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Generalized run tests for heteroscedastic time series


  • Marc Hallin
  • Jean-Marie Dufour
  • Ivan Mizera


The problem of testing for nonhomogeneous white noise (i.e. independently but possibly nonidentically distributed observations, with a common, specified or unspecified, median) against alternatives of serial dependence is considered. This problem includes as a particular case the important problem of testing for heteroscedastic white noise. When the value of the common median is specified, invariance arguments suggest basing this test on a generalized version of classical runs: the generalized runs statistics. These statistics yield a run-based correlogram concept with exact (under the hypothesis of nonhomogeneous white noise) p-values. A run-based portmanteau test is also provided. The local powers and asymptotic relative efficiencies (AREs) of run-based correlograms and the corresponding run-based tests with respect to their traditional parametric counterparts (based on classical correlograms) are investigated and explicitly computed. In practice, however, the value of the exact median of the observations is seldom specified. For such situations, we propose two different solutions. The first solution is based on the classical idea of replacing the unknown median by its empirical counterpart, yielding aligned runs statistics. The asymptotic equivalence between exact and aligned runs statistics is established under extremely mild assumptions. These assumptions do not require that the empirical median consistently estimates the exact one, so that the continuity properties usually invoked in this context are totally helpless. The proofs we are giving are of a combinatorial nature, and related to the so-called Banach match box problem. The second solution is a finite-sample, nonasymptotic one, yielding (for fixed n) strictly conservative testing procedures, irrespectively of the underlying densities. Instead of the empirical median, a nonparametric confidence interval for the unknown median is considered. Run-based correlograms can be expected to play the same role in the statistical analysis of time series with nonhomogeneous innovation process as classical correlograms in the traditional context of second-order stationary ARMA series.

Suggested Citation

  • Marc Hallin & Jean-Marie Dufour & Ivan Mizera, 1998. "Generalized run tests for heteroscedastic time series," ULB Institutional Repository 2013/2077, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:ulb:ulbeco:2013/2077
    Note: SCOPUS: ar.j

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    Cited by:

    1. Jean-Marie Dufour, 2003. "Identification, weak instruments, and statistical inference in econometrics," Canadian Journal of Economics, Canadian Economics Association, vol. 36(4), pages 767-808, November.
    2. Linton, Oliver & Whang, Yoon-Jae, 2003. "A quantilogram approach to evaluating directional predictability," LSE Research Online Documents on Economics 2112, London School of Economics and Political Science, LSE Library.
    3. Dufour, Jean-Marie, 2006. "Monte Carlo tests with nuisance parameters: A general approach to finite-sample inference and nonstandard asymptotics," Journal of Econometrics, Elsevier, vol. 133(2), pages 443-477, August.
    4. BEAULIEU, Marie-Claude & DUFOUR, Jean-Marie & KHALAF, Lynda, 2005. "Exact Multivariate Tests of Asset Pricing Models with Stable Asymmetric Distributions," Cahiers de recherche 2005-04, Universite de Montreal, Departement de sciences economiques.
    5. Dufour, Jean-Marie & Torres, Olivier, 2000. "Markovian processes, two-sided autoregressions and finite-sample inference for stationary and nonstationary autoregressive processes," Journal of Econometrics, Elsevier, vol. 99(2), pages 255-289, December.
    6. Paindaveine, Davy, 2009. "On Multivariate Runs Tests for Randomness," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1525-1538.
    7. Rainer Dyckerhoff & Christophe Ley & Davy Paindaveine, 2014. "Depth-Based Runs Tests for bivariate Central Symmetry," Working Papers ECARES ECARES 2014-03, ULB -- Universite Libre de Bruxelles.
    8. Matteo Barigozzi & Marc Hallin, 2018. "Generalized Dynamic Factor Models and Volatilities: Consistency, Rates, and Prediction Intervals," Working Papers ECARES 2018-33, ULB -- Universite Libre de Bruxelles.
    9. 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.
    10. Davy Paindaveine & Thomas Verdebout, 2013. "Universal Asymptotics for High-Dimensional Sign Tests," Working Papers ECARES ECARES 2013-40, ULB -- Universite Libre de Bruxelles.


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