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WeightedLeastSquaresApp - Weighted LeastSquares Regression...

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Weighted Least- Squares Regression A technique for correcting the problem of heteroskedasticity by log- likelihood estimation of a weight that adjusts the errors of prediction Weighted Least-Squares Regression: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
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Key Concepts ***** Weighted Least-Squares Regression OLS Parameter estimates as: Unbiased Efficient BLUE Theoretical Sampling distribution of b Standard error of b Relationship between the standard error of b and: The variance of X The residual sum of squares The sample size Gauss-Markov Theorem Assumptions about the errors (e) in regression analysis and the consequences of their violation: e is uncorrelated with X e has the same variance across all levels of X The values of e are independent of each other e is normally distributed The concepts of homoskedasticity and heteroskedasticity of the error distributions The concept of autocorrelation or serial correlation Spurious relationships Collinear relationships Intervening relationships Techniques for identifying heteroskedasticity Graphic Statistical White’s Test for heteroskedasticity Rezidualizing a variable Techniques for identifying WLS weights Theory, the literature, or prior experience Regression of e 2 on X and transformation Log-likelihood estimation of w i SPSS weight estimation procedure SPSS WLS procedure Weighted Least-squares Regression: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University 2
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Overview Theoretical sampling distribution of b Assumptions about errors in regression Identifying heteroskedasticity The concept of weighted least-squares regression Methods for estimating weights Regressing e i 2 on X Log-likelihood estimation of weights Using WLS>> command in SPSS Weighted Least-squares Regression: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University 3
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References White, Halbert (1980) A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity . Econometrica 48:817-838. Graybill, Fraklin A. and Iyer, Hariharan K. (1994) Regression Analysis: Concepts and Applications. Duxbury Press 571-592. Freund, Rudolf J. and Wilson, William J. (1998) Regression Analysis: Statistical Modeling of a Response Variable. Academic Press 378-382. McClendon, McKee J. (1994) Multiple Regression and Causal Analysis. F. E. Peacock Publishers, Inc. 138-146, 174-181, 189-197. Weighted Least-squares Regression: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University 4
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Violation of OLS Regression Assumptions Y = a + b 1 X 1 + b 2 X 2 + … + b k X k OLS regression makes various assumptions about the errors that result from a regression model. If these assumptions are met … One can assume that the estimates of the regression constant (a) and the regression coefficients (b k ) are Unbiased : Replications of the study will yield values of a and b k which will be distributed on either side of their respective parameters α and β k Efficient : The standard errors of a and b k
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