A revisit to simple DID and Generalised DID

Question

After a couple of questions being asked, I am curious about the inclusion of Post and Treat variable in a simple DID (two groups two-time period). We mainly know there are mainly two types of DID including simple and generalised.

From the genralized DID, from this discussion, we can see that Post and Treat are not included into the regression equation because they will be swallowed by group and period fixed effects (normally we call firm and year fixed effects).

I revisit the equation of simple DID in this link:

$$Y_{it} = \beta_0 + \beta_1*P_t + \beta_2*T_i + \beta_3(P_t*T_i) + u_{it}$$

While $P_t$ and $T_i$ are Post and Treat variables accordingly. My concern is

(1) whether we should not run the group and period fixed effect for simple DID because they will swallow the variables $P_t$ and $T_i$ as in the generalized case?

(2) And because not running unit and period fixed effect, now $u_{it}$ would be a mess. I mean now $u_{it}$ will include time-variant, time-invariant variables and variables vary across both over time and firms, for example, from this discussion:

$$u_{i,t} = \delta_i + \gamma_t + \chi_{i,t} $$

In generalised DID, because we control for group and period fixed effect, so we only need to add independent variables to satisfy $$ \mathbb{E}(\chi x) = 0 $$ Is it correct?

If it is the case, because it seems that we cannot control firm and year fixed effect in simple DID. Therefore, we need to add more variables to try to reach $$\mathbb{E}(\delta x) =0$$ and $$\mathbb{E}(\gamma x)=0$$ apart from $$\mathbb{E}(\chi x) = 0$$ I am wondering that apart from adding too many independent variables, which can cause multicollinearity problems, is there any other solution to solve the messy error term in simple DID?

Answer 1

(1) whether we should not run the group and period fixed effect for simple DID because they will swallow the variables $P_t$ and $T_i$ as in the generalized case?

In a simple 2 period 2 group regression, the $P_t$ and $T_i$ variables are capturing the time and group fixed effect. In fact, the regression with fixed effects is formally identical to the simple DiD specification.

To see this assume that there are two time periods $t = 0,1$ and two groups $g = 0,1$ then the estimation with fixed effects takes the form: $$ y_{i,t,g} = \alpha_0 D_i(t=0) + \alpha_1 D_i(t = 1) + \alpha_2 D_i(g = 0) + \alpha_3 D_i(g = 1) + \alpha_4 x_{i,t,g} + \varepsilon_{i,t,g}. $$ Here $D_i(t = 0)$ is a dummy that equals one if $t = 0$, etc.

If there are only two time periods, we have that $D_i(t = 0) = 1 - D_i(t = 1)$ and $D_i(g = 0) = 1 - D_i(g = 1)$. Using this substitution gives: $$ y_{i,t,g} = \underbrace{(\alpha_0 + \alpha_2)}_{\beta_0} + \underbrace{(\alpha_1 - \alpha_0)}_{\beta_1} D_i(t = 1) + \underbrace{(\alpha_3 - \alpha_2)}_{\beta_2} D_i(g = 1) + \alpha_4 x_{i,t,g} + \varepsilon_{i,t,g}. $$ If we denote $T_i = D_i(t = 1)$, $P_i = D_i(g = 1)$ and if we consider the case where $x_{i,t,g} = T_i P_i$ we obtain: $$ y_{i,t,g} = \beta_0 + \beta_1 T_i + \beta_2 P_i + \alpha_4 T_i P_i + \varepsilon_{i,t,g}. $$ So $\alpha_4$ is exactly the DiD estimator for a model with time and group fixed effects; The $\beta$-parameters are simple linear combinations of the $\alpha$-parameters.