Lithuanian Mathematical Journal

, Volume 59, Issue 2, pp 276–293 | Cite as

Wavelet estimation in time-varying coefficient models

  • Xingcai ZhouEmail author
  • Beibei Ni
  • Chunhua Zhu


The paper is concerned with the estimation of a time-varying coefficient time series model, which is used to characterize the nonlinearity and trending phenomenon. We develop the wavelet procedures to estimate the coefficient functions and the error variance. We establish asymptotic properties of the proposed wavelet estimators under the α-mixing conditions and without specifying the error distribution. These results can be used to make asymptotically valid statistical inference.


time-varying coefficient model wavelet estimation convergence rate asymptotic normality 


primary 62G05 secondary 60G07 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    A. Antoniadis, G. Gregoire, and I.W. McKeague, Wavelet methods for curve estimation, J. Am. Stat. Assoc., 89(428): 1340–1353, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Z. Cai, Trending time-varying coefficient time series models with serially correlated errors, J. Econom., 136(1):163–188, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Z. Cai and E. Masry, Nonparametric estimation in nonlinear ARX time series models: Projection and linear fitting, Econom. Theory, 16:465–501, 2000.zbMATHCrossRefGoogle Scholar
  4. 4.
    Y. Chang and E. Martinez-Chombo, Electricity demand analysis using cointegration and error-correctionmodels with time varying parameters: The Mexican case, Working paper, Department of Economics, Rice University, Houston, TX, 2003.Google Scholar
  5. 5.
    J.H. Cochrane, Asset Pricing, Princeton Univ. Press, Englewood Cliffs, NJ, 2001.Google Scholar
  6. 6.
    D.L. Donoho and I.M. Johnstone, Ideal spatial adaption by wavelet shrinkage, Biometrika, 81(3):425–455, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    P. Doukhan, Mixing: Properties and Examples, Lect. Notes Stat., Vol. 85, Springer, Berlin, 1994.Google Scholar
  8. 8.
    G.L. Fan, H.Y. Liang, and J.F. Wang, Statistical inference for partially time-varying coefficient errors-in-variables models, J. Stat. Plann. Inference, 143(3):505–519, 2013.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    J. Fan and I. Gijbels, Local polynomial modeling and its application, Chapman & Hall, London, 1996.zbMATHGoogle Scholar
  10. 10.
    J. Fan and Q. Yao, Nonlinear Time Series: Nonparametric and Parametric methods, Springer, New York, 2003.zbMATHCrossRefGoogle Scholar
  11. 11.
    J. Fan, Q. Yao, and Z. Cai, Adaptive varying-coefficient linear models, J. R. Stat. Soc., Ser. B, Stat. Methodol., 65(1): 57–80, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    J. Fan and W. Zhang, Statistical estimation in varying coefficient models, Ann. Stat., 27(5):1491–1518, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    P.J. Green and B.W. Silverman, Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach, Chapman & Hall, London, 1994.zbMATHCrossRefGoogle Scholar
  14. 14.
    P. Hall and C.C. Heyde, Martingale Limit Theory and Its Applications, Academic Press, New York, 1980.zbMATHGoogle Scholar
  15. 15.
    P. Hall and P. Patil, On wavelet methods for estimating smooth function, Bernoulli, 1(1–2):41–58, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    W. Härdle, H. Liang, and J. Gao, Partially Linear Models, Physica-Verlag, New York, 2000.zbMATHCrossRefGoogle Scholar
  17. 17.
    W. Härdle and T.M. Stoker, Investigating smooth multiple regression by the method of average derivatives, J. Am. Stat. Assoc., 84(408):986–995, 1989.MathSciNetzbMATHGoogle Scholar
  18. 18.
    T. Hastie and R. Tibshirani, Varying-coefficient model, J. R. Stat. Soc., Ser. B, Stat. Methodol., 55:757–796, 1993.MathSciNetzbMATHGoogle Scholar
  19. 19.
    T.J. Hastie and R. Tibshirani, Generalized Additive Models, Chapman & Hall, London, 1990.zbMATHGoogle Scholar
  20. 20.
    D.R. Hoover, J.A. Rice, C.O. Wu, and L.P. Yang, Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data, Biometrika, 85(4):809–822, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    J.Z. Huang, C.O.Wu, and L. Zhou, Varying-coefficient models and basis function approximations for the analysis of repeated measurements, Biometrika, 89(1):111–128, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    J.Z. Huang, C.O. Wu, and L. Zhou, Polynomial spline estimation and inference for varying coefficient models with longitudinal data, Stat. Sin., 14(3):763–788, 2004.MathSciNetzbMATHGoogle Scholar
  23. 23.
    D. Li, J. Chen, and Z. Lin, Statistical inference in partially time-varying coefficient models, J. Stat. Plann. Inference, 141(2):995–1013, 2011.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Y.M. Li, S.C. Yang, and Y. Zhou, Consistency and uniformly asymptotic normality of wavelet estimator in regression model with associated samples, Stat. Probab. Lett., 78(17):2947–2956, 2008.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    H.Y. Liang and X.Z.Wang, Convergence rate of wavelet estimator in semiparametric models with dependentMA() error process, Chin. J. Appl. Probab. Stat., 26(1):35–46, 2010.zbMATHGoogle Scholar
  26. 26.
    Z.Y. Lin and C.R. Lu, Limit Theory for Mixing Dependent Random Variables, Math. Appl., Vol. 378, Science Press/Kluwer, Beijing/Dordrecht, 1996.Google Scholar
  27. 27.
    Y. Lu and Z. Li, Wavelet estimation in varying-coefficient models, Chin. J. Appl. Probab. Stat., 25(4):409–420, 2009.MathSciNetzbMATHGoogle Scholar
  28. 28.
    S. Orbe, E. Ferreira, and J. Rodríguez-Póo, A nonparametric method to estimate time varying coefficients under seasonal constraints, J. Nonparametric Stat., 12(6):779–806, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    S. Orbe, E. Ferreira, and J. Rodríguez-Póo, Nonparametric estimation of time varying parameters under shape restrictions, J. Econom., 126(1):53–77, 2005.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    S. Orbe, E. Ferreira, and J. Rodríguez-Póo, On the estimation and testing of time varying constraints in econometric models, Stat. Sin., 16(4):1313–1333, 2006.MathSciNetzbMATHGoogle Scholar
  31. 31.
    P.C.B. Phillips, Trending time series and macroeconomic activity: Some present and future challenges, J. Econom., 100(1):21–27, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    P.M. Robinson, Nonparametric estimation of time-varying parameters, in P. Hackl (Ed.), Statistical Analysis and Forecasting of Economic Structural Change, Springer, Berlin, 1989, pp. 164–253.Google Scholar
  33. 33.
    P.M. Robinson, Time-varying nonlinear regression, in P. Hackl and A.H. Westland (Eds.), Economic Structure Change Analysis and Forecasting, Springer, Berlin, 1991, pp. 179–190.Google Scholar
  34. 34.
    Q.M. Shao and H. Yu, Weak convergence for weighted empirical processes of dependent sequences, Ann. Stat., 24(4): 2098–2127, 1996.MathSciNetzbMATHGoogle Scholar
  35. 35.
    C.J. Stone, M. Hansen, C. Kooperberg, and Y.K. Truong, Polynomial splines and their tensor products in extended linear modelling, Ann. Stat., 25(4):1371–1470, 1997.zbMATHCrossRefGoogle Scholar
  36. 36.
    R. Tsay, Analysis of Financial Time Series, Wiley, New York, 2002.zbMATHCrossRefGoogle Scholar
  37. 37.
    B. Vidakovic, Statistical Modeling by Wavelet, JohnWiley & Sons, New York, 1999.zbMATHCrossRefGoogle Scholar
  38. 38.
    V.A. Volkonskii and Y.A. Rozanov, Some limit theorems for random functions, Theory Probab. Appl., 4:178–197, 1959.MathSciNetCrossRefGoogle Scholar
  39. 39.
    G.G. Walter, Wavelets and Orthogonal Systems with Applications, CRC Press, Boca Raton, FL, 1994.zbMATHGoogle Scholar
  40. 40.
    K. Wang, Asset pricing with conditioning information: A new test, J. Finance, 58(1):161–196, 2003.CrossRefGoogle Scholar
  41. 41.
    L. Wang, H. Li, and J.Z. Huang, Variable selection in nonparametric varying coefficient models for analysis of repeated measurements, J. Am. Stat. Assoc., 103(484):1556–1569, 2008.MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    L.H. Wang and H.Y. Cai, Wavelet change-point estimation for long memory non-parametric random design models, J. Time Ser. Anal., 31(2):86–97, 2010.MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    C.O. Wu, C. Chiang, and D.R. Hoover, Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data, J. Am. Stat. Assoc., 93(444):1388–1403, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    X. Zhou and J. You, Wavelet estimation in varying-coefficient partially linear regression models, Stat. Probab. Lett., 68(1):91–104, 2004.MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    X.C. Zhou and J.G. Lin, Asymptotic properties of wavelet estimators in semiparametric regression models under dependent errors, J. Multivariate Anal., 122:251–270, 2013.MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    X.C. Zhou and J.G. Lin, On complete convergence for strong mixing sequences, Stochastics, 85(2):262–271, 2013.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Statistics and Data ScienceNanjing Audit UniversityNanjingChina

Personalised recommendations