WLS_Overview - Regression Analysis Tutorial 183 LECTURE /...

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Regression Analysis Tutorial 183 Econometrics Laboratory C University of California at Berkeley C 22-26 March 1999 LECTURE / DISCUSSION Weighted Least Squares
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Regression Analysis Tutorial 184 Econometrics Laboratory C University of California at Berkeley C 22-26 March 1999 Introduction In a regression problem with time series data (where the variables have subscript "t" denoting the time the variable was observed), it is common for the error terms to be correlated across time, but with a constant variance; this is the problem of "autocorrelated disturbances," which will be considered in the next lecture. For regressions with cross-section data (where the subscript "i" now denotes a particular individual or firm at a point in time), it is usually safe to assume the errors are uncorrelated, but often their variances are not constant across individuals. This is known as the problem of heteroskedasticity (for "unequal scatter"); the usual assumption of constant error variance is referred to as homoskedasticity . Although the mean of the dependent variable might be a linear function of the regressors, the variance of the error terms might also depend on those same regressors, so that the observations might "fan out" in a scatter diagram, as illustrated in the following diagrams.
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Regression Analysis Tutorial 185 Econometrics Laboratory C University of California at Berkeley C 22-26 March 1999 . . . . . . . . . . . . . . . X Y . . . . . . . . . . . . . . . Y X . . . . . . . . . . . . . . . . . . . . . . . . . X Y "U-shaped" heteroskedasticity Increasing heteroskedasticity Homoskedasticity
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Regression Analysis Tutorial 186 Econometrics Laboratory C University of California at Berkeley C 22-26 March 1999 Assumptions of Heteroskedastic Linear Model C [simple linear model] or y i ' % @ x i % g i [multiple regression model]; y i ' j K j ' 1 x ij @ j % g i C [zero mean error terms]; E( g i ) ' 0 C [no serial correlation]; and Cov( g i , g i ) ) ' 0i f i ú i ) C [heteroskedasticity]. Var( g i ) ' 2 i ' 2 @ h i ,s o m eh i Sometimes also assume C normally distributed [optional]. g i
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Regression Analysis Tutorial 187 Econometrics Laboratory C University of California at Berkeley C 22-26 March 1999 y is ' % @ x is % g is , with Var( g is ) ' 2 ,e t c . y s ' 1 n s j n s i ' 1 y is , x s ' 1 n s j n s i ' 1 x is . y s ' % @ x s % g s , with Var( g s ) ' 2 @ 1 n s / 2 @ h s . Examples of Heteroskedastic Models 1. Grouped (Aggregate) Data For individual "i" in group "s" (i.e., state, region, time period) However, we only observe some group averages: Then
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Regression Analysis Tutorial 188 Econometrics Laboratory C
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This note was uploaded on 02/15/2012 for the course GEO 4167 taught by Professor Staff during the Spring '12 term at University of Florida.

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WLS_Overview - Regression Analysis Tutorial 183 LECTURE /...

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