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I am a natural scientist dipping my toes into the more social, economical side of things where instead of GitHub and R, Stata is the go-to tool and almost everything is 'panel' data. Similar analyses go under different names which is fine, but sometimes analyses actually are vastly different in approach and, like now, I get perplexed.

If I had a before ($_{t=0}$) & after ($_{t=1}$) analysis where there are truly only two time points and the response is a continuous variable with its continuous pre-intervention counterpart, I would naturally fall into modelling the response as $y_{t=1} \sim y_{t=0} + x_1 + ...$ The $x_{1:n}$ could include factors such as control-treatment indicators ($ct$), continuous variables, etc. (I might be interested in an $x_i*ct$ interaction)

Econometricians, however, use difference-in-difference regressions where all $y_{0,1}$ are the response and dummies are used to express both $t$ and $ct$. The effect of the intervention is then examined in the interaction $t * ct$ for group $t=1, ct=treatment$.

I would find the the former way more intuitive and fine-grained (and better) for answering the question, as instead of looking at broad intergroup differences complicated by multiple factor levels, there would be a more specific relationship between the before and after outcome, as well as a more helpful tool to evaluate interactions. 1) Am I right? 2) If so, why is the difference-in-differences method still so dominant? (or have I just completely misunderstood it)

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This is an interesting way to think about it. The problem with this is that it violates one assumption of OLS, which is that the regressor (RHS variables) do not correlate with the error term. By having $y_{t=0}$ on the RHS, it could very well correlate with the error term.

Your expression would be correct if you said $y_{t=1} = \hat y_{t=0} + \sum x + \varepsilon$, where $\hat y$ is the predicted value of $y$ (which is then sum of $\beta_i x_i$'s). But then you could rewrite this in terms of diff-in-diff.

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