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MIT 17 871 - Lecture Notes

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Special topicsFixed effects When trying to compare apples with apples, we worry about the numerous potential differences on confounding variables If differences on confounding variables are stable over time, we can eliminate bias from them by only analyzing variation within the same unit over time E.g., breast-feeding study (unit is woman) E.g., currency unions and euro (unit is country) To only analyze variation within the same unit over time, we use fixed effects  Stata commands areg and xtreg Equivalent to adding indicator (or dummy variables) variables for units Equivalent to between subjects design (as opposed to within subjects)Intra-Country Variationxtreg [DV] [IV], fe(or)areg [DV] [IV], a(country)Inter-Country Variationxtreg [DV] [IV], be(or)collapse [DV] [IV], by(country)reg [DV] [IV]Aggregate Panel Variationreg [DV] [IV]Intra- or Inter-country Variation?(animated slide, see summary on next slide)DVIVIntra- or Inter-country Variation?DVIVAggregate Panel Variationreg [DV] [IV]DVIVFixed effects (fe)Intra-Country Variationxtreg [DV] [IV], fe(or) areg [DV] [IV], a(country)DVIVBetween effects (be)Inter-Country Variationxtreg [DV] [IV], be(or)collapse [DV] [IV], by(country)reg [DV] [IV]Fixed effects Problems Throws 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)InteractionsInteractions Interactions test whether the combination of variables affects the outcome differently than the sum of the main (or individual) effects.  Examples Interaction 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 and non-smokers. Both risk factors were not shown to be additive – a clear indication of interaction Example from problem set: how would we test whether defendants are sentenced to death more frequently when their victims are both white andstrangers than you would expect from the coefficients on white victim andon victim strangerInteractions. 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 0.600vs | .1076086 .2004193 0.54 0.593wvXvs | .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| vswv | 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- yvVariable | Obs Mean Std. Dev. Min Max-------------+--------------------------------------------------------death | 100 .49 .5024184 0 1bd | 100 .53 .5016136 0 1wv | 100 .74 .440844 0 1ac | 100 .4366667 .225705 0 1fv | 100 .31 .4648232 0 1-------------+--------------------------------------------------------vs | 100 .51 .5024184 0 1v2 | 100 .14 .3487351 0 1ms | 100 .12 .3265986 0 1yv | 100 .08 .2726599 0 1Death penalty example. reg death bd-yv , beta------------------------------------------------death | Coef. Std. Err. P>|t| Beta-------+-----------------------------------------bd | -.0869168 .1102374 0.432 -.0867775wv | .3052246 .1207463 0.013 .2678175ac | .4071931 .2228501 0.071 .1829263fv | .0790273 .1061283 0.458 .0731138vs | .3563889 .101464 0.001 .3563889v2 | .0499414 .1394044 0.721 .0346649ms | .2836468 .1517671 0.065 .1843855yv | .050356 .1773002 0.777 .027328_cons | -.1189227 .1782999 0.506 .-------------------------------------------------Importance of a variable Three potential answers Theoretical importance Level importance Dispersion importanceImportance of a variable Theoretical importance Theoretical importance = Regression coefficient (b) To compare explanatory variables, put them on the same scale E.g., vary between 0 and 1Importance of a variable Level importance: most important in particular times and places E.g., did the economy or presidential popularity matter more in congressional races in 2006? Level importance= bj* xjImportance of a variable Dispersion importance: what explains the variance on the dependent variable E.g., given that the GOP won in this particular election, why did some people vote for them and others against? Dispersion importance =  Standardized coefficients, or alternatively Regression coefficient times standard deviation of explanatory variable In bivariate case, correlationWhich to use? Depends on the research question Usually theoretical importance Sometimes level importance Dispersion importance not usually relevantPartial residual scatter plotsPartial residual scatter plots Importance of plotting your data Importance of controls


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