Can we introduce an user-defined optional lower-bound in the stationary constraint of the variance process?

[sitmo]

Would it be possible to have a feature to set an EPSILON in the stationarity constraint?

Like:

1 - sum(alpha) - sum(beta) -0.5*sum(gamma) - EPSILON >= 0

The reason it that the optimiser sometimes give solution where the constraint is equal to zero, or very close like 1E-10, and that will give a crazy large long-term variance. This seems to sometimes happen when the return data covers a period of ever increasing variance.

By adding an EPSILON to the constraint we can force the variance process to have a minimal amount of mean reversion, e.g. EPSILON=1E-3 would give a variance half life of at most 1000 days, which is reasonable to impose?

Alternatively, we could also set an upper-bound on the long-term variance?

omega / [ 1 - sum(alpha) - sum(beta) - 0.5*sum(gamma) ] < max_LTV

edit: somewhere around here in the code

arch/arch/univariate/volatility.py

Line 1078 in a61dc7d

def constraints(self) -> tuple[Float64Array, Float64Array]:

[bashtage]

I could consider something liek this, but it is hard to generalize to all ARCH-type models.

I think when you want a sensible LRV the best choice is to use variance targetting, which writes the model as sigma2-bar (1-a-b) + a r(t-1)**2 + b sigma2(t-1) which always mean reverts to sigma2-bar as long as a+b<1. Of course, it is 1 then you have the same issue.