# GARCHM_LLF - Log Likelihood Function of an GARCH-M Model

Calculates the log-likelihood function for the fitted GARCH-M model.

## Syntax

GARCHM_LLF(X, Order, mean, lambda, alphas, betas, innovation, v)
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 GARCH-M model mean (i.e. mu).
lambda
is the volatility coefficient for the mean. In finance, lambda is referenced as the risk premium.
alphas
are the parameters of the ARCH(p) component model (starting with the lowest lag).
betas
are the parameters of the GARCH(q) component model (starting with the lowest lag).
innovation
is the probability distribution model for the innovations/residuals (1=Gaussian (default), 2=t-Distribution, 3=GED).
value Description
1 Gaussian or Normal Distribution (default)
2 Student's t-Distribution
3 Generalized Error Distribution (GED)
v
is the shape parameter (or degrees of freedom) of the innovations/residuals probability distribution function.

## 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 maximum likelihood estimation (MLE) is a statistical method for fitting a model to the data and provides estimates for the model's parameters.
6. The number of parameters in the input argument - alpha - determines the order of the ARCH component model.
7. The number of parameters in the input argument - beta - determines the order of the GARCH component model.

## 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
31
32
A B C D
Date Data
January 10, 2008 -2.827 GARCH-M(1,1)
January 11, 2008 -0.947 Mean -0.076
January 12, 2008 -0.877 Lambda 0.145
January 14, 2008 1.209 Alpha_0 0.593
January 13, 2008 -1.669 Alpha_1 0.000
January 15, 2008 0.835 Beta_1 0.403
January 16, 2008 -0.266
January 17, 2008 1.361
January 18, 2008 -0.343
January 19, 2008 0.475
January 20, 2008 -1.153
January 21, 2008 1.144
January 22, 2008 -1.070
January 23, 2008 -1.491
January 24, 2008 0.686
January 25, 2008 0.975
January 26, 2008 -1.316
January 27, 2008 0.125
January 28, 2008 0.712
January 29, 2008 -1.530
January 30, 2008 0.918
January 31, 2008 0.365
February 1, 2008 -0.997
February 2, 2008 -0.360
February 3, 2008 1.347
February 4, 2008 -1.339
February 5, 2008 0.481
February 6, 2008 -1.270
February 7, 2008 1.710
February 8, 2008 -0.125
February 9, 2008 -0.940

Formula Description (Result)
=GARCHM_LLF(\$B\$2:\$B\$32,1,\$D\$3,\$D\$4,\$D\$5:\$D\$6,\$D\$7) Log-Likelihood Function for Guassian Distribution (-45.482)
=GARCHM_LLF(\$B\$2:\$B32,1,\$D\$3,\$D\$4,\$D\$5:\$D\$6,\$D\$7,2,5) Log-Likelihood Function for t-Distribution when v = 5 (-49.122)
=GARCHM_LLF(\$B\$2:\$B32,1,\$D\$3,\$D\$4,\$D\$5:\$D\$6,\$D\$7,3,2) Log-Likelihood Function for GED when v = 2 (-45.482)
=GARCHM_AIC(\$B\$2:\$B\$32,1,\$D\$3,\$D\$4,\$D\$5:\$D\$6,\$D\$7) Akaike's information criterion (96.964)
=GARCHM_CHECK(\$D\$3,\$D\$4,\$D\$5:\$D\$6,\$D\$7) The GARCH-M(1,1) model is stable? (1)