## An Alternative Proof of Minimum Trace Reconciliation

_IMF Working Papers, July 8, 2022_

## Source details

**Canonical URL:** [An Alternative Proof of Minimum Trace Reconciliation](https://www.imf.org/en/publications/wp/issues/2022/07/08/an-alternative-proof-of-minimum-trace-reconciliation-520572)

## Other formats

- [Markdown version](/en/publications/wp/issues/2022/07/08/an-alternative-proof-of-minimum-trace-reconciliation-520572/index.md)
- [Structured JSON version](/en/publications/wp/issues/2022/07/08/an-alternative-proof-of-minimum-trace-reconciliation-520572/index.json)
- [Bundle manifest](/en/publications/wp/issues/2022/07/08/an-alternative-proof-of-minimum-trace-reconciliation-520572/bundle-manifest.json)

## Bibliographic details
- Authors: Sakai Ando, Futoshi Narita
- Published: July 8, 2022
- Series: IMF Working Papers
- DOI: https://doi.org/10.5089/9798400216312.001

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### Summary
- Minimum trace reconciliation, developed by Wickramasuriya et. al. (2019), is an innovation in the literature of forecast reconciliation.
- The existing proof is indirect and not easy to extend to more general situations.
- This paper provides an alternative proof based on the first-order condition in the space of non-square matrix and argues that it is not only simpler but also can be extended to incorporate more general results on minimum weighted trace reconciliation in Panagiotelis et. al. (2021).
- The alternative proof has pedagogical value and connects results in the literature from a unified perspective.

### Key Contributions and Findings
- Provides an alternative, simpler proof of Minimum Trace Reconciliation using first-order conditions in the space of non-square matrix.
- Demonstrates extendibility to minimum weighted trace reconciliation results in Panagiotelis et. al. (2021).
- Emphasizes unification of results in forecast reconciliation literature.

### Methodology and Technical Focus
- Proof based on first-order condition in the space of non-square matrix.
- Technical concepts and terminology preserved from the source:
  - choice matrix p
  - covariance matrix
  - forecast error
  - Forecast Reconciliation
  - Hierarchical Time Series
  - lower bound function
  - lower bound minimization
  - Minimum Trace Reconciliation
  - n×m matrix
  - trace reconciliation
  - weight matrix

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## Content in this bundle

- **Working Paper**
  - [Working Paper (Markdown version)](/-/media/files/publications/wp/2022/english/wpiea2022136-print-pdf.pdf.md){rel="alternate" type="text/markdown"}
  - [Working Paper (PDF)](/-/media/files/publications/wp/2022/english/wpiea2022136-print-pdf.pdf){rel="external" type="application/pdf"}

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_Source: https://www.imf.org/en/publications/wp/issues/2022/07/08/an-alternative-proof-of-minimum-trace-reconciliation-520572_