21-Robust regression

# 21-Robust regression - WEIGHTED LEAST SQUARES - used to...

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WEIGHTED LEAST SQUARES - used to give more emphasis to selected points in the analysis What are weighted least squares? Recall, in OLS we minimize Q = = (Y - - X ) !! nn i=1 i=1 22 %" " 3 3!" 3 or Q = (Y - X ) (Y - X ) __ "" w In weighted least squares, we minimize Q = e w Y Y b X ) ij DD DD 2 2 3 33 " 3 œ cd The normal equations become b w + b wX = wY !3 "3 3 3 3 DD D b wX + b wX = wXY ! " 3 333 D i 2 For the intermediate calculations we get (where w is the weight) 3 w X , w Y , w X Y , w X , w Y , w t i j n D D DD DD DD DD DD 3 3 3 3 3 ij ij ij Calculation of the corrected sum of squares is y = w (Y Y..) = DD D D DD 2 ij ij ij (w Y ) w  DD DD 3 3 ij 2 the slope is = b = ^ " " " DD DD DD DD xy wYX x wX 3 3 3 3 3 3 ij ij Y X ) ij w 2 ( wX) w œ DD D DD DD the intercept is calculated with X and Y as usual, but these are calculated as –– Y.. = and X.. = DD DD DD DD wX ww 3 ij It the weights are 1, then all results are the same as OLS The variance is = = = 555 222 Y w 3 % 5 3 3 3 2

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WEIGHTED LEAST SQUARES handout In OLS we minimize Q = = (Y - - X ) or Q = ( - X ) ( - X ) !! nn i=1 i=1 22 %" " 3 3!" 3 w Y Y " " In weighted least squares, we minimize Q = e w Y b - b X ii DD 2 2 3 33 ! " 3 œ cd If the weights are 1, then all results are the same as OLS The normal equations become b w + b wX = wY !3 "3 3 3 3 D b wX + b wX = wXY ! " 3 333 D i 2 For the intermediate calculations we get (where w is the weight) 3 w X , w Y , w X Y , w X , w Y , w t i j ij n D D DD DD DD DD DD 3 3 3 3 3 ij ij ij Calculation of the corrected sum of squares is y = w (Y Y..) = w Y _ DD D D DD 2 ij ij ij (w Y ) w  DD DD 3 3 ij 2 the slope is = b = ^ " " " wYX wX DD DD 3 3 3 3 ij Y X ) ij w 2 ( wX) 2 w DD D DD DD the intercept is calculated with X and Y as usual, but the means are calculated as __ Y.. = and X.. = DD DD DD DD wX ww 3 ij The variance is = = = 555 222 Y w 3 % 5 3 3 3 2 For multiple regression Q = e = b b X b X ... b X ) DD DD 2 - - , i 2 3 ! "" ## : ": "  the normal equations (X WX)B = (X WY) the regression coefficients B = (X WX) (X WY) w" w - the variance-covariance matrix; = (X WX) 55 - , 5 2 w(Y Y) ^ np is estimated by MSE , based on weighted deviations; MSE = AA D 33 3 2
In a multiple regression, we minimize Q = e w Y Y b X b X ... b X

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## 21-Robust regression - WEIGHTED LEAST SQUARES - used to...

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