Yes I've used the sum of past squared ! The var measure is even more different with the GARCH modelized variance !

@RichardHardy I've edited the post !

I edited the reference. There is a lag between $S_{i,t+H}$ and $\sigma_i,t$, but I didn't know if it was enough to ensure exogeneity !

Spillovers occurs when an idiosyncratic shock in one asset is transmitted to another (through the portfolio rebalancing channel mainly). A volatility shock is a significant shift of an asset volatility. I use the Diebold-Yilmaz (2012) methodology to estimate the cross-assets spillovers, and the volatility shock is estimated, as in Bloom (2009) !

Sorry, I've edited the question with much more details, hope it's enough (and I've modified the question as the conditional volatility measured in a GARCH is clearly not exogeneous !)