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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?

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Due to the nature of the autoregressive components in GARCH, the shocks dissipate at an exponential rate and never fully.* So to answer your question, for a shock to dissipate fully would take an infinitely long time, and the long-run variance would never be reached, only approched however closely.

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.

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  • $\begingroup$ Thank you! Could you share a reference? $\endgroup$
    – John Doe
    Jun 24, 2022 at 12:23
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    $\begingroup$ @JohnDoe, I spent a good while looking for it, but I cannot find the original reference anymore. Here are some papers that employ half-life, though. (I have also used half-life in one of my empirical papers, but I do not have a reference for it in the paper, so I will not cite my paper here.) $\endgroup$ Jun 24, 2022 at 17:45

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