# GARCHM - Defining a GARCH-M Model

Returns a unique string to designate the specified GARCH-M model.

## Syntax

GARCHM(mean, lambda, alphas, betas, innovation, v)

mean is the GARCH-M model mean (i.e. mu). If missing, a default value of 0 is assumed.

lambda is the volatility coefficient for the mean (aka. the risk premium). If missing, a default value of 0.0 is assumed.

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. If missing, a default value of 5.0 is assumed.

## Remarks

1. The underlying model is described here.
2. The long-run mean can take any value or be omitted, in which case a zero value is assumed.
3. For the input argument - alpha (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).
4. For the input argument - beta (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).
5. The lambda input argument is optional. If omitted, no risk-premium is included in the mean model component (i.e. plain GARCH).
6. The shape parameter (i.e. nu) is only used for non-Gaussian distribution and is otherwise ignored.
7. For student's t-distribution, the value of the shape parameter must be greater than four.
8. For GED distribution, the value of the shape parameter must be greater than one.