Apr 24

Persistence in GARCH(1,1)

by David Harper, CFA, FRM, CIPM

FRM | Risk | Quant |

Vol_garchDecayIntro

GARCH(1,1) and the exponentially weighted moving average (EWMA) are estimates of conditional volatility. They presume today's volatility is conditional on yesterday's volatility. This presumption reflects an observation: volatility clusters. It's sticky. In regard to the financial risk manager (FRM) exam, we have two volatility themes:

EMWA estimates volatility with one parameter, GARCH(1,1) with three. The only parameter in EWMA is lambda. Depending on the author, it is known by three synonyms: smoothing constant, persistence parameter, or less often, the decay factor. We'll follow Jorian and call it persistence.

The persistence (lambda) in EMWA is analogous to the sum of alpha and beta (alpha+beta) in GARCH(1,1):

PersistenceParams2

Note this is Hull's notation. But don't get distracted by the notation. The GARCH(1,1) has three factors: long-run variance, yesterday's squared return, and yesterday's variance. Each factor has three weights: gamma, alpha, and beta. The weights must sum to one.

In the above, alpha plus beta gives the persistence of the GARCH(1,1) series just like lambda gives persistence of the EWMA series. In GARCH(1,1), the more weight we give to alpha and beta (i.e., as their sum approaches one), the less weight that's "left over" for the long-run variance. What's the effect? the less "decay" toward the long-run variance, or put another way, the greater the persistence to recent variance.

Here is a good question to test your understanding: what happens to GARCH(1,1) if we insist that alpha plus beta is equal to 1 (alpha+beta=1)?

Answer: if alpha+beta=1, then gamma must equal zero and the first term in GARCH(1,1) drops out. GARCH(1,1) under this conditions is equivalent to EMWA. That's why we say "EMWA is a special or particular case of GARCH" or, put another way, GARCH(1,1) is a generalized form of EWMA.

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