Submission: Tingjun GARCH Omega fix

[tingjun-yang]

Hi @Orrell, here is an update on the GARCH model.

An earlier version of this model utilized an incorrect definition of the intercept term ($\omega$), which led to a scaling mismatch in the long-run variance. Special thanks to @Orrell for pointing out this discrepancy.

In this current version, I have:

• Fixed the $\omega$ definition: Implemented proper Variance Targeting logic, where $\omega = V_{target} \cdot \gamma$.
• Standardized Nomenclature: Renamed parameters to follow standard GARCH(1,1) conventions ($\alpha$ for the shock component and $\beta$ for the persistence component).
• Re-optimized Parameters: Recalibrated the parameter set specifically for the 5M-day horizon to ensure maximum fit.

Despite these structural corrections, the results remain consistent with previous findings, further confirming the original hypothesis of Q-variance emergence in GARCH-governed systems.

[Orrell]

Hi @tingjun-yang This is a detail, but I noticed you are using fractional returns to compute ret_sq instead of log returns. I think in GARCH you usually use log returns, and that would be consistent with the rest of your model. Can you modify to compute ret_sq as the square of log-return?
David

[tingjun-yang]

Hi @Orrell, Yes, I will change that and update the results.

[tingjun-yang]

Hi @Orrell, After changing to log return, it is more likely to have spikes in volatility. This affects R^2 for very long periods:

I can add a cap on volatility, but that would introduce one additional parameter to the model. Would that be OK?

[Orrell]

Hi @tingjun-yang Yes, that would be good. Main message is that with that small mod GARCH can capture q-variance to good accuracy, which is a useful result. David