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AIRLINE_LLF
Calculates the loglikelohood function (LLF^{i}) of the given airline model.
Syntax
X
is the univariate time series data (one dimensional array of cells (e.g. rows or columns)).
Order
is the time order of the data series (i.e. whether the first data point corresponds to the earliest or latest date (earliest date=1 (default), latest date=0)).
Order  Description 

1  ascending (the first data point corresponds to the earliest date) 
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 firstlagged innovation (see model description).
theta2
is the coefficient of slagged innovation (see model description).
Remarks
 The underlying model is described here.
 The LogLikelihood Function (LLF) is described here.
 Warning: AIRLINE_LLF() function is deprecated as of version 1.63: use AIRLINE_GOF function instead.
 The time series is homogeneous or equally spaced.
 The time series may include missing values (e.g. #N/A) at either end.
 The standard deviation (i.e. ) of the ARMA^{i} model's residuals should be greater than zero.
 The AIC^{i} is not a test on the model in the sense of hypothesis testing, rather it is a test between models  a tool for model selection.
 Given a fixed data set, several competing models may be ranked according to their AIC, the model with the lowest AIC being the best.
 The Airline model is a special case of multiplicative seasonal ARMA model.
 The Airline model is a special (but oftenused) case of multiplicative seasonal ARIMA^{i} model, and it assumes independent and normally distributed residuals with constant variance.

Maximizing for the loglikelihood function, the AICc function for an ARMA model becomes:
Where:
 is the number of nonmissing values in the time series.
 is the order of the AR component model.
 is the order of the MA component model.
 is the standard deviation of the residuals.
Examples
Example 1:
A  B  C  D  

1  Date  Data  
2  1/1/2008  0.300 
ARMA 

3  1/2/2008  1.278  Mean  0.00258 
4  1/3/2008  0.244  Sigma  0.14 
5  1/4/2008  1.276  Phi_1  0.236 
6  1/6/2008  1.733  Theta_1  5.60E05 
7  1/7/2008  2.184  
8  1/8/2008  0.234  
9  1/9/2008  1.095  
10  1/10/2008  1.087  
11  1/11/2008  0.690  
12  1/12/2008  1.690  
13  1/13/2008  1.847  
14  1/14/2008  0.978  
15  1/15/2008  0.774 
Formula  Description (Result)  

=ARMA_AIC($B$2:$B$15,1,$D$3,$D$4,$D$5,$D$6)  1046.59  Akaike's information criterion (AIC)  
=ARMA_LLF($B$2:$B$15,1,$D$3,$D$4,$D$5,$D$6)  519.095  LogLikelihood Function  
=ARMA_CHECK($D$3,$D$4,$D$5,$D$6)  1  Is ARMA model stable? 
Files Examples
References
 Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0691042896
 Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0471690740