Say we have the following data

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Say we are concerned with a linear regression model that is $\boldsymbol{y} = \alpha + \boldsymbol{d} \beta + \boldsymbol{u}$ and that we are interested in the sampling distribution of $\alpha$ and $\beta$. ($\boldsymbol{y}$ and $\boldsymbol{u}$ are of no interest presently.)

One possibility is to resort to bootstrapping. Given that we (may) have autocorrelation at work through time, the bootstrap units are the blocks indexed along the (discrete) line of individuals, so that, e.g. one of the resampled versions of our data can be

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You may already have an idea of what my question is going to be about. Our table also contains a (boolean) variable related to whether a given subject has undergone (or not) a treatment, namely $\boldsymbol{d}$.

I am aware of techniques that deal with that kind of data – i.e. with Treatment Effects – and, most of the time, they drive us out of the statistical-inference sphere toward that of the randomization inference, e.g. MacKinnon and Webb (2017). Something I would like to know is whether the technique I am going to describe has a name, or to the extreme, just makes (no?) sense.

Given that one drawback of "naively" block-bootstrapping is the possibility of having $\boldsymbol{d}$ 100% constant, (which may lead to singularity issues and, before that, simply breaks the balanced structure of our data), would it make sense to consider all the possible treated-vs-untreated pairs as resample blocks ? In the current toy case, this would lead to "build" and consider $(4/2)^2$ blocks

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Taking care of keeping the original number of observations identical, we could rebuild $\binom{4}{2}=6$ new data.

What bothers me is that, proceeding in this way is actually equivalent to i) naivily resampling our data and ii) putting aside unbalanced (with-respect-to-$\boldsymbol{d}$) outcomes which, presented in this way, looks like selection bias, hence my question.



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