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I have a standard DID regression of the form:

Y= β0 + β1*[Time] + β2*[Treatment] + β3*[Time*Treatment] + ε

where Time is a dummy equal to 1 for period after policy change and Treatment is a dummy for the treatment variable.

Based on my results β0, β1 and β2 are all insignificant. However, the difference-in-difference estimator β3 is significant. How can I interpret these results? Does that mean that I cannot say anything about the exact behavior of the treatment and control pre or post period other than the fact that there is a significant difference between them in the period after treatment? Is there still value in this DID regression if all other coefficients are insignificant?

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  • $\begingroup$ Why does your regression include b1 and B2 in the first place? They are both useless. B1 tells you what is average difference between pre and post treatment period ignoring treatment status- hence is completely useless even worse it would clash with fixed time effects which would be more appropriate arguably and the B2 has the opposite problem it’s a difference in average Y between treatment and control group including all pre treatment time when no treatment was applied at all. Also it would clash with panel fixed effects. I recommend looking DiD chapter in MHE from Angrist and Pischke $\endgroup$ – 1muflon1 Aug 10 at 11:55
  • $\begingroup$ Their discussion mostly relates to data in two different periods. In my case my period before treatment is a time series from 1990m1 to 2000m12. Then the period after policy change is defined as 2000m1 to 2005m12. Suppose we call the period from 2000m1 to 2005m12 as Time (or Post if that is less confusing). Can you tell me how you would write down the Difference-in-Difference regression in this case (including year fixed effects)? $\endgroup$ – user29937 Aug 10 at 15:45
  • $\begingroup$ you can easily extend their two point case into case of multiple time periods. For example, you could run $Y_{it}= a_i + \gamma_t + \beta T_{it} + ... + e_{it}$ where $a_i$ are panel fixed effects, $\gamma_t$ year fixed effects, $T$ is a treatment indicator - it’s a parsimonious version of your interaction term a dummy that is 0 if no treatment is applied at time t to panel member i and 1 if the treatment is applied at time t to panel member i, eclipses stand in for all the control variables you would want to include and $e$ is the error term $\endgroup$ – 1muflon1 Aug 10 at 15:51
  • $\begingroup$ So if I understand you correctly, assuming I use my original notation and how I define the dummies I would simply estimate Y_it= β0_i + β1*[Time*Treatment] + Year FE+ ε. Is that correct? $\endgroup$ – user29937 Aug 10 at 16:04
  • $\begingroup$ yes that would work $\endgroup$ – 1muflon1 Aug 10 at 16:13
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I would heed the advice in the comments.

In settings where policy adoption is standardized and you have multiple pre- and post-treatment time periods, you could also interact your treatment indicator with separate time dummies specific to treated and untreated units. In the comments you indicate you observe multiple units $i$ across months $t$ from 1990 to 2005. Your model would look something like this:

$$ y_{it} = \gamma T_{i} + \lambda_{1} (T_{i}*\mathbf{I}_{t = \text{Jan-2000}}) + \lambda_{2} (T_{i}*\mathbf{I}_{t = \text{Feb-2000}}) + \lambda_{3} (T_{i}*\mathbf{I}_{t = \text{Mar-2000}}) + \lambda_{j} (T_{i}*\mathbf{I}_{t = j}) + \epsilon_{it}, $$

where the $\lambda_{j}'s$ represent post-exposure effects after the policy goes into effect. Note, to estimate all post-period effects up to period $j$ you must create a 'month-year' variable to distinguish, for example, between January in 2000 and January in 2001. To do this, concatenate categorical indicators for month and year together and append it to your data frame. Interacting $T_i$ with individual post-treatment time dummies results in separate estimates for each post-exposure period (e.g., Jan-2000, Feb-2000, Mar-2000, ... , Dec-2005, etc.). This approach is helpful for many reasons. First, treatment may grow or fade over time; the policy might be perceived immediately upon implementation, or with delay. Due to its flexibility, this approach allows you to assess the relevant time dependencies. Second, if there is a treatment withdrawal period, you can estimate effects in those periods beyond policy/program termination; effects might persist even in the absence of treatment.

I admit this results in 72 interactions which might be more than your data can handle. Notwithstanding this concern, it is still a useful way to model effects in the post-treatment epoch. You can still estimate unit and time fixed effects in this setting. As already noted, the unit fixed effects will absorb $T_i$ and the time fixed effects will absorb your 'month-year' main effects. But I wouldn't worry about this. Your interactions (i.e., $T_i \times \mathbf{I}_{t=j}$) should be your focus.

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