Political Science 552

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Ὀλυμπιόδωρος Κανακάρης-Ρούφος
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1 emedal easres egresso Aprl, 4 Poltcal Scece 55 emedal easres Otlers Dscardg Otlers Trcatg Otlers obst estmato o A/AD east Absolte esdals/east Absolte Devatos (qreg Stata) o S east eda east Sqares o IS Iteratvely eeghted east Sqares (rreg Stata) The Heteroscedastcty Problem

2 emedal easres egresso Aprl, 4 Weghtg as a Solto to Heteroscedastcy ( ) Varace Proportoal to k Varace Proportoal to coted

3 emedal easres egresso Aprl, 4. pt Example ed Example, coted. regress Sorce SS df S Nmber of obs F(, 8).7 odel Prob > F.44 esdal sqared.8 Adj -sqared -.37 Total oot SE.94 Coef. Std. Err. t P>t [95% Cof. Iterval] _cos predct e,resdal. toay (scatter e ), ttle(heteroscedastcty Check) Heteroscedastc esdals Example Heteroscedastcty Check esdals

4 emedal easres egresso Aprl, 4 Example, eghted reslts. trasformato for eghtg. ge star/. ge star/. regress star star Sorce SS df S Nmber of obs F(, 8) 33.4 odel Prob > F.4 esdal sqared Adj -sqared.787 Total oot SE.694 star Coef. Std. Err. t P> t [95% Cof. Iterval] star _cos predct estar,resdal. toay (scatter estar ), ttle(heteroscedastcty Check) Utrasformed ad Trasformed esdals Heteroscedastcty Check Heteroscedastcty Check esdals -4-4 esdals Orgal ad Trasformed Data Example e star star estar

5 emedal easres egresso Aprl, 4 Orgal ad Trasformed Eqatos, Example ˆ ˆ ˆ Weghted east Sqares Geeralzed ( W) Wy b ( ) y b W Varaces Weghted east Sqares s { b } ( W) { b } s ( W) s ( ˆ ) p 5

6 emedal easres egresso Aprl, 4 6 Weghted ( ) ( ) ( ) p p, exp π ( ) ( ) p p, exp π Geeralzed east Sqares W odfyg Normal for Heteroscedastcty ( ) ( ) exp, x π ( ) ( ) exp, x π

7 emedal easres egresso Aprl, 4 odelg Heteroscedastcty e (, ) exp ( x) π ( Z Z, Z ) g, 3... Z k γ γ Z γ Z γ 3Z3... γ k Zk exp 3 3 [ γ γ Z γ Z γ Z... γ Z ] zγ [ z γ] e exp k k odelg Heteroscedastcty coted (, ) exp ( x) π zγ [ z γ] e exp (, ) exp ( x) z γ πe z γ e Example odelg varace Uses rote (hetreg.sta) rtte by Charles Frakl do d:\corses\ps55\examples\hetreg.sta ml model lf hetreg (slopes:)(varace: ) ml max Coef. Std. Err. z P> z [95% Cof. Iterval] slopes _cos varace _cos

8 emedal easres egresso Aprl, 4 Estmator Compared OS WS ˆ ˆ ˆ Varaces der Homoscedastcty {} b ( ') ( ') ' { ; } { } I [ ] { }( [ ) ] ' ; ' ' [ ] [( ) ] ' ' I ' ' [ ' ] ' ' ( ' ) {} b ( ) { b} ( ) ( ) Varace der Heteroscedastcty { ; } { } Ω [ ] [( ) ] ' Ω ' ' [ ] [( ) ] ' ' Ω ' ' {} b ( ') {} b ( ) [ ] { b} ( ') 'Ω( ' ) 8

9 emedal easres egresso Aprl, 4 s Hber-Whte Estmator 'Ω or 'Ω ' Ω 'S e S e e { b} ( ') ( 'S )( ' ). regress,robst Hber-Whte example egresso th robst stadard errors Nmber of obs F(, 8).74 Prob > F.444 -sqared.8 oot SE.94 obst Coef. Std. Err. t P> t [95% Cof. Iterval] _cos Alteratve Heteroscedastcty Estmators OS WS Hber-Whte ˆ ˆ ˆ ˆ

10 emedal easres egresso Aprl, 4 Hoses Stadard Errors ft dm y95 area _cos OS WS H-W Hoses Example, Coeffcets ft dm y95 area _cos OS WS H-W ltcollearty emedes Delete varables escale varables by ceterg o Ca solve collearty problems teractos o Also orks th polyomals Data redcto by combg varables o Prcpal compoets aalyss o Factor aalyss dge regresso

11 emedal easres egresso Aprl, 4 dge egresso ( ' ) b ' y r b r y b ry b ( r ci) b ry ( r ci) ry dge Trace

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