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Suppose that I have the following two group two time Difference-in-Difference model:

$Y_{it}=\alpha_{0}+\alpha_{1}*d_{t} + \alpha_{2}*Treated_{i}+\alpha_{3}*d_{t}*Treated_{i}+\alpha_{4}*X_{it}+\epsilon_{it}$

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 $Treated_{i}$.

$\alpha_{3}$ measure the parameter of interest, the ATT.

However, I suspect tha the ATT varies with the municipaliy sizes, how can I test for this heterogeneous effect?

I can simply write the model with interactions terms like the folowing?

$Y_{it}=\alpha_{0}+\alpha_{1}*d_{t} + \alpha_{2}*Treated_{i}+\alpha_{3}*d_{t}*Treated_{i}*Size1_{it}+\alpha_{4}*d_{t}*Treated_{i}*Size2_{it}+\alpha_{5}*X_{it}+\epsilon_{it}$

Someone can indicate some paper that construct this kind of analysis?

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You're almost there, but not quite. (Sorry -- I'm not sure how to do the fancy syntax!)

As a first stage you would probably want to run the first equation separately for the different samples, s1 and s2:

(i) Yit=α0+α1∗dt+α2∗Treatedi+α3∗dt∗Treatedi+α4∗Xit+ϵit if Size==1

Assume this gives you α3_1 = 0.1

(ii) Yit=α0+α1∗dt+α2∗Treatedi+α3∗dt∗Treatedi+α4∗Xit+ϵit if Size==2

Assume this gives you α3_2 = 0.5. If your suspicion is true, then α3_1 will be significantly different from α3_2.

If you want to combine these in one regression, you would need to tread carefully. Your second -- joint -- regression,

Yit=α0+α1∗dt+α2∗Treatedi+α3∗dt∗Treatedi∗Size1it+α4∗dt∗Treatedi∗Size2it+α5∗Xit+ϵit

, 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.

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:

Yit=α0+Size∗α1∗dt+Size∗α2∗Treatedi+Size1it + Size2it + α3∗dt∗Treatedi∗Size1it+α4∗dt∗Treatedi∗Size2it+α5∗Xit+ϵit

You shall now be able to see that α3==α3_1 and α4==α3_2.

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    $\begingroup$ The answer misses the most important point about the difference in differences setting, namely the additional assumptions required for the parallel trends. Now they not only need to be parallel for the treated and control group municipalities, but instead you need parallel pre-treatment trends for the large treated, small treated, large control, and small control municipalities. $\endgroup$ – Andy Dec 4 '15 at 11:10
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I guess parallel trend assumption must hold within municipality size.

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.

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.

This is same as assuming potential outcome trend for treated and control are same within municipalities size.

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 & 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.

For regression, baseline model will be Yist=α0+α1(dtXSize)+α2Treated+α3(dtXTreatedXSize)+α4Xist+ϵit

You could also control for Size dummy and interaction of Size and Treated but i don't think that's necessary.

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