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Adaptive varying co-efficient linear models

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  • Fan, Jianqing
  • Yao, Qiwei
  • Cai, Zongwu

Abstract

Varying-coefficient linear models arise from multivariate nonparametric regression, nonlinear time series modelling and forecasting, functional data analysis, longitudinal data analysis, and others. It has been a common practice to assume that the vary-coefficients are functions of a given variable which is often called an index. A frequently asked question is which variable should be used as the index. In this paper, we explore the class of the varying-coefficient linear models in which the index is unknown and is estimated as a linear combination of regression and/or other variables. This will enlarge the modelling capacity substantially. We search for the index such that the derived varying-coefficient model provides the best approximation to the underlying unknown multi-dimensional regression function in the least square sense. The search is implemented through the newly proposed hybrid backfitting algorithm. The core of the algorithm is the alternative iteration between estimating the index through a one-step scheme and estimating coefficient functions through a one-dimensional local linear smoothing. The generalised cross-validation method for choosing bandwidth is efficiently incorporated into the algorithm. The locally significant variables are selected in terms of the combined use of t-statistic and Akaike information criterion. We further extend the algorithm for the models with two indices. Simulation shows that the proposed methodology has appreciable flexibility to model complex multivariate nonlinear structure and is practically feasible with average modern computers. The methods are further illustrated through the Canadian mink-muskrat data in 1925-1994 and the pound/dollar exchange rates in 1974-1983.

Suggested Citation

  • Fan, Jianqing & Yao, Qiwei & Cai, Zongwu, 2003. "Adaptive varying co-efficient linear models," LSE Research Online Documents on Economics 5885, London School of Economics and Political Science, LSE Library.
  • Handle: RePEc:ehl:lserod:5885
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    File URL: http://eprints.lse.ac.uk/5885/
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    References listed on IDEAS

    as
    1. LeBaron, Blake, 1999. "Technical trading rule profitability and foreign exchange intervention," Journal of International Economics, Elsevier, vol. 49(1), pages 125-143, October.
    2. Cai, Zongwu & Fan, Jianqing & Yao, Qiwei, 2000. "Functional-coefficient regression models for nonlinear time series," LSE Research Online Documents on Economics 6314, London School of Economics and Political Science, LSE Library.
    3. Barnett,William A. & Powell,James & Tauchen,George E. (ed.), 1991. "Nonparametric and Semiparametric Methods in Econometrics and Statistics," Cambridge Books, Cambridge University Press, number 9780521424318.
    4. Hart, Jeffrey D. & Wehrly, Thomas E., 1993. "Consistency of cross-validation when the data are curves," Stochastic Processes and their Applications, Elsevier, vol. 45(2), pages 351-361, April.
    5. Jianqing Fan, 2000. "Simultaneous Confidence Bands and Hypothesis Testing in Varying-coefficient Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(4), pages 715-731.
    6. Barnett,William A. & Powell,James & Tauchen,George E. (ed.), 1991. "Nonparametric and Semiparametric Methods in Econometrics and Statistics," Cambridge Books, Cambridge University Press, number 9780521370905.
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    More about this item

    Keywords

    akaike information criterion; back-fitting algorithm; generalised cross-validation; local linear regression; local significant variable selection; one-step estimation; smoothing index.;

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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