# ARMA_LLF - Log Likelihood Function of an ARMA Model

Computes the log-likelihood function (LLF) of the estimated ARMA model.

## Syntax

ARMA_LLF(X, Order, mean, sigma, phi, ARRAY)

X is the univariate time series data (a 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 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).

ARRAY are the parameters of the MA(q) component model (starting with the lowest lag).

## Remarks

1. The underlying model is described here.
2. The Log-Likelihood Function (LLF) is described here.
3. Warning: ARMA_AIC() function is deprecated as of version 1.63: use ARMA_GOF function instead.
4. The time series is homogeneous or equally spaced.
5. The time series may include missing values (e.g. #N/A) at either end.
6. The residuals/innovations standard deviation (i.e. $\sigma$) should be greater than zero.
7. 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.
8. The maximum likelihood estimation (MLE) is a statistical method for fitting a model to the data and provides estimates for the model's parameters.
9. The number of parameters in the input argument - phi - determines the order of the AR component.
10. The number of parameters in the input argument - theta - determines the order of the MA component.

## Examples

Example 1:

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

Formula Description (Result)
=ARMA_LLF($B$2:$B$30,1,$D$3,$D$4,$D$5,$D$6) Log-Likelihood Function (-2660.88)
=ARMA_CHECK($D$3,$D$4,$D$5,$D$6) Is ARMA model stable? (1)