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When separating the sample into subsamples in this topic, this answer stated that

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.

Is it a correct assumption? And how to clarify it more? It is important because if it is the case, when running subsample analysis in DID, we do not need to test the parallel trend assumption anymore.

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