# How are the constraints of the parameters of a GARCH model calculated?

For a simple GARCH(1,1) model: h(t) = K + G*h(t-1) + A*e^2(t-1),

The parameters need to be constrained such that G + A < 1. Can anyone please tell me how this is derived since I've seen this in a lot of books but there's not much explanation given about this.

Thanks!

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For a GARCH(1,1) model of the form

$\sigma_n^2 = \omega + \alpha u_{n-1}^2 + \beta\sigma_{n-1}^2$

we require that

$\alpha+\beta < 1$

for reasons of stationarity. A GARCH(1,1) process which meets these constraints is deemed to be a stable one. If these constraints are not met the process is not stable but rather explosive and running a GARCH model on it does not make much sense.

It's for similar reasons for example that when we apply a GARCH model to a stock price time series we use returns rather than prices. Returns tend to be stationary while prices do not.

You can also think about the $\alpha + \beta$ sum as a measure of how fast shocks in variance dissipate over time. If they add up to more than one then shocks don't dissipate over time. Low quality samples will tend to generate such numbers and results in such situations should not be trusted.

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