GARCH_LLF - GARCH Model's Log-Likelihood Function

Computes the log-likelihood function for the fitted model.

Syntax

GARCH_LLF ([x], order, µ, [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.

[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. For the input argument - ([α]) (parameters of the ARCH component):
   • The input argument is not optional.
   • The value in the first element must be positive.
   • The order of the parameters starts with the lowest lag.
   • One or more parameters may have missing values or error codes (i.e., #NUM!, #VALUE!, etc.).
   • In the case where alpha has one non-missing entry/element (first), no ARCH component is included.
   • The order of the ARCH component model is solely determined by the order (minus one) of the last value in the array with a numeric value (vs. missing or error).
7. For the input argument - ([β]) (parameters of the GARCH component):
   • The input argument is optional and can be omitted, in which case no GARCH component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more parameters may have missing values or error codes (i.e., #NUM!, #VALUE!, etc.).
   • The order of the GARCH component model is solely determined by the order of the last value in the array with a numeric value (vs. missing or error).

Files Examples

Related Links

• Wikipedia - Likelihood function.
• Wikipedia - Autoregressive conditional heteroskedasticity.

References

• James Douglas Hamilton; Time Series Analysis, Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896.
• Tsay, Ruey S.; Analysis of Financial Time Series, John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740.