IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2307.03594.html
   My bibliography  Save this paper

Generalised Covariances and Correlations

Author

Listed:
  • Tobias Fissler
  • Marc-Oliver Pohle

Abstract

The covariance of two random variables measures the average joint deviations from their respective means. We generalise this well-known measure by replacing the means with other statistical functionals such as quantiles, expectiles, or thresholds. Deviations from these functionals are defined via generalised errors, often induced by identification or moment functions. As a normalised measure of dependence, a generalised correlation is constructed. Replacing the common Cauchy-Schwarz normalisation by a novel Fr\'echet-Hoeffding normalisation, we obtain attainability of the entire interval $[-1, 1]$ for any given marginals. We uncover favourable properties of these new dependence measures. The families of quantile and threshold correlations give rise to function-valued distributional correlations, exhibiting the entire dependence structure. They lead to tail correlations, which should arguably supersede the coefficients of tail dependence. Finally, we construct summary covariances (correlations), which arise as (normalised) weighted averages of distributional covariances. We retrieve Pearson covariance and Spearman correlation as special cases. The applicability and usefulness of our new dependence measures is illustrated on demographic data from the Panel Study of Income Dynamics.

Suggested Citation

  • Tobias Fissler & Marc-Oliver Pohle, 2023. "Generalised Covariances and Correlations," Papers 2307.03594, arXiv.org, revised Sep 2023.
  • Handle: RePEc:arx:papers:2307.03594
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2307.03594
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-1054, July.
    2. Victor Chernozhukov & Iván Fernández‐Val & Blaise Melly, 2013. "Inference on Counterfactual Distributions," Econometrica, Econometric Society, vol. 81(6), pages 2205-2268, November.
    3. Genest, Christian & Nešlehová, Johanna, 2007. "A Primer on Copulas for Count Data," ASTIN Bulletin, Cambridge University Press, vol. 37(2), pages 475-515, November.
    4. Linton, O. & Whang, Yoon-Jae, 2007. "The quantilogram: With an application to evaluating directional predictability," Journal of Econometrics, Elsevier, vol. 141(1), pages 250-282, November.
    5. Joe, H., 1993. "Parametric Families of Multivariate Distributions with Given Margins," Journal of Multivariate Analysis, Elsevier, vol. 46(2), pages 262-282, August.
    6. Guodong Li & Yang Li & Chih-Ling Tsai, 2015. "Quantile Correlations and Quantile Autoregressive Modeling," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 246-261, March.
    7. Han, Heejoon & Linton, Oliver & Oka, Tatsushi & Whang, Yoon-Jae, 2016. "The cross-quantilogram: Measuring quantile dependence and testing directional predictability between time series," Journal of Econometrics, Elsevier, vol. 193(1), pages 251-270.
    8. Natalia Nolde & Johanna F. Ziegel, 2016. "Elicitability and backtesting: Perspectives for banking regulation," Papers 1608.05498, arXiv.org, revised Feb 2017.
    9. Gert Stulp & Abraham P Buunk & Thomas V Pollet & Daniel Nettle & Simon Verhulst, 2013. "Are Human Mating Preferences with Respect to Height Reflected in Actual Pairings?," PLOS ONE, Public Library of Science, vol. 8(1), pages 1-7, January.
    10. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    11. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    12. Sourav Chatterjee, 2021. "A New Coefficient of Correlation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(536), pages 2009-2022, October.
    13. Tjøstheim, Dag & Hufthammer, Karl Ove, 2013. "Local Gaussian correlation: A new measure of dependence," Journal of Econometrics, Elsevier, vol. 172(1), pages 33-48.
    14. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Sander Barendse, 2017. "Interquantile Expectation Regression," Tinbergen Institute Discussion Papers 17-034/III, Tinbergen Institute.
    2. Tobias Fissler & Jana Hlavinová & Birgit Rudloff, 2021. "Elicitability and identifiability of set-valued measures of systemic risk," Finance and Stochastics, Springer, vol. 25(1), pages 133-165, January.
    3. Donald Lien & Zijun Wang, 2019. "Quantile information share," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 39(1), pages 38-55, January.
    4. Tobias Fissler & Michael Merz & Mario V. Wuthrich, 2021. "Deep Quantile and Deep Composite Model Regression," Papers 2112.03075, arXiv.org.
    5. Jozef Baruník & Tobias Kley, 2019. "Quantile coherency: A general measure for dependence between cyclical economic variables," The Econometrics Journal, Royal Economic Society, vol. 22(2), pages 131-152.
    6. Corbet, Shaen & Katsiampa, Paraskevi & Lau, Chi Keung Marco, 2020. "Measuring quantile dependence and testing directional predictability between Bitcoin, altcoins and traditional financial assets," International Review of Financial Analysis, Elsevier, vol. 71(C).
    7. Parente, Paulo M.D.C. & Smith, Richard J., 2011. "Gel Methods For Nonsmooth Moment Indicators," Econometric Theory, Cambridge University Press, vol. 27(1), pages 74-113, February.
    8. Davis, Richard & Drees, Holger & Segers, Johan & Warchol, Michal, 2018. "Inference on the tail process with application to financial time series modelling," LIDAM Discussion Papers ISBA 2018002, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    9. Lee, Ji Hyung, 2016. "Predictive quantile regression with persistent covariates: IVX-QR approach," Journal of Econometrics, Elsevier, vol. 192(1), pages 105-118.
    10. Zhao, Yixiu & Upreti, Vineet & Cai, Yuzhi, 2021. "Stock returns, quantile autocorrelation, and volatility forecasting," International Review of Financial Analysis, Elsevier, vol. 73(C).
    11. Ruodu Wang & Yunran Wei, 2020. "Risk functionals with convex level sets," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1337-1367, October.
    12. Riza Demirer & Rangan Gupta & Hossein Hassani & Xu Huang, 2020. "Time-Varying Risk Aversion and the Profitability of Carry Trades: Evidence from the Cross-Quantilogram," Economies, MDPI, vol. 8(1), pages 1-12, March.
    13. Bonaccolto, Giovanni & Caporin, Massimiliano & Maillet, Bertrand B., 2022. "Dynamic large financial networks via conditional expected shortfalls," European Journal of Operational Research, Elsevier, vol. 298(1), pages 322-336.
    14. Naeem, Muhammad Abubakr & Qureshi, Fiza & Arif, Muhammad & Balli, Faruk, 2021. "Asymmetric relationship between gold and Islamic stocks in bearish, normal and bullish market conditions," Resources Policy, Elsevier, vol. 72(C).
    15. Maria Marino & Alessio Farcomeni, 2015. "Linear quantile regression models for longitudinal experiments: an overview," METRON, Springer;Sapienza Università di Roma, vol. 73(2), pages 229-247, August.
    16. Kneib, Thomas & Silbersdorff, Alexander & Säfken, Benjamin, 2023. "Rage Against the Mean – A Review of Distributional Regression Approaches," Econometrics and Statistics, Elsevier, vol. 26(C), pages 99-123.
    17. Alomari, Mohammad & Mensi, Walid & Vo, Xuan Vinh & Kang, Sang Hoon, 2022. "Extreme return spillovers and connectedness between crude oil and precious metals futures markets: Implications for portfolio management," Resources Policy, Elsevier, vol. 79(C).
    18. Uribe, Jorge M. & Guillen, Montserrat & Mosquera-López, Stephania, 2018. "Uncovering the nonlinear predictive causality between natural gas and electricity prices," Energy Economics, Elsevier, vol. 74(C), pages 904-916.
    19. David Happersberger & Harald Lohre & Ingmar Nolte, 2020. "Estimating portfolio risk for tail risk protection strategies," European Financial Management, European Financial Management Association, vol. 26(4), pages 1107-1146, September.
    20. Timo Dimitriadis & Tobias Fissler & Johanna Ziegel, 2022. "Characterizing M-estimators," Papers 2208.08108, arXiv.org.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2307.03594. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.