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Simultaneous Clustered Orthogonalization

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  • Bastien Buchwalter
  • Francis X. Diebold
  • Kamil Yilmaz

Abstract

Identification in vector autoregressions involves two distinct choices: how extensively to orthogonalize shocks and whether orthogonality is imposed sequentially or simultaneously. Generalized identification imposes no orthogonality; Sims (1980} imposes full orthogonality sequentially; Francis et al. (2026) impose full orthogonality simultaneously; and Buchwalter et al. (2026a) provide clustered partial orthogonalization sequentially. We fill the remaining case by developing simultaneous clustered orthogonalization (SCO). SCO preserves contemporaneous dependence within economically meaningful clusters while imposing orthogonality across clusters jointly, thereby eliminating dependence on cluster ordering. We formulate the associated correlation-maximizing identification problem and show that, in correlation space, it reduces to a quadratic matrix equation. This yields a closed-form solution, and we prove that the associated identification matrix is the unique global maximizer. SCO is order- and scale-invariant and nests generalized identification and full simultaneous orthogonalization as special cases, yielding a flexible family of structural decompositions indexed by the number and composition of clusters.

Suggested Citation

  • Bastien Buchwalter & Francis X. Diebold & Kamil Yilmaz, 2026. "Simultaneous Clustered Orthogonalization," Papers 2609.12280, arXiv.org.
    • Handle: RePEc:arx:papers:2609.12280
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    References listed on IDEAS

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    1. Sims, Christopher A, 1980. "Macroeconomics and Reality," Econometrica, Econometric Society, vol. 48(1), pages 1-48, January.
    2. Kristin J. Forbes & Roberto Rigobon, 2002. "No Contagion, Only Interdependence: Measuring Stock Market Comovements," Journal of Finance, American Finance Association, vol. 57(5), pages 2223-2261, October.
    3. Diebold, Francis X. & Yilmaz, Kamil, 2012. "Better to give than to receive: Predictive directional measurement of volatility spillovers," International Journal of Forecasting, Elsevier, vol. 28(1), pages 57-66.
    4. Pesaran, H. Hashem & Shin, Yongcheol, 1998. "Generalized impulse response analysis in linear multivariate models," Economics Letters, Elsevier, vol. 58(1), pages 17-29, January.
    5. Diebold, Francis X. & Yılmaz, Kamil, 2014. "On the network topology of variance decompositions: Measuring the connectedness of financial firms," Journal of Econometrics, Elsevier, vol. 182(1), pages 119-134.
    6. Neville Francis & Peter Reinhard Hansen & Chen Tong, 2025. "Principled Identification of Structural Dynamic Models," Papers 2512.17005, arXiv.org, revised Apr 2026.
    7. Koop, Gary & Pesaran, M. Hashem & Potter, Simon M., 1996. "Impulse response analysis in nonlinear multivariate models," Journal of Econometrics, Elsevier, vol. 74(1), pages 119-147, September.
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