# GARCH model: convergence of the conditional variance to the unconditional variance

Suppose a monthly, stationary time series. The series seems to have some ARCH effects and I model its variance as a GARCH process. I obtain the following output of a GARCH(1,1) model:

• alpha ($$\alpha$$): 0,07
• beta ($$\beta$$): 0,7
• intercept ($$c$$): 0,00008

I obtain the unconditional variance of the GARCH(1,1) model by taking the expectation of the GARCH equation:

$$E(V_t) = E(c + \alpha\epsilon^2_t+\beta V_{t-1})$$ $$V = c + \alpha V + \beta V$$ $$V = c/(1-\alpha - \beta)$$

But now I would like to assess how many months would it take to a regular shock (e.g., one standard deviation?) to dissipate or the conditional variance converge into its unconditional variance. How would someone do this?

Therefore, people study half-life instead: $$\ell:=\frac{\ln(0.5)}{\ln(\sum_{i=1}^s \alpha_i+\sum_{\beta_j}^r \beta_j)}$$ where $$\alpha$$s and $$\beta$$s are the GARCH model coefficients. In the case of a GARCH(1,1), $$\ell=\frac{\ln(0.5)}{\ln(\alpha_1+\beta_1)}.$$ Half-life tells you after how many periods half of the shock has dissipated.
In your case, $$\ell=\frac{\ln(0.5)}{\ln(0.07+0.70)}\approx 2.65$$.
*This is in contrast to an ARCH(s) model where shocks completely dissipate after $$s$$ periods.