Fixed effect difference-in-differences with count data According to my understanding about the difference-in-differences (DID) model with fixed effects, there are two specifications


*

*y=a0 + a1*TREAT + a2*POST + a3*TREAT_POST + e

*y=a0 + a1*TREAT_POST + timeFE + individualFE + e


Here, the dependent variable is a count variable and TREAT is an indicator variable that represents multiple groups of individuals affected by an event/treatment. The treatment affects groups of individuals at different calendar years, so yrsfromtreatment is a dummy variable for year pre- and post-treatment.
TREAT represent 1 for treated individuals (regardless of group) and 0 for matched control individuals.
POST represent 1 for all years after the treatment is introduced.
TREAT_POST=1 for the treated individuals in the post-treatment period.
So, Model 2 is better if there are possible omitted time-invariant and time-specific variables. And TREAT and POST indicator variables should usually be dropped in Model 2.
However, I find that I can actually run a fixed-effect negative binomial regression with calendar year (i.year) and treatment year (i.yrsfromtreatment) dummies in Stata:
xtset panelid yrsfromtreatment
xtnbreg y i.TREAT##i.POST i.yrsfromtreatment i.year, fe robust

Stata reports positive signs and significance in TREAT and POST. 
How do I explain the coefficients on TREAT and POST when fixed effects are included? Is this model incorrect?
 A: I'm a bit confused. Shouldn't your regression be collinear?
Your command is:
xtnbreg y i.TREAT##i.POST i.yrsfromtreatment i.year, fe robust
Let:


*

*$I_{i,t,\tau}$ be a variable indicating whether individual $i$ is $\tau$ years from treatment at time $t$.

*$\mathrm{Treat}_i$ is an indicator as to whether $i$ at any point receives treatment.

*$\mathrm{Post}_{it}$ is an indicator as to whether $i$ has received treatment at time $t$ or in prior periods.


It looks that command is running the regression:
$$ y_{it} = b \mathrm{Treat}_{i}\mathrm{Post}_{it}+ \sum_{\tau=0}^\bar{\tau} c_\tau I_{i,t,\tau }+ u_i + v_t + \epsilon_{it} $$
What confuses me is that shouldn't this be collinear? If $\tau$ takes discrete values $0$, $1$, $2$, etc... then shouldn't
$$ \mathrm{Treat}_{i}\mathrm{Post}_{it} = \sum_{\tau = 0}^\bar{\tau} I_{i,t,\tau}$$
That is, if individual $i$ had treatment prior to or in year $t$ (i.e. $\mathrm{Treat}_{i}\mathrm{Post}_{it} = 1$), then they were either treated this year (i.e. $I_{i,t,0}$ = 1), or they were treated last year  (i.e. $I_{i,t,1} = 1$), or treated the year before that ($I_{i,t,2} = 1$), etc....
