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Say you are running a diff in diff, where some treatment occured nationally at time t=0, and you use some continuous measure of pre-existing characteristics (or some continuous instrument) to capture 'intensity' of treatment proportional to that continuous variable, call it X. I.e. running:

$Y_{i,t} = \mu_i + \lambda_t + \sum_{t\neq-1}$$\beta$$_t$X$_{i,t}$ + $\epsilon$$_{i,t}$

where I omit base year -1, the year prior to treatment, and X is a continuous variable. Say I ha

is the coefficient $\beta$$_t$, for some arbitrary t post treatment, taking the marginal effect of y on x in year t relative to the marginal effect in year t=-1? so similar to how you would interpret it as a discrete treatment -event study? I am Am just confused how to put the coefficient in words.

Steve
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  • How is your question different from this question: https://stats.stackexchange.com/questions/152684/how-do-i-interpret-a-difference-in-differences-model-with-continuous-treatment – Jesper for President Apr 03 '20 at 07:26
  • I guess I was thinking maybe the definition is a little different due to many time periods and doing the event study rather than the pure diff in diff.. but would the interpretation be the same except for the different base year? so whether areas with higher values of x have higher/lower values of y from period t = -1 to time t? – Steve Apr 03 '20 at 15:00
  • Maybe you could say how many periods you have, because I was not sure. Also you say treatment is time t but then you say t=-1 is the period before treatment which means treatment is t=0. You could also clarify whether the group effects are individual so one group effect for each $i$ for example written $mu_i$. – Jesper for President Apr 03 '20 at 17:09
  • Thank you for the clarifications, I edited my post, hopefully it makes sense now – Steve Apr 03 '20 at 20:34

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