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On the equivalence of the weighted least squares and the generalised least squares estimators, with applications to kernel smoothing

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  • Alessandra Luati

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  • Tommaso Proietti

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Abstract

The paper establishes the conditions under which the generalised least squares estimator of the regression parameters is equivalent to the weighted least squares estimator. The equivalence conditions have interesting applications in local polynomial regression and kernel smoothing. Specifically, they enable to derive the optimal kernel associated with a particular covariance structure of the measurement error, where optimality has to be intended in the Gauss-Markov sense. For local polynomial regression it is shown that there is a class of covariance structures, associated with non-invertible moving average processes of given orders which yield the the Epanechnikov and the Henderson kernels as the optimal kernels.
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Suggested Citation

  • Alessandra Luati & Tommaso Proietti, 2011. "On the equivalence of the weighted least squares and the generalised least squares estimators, with applications to kernel smoothing," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(4), pages 851-871, August.
  • Handle: RePEc:spr:aistmt:v:63:y:2011:i:4:p:851-871
    DOI: 10.1007/s10463-009-0267-8
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    File URL: http://hdl.handle.net/10.1007/s10463-009-0267-8
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    References listed on IDEAS

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    1. Kramer, Walter & Hassler, Uwe, 1998. "Limiting efficiency of OLS vs. GLS when regressors are fractionally integrated," Economics Letters, Elsevier, vol. 60(3), pages 285-290, September.
    2. McAleer, Michael, 1992. "Efficient Estimation: The Rao-Zyskind Condition, Kruskal's Theorem and Ordinary Least Squares," The Economic Record, The Economic Society of Australia, vol. 68(200), pages 65-72, March.
    3. Frederick R. Macaulay, 1931. "Introduction to "The Smoothing of Time Series"," NBER Chapters,in: The Smoothing of Time Series, pages 17-30 National Bureau of Economic Research, Inc.
    4. Wallis, Kenneth F, 1981. "Models for X-11 and 'X-11-Forecast' Procedures for Preliminary and Revised Seasonal Adjustments," The Warwick Economics Research Paper Series (TWERPS) 198, University of Warwick, Department of Economics.
    5. Frederick R. Macaulay, 1931. "Appendices to "The Smoothing of Time Series"," NBER Chapters,in: The Smoothing of Time Series, pages 118-169 National Bureau of Economic Research, Inc.
    6. Peter C.B. Phillips & Joon Y. Park, 1986. "Asymptotic Equivalence of OLS and GLS in Regressions with Integrated Regressors," Cowles Foundation Discussion Papers 802, Cowles Foundation for Research in Economics, Yale University.
    7. Frederick R. Macaulay, 1931. "The Smoothing of Economic Time Series, Curve Fitting and Graduation," NBER Chapters,in: The Smoothing of Time Series, pages 31-42 National Bureau of Economic Research, Inc.
    8. Tian, Yongge & Wiens, Douglas P., 2006. "On equality and proportionality of ordinary least squares, weighted least squares and best linear unbiased estimators in the general linear model," Statistics & Probability Letters, Elsevier, vol. 76(12), pages 1265-1272, July.
    9. Phillips, Peter C.B., 1992. "Geometry of the Equivalence of OLS and GLS in the Linear Model," Econometric Theory, Cambridge University Press, vol. 8(01), pages 158-159, March.
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    More about this item

    Keywords

    Epanechnikov kernel; Local polynomial regression; Non-invertible moving average processes;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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