# diff in diff using time dummies instead of post

$$y_{it} = \beta0\ + \beta1\cdot treat\ + beta2\cdot post+\sum_{j\neq k} \delta_j \left( \text{treat}_i \cdot year(t=j) \right) + X'\gamma + \epsilon_t$$

time periods t=1,2,...,k,...,T periods where the treatment happens between k and k+1 (so time k is my last pre-treatment period)

I'm trying regression as mentioned above and I have few questions:

1) First is it possible to see the control group trend through this regression results?

2) Is it possible to get the total impact on Y? from the regression I get the results of T-k different coefficient, can I say what is the total impact of treatment over Y?

• Please start by considering whether the model equation is correctly specified. You can find some guidance here stats.stackexchange.com/questions/76058/…. I can spot several potential problems with the specification. Apr 12 '20 at 14:55
• @ytz Hi. I agree. Did you reproduce this equation from a paper? If so, please provide a reference. Apr 14 '20 at 3:25
• Hi, its not from any paper its for a work i'm doing
– XYZ
Apr 14 '20 at 9:22

Your notation is a bit clunky but I can work with it for the purposes of this answer. This is the standard difference-in-differences (DiD) estimator with multiple pre- and post-exposure periods. Instead of indexing the entire post-treatment epoch with one dummy, it appears you want to interact your treatment dummy with separate post-treatment year indicators. Here is what I think you want to do:

$$y_{it} = \beta_{0} + \beta_{1}\text{Treat}_{i} + \sum_{j \neq k} \lambda_{j} \text{Year}_{t=j} + \sum_{j \neq k} \delta_j \left( \text{Treat}_i \cdot \text{Year}_{t=j} \right) + X_{it}'\gamma + \epsilon_{it},$$

where the $$\delta_{j}$$'s are separate estimates of your treatment effect for each individual treatment year. As per your post, you consider $$j \leq k$$ as your pretreatment epoch. All periods $$j$$ not equal to $$k$$ are thus representative of post-treatment period dummies. Each coefficient on $$\delta_{j}$$ is an estimate of the $$j$$-th additive yearly treatment effect.

Note, I replaced the variable $$\text{Post}_{t}$$ with a series of post-treatment year dummies. This is not a full set of $$T - 1$$ dummies for years; rather, they are separate dummies for post-exposure years. In standard software, interacting $$\text{Treat}_{i}$$ with a series of post-treatment indicators will automatically result in the estimation of the year main effects as well. If you incorporate $$\text{Post}_{t}$$ in your specification before $$\text{Year}_{t}$$, your model will likely exclude one year to allow for the estimation of the post-treatment variable. However, if you include $$\text{Post}_{t}$$ after the individual year dummies, then software will likely drop the $$\text{Post}_{t}$$ variable entirely. The variable $$\text{Post}_{t}$$ is a linear combination of the post-treatment year dummies, and most software packages are smart enough to have quick fixes for it. In R for instance, variable ordering matters when faced with collinearity.

I would drop $$\text{Post}_{t}$$ entirely and replace it with separate indicators for years (i.e., post-exposure dummies).

1) First is it possible to see the control group trend through this regression results?

I am not sure what you mean when you say "see" the control group trend. Ideally, you should have plotted the evolution of the trends in your treatment and control group to assess the validity of this approach. If you are referring to the point estimates, then I believe you are referring to the individual year dummies (i.e., post-treatment dummies). Because this is an interaction model, the time dummies represent the individual pre-post differences in units not exposed to treatment (i.e., $$\text{Treat}_{i} = 0$$). Put more simply, the classical DiD time variable is the time trend in the control group. In most applications, $$\delta_{j}$$ should be your focus; this is your DiD coefficient(s).

2) Is it possible to get the total impact on Y? from the regression I get the results of T-k different coefficient, can I say what is the total impact of treatment over Y?

The total impact of the treatment is the interaction of your treatment dummy with $$one$$ post-treatment indicator. Your formulation is now more concise:

$$y_{it} = \beta_{0} + \beta_{1}\text{Treat}_{i} + \lambda \text{Post}_{t} + \delta (\text{Treat}_i \cdot \text{Post}_{t}) + X_{it}'\gamma + \epsilon_{it},$$

where $$\text{Post}_{t}$$ is no longer representative of individual dummies. It is one unique dummy indexing all post-treatment periods. To put this in perspective, suppose you observe each cross-sectional unit from 2010 to the present year. And, suppose treatment begins in 2016 and remains in place for the entire observation period. A single post-treatment dummy is equal to 1 in all the years treatment is in effect in both treatment and control groups. This is a dummy equal to 1 from 2016 onward, irrespective of a unit's group status. However, in the previous formulation, we included multiple additive year effects: a dummy for 2016, a dummy for 2017, a dummy for 2018, so on and so forth. Each is uniquely interacted with the treatment dummy.

The second specification is typically where you should start. Your estimate of $$\delta$$ is the total effect of the treatment/intervention. The first equation can be viewed as an extension of the second equation, whereby we investigate possible effect heterogeneity throughout the post-treatment period. Effects may grow or fade over time.

• I have performed the first resgression as suggested: 1. What is the meaning of the coefficient b0 and b1? I know that the meaning in regular pre-post regression but not sure what is the meaning in this case. 2. Just to be sure, there is no way I can sum what is the total influence of the treatment over Y? I can just tell what is the treatment effect in a specific year? 3. What is the meaning of gamma? how the specific characteristic I added effected Y?
– XYZ
May 1 '20 at 19:17