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Borusyak, 2021 has a sentence

We show that this estimator takes a particularly transparent form under unrestricted treatment effect heterogeneity, while our construction also yields efficiency when some restrictions on treatment effects are imposed.

I am wondering what does under unrestricted treatment effect heterogeneity mean? I translate it to that "when the treatment effect heterogeneity is not restricted", and it also means that this situation works in both cases "treatment effect heterogeneity" and "treatment effect homogeneity". Is it a correct thought?

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    The sentence does not seem grammatical to me, unless "treatment effect heterogeneity" is a concept all of its own. Even then I would expect there to be a hyphen in the phrase. I cannot understand the sentence.
    – randomhead
    Sep 16 '21 at 0:45
  • @randomhead, it is a phrase and I cited directly from this paper. Many thanks and warm regards.
    – Louise
    Sep 16 '21 at 0:46
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    I think there are technical terms that take this beyond a strictly English language question. The only way I can make sense of it is (under unrestricted/treatment effect/heterogeneity).
    – user3169
    Sep 16 '21 at 1:21
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I think this is better asked of people with domain-specific knowledge, as understanding the phrase depends more on that knowledge than general grammatical knowledge.

It may be easier to understand if we start with "our construction also yields efficiency when some restrictions on treatment effects are imposed." This is saying that there is some possible restriction on treatment effects. I can only guess what those may be; perhaps effects beyond a certain bounds are discarded from the data set. When those restrictions are imposed, their construction yields efficiency.

"Heterogeneity" means "variation", so "unrestricted effect heterogeneity" is the opposite of "restrictions on treatment effects". In this context, "under" means "as a result of this condition". So "particularly transparent form under unrestricted treatment effect heterogeneity" means "particularly transparent form if effect heterogeneity is not restricted." In other words, "particularly transparent form if there is no restriction placed on the variation of effects."

There is some ambiguity as to what the comparison is being made to. It could be interpreted as "if no restrictions are placed on the effects, then this estimator takes a form that is more transparent than if restrictions are placed" and "if restrictions are placed on the effects, then our construction is more efficient than if no restrictions are placed."

However, I am more inclined to interpret it as "the transparency of our estimator when there are no restrictions is greater than the transparency of other people's estimator when there are no restrictions" and "the efficiency of our construction when there are restrictions is greater than the efficiency of other people's construction when there are restrictions."

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