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Garson, G. D. (2013). Weighted Least Squares Regression. Asheboro, NC: Statistical Associates Publishers.

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ISBN: 978-1-62638-017-2
@c 2013 by G. David Garson and Statistical Associates Publishers. worldwide rights reserved in all languages and on all media. Permission is not granted to copy, distribute, or post e-books or passwords.


A graduate-level introduction and illustrated tutorial on weighted least squares regression (WLS) using SPSS, SAS, or Stata. WLS addresses the heteroscedasticity problem in OLS. In the face of heteroscedasticity, ordinary regression computes erroneous standard errors. This in turn makes significance tests incorrect.

Why we think it's important: Heteroscedasticity is a common regression problem which causes significance tests to be in error. Moreover, contrary to widely held belief, regression with robust standard errors does not substitute for WLS, which is rarely covered in general texts on multivariate analysis.

New in the 2013 edition:

Over twice as much depth (now 54 pp. compared to 19 in the 2012 edition)

  • Covers SPSS, SAS, and Stata
  • Discussion of a wide variety of weighting functions.
  • Explains why robust standard errors do not substitute for WLS
  • 25 new illustrations

    The full content is now available from Statistical Associates Publishers. Click here.

    Below is the unformatted table of contents.

    Table of Contents
    Overview of WLS	5
    What the researcher can expect if WLS regression is needed	6
    Are robust standard errors a substitute for WLS?	6
    Weighting with replicates	7
    Weight estimation functions	8
    Data example	9
    Key Terms and Concepts	9
    The homoscedasticity assumption in regression	9
    Weighted cases	11
    WLS in SPSS	11
    SPSS overview	11
    Testing for heteroscedasticity in SPSS	11
    The graphical method	11
    Statistical tests for heteroscedasticity	12
    Park test	13
    Breusch-Pagan test	15
    White's test	16
    Goldfeld-Quandt test	16
    Glejser test	17
    Weighting cases in SPSS	17
    Weight estimation input: Weighting with powers	17
    Weight estimation output: The log-likelihood values table	19
    Output from SPSS Weight Estimation	21
    SPSS OLS regression on weighted cases	23
    SPSS input	23
    SPSS output	24
    WLS in SAS	28
    Overview	28
    SAS input	29
    SAS output	34
    OLS output	34
    WLS output	38
    WLS in Stata	39
    Stata overview	39
    Stata input	39
    Stata output	42
    Unweighted linear regression	42
    Weighted least squares regression	44
    Assumptions	45
    Proper specification	45
    Proper weighting	46
    Data level	46
    Multivariate normality	46
    Linearity	46
    Independence	46
    Predictable variance	47
    Frequently Asked Questions	47
    Is WLS regression something that could be used with regression models other than OLS?	47
    What is SPSS syntax for WLS?	47
    How can one get OLS regression with robust standard errors in SAS?	48
    How does PROC  ROBUSTREG in SAS work?	48
    Heteroskedasticity or heteroscedasticity?	51
    Acknowledgment	52
    Bibliography	52
    Pagecount: 54