# AIRLINE_GOF - Goodness of fit of an Airline Model

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

## Syntax

AIRLINE_GOF(X, Order, mean, sigma, s, theta, theta2, Type)

X is the univariate time series data (one dimensional array of cells (e.g. rows or columns)).

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)).

Order Description
1 ascending (the first data point corresponds to the earliest date) (default)
0 descending (the first data point corresponds to the latest date)

mean is the model mean (i.e. mu).

sigma is the standard deviation of the model's residuals/innovations.

s is the length of seasonality (expressed in terms of lags, where s > 1).

theta is the coefficient of non-seasonal MA component (see model description).

theta2 is the coefficient of seasonal MA component (see model description).

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

1. The underlying model is described here.
2. The Log-Likelihood Function (LLF) is described here.
3. Akaike's Information Criterion (AIC) is described here.
4. Bayesian (Schwartz) Information Criterion (BIC) is described here.
5. "Hannan-Quinn Information Criterion (HQC) is described here.
6. The time series is homogeneous or equally spaced.
7. The time series may include missing values (e.g. #N/A) at either end.
8. The airline model with order $s$ has 4 parameters: $\mu\,,\sigma\,\,,\theta\,,\mathit{and} \Theta$
9. The Airline model is a special case of multiplicative seasonal ARIMA model and it assumes independent and normally distributed residuals with constant variance.
10. The function was added in version 1.63 SHAMROCK.

## Examples

Example 1:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A B C D
Date Data
1/1/2008 -0.300 AIRLINE
1/2/2008 -1.278 Mean 0.0481
1/3/2008 0.244 theta(1) 0.14
1/4/2008 1.276 theta(2) 0.30
1/6/2008 1.733 Sigma 2.74127
1/7/2008 -2.184 s 2
1/8/2008 -0.234
1/9/2008 1.095
1/10/2008 -1.087
1/11/2008 -0.690
1/12/2008 -1.690
1/13/2008 -1.847
1/14/2008 -0.978
1/15/2008 -0.774

Formula Description (Result)
=AIRLINE_AIC(Sheet1!$B$2:$B$15,1,$D$3,$D$6,$D$7,$D$4,$D$5) 65.6 Akaike's information criterion (AIC)
=AIRLINE_LLF(Sheet1!$B$2:$B$15,1,$D$3,$D$6,$D$7,$D$4,$D$5) -25.47 Log-Likelihood Function
=AIRLINE_CHECK($D$3,$D$6,$D$7,$D$4,$D$5) 1 Is the AIRLINE model stable?