Why post main effect being subsumed into the time fixed effects In the DD equation, I have Treat indicator and Post indicator. The model also includes state fixed effects and year fixed effects. Is it necessary to omit Post variable (thus only Treat and Treat*Post)?
The paper said, "Post main effect being subsumed into the time fixed effects."
I am always confused on this point.
The DD model is:
$$Y_{ist}=\beta_0+\beta_1Treat_{ist}+\beta_2Treat_{ist}*Post_t+\gamma'X_{ist}+\rho'C_{st}+\delta_s+\phi_t+\epsilon_{ist}$$
where the outcome $Y_{ist}$ is an indicator equal to one if individual i living in state s who is surveyed in year t is on leave from work in the survey reference week and zero otherwise. (This paper is on paid family leave policy.)
The dummy variable $Treat_{ist}$ is equal to one for California fathers of infants;
$Post_t$ is an indicator equal to one if the individual is surveyed in 2005 or later. (2005 is the first year of the policy.)
The vector $X_{ist}$ contains the following individual-level indicator variables: father's age in bins, race,,,,(omit here)
vector $C_{st}$: state-year controls to account for labor market conditions and other state-specific factors affecting the decision to work: unemployment rate, poverty rate,,,,(omit here)
State and year fixed effects are captured by $\delta_s$ and $\phi_t$, respectively, with the $Post_t$ main effect being subsumed into the time fixed effects.
 A: I just figured it out today. Let's review the concept of Perfect multicollinearity first. Perfect multicollinearity occurs when two or more independent variables in a regression model exhibit a deterministic (perfectly predictable or containing no randomness) linear relationship. Suppose you have a model that $$Y_i=\beta_o+\beta_1X_{i1}+\beta_2X_{i2}+\epsilon_i$$ and
$$X_{i2}=\alpha_o+\alpha_1X_{i1}$$
By doing easy substitution, you then get $$Y_i=\beta_o+\beta_2\alpha_o+(\beta_1+\beta_2\alpha_1)X_{i1}+\epsilon_i$$
You can find that now the equation only contains $X_{i1}$. Obtaining individual regression coefficients (for example, $X_{i2}$) for every variable is impossible if you have perfect multicollinearity.
There are 2 cases in econometrics that could have collinearity: among each other, or with the fixed-effects. The above example is for the former case. Your questions is on the latter case.
Why the above example have collinearity problem? Because in the equation, some independent variable(s) can perfectly predictable other independent variable(s). Once I know $X_{i1}$, I can definitely know $X_{i2}$. That's what I say "perfectly predictable". Once you include fixed effects, then such "perfectly predictable" could happen.
In my example, the paper includes time (year) fixed effects, which is equivalent to you have a set of year dummy variables to denote every single year in your model. Thus, you can imagine there is one variable, say,  $X_{i1}$ means that whether this unit is year 2005. Once you know it is the year 2005, then you know whether this observation is post or before. Because Post is fixed within every-year level. This is exactly what I call "perfectly predictable". But if this variable is changing with each year, then there will be no problem at all. The take-away is Anything that does not vary within the levels (could be individual, time, etc.) is taken care of and cannot be included.
