Speed of Reversion in the GARCH(1, 1) Model.

[Adzi]

Hi David,

Could you please elaborate on aspects speed or slow reversion in the GARCH(1, 1) model
(variance estimate = omega + alpha*lagged return^2 + beta*lagged variance)?

If possible, please, provide few GARCH(1, 1) models with different persistence levels (weights) and show which one of the models has a slowest or speediest reversion to the long-run variance.

Thanks in advance for your help!

 

[David Harper CFA FRM]

Hi Denis,

omega (a constant) is a product of gamma (a weight) and the unconditional long-run variance to which the series mean reverts. Greater mean reversion is therefore associated with a higher gamma, and since the three weights must sum to 1.0, gamma = 1.0 - alpha - beta. So, the model with the highest (1 - alpha - beta) will, in relative terms, exhibit the greatest mean reversion. I don't think you'll need examples given this "test" is straightforward?

Thanks, David

 

[Adzi]

Hi David,

To summarise what you said, if I have 2 Garch(1, 1) models at hand, the one with the higher gamma will have the speediest reversion to the LR variance. And the other model with a lower gamma will revert slowly to the LR variance.

Thank you very much!

 

[David Harper CFA FRM]

Hi Denis:

Yes, exactly. To illustrate concretely compare two GARCH(1,1) where the first (a) has greater mean reversion despite a smaller omega:
a. var(t) = 0.0000060 + 0.90 * var(t-1) + 0.04 * r^2
b. var(t) = 0.0000160 + 0.90 * var(t-1) + 0.06 * r^2

Gamma weight:
a. gamma weight = 1 - 0.90 - 0.04 = 6%; greater reversion to long-run average (unconditional) variance
b. gamma weight = 1 - 0.90 - 0.06 = 4%; lesser reversion

Long-run average variance (please note: this is a common test question!):
a. LR var = 0.0000060 / gamma = 0.0000060 / (1 - alpha - beta) = 0.0000060 / 6% = 0.0001; LR vol = 1%
b. LR var = 0.0000160 / gamma = 0.0000160 / (1 - alpha - beta) = 0.0000160 / 4% = 0.0004; LR vol = 2%

i.e., omega = LR var * gamma weight.
a. omega = 0.0000060 = 1%^2*6%
b. omega = 0.0000160 = 2%^2*4%

Hope that helps, David