GARCH_VL - Long-run Volatility of the GARCH Model

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).

v is the shape parameter (or degrees of freedom) of the innovations/residuals probability distribution function.

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.

Example 1:

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