# EGARCH_LLF - Log Likelihood Function of an EGARCH Model

Computes the log-likelihood function for the fitted model.

## Syntax

EGARCH_LLF(X, Order, mean, alphas, gammas, 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 E-GARCH model mean (i.e. mu).

alphas are the parameters of the ARCH(p) component model (starting with the lowest lag).

gammas are the leverage parameters (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 number of gamma-coefficients must match the number of alpha-coefficients.
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.
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.

## 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 EGARCH(1,1)
January 11, 2008 -0.947 Mean -0.266
January 12, 2008 -0.877 Alpha_0 1.583
January 14, 2008 1.209 Alpha_1 -1.755
January 13, 2008 -1.669 Gamma_1 0.286
January 15, 2008 0.835 Beta_1 0.470
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)
=EGARCH_LLF($B$2:$B$32,1,$D$3,$D$4:$D$5,$D$6,$D$7) Log-Likelihood function FOR Normal Distribution (-42.235)
=EGARCH_LLF($B$2:$B$32,1,$D$3,$D$4:$D$5,$D$6,$D$7,2,5) Log-Likelihood function FOR t-Distribution with freedom = 5 (-44.695)
=EGARCH_LLF($B$2:$B$32,1,$D$3,$D$4:$D$5,$D$6,$D$7,3,10) Log-Likelihood function FOR GED with freedom = 10 (-40.685)