Transforming DID into TWFE

[Zariab Hossain]

Dear Stata users,

I am working on a project that uses DID with continuous treatment variable. I had the following DID specification:

Is the TWFE version of the above specification is like below:

Here \delta is the continuous treatment variable which varies over time. Is the above specification correct? X is a vector of covariates, \lambda_{j} and \lambda_{t} are caseworker and referral year fixed effects.

Previously, I estimated the event-study version of equation-1 in Stata as:

   summ     event_time
    g         t = event_time - r(min)
    summ     t if event_time == -1
    local     true_neg = r(mean)
    
    reghdfe        breadth_search c.delta##ib`true_neg'.t  ///
                LWbas_male age LWbas_married LWbas_immigrant LWbas_child LWbas_pre LWbas_pre9 ///
                LWbas_s_high LWbas_l_high LWbas_s_univ LWbas_l_univ LW5_emp_mon_last5y ///
                LWhist_empbef1 LW5_av_dur LW5_last_dur LWhist_income_bef1 LW5_welfare5 , ///
                absorb(referal_year referal_CWid) ///
                cluster(referal_CWid)

Here I used event time == -1 as a reference time.

My question is how to do the event-study version of the TWFE specification?

Any help or suggestion would be highly appreciated.

Regards,
Zariab
Uppsala University