Skip to main content

Nonparametric tests in linear model with autoregressive errors


In the linear regression model with possibly autoregressive errors, we construct a family of nonparametric tests for significance of regression, under a nuisance autoregression of model errors. The tests avoid an estimation of nuisance parameters, in contrast to the tests proposed in the literature. A simulation study illustrate their good performance.

This is a preview of subscription content, access via your institution.


  • Alpuim T, El-Shaarawi A (2008) On the efficiency of regression analysis with AR(p) errors. J Appl Stat 35:717–737

    MathSciNet  Article  Google Scholar 

  • Brockwell PJ, Davis RA (1991) Time series: theory and methods, 2nd edn. Springer, New York

    Book  Google Scholar 

  • El Bantli F, Hallin M (2001) Kolmogorov–Smirnov tests for AR models based on autoregression rank scores. Lecture notes-monograph series, selected proceedings of the symposium on inference for stochastic processes, vol 37, pp 111–124

  • Güney Y, Tuaç Y, Özdemir Ş, Arslan O (2020) Conditional maximum \(L_q\)-likelihood estimation for regression model with autoregressive error terms. Metrika 39:47

    MATH  Google Scholar 

  • Güney Y, Jurečková J, Arslan O (2020b) Averaged autoregression quantiles in autoregressive model. In: Maciak M et al (eds) Analytical methods in statistics, Springer proceedings in mathematics and statistics, vol.329. ISBN 978-3-030-48813-0

  • Gutenbrunner C, Jurečková J (1992) Regression rank scores and regression quantiles. Ann Stat 20:305–330

    MathSciNet  Article  Google Scholar 

  • Gutenbrunner C, Jurečková J, Koenker R, Portnoy S (1993) Tests of linear hypotheses based on regression rank scores. J Nonpar Stat 2:307–331

    MathSciNet  Article  Google Scholar 

  • Hallin M, Jurečková J (1999) Optimal tests for autoregressive models based on autoregression rank scores. Ann Stat 27:1385–1414

    MathSciNet  Article  Google Scholar 

  • Hallin M, Jurečková Picek J, Zahaf T (1999) Nonparametric tests of independence of two autoregressive time series based on autoregression rank scores. J Stat Plann Infer 75:319–330

    MathSciNet  Article  Google Scholar 

  • Jurečková J (1991) Tests of Kolmogorov–Smirnov type based on regression rank scores. In: Víšek JÁ (ed) 11th Prague international conference on information theory and statistics, decision functions and random processes, Academia, Prague, pp 41–49

  • Koenker R (2019) quantreg: quantile regression. R package version 5.54.

  • Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46:33–50

    MathSciNet  Article  Google Scholar 

  • Koenker R, Xiao Z (2004) Unit root quantile autoregression inference. J Am Stat Assoc 99:775–787

    MathSciNet  Article  Google Scholar 

  • Koenker R, Xiao Z, Autoregression Q (2006) Quantile autoregression. J Ame Stat Assoc 101:980–990

    MathSciNet  Article  Google Scholar 

  • Koul HL, Saleh AKME (1995) Autoregression quantiles and related rank scores processes. Ann Stat 23:670–689

    MathSciNet  Article  Google Scholar 

  • McKnight S, McKean J, Huitema B (2000) A double bootstrap method to analyze linear models with autoregressive error terms. Psychol Methods 5:87–101

    Article  Google Scholar 

  • Puri ML, Sen PK (1985) Nonparametric methods in general linear models. Wiley, New York

    MATH  Google Scholar 

  • R Core Team R (2020) A language and environment for statistical. R Foundation for Statistical Computing, Vienna, Austria

  • Tuaç Y, Güney Y, Senoglu B, Arslan O (2018) Robust parameter estimation of regression model with AR(p) error terms. Commun Stat Simul Comput 47:2343–2359

    MathSciNet  Article  Google Scholar 

  • Tuaç Y, Güney Y, Arslan O (2020) Parameter estimation of regression model with AR(p) error terms based on skew distributions with EM algorithm. Soft Comput 24:3309–3330

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Jana Jurečková.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research of J. Jurečková, J. Picek and M. Schindler was supported by the Grant GAčR 22-036036S.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jurečková, J., Arslan, O., Güney, Y. et al. Nonparametric tests in linear model with autoregressive errors. Metrika (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


  • Autoregression
  • Autoregression rank scores
  • Linear regression
  • Rank test
  • Regression rank scores