described here.
2. The Log-Likelihood Function (LLF) is described here.
3. Each column in the explanatory factors input matrix (i.e. X) corresponds to a separate variable.
4. Each row in the explanatory factors input matrix (i.e. X) corresponds to an observation.
5. Observations (i.e. rows) with missing values in X or Y are assumed missing.
6. The number of rows of the explanatory variable (X) must be at equal to the number of rows of the response variable (Y).
7. The time series is homogeneous or equally spaced.
8. The time series may include missing values (e.g. #N/A) at either end.
9. The residuals/innovations standard deviation (i.e. $\sigma$) should be greater than zero.
10. 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.
11. The value of the input argument - period - must be greater than one, or the function returns #VALUE!.
12. The value of the seasonal difference argument - sD - must be greater than one, or the function returns #VALUE!.
13. The maximum likelihood estimation (MLE) is a statistical method for fitting a model to the data and provides estimates for the model's parameters.
14. The intercept or the regression constant term input argument is optional. If omitted, a zero value is assumed.
15. 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 SARIMA).
    • The order of the parameters defines how the exogenous factor input arguments are passed.
    • One or more parameters may have missing value or an error code(i.e. #NUM!, #VALUE!, etc.).
16. The long-run mean argument (mean) of the differenced regression residuals can take any value. If omitted, a zero value is assumed.
17. The residuals/innovations standard deviation (sigma) must greater than zero.
18. For the input argument - phi (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 value or an error code(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).
19. For the input argument - theta (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).
20. For the input argument - sPhi (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 value or an error code(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).
21. For the input argument - sTheta (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).
22. The non-seasonal integration order - d - is optional and can be omitted, in which case d is assumed zero.
23. The seasonal integration order - sD - is optional and can be omitted, in which case sD is assumed zero.
24. The season length - s - is optional and can be omitted, in which case s is assumed zero (i.e. Plain ARIMA).
25. The function was added in version 1.63 SHAMROCK.

Files Examples

Related Links

• Wikipedia - Likelihood function
• Wikipedia - Likelihood principle
• Wikipedia - Autoregressive integrated 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