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The Generalized Fisher Transformation: Finite-Sample Properties and Inference

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  • Ilya Archakov
  • Peter Reinhard Hansen

Abstract

We study the finite-sample behavior of the Generalized Fisher Transformation (GFT), the parametrization of a correlation matrix $C$ by $\gamma(C)=\operatorname{vecl}\log C$. The GFT coordinates extend Fisher's transformation to dimension $n>2$: for elliptical data their finite-sample distributions are close to Gaussian. More strikingly, the coordinates are nearly uncorrelated and their covariance is largely invariant to $C$. This approximate orthogonality and invariance make GFT-based inference far better behaved in finite samples than inference based on sample correlations or element-wise Fisher transformed correlations, yielding estimation errors that are approximately Gaussian, weakly dependent, and nearly pivotal.

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  • Ilya Archakov & Peter Reinhard Hansen, 2026. "The Generalized Fisher Transformation: Finite-Sample Properties and Inference," Papers 2606.13864, arXiv.org.
    • Handle: RePEc:arx:papers:2606.13864
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    References listed on IDEAS

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    1. Andersen T. G & Bollerslev T. & Diebold F. X & Labys P., 2001. "The Distribution of Realized Exchange Rate Volatility," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 42-55, March.
    2. Ilya Archakov & Peter Reinhard Hansen & Yiyao Luo, 2024. "A new method for generating random correlation matrices," The Econometrics Journal, Royal Economic Society, vol. 27(2), pages 188-212.
    3. Michael W. McCracken & Serena Ng, 2016. "FRED-MD: A Monthly Database for Macroeconomic Research," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 34(4), pages 574-589, October.
    4. Ilya Archakov & Peter Reinhard Hansen, 2021. "A New Parametrization of Correlation Matrices," Econometrica, Econometric Society, vol. 89(4), pages 1699-1715, July.
    5. Andersen, Torben G. & Bollerslev, Tim & Diebold, Francis X. & Ebens, Heiko, 2001. "The distribution of realized stock return volatility," Journal of Financial Economics, Elsevier, vol. 61(1), pages 43-76, July.
    6. Peter Reinhard Hansen & Yiyao Luo, 2023. "Robust Estimation of Realized Correlation: New Insight about Intraday Fluctuations in Market Betas," Papers 2310.19992, arXiv.org.
    7. Linton, Oliver & McCrorie, J. Roderick, 1995. "Differentiation of an Exponential Matrix Function," Econometric Theory, Cambridge University Press, vol. 11(05), pages 1182-1185, October.
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