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Identification through Heteroskedasticity in One Lemma: Partial Identification, an Eigenvalue Construction, and a Caution on Rank Tests

  • George Milunovich
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Abstract

This teaching note unifies known results on identification through heteroskedasticity, under volatility regimes, GARCH, stochastic volatility and covariate-driven variances, in one lemma with a short linear-algebra proof. A shock is identified up to sign and labelling if and only if no other shock shares its variance path, and shocks that share a path are identified up to a rotation. The rows of the structural coefficient matrix solve one generalised eigenvalue problem that nests Rigobon's (2003) two-regime solution. Rank tests on second moments of the errors count at most linearly independent centred variance paths, not identified shocks. A simulation illustrates each point.

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Cite

BibTeXRIS
BibTeX
@misc{milunovich2026identification,
  title = {{Identification through Heteroskedasticity in One Lemma: Partial Identification, an Eigenvalue Construction, and a Caution on Rank Tests}},
  author = {George Milunovich},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41361},
  eprint = {2610.41361},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41361},
}
RIS
TY  - UNPB
TI  - Identification through Heteroskedasticity in One Lemma: Partial Identification, an Eigenvalue Construction, and a Caution on Rank Tests
AU  - George Milunovich
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41361
UR  - https://arxiv.org/abs/2610.41361
AB  - This teaching note unifies known results on identification through heteroskedasticity, under volatility regimes, GARCH, stochastic volatility and covariate-driven variances, in one lemma with a short linear-algebra proof. A shock is identified up to sign and labelling if and only if no other shock shares its variance path, and shocks that share a path are identified up to a rotation. The rows of the structural coefficient matrix solve one generalised eigenvalue problem that nests Rigobon's (2003) two-regime solution. Rank tests on second moments of the errors count at most linearly independent centred variance paths, not identified shocks. A simulation illustrates each point.
ER  -