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Goodness-of-Fit tests with Dependent Observations

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  • Remy Chicheportiche
  • Jean-Philippe Bouchaud

Abstract

We revisit the Kolmogorov-Smirnov and Cram\'er-von Mises goodness-of-fit (GoF) tests and propose a generalisation to identically distributed, but dependent univariate random variables. We show that the dependence leads to a reduction of the "effective" number of independent observations. The generalised GoF tests are not distribution-free but rather depend on all the lagged bivariate copulas. These objects, that we call "self-copulas", encode all the non-linear temporal dependences. We introduce a specific, log-normal model for these self-copulas, for which a number of analytical results are derived. An application to financial time series is provided. As is well known, the dependence is to be long-ranged in this case, a finding that we confirm using self-copulas. As a consequence, the acceptance rates for GoF tests are substantially higher than if the returns were iid random variables.

Suggested Citation

  • Remy Chicheportiche & Jean-Philippe Bouchaud, 2011. "Goodness-of-Fit tests with Dependent Observations," Papers 1106.3016, arXiv.org, revised Aug 2011.
    • Handle: RePEc:arx:papers:1106.3016
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    Cited by:

    1. Vance Martin & Yoshihiko Nishiyama & John Stachurski, 2011. "A Goodness of Fit Test for Ergodic Markov Processes," ANU Working Papers in Economics and Econometrics 2011-557, Australian National University, College of Business and Economics, School of Economics.
    2. R'emy Chicheportiche & Jean-Philippe Bouchaud, 2013. "Some applications of first-passage ideas to finance," Papers 1306.3110, arXiv.org.
    3. Zhang, Hong & Wu, Zheyang, 2022. "The general goodness-of-fit tests for correlated data," Computational Statistics & Data Analysis, Elsevier, vol. 167(C).
    4. Partida, Alberto & Gerassis, Saki & Criado, Regino & Romance, Miguel & Giráldez, Eduardo & Taboada, Javier, 2022. "The chaotic, self-similar and hierarchical patterns in Bitcoin and Ethereum price series," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    5. Michele Caraglio & Fulvio Baldovin & Attilio L. Stella, 2021. "How Fast Does the Clock of Finance Run?—A Time-Definition Enforcing Stationarity and Quantifying Overnight Duration," JRFM, MDPI, vol. 14(8), pages 1-15, August.
    6. Chicheportiche, Rémy & Chakraborti, Anirban, 2017. "A model-free characterization of recurrences in stationary time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 474(C), pages 312-318.
    7. Alex D Washburne & Joshua W Burby & Daniel Lacker, 2016. "Novel Covariance-Based Neutrality Test of Time-Series Data Reveals Asymmetries in Ecological and Economic Systems," PLOS Computational Biology, Public Library of Science, vol. 12(9), pages 1-14, September.
    8. Morales, Raffaello & Di Matteo, T. & Gramatica, Ruggero & Aste, Tomaso, 2012. "Dynamical generalized Hurst exponent as a tool to monitor unstable periods in financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(11), pages 3180-3189.
    9. Jean-Philippe Bouchaud, 2021. "Radical Complexity," Papers 2103.09692, arXiv.org.
    10. Dangxing Chen, 2019. "Does the leverage effect affect the return distribution?," Papers 1909.08662, arXiv.org, revised Sep 2019.

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