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ARMA_LLF
Computes the loglikelihood function (LLF^{i}) of the estimated ARMA^{i} model.
Syntax
X
is the univariate time series data (a 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 ARMA model mean (i.e. mu).
sigma
is the standard deviation of the model's residuals/innovations.
phi
are the parameters of the AR(p) component model (starting with the lowest lag^{i}).
theta
are the parameters of the MA(q) component model (starting with the lowest lag).
Remarks
 Warning: ARMA_LLF() function is deprecated as of version 1.63: use ARMA_GOF function instead.
 The underlying model is described here.
 The LogLikelihood Function (LLF) is described here.
 The time series is homogeneous or equally spaced.
 The time series may include missing values (e.g. #N/A) at either end.
 The residuals/innovations standard deviation (i.e. ) should be greater than zero.

ARMA model has independent and normally distributed residuals with constant variance. The ARMA loglikelihood function becomes:
Where:
 is the standard deviation of the residuals.
 The maximum likelihood estimation (MLE) is a statistical method for fitting a model to the data and provides estimates for the model's parameters.
 The number of parameters in the input argument  phi  determines the order of the AR component.
 The number of parameters in the input argument  theta  determines the order of the MA component.
Examples
Example 1:
A  B  C  D  

1  Date  Data  
2  January 10, 2008  0.30 
ARMA 

3  January 11, 2008  1.28  Mean  1.2 
4  January 12, 2008  0.24  Sigma  0.086 
5  January 13, 2008  1.28  Phi_1  0.0014 
6  January 14, 2008  1.20  Theta  0.36 
7  January 15, 2008  1.73  
8  January 16, 2008  2.18  
9  January 17, 2008  0.23  
10  January 18, 2008  1.10  
11  January 19, 2008  1.09  
12  January 20, 2008  0.69  
13  January 21, 2008  1.69  
14  January 22, 2008  1.85  
15  January 23, 2008  0.98  
16  January 24, 2008  0.77  
17  January 25, 2008  0.30  
18  January 26, 2008  1.28  
19  January 27, 2008  0.24  
20  January 28, 2008  1.28  
21  January 29, 2008  1.20  
22  January 30, 2008  1.73  
23  January 31, 2008  2.18  
24  February 1, 2008  0.23  
25  February 2, 2008  1.10  
26  February 3, 2008  1.09  
27  February 4, 2008  0.69  
28  February 5, 2008  1.69  
29  February 6, 2008  1.85  
30  February 7, 2008  0.98 
Formula  Description (Result)  

=ARMA_LLF($B$2:$B$30,1,$D$3,$D$4,$D$5,$D$6)  LogLikelihood Function (2660.88)  
=ARMA_CHECK($D$3,$D$4,$D$5,$D$6)  Is ARMA model stable? (1) 
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