# 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, theta)
X
= the univariate time series data (a one-dimensional array of cells (e.g. rows or columns)).
Order
= 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
= the ARMA model mean (i.e., mu).
sigma
= the standard deviation of the model's residuals/innovations.
phi
= the parameters of the AR(p) component model (starting with the lowest lag).
theta
= the parameters of the MA(q) component model (starting with the lowest lag).

#### Warning

ARMA_LLF() function is deprecated as of version 1.63: use the ARMA_GOF function instead.

## Remarks

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

## Examples

Example 1:

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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 the ARMA model stable? (1)