Political Science 552

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Eleanor Stevens
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1 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 Party ID

2 Dagnostcs and Smple Remedaton February 9, 4 Large Sample Plots FT-Bush PartyID Plots wth Jtterng FT-Bush PartyID Resduals as Dagnostc Tool e Y Y $ e* e MSE

3 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 Y X Seeng Is Recognzng

4 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 PartyID 4

5 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 Nonlnear Improvement >. Error 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 pd v Resdual Total Grad GPA Outlers y.7x.4x.5x r GRE-Total 5

6 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

7 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 d combned dff Degrees of freedom: 38 Ho: mean(d) - mean(d) dff Ha: dff < Ha: dff! Ha: dff > t t t 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 _cons hettest Breusch-Pagan / Cook-Wesberg test for heteroskedastcty Ho: Constant varance Varables: ftted values of grad_gpa ch().8 Prob > ch

8 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] n Emprcal P[] /(N+) The Correlaton Test (obs4) eresd resd eresd. resd.943. TABLE B.6 n 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

9 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. v _cons 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. v _cons 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. v _cons 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 _cons 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

10 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

11 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 Lhours Coef. Std. Err. t stakes _cons gen sqstakessqrt(stakes). regress lhours sqstakes Number of obs 34 R-squared.3443 Root MSE 3.8 lhours Coef. Std. Err. t sqstakes _cons Box Cox Example. boxcox lhours stakes,model(rhs) Number of obs 34 LR ch() 47.5 Prob > ch. lhours Coef. Std. Err. z /lambda Estmates of scale-varant parameters Coef. Notrans _cons Trans stakes.795 /sgma boxcox lhours stakes,model(theta) Number of obs 34 LR ch() Prob > ch. lhours Coef. Std. Err. z /lambda /theta Estmates of scale-varant parameters Coef. Notrans _cons Trans stakes.695 /sgma

12 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 stakes bandwdth.8 Lowess smoother lhours 3 Lowess smoother lhours stakes bandwdth stakes bandwdth.8

Y = β 0 + β 1 X 1 + β 2 X β k X k + ε

Y = β 0 + β 1 X 1 + β 2 X β k X k + ε Chapter 3 Secton 3.1 Model Assumptons: Multple Regresson Model Predcton Equaton Std. Devaton of Error Correlaton Matrx Smple Lnear Regresson: 1.) Lnearty.) Constant Varance 3.) Independent Errors 4.) Normalty