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I ran a DID regression and found my estimate on the DID coefficient to be .022. The units of time I am using are days, and at a certain day around halfway through my data, the treatment group was introduced to a specific policy change. I am just wondering how to interpret this .022 number. Is this saying that relative to the counterfactual, the treatment group increased by .022 units per day or just .022 units total over the time from the treatment date until where the data ends.

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  • $\begingroup$ It would be helpful for you to write down the specific regression specification you estimated. I'm guessing you estimated a model like $y_{it} = \beta_0 + \beta_1 \cdot post_t + \beta_2 \cdot treated_i + \beta_3 \cdot post_t\times treated_i + \varepsilon_{it}$? $\endgroup$ May 1, 2022 at 17:07
  • $\begingroup$ Yes, this is correct. Here are the specific numbers I estimated: ๐ด๐‘ฃ๐‘”๐‘†๐‘๐‘’๐‘›๐‘‘๐‘–๐‘›๐‘”=โˆ’.01+โˆ’.92๐‘ƒ๐‘œ๐‘ ๐‘ก + .024๐‘‡๐‘Ÿ๐‘ก๐‘š๐‘›๐‘ก + .022(๐‘ƒ๐‘œ๐‘ ๐‘กโˆ—๐‘‡๐‘Ÿ๐‘ก๐‘š๐‘›๐‘ก) $\endgroup$
    – Abby S
    May 1, 2022 at 20:04
  • $\begingroup$ Sorry, in the original post I meant to put .022 instead of .044. $\endgroup$
    – Abby S
    May 2, 2022 at 0:02

1 Answer 1

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To help interpret your DiD specification, it is helpful to consider the predicted value of $y_{it}$ for each of the 4 different combinations of the variables $post_{t}$ and $treated_i$.

First, consider $post_t = 0$ and $treated_i = 0$. This corresponds to an observation from the period before treatment occurred and where you are looking at a unit which is never treated. Since the predictors multiplying the $\beta$'s are all 0, your prediction for $y_{it}$ is just $\beta_0$. This shows that $beta_0$ can be interpreted as the mean value of the outcome on that single day for units in the control group on days that occur pre-treatment.

Second, consider the case where $post_t = 0$ and $treated_i = 1$. This corresponds to the case where you are looking at the period before treatment occurred and you are looking at a unit which eventually gets treated (but not yet). Here, only the independent variable multiplying $\beta_2$ is non-zero. Thus, for this group, your prediction for $y_{it}$ would be $\beta_0+\beta_2$, and hence, $\beta_2$ corresponds to the difference between the average per day value of the outcome when comparing treated with control units pre-treatment.

Third, we have the case where $post_t = 1$ and $treated_i = 0$. By the same logic as above, we have that $\beta_0 + \beta_1$ is interpreted as the mean value of $y_{it}$ on days post-treatment for units in the control group. Thus $\beta_1$ is interpreted as the difference between the per-day value of $y_{it}$ post versus pre treatment among control units.

Finally, we have the case where $post_t = 1$ and $treated_i = 1$. Again, we apply the same logic as before to get that $\beta_0 + \beta_1 + \beta_2 + \beta_3$ is the mean value of $y_{it}$ on days post-treatment for treated units. This shows that $\beta_1+\beta_3$ can be interpreted as the difference between the difference in the average day-level value of $y_{it}$ within the treated units comparing pre/post treatment. Then, subtracting off $\beta_1$, this shows that $\beta_3$ is interpreted as the difference in this pre/post difference when comparing treated vs control units.

Summarizing, we have that $$\begin{aligned}\beta_3 &= \left(\{\text{avg. }y/\text{day when post and treated}\} - \{\text{avg. }y/\text{day when pre and treated}\}\right) \\&- \left(\{\text{avg. }y/\text{day when post and control}\} - \{\text{avg. }y/\text{day when pre and control}\}\right)\end{aligned}$$ Note that this is subtly different from either of your interpretations. It is saying that the difference in $y_{it}$ between treatment and control on a fixed day post-treatment is on average $\beta_3$ more than the difference in $y_{it}$ between treatment and control on a fixed day pre-treatment.

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