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

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


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

Required. Is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
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).
Optional. Is the GARCH model long-run mean (i.e., mu). If missing, the process mean is assumed to be zero.
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.
Required. Are the parameters of the ARCH(p) component model: [αo α1, α2 … αp] (starting with the lowest lag).
Optional. Are the parameters of the GARCH(q) component model: [β1, β2 … βq] (starting with the lowest lag).
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).
Optional. Is the shape parameter (or degrees of freedom) of the innovations/residuals’ probability distribution function.


  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.

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