Arkhangelsky et al.'s estimator for units adopting a policy at one date: comparison units weighted so their pre-period path parallels the treated units', pre-periods weighted so they predict the post-period, and the difference in differences taken with both weights.
Arguments
- data
A data frame, one row per unit and period.
- unit, time, y
Column names: the unit, the period (numeric, for example a year or a month index) and the outcome.
- ftreat
Column name of the period a unit is first treated (
NA, 0 orInffor a unit never treated). Treatment is taken as staying on.- reps
Placebo replications (default 200).
- seed
Optional seed for the placebo draws.
- conflvl
Confidence level of the intervals reported (default 0.95).
Value
A list of class rmbl_synthdid: estimate, se,
lower, upper, unitwts, tmwts, method
("placebo" or "jackknife"), paths (treated and
weighted comparison means by period). plot() draws the paths.
Details
With \(N_0\) comparison units, \(N_1\) treated, \(T_0\)
pre-periods and \(T_1\) post-periods, the unit weights solve
$$\min_{\omega_0, \omega \in \Delta} \frac{1}{T_0}\sum_{t \le
T_0}\Big(\omega_0 + \sum_j \omega_j Y_{jt} - \bar Y_{1t}\Big)^2 + \zeta^2
\|\omega\|^2,$$ with \(\zeta = (N_1 T_1)^{1/4} \hat\sigma\) and \(\hat\sigma\)
the standard deviation of the comparison units' first differences before
adoption; the time weights solve the same problem across comparison units,
predicting each unit's post-period mean from its pre-periods, with \(\zeta
= 10^{-6}\hat\sigma\). The estimate is $$\hat\tau = \Big(\bar Y_{1,post} -
\sum_t \lambda_t \bar Y_{1t}\Big) - \sum_j \omega_j \Big(\bar Y_{j,post} -
\sum_t \lambda_t Y_{jt}\Big).$$ The weights are the exact solutions of these
convex problems (the synthdid package runs Frank-Wolfe and then
sparsifies, so its estimate differs slightly). The standard error is the
placebo one when there is one treated unit (each set of \(N_1\) comparison
units in turn treated as if adopting, weights re-estimated), else the
jackknife over units with the weights held fixed (Arkhangelsky et al. 2021,
algorithms 3 and 4).
References
Arkhangelsky D, Athey S, Hirshberg DA, Imbens GW, Wager S (2021). Synthetic difference-in-differences. American Economic Review 111(12):4088-4118. doi:10.1257/aer.20190159
Examples
p <- polex()
p1 <- p[is.na(p$refmonth) | p$refmonth == 13, ]
syndid(p1, "district", "month", "crmrate", "refmonth", reps = 50, seed = 1)
#> ╭── Synthetic difference in differences ─────────────────────────────────╮
#> │ Estimate -3.126 │
#> │ Standard error 0.2474 (jackknife) │
#> │ Interval [-3.611, -2.641] │
#> │ Adoption 13 │
#> │ Comparison units 5 │
#> │ With weight 5 │
#> │ Pre-periods weighted 12 │
#> │ │
#> │ Unit weights │
#> │ unit weight │
#> │ ---- ------ │
#> │ D12 0.2031 │
#> │ D16 0.1999 │
#> │ D08 0.1998 │
#> │ D20 0.1991 │
#> │ D04 0.1981 │
#> │ │
#> │ Weighting the comparison units and the pre-periods to match the │
#> │ treated units, the effect is -3.126 (its interval [-3.611, -2.641]): │
#> │ the interval excludes 0, so the data distinguish it from no effect. │
#> ╰────────────────────────────────────────────────────────────────────────╯
# 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$income + sin(seq_len(80))
own1 <- own[is.na(own$first) | own$first == 2019, ]
syndid(own1, "town", "year", "rate", "first", reps = 50, seed = 1)
#> ╭── Synthetic difference in differences ─────────────────────────────────╮
#> │ Estimate -4.028 │
#> │ Standard error 0.02014 (jackknife) │
#> │ Interval [-4.067, -3.988] │
#> │ Adoption 2019 │
#> │ Comparison units 4 │
#> │ With weight 4 │
#> │ Pre-periods weighted 4 │
#> │ │
#> │ Unit weights │
#> │ unit weight │
#> │ ---- ------ │
#> │ G 0.2881 │
#> │ F 0.2662 │
#> │ E 0.2303 │
#> │ H 0.2154 │
#> │ │
#> │ Weighting the comparison units and the pre-periods to match the │
#> │ treated units, the effect is -4.028 (its interval [-4.067, -3.988]): │
#> │ the interval excludes 0, so the data distinguish it from no effect. │
#> ╰────────────────────────────────────────────────────────────────────────╯