# GARCHM_LLF – Log-Likelihood Function of a GARCH-M Model

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

## Syntax

GARCHM_LLF ([x], order, µ, λ, [α], [β], f, ν)

[X]
Required. Is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
Order
Optional. Is the time order in the data series (i.e., the first data point's corresponding date (earliest date = 1 (default), latest date = 0)).
Value Order
1 Ascending (the first data point corresponds to the earliest date) (default).
0 Descending (the first data point corresponds to the latest date).
µ
Optional. Is the GARCH model long-run mean (i.e., mu). If missing, the process mean is assumed to be zero.
λ
Optional. Is the volatility coefficient for the mean. In finance, lambda is referenced as the risk premium. If missing, a default of 0 is assumed.
[α]
Required. Are the parameters of the ARCH(p) component model: [αo α1, α2 … αp] (starting with the lowest lag).
[β]
Optional. Are the parameters of the GARCH(q) component model: [β1, β2 … βq] (starting with the lowest lag).
F
Optional. Is the probability distribution function of the innovations/residuals (1 = Gaussian (default), 2 = t-Distribution, 3 = GED).
Value Probability Distribution
1 Gaussian or Normal Distribution (default).
2 Student's t-Distribution.
3 Generalized Error Distribution (GED).
ν
Optional. 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 - [αo α1, α2 … αp] - determines the order of the ARCH component model.
7. The number of parameters in the input argument - [β1, β2 … βq] - determines the order of the GARCH component model.