19.1 Notes
For outcomes with 0s, we can’t use log-like transformation, because it’s sensitive to outcome unit (J. Chen and Roth 2023). For info on this issue, check [Zero-valued Outcomes]. We should use:
Percentage changes in the Average: by using Poisson QMLE, we can interpret the coefficients of the effect of treatment on the treated group relative to the mean of the control group.
Extensive vs. Intensive Margins: Distinguish the treatment effect on the intensive (outcome: 10 to 11) vs. extensive margins (outcome: 0 to 1).
- To get the bounds for the intensive-margin, use Lee (2009) (assuming that treatment has a monotonic effect on outcome)
set.seed(123) # For reproducibility
library(tidyverse)
n <- 1000 # Number of observations
p_treatment <- 0.5 # Probability of being treated
# Step 1: Generate the treatment variable D
D <- rbinom(n, 1, p_treatment)
# Step 2: Generate potential outcomes
# Untreated potential outcome (mostly zeroes)
Y0 <- rnorm(n, mean = 0, sd = 1) * (runif(n) < 0.3)
# Treated potential outcome (shifting both the probability of being positive - extensive margin and its magnitude - intensive margin)
Y1 <- Y0 + rnorm(n, mean = 2, sd = 1) * (runif(n) < 0.7)
# Step 3: Combine effects based on treatment
Y_observed <- (1 - D) * Y0 + D * Y1
# Add explicit zeroes to model situations with no effect
Y_observed[Y_observed < 0] <- 0
data <-
data.frame(
ID = 1:n,
Treatment = D,
Outcome = Y_observed,
X = rnorm(n)
) |>
# whether outcome is positive
dplyr::mutate(positive = Outcome > 0)
# Viewing the first few rows of the dataset
head(data)
#> ID Treatment Outcome X positive
#> 1 1 0 0.0000000 1.4783345 FALSE
#> 2 2 1 2.2369379 -1.4067867 TRUE
#> 3 3 0 0.0000000 -1.8839721 FALSE
#> 4 4 1 3.2192276 -0.2773662 TRUE
#> 5 5 1 0.6649693 0.4304278 TRUE
#> 6 6 0 0.0000000 -0.1287867 FALSE
hist(data$Outcome)
- Percentage changes in the Average
library(fixest)
res_pois <-
fepois(
fml = Outcome ~ Treatment + X,
data = data,
vcov = "hetero"
)
etable(res_pois)
#> res_pois
#> Dependent Var.: Outcome
#>
#> Constant -2.223*** (0.1440)
#> Treatment 2.579*** (0.1494)
#> X 0.0235 (0.0406)
#> _______________ __________________
#> S.E. type Heteroskedas.-rob.
#> Observations 1,000
#> Squared Cor. 0.33857
#> Pseudo R2 0.26145
#> BIC 1,927.9
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1To calculate the proportional effect
# proportional effect
exp(coefficients(res_pois)["Treatment"]) - 1
#> Treatment
#> 12.17757
# SE
exp(coefficients(res_pois)["Treatment"]) *
sqrt(res_pois$cov.scaled["Treatment", "Treatment"])
#> Treatment
#> 1.968684Hence, we conclude that the treatment effect is 1215% higher for the treated group as compared to the control group.
- Extensive vs. Intensive Margins
Here, we can estimate the intensive-margin treatment effect (i.e., the treatment effect for “always-takers”).
res <- causalverse::lee_bounds(
df = data,
d = "Treatment",
m = "positive",
y = "Outcome",
numdraws = 10
) |>
causalverse::nice_tab(2)
print(res)
#> term estimate std.error
#> 1 Lower bound -0.22 0.09
#> 2 Upper bound 2.77 0.14Since in this case, the bounds contains 0, we can’t say much about the intensive margin for always-takers.
If we aim to examine the sensitivity of always-takers, we should consider scenarios where the average outcome of compliers are \(100 \times c\%\) lower or higher than that of always-takers.
We assume that \(E(Y(1)|Complier) = (1-c)E(Y(1)|Always-taker)\)
set.seed(1)
c_values = c(.1, .5, .7)
combined_res <- bind_rows(lapply(c_values, function(c) {
res <- causalverse::lee_bounds(
df = data,
d = "Treatment",
m = "positive",
y = "Outcome",
numdraws = 10,
c_at_ratio = c
)
res$c_value <- as.character(c)
return(res)
}))
combined_res |>
dplyr::select(c_value, everything()) |>
causalverse::nice_tab()
#> c_value term estimate std.error
#> 1 0.1 Point estimate 6.60 0.71
#> 2 0.5 Point estimate 2.54 0.13
#> 3 0.7 Point estimate 1.82 0.08If we assume \(c = 0.1\) (i.e., under treatment, compliers would have an outcome equal to 10% of the outcome of always-takers), then the intensive-margin effect for always-takers is 6.6 more in the unit of the outcome.
If we assume \(c = 0.5\) (i.e., under treatment, compliers would have an outcome equal to 50% of the outcome of always-takers), then the intensive-margin effect for always-takers is 2.54 more in the unit of the outcome.