I have a bunch of component processes $y_{it}$, where $i=1..n$. I can build reasonable time series models $y_{it}=f_i(y_{i,s<t},X_t)$, where $X_t$ - exogenous variables. These could be ARIMAX processes for each component $i$. I could also build vector autoregression (VAR) models $y_t=Ay_{t-1}+BX_t$ etc.
However, I have an additional piece of information available: the aggregate level $Y_t=\sum_{i=1}^nw_{it}y_{it}$ for each time $t$ both in past and future, where weights $w_{it}$ are known in the past. The weights are not known in future, but I could probably get away with assuming constant weights in future, so it's not a big deal.
My problem is how to incorporate $Y_t$ into the forecast of $y_{it}$? It's basically disaggregation problem in some way.
Nuance. When I say "$Y_t$ is known" I'm not being precise. I don't really know $Y_t$ in future, so I should say "Have I known $Y_t$ what would have been my best forecast of $y_{it}$?" In other words, I'm looking for the best forecast conditional on knowledge of the aggregate level $\hat{y}_{i,t+h}=f_i(y_t,X_{t+h},Y_{t+h})$
