EGARCH_VL - Long-Run Volatility of the EGARCH Model

Jacquie Nesbitt

October 28, 2016 00:44

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.

The underlying model is described here.
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: $$\eta=\frac{\sqrt{\nu-2}\times\Gamma(\frac{\nu-1}{2})}{\sqrt{\pi}\times\Gamma(\frac{\nu}{2})}$$
The time series is homogeneous or equally spaced.
The number of gamma coefficients must match the number of alpha coefficients (minus one).
The number of parameters in the input argument - [α_oα_1, α₂… α_p] - determines the order of the ARCH component model.
The number of parameters in the input argument - [β_1, β₂… β_q] - determines the order of the GARCH component model.
EGARCH_VL examines the model's coefficients for:

James Douglas Hamilton; Time Series Analysis, Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896.
Tsay, Ruey S.; Analysis of Financial Time Series, John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740.