# GARCH_VL - Long-run Volatility of the GARCH Model

Calculates the long-run average volatility for the given GARCH model.

## Syntax

GARCH_VL ([α], [β], f, ν)

[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 long-run variance of a GARCH process is defined as follows:$$\sigma_{\infty}^2 \rightarrow V_L=\frac{\alpha_o}{1-\sum_{i=1}^{max(p,q)}\left(\alpha_i+\beta_i\right)}$$
3. The long-run variance is not affected by our choice of shock/innovation distribution.
4. 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).
5. 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).