The Goodman-Bacon decomposition of a two-way fixed-effects estimate
Source:R/policy_evaluation.R
bacondcmp.RdSplits the two-way fixed-effects difference-in-differences estimate of a staggered adoption into the two-group, two-period comparisons it averages, with their weights: treated against never treated, earlier against later treated, and later against earlier treated (the comparisons that use already-treated units as controls).
Value
A data frame of class rmbl_bacon: type ("treated
vs never", "earlier vs later", "later vs earlier"),
treated, control (the adoption periods; Inf for
never), estimate, weight; attribute twfe (the weighted
sum, equal to the two-way estimate).
Details
For a balanced panel with treatment that stays on and no covariates,
the estimate of plndid is exactly \(\hat\beta = \sum_{k}
s_{kU}\hat\beta_{kU} + \sum_{k \ne l} s_{kl}\hat\beta_{kl}\), with weights
that sum to one and grow with the groups' sizes \(n\) and the variance of
their treatment over time \(\bar D(1 - \bar D)\) (Goodman-Bacon 2021,
theorem 1). Units treated from the first period act as an already-treated
group. A large weight on “later vs earlier” comparisons, together with
effects that grow over time, is the warning sign that the two-way estimate is
biased.
References
Goodman-Bacon A (2021). Difference-in-differences with variation in treatment timing. Journal of Econometrics 225(2):254-277. doi:10.1016/j.jeconom.2021.03.014
Examples
p <- polex()
b <- bacondcmp(p, "district", "month", "crmrate", "reformed")
b
#> ╭── Goodman-Bacon decomposition ──────────────────────────────────────────╮
#> │ Two-way estimate -3.232 │
#> │ Comparisons 9 │
#> │ Weight on later vs earlier 20.9% │
#> │ │
#> │ By kind of comparison │
#> │ type comparisons weight estimate │
#> │ ---------------- ----------- ------ -------- │
#> │ treated vs never 3 0.5814 -3.048 │
#> │ earlier vs later 3 0.2093 -3.558 │
#> │ later vs earlier 3 0.2093 -3.414 │
#> │ │
#> │ Each comparison (control Inf = never treated) │
#> │ type treated control estimate weight │
#> │ ---------------- ------- ------- -------- ------- │
#> │ treated vs never 13 Inf -3.181 0.18605 │
#> │ treated vs never 19 Inf -3.137 0.20930 │
#> │ treated vs never 25 Inf -2.817 0.18605 │
#> │ earlier vs later 13 19 -3.650 0.04651 │
#> │ later vs earlier 19 13 -3.624 0.06977 │
#> │ earlier vs later 13 25 -3.506 0.09302 │
#> │ later vs earlier 25 13 -3.332 0.09302 │
#> │ earlier vs later 19 25 -3.568 0.06977 │
#> │ later vs earlier 25 19 -3.261 0.04651 │
#> │ │
#> │ The two-way estimate -3.232 is the weighted average of these 9 │
#> │ two-group comparisons. 20.9% of the weight is on comparisons that use │
#> │ already-treated units as controls; if the effect grows over time, │
#> │ that part pulls the two-way estimate away from the true effect. │
#> ╰─────────────────────────────────────────────────────────────────────────╯
tapply(b$weight, b$type, sum)
#> earlier vs later later vs earlier treated vs never
#> 0.2093023 0.2093023 0.5813953
# your own panel: eight towns over ten years, four adopting a by-law (two
# in 2019, two in 2021), with each town's population, county and income
own <- data.frame(
town = rep(c("A", "B", "C", "D", "E", "F", "G", "H"), each = 10),
year = rep(2015:2024, times = 8)
)
i <- match(own$town, unique(own$town))
own$first <- c(2019, 2019, 2021, 2021, NA, NA, NA, NA)[i]
own$bylaw <- as.integer(!is.na(own$first) & own$year >= own$first)
own$pop <- c(12, 30, 8, 22, 15, 40, 9, 18)[i] * 1000
own$county <- rep(c("North", "South"), 4)[i]
own$income <- 50 + own$year - 2015 + cos(seq_len(80))
own$rate <- 30 + i + 0.5 * (own$year - 2015) - 4 * own$bylaw +
0.2 * own$income + sin(seq_len(80))
bacondcmp(own, "town", "year", "rate", "bylaw")
#> ╭── Goodman-Bacon decomposition ─────────────────────────────────────────╮
#> │ Two-way estimate -4.039 │
#> │ Comparisons 4 │
#> │ Weight on later vs earlier 7.1% │
#> │ │
#> │ By kind of comparison │
#> │ type comparisons weight estimate │
#> │ ---------------- ----------- ------- -------- │
#> │ treated vs never 2 0.85714 -4.026 │
#> │ earlier vs later 1 0.07143 -4.136 │
#> │ later vs earlier 1 0.07143 -4.096 │
#> │ │
#> │ Each comparison (control Inf = never treated) │
#> │ type treated control estimate weight │
#> │ ---------------- ------- ------- -------- ------ │
#> │ treated vs never 2,019 Inf -4.240 0.4286 │
#> │ treated vs never 2,021 Inf -3.813 0.4286 │
#> │ earlier vs later 2,019 2,021 -4.136 0.0714 │
#> │ later vs earlier 2,021 2,019 -4.096 0.0714 │
#> │ │
#> │ The two-way estimate -4.039 is the weighted average of these 4 │
#> │ two-group comparisons. 7.1% of the weight is on comparisons that use │
#> │ already-treated units as controls; if the effect grows over time, │
#> │ that part pulls the two-way estimate away from the true effect. │
#> ╰────────────────────────────────────────────────────────────────────────╯