# GARCH_VL - Long-run Volatility of the GARCH Model

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

## Syntax

GARCH_VL(alphas, betas, innovation, v)

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.

## Remarks

1. The underlying model is described here.
2. The long-run variance of a GARCH process is defined as follow:
1. $$\sigma_{\infty}^2 \rightarrow V_L=\frac{\alpha_o}{1-\sum_{i=1}^{max(p,q)}\left(\alpha_i+\beta_i\right)}$$
3. The long-run variance is not affected by our choice of shock/innovation distribution.
4. The number of parameters in the input argument - alpha - determines the order of the ARCH component model.
5. The number of parameters in the input argument - beta - determines the order of the GARCH component model.

## Examples

Example 1:

 1 2 3 4 5
A B
GARCH(1,1)
Mean -0.160
Alpha_0 0.608
Alpha_1 0.000
Beta_1 0.391

Formula Description (Result)
=GARCH_VL($B$3:$B$4,$B$5) GARCH long-run average volatility (0.999)