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Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors


  • Hallin, Marc
  • Paindaveine, Davy


We develop optimal rank-based procedures for testing affine-invariant linear hypotheses on the parameters of a multivariate general linear model with elliptical VARMA errors. We propose a class of optimal procedures that are based either on residual (pseudo-)Mahalanobis signs and ranks, or on absolute interdirections and lift-interdirection ranks, i.e., on hyperplane-based signs and ranks. The Mahalanobis versions of these procedures are strictly affine-invariant, while the hyperplane-based ones are asymptotically affine-invariant. Both versions generalize the univariate signed rank procedures proposed by Hallin and Puri (J. Multivar. Anal. 50 (1994) 175), and are locally asymptotically most stringent under correctly specified radial densities. Their AREs with respect to Gaussian procedures are shown to be convex linear combinations of the AREs obtained in Hallin and Paindaveine (Ann. Statist. 30 (2002) 1103; Bernoulli 8 (2002) 787) for the pure location and purely serial models, respectively. The resulting test statistics are provided under closed form for several important particular cases, including multivariate Durbin-Watson tests, VARMA order identification tests, etc. The key technical result is a multivariate asymptotic linearity result proved in Hallin and Paindaveine (Asymptotic linearity of serial and nonserial multivariate signed rank statistics, submitted).

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  • Hallin, Marc & Paindaveine, Davy, 2005. "Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors," Journal of Multivariate Analysis, Elsevier, vol. 93(1), pages 122-163, March.
  • Handle: RePEc:eee:jmvana:v:93:y:2005:i:1:p:122-163

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    References listed on IDEAS

    1. Marc Hallin & Madan Lal Puri, 1994. "Aligned rank tests for linear models with autocorrelated errors," ULB Institutional Repository 2013/2045, ULB -- Universite Libre de Bruxelles.
    2. Marc Hallin & Madan Lal Puri, 1995. "A multivariate Wald-Wolfowitz rank test against serial dependence," ULB Institutional Repository 2013/2051, ULB -- Universite Libre de Bruxelles.
    3. Marc Hallin & Bas Werker, 2003. "Semiparametric efficiency, distribution-freeness, and invariance," ULB Institutional Repository 2013/2119, ULB -- Universite Libre de Bruxelles.
    4. Hallin, Marc & Ingenbleek, Jean-Francois & Puri, Madan L., 1989. "Asymptotically most powerful rank tests for multivariate randomness against serial dependence," Journal of Multivariate Analysis, Elsevier, vol. 30(1), pages 34-71, July.
    5. Bernard Garel & Marc Hallin, 1995. "Local asymptotic normality of multivariate ARMA processes with a linear trend," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(3), pages 551-579, September.
    6. Hannu Oja, 1999. "Affine Invariant Multivariate Sign and Rank Tests and Corresponding Estimates: a Review," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(3), pages 319-343.
    7. Hallin, M. & Puri, M. L., 1994. "Aligned Rank Tests for Linear Models with Autocorrelated Error Terms," Journal of Multivariate Analysis, Elsevier, vol. 50(2), pages 175-237, August.
    8. Marc Hallin & Bernard Garel, 1999. "Rank-based AR order identification," ULB Institutional Repository 2013/2087, ULB -- Universite Libre de Bruxelles.
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    Cited by:

    1. Hallin, M. & van den Akker, R. & Werker, B.J.M., 2012. "Rank-based Tests of the Cointegrating Rank in Semiparametric Error Correction Models," Discussion Paper 2012-089, Tilburg University, Center for Economic Research.
    2. Paindaveine, Davy, 2006. "A Chernoff-Savage result for shape:On the non-admissibility of pseudo-Gaussian methods," Journal of Multivariate Analysis, Elsevier, vol. 97(10), pages 2206-2220, November.
    3. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434,, revised Sep 2015.
    4. Bramati, Maria Caterina, 2013. "Optimal rank-based tests for block exogeneity in vector autoregressions," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 141-162.
    5. Hallin, M. & Werker, B.J.M. & van den Akker, R., 2015. "Optimal Pseudo-Gaussian and Rank-based Tests of the Cointegration Rank in Semiparametric Error-correction Models," Discussion Paper 2015-001, Tilburg University, Center for Economic Research.
    6. Marc Hallin & Davide La Vecchia, 2017. "A Simple R-Estimation Method for Semiparametric Duration Models," Working Papers ECARES ECARES 2017-01, ULB -- Universite Libre de Bruxelles.
    7. Francq, C. & Jiménez-Gamero, M.D. & Meintanis, S.G., 2017. "Tests for conditional ellipticity in multivariate GARCH models," Journal of Econometrics, Elsevier, vol. 196(2), pages 305-319.
    8. Paindaveine, Davy, 2009. "On Multivariate Runs Tests for Randomness," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1525-1538.
    9. Hallin, Marc & van den Akker, Ramon & Werker, Bas J.M., 2016. "Semiparametric error-correction models for cointegration with trends: Pseudo-Gaussian and optimal rank-based tests of the cointegration rank," Journal of Econometrics, Elsevier, vol. 190(1), pages 46-61.
    10. Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2009. "Optimal rank-based testing for principal component," Working Papers ECARES 2009_013, ULB -- Universite Libre de Bruxelles.
    11. Marc Hallin & Yves-Caoimhin Swan & Thomas Verdebout, 2013. "A Serial Version of Hodges and Lehmann's "6/pi Result"," Working Papers ECARES ECARES 2013-17, ULB -- Universite Libre de Bruxelles.
    12. Serfling, Robert & Wang, Yunfei, 2016. "On Liu’s simplicial depth and Randles’ interdirections," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 235-247.


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