EGARCH_VL - Long-run Volatility of the EGARCH Model

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

Syntax

EGARCH_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	Innovation
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 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.
5. The number of parameters in the input argument - alpha - determines the order of the ARCH component model.
6. The number of parameters in the input argument - beta - determines the order of the GARCH component model.
7. EGARCH_CHECK examines the model's coefficients for:
   • Coefficients are all positive.

Files Examples

Related Links

• Wikipedia - Autoregressive conditional heteroskedasticity.

References

• Hamilton, J.D.; Time Series Analysis, Princeton University Press (1994), ISBN 0-691-04289-6.
• Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740.