Dagnostcs and Smple Remedaton February 9, 4 Poltcal Scence 55 Volatng Assumptons Key Assumptons E { Y; X } β + β X E (, ) ε ~ nor σ { ε,ε } for j j X s measured wthout error Small Sample Plot Feelng Thermometer-Bush 8 7 6 5 4 3 3 4 5 6 Party ID
Dagnostcs and Smple Remedaton February 9, 4 Large Sample Plots FT-Bush 4 6 8 4 6 PartyID Plots wth Jtterng FT-Bush 4 6 8 4 6 PartyID Resduals as Dagnostc Tool e Y Y $ e* e MSE
Dagnostcs and Smple Remedaton February 9, 4 Prototypcal Resdual Plots Data Ambguty Data Set Varable Obs. (a)-(c) X (a) Y (b) Y (c) Y (d) X (d) Y. 4 9.4 7.46 6.58 6.95 8.4 6.77 5.76 3 3. 7.58 8.74.74 7.7 4 5 9.. 8.8 8.33 8.77 9.6 7. 7.8 8.84 8.47 Y 3 +. 5 X 6 4. 9.96 8.84 7.4 7 6. 7.4 6.3 6.8 5.5 8 4. 4.6 3. 5.39 9..5 9..84 9.3 8.5 5.56 7. 4.8 7.6 6.4 7.9 5. 5.68 4.74 5.73 6.89 Seeng Is Recognzng 5 4 3 - - -3 5 5 5 4 3 - - -3 5 5 5 4 3 - - 5 5 5 4 3 - - -3 5 5 3
Dagnostcs and Smple Remedaton February 9, 4 Polynomal Relatonshp Testng for Lnearty SSTO SSE r SSTO SSTO SSE E SSTO SSR SSTO SSG SSTO Nonlnear Relatonshp FT-Bush 4 6 8 4 6 PartyID 4
Dagnostcs and Smple Remedaton February 9, 4 ANOVA for Nonlnearty Source SS df MS F Total SSTO n- Lnear r*ssto SSR/ Nonlnear E*SSTO k- SSG/k- Addtonal (E-r)*SSTO k- SSad/k- (E-r)/(k-) Error (-E)*SSTO n-k- SSE/n-k- (-E)/(n-k- SOURCE DF SS MS F p Regresson 38. 38. 7.49. Nonlnear 6 393.7 655. 3.98. Improvement 5 3.7 6..56 >. Error 4 87.5 46.9 Total 48. E-sq.954 (E-sq - R-sq).3 df 5 R-sq.93 ( - E-sq).46 df 4 F (.3/5)/(.46/4.46/4).6/.5.54 Lnearty Test n Stata. generate pdv53. anova v36 pd v53, contnuous(pd) sequental anova Number of obs 496 R-squared.344 Root MSE.53 Adj R-squared.345 Source Seq. SS df MS F Prob > F -----------+---------------------------------------------------- Model 3449.397 6 5348.38 3.. pd 3535.659 3535.659 744.37. v53 53.7376 5 34.7475 7.37. Resdual 676.587 489 4.8395 -----------+---------------------------------------------------- Total 93.983 495 6.8836 Grad GPA 4.5 4. 3.5 3..5 Outlers y.7x.4x.5x +.8667 3.33 3.66 r.348.37.4 5 7 9 3 5 7 GRE-Total 5
Dagnostcs and Smple Remedaton February 9, 4 Heteroscedastcty Modfed Levene Test e~ and ~ e d ~e e d e ~e t d s d { d d } Modfed Levene: Stata Commands * Modfed Levene Test drop _all use "D:\COURSES\PS55\EXAMPLES\grades.dta", clear * get medan for predctor and create splt for sample summarze gre_total, detal generate splt(gre_tot>) * do regresson, get resdual, and get medans regress grad_gpa gre_tot predct resd,r bysort splt: summarze resd, detal * get devatons separately for the two groups gen dabs(resd-.3335) f splt gen dabs(resd-.65) f splt ttest d d, unpared 6
Dagnostcs and Smple Remedaton February 9, 4 Modfed Levene: Stata Output. ttest d d, unpared Two-sample t test wth equal varances Varable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] d.69.457.5948.594377.6948 d 9.34855.39844.73547.5784.8475 combned 4.545.6667.43357.76367.683 dff -.435.45866 -.6886.68656 Degrees of freedom: 38 Ho: mean(d) - mean(d) dff Ha: dff < Ha: dff! Ha: dff > t -.568 t -.568 t -.568 P < t.37 P > t.64 P > t.6993 Breusch-Pagan Test log eσ γ + γ X SSE usual regresson SSE SSR * SSR from regresson on squared resduals SSR SSE χ * BP n Breusch-Pagan n Stata. regress grad_gpa gre_tot grad_gpa Coef. Std. Err. t P>t [95% Conf.Interval] gre_total.38.89..4.58.748 _cons 3.3589.36 4.77..84994 3.75564. hettest Breusch-Pagan / Cook-Wesberg test for heteroskedastcty Ho: Constant varance Varables: ftted values of grad_gpa ch().8 Prob > ch.7758 7
Dagnostcs and Smple Remedaton February 9, 4 Normalty Test/Normalty Plots E.375 { e } MSE z th of n STATA COMMANDS regress grad_gpa gre_total predct resd pnorm resd sort resd gen pctle(_n-.375)/(_n+.5) gen eresde(rmse)*nvnorm(pctle) correlate eresd resd (obs4) eresd resd eresd. resd.943. Normal F[(resd-m)/s]..5.5.75. n +.5..5.5.75. Emprcal P[] /(N+) The Correlaton Test (obs4) eresd resd eresd. resd.943. TABLE B.6 n.5 5.88 6.888 7.898 8.96 9.9.98.95 3.964 5.977.987..86.838.85.86.87.879.96.947.966.98 Shapro Wlk test for Normalty W ( ) ( X X ) a X (). swlk resd Shapro-Wlk W test for normal data Varable Obs W V z Prob>z -------------+----------------------------------------------------------- resd 4.85894 5.576 3.66.5 8
Dagnostcs and Smple Remedaton February 9, 4 Error n Varables (Y). drop _all. use "D:\krtzervlle.DTA".. drop f v53>6 (9 observatons deleted). regress v36 v53 Number of obs 496 R-squared.379 Adj R-squared.374 Root MSE.468 v36 Coef. Std. Err. v53 6.8563.53965 _cons 37.35356.8848 gen FTv36+*nvnorm(unform()) (3 mssng values generated). regress FT v53 Number of obs 496 R-squared.699 Adj R-squared.694 Root MSE.43 FT Coef. Std. Err. v53 6.5387.787 _cons 38.33388.9685734 Error n Varables (X). drop _all. use "D:\krtzervlle.DTA".. drop f v53>6 (9 observatons deleted). regress v36 v53 Number of obs 496 R-squared.379 Adj R-squared.374 Root MSE.468 v36 Coef. Std. Err. v53 6.8563.53965 _cons 37.35356.8848. gen PIDv53+nvnorm(unform()). regress v36 PID Number of obs 496 R-squared.64 Adj R-squared.599 Root MSE.47 v36 Coef. Std. Err. PID 5.54367.469 _cons 4.934.8734 x Error n X Y x b where x X X b Y ( x + k ) Y x + Y k ( x + ) x + x k + k k b x Y x + k 9
Dagnostcs and Smple Remedaton February 9, 4 Transformng X Transformng Y Box-Cox λ Y ' Y Y ' log Y when λ λ Y Y β + βx n L( β, β, σ ) n exp ( Y β β X) ( πσ ) σ n λ L( β, β, σ, λ) n exp ( Y β β X) ( πσ ) σ e
Dagnostcs and Smple Remedaton February 9, 4 Generalzed Box-Cox n Stata θ Y β + β X λ use "D:\COURSES\PS55\EXAMPLES\clrp.dta", clear regress lhours stakes gen sqstakessqrt(stakes) regress lhours sqstakes replace lhours. f lhours boxcox lhours stakes boxcox lhours stakes,model(rhs) boxcox lhours stakes,model(theta) Nonlnear Example. regress lhours stakes Number of obs 34 R-squared.974 Root MSE 35.89 Lhours Coef. Std. Err. t stakes.488.47. _cons 54.36698 7.53458 7.3. gen sqstakessqrt(stakes). regress lhours sqstakes Number of obs 34 R-squared.3443 Root MSE 3.8 lhours Coef. Std. Err. t sqstakes.68.46686 3.36 _cons -4.849369 9.8979 -.53 Box Cox Example. boxcox lhours stakes,model(rhs) Number of obs 34 LR ch() 47.5 Prob > ch. lhours Coef. Std. Err. z /lambda.59465.573964.36 Estmates of scale-varant parameters Coef. Notrans _cons 4.79844 Trans stakes.795 /sgma 3.3397. boxcox lhours stakes,model(theta) Number of obs 34 LR ch() 97.49 Prob > ch. lhours Coef. Std. Err. z /lambda.479974.65778 3.77 /theta.5995.736 7.34 Estmates of scale-varant parameters Coef. Notrans _cons.56594 Trans stakes.695 /sgma.998667
Dagnostcs and Smple Remedaton February 9, 4 Lowess lowess lhours stakes lowess lhours stakes, f stakes<5 lowess lhours stakes, f stakes< lhours 5 5 Lowess smoother 5 5 5 stakes bandwdth.8 Lowess smoother lhours 3 Lowess smoother lhours 3 4 3 4 5 stakes bandwdth.8 4 6 8 stakes bandwdth.8