An Alternative Proof of Minimum Trace Reconciliation
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe alternative proof for Theorem 1 in Wickramasuriya et al. (2019) contains a more general situation (i.e., eq. 31) than the special case considered in the original JASA paper. I suggest considering a real-world dataset to illustrate the necessity of including various weights (i.e., $\omega \neq I_m$ as in eq. 31).
For example, if some of the bottom-level observations in the hierarchy contain significant noise due to data collection errors, we would intentionally add weights to minimise the impact of such "bad" observations on the reconciled forecasts. In this scenario, the important assumption of additive errors in Wickramasuriya et al. (2019) may not hold. The discussion in Section 5 is a good starting point. However, without a real data application, it is difficult to convince people that such "weights" $\omega$ are necessary in practice.
Author Response
Comments and Suggestions for Authors
The alternative proof for Theorem 1 in Wickramasuriya et al. (2019) contains a more general situation (i.e., eq. 31) than the special case considered in the original JASA paper. I suggest considering a real-world dataset to illustrate the necessity of including various weights (i.e., $\omega \neq I_m$ as in eq. 31). For example, if some of the bottom-level observations in the hierarchy contain significant noise due to data collection errors, we would intentionally add weights to minimise the impact of such "bad" observations on the reconciled forecasts. In this scenario, the important assumption of additive errors in Wickramasuriya et al. (2019) may not hold. The discussion in Section 5 is a good starting point. However, without a real data application, it is difficult to convince people that such "weights" $\omega$ are necessary in practice.
Please note that the general situation with a weighted sum of forecast error variance is to emphasize the un-necessity, rather than the necessity, to consider weights. This is because the solution is the same irrespective of the weight, and thus, whether to use real data to estimate the weight or not does not affect the resulting reconciled forecast.
We have clarified this point in the introduction and right before proposition 1 by saying “one does not need to exercise judgment or estimate the weights.”
We have also added the implication of the irrelevance of weight at the end of section 5 as below.
In summary, the extension to allow a general weight highlights two observations. First, the irrelevance of weight implies that the objective function being the trace of the forecast error covariance matrix is not essential, although (9) is called minimum trace reconciliation in the literature. What is essential is the unbiasedness assumption, and thus, it could alternatively be called an optimal unbiased reconciliation. Second, the irrelevance of weight suggests that (9) reconciles the base forecast as if the forecast error variance of each variable can be minimized independently, but at the same time, (9) can be obtained by minimizing the variance of only the bottom-level variables. The extension suggests that these two apparently contradictory interpretations can coexist.
Reviewer 2 Report
Comments and Suggestions for AuthorsThe authors highlight a gap in the proof of Minimum Trace Reconciliation by Wicramasuriya et al. (2019). They have successfully identified and resolved this gap. I find the paper well-written and important for the community, given the significance of the subject. However, I suggest the following improvements before it can be accepted for publication:
- The introduction section is extremely concise and could be better developed to present the problem more clearly.
- Line 60 is not formatted according to the standard.
- "Weyl’s inequalities" lack an associated reference (line 104).
- "Frobenius inner" lacks an associated reference (line 137).
- Please clarify where the proof of Equation 10 is located in the section "4. An Alternative Proof of (9)."
- The section "5. An Extension of the Alternative Proof" seems unnecessarily long as it essentially repeats the previous proof with a "practical" application.
Author Response
Comments and Suggestions for Authors
The authors highlight a gap in the proof of Minimum Trace Reconciliation by Wicramasuriya et al. (2019). They have successfully identified and resolved this gap. I find the paper well-written and important for the community, given the significance of the subject. However, I suggest the following improvements before it can be accepted for publication:
- The introduction section is extremely concise and could be better developed to present the problem more clearly.
The introduction is expanded to explain the gap in Wickramasuriya et al. (2019)
- Line 60 is not formatted according to the standard.
The format is updated.
- "Weyl’s inequalities" lack an associated reference (line 104).
The Weyl’s inequality is not essential for the argument, and thus, removed.
- "Frobenius inner" lacks an associated reference (line 137).
A reference is added.
- Please clarify where the proof of Equation 10 is located in the section "4. An Alternative Proof of (9)."
To clarify the location of (10)’s proof, we have added “in its online appendix A2.”
We have also clarified the location where the equivalence of equation (9) and (11) is discussed by creating proposition 2 in section 5.
- The section “5. An Extension of the Alternative Proof” seems unnecessarily long as it essentially repeats the previous proof with a “practical” application.
The proof for proposition 1 is shortened by removing the detailed explanation of the first order condition.
Reviewer 3 Report
Comments and Suggestions for AuthorsPlease see the review report.
Comments for author File: 
Comments.pdf
Minor editing of English language required.
Author Response
Review report for Forecasting-An Alternative Proof of Minimum Trace Reconciliation
The manuscript points out the gap of proof in Wickramasuriya et al. (2019) and tries to fill it by providing alternative proof based on the first-order condition. The authors also attempt to show that both the trace and weighted trace can be analyzed from a unified perspective. I have the following major and minor concerns.
Major comments
The claim of the non-trivial gap in the Proof of Wickramasuriya et al. (2019) provided in Section 3 is valid, but some doubts exist in the alternative proof given in Section 4. The Frechet derivative of the Lagrangian function in Equation (16) is not trivial. For example, how do we solve the Frechet derivative of a trace function? This proof generalizes the vector form in the theorem of Luenberger (1969, p.243) to the matrix form, which could be more apparent. This claim would have been more convincing to the readers if the authors had provided some detailed mathematical proof process.
We have elaborated the proof by explaining that the first order condition of the trace can be taken by a direct calculation, as in equation (17).
Equation 32 of the article shows that both traces are identical, which is the objective function when the assumption of unbiasedness is satisfied. Still, this conclusion needs to sound more convincing since the weights of the objective function are changed, while the optimization result does not change. Please clarify.
Please note that the same unbiasedness assumption is still in place. We have clarified the point by adding “Note that the constraint in (25) is the same as that in (9), so the same unbiasedness assumption is still imposed. The same unbiasedness assumption is also reflected in the objective function as in (9).” at the end of the paragraph after (25).
Minor comments
Please provide complete information on the references, including issues and volumes.
Corrected. Please note that the issues and volumes for Ando and Kim (2013) is not available.
When an equation ends with a comma, no indent should exist at the beginning of the following line.
Corrected.
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsThank you for considering my previous suggestions.
Reviewer 3 Report
Comments and Suggestions for AuthorsThe authors have addressed my concerns.
