Bounds on the effects of an evstudy when the trend after
adoption may depart from parallel by up to \(\bar M\) times the largest
departure seen before it (Rambachan and Roth's relative magnitudes), with a
conservative interval and the "breakdown" \(\bar M\) at which the
interval first includes zero.
Usage
tse(
x,
mbar = c(0, 0.5, 1, 1.5, 2),
target = "average",
draws = 5000L,
seed = NULL,
conflvl = 0.95
)Arguments
- x
An
evstudyresult with one lead at least.- mbar
The values of \(\bar M\) (default 0, 0.5, 1, 1.5, 2).
- target
"average"(the mean of the post-period effects, default) or an event time.- draws
Normal draws for the interval (default 5000).
- seed
Optional seed for the bootstrap draws.
- conflvl
Confidence level of the robust interval, a number strictly between 0 and 1 (default 0.95: the 2.5\ drawn bounds).
Value
A data frame of class rmbl_sensitivity: mbar,
lowerbnd, upbound (the identified set at the estimates),
lower, upper (the interval); attribute breakdown (the
smallest \(\bar M\) whose interval includes 0, found by bisection;
Inf if none up to 10).
Details
Let \(\delta_e\) be the bias of the event-time coefficient
\(\beta_e\), with \(\delta_{-1} = 0\) at the reference period and the
pre-period biases identified by the leads. The relative-magnitudes
restriction is \(|\delta_{e+1} - \delta_e| \le \bar M D\), with \(D =
\max_{s < 0} |\delta_{s+1} - \delta_s|\) the largest change between
consecutive pre-periods. The bias at post-period \(e \ge 0\) is then at
most \((e + 1)\bar M D\) in absolute value, so the identified set for the
effect \(\tau_e\) is \(\beta_e \pm (e + 1)\bar M D\), and for the average
post-period effect the average of those. The interval draws \((\beta^*)\)
from the normal distribution of the estimates (with the event study's
cluster-robust covariance), computes the bounds for each, and takes the
\(\alpha/2\) quantile of the lower bound and the \(1 - \alpha/2\)
quantile of the upper: by Bonferroni it covers the identified set, and so the
effect, with probability at least conflvl. It is conservative;
Rambachan and Roth's conditional and hybrid intervals (the HonestDiD
package) are shorter.
References
Rambachan A, Roth J (2023). A more credible approach to parallel trends. Review of Economic Studies 90(5):2555-2591. doi:10.1093/restud/rdad018
Examples
p <- polex()
es <- evstudy(p, "district", "month", "crmrate", "refmonth",
leads = 3, lags = 4)
tse(es, seed = 1)
#> ╭── Sensitivity to parallel trends ──────────────────────────────────────╮
#> │ Method relative magnitudes (Rambachan and Roth) │
#> │ Target average post-period effect │
#> │ Breakdown Mbar = 0.6214 │
#> │ Robust interval level 95% │
#> │ │
#> │ mbar identified_set robust_interval │
#> │ ----- ---------------- ---------------- │
#> │ 0.000 [-3.166, -3.166] [-3.916, -2.400] │
#> │ 0.500 [-3.865, -2.467] [-4.980, -0.479] │
#> │ 1.000 [-4.565, -1.768] [-6.788, 1.519] │
#> │ 1.500 [-5.264, -1.069] [-8.811, 3.532] │
#> │ 2.000 [-5.963, -0.370] [-10.842, 5.572] │
#> │ │
#> │ The negative effect stays clear of zero while the trend after │
#> │ adoption departs from parallel by up to 0.621 times the largest │
#> │ departure before it. A departure as large as the pre-period's (Mbar │
#> │ 1) is enough to lose it: the conclusion leans on parallel trends. │
#> │ │
#> │ Mbar bounds how far the trend after adoption may depart from │
#> │ parallel, as a multiple of the largest departure seen before it; the │
#> │ effect is distinguishable from zero until the robust interval first │
#> │ includes 0, at the breakdown value. │
#> ╰────────────────────────────────────────────────────────────────────────╯
# 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))
es <- evstudy(own, "town", "year", "rate", "first", leads = 3, lags = 3)
# a 90 percent robust interval
tse(es, seed = 1, conflvl = 0.9)
#> ╭── Sensitivity to parallel trends ──────────────────────────────────────╮
#> │ Method relative magnitudes (Rambachan and Roth) │
#> │ Target average post-period effect │
#> │ Breakdown Mbar = 0.9629 │
#> │ Robust interval level 90% │
#> │ │
#> │ mbar identified_set robust_interval │
#> │ ----- ---------------- ---------------- │
#> │ 0.000 [-4.016, -4.016] [-5.236, -2.802] │
#> │ 0.500 [-4.052, -3.979] [-6.637, -1.345] │
#> │ 1.000 [-4.089, -3.943] [-8.056, 0.108] │
#> │ 1.500 [-4.125, -3.907] [-9.506, 1.600] │
#> │ 2.000 [-4.162, -3.870] [-11.020, 3.300] │
#> │ │
#> │ The negative effect stays clear of zero while the trend after │
#> │ adoption departs from parallel by up to 0.963 times the largest │
#> │ departure before it. A departure as large as the pre-period's (Mbar │
#> │ 1) is enough to lose it: the conclusion leans on parallel trends. │
#> │ │
#> │ Mbar bounds how far the trend after adoption may depart from │
#> │ parallel, as a multiple of the largest departure seen before it; the │
#> │ effect is distinguishable from zero until the robust interval first │
#> │ includes 0, at the breakdown value. │
#> ╰────────────────────────────────────────────────────────────────────────╯