, Volume 27, Issue 1, pp 70–94 | Cite as

Testing the adequacy of semiparametric transformation models

  • J. S. Allison
  • M. Hušková
  • S. G. Meintanis
Original Paper


We consider a semiparametric model whereby the response variable following a transformation can be expressed by means of a regression model. In this model, the form of the transformation is specified analytically (up to an unknown transformation parameter), while the regression function is completely unknown. We develop testing procedures for the null hypothesis that this semiparametric model adequately describes the data at hand. In doing so, the test statistic is formulated on the basis of Fourier-type conditional expectations, an idea first put forward by Bierens (J Econom 20:105–134, 1982). The asymptotic distribution of the test statistic is obtained under the null as well as under alternative hypotheses. Since the limit null distribution is nonstandard, a bootstrap version is utilized in order to actually carry out the test procedure. Monte Carlo results are included that illustrate the finite-sample properties of the new method.


Transformation model Goodness-of-fit test Nonparametric regression Bootstrap test 

Mathematics Subject Classification

62G08 62G09 62G10 



Simos Meintanis acknowledges support by the Special Account for Research Grants \((\hbox {E}\mathrm{{\Lambda }}\hbox {KE})\) (Research Grant 11699) of the National and Kapodistrian University of Athens. M. Hušková acknowledges support from Grant GACR 15-096635S. James Allison thanks the National Research Foundation of South Africa for financial support. The authors would also like to thank the referees for their constructive comments that led to an improvement of the paper.

Supplementary material

11749_2017_544_MOESM1_ESM.pdf (224 kb)
Supplementary material 1 (pdf 224 KB)


  1. Bierens HJ (1982) Consistent model specification tests. J Econom 20:105–134MathSciNetCrossRefMATHGoogle Scholar
  2. Breiman L, Friedman JH (1985) Estimating optimal transformations for multiple regression and correlation. J Am Stat Assoc 80:580–598MathSciNetCrossRefMATHGoogle Scholar
  3. Carrasco M, Florens JP (2000) Generalization of GMM to a continuum of moment conditions. Econom Theory 16:797–834MathSciNetCrossRefMATHGoogle Scholar
  4. Colling B, van Keilegom I (2016) Goodness-of-fit tests in semiparametric transformation models. Test 25:291–308Google Scholar
  5. Colling B, Heuchenne C, Samb R, Van Keilegom I (2015) Estimation of the error density in a semiparametric transformation model. Ann Inst Stat Math 67:1–18MathSciNetCrossRefMATHGoogle Scholar
  6. Davidson R, MacKinnon JG (1993) Estimation and inference in econometrics. Oxford University Press, New YorkMATHGoogle Scholar
  7. De Jong RM (1996) Bierens’ test under data dependence. J Econom 72:1–32MathSciNetCrossRefMATHGoogle Scholar
  8. Delgado M, González-Manteiga W (2001) Significance testing in nonparametric regression based on the bootstrap. Ann Stat 29:1469–1507MathSciNetCrossRefMATHGoogle Scholar
  9. Delgado MA, Fiteni I (2002) External bootstrap tests for parameter stability. J Econom 109:275–303MathSciNetCrossRefMATHGoogle Scholar
  10. Delgado M, Domínguez M, Lavergne P (2006) Consistent tests of conditional moment restrictions. Ann Econom Stat 81:33–67Google Scholar
  11. De Wet T, Venter JH (1973) Asymptotic distributions for quadratic forms with applications to tests of fit. Ann Stat 1:380–387MathSciNetCrossRefMATHGoogle Scholar
  12. Epps TW (1999) Limit behavior of the ICF test for normality under Graham Charlier alternatives. Stat Probab Lett 42:175–184CrossRefMATHGoogle Scholar
  13. Giacomini R, Politis DN, White H (2013) A warp-speed method for conducting Monte Carlo experiments involving bootstrap estimators. Econom Theory 29:567–589MathSciNetCrossRefMATHGoogle Scholar
  14. González-Manteiga W, Crujeiras RM (2013) An updated review of goodness-of-fit tests for regression models. Test 22:361–411MathSciNetCrossRefMATHGoogle Scholar
  15. Hlávka Z, Hušková M, Kirch C, Meintanis SG (2017) Fourier-type tests involving martingale difference processes. Econom Rev 36:468–492MathSciNetCrossRefGoogle Scholar
  16. Horowitz JL (2009) Transformation models. In: Semiparametric and non parametric methods in econometrics. Springer series in statistics. Springer, New York, pp 189–232Google Scholar
  17. Kasparis I (2010) The Bierens test for certain nonstationary models. J Econom 158:221–230MathSciNetCrossRefMATHGoogle Scholar
  18. Lavergne P, Patilea V (2013) Smooth minimum distance estimation and testing with conditional estimating equations: uniform in bandwidth theory. J Econom 177:47–59MathSciNetCrossRefMATHGoogle Scholar
  19. Linton OB, Chen R, Wang N, Härdle W (1997) An analysis of transformations for additive nonparametric regression. J Am Stat Assoc 92:1512–1521Google Scholar
  20. Linton OB, Sperlich S, van Keilegom I (2008) Estimation of a semiparametric transformation model. Ann Stat 36:686–718MathSciNetCrossRefMATHGoogle Scholar
  21. Mai Q, Zou H (2015) The fused Kolmogorov filter: a nonparametric model-free screening method. Ann Stat 4:1471–1497MathSciNetCrossRefMATHGoogle Scholar
  22. Mammen E (1993) Bootstrap and wild bootstrap for high dimensional linear models. Ann Stat 21:255–285MathSciNetCrossRefMATHGoogle Scholar
  23. Matteson DS, James NA (2014) A nonparametric approach for multiple change point analysis of multivariate data. J Am Stat Assoc 505:334–345MathSciNetCrossRefMATHGoogle Scholar
  24. Meintanis SG, Einbeck J (2015) Validation tests for semiparametric models. J Stat Comput Simul 85:131–146MathSciNetCrossRefGoogle Scholar
  25. Neumeyer N, Noh H, van Keilegom I (2016) Heteroscedastic semiparametric transformation models: estimation and testing for validity. Stat Sin 26:925–954MathSciNetMATHGoogle Scholar
  26. Nolan JP (2013) Multivariate elliptically contoured stable distribution: theory and estimation. Comput Stat 28:2067–2089MathSciNetCrossRefMATHGoogle Scholar
  27. Stute W, Zhu LX (2005) Nonparametric checks for single-index models. Ann Stat 33:1048–1083MathSciNetCrossRefMATHGoogle Scholar
  28. Stute W, González Manteiga W, Presedo Quindimil M (1998) Bootstrap approximations in model checks for regression. J Amer Statist Assoc 93:141–149Google Scholar
  29. Székely G, Rizzo M (2005) Hierarchical clustering via joint between-within distances: extending Ward’s minimum variance method. J Classif 22:151–183MathSciNetCrossRefMATHGoogle Scholar
  30. Székely GJ, Rizzo ML (2013) Energy statistics: a class of statistics based on distances. J Stat Plan Inference 143:1249–1272MathSciNetCrossRefMATHGoogle Scholar
  31. Tenreiro C (2009) On the choice of the smoothing parameter for the BHEP goodness-of-fit test. Comput Stat Data Anal 53:1038–1053MathSciNetCrossRefMATHGoogle Scholar
  32. Wu CFJ (1986) Jacknife, bootstrap and other resampling methods in regression analysis. Ann Stat 14:1261–1295CrossRefMATHGoogle Scholar
  33. Whang YJ (2000) Consistent bootstrap tests of parametric regression functions. J Econom 98:27–46MathSciNetCrossRefMATHGoogle Scholar
  34. Yeo I, Johnson R (2000) A new family of power transformations to improve normality or symmetry. Biometrika 87:954–959MathSciNetCrossRefMATHGoogle Scholar
  35. Zhu LX (2005) Checking the adequacy of a varying coefficients model. In: Nonparametric Monte Carlo tests and their applications. Lecture notes in statistics, vol 182. Springer, New York, pp 123–139Google Scholar
  36. Zhu LX, Ng KW (2003) Checking the adequacy of a partial linear model. Stat Sin 13:763–781MathSciNetMATHGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2017

Authors and Affiliations

  • J. S. Allison
    • 1
  • M. Hušková
    • 2
  • S. G. Meintanis
    • 1
    • 3
  1. 1.Unit for Business Mathematics and InformaticsNorth-West UniversityPotchefstroomSouth Africa
  2. 2.Department of Statistics, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  3. 3.Department of EconomicsNational and Kapodistrian University of AthensAthensGreece

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