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DiD.Rmd
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---
title: 'Difference-in-Differences: What it DiD?'
author: "Andrew Baker"
institute: "Stanford University"
date: "`r Sys.Date()`"
output:
xaringan::moon_reader:
lib_dir: libs
nature:
highlightStyle: github
highlightLines: true
countIncrementalSlides: false
extra_dependencies: ["xcolor"]
ratio: "16:9"
---
```{r setup, include=FALSE}
knitr::opts_chunk$set(fig.retina = 2)
```
```{r, message = FALSE, warning = FALSE, echo = FALSE}
library(tidyverse)
library(kableExtra)
library(here)
library(ggthemes)
library(lfe)
library(did2)
library(xaringan)
library(patchwork)
library(bacondecomp)
library(multcomp)
library(fastDummies)
library(magrittr)
library(MCPanel)
select <- dplyr::select
theme_set(theme_clean() + theme(plot.background = element_blank()))
```
# .center.pull[Outline of Talk]
$\hspace{2cm}$
1. Overview of DiD
2. Problems with Staggered DiD
3. Simulation Results
4. Some Alternative Methods
5. Application
---
# .center.pull[Difference-in-Differences]
$\hspace{2cm}$
- Think Card and Krueger minimum wage study comparing NJ and PA.
- 2 units and 2 time periods.
- 1 unit (T) is treated, and receives treatment in the second period. The control unit (C) is never treated.
---
# .center.pull[Difference-in-Differences]
```{r d1, echo = FALSE, fig.align = 'center', fig.width = 10, cache = TRUE}
# make data
data <- tibble(
Y = c(2, 5, 1, 2),
Unit = c("Treat", "Treat", "Control", "Control"),
T = c(0, 1, 0, 1)
)
# plot
data %>%
ggplot(aes(x = T, y = Y, group = Unit, color = Unit)) + geom_line(size = 2) +
labs(x = "Time", y = "Outcome") +
scale_x_continuous(breaks = c(0, 1)) +
scale_colour_brewer(palette = 'Set1') +
theme(axis.title = element_text(size = 18),
axis.text = element_text(size = 16),
legend.position = 'bottom',
legend.title = element_blank(),
legend.text = element_text(size = 16),
plot.background = element_blank())
```
---
# .center.pull[Difference-in-Differences]
- Building upon $\color{blue}{\text{Angrist & Pischke (2008, p. 228)}}$ we can think of these simple 2x2 DiDs as a fixed effects estimator.
- Potential Outcomes
- $Y_{i, t}^1$ = value of dependent variable for unit $i$ in period $t$ with treatment.
- $Y_{i, t}^0$ = value of dependent variable for unit $i$ in period $t$ without treatment.
- The expected outcome is a *linear function* of unit and time fixed effects:
$$E[{Y_{i, t}^0}] =\alpha_i + \alpha_t$$
$$E[{Y_{i, t}^1}] =\alpha_i + \alpha_t + \delta D_{st}$$
- Goal of DiD is to get an unbiased estimate of the treatment effect $\delta$.
---
# .center.pull[Difference-in-Differences as Solving System of Equations for Unknown Variable]
- Difference in expectations for the *control* unit times t = 1 and t = 0:
$$\begin{align*} E[Y_{C, 1}^0] & = \alpha_1 + \alpha_C \\ E[Y_{C, 0}^0] & = \alpha_0 + \alpha_C \\ E[Y_{C, 1}^0] - E[Y_{C, 0}^0] & = \alpha_1 - \alpha_0 \end{align*}$$
- Now do the same thing for the *treated* unit:
$$\begin{align*} E[Y_{T, 1}^1] & = \alpha_1 + \alpha_T + \delta \\ E[Y_{T, 0}^1] & = \alpha_0 + \alpha_T \\ E[Y_{T, 1}^1] - E[Y_{T, 0}^1] & = \alpha_1 - \alpha_0 + \delta \end{align*}$$
- If we assume the linear structure of DiD, then unbiased estimate of $\delta$ is:
$$\delta=
\begin{align*} & \left( E[Y_{T, 1}^1] - E[Y_{T, 0}^1] \right) - \left( E[Y_{C, 1}^0] - E[Y_{C, 0}^0] \right) \end{align*}$$
---
# .center.pull[Two-Way Differencing]
```{r d2, echo = FALSE, fig.align = 'center', fig.width = 10, cache = TRUE}
data %>%
group_by(Unit) %>%
mutate(Y2 = Y - Y[which(T == 0)]) %>%
ggplot(aes(x = T, y = Y, group = Unit, color = Unit)) +
geom_line(size = 2) +
geom_line(aes(x = T, y = Y2, group = Unit, color = Unit), linetype = "dashed", size = 2) +
labs(x = "Time", y = "Outcome") +
scale_x_continuous(breaks = c(1, 2)) +
scale_colour_brewer(palette = 'Set1') +
theme(axis.title = element_text(size = 18),
axis.text = element_text(size = 16),
legend.position = 'bottom',
legend.title = element_blank(),
legend.text = element_text(size = 16),
plot.background = element_blank()) +
annotate("label", y = 2, x = 0.85, label = "Treatment \n Effect") +
annotate("segment", x = 1, xend = 1, y = 1, yend = 3, color = "black", size = 1) +
annotate("segment", x = 0.92, xend = 1, y = 2, yend = 3, color = "black",
linetype = "dashed", size = 2) +
annotate("segment", x = 0.92, xend = 1, y = 2, yend = 1, color = "black",
linetype = "dashed", size = 2) +
annotate("segment", x = 0, xend = 0, y = 2, yend = 0, color = "#377EB8",
arrow = arrow(length = unit(0.1, "inches")), size = 2) +
annotate("segment", x = 0.5, xend = 0.5, y = 1.5, yend = 0.5, color = "#E41A1C",
arrow = arrow(length = unit(0.1, "inches")), size = 2)
```
---
# .center.pull[Regression DiD]
The DiD can be estimated through linear regression of the form:
$$\tag{1} y_{it} = \alpha + \beta_1 TREAT_i + \beta_2 POST_t + \delta (TREAT_i \cdot POST_t) + \epsilon_{it}$$
The coefficients from the regression estimate in (1) recover the same parameters as the double-differencing performed above:
$$\begin{align*}
\alpha &= E[y_{it} | i = C, t = 0] = \alpha_0 + \alpha_C \\
\beta_1 &= E[y_{it} | i = T, t = 0] - E[y_{it} | i = C, t= 0] \\
&= (\alpha_0 + \alpha_T) - (\alpha_0 + \alpha_C) = \alpha_T - \alpha_C \\
\beta_2 &= E[y_{it} | i = C, t = 1] - E[y_{it} | i = C, t = 0] \\
&= (\alpha_1 + \alpha_C) - (\alpha_0 + \alpha_C) = \alpha_1 - \alpha_0 \\
\delta &= \left(E[y_{it} | i = T, t = 1] - E[y_{it} | i = T, t = 0] \right) - \\
&\hspace{.5cm} \left(E[y_{it} | i = C, t = 1] - E[y_{it} | i = C t = 0] \right) = \delta
\end{align*}$$
---
# .center.pull[Regression DiD]
<center>
$\hspace{2cm}$
---
# .center.pull[Regression DiD - The Workhorse Model]
- Advantage of regression DiD - it provides both estimates of $\delta$ and standard errors for the estimates.
- $\color{blue}{\text{Angrist & Pischke (2008)}}$:
- "It's also easy to add additional (units) or periods to the regression setup... [and] it's easy to add additional covariates."
- Two-way fixed effects estimator:
$$y_{it} = \alpha_i + \alpha_t + \delta^DD D_{it} + \epsilon_{it}$$
- $\alpha_i$ and $\alpha_t$ are unit and time fixed effects, $D_{it}$ is the unit-time indicator for treatment.
- $TREAT_i$ and $POST_t$ now subsumed by the fixed effects.
- can be easily modified to include covariate matrix $X_{it}$, time trends, dynamic treatment effects estimation, etc.
---
# .center.pull[Where It Goes Wrong]
- Developed literature now on the issues with TWFE DiD with "staggered treatment timing" <span style="color:blue"> (Abraham and Sun (2018), Borusyak and Jaravel (2018), Callaway and Sant'Anna (2019), Goodman-Bacon (2019), Strezhnev (2018), Athey and Imbens (2018))<span>
- Different units receive treatment at different periods in time.
- Probably the most common use of DiD today. If done right can increase amount of cross-sectional variation.
- Without digging into the literature:
- $\delta^{DD}$ with staggered treatment timing is a *weighted average of many different treatment effects*.
- We know little about how it measures when treatment timing varies, how it compares means across groups, or why different specifications change estimates.
- The weights are often negative and non-intuitive.
---
# .center.pull[Bias with TWFE - Goodman-Bacon (2019)]
- $\color{blue}{\text{Goodman-Bacon (2019)}}$ provides a clear graphical intuition for the bias. Assume three treatment groups - never treated units (U), early treated units (k), and later treated units (l).
```{r d3, echo = FALSE, warning = FALSE, fig.align = 'center', fig.height = 5, cache = TRUE}
data <- tibble(
time = 0:100,
U = seq(5, 12, length.out = 101),
l = seq(10, 17, length.out = 101) + c(rep(0, 85), rep(15, 16)),
k = seq(18, 25, length.out = 101) + c(rep(0, 34), rep(10, 67))
) %>%
pivot_longer(-time, names_to = "series", values_to = "value")
data %>%
ggplot(aes(x = time, y = value, group = series, color = series, shape = series)) +
geom_line(size = 2) + geom_point(size = 2) +
geom_vline(xintercept = c(34, 85)) +
labs(x = "Time", y = "Units of y") +
scale_x_continuous(limits = c(0, 100), breaks = c(34, 85),
labels = c(expression('t'['k']^'*'), expression('t'['l']^'*')),
expand = c(0, 0)) +
annotate("text", x = 10, y = 21, label = expression('y'['it']^'k'), size = 9) +
annotate("text", x = 50, y = 16, label = expression('y'['it']^'l'), size = 9) +
annotate("text", x = 90, y = 14, label = expression('y'['it']^'U'), size = 9) +
annotate('label', x = 17, y = 3, label = 'PRE(k)') +
annotate('label', x = 60, y = 3, label = 'MID(k, l)') +
annotate('label', x = 93, y = 3, label = 'POST(l)') +
annotate("segment", x = 1, xend = 33, y = 2, yend = 2, color = "black",
arrow = arrow(length = unit(0.1, "inches"))) +
annotate("segment", x = 33, xend = 1, y = 2, yend = 2, color = "black",
arrow = arrow(length = unit(0.1, "inches"))) +
annotate("segment", x = 35, xend = 84, y = 2, yend = 2, color = "black",
arrow = arrow(length = unit(0.1, "inches"))) +
annotate("segment", x = 84, xend = 35, y = 2, yend = 2, color = "black",
arrow = arrow(length = unit(0.1, "inches"))) +
annotate("segment", x = 86, xend = 99, y = 2, yend = 2, color = "black",
arrow = arrow(length = unit(0.1, "inches"))) +
annotate("segment", x = 99, xend = 86, y = 2, yend = 2, color = "black",
arrow = arrow(length = unit(0.1, "inches"))) +
scale_y_continuous(limits = c(0, 40), expand = c(0, 0)) +
scale_colour_brewer(palette = 'Set1') +
theme(axis.ticks.x = element_blank(),
legend.position = 'none',
panel.grid = element_blank(),
axis.title = element_text(size = 18),
axis.text = element_text(size = 16),
plot.background = element_blank())
```
---
# .center.pull[Bias with TWFE - Goodman-Bacon (2019)]
- $\color{blue}{\text{Goodman-Bacon (2019)}}$ shows that we can form four different 2x2 groups in this setting, where the effect can be estimated using the simple regression DiD in each group:
```{r d4, echo = FALSE, warning = FALSE, fig.align = 'center', fig.height = 5, cache = TRUE}
# function to make subplots
make_subplot <- function(omit, keep_dates, colors, breaks, break_expressions, series,
series_x, series_y, break_names, break_loc, arrow_start, arrow_stop, title){
data %>%
filter(series != omit & time >= keep_dates[1] & time <= keep_dates[2]) %>%
ggplot(aes(x = time, y = value, group = series, color = series, shape = series)) + geom_line() + geom_point() +
geom_vline(xintercept = breaks) +
labs(x = "Time", y = "Units of y") +
scale_x_continuous(limits = c(0, 105), breaks = breaks,
labels = break_expressions,
expand = c(0, 0)) +
annotate("text", x = series_x[1], y = series_y[1], label = series[1]) +
annotate("text", x = series_x[2], y = series_y[2], label = series[2]) +
annotate('label', x = break_loc[1], y = 5, label = break_names[1]) +
annotate('label', x = break_loc[2], y = 5, label = break_names[2]) +
annotate("segment", x = arrow_start[1], xend = arrow_stop[1], y = 2, yend = 2, color = "black",
arrow = arrow(length = unit(0.1, "inches"))) +
annotate("segment", x = arrow_stop[1], xend = arrow_start[1], y = 2, yend = 2, color = "black",
arrow = arrow(length = unit(0.1, "inches"))) +
annotate("segment", x = arrow_start[2], xend = arrow_stop[2], y = 2, yend = 2, color = "black",
arrow = arrow(length = unit(0.1, "inches"))) +
annotate("segment", x = arrow_stop[2], xend = arrow_start[2], y = 2, yend = 2, color = "black",
arrow = arrow(length = unit(0.1, "inches"))) +
scale_y_continuous(limits = c(0, 40), expand = c(0, 0)) +
scale_color_manual(values = c(colors[1], colors[2])) +
ggtitle(title) +
theme(axis.ticks.x = element_blank(),
legend.position = 'none',
panel.grid = element_blank(),
plot.title = element_text(hjust = 0.5, face = "plain"),
plot.background = element_blank())
}
p1 <- make_subplot(omit = "l", keep_dates = c(0, 100), colors = c('#E41A1C', '#377EB8'), breaks = 34,
break_expressions = expression('t'['k']^'*'),
series = c(expression('y'['it']^'k'), expression('y'['it']^'U')),
series_x = c(10, 90), series_y = c(23, 16),
break_names = c('Pre(k)', 'Post(k)'), break_loc = c(17, 66),
arrow_start = c(1, 35), arrow_stop = c(33, 99),
title = paste('A. Early Group vs. Untreated Group'))
p2 <- make_subplot(omit = "k", keep_dates = c(0, 100), colors = c('#4DAF4A', '#377EB8'), breaks = 85,
break_expressions = expression('t'['l']^'*'),
series = c(expression('y'['it']^'l'), expression('y'['it']^'U')),
series_x = c(50, 90), series_y = c(18, 16),
break_names = c('Pre(l)', 'Post(l)'), break_loc = c(50, 95),
arrow_start = c(1, 86), arrow_stop = c(84, 99),
title = paste('B. Late Group vs. Untreated Group'))
p3 <- make_subplot(omit = "U", keep_dates = c(0, 84), colors = c('#E41A1C', '#4DAF4A'), breaks = c(34, 85),
break_expressions = c(expression('t'['k']^'*'), expression('t'['l']^'*')),
series = c(expression('y'['it']^'k'), expression('y'['it']^'l')),
series_x = c(10, 50), series_y = c(23, 18),
break_names = c('Pre(k)', 'Mid(k, l)'), break_loc = c(17, 60),
arrow_start = c(1, 35), arrow_stop = c(33, 84),
title = bquote(paste('C. Early Group vs. Late Group, before ', 't'['l']^'*', sep = " ")))
p4 <- make_subplot(omit = "U", keep_dates = c(34, 100), colors = c('#E41A1C', '#4DAF4A'), breaks = c(34, 85),
break_expressions = c(expression('t'['k']^'*'), expression('t'['l']^'*')),
series = c(expression('y'['it']^'k'), expression('y'['it']^'l')),
series_x = c(60, 50), series_y = c(36, 18),
break_names = c('Mid(k, l)', 'Post(l)'), break_loc = c(60, 95),
arrow_start = c(35, 86), arrow_stop = c(84, 99),
title = bquote(paste('D. Late Group vs. Early Group, after ', 't'['k']^'*', sep = " ")))
# combine plots
p1 + p2 + p3 + p4 + plot_layout(nrow = 2)
```
---
# .center.pull[Bias with TWFE - Goodman-Bacon (2019)]
- Important Insights
- $\delta^{DD}$ is just the weighted average of the four 2x2 treatment effects. The weights are a function of the size of the subsample, relative size of treatment and control units, and the timing of treatment in the sub sample.
- Already-treated units act as controls even though they are treated.
- Given the weighting function, panel length alone can change the DiD estimates substantially, even when each $\delta^{DD}$ does not change.
- Groups treated closer to middle of panel receive higher weights than those treated earlier or later.
---
# .center.pull[Simulation Exercise]
- Can show how easily $\delta^{DD}$ goes awry up through a simulation exercise.
- Consider two sets of DiD estimates - one where the treatment occurs in one period, and one where the treatment is staggered.
- The data generating process is linear: $y_{it} = \alpha_i + \alpha_t + \delta_{it} + \epsilon_{it}$.
- $\alpha_i, \alpha_t \sim N(0, 1)$
- $\epsilon_{i, t} \sim N\left(0, \left(\frac{1}{2}\right)^2\right)$
- We will consider two different treatment assignment set ups for $\delta_{it}$.
---
# .center.pull[Simulation 1 - 1 Period Treatment]
- There are 40 states $s$, and 1000 units $i$ randomly drawn from the 40 states.
- Data covers years 1980 to 2010, and half the states receive "treatment" in 1995.
- For every unit incorporated in a treated state, we pull a unit-specific treatment effect from $\mu_i \sim N(0.3, (1/5)^2)$.
- Treatment effects here are trend breaks rather than unit shifts: the accumulated treatment effect $\delta_{it}$ is $\mu_i \times (year - 1995 + 1)$ for years after 1995.
- We then estimate the average treatment effect as $\hat{\delta}$ from:
$$y_{it} = \hat{\alpha_i} + \hat{\alpha_t} + \hat{\delta} D_{it}$$
- Simulate this data 1,000 and plot the distribution of estimates $\hat{delta}$ and the true effect (red line).
---
# .center.pull[Simulation 1 - 1 Period Treatment]
```{r d5, echo = FALSE, warning = FALSE, message = FALSE, fig.align = 'center', cache = TRUE, fig.width = 10}
# Make Data ---------------------------------------------
# loop function for one-time DiD shock
DID_onetime <- function(...) {
# Fixed Effects ------------------------------------------------
# unit fixed effects
unit <- tibble(
unit = 1:1000,
unit_fe = rnorm(1000, 0, 1),
# generate state
state = sample(1:40, 1000, replace = TRUE),
# generate treatment effect
mu = rnorm(1000, 0.3, 0.2))
# year fixed effects
year <- tibble(
year = 1980:2010,
year_fe = rnorm(31, 0, 1))
# Trend Break -------------------------------------------------------------
# Put the states into treatment groups
treat_taus <- tibble(
# sample the states randomly
state = sample(1:40, 40, replace = FALSE),
# place the randomly sampled states into treatment and control states
treated_unit = sample(c(rep(1, 20), rep(0, 20)), 40, replace = FALSE))
# make main dataset
# full interaction of unit X year
data <- expand_grid(unit = 1:1000, year = 1980:2010) %>%
left_join(., unit) %>%
left_join(., year) %>%
left_join(., treat_taus) %>%
# make error term and get treatment indicators and treatment effects
mutate(error = rnorm(31000, 0, 0.5),
treat = ifelse(year >= 1995 & treated_unit == 1, 1, 0),
tau = ifelse(treat == 1, mu, 0)) %>%
# calculate cumulative treatment effects
group_by(unit) %>%
mutate(tau_cum = cumsum(tau)) %>%
ungroup() %>%
# calculate the dep variable
mutate(dep_var = unit_fe + year_fe + tau_cum + error)
# run the DID and get the treatment effect estimates
broom::tidy(felm(dep_var ~ treat | unit + year | 0 | state, data = data,
exactDOF = TRUE, cmethod = "reghdfe"))
}
# estimate 1000 times
# set seed
set.seed(2140851)
DID_one_data <- map_dfr(1:1000, DID_onetime)
# plot DID estimates
DID_one_data %>%
ggplot(aes(x = estimate)) + geom_density(fill = "gray", alpha = 1/2) +
geom_vline(xintercept = 8.5*.3, color = "red", size = 2) +
labs(x = "Estimate Size", y = "Density") +
theme(axis.title = element_text(size = 14))
```
---
# .center.pull[Simulation 1 - 1 Period Treatment]
<center>
$\hspace{2cm}$
---
# .center.pull[Simulation 2 - Staggered Treatment]
- Run similar analysis with staggered treatment.
- The 40 states are randomly assigned into four treatment cohorts of size 250 depending on year of treatment assignment (1986, 1992, 1998, and 2004)
- DGP is identical, except that now $\delta_{it}$ is equal to $\mu_i \times (year - \tau_g + 1)$ where $\tau_g$ is the treatment assignment year.
- Estimate the treatment effect using TWFE and compare to the analytically derived true $\delta$ (red line).
---
# .center.pull[Simulation 2 - Staggered Treatment]
```{r d6, echo = FALSE, warning = FALSE, message = FALSE, fig.align = 'center', cache = TRUE, fig.width = 10}
## estimate DID model 2 - 250 firms are treated every period, with the treatment effect still = 0.3 on average
# loop function for one-time DiD shock
DID_multiple <- function(...) {
# Fixed Effects ------------------------------------------------
# unit fixed effects
unit <- tibble(
unit = 1:1000,
unit_fe = rnorm(1000, 0, 1),
# generate state
state = sample(1:40, 1000, replace = TRUE),
# generate treatment effect
mu = rnorm(1000, 0.3, 0.2))
# year fixed effects
year <- tibble(
year = 1980:2010,
year_fe = rnorm(31, 0, 1))
# Trend Break -------------------------------------------------------------
# Put the states into treatment groups
treat_taus <- tibble(
# sample the states randomly
state = sample(1:40, 40, replace = FALSE),
# place the randomly sampled states into five treatment groups G_g
cohort_year = sort(rep(c(1986, 1992, 1998, 2004), 10)))
# make main dataset
# full interaction of unit X year
data <- expand_grid(unit = 1:1000, year = 1980:2010) %>%
left_join(., unit) %>%
left_join(., year) %>%
left_join(., treat_taus) %>%
# make error term and get treatment indicators and treatment effects
mutate(error = rnorm(31000, 0, 0.5),
treat = ifelse(year >= cohort_year, 1, 0),
tau = ifelse(treat == 1, mu, 0)) %>%
# calculate cumulative treatment effects
group_by(unit) %>%
mutate(tau_cum = cumsum(tau)) %>%
ungroup() %>%
# calculate the dep variable
mutate(dep_var = unit_fe + year_fe + tau_cum + error)
# run the DID and get the treatment effect estimates
broom::tidy(felm(dep_var ~ treat | unit + year | 0 | state, data = data,
exactDOF = TRUE, cmethod = "reghdfe"))
}
# set seed
set.seed(2140851)
DID_multiple <- map_dfr(1:1000, DID_multiple)
# plot DID estimates
DID_multiple %>%
ggplot(aes(x = estimate)) + geom_density(fill = "gray", alpha = 1/2) +
geom_vline(xintercept = 8.5*.3, color = "red", size = 2) +
labs(x = "Estimate Size", y = "Density") +
theme(axis.title = element_text(size = 14))
```
---
# .center.pull[Simulation 2 - Staggered Treatment]
<center>
$\hspace{2cm}$
---
# .center.pull[Simulation 2 - Staggered Treatment]
- Main problem - we use prior treated units as controls.
- When the treatment effect is "dynamic", i.e. takes more than one period to be incorporated into your dependent variable, you are *subtracting* the treatment effects from prior treated units from the estimate of future control units.
- This biases your estimates towards zero when all the treatment effects are the same.
---
# .center.pull[Another Simulation]
- Can we actually get estimates for $\delta$ that are of the *wrong sign*? Yes, if treatment effects for early treated units are larger (in absolute magnitude) than the treatment effects on later treated units.
- Here firms are randomly assigned to one of 50 states. The 50 states are randomly assigned into one of 5 treatment groups $G_g$ based on treatment being initiated in 1985, 1991, 1997, 23003, and 2009.
- All treated firms incorporated in a state in treatment group $G_g$ receive a treatment effect $\delta_i \sim N(\delta_g, .2^2)$.
- The treatment effect is cumulative or dynamic - $\delta_{it} = \delta_i \times (year - G_g)$.
---
# .center.pull[Another Simulation]
- The average treatment effect multiple decreases over time:
$\hspace{2cm}$
```{r d7, message = FALSE, error = FALSE, echo = FALSE, cache = TRUE, results = 'asis'}
treats <- tibble(
"$G_g$" = c(1985, 1991, 1997, 2003, 2009),
"$\\delta_g$" = c(.5, .4, .3, .2, .1)
)
kable(treats, format = "html", align = 'c',
booktabs = T, caption = "Treatment Effect Averages") %>%
kable_styling(position = "center")
```
---
# .center.pull[Another Simulation]
- First let's look at the distribution of $\delta^{DD}$ using TWFE estimation with this simulated sample:
```{r d8, echo = FALSE, warning = FALSE, message = FALSE, fig.align = 'center', cache = TRUE, fig.height = 6, fig.width = 10}
# set seed
set.seed(2140851)
runreg <- function(i){
# Fixed Effects ------------------------------------------------
# unit fixed effects
unit <- tibble(
unit = 1:1000,
unit_fe = rnorm(1000, 0, 1),
# generate state
state = sample(1:50, 1000, replace = TRUE))
# year fixed effects
year <- tibble(
year = 1980:2015,
year_fe = rnorm(36, 0, 1))
# Trend Break -------------------------------------------------------------
# Put the states into treatment groups
treat_taus <- tibble(
# sample the states randomly
state = sample(1:50, 50, replace = FALSE),
# place the randomly sampled states into five treatment groups G_g
cohort_year = sort(rep(c(1985, 1991, 1997, 2003, 2009), 10)),
# assign them a mean treatment effect from 0.5 to 0.1
mu = sort(rep(c(.5, .4, .3, .2, .1), 10), decreasing = TRUE))
# make main dataset
# full interaction of unit X year
data <- expand_grid(unit = 1:1000, year = 1980:2015) %>%
left_join(., unit) %>%
left_join(., year)
# bring in the treatment indicators and values
get_treat <- function(u) {
# get the state for the unit
st <- unit %>% filter(unit == u) %>% pull(state)
# find the treatment year for the state
treat_yr <- treat_taus %>% filter(state == st) %>% pull(cohort_year)
# treatment effect tau_g
mu <- treat_taus %>% filter(state == st) %>% pull(mu)
# Make a data set with the results
tibble(unit = rep(u, 36),
year = 1980:2015,
# get a treatment cohort indicator
# make treatment indicator
treat = ifelse(year < treat_yr, 0, 1),
# get the treatment effect \tau_i for post-treatment years
cohort_year = treat_yr,
static_tau = rep(rnorm(1, mu, .2), 36),
tau = ifelse(year < treat_yr, 0, static_tau),
# cumulate the effect
tau_cum = cumsum(tau))
}
# call the function over our 1000 firms
treatments <- map_dfr(1:1000, get_treat)
# merge in the treatment effect data
data <- left_join(data, treatments) %>%
# simulate error and generate the dependent variable
mutate(error = rnorm(36000, 0, 0.5),
dep_var = unit_fe + year_fe + tau_cum + error)
broom::tidy(felm(dep_var ~ treat | unit + year | 0 | state, data = data,
exactDOF = TRUE, cmethod = "reghdfe"))
}
simdata <- map_dfr(1:300, runreg)
simdata %>%
ggplot(aes(x = estimate)) + geom_density(fill = "gray", alpha = 1/2) +
geom_vline(xintercept = 0, color = "red", size = 2) +
labs(x = "Estimate Size", y = "Density") +
theme(axis.title = element_text(size = 14))
```
---
# .center.pull[Goodman-Bacon Decomposition]
```{r d9, echo = FALSE, warning = FALSE, message = FALSE, fig.align = 'center', cache = TRUE, fig.width = 10}
# unit fixed effects
unit <- tibble(
unit = 1:1000,
unit_fe = rnorm(1000, 0, 1),
# generate state
state = sample(1:50, 1000, replace = TRUE))
# year fixed effects
year <- tibble(
year = 1980:2015,
year_fe = rnorm(36, 0, 1))
# Trend Break -------------------------------------------------------------
# Put the states into treatment groups
treat_taus <- tibble(
# sample the states randomly
state = sample(1:50, 50, replace = FALSE),
# place the randomly sampled states into five treatment groups G_g
cohort_year = sort(rep(c(1985, 1991, 1997, 2003, 2009), 10)),
# assign them a mean treatment effect from 0.5 to 0.1
mu = sort(rep(c(.5, .4, .3, .2, .1), 10), decreasing = TRUE))
# make main dataset
# full interaction of unit X year
data <- expand_grid(unit = 1:1000, year = 1980:2015) %>%
left_join(., unit) %>%
left_join(., year)
# bring in the treatment indicators and values
get_treat <- function(u) {
# get the state for the unit
st <- unit %>% filter(unit == u) %>% pull(state)
# find the treatment year for the state
treat_yr <- treat_taus %>% filter(state == st) %>% pull(cohort_year)
# treatment effect tau_g
mu <- treat_taus %>% filter(state == st) %>% pull(mu)
# Make a data set with the results
tibble(unit = rep(u, 36),
year = 1980:2015,
# get a treatment cohort indicator
# make treatment indicator
treat = ifelse(year < treat_yr, 0, 1),
# get the treatment effect \tau_i for post-treatment years
cohort_year = treat_yr,
static_tau = rep(rnorm(1, mu, .2), 36),
tau = ifelse(year < treat_yr, 0, static_tau),
# cumulate the effect
tau_cum = cumsum(tau))
}
# call the function over our 1000 firms
treatments <- map_dfr(1:1000, get_treat)
# merge in the treatment effect data
data <- left_join(data, treatments) %>%
# simulate error and generate the dependent variable
mutate(error = rnorm(36000, 0, 0.5),
dep_var = unit_fe + year_fe + tau_cum + error)
# calculate the bacon decomposition without covariates
bacon_out <- bacon(dep_var ~ treat,
data = data,
id_var = "unit",
time_var = "year")
bacon_out %>%
ggplot(aes(x = weight, y = estimate, shape = factor(type), color = factor(type))) +
geom_point(size = 3) +
geom_hline(yintercept = 0) +
scale_colour_brewer(palette = 'Set1') +
labs(x = "Weight", y = "Estimate") +
theme(legend.position = "bottom",
legend.title = element_blank(),
axis.title = element_text(size = 18),
axis.text = element_text(size = 16))
```
---
# .center.pull[Callaway & Sant'Anna]
- Inverse propensity weighted long-difference in cohort-specific average treatment effects between treated and untreated units for a given treatment cohort.
$$\begin{equation} ATT(g, t) = \mathbb{E} \left[\left( \frac{G_g}{\mathbb{E}[G_g]} - \frac{\frac{p_g(X)C}{1 - p_g(X)}}{\mathbb{E}\left[\frac{p_g(X)C}{1 - p_g(X)} \right]} \right) \left(Y_t - T_{g - 1}\right)\right] \end{equation}$$
- Without covariates, as in the simulated example here, it calculates the simple long difference between all treated units $i$ in relative year $k$ with all potential control units that have not yet been treated by year $k$.
---
# .center.pull[Callaway & Sant'Anna]
```{r d10, echo = FALSE, warning = FALSE, message = FALSE, fig.align = 'center', cache = TRUE, fig.width = 10}
# create a lead/lag indicators
data <- data %>%
# variable with relative year from treatment
mutate(rel_year = year - cohort_year) %>%
# drop observations after 2008 bc all treated
filter(year <= 2008) %>%
dplyr::arrange(cohort_year, unit, year)
# first get percentage contribution to each lead/lag indicator by treatment cohort for weights
# we will need this for the Abraham/Sun method, as well as the true treatment indicator
# calculate weights
weights <- data %>%
mutate(rel_year = year - cohort_year) %>%
# drop covariates for 2009 adopters
filter(cohort_year != 2009) %>%
group_by(cohort_year, rel_year) %>%
count %>%
ungroup() %>%
group_by(rel_year) %>%
mutate(total = sum(n),
perc = n / total) %>%
# keep just the variables we need
select(rel_year, cohort_year, perc) %>%
ungroup() %>%
rowwise() %>%
# add variable equal to coefficient from regression
mutate(term = paste("cohort_year_", cohort_year, "_", rel_year + 29, sep = "")) %>%
ungroup()
# make a dataset with the theoretical values to merge in
true_effect <- weights %>%
# add in the multiples
mutate(
multiple = case_when(
rel_year < 0 ~ 0,
rel_year >= 0 ~ rel_year + 1),
# add in the tau_g values
tau_g = case_when(
cohort_year == 1985 ~ .5,
cohort_year == 1991 ~ .4,
cohort_year == 1997 ~ .3,
cohort_year == 2003 ~ .2),
# multiply the two
effect = multiple*tau_g) %>%
#collapse by time period
group_by(rel_year) %>%
summarize(true_tau = weighted.mean(effect, w = perc)) %>%
# make the time variable for merging
mutate(t = rel_year)
# run the CS algorithm
CS_out <- att_gt("dep_var", data = data,
first.treat.name="cohort_year",
idname="unit", tname="year", aggte = T,
clustervars = "state",
bstrap=T, cband=T,
maxe = 6,
mine = -4,
nevertreated = F,
printdetails = F)
# plot
tibble(
t = -5:5,
estimate = CS_out$aggte$dynamic.att.e,
se = CS_out$aggte$dynamic.se.e,
conf.low = estimate - 1.96*se,
conf.high = estimate + 1.96*se,) %>%
left_join(true_effect) %>%
# split the error bands by pre-post
mutate(band_groups = case_when(
t < -1 ~ "Pre",
t >= 0 ~ "Post",
t == -1 ~ ""
)) %>%
# plot
ggplot(aes(x = t, y = estimate)) +
geom_line(aes(x = t, y = true_tau, color = "True Effect"), size = 1.5, linetype = "dashed") +
geom_ribbon(aes(ymin = conf.low, ymax = conf.high, group = band_groups),
color = "lightgrey", alpha = 1/4) +
#geom_point(aes(color = "Estimated Effect")) +
geom_pointrange(aes(ymin = conf.low, ymax = conf.high, color = "Estimated Effect"), show.legend = FALSE) +
geom_hline(yintercept = 0) +
geom_vline(xintercept = -0.5, linetype = "dashed") +
scale_x_continuous(breaks = -5:5) +
labs(x = "Relative Time", y = "Estimate") +
scale_color_brewer(palette = 'Set1') +
theme(legend.position = "bottom",
legend.title = element_blank(),
axis.title = element_text(size = 18),
axis.text = element_text(size = 16))
```
---
# .center.pull[Abraham and Sun]
- A relatively straightforward extension of the standard event-study TWFE model:
$$y_{it} = \alpha_i + \alpha_t + \sum_e \sum_{l \neq -1} \delta_{el}(1\{E_i = e\} \cdot D_{it}^l) + \epsilon_{it}$$
- You saturate the relative time indicators (i.e. t = -2, -1, ...) with indicators for the treatment initiation year group, and aggregate to overall aggregate relative time indicators by cohort size.
- In the case of no covariates, this gives you the same estimate as Callaway & Sant'Anna if you *fully saturate* the model with time indicators (leaving only two relative year identifiers missing).
- The authors don't claim that it can be used with covariates, but it seemingly follows if we think it is okay with normal TWFE DiD.
---
# .center.pull[Abraham and Sun]
```{r d11, echo = FALSE, warning = FALSE, message = FALSE, fig.align = 'center', cache = TRUE, fig.width = 10}
## Make cohort-relative time dummies
# relative year dummies
rel_year <- data %>% select(rel_year) %>%
dummy_cols %>% select(-1) %>%
set_colnames(as.numeric(str_remove(colnames(.), "rel_year_")) + 29) %>%
as.data.frame
# cohort dummies
cohorts <- data %>% select(cohort_year) %>%
dummy_cols %>% select(-1) %>%
as.data.frame
# combine matrix functions
combine_mat <- function(i) {
cohorts[, i] * rel_year %>%
set_colnames(paste(colnames(cohorts)[i], colnames(rel_year), sep = "_"))
}
# combine dummies and merge into our data
dummies <- map_dfc(1:4, combine_mat)
data <- data %>% bind_cols(dummies)
# put the covariates into a vector form
covs <- paste("cohort_year_", rep(c(1985, 1991, 1997, 2003), 51), "_", c(-28:-2, 0:23) + 29, sep = "")
# estimate the saturated model
fit <- felm(as.formula(paste("dep_var ~ ", paste(covs, collapse = "+"), "| unit + year | 0 | state")),
data = data, exactDOF = TRUE)
# rerun without the NA covariates because glmt won't run otherwise
# new set of covariates without the na
covs <- broom::tidy(fit) %>% filter(!is.na(estimate)) %>% pull(term)
fit <- felm(as.formula(paste("dep_var ~ ", paste(covs, collapse = "+"), "| unit + year | 0 | state")),
data = data, exactDOF = TRUE)
# get the coefficients and make a dataset for plotting
coefs <- fit$coefficients %>%
# add in coefficient name to tibble
as_tibble(rownames = "term") %>%
# bring in weights
left_join(., weights)
# get the relevant coefficients and weights into a string to get the linear combination
get_lincom <- function(ll) {
# get just the coefficients for a specific lead lag
cf2 <- coefs %>% filter(rel_year == ll)
# paste the function that goes into the linear combination function
F <- paste(paste(cf2$perc, cf2$term, sep = " * ", collapse = " + "), " = 0")
# take linear combination and put into a data frame
broom::tidy(
confint(glht(fit, linfct = F)),
conf.int = TRUE
) %>% mutate(rel_year = ll)
}
# run over all lead/lags
AS_plot <- map_df(c(-5:-2, 0:5), get_lincom) %>%
# add time variable
mutate(t = c(-5:-2, 0:5))
#Plot the results
AS_plot %>%
select(t, estimate, conf.low, conf.high) %>%
# add in data for year -1
bind_rows(tibble(t = -1, estimate = 0,
conf.low = 0, conf.high = 0
)) %>%
left_join(true_effect) %>%
# split the error bands by pre-post
mutate(band_groups = case_when(
t < -1 ~ "Pre",
t >= 0 ~ "Post",
t == -1 ~ ""
)) %>%
# plot
ggplot(aes(x = t, y = estimate)) +
geom_line(aes(x = t, y = true_tau, color = "True Effect"), size = 1.5, linetype = "dashed") +
geom_ribbon(aes(ymin = conf.low, ymax = conf.high, group = band_groups),
color = "lightgrey", alpha = 1/4) +
#geom_point(aes(color = "Estimated Effect")) +
geom_pointrange(aes(ymin = conf.low, ymax = conf.high, color = "Estimated Effect"), show.legend = FALSE) +
geom_hline(yintercept = 0) +
geom_vline(xintercept = -0.5, linetype = "dashed") +
scale_x_continuous(breaks = -5:5) +
labs(x = "Relative Time", y = "Estimate") +
scale_color_brewer(palette = 'Set1') +
theme(legend.position = "bottom",
legend.title = element_blank(),
axis.title = element_text(size = 18),
axis.text = element_text(size = 16))
```
---
# .center.pull[Cengiz et al. (2019)]
- Similar to the standard TWFE DiD, but we ensure that no previously treated units enter as controls by trimming the sample.
- For each treatment cohort $G_g$, get all treated units, and all units that are not treated by year $g + k$ where $g$ is the treatment year and $k$ is the outer most relative year that you want to test (e.g. if you do an event study plot from -5 to 5, $k$ would equal 5).
- Keep only observations within years $g - k$ and $g + k$ for each cohort-specific dataset, and then stack them in relative time.
- Run the same TWFE estimates as in standard DiD, but include interactions for the cohort-specific dataset with all of the fixed effects, controls, and clusters.
---
# .center.pull[Cengiz et al. (2019)]
```{r d12, echo = FALSE, warning = FALSE, message = FALSE, fig.align = 'center', cache = TRUE, fig.width = 10}
# get the cohort years
obs <- data %>%
filter(cohort_year != 2009) %>%
pull(cohort_year) %>%
unique()
# make fomula to run within our FE specification
# get the lead lags in one set of covariates
leadlags <- c("lag_5", "lag_4", "lag_3", "lag_2", "date_0",
"lead_1", "lead_2", "lead_3", "lead_4", "lead_5")
# add in lead lags
data <- data %>%
# variable with relative year from treatment
mutate(rel_year = year - cohort_year,