Event-study DiD with staggered treatment design

August 2, 2020

This is another post in my series of attempts to learn the data.table package for R and to get more comfortable with base R graphics. Today, I reimplement the data generating process and one of the models in this excellent blog post by Andrew Baker: How to create relative time indicators. I also strongly recommend watching Andrew’s talk on Difference-in-Differences staggered treatment designs.

For this exercise, we will only use two libraries:

library(data.table)
library(fixest)
package ‘data.table’ was built under R version 4.5.2

As in the original post, the data generating process includes unit and time fixed effects, as well as unit-specific treatment effects.

$$y_{it}=\alpha_i + \alpha_t + \tau_{it} + \varepsilon_{it}$$ $$\alpha_i,\alpha_t \sim N(0,1)$$ $$\varepsilon_{it} \sim N(0,0.5)$$ $$\mu_{it} \sim N(0.3,0.2^2)$$

and $\tau_{it}$ is equal to the sum of each firm-specific $\mu_{it}$ since the start of the treatment period.

We have 1000 firms split into 4 cohorts which receive treatment in 1986, 1992, 1998, and 2004.

make_data = function() {

  # unit fixed effects
  unit = data.table(
    unit    = 1:1000,
    unit_fe = rnorm(1000, 0, 1),                # unit fe
    state   = sample(1:40, 1000, replace=TRUE), # state id
    mu      = rnorm(1000, .3, .2)               # treatment effect
  )                

  # year fixed effects
  year = data.table(
    year    = 1980:2010,
    year_fe = rnorm(31, 0, 1)
  )

  # treatment groups by state
  treat_taus = data.table(
    state       = sample(1:40, 40),
    cohort_year = sort(rep(c(1986, 1992, 1998, 2004), 10))
  )

  # full interaction of unit X year
  out = CJ(
    unit = 1:1000,
    year = 1980:2010
  )
  out = merge(out, unit, by="unit")
  out = merge(out, year, by="year")
  out = merge(out, treat_taus, by="state")

  # error term, treatment indicator, and treatment effects
  out[, error := rnorm(31000, 0, .5)][
      , treat := fifelse(year >= cohort_year, 1, 0)][
      , tau   := fifelse(treat == 1, mu, 0)]

  # cumulative treatment effects
  setkey(out, unit, year)
  out[, tau_cum := cumsum(tau), by="unit"]

  # dependent variable
  out[, dep_var := unit_fe + year_fe + tau_cum + error]

  return(out)

}

Plot data from a single simulation

dat = make_data()

# empty plot window
plot(0,
     type = "n",
     ylim = c(-8, 20),
     xlim = c(1980, 2010),
     xlab = "",
     ylab = "Outcome variable")

# one line per firm
for (i in 1:1000) {
  lines(x   = dat[unit==i]$year,
        y   = dat[unit==i]$dep_var,
        col = adjustcolor("grey", alpha=.1))
}

# means by treatment cohort
cohorts = dat[, .(dep_var = mean(dep_var)), by=.(cohort_year, year)]

years <- c(
  "#CC6677" = 1986,
  "#332288" = 1992,
  "#DDCC77" = 1998,
  "#117733" = 2004
) 

for (y in years) {
  abline(v   = y,
         col = names(years)[years==y],
         lwd = 3)
  lines(x   = cohorts[cohort_year==y]$year,
        y   = cohorts[cohort_year==y]$dep_var,
        col = names(years)[years==y],
        lwd = 3)
}
# simulate data
dat = make_data()

# drop 2004 treatment
dat = dat[year <= 2003]

# years since/to treatment
dat[, rel_year := year - cohort_year]
> Key: 
>        state  year  unit    unit_fe        mu     year_fe cohort_year
>                                   
>     1:    17  1980     1 -0.8350648 0.3592835  0.58993428        2004
>     2:    17  1981     1 -0.8350648 0.3592835 -0.05174812        2004
>     3:    17  1982     1 -0.8350648 0.3592835  0.61722359        2004
>     4:    17  1983     1 -0.8350648 0.3592835 -1.02623512        2004
>     5:    17  1984     1 -0.8350648 0.3592835 -0.04833219        2004
>    ---                                                               
> 23996:     2  1999  1000 -0.6987034 0.2791671  0.70941625        1986
> 23997:     2  2000  1000 -0.6987034 0.2791671 -1.32852478        1986
> 23998:     2  2001  1000 -0.6987034 0.2791671 -0.04398506        1986
> 23999:     2  2002  1000 -0.6987034 0.2791671  1.01043319        1986
> 24000:     2  2003  1000 -0.6987034 0.2791671  0.10484482        1986
>             error treat       tau  tau_cum     dep_var rel_year
>                                  
>     1:  0.1527800     0 0.0000000 0.000000 -0.09235047      -24
>     2: -0.4113322     0 0.0000000 0.000000 -1.29814509      -23
>     3:  0.8993860     0 0.0000000 0.000000  0.68154483      -22
>     4: -0.6836207     0 0.0000000 0.000000 -2.54492061      -21
>     5: -0.2962390     0 0.0000000 0.000000 -1.17963595      -20
>    ---                                                         
> 23996:  1.0399516     1 0.2791671 3.908340  4.95900410       13
> 23997:  0.5727518     1 0.2791671 4.187507  2.73303045       14
> 23998:  0.2524643     1 0.2791671 4.466674  3.97644978       15
> 23999:  0.2064968     1 0.2791671 4.745841  5.26406762       16
> 24000: -0.1569725     1 0.2791671 5.025008  4.27417715       17
min_year = dat[cohort_year==2004, min(rel_year)]
max_year = dat[cohort_year==2004, max(rel_year)]

# identification requires dropping two dummies
# dat[, rel_year := fifelse(!rel_year %in% c(min_year, -1), as.character(rel_year), "omitted")][
    # , rel_year := factor(rel_year)][
    # , rel_year := relevel(rel_year, ref=c("omitted")]
dat[, rel_year := factor(rel_year, sort(unique(rel_year)))]
> Key: 
> Index: 
>        state  year  unit    unit_fe        mu     year_fe cohort_year
>                                   
>     1:    17  1980     1 -0.8350648 0.3592835  0.58993428        2004
>     2:    17  1981     1 -0.8350648 0.3592835 -0.05174812        2004
>     3:    17  1982     1 -0.8350648 0.3592835  0.61722359        2004
>     4:    17  1983     1 -0.8350648 0.3592835 -1.02623512        2004
>     5:    17  1984     1 -0.8350648 0.3592835 -0.04833219        2004
>    ---                                                               
> 23996:     2  1999  1000 -0.6987034 0.2791671  0.70941625        1986
> 23997:     2  2000  1000 -0.6987034 0.2791671 -1.32852478        1986
> 23998:     2  2001  1000 -0.6987034 0.2791671 -0.04398506        1986
> 23999:     2  2002  1000 -0.6987034 0.2791671  1.01043319        1986
> 24000:     2  2003  1000 -0.6987034 0.2791671  0.10484482        1986
>             error treat       tau  tau_cum     dep_var rel_year
>                                 
>     1:  0.1527800     0 0.0000000 0.000000 -0.09235047      -24
>     2: -0.4113322     0 0.0000000 0.000000 -1.29814509      -23
>     3:  0.8993860     0 0.0000000 0.000000  0.68154483      -22
>     4: -0.6836207     0 0.0000000 0.000000 -2.54492061      -21
>     5: -0.2962390     0 0.0000000 0.000000 -1.17963595      -20
>    ---                                                         
> 23996:  1.0399516     1 0.2791671 3.908340  4.95900410       13
> 23997:  0.5727518     1 0.2791671 4.187507  2.73303045       14
> 23998:  0.2524643     1 0.2791671 4.466674  3.97644978       15
> 23999:  0.2064968     1 0.2791671 4.745841  5.26406762       16
> 24000: -0.1569725     1 0.2791671 5.025008  4.27417715       17

# regression model
f = dep_var ~ rel_year | unit + year
mod = feols(f, data=dat)

# clean results
out = data.table("term" = names(coef(mod)), "estimate" = coef(mod))
out = out[ , term := as.numeric(gsub(".*year", "", term))]
out = out[term %in% -5:5]
The variables 'rel_year12', 'rel_year13', 'rel_year14', 'rel_year15',
'rel_year16' and 'rel_year17' have been removed because of collinearity (see
$collin.var).

Plot results

# event study estimates
plot(x    = out$term,
     y    = out$estimate, 
     pch=20,
     xlim = c(-5, 5),
     ylim = c(-.5, 1.5),
     xlab = "Relative time",
     ylab = "Estimate")

# truth
truth = data.table(term=-5:5)
truth[, estimate := fifelse(term >= 0, (term + 1) * .3, 0)]
>      term estimate
>         
>  1:    -5      0.0
>  2:    -4      0.0
>  3:    -3      0.0
>  4:    -2      0.0
>  5:    -1      0.0
>  6:     0      0.3
>  7:     1      0.6
>  8:     2      0.9
>  9:     3      1.2
> 10:     4      1.5
> 11:     5      1.8
lines(x   = truth$term,
      y   = truth$estimate,
      col = "#CC6677",
      lty = "dashed")
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