Special topicsFixed effectsIntra- or Inter-country Variation? (animated slide, see summary on next slide)Slide 4Slide 5InteractionsInteractionsSlide 8Slide 9Importance of a variableDeath penalty exampleSlide 12Slide 13Slide 14Slide 15Slide 16Which to use?Partial residual scatter plotsPartial residual scatter plotsMIT students rated Mexican candidates faces on “competence”Slide 21RegressionReminder: ResidualsCalculating partial residualsSlide 25Slide 26Avplot & cprplotImputing missing data (on controls)Imputing missing dataImputation exampleStata imputation commandRules about imputingSpecial topicsFixed effectsWhen trying to compare apples with apples, we worry about the numerous potential differences on confounding variablesIf differences on confounding variables are stable over time, we can eliminate bias from them by only analyzing variation within the same unit over timeE.g., breast-feeding study (unit is woman)E.g., income and partisanship (unit is state)To only analyze variation within the same unit over time, we use fixed effects Stata commands areg and xtregEquivalent to adding indicator (or dummy variables) variables for unitsEquivalent to between subjects design (as opposed to within subjects)Inter-State Variationxtreg [DV] [IV], be(or)collapse [DV] [IV], by(state)reg [DV] [IV]Intra-State Variationxtreg [DV] [IV], fe(or)areg [DV] [IV], a(state)Aggregate Panel Variationreg [DV] [IV]Intra- or Inter-country Variation?(animated slide, see summary on next slide)DVIVe.g., incomee.g., partisanship (Democrat %)Intra- or Inter-State(Person/Firm) Variation?DVIVAggregate Panel Variationreg [DV] [IV]DVIVFixed effects (fe)Intra-State Variationxtreg [DV] [IV], fe (or) areg [DV] [IV], a(state)DVIVBetween effects (be)Inter-State Variationxtreg [DV] [IV], be(or)collapse [DV] [IV], by(state)reg [DV] [IV]Fixed effectsProblemsThrows away potentially relevant variation (alternative: random effects)Variation over time may be primarily from random measurement error (e.g., unions and wages)Unusual factors may drive changes in explanatory variables over time and also influence the dependent variable (e.g., currency unions)InteractionsInteractionsInteractions test whether the combination of variables affects the outcome differently than the sum of the main (or individual) effects. ExamplesInteraction between adding sugar to coffee and stirring the coffee. Neither of the two individual variables has much effect on sweetness but a combination of the two does.Interaction between smoking and inhaling asbestos fibres: Both raise lung carcinoma risk, but exposure to asbestos multiplies the cancer risk in smokers.Example from problem setHow would we test whether defendants are sentenced to death more frequently when their victims are both white and strangers than you would expect from the coefficients on white victim and on victim stranger.Interactions. g wvXvs = wv* vs. reg death bd yv ac fv v2 ms wv vs wvXvs ----------------------------------------------- death | Coef. Std. Err. t P>|t|-------+--------------------------------------- (omitted) wv | .0985493 .1873771 0.53 0.600 vs | .1076086 .2004193 0.54 0.593 wvXvs | .3303334 .2299526 1.44 0.154 _cons | .0558568 .2150039 0.26 0.796-----------------------------------------------•To interpret interactions, substitute the appropriate values for each variable•E.g., what’s the effect for • .099 wv+.108 vs+.330 wvXvs •White, non-stranger: .099(1)+.108(0)+.330(1)*(0) = .099•White, stranger: .099(1)+.108(1)+.330(1)*(1) = .537•Black, non-stranger: .099(0)+.108(0)+.330(0)*(0) = comparison•Black, stranger: .099(0)+.108(1)+.330(1)*(0) = .108Interactions. tab wv vs, sum(death)Means, Standard Deviations and Frequencies of death | vs wv | 0 1 | Total-----------+----------------------+---------- 0 | .16666667 .28571429 | .23076923 | .38924947 .46880723 | .42966892 | 12 14 | 26-----------+----------------------+---------- 1 | .40540541 .75675676 | .58108108 | .49774265 .43495884 | .4967499 | 37 37 | 74-----------+----------------------+---------- Total | .34693878 .62745098 | .49 | .48092881 .48829435 | .50241839 | 49 51 | 100Importance of a variableDeath penalty example. sum death bd- yv Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- death | 100 .49 .5024184 0 1 bd | 100 .53 .5016136 0 1 wv | 100 .74 .440844 0 1 ac | 100 .4366667 .225705 0 1 fv | 100 .31 .4648232 0 1-------------+-------------------------------------------------------- vs | 100 .51 .5024184 0 1 v2 | 100 .14 .3487351 0 1 ms | 100 .12 .3265986 0 1 yv | 100 .08 .2726599 0 1Death penalty example. reg death bd-yv , beta------------------------------------------------ death | Coef. Std. Err. P>|t| Beta-------+----------------------------------------- bd | -.0869168 .1102374 0.432 -.0867775 wv | .3052246 .1207463 0.013 .2678175 ac | .4071931 .2228501 0.071 .1829263 fv | .0790273 .1061283 0.458 .0731138 vs | .3563889 .101464 0.001 .3563889 v2 | .0499414 .1394044 0.721 .0346649 ms | .2836468 .1517671 0.065 .1843855 yv | .050356 .1773002 0.777 .027328 _cons | -.1189227 .1782999 0.506 .-------------------------------------------------Importance of a variableThree potential answersTheoretical importanceLevel importanceDispersion importanceImportance of a variableTheoretical importanceTheoretical importance = Regression coefficient (b) To compare explanatory variables, put them on the same scaleE.g., vary between 0 and 1Importance of a variableLevel importance: most important in particular times and placesE.g., did the economy or presidential popularity matter more in congressional races in 2006?Level importance= bj* xjImportance of a variableDispersion importance:
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