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Considering a Difference-in-Differences (DiD) equation with a dummy variable:

$$ y_i = \alpha + \beta D_i + Xit + \varepsilon_i. $$ While $$Xit$$ is a set of covariates. Let's say the DiD here has the frequency is year and unit level is country. The outcome variable is number of rich people per million (yearly data).

Can I explain that $\beta$ as below: Let's say $\beta$= -0.5

The number of rich people in the treatment group decrease 0.5 per million of people per year (annually average) compared to that of control group after the event date?

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No, here the correct interpretation would be that on the average the treatment led to 0.5 decrease in number of rich people per million, conditional on all other covariates. Here (I am adding t subscript because I think you must have omitted it):

$$-0.5=\beta = E[Y_{it1} -Y_{it0}]$$

There is no per year there, this is one off effect that reduces the amount of rich people after the treatment is implemented not a negative growth that will decrease the number of rich people period after period.

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  • $\begingroup$ I thought it would be we calculate the ATT(Average treatment effect on treated), and we account for the average effect of all years later, so the result would be yearly average, I may fall into a fallacy, please let me know then. $\endgroup$
    – Phil Nguyen
    Sep 21, 2021 at 9:48
  • $\begingroup$ If saying that "on the average the treatment led to 0.5 decrease in number of rich people per million", meaning that we does not mention anything about time then? $\endgroup$
    – Phil Nguyen
    Sep 21, 2021 at 9:49
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    $\begingroup$ @Louise the raw output from DiD is ATE $\beta=E[Y_{it1}-Y_{it0}]$ under some conditions you can show that ATE=ATT but the raw output is just ATE, also neither ATT or ATE here would tell you what is happening per year, they only tell you how Y changed on average over all subsequent time periods within your study not how it changed per year $\endgroup$
    – 1muflon1 ♦
    Sep 21, 2021 at 10:12
  • $\begingroup$ ah, I see, becuase when calculating the DiD, we already calculated the average first then we derive the difference in average of outcome before and after of treatment by that of control, so it is a one-off effect. Is it a correct explanation? Many thanks $\endgroup$
    – Phil Nguyen
    Sep 21, 2021 at 10:19
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    $\begingroup$ @Louise yes you could also get an effect on growth rate of rich people but then you need to use $\Delta Y$ - however in that case you need to recheck all assumptions again because there might be parallel trend when u use Y and not $\Delta Y$ and vice versa $\endgroup$
    – 1muflon1 ♦
    Sep 21, 2021 at 10:23

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