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
emedal easres egresso Aprl, 4 Weghtg as a Solto to Heteroscedastcy ( ) Varace Proportoal to k Varace Proportoal to coted
emedal easres egresso Aprl, 4. pt Example. 4. 5 3. 4 4. 6 5. 3 3 6. 5 3 7. 7 3 8. 3 4 9. 6 4. 9 4. ed Example, coted. regress Sorce SS df S Nmber of obs F(, 8).7 odel.5973554.5973554 Prob > F.44 esdal 9.8645 8 3.66336 -sqared.8 Adj -sqared -.37 Total 3.6 9 3.5 oot SE.94 Coef. Std. Err. t P>t [95% Cof. Iterval].46899.54743.85.4 -.79955.757 _cos 3.9543.595999.48.38.736 7.63794. predct e,resdal. toay (scatter e ), ttle(heteroscedastcty Check) Heteroscedastc esdals Example Heteroscedastcty Check esdals -4-4 3 4 3
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.86563.86563 Prob > F.4 esdal 3.6565995 8.3837494 -sqared.869 ------------------------------------------- Adj -sqared.787 Total 5.873 9.763585 oot SE.694 star Coef. Std. Err. t P> t [95% Cof. Iterval] ----------------------------------------------------------------------------- star 4.9643.779533 5.78..46 5.7586 _cos.459945.3896.4.37 -.49856.3375. predct estar,resdal. toay (scatter estar ), ttle(heteroscedastcty Check) Utrasformed ad Trasformed esdals Heteroscedastcty Check Heteroscedastcty Check esdals -4-4 esdals - -.5.5 3 4 3 4 Orgal ad Trasformed Data Example e star star estar. 4 -.43 4.. -.499. 5.587 5...5 3. 4 -.876..5 -.45 4. 6.4 3..5.548 5. 3 3 -.339..333 -.77 6. 3 5 -.339.667.333 -.4 7. 3 7.66.333.333.563 8. 4 3 -.8.75.5 -.679 9. 4 6.98.5.5.7. 4 9 3.98.5.5.8 4
emedal easres egresso Aprl, 4 Orgal ad Trasformed Eqatos, Example ˆ 3.95.463.596.554.8 ˆ.46 4.93.389.78.869 ˆ 4.93.46.78.389.8 Weghted east Sqares Geeralzed ( W) Wy b ( ) y b W Varaces Weghted east Sqares s { b } ( W) { b } s ( W) s ( ˆ ) p 5
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
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.385457.36433.6.87 -.34578.9539 _cos 4.388.64885 6.36..8595 5.3948 ----------------------------------------------------------------------------- varace.435.4335.4.6.945566.89945 _cos -.358.5755 -.79.74-4.6885.8865 7
emedal easres egresso Aprl, 4 Estmator Compared OS WS ˆ 3.95.463.596.554 ˆ 4.93.46.78.389 ˆ 4..385.648.36 Varaces der Homoscedastcty {} b ( ') ( ') ' { ; } { } I [ ] { }( [ ) ] ' ; ' ' [ ] [( ) ] ' ' I ' ' [ ' ] ' ' ( ' ) {} b ( ) { b} ( ) ( ) Varace der Heteroscedastcty { ; } { } Ω [ ] [( ) ] ' Ω ' ' [ ] [( ) ] ' ' Ω ' ' {} b ( ') {} b ( ) [ ] { b} ( ') 'Ω( ' ) 8
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] -----------------------------------------------------------------------------.46899.537694.86.44 -.776946.756 _cos 3.9543.69687 3.69.6.4837 6.476 Alteratve Heteroscedastcty Estmators OS WS Hber-Whte ˆ 3.95.463.596.554 ˆ 4.93.46.78.389 ˆ 4..385.648.36 ˆ 3.95.463.7.537 9
emedal easres egresso Aprl, 4 Hoses Stadard Errors ft dm y95 area _cos OS 3. 89.38 783.5 994.46 5.7 WS.3 85.6 6784.5 8449. 958.56 H-W 3.8 96.5 7436.93 793.35 76.6.6 73.97 543.5 669.63 7935.9 Hoses Example, Coeffcets ft dm y95 area _cos OS 63. -9.67-38.47 45.39 49634.3 WS 6.87-6.6 66. 34.45 5386.55 H-W 63. -9.67-38.47 45.39 49634.3 56.6-55.43 5356.58 434.6 655.3 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
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