Proportional ATTs for Poisson Regressions

[frederickluser]

Dear @grantmcdermott , Dear @vincentarelbundock - a question regarding the marginal effects in etwfe.

I want to use etwfe with a Poisson specification, reporting the overall ATT as a percentage change (i.e., "Employment decreased by x% due to the treatment"). Here is a super simple example with only one coefficient:

# Load the data
data("mpdta")
mpdta$emp <- exp(mpdta$lemp)

# Regression
(mod = etwfe(
        fml  = emp ~ 0,
        tvar = year,     
        gvar = first.treat, 
        family = "quasipoisson",
        data = mpdta %>% dplyr::filter(year %in% c(2004, 2005), first.treat %in% c(2004, 2005, 2006)),     
        vcov = ~countyreal
    ))

# Output
GLM estimation, family = quasipoisson, Dep. Var.: emp
Observations: 120 
Fixed-effects: first.treat: 2,  year: 2
Standard-errors: Clustered (countyreal) 
                                      Estimate Std. Error  t value Pr(>|t|)    
.Dtreat:first.treat::2004:year::2005 -0.050079    0.02777 -1.80335  0.07644 .  

# Aggregation / Marginal Effect
mod2 <- emfx(mod)
  Term                 Contrast .Dtreat Estimate Std. Error     z Pr(>|z|)   S 2.5 % 97.5 %
 .Dtreat mean(TRUE) - mean(FALSE)    TRUE    -36.9         20 -1.85   0.0644 4.0 -76.1   2.21

I want to get the marginal effects as percentage changes. I was hoping I could simply relate the marginal ATT reported by emfx to the mean of the control group pre-treatment. In this example, I think it should be:

dat <- data.table::as.data.table(eval(mod$call$data, mod$call_env))
mod2$estimate / ( dat%>% dplyr::filter(year == 2004, first.treat == 2006) %>% pull(emp) %>% mean() )

But then, could in general the following be correct?

dat <- data.table::as.data.table(eval(mod$call$data, mod$call_env))
mod2$estimate / (dat %>% dplyr::filter(year < first.treat) %>% pull(emp) %>% mean() )

Does that make any sense? Any feedback is highly appreciated.