, Volume 27, Issue 1, pp 3–26 | Cite as

Serial independence tests for innovations of conditional mean and variance models

  • Kilani Ghoudi
  • Bruno Rémillard
Original Paper


In this paper, one studies the asymptotic behavior of empirical processes based on consecutive residuals of univariate conditional mean and variance models. These processes are then used to develop tests of serial independence of the innovations. Even if the limiting distributions of the empirical processes depend on unknown parameters, it is shown that a Monte Carlo method based on the so-called multipliers can be applied to estimate the P values of the proposed test statistics. A simulation study is carried out to demonstrate the effectiveness of the proposed tests and the behavior of the statistics is also studied under contiguous alternatives.


Independence tests Serial independence Randomness GARCH models Residuals Squared residuals Empirical processes Empirical copula Multipliers Bootstrap 

Mathematics Subject Classification

Primary 60F05 Secondary 62G09 62G30 

Supplementary material

11749_2016_521_MOESM1_ESM.pdf (290 kb)
Supplementary material 1 (pdf 289 KB)


  1. Andreou E, Werker BJ (2015) Residual-based rank specification tests for AR-GARCH type models. J Econ 185:305–331MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bai J (2003) Testing parametric conditional distributions of dynamic models. Rev Econ Stat 85:531–549CrossRefGoogle Scholar
  3. Berkes I, Horváth L, Kokoszka P (2003) Asymptotics for GARCH squared residual correlations. Econ Theory 19:515–540MathSciNetGoogle Scholar
  4. Delgado MA (1996) Testing serial independence using the sample distribution function. J Time Ser Anal 17:271–285MathSciNetCrossRefzbMATHGoogle Scholar
  5. Du Z (2016) Nonparametric bootstrap tests for independence of generalized errors. Econ J 19:55–83MathSciNetGoogle Scholar
  6. Du Z, Escanciano JC (2015) A nonparametric distribution-free test for serial independence of errors. Econ Rev 34(6–10):1011–1034MathSciNetCrossRefGoogle Scholar
  7. Duchesne P, Ghoudi K, Rémillard B (2012) On testing for independence between the innovations of several time series. Can J Stat 40:447–479MathSciNetCrossRefzbMATHGoogle Scholar
  8. Durrett R (1996) Probability: theory and examples. Duxbury Press, BelmontzbMATHGoogle Scholar
  9. Escanciano JC, Lobato IN (2009) An automatic portmanteau test for serial correlation. J Econ 151(2):140–149MathSciNetCrossRefzbMATHGoogle Scholar
  10. Francq C, Zakoïan J-M (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10:605–637MathSciNetCrossRefzbMATHGoogle Scholar
  11. Genest C, Ghoudi K, Rémillard B (2007) Rank-based extensions of the Brock Dechert Scheinkman test for serial dependence. J Am Stat Assoc 102:1363–1376Google Scholar
  12. Genest C, Quessy J-F, Rémillard B (2007) Asymptotic local efficiency of Cramér–von Mises tests for multivariate independence. Ann Stat 35:166–191Google Scholar
  13. Genest C, Rémillard B (2004) Tests of independence or randomness based on the empirical copula process. Test 13:335–369MathSciNetCrossRefzbMATHGoogle Scholar
  14. Ghoudi K, Kulperger RJ, Rémillard B (2001) A nonparametric test of serial independence for time series and residuals. J Multivar Anal 79:191–218MathSciNetCrossRefzbMATHGoogle Scholar
  15. Ghoudi K, Rémillard B (2014) Comparison of specification tests for GARCH models. Comput Stat Data Anal 76:291–300MathSciNetCrossRefGoogle Scholar
  16. Ghoudi K, Rémillard B (2015) Diagnostic tests for innovations of ARMA models using empirical processes of residuals. In: Dawson D, Kulik R, Ould Haye M, Szyszkowicz B, Zhao Y (eds) Asymptotic laws and methods in stochastics, Fields Institute communications. Springer, New York, pp 239–282Google Scholar
  17. Hong Y, White H (2005) Asymptotic distribution theory for nonparametric entropy measures of serial dependence. Econometrica 73:837–901MathSciNetCrossRefzbMATHGoogle Scholar
  18. Horváth L, Kokoszka P, Teyssière G (2004) Bootstrap misspecification tests for ARCH based on the empirical process of squared residuals. J Stat Comput Simul 74:469–485MathSciNetCrossRefzbMATHGoogle Scholar
  19. Iqbal F (2013) Diagnostic checking for GARCH-type models. Commun Stat Theory Methods 42(6):934–953MathSciNetCrossRefzbMATHGoogle Scholar
  20. Li WK, Mak TK (1994) On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity. J Time Ser Anal 15:627–636MathSciNetCrossRefzbMATHGoogle Scholar
  21. Ling S, Li WK (1997) On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. J Am Stat Assoc 92:1184–1194MathSciNetCrossRefzbMATHGoogle Scholar
  22. Littell RC, Folks JL (1973) Asymptotic optimality of Fisher’s method of combining independent tests. II. J. Am. Stat. Assoc. 68:193–194MathSciNetCrossRefzbMATHGoogle Scholar
  23. McLeod AI, Li WK (1983) Diagnostic checking ARMA time series models using squared-residual autocorrelations. J Time Ser Anal 4:269–273MathSciNetCrossRefzbMATHGoogle Scholar
  24. Nelsen RB (1999) An introduction to copulas. In: Lecture notes in statistics, vol 139. Springer, New YorkGoogle Scholar
  25. Rémillard B (2012) Specification tests for dynamic models using multipliers. In: SSRN working paper series no. 2028558Google Scholar
  26. Rémillard B (2013) Statistical methods for financial engineering. Taylor & Francis, Chapman and Hall/CRC Financial Mathematics Series, New YorkGoogle Scholar
  27. Skaug HJ, Tjøstheim D (1993) A nonparametric test of serial independence based on the empirical distribution function. Biometrika 80:591–602MathSciNetCrossRefzbMATHGoogle Scholar
  28. van der Vaart AW, Wellner JA (1996) Springer series in statistics., Weak convergence and empirical processesSpringer, New YorkGoogle Scholar
  29. Wong H, Ling S (2005) Mixed portmanteau tests for time-series models. J Time Ser Anal 26:569–579MathSciNetCrossRefzbMATHGoogle Scholar
  30. Zhu K (2013) A mixed portmanteau test for ARMA-GARCH models by the quasi-maximum exponential likelihood estimation approach. J Time Ser Anal 34:230–237MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2016

Authors and Affiliations

  1. 1.Department of StatisticsUnited Arab Emirates UniversityAl AinUnited Arab Emirates
  2. 2.CRM, GERAD, Department of Decision SciencesHEC MontréalMontrealCanada

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