WLS: WEIGHTED LEAST SQUARES
One of the critical assumptions of ordinary least squares regression is homoscedasticity: that the variance of residual error should be constant for all values of the independent(s). If the independent(s) has/have different error variance at different ranges of their values, then the estimates of the regression coefficients will have unduly large standard errors for some ranges of the dependent and too small for other ranges. The power of significance tests will be reduced, which is to say regression estimates will be inefficient. Weighted least squares (WLS) regression compensates for violation of the homoscedasticity assumption by weighting cases differentially: cases whose value on the dependent variable corresponds to large variances on the independent variable(s) count less and those with small variances count more in estimating the regression coefficients. That is, cases with greater weights contribute more to the fit of the regression line. The result is that the estimated coefficients are usually very close to what they would be in OLS regression, but under WLS regression their standard errors are smaller.
Apart from its main function in correcting for heteroscedasticity, WLS regression is sometimes also used to adjust fit to give less weight to distant points and outliers, or to give less weight to observations thought to be less reliable.

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Below is the unformatted table of contents.

Table of Contents WLS Weighted Least Squares Regression 5 Overview 5 Key Terms and Concepts 6 The homoscedasticity assumption in regression 6 Violation of homoscedasticity 6 Weighted cases 7 WLS in SPSS 7 SPSS overview 7 Weighting cases in SPSS 8 Weighting with powers 8 The log-likelihood values table 9 Weighting with replicates 10 WLS regression output from SPSS Analyze, Regression, Weight Estimation 11 Running OLS regression on weighted cases 12 SPSS 12 WLS regression output from SPSS Analyze, Regression, Linear 13 Weighted predicted/residual plots 14 Obtaining weighted predicted/residual plots in SPSS 15 Assumptions 17 Proper specification 17 Data level 17 Multivariate normality 17 Linearity 17 Independence 17 Predictable variance 17 Frequently Asked Questions 18 Is WLS regression something that could be used with regression models other than OLS? 18 Can you do things in SPSS syntax not available in the Weight Estimation and WLS dialogs? 18 Bibliography 18