Difference-in-Difference with Heterogeneous Effects - Cross Validated most recent 30 from stats.stackexchange.com 2019-07-18T20:13:21Z https://stats.stackexchange.com/feeds/question/168607 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://stats.stackexchange.com/q/168607 2 Difference-in-Difference with Heterogeneous Effects Ricardo Carvalho https://stats.stackexchange.com/users/70356 2015-08-24T21:34:25Z 2017-10-05T18:49:45Z <p>Suppose that I have the following two group two time Difference-in-Difference model:</p> <p>$Y_{it}=\alpha_{0}+\alpha_{1}*d_{t} + \alpha_{2}*Treated_{i}+\alpha_{3}*d_{t}*Treated_{i}+\alpha_{4}*X_{it}+\epsilon_{it}$</p> <p>The objective is to infer a causal relationship between the outcome variable $Y_{it}$ (that represent the municipality "i" per capita expenditures) and a policy status, represented by $Treated_{i}$.</p> <p>$\alpha_{3}$ measure the parameter of interest, the ATT.</p> <p>However, I suspect tha the ATT varies with the municipaliy sizes, how can I test for this heterogeneous effect? </p> <p>I can simply write the model with interactions terms like the folowing?</p> <p>$Y_{it}=\alpha_{0}+\alpha_{1}*d_{t} + \alpha_{2}*Treated_{i}+\alpha_{3}*d_{t}*Treated_{i}*Size1_{it}+\alpha_{4}*d_{t}*Treated_{i}*Size2_{it}+\alpha_{5}*X_{it}+\epsilon_{it}$</p> <p>Someone can indicate some paper that construct this kind of analysis?</p> https://stats.stackexchange.com/questions/168607/-/184981#184981 3 Answer by anders osterling for Difference-in-Difference with Heterogeneous Effects anders osterling https://stats.stackexchange.com/users/97075 2015-12-04T09:37:46Z 2015-12-04T09:37:46Z <p>You're almost there, but not quite. (Sorry -- I'm not sure how to do the fancy syntax!)</p> <p>As a first stage you would probably want to run the first equation separately for the different samples, s1 and s2:</p> <p>(i) Yit=α0+α1∗dt+α2∗Treatedi+α3∗dt∗Treatedi+α4∗Xit+ϵit if Size==1</p> <p>Assume this gives you α3_1 = 0.1</p> <p>(ii) Yit=α0+α1∗dt+α2∗Treatedi+α3∗dt∗Treatedi+α4∗Xit+ϵit if Size==2</p> <p>Assume this gives you α3_2 = 0.5. If your suspicion is true, then α3_1 will be significantly different from α3_2.</p> <p>If you want to combine these in one regression, you would need to tread carefully. Your second -- joint -- regression,</p> <p>Yit=α0+α1∗dt+α2∗Treatedi+α3∗dt∗Treatedi∗Size1it+α4∗dt∗Treatedi∗Size2it+α5∗Xit+ϵit</p> <p>, only gives α3==α3_1 and α4==α3_2 if the other effects (other α's) are exactly the same across the Sizes. This will not be the case in general.</p> <p>The solution is to allow the other α's vary across the sizes too. You also need separate dummies for Size1 and Size2. Assuming that Size==1 if it's size1 and ==0 if it's size2, the full equation is:</p> <p>Yit=α0+Size∗α1∗dt+Size∗α2∗Treatedi+Size1it + Size2it + α3∗dt∗Treatedi∗Size1it+α4∗dt∗Treatedi∗Size2it+α5∗Xit+ϵit</p> <p>You shall now be able to see that α3==α3_1 and α4==α3_2.</p> https://stats.stackexchange.com/questions/168607/-/306490#306490 0 Answer by HavingSomeQuestions for Difference-in-Difference with Heterogeneous Effects HavingSomeQuestions https://stats.stackexchange.com/users/134618 2017-10-05T18:49:45Z 2017-10-05T18:49:45Z <p>I guess parallel trend assumption must hold within municipality size.</p> <p>DID with homogeneous effect typically assumes E[y0it|i,t]=A(i)+B(t), where A(i) denotes municipality fixed effects and B(t) denotes time fixed effects.</p> <p>However, assuming heterogeneous effect across municipality size, E[y0ist|i,s,t]=A(i)+B(s,t) seems more adequate. Where s=Big,small denotes size.</p> <p>This is same as assuming potential outcome trend for treated and control are same within municipalities size. </p> <p>For instance, check how potential outcome changes as time passes. E[y0ist|i=treated,s=small,t=post]-E[y0ist|i=treated,s=small,t=pre] ={A(treated)+B(small,post)}-{A(treated)+B(small,pre)}=B(small,post)-B(small,pre) and potential outcome trend for small &amp; control will be also B(small,post)-B(small,pre). They have same trend. Doing same thing for the big, we get same trend B(Big,post)-B(Big,pre) for both treated and control groups.</p> <p>For regression, baseline model will be Yist=α0+α1(dtXSize)+α2Treated+α3(dtXTreatedXSize)+α4Xist+ϵit </p> <p>You could also control for Size dummy and interaction of Size and Treated but i don't think that's necessary.</p>