# EGARCH_VL - Long-Run Volatility of the EGARCH Model

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

## Syntax

EGARCH_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 EGARCH long-run average log variance is defined as: $$\ln V_L=\frac{\alpha_o+\eta \times \sum_{i=1}^p\alpha_i}{1-\sum_{j=1}^q\beta_j}$$

Where:

• Gaussian distributed innovations/shocks: $$\eta=\sqrt{\frac{\pi}{2}}$$
• GED distributed innovations/shocks: $$\eta=\frac{\Gamma(2/\nu)}{\sqrt{\Gamma(1/\nu)\times\Gamma(3/\nu)}}$$
• Student's t-Distributed innovations/shocks: $$\eta=\frac{\sqrt{\nu-2}\times\Gamma(\frac{\nu-1}{2})}{\sqrt{\pi}\times\Gamma(\frac{\nu}{2})}$$
3. The time series is homogeneous or equally spaced.
4. The number of gamma coefficients must match the number of alpha coefficients (minus one).
5. The number of parameters in the input argument - [αo α1, α2 … αp] - determines the order of the ARCH component model.
6. The number of parameters in the input argument - [β1, β2 … βq] - determines the order of the GARCH component model.
7. EGARCH_VL examines the model's coefficients for:
• Coefficients are all positive.