The effect of a policy on a unit-by-period panel (municipalities by month, blocks by quarter, courts by year), from a regression of the outcome on the treatment indicator with unit and period fixed effects, and standard errors clustered by unit or by any coarser grouping.
Usage
plndid(
data,
unit,
time,
y,
trtd,
covs = NULL,
cl = NULL,
fe = c("unit", "time"),
w = NULL,
comb = NULL,
conflvl = 0.95
)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.
- trtd
Column name of the treatment indicator (0 or 1).
- covs
Optional column names of time-varying controls.
- cl
Column name of the clusters (default the unit), or two column names for two-way clustering (Cameron, Gelbach and Miller 2011): \(V_a + V_b - V_{ab}\), each term with the small-sample factor of the dimension with fewer clusters, and t on its \(G - 1\) degrees of freedom. A composite cluster (district by month, say) is one column made with
interaction(). Fewer than ten clusters give a warning (CR1 errors are unreliable with so few;wcbis made for that case), and so does a cluster that changes within a unit.- fe
c("unit", "time")(default) or"time".- w
Optional column name of non-negative weights, such as population or matching weights, for weighted least squares with weighted within-means. Rows of weight 0 take no part in the fit.
- comb
Optional linear combination of the terms, named by them, reported with its delta-method standard error as the attribute
combined. For a phase-in model with an exposure term \(E_{it} = \min(t - G_i, K)\),comb = c(treated = 1, exposure = 12)is the effect after twelve periods, with its error, and as a percentage of the comparison units' pre-period mean (that mean taken as fixed).- conflvl
Confidence level of the intervals reported (default 0.95).
Value
A data frame of class rmbl_did, one row per term (the
treatment first, then the covariates): term, estimate,
se, lower, upper, t, p_value.
Attributes: n, clusters, ctlmean (the pre-period mean
of the comparison units), releff (the estimate over it),
vcov, and with comb also combined.
Details
The model is $$Y_{it} = \beta T_{it} + X_{it}'\gamma + \delta_i
+ \mu_t + \varepsilon_{it},$$
fitted by ordinary least squares after the within transformation (alternating
projections, so an unbalanced panel is fine). The variance is the one-way
cluster-robust CR1 estimator, $$\hat V = \frac{G}{G - 1}\frac{N - 1}{N -
K} (\tilde X'\tilde X)^{-1} \Big(\sum_g \tilde X_g'\hat u_g \hat u_g'\tilde
X_g\Big) (\tilde X'\tilde X)^{-1},$$ with \(K\) counting the regressors and
the fixed effects not nested in the clusters, and tests and intervals use
Student's t on \(G - 1\) degrees of freedom. These are the conventions of
Stata's reghdfe and R's fixest.
The relative effect is \(\hat\beta\) over the mean outcome of the units never treated (or not yet treated) in the periods before the first treatment, the way an effect on a rate is usually reported ("a 12% fall").
With fe = "time" the unit effects are left out, and the model is the
cross-section with period effects of a comparison between kinds of units (for
example blocks in dynastic and in other wards).
When treatment starts at different times, this estimator weights
earlier-treated units as controls for later ones, and can be biased if the
effect changes over time: see bacondcmp for the weights and
stgatt for estimators that avoid them.
References
Bertrand M, Duflo E, Mullainathan S (2004). How much should we trust differences-in-differences estimates? Quarterly Journal of Economics 119(1):249-275. doi:10.1162/003355304772839588
Cameron AC, Gelbach JB, Miller DL (2011). Robust inference with multiway clustering. Journal of Business and Economic Statistics 29(2):238-249. doi:10.1198/jbes.2010.07136
Cameron AC, Miller DL (2015). A practitioner's guide to cluster-robust inference. Journal of Human Resources 50(2):317-372. doi:10.3368/jhr.50.2.317
Applied to a staggered criminal-procedure reform, municipality by month: Acosta C, Mejia D, Zorro Medina A (2023). On the tension between due process protection and public safety: the case of an extensive procedural reform in Colombia. doi:10.2139/ssrn.4585682
Examples
p <- polex()
plndid(p, "district", "month", "crmrate", "reformed")
#> ╭── Difference in differences ────────────────────────────────────────────────╮
#> │ Method fixed-effects least squares, cluster-robust (CR1) errors │
#> │ Observations 720 │
#> │ Clusters 20 │
#> │ Confidence 95% │
#> │ Comparison mean 26.52 │
#> │ Relative effect -12.2% of the comparison mean │
#> │ │
#> │ Estimates │
#> │ term estimate se 95% CI t p_value │
#> │ ------- -------- ------ ---------------- ------ ------- │
#> │ treated -3.232 0.1164 [-3.475, -2.988] -27.76 < 0.001 │
#> │ │
#> │ The policy's effect on the outcome is -3.232 (95% interval [-3.475, │
#> │ -2.988]): the interval excludes 0, so the data distinguish it from no │
#> │ effect. │
#> │ │
#> │ The comparison mean is the pre-period mean outcome of the units not │
#> │ (yet) treated. │
#> ╰─────────────────────────────────────────────────────────────────────────────╯
# clustered by region instead of district: only four regions, so the
# package warns, and wcb() is the tool for so few clusters
plndid(p, "district", "month", "crmrate", "reformed", cl = "region")
#> Warning: plndid: `cl` (region) gives only 4 clusters: cluster-robust (CR1) standard errors are unreliable with so few; wcb() is made for this case
#> ╭── Difference in differences ────────────────────────────────────────────────╮
#> │ Method fixed-effects least squares, cluster-robust (CR1) errors │
#> │ Observations 720 │
#> │ Clusters 4 │
#> │ Confidence 95% │
#> │ Comparison mean 26.52 │
#> │ Relative effect -12.2% of the comparison mean │
#> │ │
#> │ Estimates │
#> │ term estimate se 95% CI t p_value │
#> │ ------- -------- ------ ---------------- ------ ------- │
#> │ treated -3.232 0.1656 [-3.759, -2.705] -19.52 < 0.001 │
#> │ │
#> │ The policy's effect on the outcome is -3.232 (95% interval [-3.759, │
#> │ -2.705]): the interval excludes 0, so the data distinguish it from no │
#> │ effect. │
#> │ │
#> │ The comparison mean is the pre-period mean outcome of the units not │
#> │ (yet) treated. │
#> ╰─────────────────────────────────────────────────────────────────────────────╯
# 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))
# the effect of the by-law, with income as a control, clustered by town
# (the default), weighted by population
plndid(own, "town", "year", "rate", "bylaw", covs = "income", w = "pop")
#> ╭── Difference in differences ────────────────────────────────────────────────╮
#> │ Method fixed-effects least squares, cluster-robust (CR1) errors │
#> │ Observations 80 │
#> │ Clusters 8 │
#> │ Confidence 95% │
#> │ Comparison mean 47.43 │
#> │ Relative effect -8.3% of the comparison mean │
#> │ │
#> │ Estimates │
#> │ term estimate se 95% CI t p_value │
#> │ ------- -------- ------ ---------------- ------ ------- │
#> │ treated -3.914 0.1702 [-4.316, -3.511] -22.99 < 0.001 │
#> │ income 0.227 0.0367 [0.140, 0.313] 6.17 < 0.001 │
#> │ │
#> │ The policy's effect on the outcome is -3.914 (95% interval [-4.316, │
#> │ -3.511]): the interval excludes 0, so the data distinguish it from no │
#> │ effect. │
#> │ │
#> │ The comparison mean is the pre-period mean outcome of the units not │
#> │ (yet) treated. │
#> ╰─────────────────────────────────────────────────────────────────────────────╯