{
  "title": "An Alternative Proof of Minimum Trace Reconciliation",
  "publication": "IMF Working Papers, July 8, 2022",
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  "summary": "Minimum trace reconciliation, developed by Wickramasuriya et. al. (2019), is an innovation in the literature of forecast reconciliation. The proof, however, is indirect and not easy to extend to more general situations.",
  "sections": [
    {
      "heading": "Summary",
      "content": "- Minimum trace reconciliation, developed by Wickramasuriya et. al. (2019), is an innovation in the literature of forecast reconciliation.\n- The existing proof is indirect and not easy to extend to more general situations.\n- 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).\n- The alternative proof has pedagogical value and connects results in the literature from a unified perspective."
    },
    {
      "heading": "Key Contributions and Findings",
      "content": "- Provides an alternative, simpler proof of Minimum Trace Reconciliation using first-order conditions in the space of non-square matrix.\n- Demonstrates extendibility to minimum weighted trace reconciliation results in Panagiotelis et. al. (2021).\n- Emphasizes unification of results in forecast reconciliation literature."
    },
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      "heading": "Methodology and Technical Focus",
      "content": "- Proof based on first-order condition in the space of non-square matrix.\n- Technical concepts and terminology preserved from the source:\n  - choice matrix p\n  - covariance matrix\n  - forecast error\n  - Forecast Reconciliation\n  - Hierarchical Time Series\n  - lower bound function\n  - lower bound minimization\n  - Minimum Trace Reconciliation\n  - n×m matrix\n  - trace reconciliation\n  - weight matrix\n\n---\n\n Content in this bundle\n\n- Working Paper\n  - Working Paper (Markdown version){rel=\"alternate\" type=\"text/markdown\"}\n  - Working Paper (PDF){rel=\"external\" type=\"application/pdf\"}\n\n---\n\nSource: https://www.imf.org/en/publications/wp/issues/2022/07/08/an-alternative-proof-of-minimum-trace-reconciliation-520572"
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    "[Markdown version](/en/publications/wp/issues/2022/07/08/an-alternative-proof-of-minimum-trace-reconciliation-520572/index.md)",
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    "Authors: Sakai Ando, Futoshi Narita",
    "Published: July 8, 2022",
    "Series: IMF Working Papers",
    "DOI: https://doi.org/10.5089/9798400216312.001",
    "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.",
    "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.",
    "Proof based on first-order condition in the space of non-square matrix.",
    "Technical concepts and terminology preserved from the source:",
    "**Working Paper**"
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