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Can someone give me an intuitive explanation of how Optimal Combination Forecasts work for hierarchical time series? I've been reading Rob Hyndman's paper and understand that the resolved forecasts can be written as $\widetilde{Y}_h = SP\hat{Y}_n(h)$. This is pretty simple and intuitive when considering a top-down method. But how exactly does optimal combination work with and without weights? I don't see how $P = (S'\Sigma_h^{\dagger}S)^{-1}S'\Sigma_h^{\dagger}$ resolves the forecasts so they sum to the proper amount at higher levels.

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Careful there. $P$ is not the summation matrix. $P$ is only involved in estimating the new bottom-level forecasts $\tilde{Y}_n$ based on the original bottom level forecasts $\hat{Y}_n$:

$$ \tilde{Y}_n := P\hat{Y}_n.$$

After that, the optimally combined hierarchical forecasts $\tilde{Y}_h$ are calculated by applying the original summation matrix $S$ to $\tilde{Y}_n$:

$$ \tilde{Y}_h := S\tilde{Y}_n.$$

(And these are consistent, because $S$ simply encodes the consistency conditions.)

The somewhat confusing matrix $P$ is used instead of the more common OLS solution matrix to account for heteroskedasticity, and because the estimated covariance matrix may be near-singular because of the aggregation.

I'd recommend reading the original manuscript - a slightly different exposition to the published paper may be helpful. Or the original presentation Rob Hyndman gave at ISF 2016 in Santander. (BTW, the ISF will return to Santander this year - very much recommended.)

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  • $\begingroup$ So this solution to P is using GLS, right? And it also states in the paper that the weights can be calculated directly from the hierarchy. Why exactly is this true? Also, could you calculate the weights as 1/var(residuals)? $\endgroup$ – Kristofersen May 4 '16 at 22:26
  • $\begingroup$ It seems to me like you could calculate the weights (i.e., the estimation matrix $P$) directly from the hierarchy if you didn't care about heteroskedasticity and near-singularity, because then $P$ only depends on $S$, which in turn only depends on the topology of the hierarchy. In which case you wouldn't be using GLS any more. And yes, the precision (the reciprocal of the variance) makes sense as weights (along with $S$). $\endgroup$ – Stephan Kolassa May 5 '16 at 8:38

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