Interpretation of Difference-in-Differences Regression Results when Only the DID Coefficient is Significant 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?
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.