Efficient way to estimate cohort-specific event-study regression with many cohorts

[Heike Waechter]

Hello everyone,

I am estimating an event-study regression with a dependent variable y, an event-time indicator t_idx that takes values from 1 to 16, and a cohort variable. The goal is to estimate cohort-specific event-time effects, using the period where t_idx == 5 as the baseline within each cohort.

In a smaller setting (with only a few cohorts), I implemented it as follows:

*define cohort
local cohort languageregion
local coh3 = substr("`cohort'", 1, 3)
local coh9 = substr("`cohort'", 1, 9)

*create interaction dummies
xi i.t_idx*i.`cohort', noomit
drop _It_iX`coh3'_5_* _It_idx* _I`coh9'_*

* drop empty interactions
local vardrop
foreach var of varlist _I* {
    quietly summarize `var', meanonly
    if r(mean) == 0 {
        di "`var' --> delete"
        local vardrop `vardrop' `var'
    }
}
capture drop `vardrop'

* regression
reg y _It_iX`coh3'_* i.age i.statyear, r

Now I would like to replace the cohort variable with municipality, which has more than 2,000 unique values. With this approach, Stata would have to generate roughly 15 × 2,000 = 30'000 interaction dummies, which is obviously not efficient.

I was wondering if there is a more efficient way to estimate this event-study specification. For example i thought about using Stata’s factor-variable notation (# or ##), but I haven’t figured out how to replicate the structure above — specifically, how to ensure that t_idx == 5 serves as the baseline within each cohort.

Any suggestions or references to more efficient approaches would be very welcome.

Thank you for your time and help,
Heike Waechter

[Joseph Coveney]

I am estimating an event-study regression . . . with municipality, which has more than 2,000 unique values. . . . I was wondering if there is a more efficient way to estimate this event-study specification.

No one's yet taken you up on this, and so I'll give it try: if efficiency is what you're after, then how about modeling municipality as a random effect?

In mixed-effects linear regression models (mixed) and mixed-effects generalized linear regression models (meglm and other, more-specific, estimation commands), you can include time as a random slope, which is the same as a time × municipality interaction term. For individual municipality time slopes, you can examine the best linear unbiased estimates (BLUPs) after mixed and corresponding so-called empirical Bayes predictions after meglm using the postestimation command predict double . . . , reffects.

As an aside, I don't recall seeing a linear model fitted to an event study's data. I assume that you've considered what's offered in

help st

and have opted for regress despite what's shown there.

[Heike Waechter]

First, thank you for your answer — it’s greatly appreciated.

I may have overemphasized the efficiency aspect a bit in my initial post. I’m perfectly fine with a slower program; it’s just that my current version runs for days without making progress on the dummy creation. By efficiency I mainly meant computationally feasible — it doesn’t need to be lightning fast, just something that runs in hours rather than days. In that light, do you think it’s not possible to make the current approach work (for example by using Stata’s # or ## operators correctly)?

I don’t know much about mixed-effects beyond what a quick search told me, but I’m worried that it’s not conceptually the same. If I understand correctly, the random-slope model gives an overall pattern plus deviations, rather than a separate coefficient for every municipality × event-time combination.

Do I understand that right? And if so, is there a way to stay closer to the explicit interaction setup but still make it computationally feasible?

[Joseph Coveney]

I may have overemphasized the efficiency aspect a bit in my initial post. I’m perfectly fine with a slower program . . . By efficiency I mainly meant computationally feasible . . .

I wasn't using the word efficiency in that sense; regardless, you might not get anything tractable with a fixed-effect interaction approach when one factor has more than two thousand levels.

If I understand correctly, the random-slope model gives an overall pattern plus deviations, rather than a separate coefficient for every municipality × event-time combination.

Do I understand that right?

The random slopes are estimable individually, and so you can add them to the fixed-effect coefficient for time to get the municipality's overall prediction, which will be analogous to your separate coefficients, albeit likely with some so-called shrinkage.

It seems that i misconstrued what you mean by "event-study", perhaps in a manner similar in another recent thread

[Andrew Musau]

With this approach, Stata would have to generate roughly 15 × 2,000 = 30'000 interaction dummies

What do you intend to do with this output? Can you really make sense of all 30,000 dummy coefficients?

I was wondering if there is a more efficient way to estimate this event-study specification. For example i thought about using Stata’s factor-variable notation (# or ##), but I haven’t figured out how to replicate the structure above — specifically, how to ensure that t_idx == 5 serves as the baseline within each cohort.

There is some efficiency to be gained by not manually creating the indicators, but I suspect the main issue is the number of parameters you want to estimate. OLS has to invert a matrix, so this brute-force approach of estimating all dummy variable coefficients is not recommended. You can also absorb the age and state-year dummies if their estimates are of no immediate interest. If you provide a reproducible example with only a few levels of the municipality/cohort variable, I can illustrate how to achieve what you want using factor-variable notation.

[Heike Waechter]

Thank you both for your helpful input — and apologies for the delay over the weekend.

My event-study is on measuring “child penalties”: in year t_idx=6 the child is born, and t_idx=5 serves as the baseline (i.e., income one year before the birth).

What I plan to do with the estimates is to use a “movers design”. Therefore i will ultimately average the child-penalties within municipalities. Nevertheless, it remains of interest to me to observe the variance across municipalities for the different t_idx before averaging — and hence I would like to estimate all the municipality × event-time interactions. That also adds consistency with previous subgroup analyses I’ve done (by language region, degree of urbanisation etc.).

That said, if estimating every municipality’s full interaction proves intractable, estimating the average effect may indeed be the best course of action. Would that be achieved by the random slope approach?

A question for you, Andrew: when you refer to a “reproducible example”, could you clarify what exactly you need? Would it suffice for me to provide my full code (with anonymised/synthetic data) or do you need actual data?

Thanks again, and I look forward to any suggestions you may have for feasibly estimating this many interactions (or moving to a pooled/random‐slope alternative).

[Andrew Musau]

A question for you, Andrew: when you refer to a “reproducible example”, could you clarify what exactly you need? Would it suffice for me to provide my full code (with anonymised/synthetic data) or do you need actual data?

Yes, e.g., data that can be used to run what you have in #2:

*define cohort
local cohort languageregion
local coh3 = substr("`cohort'", 1, 3)
local coh9 = substr("`cohort'", 1, 9)

*create interaction dummies
xi i.t_idx*i.`cohort', noomit
drop _It_iX`coh3'_5_* _It_idx* _I`coh9'_*

* drop empty interactions
local vardrop
foreach var of varlist _I* {
quietly summarize `var', meanonly
if r(mean) == 0 {
di "`var' --> delete"
local vardrop `vardrop' `var'
}
}
capture drop `vardrop'

* regression
reg y _It_iX`coh3'_* i.age i.statyear, r

Would that be achieved by the random slope approach?

For observational data, it would be difficult to justify a random effects model. I would not consider this.

Therefore i will ultimately average the child-penalties within municipalities. Nevertheless, it remains of interest to me to observe the variance across municipalities for the different t_idx before averaging — and hence I would like to estimate all the municipality × event-time interactions.

The predicted residuals obtained after an FE transformation would provide the same information, so it would appear unnecessary to estimate all the interaction terms and then average them subsequently. There is some discussion here: https://www.statalist.org/forums/for...cts-with-xtreg. This could also be illustrated with a data example.

[Heike Waechter]

Thank you for your willingness to take a look.

I’ve prepared code that generates a synthetic dataset and runs an analysis replicating the structure of my actual setup — an event-study design with multiple municipalities (cohorts), an event-time index, age, calendar year, and an outcome variable.

// ===== Synthetic dataset for reproducible example =====
clear  
set seed 12345  

// Parameters
local municipality     = 100        // number of municipalities/cohorts
local Npersons   = 50          // number of individuals per municipality
local Tmin       = 1          // event-time baseline
local Tmax       = 17        // event-time max
local Tobs       = `Tmax' - `Tmin'
local obs        = `municipality' * `Npersons' * `Tobs'
display `obs'

// Create panel: each municipality × each individual × each t_idx
set obs `obs'

// Define identifiers
gen mun_id = ceil(_n / (`Npersons' * `Tobs'))
gen person_in_unit = mod(ceil(_n / `Tobs') - 1, `Npersons') + 1
gen id = mun_id * 10000 + person_in_unit   // individual identifier

// Create t_idx
bysort mun_id person_in_unit: gen t_idx = `Tmin' + _n - 1

// Cohort variable (e.g., language region)
gen str3 language = cond(mod(mun_id,3)==0, "GER", ///
                      cond(mod(mun_id,3)==1, "FRE", "ITA"))

// We first generate a random base statyear for each individual
bysort id: gen statyear0 = 1995 + floor(runiform()*10)   // base year between 1995 and 2004
// Then statyear for each row = statyear0 + t_idx
gen statyear = statyear0 + t_idx

// Age: define starting age at t_idx == 0, then age = start_age + t_idx
bysort id: gen start_age = 25 + int(runiform()*20)   // starting age between ~25 and ~44 at t_idx = 0
gen age = start_age + t_idx

// Outcome y: municipality + individual + event time effects + noise
gen u_intercept = rnormal(0, 5)
gen u_slope     = rnormal(0, 0.5)
gen event_effect = cond(t_idx>=0, -10 + 0.5*t_idx, 0)
gen y = 100 + u_intercept + u_slope * t_idx + event_effect + rnormal(0, 10)

// Clean up interim vars
drop u_intercept u_slope start_age statyear0 event_effect

label var id            "Individual identifier"
label var mun_id       "Municipality / cohort id"
label var person_in_unit "Person within municipality"
label var cohort        "Cohort (language region, fictitious)"
label var t_idx         "Event time index (years relative)"
label var age           "Age of individual (synthetic)"
label var statyear      "Calendar year of observation"
label var y             "Outcome (synthetic)"

* analysis

*define cohort
local cohort unit_id
local coh3 = substr("`cohort'", 1, 3)
local coh9 = substr("`cohort'", 1, 9)

// Create interaction dummies as in your original code
*create interaction dummies
xi i.t_idx*i.`cohort', noomit
drop _It_iX`coh3'_5_* _It_idx* _I`coh9'_*

* drop empty interactions
local vardrop
foreach var of varlist _I* {
    quietly summarize `var', meanonly
    if r(mean) == 0 {
        di "`var' --> delete"
        local vardrop `vardrop' `var'
    }
}
capture drop `vardrop'

* regression
reg y _It_iX`coh3'_* i.age i.statyear, r

Is this sufficient for you to show me how to run with interactions aswell as predict residuals to get the averages directly? When talking about averaging the interaction terms it is important to note that i would only average over the post periods (t_idx from 6 to 16). I am not sure thats really captured in the post you linked where you (as far as i understand) recommend to run a regression like "reg y eventtime i.age i.year i.mun_id" and then extract the fe for the mun_ids, right?

Thanks for helping me out and if you need any additional info, just let me know!

[Andrew Musau]

I’ve prepared code that generates a synthetic dataset and runs an analysis replicating the structure of my actual setup — an event-study design with multiple municipalities (cohorts), an event-time index, age, calendar year, and an outcome variable.

Thanks. Please check the example again. When I run the code, I get the error:

. label var cohort        "Cohort (language region, fictitious)"
variable cohort not found
r(111);

I don't see that you have generated a variable named "cohort" anywhere.

[Heike Waechter]

ah sry, i chose to change the naming and forgot that one. here you go:

// ===== Synthetic dataset for reproducible example =====
clear  
set seed 12345  

// Parameters
local municipality     = 100        // number of municipalities/cohorts
local Npersons   = 50          // number of individuals per municipality
local Tmin       = 1          // event-time baseline
local Tmax       = 17        // event-time max
local Tobs       = `Tmax' - `Tmin' 
local obs        = `municipality' * `Npersons' * `Tobs'
display `obs' 

// Create panel: each municipality × each individual × each t_idx
set obs `obs'

// Define identifiers
gen mun_id = ceil(_n / (`Npersons' * `Tobs'))
gen person_in_unit = mod(ceil(_n / `Tobs') - 1, `Npersons') + 1
gen id = mun_id * 10000 + person_in_unit   // individual identifier

// Create t_idx
bysort mun_id person_in_unit: gen t_idx = `Tmin' + _n - 1

// Cohort variable (e.g., language region)
gen str3 language = cond(mod(mun_id,3)==0, "GER", ///
                      cond(mod(mun_id,3)==1, "FRE", "ITA"))

// We first generate a random base statyear for each individual
bysort id: gen statyear0 = 1995 + floor(runiform()*10)   // base year between 1995 and 2004
// Then statyear for each row = statyear0 + t_idx
gen statyear = statyear0 + t_idx

// Age: define starting age at t_idx == 0, then age = start_age + t_idx
bysort id: gen start_age = 25 + int(runiform()*20)   // starting age between ~25 and ~44 at t_idx = 0
gen age = start_age + t_idx

// Outcome y: municipality + individual + event time effects + noise
gen u_intercept = rnormal(0, 5)
gen u_slope     = rnormal(0, 0.5)
gen event_effect = cond(t_idx>=0, -10 + 0.5*t_idx, 0)
gen y = 100 + u_intercept + u_slope * t_idx + event_effect + rnormal(0, 10)

// Clean up interim vars
drop u_intercept u_slope start_age statyear0 event_effect

label var id            "Individual identifier"
label var mun_id       "Municipality / cohort id"
label var person_in_unit "Person within municipality"
label var language        "language region"
label var t_idx         "Event time index (years relative)"
label var age           "Age of individual (synthetic)"
label var statyear      "Calendar year of observation"
label var y             "Outcome (synthetic)"

* analysis

*define cohort
local cohort language // choose mun_id or language
local coh3 = substr("`cohort'", 1, 3)
local coh9 = substr("`cohort'", 1, 9)

// Create interaction dummies as in your original code
*create interaction dummies
xi i.t_idx*i.`cohort', noomit
drop _It_iX`coh3'_5_* _It_idx* _I`coh9'_*

* drop empty interactions
local vardrop
foreach var of varlist _I* {
    quietly summarize `var', meanonly
    if r(mean) == 0 {
        di "`var' --> delete"
        local vardrop `vardrop' `var'
    }
}
capture drop `vardrop'

* regression
reg y _It_iX`coh3'_* i.age i.statyear, r

[Andrew Musau]

Thank you. This works. Here's how to run the regressions using factor variable notation. I'm teaching all day, so I'll post the solution that involves absorbing the interactions and then predicting the residuals later in the day.

// ===== Synthetic dataset for reproducible example =====
clear  
set seed 12345  

// Parameters
local municipality     = 100        // number of municipalities/cohorts
local Npersons   = 50          // number of individuals per municipality
local Tmin       = 1          // event-time baseline
local Tmax       = 17        // event-time max
local Tobs       = `Tmax' - `Tmin' 
local obs        = `municipality' * `Npersons' * `Tobs'
display `obs' 

// Create panel: each municipality × each individual × each t_idx
set obs `obs'

// Define identifiers
gen mun_id = ceil(_n / (`Npersons' * `Tobs'))
gen person_in_unit = mod(ceil(_n / `Tobs') - 1, `Npersons') + 1
gen id = mun_id * 10000 + person_in_unit   // individual identifier

// Create t_idx
bysort mun_id person_in_unit: gen t_idx = `Tmin' + _n - 1

// Cohort variable (e.g., language region)
gen str3 language = cond(mod(mun_id,3)==0, "GER", ///
                      cond(mod(mun_id,3)==1, "FRE", "ITA"))

// We first generate a random base statyear for each individual
bysort id: gen statyear0 = 1995 + floor(runiform()*10)   // base year between 1995 and 2004
// Then statyear for each row = statyear0 + t_idx
gen statyear = statyear0 + t_idx

// Age: define starting age at t_idx == 0, then age = start_age + t_idx
bysort id: gen start_age = 25 + int(runiform()*20)   // starting age between ~25 and ~44 at t_idx = 0
gen age = start_age + t_idx

// Outcome y: municipality + individual + event time effects + noise
gen u_intercept = rnormal(0, 5)
gen u_slope     = rnormal(0, 0.5)
gen event_effect = cond(t_idx>=0, -10 + 0.5*t_idx, 0)
gen y = 100 + u_intercept + u_slope * t_idx + event_effect + rnormal(0, 10)

// Clean up interim vars
drop u_intercept u_slope start_age statyear0 event_effect

label var id            "Individual identifier"
label var mun_id       "Municipality / cohort id"
label var person_in_unit "Person within municipality"
label var language        "language region"
label var t_idx         "Event time index (years relative)"
label var age           "Age of individual (synthetic)"
label var statyear      "Calendar year of observation"
label var y             "Outcome (synthetic)"

* analysis

*define cohort
local cohort language // choose mun_id or language
local coh3 = substr("`cohort'", 1, 3)
local coh9 = substr("`cohort'", 1, 9)

// Create interaction dummies as in your original code
*create interaction dummies
xi i.t_idx*i.`cohort', noomit
drop _It_iX`coh3'_5_* _It_idx* _I`coh9'_*

* drop empty interactions
local vardrop
foreach var of varlist _I* {
    quietly summarize `var', meanonly
    if r(mean) == 0 {
        di "`var' --> delete"
        local vardrop `vardrop' `var'
    }
}
capture drop `vardrop'

* regression
reg y _It_iX`coh3'_* i.age i.statyear, r


egen lang= group(language)

local omitted 
qui levelsof lang, local(levs)
foreach l of local levs{
    local omitted `omitted' o5.t_idx#o.`l'.lang
}

reg y ibn.t_idx#ibn.lang `omitted' i.age i.statyear, r noomit

Res.:

. * regression
. reg y _It_iX`coh3'_* i.age i.statyear, r

Linear regression                               Number of obs     =     80,000
                                                F(103, 79896)     =      27.87
                                                Prob > F          =     0.0000
                                                R-squared         =     0.0348
                                                Root MSE          =     12.162

--------------------------------------------------------------------------------
               |               Robust
             y | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
---------------+----------------------------------------------------------------
 _It_iXlan_1_1 |  -2.368133   .3336004    -7.10   0.000    -3.021988   -1.714278
 _It_iXlan_1_2 |  -2.310124   .3433546    -6.73   0.000    -2.983097   -1.637151
 _It_iXlan_1_3 |  -2.030665   .3387226    -6.00   0.000    -2.694559    -1.36677
 _It_iXlan_2_1 |  -1.670071   .3224575    -5.18   0.000    -2.302085   -1.038056
 _It_iXlan_2_2 |  -1.862749   .3324816    -5.60   0.000    -2.514411   -1.211087
 _It_iXlan_2_3 |  -1.583652   .3354648    -4.72   0.000    -2.241161   -.9261431
 _It_iXlan_3_1 |  -1.563066   .3132553    -4.99   0.000    -2.177044   -.9490873
 _It_iXlan_3_2 |  -1.576734   .3279649    -4.81   0.000    -2.219543   -.9339246
 _It_iXlan_3_3 |  -1.193576   .3323675    -3.59   0.000    -1.845014   -.5421378
 _It_iXlan_4_1 |  -.7413887   .3187269    -2.33   0.020    -1.366091   -.1166859
 _It_iXlan_4_2 |  -.4857789   .3178338    -1.53   0.126    -1.108731    .1371733
 _It_iXlan_4_3 |  -.6801947   .3245295    -2.10   0.036     -1.31627   -.0441189
 _It_iXlan_6_1 |   .4780968   .3246934     1.47   0.141    -.1583003    1.114494
 _It_iXlan_6_2 |   .6149669    .331429     1.86   0.064    -.0346318    1.264566
 _It_iXlan_6_3 |   .2160518   .3288104     0.66   0.511    -.4284144     .860518
 _It_iXlan_7_1 |   1.535011   .3354032     4.58   0.000     .8776232      2.1924
 _It_iXlan_7_2 |   .9206809   .3432816     2.68   0.007     .2478512    1.593511
 _It_iXlan_7_3 |    .952353   .3310848     2.88   0.004     .3034288    1.601277
 _It_iXlan_8_1 |   .7677342   .3368691     2.28   0.023     .1074729    1.427995
 _It_iXlan_8_2 |   .7492054   .3397146     2.21   0.027      .083367    1.415044
 _It_iXlan_8_3 |    1.19948   .3471386     3.46   0.001     .5190909     1.87987
 _It_iXlan_9_1 |   1.656851   .3458675     4.79   0.000     .9789526    2.334749
 _It_iXlan_9_2 |   2.412133   .3466938     6.96   0.000     1.732615    3.091651
 _It_iXlan_9_3 |   1.538266   .3481109     4.42   0.000     .8559711    2.220561
_It_iXlan_10_1 |   1.955685   .3525141     5.55   0.000      1.26476    2.646611
_It_iXlan_10_2 |   2.651968   .3606623     7.35   0.000     1.945072    3.358864
_It_iXlan_10_3 |    2.01723     .36384     5.54   0.000     1.304106    2.730354
_It_iXlan_11_1 |   2.862153   .3619068     7.91   0.000     2.152817    3.571488
_It_iXlan_11_2 |   3.034875   .3701407     8.20   0.000     2.309402    3.760348
_It_iXlan_11_3 |   3.112749    .372354     8.36   0.000     2.382937     3.84256
_It_iXlan_12_1 |   3.044052   .3783696     8.05   0.000      2.30245    3.785654
_It_iXlan_12_2 |   3.347627   .3806228     8.80   0.000     2.601609    4.093646
_It_iXlan_12_3 |   3.188872   .3817797     8.35   0.000     2.440587    3.937158
_It_iXlan_13_1 |   3.966754   .3872396    10.24   0.000     3.207767    4.725741
_It_iXlan_13_2 |   3.903928   .3943237     9.90   0.000     3.131056      4.6768
_It_iXlan_13_3 |   3.539779   .3922857     9.02   0.000     2.770901    4.308656
_It_iXlan_14_1 |   4.276604   .4006579    10.67   0.000     3.491317    5.061891
_It_iXlan_14_2 |   4.516788   .4068208    11.10   0.000     3.719422    5.314154
_It_iXlan_14_3 |    4.23396    .405333    10.45   0.000      3.43951     5.02841
_It_iXlan_15_1 |   5.216766   .4113119    12.68   0.000     4.410597    6.022935
_It_iXlan_15_2 |   4.904253   .4078983    12.02   0.000     4.104775    5.703731
_It_iXlan_15_3 |   5.309786    .419863    12.65   0.000     4.486857    6.132715
_It_iXlan_16_1 |   4.722023    .427838    11.04   0.000     3.883463    5.560583
_It_iXlan_16_2 |   5.363941   .4285051    12.52   0.000     4.524074    6.203809
_It_iXlan_16_3 |   5.330657   .4290629    12.42   0.000     4.489697    6.171618
               |
           age |
           27  |   .7325874   .9076534     0.81   0.420    -1.046408    2.511582
           28  |   .3208175   .8504925     0.38   0.706    -1.346142    1.987778
           29  |    .191439   .8339289     0.23   0.818    -1.443056    1.825934
           30  |  -.4719228   .8236589    -0.57   0.567    -2.086289    1.142444
           31  |   -.142288   .8137081    -0.17   0.861    -1.737151    1.452575
           32  |   .1990107   .8054323     0.25   0.805    -1.379632    1.777653
           33  |  -.2738529   .7992741    -0.34   0.732    -1.840425    1.292719
           34  |  -.1529101   .7961833    -0.19   0.848    -1.713424    1.407604
           35  |  -.1288967    .793197    -0.16   0.871    -1.683558    1.425764
           36  |   .0588335   .7908345     0.07   0.941    -1.491197    1.608864
           37  |   .3921179   .7880807     0.50   0.619    -1.152515    1.936751
           38  |   .0813392   .7881708     0.10   0.918    -1.463471    1.626149
           39  |  -.1347576   .7870812    -0.17   0.864    -1.677432    1.407917
           40  |  -.4748054   .7847014    -0.61   0.545    -2.012815    1.063205
           41  |  -.0754757   .7830682    -0.10   0.923    -1.610284    1.459333
           42  |   .2493342   .7840692     0.32   0.750    -1.287437    1.786105
           43  |  -.1051961   .7849101    -0.13   0.893    -1.643615    1.433223
           44  |  -.1332997   .7836731    -0.17   0.865    -1.669294    1.402695
           45  |  -.0336313   .7837356    -0.04   0.966    -1.569748    1.502485
           46  |   -.077499   .7880767    -0.10   0.922    -1.622124    1.467126
           47  |  -.1240083   .7906083    -0.16   0.875    -1.673596    1.425579
           48  |  -.1406644   .7925696    -0.18   0.859    -1.694096    1.412767
           49  |   -.311831   .7964584    -0.39   0.695    -1.872884    1.249222
           50  |   .0795048   .8018629     0.10   0.921    -1.492141    1.651151
           51  |    -.03987   .8063175    -0.05   0.961    -1.620247    1.540507
           52  |    .175123   .8123457     0.22   0.829    -1.417069    1.767315
           53  |  -.2160892   .8206156    -0.26   0.792    -1.824491    1.392312
           54  |   .1508225   .8302669     0.18   0.856    -1.476495     1.77814
           55  |  -.1588468   .8413972    -0.19   0.850     -1.80798    1.490286
           56  |  -.4077025   .8533693    -0.48   0.633    -2.080301    1.264896
           57  |   .1506475   .8806643     0.17   0.864    -1.575449    1.876744
           58  |  -.5995126   .9128689    -0.66   0.511     -2.38873    1.189705
           59  |    .000663   .9649573     0.00   0.999    -1.890647    1.891973
           60  |   .1268308   1.179696     0.11   0.914    -2.185366    2.439027
               |
      statyear |
         1997  |   .6893219   .6308627     1.09   0.275     -.547165    1.925809
         1998  |   .2284195   .5983482     0.38   0.703    -.9443392    1.401178
         1999  |   .1217415    .578479     0.21   0.833    -1.012074    1.255557
         2000  |  -.2051283   .5716441    -0.36   0.720    -1.325547    .9152904
         2001  |   .1752776   .5685706     0.31   0.758    -.9391171    1.289672
         2002  |   .3832778   .5644336     0.68   0.497    -.7230086    1.489564
         2003  |    .053925   .5606968     0.10   0.923    -1.045037    1.152887
         2004  |   .4116798   .5586755     0.74   0.461    -.6833205     1.50668
         2005  |   .0171829   .5558482     0.03   0.975    -1.072276    1.106642
         2006  |  -.2911823    .565337    -0.52   0.607    -1.399239    .8168745
         2007  |   .0164182   .5695602     0.03   0.977    -1.099916    1.132753
         2008  |   .0714586   .5724327     0.12   0.901    -1.050506    1.193423
         2009  |  -.1116666   .5763015    -0.19   0.846    -1.241214    1.017881
         2010  |   .0422877   .5792369     0.07   0.942    -1.093013    1.177588
         2011  |   .2263829   .5824522     0.39   0.698    -.9152197    1.367985
         2012  |   .2443245   .5900713     0.41   0.679    -.9122116    1.400861
         2013  |   .3853788   .5961326     0.65   0.518    -.7830373    1.553795
         2014  |  -.0020605   .6056642    -0.00   0.997    -1.189159    1.185038
         2015  |   .0453184   .6138172     0.07   0.941    -1.157759    1.248396
         2016  |  -.0543527   .6290408    -0.09   0.931    -1.287269    1.178563
         2017  |   .2243876   .6525029     0.34   0.731    -1.054514    1.503289
         2018  |  -.6865201    .677269    -1.01   0.311    -2.013963     .640923
         2019  |  -.5075994   .7313385    -0.69   0.488    -1.941018    .9258194
         2020  |   .3835909   .8298596     0.46   0.644    -1.242929     2.01011
               |
         _cons |   92.67639   .9388149    98.72   0.000     90.83632    94.51646
--------------------------------------------------------------------------------

. 
. 
. egen lang= group(language)

. 
. local omitted 

. qui levelsof lang, local(levs)

. foreach l of local levs{
  2.     local omitted `omitted' o5.t_idx#o.`l'.lang
  3. }

. 
. reg y ibn.t_idx#ibn.lang `omitted' i.age i.statyear, r noomit

Linear regression                               Number of obs     =     80,000
                                                F(103, 79896)     =      27.87
                                                Prob > F          =     0.0000
                                                R-squared         =     0.0348
                                                Root MSE          =     12.162

------------------------------------------------------------------------------
             |               Robust
           y | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
  t_idx#lang |
        1 1  |  -2.368133   .3336004    -7.10   0.000    -3.021988   -1.714278
        1 2  |  -2.310124   .3433546    -6.73   0.000    -2.983097   -1.637151
        1 3  |  -2.030665   .3387226    -6.00   0.000    -2.694559    -1.36677
        2 1  |  -1.670071   .3224575    -5.18   0.000    -2.302085   -1.038056
        2 2  |  -1.862749   .3324816    -5.60   0.000    -2.514411   -1.211087
        2 3  |  -1.583652   .3354648    -4.72   0.000    -2.241161   -.9261431
        3 1  |  -1.563066   .3132553    -4.99   0.000    -2.177044   -.9490873
        3 2  |  -1.576734   .3279649    -4.81   0.000    -2.219543   -.9339246
        3 3  |  -1.193576   .3323675    -3.59   0.000    -1.845014   -.5421378
        4 1  |  -.7413887   .3187269    -2.33   0.020    -1.366091   -.1166859
        4 2  |  -.4857789   .3178338    -1.53   0.126    -1.108731    .1371733
        4 3  |  -.6801947   .3245295    -2.10   0.036     -1.31627   -.0441189
        6 1  |   .4780968   .3246934     1.47   0.141    -.1583003    1.114494
        6 2  |   .6149669    .331429     1.86   0.064    -.0346318    1.264566
        6 3  |   .2160518   .3288104     0.66   0.511    -.4284144     .860518
        7 1  |   1.535011   .3354032     4.58   0.000     .8776232      2.1924
        7 2  |   .9206809   .3432816     2.68   0.007     .2478512    1.593511
        7 3  |    .952353   .3310848     2.88   0.004     .3034288    1.601277
        8 1  |   .7677342   .3368691     2.28   0.023     .1074729    1.427995
        8 2  |   .7492054   .3397146     2.21   0.027      .083367    1.415044
        8 3  |    1.19948   .3471386     3.46   0.001     .5190909     1.87987
        9 1  |   1.656851   .3458675     4.79   0.000     .9789526    2.334749
        9 2  |   2.412133   .3466938     6.96   0.000     1.732615    3.091651
        9 3  |   1.538266   .3481109     4.42   0.000     .8559711    2.220561
       10 1  |   1.955685   .3525141     5.55   0.000      1.26476    2.646611
       10 2  |   2.651968   .3606623     7.35   0.000     1.945072    3.358864
       10 3  |    2.01723     .36384     5.54   0.000     1.304106    2.730354
       11 1  |   2.862153   .3619068     7.91   0.000     2.152817    3.571488
       11 2  |   3.034875   .3701407     8.20   0.000     2.309402    3.760348
       11 3  |   3.112749    .372354     8.36   0.000     2.382937     3.84256
       12 1  |   3.044052   .3783696     8.05   0.000      2.30245    3.785654
       12 2  |   3.347627   .3806228     8.80   0.000     2.601609    4.093646
       12 3  |   3.188872   .3817797     8.35   0.000     2.440587    3.937158
       13 1  |   3.966754   .3872396    10.24   0.000     3.207767    4.725741
       13 2  |   3.903928   .3943237     9.90   0.000     3.131056      4.6768
       13 3  |   3.539779   .3922857     9.02   0.000     2.770901    4.308656
       14 1  |   4.276604   .4006579    10.67   0.000     3.491317    5.061891
       14 2  |   4.516788   .4068208    11.10   0.000     3.719422    5.314154
       14 3  |    4.23396    .405333    10.45   0.000      3.43951     5.02841
       15 1  |   5.216766   .4113119    12.68   0.000     4.410597    6.022935
       15 2  |   4.904253   .4078983    12.02   0.000     4.104775    5.703731
       15 3  |   5.309786    .419863    12.65   0.000     4.486857    6.132715
       16 1  |   4.722023    .427838    11.04   0.000     3.883463    5.560583
       16 2  |   5.363941   .4285051    12.52   0.000     4.524074    6.203809
       16 3  |   5.330657   .4290629    12.42   0.000     4.489697    6.171618
             |
         age |
         27  |   .7325874   .9076534     0.81   0.420    -1.046408    2.511582
         28  |   .3208175   .8504925     0.38   0.706    -1.346142    1.987778
         29  |    .191439   .8339289     0.23   0.818    -1.443056    1.825934
         30  |  -.4719228   .8236589    -0.57   0.567    -2.086289    1.142444
         31  |   -.142288   .8137081    -0.17   0.861    -1.737151    1.452575
         32  |   .1990107   .8054323     0.25   0.805    -1.379632    1.777653
         33  |  -.2738529   .7992741    -0.34   0.732    -1.840425    1.292719
         34  |  -.1529101   .7961833    -0.19   0.848    -1.713424    1.407604
         35  |  -.1288967    .793197    -0.16   0.871    -1.683558    1.425764
         36  |   .0588335   .7908345     0.07   0.941    -1.491197    1.608864
         37  |   .3921179   .7880807     0.50   0.619    -1.152515    1.936751
         38  |   .0813392   .7881708     0.10   0.918    -1.463471    1.626149
         39  |  -.1347576   .7870812    -0.17   0.864    -1.677432    1.407917
         40  |  -.4748054   .7847014    -0.61   0.545    -2.012815    1.063205
         41  |  -.0754757   .7830682    -0.10   0.923    -1.610284    1.459333
         42  |   .2493342   .7840692     0.32   0.750    -1.287437    1.786105
         43  |  -.1051961   .7849101    -0.13   0.893    -1.643615    1.433223
         44  |  -.1332997   .7836731    -0.17   0.865    -1.669294    1.402695
         45  |  -.0336313   .7837356    -0.04   0.966    -1.569748    1.502485
         46  |   -.077499   .7880767    -0.10   0.922    -1.622124    1.467126
         47  |  -.1240083   .7906083    -0.16   0.875    -1.673596    1.425579
         48  |  -.1406644   .7925696    -0.18   0.859    -1.694096    1.412767
         49  |   -.311831   .7964584    -0.39   0.695    -1.872884    1.249222
         50  |   .0795048   .8018629     0.10   0.921    -1.492141    1.651151
         51  |    -.03987   .8063175    -0.05   0.961    -1.620247    1.540507
         52  |    .175123   .8123457     0.22   0.829    -1.417069    1.767315
         53  |  -.2160892   .8206156    -0.26   0.792    -1.824491    1.392312
         54  |   .1508225   .8302669     0.18   0.856    -1.476495     1.77814
         55  |  -.1588468   .8413972    -0.19   0.850     -1.80798    1.490286
         56  |  -.4077025   .8533693    -0.48   0.633    -2.080301    1.264896
         57  |   .1506475   .8806643     0.17   0.864    -1.575449    1.876744
         58  |  -.5995126   .9128689    -0.66   0.511     -2.38873    1.189705
         59  |    .000663   .9649573     0.00   0.999    -1.890647    1.891973
         60  |   .1268308   1.179696     0.11   0.914    -2.185366    2.439027
             |
    statyear |
       1997  |   .6893219   .6308627     1.09   0.275     -.547165    1.925809
       1998  |   .2284195   .5983482     0.38   0.703    -.9443392    1.401178
       1999  |   .1217415    .578479     0.21   0.833    -1.012074    1.255557
       2000  |  -.2051283   .5716441    -0.36   0.720    -1.325547    .9152904
       2001  |   .1752776   .5685706     0.31   0.758    -.9391171    1.289672
       2002  |   .3832778   .5644336     0.68   0.497    -.7230086    1.489564
       2003  |    .053925   .5606968     0.10   0.923    -1.045037    1.152887
       2004  |   .4116798   .5586755     0.74   0.461    -.6833205     1.50668
       2005  |   .0171829   .5558482     0.03   0.975    -1.072276    1.106642
       2006  |  -.2911823    .565337    -0.52   0.607    -1.399239    .8168745
       2007  |   .0164182   .5695602     0.03   0.977    -1.099916    1.132753
       2008  |   .0714586   .5724327     0.12   0.901    -1.050506    1.193423
       2009  |  -.1116666   .5763015    -0.19   0.846    -1.241214    1.017881
       2010  |   .0422877   .5792369     0.07   0.942    -1.093013    1.177588
       2011  |   .2263829   .5824522     0.39   0.698    -.9152197    1.367985
       2012  |   .2443245   .5900713     0.41   0.679    -.9122116    1.400861
       2013  |   .3853788   .5961326     0.65   0.518    -.7830373    1.553795
       2014  |  -.0020605   .6056642    -0.00   0.997    -1.189159    1.185038
       2015  |   .0453184   .6138172     0.07   0.941    -1.157759    1.248396
       2016  |  -.0543527   .6290408    -0.09   0.931    -1.287269    1.178563
       2017  |   .2243876   .6525029     0.34   0.731    -1.054514    1.503289
       2018  |  -.6865201    .677269    -1.01   0.311    -2.013963     .640923
       2019  |  -.5075994   .7313385    -0.69   0.488    -1.941018    .9258194
       2020  |   .3835909   .8298596     0.46   0.644    -1.242929     2.01011
             |
       _cons |   92.67639   .9388149    98.72   0.000     90.83632    94.51646
------------------------------------------------------------------------------

[Andrew Musau]

Here is a way to recover the dummy coefficients from the predicted fixed effects. At present, the official estimators do not allow you to absorb interactions, but reghdfe from https://github.com/sergiocorreia/reghdfe does.

// ===== Synthetic dataset for reproducible example =====
clear  
set seed 12345  

// Parameters
local municipality     = 100        // number of municipalities/cohorts
local Npersons   = 50          // number of individuals per municipality
local Tmin       = 1          // event-time baseline
local Tmax       = 17        // event-time max
local Tobs       = `Tmax' - `Tmin' 
local obs        = `municipality' * `Npersons' * `Tobs'
display `obs' 

// Create panel: each municipality × each individual × each t_idx
set obs `obs'

// Define identifiers
gen mun_id = ceil(_n / (`Npersons' * `Tobs'))
gen person_in_unit = mod(ceil(_n / `Tobs') - 1, `Npersons') + 1
gen id = mun_id * 10000 + person_in_unit   // individual identifier

// Create t_idx
bysort mun_id person_in_unit: gen t_idx = `Tmin' + _n - 1

// Cohort variable (e.g., language region)
gen str3 language = cond(mod(mun_id,3)==0, "GER", ///
                      cond(mod(mun_id,3)==1, "FRE", "ITA"))

// We first generate a random base statyear for each individual
bysort id: gen statyear0 = 1995 + floor(runiform()*10)   // base year between 1995 and 2004
// Then statyear for each row = statyear0 + t_idx
gen statyear = statyear0 + t_idx

// Age: define starting age at t_idx == 0, then age = start_age + t_idx
bysort id: gen start_age = 25 + int(runiform()*20)   // starting age between ~25 and ~44 at t_idx = 0
gen age = start_age + t_idx

// Outcome y: municipality + individual + event time effects + noise
gen u_intercept = rnormal(0, 5)
gen u_slope     = rnormal(0, 0.5)
gen event_effect = cond(t_idx>=0, -10 + 0.5*t_idx, 0)
gen y = 100 + u_intercept + u_slope * t_idx + event_effect + rnormal(0, 10)

// Clean up interim vars
drop u_intercept u_slope start_age statyear0 event_effect

label var id            "Individual identifier"
label var mun_id       "Municipality / cohort id"
label var person_in_unit "Person within municipality"
label var language        "language region"
label var t_idx         "Event time index (years relative)"
label var age           "Age of individual (synthetic)"
label var statyear      "Calendar year of observation"
label var y             "Outcome (synthetic)"

* analysis


egen lang= group(language)

local omitted 
qui levelsof lang, local(levs)
foreach l of local levs{
    local omitted `omitted' o5.t_idx#o.`l'.lang
}

reg y ibn.t_idx#ibn.lang `omitted' i.age i.statyear, r noomit 

qui reghdfe y i.age i.statyear, absorb(i.t_idx#i.lang, savefe)  vce(robust)
egen mean_t5= mean(cond(t_idx==5, __hdfe1__, .))
egen base= max(cond(t_idx==5, mean_t5, .))
gen coef= __hdfe1__- base 
table (t_idx lang), statistic(mean coef) nototal

Res.: (Selection highlighted)

. reg y ibn.t_idx#ibn.lang `omitted' i.age i.statyear, r noomit 

Linear regression                               Number of obs     =     80,000
                                                F(103, 79896)     =      27.87
                                                Prob > F          =     0.0000
                                                R-squared         =     0.0348
                                                Root MSE          =     12.162

------------------------------------------------------------------------------
             |               Robust
           y | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
  t_idx#lang |
        1 1  |  -2.368133   .3336004    -7.10   0.000    -3.021988   -1.714278
        1 2  |  -2.310124   .3433546    -6.73   0.000    -2.983097   -1.637151
        1 3  |  -2.030665   .3387226    -6.00   0.000    -2.694559    -1.36677
        2 1  |  -1.670071   .3224575    -5.18   0.000    -2.302085   -1.038056
        2 2  |  -1.862749   .3324816    -5.60   0.000    -2.514411   -1.211087
        2 3  |  -1.583652   .3354648    -4.72   0.000    -2.241161   -.9261431
        3 1  |  -1.563066   .3132553    -4.99   0.000    -2.177044   -.9490873
        3 2  |  -1.576734   .3279649    -4.81   0.000    -2.219543   -.9339246
        3 3  |  -1.193576   .3323675    -3.59   0.000    -1.845014   -.5421378
        4 1  |  -.7413887   .3187269    -2.33   0.020    -1.366091   -.1166859
        4 2  |  -.4857789   .3178338    -1.53   0.126    -1.108731    .1371733
        4 3  |  -.6801947   .3245295    -2.10   0.036     -1.31627   -.0441189
        6 1  |   .4780968   .3246934     1.47   0.141    -.1583003    1.114494
        6 2  |   .6149669    .331429     1.86   0.064    -.0346318    1.264566
        6 3  |   .2160518   .3288104     0.66   0.511    -.4284144     .860518
        7 1  |   1.535011   .3354032     4.58   0.000     .8776232      2.1924
        7 2  |   .9206809   .3432816     2.68   0.007     .2478512    1.593511
        7 3  |    .952353   .3310848     2.88   0.004     .3034288    1.601277
        8 1  |   .7677342   .3368691     2.28   0.023     .1074729    1.427995
        8 2  |   .7492054   .3397146     2.21   0.027      .083367    1.415044
        8 3  |    1.19948   .3471386     3.46   0.001     .5190909     1.87987
        9 1  |   1.656851   .3458675     4.79   0.000     .9789526    2.334749
        9 2  |   2.412133   .3466938     6.96   0.000     1.732615    3.091651
        9 3  |   1.538266   .3481109     4.42   0.000     .8559711    2.220561
       10 1  |   1.955685   .3525141     5.55   0.000      1.26476    2.646611
       10 2  |   2.651968   .3606623     7.35   0.000     1.945072    3.358864
       10 3  |    2.01723     .36384     5.54   0.000     1.304106    2.730354
       11 1  |   2.862153   .3619068     7.91   0.000     2.152817    3.571488
       11 2  |   3.034875   .3701407     8.20   0.000     2.309402    3.760348
       11 3  |   3.112749    .372354     8.36   0.000     2.382937     3.84256
       12 1  |   3.044052   .3783696     8.05   0.000      2.30245    3.785654
       12 2  |   3.347627   .3806228     8.80   0.000     2.601609    4.093646
       12 3  |   3.188872   .3817797     8.35   0.000     2.440587    3.937158
       13 1  |   3.966754   .3872396    10.24   0.000     3.207767    4.725741
       13 2  |   3.903928   .3943237     9.90   0.000     3.131056      4.6768
       13 3  |   3.539779   .3922857     9.02   0.000     2.770901    4.308656
       14 1  |   4.276604   .4006579    10.67   0.000     3.491317    5.061891
       14 2  |   4.516788   .4068208    11.10   0.000     3.719422    5.314154
       14 3  |    4.23396    .405333    10.45   0.000      3.43951     5.02841
       15 1  |   5.216766   .4113119    12.68   0.000     4.410597    6.022935
       15 2  |   4.904253   .4078983    12.02   0.000     4.104775    5.703731
       15 3  |   5.309786    .419863    12.65   0.000     4.486857    6.132715
       16 1  |   4.722023    .427838    11.04   0.000     3.883463    5.560583
       16 2  |   5.363941   .4285051    12.52   0.000     4.524074    6.203809
       16 3  |   5.330657   .4290629    12.42   0.000     4.489697    6.171618
             |
         age |
         27  |   .7325874   .9076534     0.81   0.420    -1.046408    2.511582
         28  |   .3208175   .8504925     0.38   0.706    -1.346142    1.987778
         29  |    .191439   .8339289     0.23   0.818    -1.443056    1.825934
         30  |  -.4719228   .8236589    -0.57   0.567    -2.086289    1.142444
         31  |   -.142288   .8137081    -0.17   0.861    -1.737151    1.452575
         32  |   .1990107   .8054323     0.25   0.805    -1.379632    1.777653
         33  |  -.2738529   .7992741    -0.34   0.732    -1.840425    1.292719
         34  |  -.1529101   .7961833    -0.19   0.848    -1.713424    1.407604
         35  |  -.1288967    .793197    -0.16   0.871    -1.683558    1.425764
         36  |   .0588335   .7908345     0.07   0.941    -1.491197    1.608864
         37  |   .3921179   .7880807     0.50   0.619    -1.152515    1.936751
         38  |   .0813392   .7881708     0.10   0.918    -1.463471    1.626149
         39  |  -.1347576   .7870812    -0.17   0.864    -1.677432    1.407917
         40  |  -.4748054   .7847014    -0.61   0.545    -2.012815    1.063205
         41  |  -.0754757   .7830682    -0.10   0.923    -1.610284    1.459333
         42  |   .2493342   .7840692     0.32   0.750    -1.287437    1.786105
         43  |  -.1051961   .7849101    -0.13   0.893    -1.643615    1.433223
         44  |  -.1332997   .7836731    -0.17   0.865    -1.669294    1.402695
         45  |  -.0336313   .7837356    -0.04   0.966    -1.569748    1.502485
         46  |   -.077499   .7880767    -0.10   0.922    -1.622124    1.467126
         47  |  -.1240083   .7906083    -0.16   0.875    -1.673596    1.425579
         48  |  -.1406644   .7925696    -0.18   0.859    -1.694096    1.412767
         49  |   -.311831   .7964584    -0.39   0.695    -1.872884    1.249222
         50  |   .0795048   .8018629     0.10   0.921    -1.492141    1.651151
         51  |    -.03987   .8063175    -0.05   0.961    -1.620247    1.540507
         52  |    .175123   .8123457     0.22   0.829    -1.417069    1.767315
         53  |  -.2160892   .8206156    -0.26   0.792    -1.824491    1.392312
         54  |   .1508225   .8302669     0.18   0.856    -1.476495     1.77814
         55  |  -.1588468   .8413972    -0.19   0.850     -1.80798    1.490286
         56  |  -.4077025   .8533693    -0.48   0.633    -2.080301    1.264896
         57  |   .1506475   .8806643     0.17   0.864    -1.575449    1.876744
         58  |  -.5995126   .9128689    -0.66   0.511     -2.38873    1.189705
         59  |    .000663   .9649573     0.00   0.999    -1.890647    1.891973
         60  |   .1268308   1.179696     0.11   0.914    -2.185366    2.439027
             |
    statyear |
       1997  |   .6893219   .6308627     1.09   0.275     -.547165    1.925809
       1998  |   .2284195   .5983482     0.38   0.703    -.9443392    1.401178
       1999  |   .1217415    .578479     0.21   0.833    -1.012074    1.255557
       2000  |  -.2051283   .5716441    -0.36   0.720    -1.325547    .9152904
       2001  |   .1752776   .5685706     0.31   0.758    -.9391171    1.289672
       2002  |   .3832778   .5644336     0.68   0.497    -.7230086    1.489564
       2003  |    .053925   .5606968     0.10   0.923    -1.045037    1.152887
       2004  |   .4116798   .5586755     0.74   0.461    -.6833205     1.50668
       2005  |   .0171829   .5558482     0.03   0.975    -1.072276    1.106642
       2006  |  -.2911823    .565337    -0.52   0.607    -1.399239    .8168745
       2007  |   .0164182   .5695602     0.03   0.977    -1.099916    1.132753
       2008  |   .0714586   .5724327     0.12   0.901    -1.050506    1.193423
       2009  |  -.1116666   .5763015    -0.19   0.846    -1.241214    1.017881
       2010  |   .0422877   .5792369     0.07   0.942    -1.093013    1.177588
       2011  |   .2263829   .5824522     0.39   0.698    -.9152197    1.367985
       2012  |   .2443245   .5900713     0.41   0.679    -.9122116    1.400861
       2013  |   .3853788   .5961326     0.65   0.518    -.7830373    1.553795
       2014  |  -.0020605   .6056642    -0.00   0.997    -1.189159    1.185038
       2015  |   .0453184   .6138172     0.07   0.941    -1.157759    1.248396
       2016  |  -.0543527   .6290408    -0.09   0.931    -1.287269    1.178563
       2017  |   .2243876   .6525029     0.34   0.731    -1.054514    1.503289
       2018  |  -.6865201    .677269    -1.01   0.311    -2.013963     .640923
       2019  |  -.5075994   .7313385    -0.69   0.488    -1.941018    .9258194
       2020  |   .3835909   .8298596     0.46   0.644    -1.242929     2.01011
             |
       _cons |   92.67639   .9388149    98.72   0.000     90.83632    94.51646
------------------------------------------------------------------------------

. 
. qui reghdfe y i.age i.statyear, absorb(i.t_idx#i.lang, savefe)  vce(robust)

. egen mean_t5= mean(cond(t_idx==5, __hdfe1__, .))

. egen base= max(cond(t_idx==5, mean_t5, .))

. gen coef= __hdfe1__- base 

. table (t_idx lang), statistic(mean coef) nototal

----------------------------------------------
                                  |       Mean
----------------------------------+-----------
Event time index (years relative) |           
  1                               |           
    group(language)               |           
      1                           |  -2.368179
      2                           |  -2.310183
      3                           |  -2.030731
  2                               |           
    group(language)               |           
      1                           |  -1.670031
      2                           |  -1.862692
      3                           |  -1.583619
  3                               |           
    group(language)               |           
      1                           |  -1.563028
      2                           |  -1.576714
      3                           |  -1.193569
  4                               |           
    group(language)               |           
      1                           |   -.741419
      2                           |  -.4857906
      3                           |  -.6802034
  5                               |           
    group(language)               |           
      1                           |  -.1208417
      2                           |  -.0481905
      3                           |   .1726941
  6                               |           
    group(language)               |           
      1                           |   .4780338
      2                           |   .6149135
      3                           |   .2160078
  7                               |           
    group(language)               |           
      1                           |   1.535095
      2                           |   .9207565
      3                           |   .9523664
  8                               |           
    group(language)               |           
      1                           |   .7677443
      2                           |   .7492517
      3                           |   1.199506
  9                               |           
    group(language)               |           
      1                           |   1.656813
      2                           |   2.412107
      3                           |   1.538208
  10                              |           
    group(language)               |           
      1                           |   1.955603
      2                           |   2.651851
      3                           |   2.017146
  11                              |           
    group(language)               |           
      1                           |   2.862133
      2                           |    3.03486
      3                           |   3.112699
  12                              |           
    group(language)               |           
      1                           |   3.043885
      2                           |   3.347464
      3                           |   3.188691
  13                              |           
    group(language)               |           
      1                           |   3.966613
      2                           |   3.903773
      3                           |   3.539629
  14                              |           
    group(language)               |           
      1                           |   4.276559
      2                           |   4.516722
      3                           |   4.233897
  15                              |           
    group(language)               |           
      1                           |   5.216743
      2                           |   4.904229
      3                           |   5.309759
  16                              |           
    group(language)               |           
      1                           |   4.721995
      2                           |   5.363917
      3                           |   5.330634
----------------------------------------------

[Heike Waechter]

That’s great, thank you so much.

Just to make sure I understand and tie everything up: the two approaches are conceptually equivalent, but using reghdfe plus a prediction step is computationally faster? So if that’s the case, there is no need for me ever to use the full interaction-dummy approach — whether I intend to average across municipalities or not, right?

Thanks again for all your help — please let me know if I have misunderstood anything.

[Andrew Musau]

That's correct: the least squares dummy variable (LSDV) regression and the "within estimator" (as implemented by reghdfe or similar commands) are algebraically equivalent by the Frisch–Waugh theorem. This result is detailed in Frisch and Waugh (Econometrica, 1933). Including a full set of fixed effects via dummy variables produces the same coefficient estimates as using reghdfe, which partials out the fixed effects before estimation. The difference lies purely in computation: reghdfe (or the within transformation more generally) is much more efficient, especially when you have many fixed effects or large datasets. In practice, if you don't need the standard errors of the individual dummy coefficients, there's no reason to use the full interaction-dummy approach. For all other purposes (including averaging across municipalities), the within estimator yields the same results far more efficiently.

Reference:
Frisch, R., & Waugh, F. V. (1933). Partial Time Regressions as Compared with Individual Trends. Econometrica, 1(4), 387–401.

[Heike Waechter]

Thank you so much for your incredible help! I really appreciate it.

Best, Heike