26 Differenceindifferences
Examples in marketing
 (Liaukonyte, Teixeira, and Wilbur 2015): TV ad on online shopping
 (Wang, Lewis, and Schweidel 2018): political ad source and message tone on vote shares and turnout using discontinuities in the level of political ads at the borders
 (Datta, Knox, and Bronnenberg 2018): streaming service on total music consumption using timing of users adoption of a music streaming service
 (Janakiraman, Lim, and Rishika 2018): data breach announcement affect customer spending using timing of data breach and variation whether customer info was breached in that event
 (Israeli 2018): digital monitoring and enforcement on violations using enforcement of min ad price policies
 (Ramani and Srinivasan 2019): firms respond to foreign direct investment liberalization using India’s reform in 1991.
 (Pattabhiramaiah, Sriram, and Manchanda 2019): paywall affects readership
 (Akca and Rao 2020): aggregators for airlines business effect
 (Lim et al. 2020): nutritional labels on nutritional quality for other brands in a category using variation in timing of adoption of nutritional labels across categories
 (Guo, Sriram, and Manchanda 2020): payment disclosure laws effect on physician prescription behavior using Timing of the Massachusetts open payment law as the exogenous shock
 (He, Hollenbeck, and Proserpio 2022): using Amazon policy change to examine the causal impact of fake reviews on sales, average ratings.
 (Peukert et al. 2022): using European General data protection Regulation, examine the impact of policy change on website usage.
Show the mechanism via
Mediation Under DiD analysis: see (Habel, Alavi, and Linsenmayer 2021)
Moderation analysis: see (Goldfarb and Tucker 2011)
26.1 Visualization
library(panelView)
library(fixest)
library(tidyverse)
base_stagg < fixest::base_stagg >
# treatment status
dplyr::mutate(treat_stat = dplyr::if_else(time_to_treatment < 0, 0, 1)) >
select(id, year, treat_stat, y)
head(base_stagg)
#> id year treat_stat y
#> 2 90 1 0 0.01722971
#> 3 89 1 0 4.58084528
#> 4 88 1 0 2.73817174
#> 5 87 1 0 0.65103066
#> 6 86 1 0 5.33381664
#> 7 85 1 0 0.49562631
panelView::panelview(
y ~ treat_stat,
data = base_stagg,
index = c("id", "year"),
xlab = "Year",
ylab = "Unit",
display.all = F,
gridOff = T,
by.timing = T
)
26.2 Simple Difndif
A tool developed intuitively to study “natural experiment”, but its uses are much broader.
Fixed Effects Estimator is the foundation for DID

Why is difindif attractive? Identification strategy: Intertemporal variation between groups
Crosssectional estimator helps avoid omitted (unobserved) common trends
Timeseries estimator helps overcome omitted (unobserved) crosssectional differences
Consider
\(D_i = 1\) treatment group
\(D_i = 0\) control group
\(T= 1\) After the treatment
\(T =0\) Before the treatment
After (T = 1)  Before (T = 0)  

Treated \(D_i =1\)  \(E[Y_{1i}(1)D_i = 1]\)  \(E[Y_{0i}(0)D)i=1]\) 
Control \(D_i = 0\)  \(E[Y_{0i}(1) D_i =0]\)  \(E[Y_{0i}(0)D_i=0]\) 
missing \(E[Y_{0i}(1)D=1]\)
The Average Treatment Effect on Treated (ATT)
\[ \begin{aligned} E[Y_1(1)  Y_0(1)D=1] &= \{E[Y(1)D=1]  E[Y(1)D=0] \} \\ & \{E[Y(0)D=1]  E[Y(0)D=0] \} \end{aligned} \]
More elaboration:
 For the treatment group, we isolate the difference between being treated and not being treated. If the untreated group would have been affected in a different way, the DiD design and estimate would tell us nothing.
 Alternatively, because we can’t observe treatment variation in the control group, we can’t say anything about the treatment effect on this group.
Extension
 More than 2 groups (multiple treatments and multiple controls), and more than 2 period (pre and post)
\[ Y_{igt} = \alpha_g + \gamma_t + \beta I_{gt} + \delta X_{igt} + \epsilon_{igt} \]
where
\(\alpha_g\) is the groupspecific fixed effect
\(\gamma_t\) = time specific fixed effect
\(\beta\) = difindif effect
\(I_{gt}\) = interaction terms (n treatment indicators x n posttreatment dummies) (capture effect heterogeneity over time)
This specification is the “twoway fixed effects DiD”  TWFE (i.e., 2 sets of fixed effects: group + time).
 However, if you have Staggered Difndif (i.e., treatment is applied at different times to different groups). TWFE is really bad.
 Longterm Effects
To examine the dynamic treatment effects (that are not under rollout/staggered design), we can create a centered time variable,
Centered Time Variable  Period 

…  
\(t = 1\)  2 periods before treatment period 
\(t = 0\) 
Last period right before treatment period Remember to use this period as reference group 
\(t = 1\)  Treatment period 
… 
By interacting this factor variable, we can examine the dynamic effect of treatment (i.e., whether it’s fading or intensifying)
\[ \begin{aligned} Y &= \alpha_0 + \alpha_1 Group + \alpha_2 Time \\ &+ \beta_{T_1} Treatment+ \beta_{(T_1 1)} Treatment + \dots + \beta_{1} Treatment \\ &+ \beta_1 + \dots + \beta_{T_2} Treatment \end{aligned} \]
where
\(\beta_0\) is used as the reference group (i.e., drop from the model)
\(T_1\) is the pretreatment period
\(T_2\) is the posttreatment period
With more variables (i.e., interaction terms), coefficients estimates can be less precise (i.e., higher SE).
 DiD on the relationship, not levels. Technically, we can apply DiD research design not only on variables, but also on coefficients estimates of some other regression models with before and after a policy is implemented.
Goal:
 Pretreatment coefficients should be nonsignificant \(\beta_{T_1}, \dots, \beta_{1} = 0\) (similar to the Placebo Test)
 Posttreatment coefficients are expected to be significant \(\beta_1, \dots, \beta_{T_2} \neq0\)
 You can now examine the trend in posttreatment coefficients (i.e., increasing or decreasing)
library(tidyverse)
library(fixest)
od < causaldata::organ_donations %>%
# Treatment variable
dplyr::mutate(California = State == 'California') %>%
# centered time variable
dplyr::mutate(center_time = as.factor(Quarter_Num  3))
# where 3 is the reference period precedes the treatment period
class(od$California)
#> [1] "logical"
class(od$State)
#> [1] "character"
cali < feols(Rate ~ i(center_time, California, ref = 0) 
State + center_time,
data = od)
etable(cali)
#> cali
#> Dependent Var.: Rate
#>
#> California x center_time = 2 0.0029 (0.0051)
#> California x center_time = 1 0.0063** (0.0023)
#> California x center_time = 1 0.0216*** (0.0050)
#> California x center_time = 2 0.0203*** (0.0045)
#> California x center_time = 3 0.0222* (0.0100)
#> FixedEffects: 
#> State Yes
#> center_time Yes
#> _____________________________ ___________________
#> S.E.: Clustered by: State
#> Observations 162
#> R2 0.97934
#> Within R2 0.00979
#> 
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
iplot(cali, pt.join = T)
coefplot(cali)
26.3 Notes

Match treatment and control based on pretreatment observables
Modify SEs appropriately (James J. Heckman, Ichimura, and Todd 1997). It’s might be easier to just use the Doubly Robust DiD (Sant’Anna and Zhao 2020) where you just need either matching or regression to work in order to identify your treatment effect
Whereas the group fixed effects control for the group timeinvariant effects, it does not control for selection bias (i.e., certain groups are more likely to be treated than others). Hence, with these backdoor open (i.e., selection bias) between (1) propensity to be treated and (2) dynamics evolution of the outcome posttreatment, matching can potential close these backdoor.
Be careful when matching timevarying covariates because you might encounter “regression to the mean” problem, where pretreatment periods can have an unusually bad or good time (that is out of the ordinary), then the posttreatment period outcome can just be an artifact of the regression to the mean (Daw and Hatfield 2018). This problem is not of concern to timeinvariant variables.
Matching and DiD can use pretreatment outcomes to correct for selection bias. From real world data and simulation, (ChabéFerret 2015) found that matching generally underestimates the average causal effect and gets closer to the true effect with more number of pretreatment outcomes. When selection bias is symmetric around the treatment date, DID is still consistent when implemented symmetrically (i.e., the same number of period before and after treatment). In cases where selection bias is asymmetric, the MC simulations show that Symmetric DiD still performs better than Matching.

It’s always good to show results with and without controls because
If the controls are fixed within group or within time, then those should be absorbed under those fixed effects
If the controls are dynamic across group and across, then your parallel trends assumption is not plausible.
Under causal inference, \(R^2\) is not so important.
26.4 Standard Errors
Serial correlation is a big problem in DiD because (Bertrand, Duflo, and Mullainathan 2004)
 DiD often uses long time series
 Outcomes are often highly positively serially correlated
 Minimal variation in the treatment variable over time within a group (e.g., state).
To overcome this problem:
 Using parametric correction (standard AR correction) is not good.
 Using nonparametric (e.g., block bootstrap keep all obs from the same group such as state together) is good when number of groups is large.
 Remove time series dimension (i.e., aggregate data into 2 periods: pre and post). This still works with small number of groups (See (Donald and Lang 2007) for more notes on smallsample aggregation).
 Empirical and arbitrary variancecovariance matrix corrections work only in large samples.
26.5 Examples
Example by Philipp Leppert replicating Card and Krueger (1994)
Example by Anthony Schmidt
26.5.1 Example by Doleac and Hansen (2020)
The purpose of banning a checking box for excriminal was banned because we thought that it gives more access to felons
Even if we ban the box, employers wouldn’t just change their behaviors. But then the unintended consequence is that employers statistically discriminate based on race
3 types of ban the box
 Public employer only
 Private employer with government contract
 All employers
Main identification strategy
 If any county in the Metropolitan Statistical Area (MSA) adopts ban the box, it means the whole MSA is treated. Or if the state adopts “ban the ban,” every county is treated
Under Simple Difndif
\[ Y_{it} = \beta_0 + \beta_1 Post_t + \beta_2 treat_i + \beta_2 (Post_t \times Treat_i) + \epsilon_{it} \]
But if there is no common post time, then we should use Staggered Difndif
\[ \begin{aligned} E_{imrt} &= \alpha + \beta_1 BTB_{imt} W_{imt} + \beta_2 BTB_{mt} + \beta_3 BTB_{mt} H_{imt}\\ &+ \delta_m + D_{imt} \beta_5 + \lambda_{rt} + \delta_m\times f(t) \beta_7 + e_{imrt} \end{aligned} \]
where
\(i\) = person; \(m\) = MSA; \(r\) = region (US regions e.g., Midwest) ; \(r\) = region; \(t\) = year
\(W\) = White; \(B\) = Black; \(H\) = Hispanic
\(\beta_1 BTB_{imt} W_{imt} + \beta_2 BTB_{mt} + \beta_3 BTB_{mt} H_{imt}\) are the 3 difndif variables (\(BTB\) = “ban the box”)
\(\delta_m\) = dummy for MSI
\(D_{imt}\) = control for people
\(\lambda_{rt}\) = region by time fixed effect
\(\delta_m \times f(t)\) = linear time trend within MSA (but we should not need this if we have good pretrend)
If we put \(\lambda_r  \lambda_t\) (separately) we will more broad fixed effect, while \(\lambda_{rt}\) will give us deeper and narrower fixed effect.
Before running this model, we have to drop all other races. And \(\beta_1, \beta_2, \beta_3\) are not collinear because there are all interaction terms with \(BTB_{mt}\)
If we just want to estimate the model for black men, we will modify it to be
\[ E_{imrt} = \alpha + BTB_{mt} \beta_1 + \delta_m + D_{imt} \beta_5 + \lambda_{rt} + (\delta_m \times f(t)) \beta_7 + e_{imrt} \]
\[ \begin{aligned} E_{imrt} &= \alpha + BTB_{m (t  3t)} \theta_1 + BTB_{m(t2)} \theta_2 + BTB_{mt} \theta_4 \\ &+ BTB_{m(t+1)}\theta_5 + BTB_{m(t+2)}\theta_6 + BTB_{m(t+3t)}\theta_7 \\ &+ [\delta_m + D_{imt}\beta_5 + \lambda_r + (\delta_m \times (f(t))\beta_7 + e_{imrt}] \end{aligned} \]
We have to leave \(BTB_{m(t1)}\theta_3\) out for the category would not be perfect collinearity
So the year before BTB (\(\theta_1, \theta_2, \theta_3\)) should be similar to each other (i.e., same pretrend). Remember, we only run for places with BTB.
If \(\theta_2\) is statistically different from \(\theta_3\) (baseline), then there could be a problem, but it could also make sense if we have pretrend announcement.
26.5.2 Example from Princeton
library(foreign)
mydata = read.dta("http://dss.princeton.edu/training/Panel101.dta") %>%
# create a dummy variable to indicate the time when the treatment started
dplyr::mutate(time = ifelse(year >= 1994, 1, 0)) %>%
# create a dummy variable to identify the treatment group
dplyr::mutate(treated = ifelse(country == "E" 
country == "F"  country == "G" ,
1,
0)) %>%
# create an interaction between time and treated
dplyr::mutate(did = time * treated)
estimate the DID estimator
didreg = lm(y ~ treated + time + did, data = mydata)
summary(didreg)
#>
#> Call:
#> lm(formula = y ~ treated + time + did, data = mydata)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> 9.768e+09 1.623e+09 1.167e+08 1.393e+09 6.807e+09
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>t)
#> (Intercept) 3.581e+08 7.382e+08 0.485 0.6292
#> treated 1.776e+09 1.128e+09 1.575 0.1200
#> time 2.289e+09 9.530e+08 2.402 0.0191 *
#> did 2.520e+09 1.456e+09 1.731 0.0882 .
#> 
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 2.953e+09 on 66 degrees of freedom
#> Multiple Rsquared: 0.08273, Adjusted Rsquared: 0.04104
#> Fstatistic: 1.984 on 3 and 66 DF, pvalue: 0.1249
The did
coefficient is the differencesindifferences estimator. Treat has a negative effect
26.5.3 Example by Card and Krueger (1993)
found that increase in minimum wage increases employment
Experimental Setting:
New Jersey (treatment) increased minimum wage
Penn (control) did not increase minimum wage
After  Before  

Treatment  NJ  A  B  A  B 
Control  PA  C  D  C  D 
A  C  B  D  (A  B)  (C  D) 
where
A  B = treatment effect + effect of time (additive)
C  D = effect of time
(A  B)  (C  D) = difndif
The identifying assumptions:
Can’t have switchers

PA is the control group
is a good counter factual
is what NJ would look like if they hadn’t had the treatment
\[ Y_{jt} = \beta_0 + NJ_j \beta_1 + POST_t \beta_2 + (NJ_j \times POST_t)\beta_3+ X_{jt}\beta_4 + \epsilon_{jt} \]
where
\(j\) = restaurant
\(NJ\) = dummy where \(1 = NJ\), and \(0 = PA\)
\(POST\) = dummy where \(1 = post\), and \(0 = pre\)
Notes:
We don’t need \(\beta_4\) in our model to have unbiased \(\beta_3\), but including it would give our coefficients efficiency
If we use \(\Delta Y_{jt}\) as the dependent variable, we don’t need \(POST_t \beta_2\) anymore
Alternative model specification is that the authors use NJ high wage restaurant as control group (still choose those that are close to the border)
The reason why they can’t control for everything (PA + NJ high wage) is because it’s hard to interpret the causal treatment

Difndif utilizes similarity in pretrend of the dependent variables. However, this is neither a necessary nor sufficient for the identifying assumption.
It’s not sufficient because they can have multiple treatments (technically, you could include more control, but your treatment can’t interact)
It’s not necessary because trends can be parallel after treatment
However, we can’t never be certain; we just try to find evidence consistent with our theory so that difndif can work.
Notice that we don’t need before treatment the levels of the dependent variable to be the same (e.g., same wage average in both NJ and PA), difndif only needs pretrend (i.e., slope) to be the same for the two groups.
26.5.4 Example by Butcher, McEwan, and Weerapana (2014)
Theory:

Highest achieving students are usually in hard science. Why?
Hard to give students students the benefit of doubt for hard science
How unpleasant and how easy to get a job. Degrees with lower market value typically want to make you feel more pleasant
Under OLS
\[ E_{ij} = \beta_0 + X_i \beta_1 + G_j \beta_2 + \epsilon_{ij} \]
where
\(X_i\) = student attributes
\(\beta_2\) = causal estimate (from grade change)
\(E_{ij}\) = Did you choose to enroll in major \(j\)
\(G_j\) = grade given in major \(j\)
Examine \(\hat{\beta}_2\)
Negative bias: Endogenous response because department with lower enrollment rate will give better grade
Positive bias: hard science is already having best students (i.e., ability), so if they don’t their grades can be even lower
Under difndif
\[ Y_{idt} = \beta_0 + POST_t \beta_1 + Treat_d \beta_2 + (POST_t \times Treat_d)\beta_3 + X_{idt} + \epsilon_{idt} \]
where
 \(Y_{idt}\) = grade average
Intercept  Treat  Post  Treat*Post  

Treat Pre  1  1  0  0 
Treat Post  1  1  1  1 
Control Pre  1  0  0  0 
Control Post  1  0  1  0 
Average for precontrol \(\beta_0\) 
A more general specification of the difndif is that
\[ Y_{idt} = \alpha_0 + (POST_t \times Treat_d) \alpha_1 + \theta_d + \delta_t + X_{idt} + u_{idt} \]
where
\((\theta_d + \delta_t)\) richer , more df than \(Treat_d \beta_2 + Post_t \beta_1\) (because fixed effects subsume Post and treat)
\(\alpha_1\) should be equivalent to \(\beta_3\) (if your model assumptions are correct)
26.6 One Difference
The regression formula is as follows (Liaukonytė, Tuchman, and Zhu 2023):
\[ y_{ut} = \beta \text{Post}_t + \gamma_u + \gamma_w(t) + \gamma_l + \gamma_g(u)p(t) + \epsilon_{ut} \]
where
 \(y_{ut}\): Outcome of interest for unit u in time t.
 \(\text{Post}_t\): Dummy variable representing a specific postevent period.
 \(\beta\): Coefficient measuring the average change in the outcome after the event relative to the preperiod.
 \(\gamma_u\): Fixed effects for each unit.
 \(\gamma_w(t)\): Timespecific fixed effects to account for periodic variations.
 \(\gamma_l\): Dummy variable for a specific significant period (e.g., a major event change).
 \(\gamma_g(u)p(t)\): Group x period fixed effects for flexible trends that may vary across different categories (e.g., geographical regions) and periods.
 \(\epsilon_{ut}\): Error term.
This model can be used to analyze the impact of an event on the outcome of interest while controlling for various fixed effects and timespecific variations, but using units themselves pretreatment as controls.
26.7 Twoway Fixedeffects
A generalization of the difndif model is the twoway fixedeffects models where you have multiple groups and time effects. But this is not a designedbased, nonparametric causal estimator (Imai and Kim 2021)
When applying TWFE to multiple groups and multiple periods, the supposedly causal coefficient is the weighted average of all twogroup/twoperiod DiD estimators in the data where some of the weights can be negative. More specifically, the weights are proportional to group sizes and treatment indicator’s variation in each pair, where units in the middle of the panel have the highest weight.
The canonical/standard TWFE only works when

Effects are homogeneous across units and across time periods (i.e., no dynamic changes in the effects of treatment). See (GoodmanBacon 2021; Clément De Chaisemartin and d’Haultfoeuille 2020; L. Sun and Abraham 2021; Borusyak, Jaravel, and Spiess 2021) for details. Similarly, it relies on the assumption of linear additive effects (Imai and Kim 2021)
Have to argue why treatment heterogeneity is not a problem (e.g., plot treatment timing and decompose treatment coefficient using GoodmanBacon Decomposition) know the percentage of observation are never treated (because as the nevertreated group increases, the bias of TWFE decreases, with 80% sample to be nevertreated, bias is negligible). The problem is worsen when you have longrun effects.
Need to manually drop two relative time periods if everyone is eventually treated (to avoid multicollinearity). Programs might do this randomly and if it chooses to drop a posttreatment period, it will create biases. The choice usually 1, and 2 periods.
Treatment heterogeneity can come in because (1) it might take some time for a treatment to have measurable changes in outcomes or (2) for each period after treatment, the effect can be different (phase in or increasing effects).
2 time periods.
Within this setting, TWFE works because, using the baseline (e.g., control units where their treatment status is unchanged across time periods), the comparison can be

Good for
Newly treated units vs. control
Newly treated units vs notyet treated

Bad for
 Newly treated vs. already treated (because already treated cannot serve as the potential outcome for the newly treated).
 Strict exogeneity (i.e., timevarying confounders, feedback from past outcome to treatment) (Imai and Kim 2019)
 Specific functional forms (i.e., treatment effect homogeneity and no carryover effects or anticipation effects) (Imai and Kim 2019)
Note: Notation for this section is consistent with (2020)
\[ Y_{it} = \alpha_i + \lambda_t + \tau W_{it} + \beta X_{it} + \epsilon_{it} \]
where
\(Y_{it}\) is the outcome
\(\alpha_i\) is the unit FE
\(\lambda_t\) is the time FE
\(\tau\) is the causal effect of treatment
\(W_{it}\) is the treatment indicator
\(X_{it}\) are covariates
When \(T = 2\), the TWFE is the traditional DiD model
Under the following assumption, \(\hat{\tau}_{OLS}\) is unbiased:
 homogeneous treatment effect
 parallel trends assumptions
 linear additive effects (Imai and Kim 2021)
Remedies for TWFE’s shortcomings
(GoodmanBacon 2021): diagnostic robustness tests of the TWFE DiD and identify influential observations to the DiD estimate (GoodmanBacon Decomposition)

(Callaway and Sant’Anna 2021): 2step estimation with a bootstrap procedure that can account for autocorrelation and clustering,
the parameters of interest are the grouptime average treatment effects, where each group is defined by when it was first treated (Multiple periods and variation in treatment timing)
Comparing posttreatment outcomes fo groups treated in a period against a similar group that is never treated (using matching).
Treatment status cannot switch (once treated, stay treated for the rest of the panel)
Package:
did

(L. Sun and Abraham 2021): a specialization of (Callaway and Sant’Anna 2021) in the eventstudy context.
They include lags and leads in their design
have cohortspecific estimates (similar to grouptime estimates in (Callaway and Sant’Anna 2021)
They propose the “interactionweighted” estimator.
Package:
fixest

Different from (Callaway and Sant’Anna 2021) because they allow units to switch in and out of treatment.
Based on matching methods, to have weighted TWFE
Package:
wfe
andPanelMatch

(Gardner 2022): twostage DiD
did2s
In cases with an unaffected unit (i.e., nevertreated), using the exposureadjusted differenceindifferences estimators can recover the average treatment effect (Clément De Chaisemartin and d’Haultfoeuille 2020). However, if you want to see the treatment effect heterogeneity (in cases where the true heterogeneous treatment effects vary by the exposure rate), exposureadjusted did still fails (L. Sun and Shapiro 2022).
(2020): see below
To be robust against
 time and unitvarying effects
We can use the reshaped inverse probability weighting (RIPW) TWFE estimator
With the following assumptions:
SUTVA
Binary treatment: \(\mathbf{W}_i = (W_{i1}, \dots, W_{it})\) where \(\mathbf{W}_i \sim \mathbf{\pi}_i\) generalized propensity score (i.e., each person treatment likelihood follow \(\pi\) regardless of the period)
Then, the unittime specific effect is \(\tau_{it} = Y_{it}(1)  Y_{it}(0)\)
Then the Doubly Average Treatment Effect (DATE) is
\[ \tau(\xi) = \sum_{T=1}^T \xi_t \left(\frac{1}{n} \sum_{i = 1}^n \tau_{it} \right) \]
where
\(\frac{1}{n} \sum_{i = 1}^n \tau_{it}\) is the unweighted effect of treatment across units (i.e., timespecific ATE).
\(\xi = (\xi_1, \dots, \xi_t)\) are userspecific weights for each time period.
This estimand is called DATE because it’s weighted (averaged) across both time and units.
A special case of DATE is when both time and unitweights are equal
\[ \tau_{eq} = \frac{1}{nT} \sum_{t=1}^T \sum_{i = 1}^n \tau_{it} \]
Borrowing the idea of inverse propensityweighted least squares estimator in the crosssectional case that we reweight the objective function via the treatment assignment mechanism:
\[ \hat{\tau} \triangleq \arg \min_{\tau} \sum_{i = 1}^n (Y_i \mu  W_i \tau)^2 \frac{1}{\pi_i (W_i)} \]
where
the first term is the least squares objective
the second term is the propensity score
In the panel data case, the IPW estimator will be
\[ \hat{\tau}_{IPW} \triangleq \arg \min_{\tau} \sum_{i = 1}^n \sum_{t =1}^T (Y_{i t}\alpha_i  \lambda_t  W_{it} \tau)^2 \frac{1}{\pi_i (W_i)} \]
Then, to have DATE that users can specify the structure of time weight, we use reshaped IPW estimator (2020)
\[ \hat{\tau}_{RIPW} (\Pi) \triangleq \arg \min_{\tau} \sum_{i = 1}^n \sum_{t =1}^T (Y_{i t}\alpha_i  \lambda_t  W_{it} \tau)^2 \frac{\Pi(W_i)}{\pi_i (W_i)} \]
where it’s a function of a dataindependent distribution \(\Pi\) that depends on the support of the treatment path \(\mathbb{S} = \cup_i Supp(W_i)\)
This generalization can transform to
IPWTWFE estimator when \(\Pi \sim Unif(\mathbb{S})\)
randomized experiment when \(\Pi = \pi_i\)
To choose \(\Pi\), we don’t need to data, we just need possible assignments in your setting.
For most practical problems (DiD, staggered, transient), we have closed form solutions
For generic solver, we can use nonlinear programming (e..g, BFGS algorithm)
As argued in (Imai and Kim 2021) that TWFE is not a nonparametric approach, it can be subjected to incorrect model assumption (i.e., model dependence).
Hence, they advocate for matching methods for timeseries crosssectional data (Imai and Kim 2021)
Use
wfe
andPanelMatch
to apply their paper.
This package is based on (Somaini and Wolak 2016)
# devtools::install_github("paulosomaini/xtreg2way")
library(xtreg2way)
# output < xtreg2way(y,
# data.frame(x1, x2),
# iid,
# tid,
# w,
# noise = "1",
# se = "1")
# equilvalently
output < xtreg2way(l_homicide ~ post,
df,
iid = df$state, # group id
tid = df$year, # time id
# w, # vector of weight
se = "1")
output$betaHat
#> [,1]
#> l_homicide 0.08181162
output$aVarHat
#> [,1]
#> [1,] 0.003396724
# to save time, you can use your structure in the
# last output for a new set of variables
# output2 < xtreg2way(y, x1, struc=output$struc)
Standard errors estimation options
Set  Estimation 

se = "0" 
Assume homoskedasticity and no within group correlation or serial correlation 
se = "1" (default) 
robust to heteroskadasticity and serial correlation (Arellano 1987) 
se = "2" 
robust to heteroskedasticity, but assumes no correlation within group or serial correlation 
se = "11" 
Aerllano SE with df correction performed by Stata xtreg (Somaini and Wolak 2021) 
Alternatively, you can also do it manually or with the plm
package, but you have to be careful with how the SEs are estimated
library(multiwayvcov) # get vcov matrix
library(lmtest) # robust SEs estimation
# manual
output3 < lm(l_homicide ~ post + factor(state) + factor(year),
data = df)
# get variancecovariance matrix
vcov_tw < multiwayvcov::cluster.vcov(output3,
cbind(df$state, df$year),
use_white = F,
df_correction = F)
# get coefficients
coeftest(output3, vcov_tw)[2,]
#> Estimate Std. Error t value Pr(>t)
#> 0.08181162 0.05671410 1.44252696 0.14979397
# using the plm package
library(plm)
output4 < plm(l_homicide ~ post,
data = df,
index = c("state", "year"),
model = "within",
effect = "twoways")
# get coefficients
coeftest(output4, vcov = vcovHC, type = "HC1")
#>
#> t test of coefficients:
#>
#> Estimate Std. Error t value Pr(>t)
#> post 0.081812 0.057748 1.4167 0.1572
As you can see, differences stem from SE estimation, not the coefficient estimate.
26.8 Multiple periods and variation in treatment timing
This is an extension of the DiD framework to settings where you have
more than 2 time periods
different treatment timing
When treatment effects are heterogeneous across time or units, the standard Twoway Fixedeffects is inappropriate.
Notation is consistent with did
package (Callaway and Sant’Anna 2021)
\(Y_{it}(0)\) is the potential outcome for unit \(i\)
\(Y_{it}(g)\) is the potential outcome for unit \(i\) in time period \(t\) if it’s treated in period \(g\)
\(Y_{it}\) is the observed outcome for unit \(i\) in time period \(t\)
\[ Y_{it} = \begin{cases} Y_{it} = Y_{it}(0) & \forall i \in \text{nevertreated group} \\ Y_{it} = 1\{G_i > t\} Y_{it}(0) + 1\{G_i \le t \}Y_{it}(G_i) & \forall i \in \text{other groups} \end{cases} \]
\(G_i\) is the time period when \(i\) is treated
\(C_i\) is a dummy when \(i\) belongs to the nevertreated group
\(D_{it}\) is a dummy for whether \(i\) is treated in period \(t\)
Assumptions:
Staggered treatment adoption: once treated, a unit cannot be untreated (revert)

Parallel trends assumptions (conditional on covariates):

Based on nevertreated units: \(E[Y_t(0) Y_{t1}(0)G= g] = E[Y_t(0)  Y_{t1}(0)C=1]\)
 Without treatment, the average potential outcomes for group \(g\) equals the average potential outcomes for the nevertreated group (i.e., control group), which means that we have (1) enough data on the nevertreated group (2) the control group is similar to the eventually treated group.

Based on notyet treated units: \(E[Y_t(0)  Y_{t1}(0)G = g] = E[Y_t(0)  Y_{t1}(0)D_s = 0, G \neq g]\)
Notyet treated units by time \(s\) ( \(s \ge t\)) can be used as comparison groups to calculate the average treatment effects for the group first treated in time \(g\)
Additional assumption: pretreatment trends across groups (Marcus and Sant’Anna 2021)

Random sampling
Irreversibility of treatment (once treated, cannot be untreated)
Overlap (the treatment propensity \(e \in [0,1]\))
GroupTime ATE
 This is the equivalent of the average treatment effect in the standard case (2 groups, 2 periods) under multiple time periods.
\[ ATT(g,t) = E[Y_t(g)  Y_t(0) G = g] \]
which is the average treatment effect for group \(g\) in period \(t\)

Identification: When the parallel trends assumption based on
Nevertreated units: \(ATT(g,t) = E[Y_t  Y_{g1} G = g]  E[Y_t  Y_{g1}C=1] \forall t \ge g\)
Notyettreated units: \(ATT(g,t) = E[Y_t  Y_{g1}G= g]  E[Y_t  Y_{g1}D_t = 0, G \neq g] \forall t \ge g\)

Identification: when the parallel trends assumption only holds conditional on covariates and based on
Nevertreated units: \(ATT(g,t) = E[Y_t  Y_{g1} X, G = g]  E[Y_t  Y_{g1}X, C=1] \forall t \ge g\)
Notyettreated units: \(ATT(g,t) = E[Y_t  Y_{g1}X, G= g]  E[Y_t  Y_{g1}X, D_t = 0, G \neq g] \forall t \ge g\)
This is plausible when you have suspected selection bias that can be corrected by using covariates (i.e., very much similar to matching methods to have plausible parallel trends).
Possible parameters of interest are:
 Average treatment effect per group
\[ \theta_S(g) = \frac{1}{\tau  g + 1} \sum_{t = 2}^\tau \mathbb{1} \{ \le t \} ATT(g,t) \]
 Average treatment effect across groups (that were treated) (similar to average treatment effect on the treated in the canonical case)
\[ \theta_S^O := \sum_{g=2}^\tau \theta_S(g) P(G=g) \]
 Average treatment effect dynamics (i.e., average treatment effect for groups that have been exposed to the treatment for \(e\) time periods):
\[ \theta_D(e) := \sum_{g=2}^\tau \mathbb{1} \{g + e \le \tau \}ATT(g,g + e) P(G = gG + e \le \tau) \]
 Average treatment effect in period \(t\) for all groups that have treated by period \(t\))
\[ \theta_C(t) = \sum_{g=2}^\tau \mathbb{1}\{g \le t\} ATT(g,t) P(G = gg \le t) \]
 Average treatment effect by calendar time
\[ \theta_C = \frac{1}{\tau1}\sum_{t=2}^\tau \theta_C(t) \]
26.9 Staggered Difndif
Recommendations by Baker, Larcker, and Wang (2022)
TWFE DiD regressions are suitable for single treatment periods or when treatment effects are homogeneous, provided there’s a solid rationale for effect homogeneity.
For TWFE staggered DiD, researchers should evaluate bias risks, plot treatment timings to check for variations, and use decompositions like GoodmanBacon (2021) when possible. If decompositions aren’t feasible (e.g., unbalanced panel), the percentage of nevertreated units can indicate bias severity. Expected treatment effect variability should also be discussed.
In TWFE staggered DiD event studies, avoid binning time periods without evidence of uniform effects. Use full relativetime indicators, justify reference periods, and be wary of multicollinearity causing bias.
To address treatment timing and bias concerns, use alternative estimators like stacked regressions, L. Sun and Abraham (2021), Callaway and Sant’Anna (2021), or separate regressions for each event with “clean” controls.
Justify the selection of comparison groups (notyet treated, last treated, never treated) and ensure the paralleltrends assumption holds, especially when anticipating no effects for certain groups.
Notes:
 When subjects are treated at different point in time (variation in treatment timing across units), we have to use staggered DiD (also known as DiD event study or dynamic DiD).
 For design where a treatment is applied and units are exposed to this treatment at all time afterward, see (Athey and Imbens 2022)
For example, basic design (Stevenson and Wolfers 2006)
\[ \begin{aligned} Y_{it} &= \sum_k \beta_k Treatment_{it}^k + \sum_i \eta_i State_i \\ &+ \sum_t \lambda_t Year_t + Controls_{it} + \epsilon_{it} \end{aligned} \]
where
\(Treatment_{it}^k\) is a series of dummy variables equal to 1 if state \(i\) is treated \(k\) years ago in period \(t\)
SE is usually clustered at the group level (occasionally time level).
To avoid collinearity, the period right before treatment is usually chosen to drop.
The more general form of TWFE (L. Sun and Abraham 2021):
First, define the relative period bin indicator as
\[ D_{it}^l = \mathbf{1}(t  E_i = l) \]
where it’s an indicator function of unit \(i\) being \(l\) periods from its first treatment at time \(t\)
 Static specificaiton
\[ Y_{it} = \alpha_i + \lambda_t + \mu_g \sum_{l \ge0} D_{it}^l + \epsilon_{it} \]
where
\(\alpha_i\) is the the unit FE
\(\lambda_t\) is the time FE
\(\mu_g\) is the coefficient of interest \(g = [0,T)\)
we exclude all periods before first adoption.
 Dynamic specification
\[ Y_{it} = \alpha_i + \lambda_t + \sum_{\substack{l = K \\ l \neq 1}}^{L} \mu_l D_{it}^l + \epsilon_{it} \]
where we have to exclude some relative peridos to avoid multicollinearity problem (e.g., either period right before treatment, or the treatment period).
In this setting, we try to show that the treatment and control groups are not statistically different (i.e., the coefficient estimates before treatment are not different from 0) to show pretreatment parallel trends.
However, this twoway fixed effects design has been criticized by L. Sun and Abraham (2021); Callaway and Sant’Anna (2021); GoodmanBacon (2021). When researchers include leads and lags of the treatment to see the longterm effects of the treatment, these leads and lags can be biased by effects from other periods, and pretrends can falsely arise due to treatment effects heterogeneity.
Applying the new proposed method, finance and accounting researchers find that in many cases, the causal estimates turn out to be null (Baker, Larcker, and Wang 2022).
Robustness Check
 The tripledifference strategy involves examining the interaction between the treatment variable and the probability of being affected by the program, and the grouplevel participation rate. The identification assumption is that there are no differential trends between high and low participation groups in early versus late implementing countries.
26.9.1 Assumptions

Rollout Exogeneity (i.e., exogeneity of treatment adoption): if the treatment is randomly implemented over time (i.e., unrelated to variables that could also affect our dependent variables)
 Evidence: Regress adoption on pretreatment variables. And if you find evidence of correlation, include linear trends interacted with pretreatment variables (Hoynes and Schanzenbach 2009)
 Evidence: (Deshpande and Li 2019, 223)
 Treatment is random: Regress treatment status at the unit level to all pretreatment observables. If you have some that are predictive of treatment status, you might have to argue why it’s not a worry. At best, you want this.
 Treatment timing is random: Conditional on treatment, regress timing of the treatment on pretreatment observables. At least, you want this.
No confounding events

Exclusion restrictions
Noanticipation assumption: future treatment time do not affect current outcomes
Invariancetohistory assumption: the time a unit under treatment does not affect the outcome (i.e., the time exposed does not matter, just whether exposed or not). This presents causal effect of early or late adoption on the outcome.
And all the assumptions in listed in the Multiple periods and variation in treatment timing

Auxiliary assumptions:
Constant treatment effects across units
Constant treatment effect over time
Random sampling
Effect Additivity
Remedies for staggered DiD (Baker, Larcker, and Wang 2022):

Each treated cohort is compared to appropriate controls (notyettreated, nevertreated)

(Callaway and Sant’Anna 2021) consistent for average ATT. more complicated but also more flexible than (L. Sun and Abraham 2021)
 (L. Sun and Abraham 2021) (a special case of (Callaway and Sant’Anna 2021))

Stacked DID (biased but simple):
26.9.2 Stacked DID
Notations following these slides
\[ Y_{it} = \beta_{FE} D_{it} + A_i + B_t + \epsilon_{it} \]
where
\(A_i\) is the group fixed effects
\(B_t\) is the period fixed effects
Steps
 Choose Event Window
 Enumerate Subexperiments
 Define Inclusion Criteria
 Stack Data
 Specify Estimating Equation
Event Window
Let
\(\kappa_a\) be the length of the preevent window
\(\kappa_b\) be the length of the postevent window
By setting a common event window for the analysis, we essentially exclude all those events that do not meet this criteria.
Subexperiments
Let \(T_1\) be the earliest period in the dataset
\(T_T\) be the last period in the dataset
Then, the collection of all policy adoption periods that are under our event window is
\[ \Omega_A = \{ A_i T_1 + \kappa_a \le A_i \le T_T  \kappa_b\} \]
where these events exist
at least \(\kappa_a\) periods after the earliest period
at least \(\kappa_b\) periods before the last period
Let \(d = 1, \dots, D\) be the index column of the subexperiments in \(\Omega_A\)
and \(\omega_d\) be the event date of the dth subexperiment (e.g., \(\omega_1\) = adoption date of the 1st experiment)
Inclusion Criteria
 Valid treated Units
Within subexperiment \(d\), all treated units have the same adoption date
This makes sure a unit can only serve as a treated unit in only 1 subexperiment
 Clean controls
Only units satisfying \(A_i >\omega_d + \kappa_b\) are included as controls in subexperiment d

This ensures controls are only
never treated units
units that are treated in far future
But a unit can be control unit in multiple subexperiments (need to correct SE)
 Valid Time Periods
All observations within subexperiment d are from time periods within the subexperiment’s event window
This ensures in subexperiment d, only observations satisfying \(\omega_d  \kappa_a \le t \le \omega_d + \kappa_b\) are included
library(did)
library(tidyverse)
library(fixest)
data(base_stagg)
# first make the stacked datasets
# get the treatment cohorts
cohorts < base_stagg %>%
select(year_treated) %>%
# exclude nevertreated group
filter(year_treated != 10000) %>%
unique() %>%
pull()
# make formula to create the subdatasets
getdata < function(j, window) {
#keep what we need
base_stagg %>%
# keep treated units and all units not treated within 5 to 5
# keep treated units and all units not treated within window to window
filter(year_treated == j  year_treated > j + window) %>%
# keep just year window to window
filter(year >= j  window & year <= j + window) %>%
# create an indicator for the dataset
mutate(df = j)
}
# get data stacked
stacked_data < map_df(cohorts, ~ getdata(., window = 5)) %>%
mutate(rel_year = if_else(df == year_treated, time_to_treatment, NA_real_)) %>%
fastDummies::dummy_cols("rel_year", ignore_na = TRUE) %>%
mutate(across(starts_with("rel_year_"), ~ replace_na(., 0)))
# get stacked value
stacked <
feols(
y ~ `rel_year_5` + `rel_year_4` + `rel_year_3` +
`rel_year_2` + rel_year_0 + rel_year_1 + rel_year_2 + rel_year_3 +
rel_year_4 + rel_year_5 
id ^ df + year ^ df,
data = stacked_data
)$coefficients
stacked_se = feols(
y ~ `rel_year_5` + `rel_year_4` + `rel_year_3` +
`rel_year_2` + rel_year_0 + rel_year_1 + rel_year_2 + rel_year_3 +
rel_year_4 + rel_year_5 
id ^ df + year ^ df,
data = stacked_data
)$se
# add in 0 for omitted 1
stacked < c(stacked[1:4], 0, stacked[5:10])
stacked_se < c(stacked_se[1:4], 0, stacked_se[5:10])
cs_out < att_gt(
yname = "y",
data = base_stagg,
gname = "year_treated",
idname = "id",
# xformla = "~x1",
tname = "year"
)
cs <
aggte(
cs_out,
type = "dynamic",
min_e = 5,
max_e = 5,
bstrap = FALSE,
cband = FALSE
)
res_sa20 = feols(y ~ sunab(year_treated, year) 
id + year, base_stagg)
sa = tidy(res_sa20)[5:14, ] %>% pull(estimate)
sa = c(sa[1:4], 0, sa[5:10])
sa_se = tidy(res_sa20)[6:15, ] %>% pull(std.error)
sa_se = c(sa_se[1:4], 0, sa_se[5:10])
compare_df_est = data.frame(
period = 5:5,
cs = cs$att.egt,
sa = sa,
stacked = stacked
)
compare_df_se = data.frame(
period = 5:5,
cs = cs$se.egt,
sa = sa_se,
stacked = stacked_se
)
compare_df_longer < compare_df_est %>%
pivot_longer(!period, names_to = "estimator", values_to = "est") %>%
full_join(compare_df_se %>%
pivot_longer(!period, names_to = "estimator", values_to = "se")) %>%
mutate(upper = est + 1.96 * se,
lower = est  1.96 * se)
ggplot(compare_df_longer) +
geom_ribbon(aes(
x = period,
ymin = lower,
ymax = upper,
group = estimator
)) +
geom_line(aes(
x = period,
y = est,
group = estimator,
col = estimator
),
linewidth = 1) +
causalverse::ama_theme()
Stack Data
Estimating Equation
\[ Y_{itd} = \beta_0 + \beta_1 T_{id} + \beta_2 P_{td} + \beta_3 (T_{id} \times P_{td}) + \epsilon_{itd} \]
where
\(T_{id}\) = 1 if unit \(i\) is treated in subexperiment \(d\), 0 if control
\(P_{td}\) = 1 if it’s the period after the treatment in subexperiment \(d\)
Equivalently,
\[ Y_{itd} = \beta_3 (T_{id} \times P_{td}) + \theta_{id} + \gamma_{td} + \epsilon_{itd} \]
\(\beta_3\) averages all the timevarying effects into a single number (can’t see the timevarying effects)
Stacked Event Study
Let \(YSE_{td} = t  \omega_d\) be the “time since event” variable in subexperiment \(d\)
Then, \(YSE_{td} = \kappa_a, \dots, 0, \dots, \kappa_b\) in every subexperiment
In each subexperiment, we can fit
\[ Y_{it}^d = \sum_{j = \kappa_a}^{\kappa_b} \beta_j^d \times 1(TSE_{td} = j) + \sum_{m = \kappa_a}^{\kappa_b} \delta_j^d (T_{id} \times 1 (TSE_{td} = j)) + \theta_i^d + \epsilon_{it}^d \]
 Different set of event study coefficients in each subexperiment
\[ Y_{itd} = \sum_{j = \kappa_a}^{\kappa_b} \beta_j \times 1(TSE_{td} = j) + \sum_{m = \kappa_a}^{\kappa_b} \delta_j (T_{id} \times 1 (TSE_{td} = j)) + \theta_{id} + \epsilon_{itd} \]
Clustering
Clustered at the unit x subexperiment level (Cengiz et al. 2019)
Clustered at the unit level (Deshpande and Li 2019)
26.9.3 GoodmanBacon Decomposition
Paper: (GoodmanBacon 2021)
For an excellent explanation slides by the author, see
Takeaways:
A pairwise DID (\(\tau\)) gets more weight if the change is close to the middle of the study window
A pairwise DID (\(\tau\)) gets more weight if it includes more observations.
Code from bacondecomp
vignette
library(bacondecomp)
library(tidyverse)
data("castle")
castle < castle %>%
select(l_homicide, post, state, year)
head(castle)
#> l_homicide post state year
#> 1 2.027356 0 Alabama 2000
#> 2 2.164867 0 Alabama 2001
#> 3 1.936334 0 Alabama 2002
#> 4 1.919567 0 Alabama 2003
#> 5 1.749841 0 Alabama 2004
#> 6 2.130440 0 Alabama 2005
df_bacon < bacon(
l_homicide ~ post,
data = castle,
id_var = "state",
time_var = "year"
)
#> type weight avg_est
#> 1 Earlier vs Later Treated 0.05976 0.00554
#> 2 Later vs Earlier Treated 0.03190 0.07032
#> 3 Treated vs Untreated 0.90834 0.08796
# weighted average of the decomposition
sum(df_bacon$estimate * df_bacon$weight)
#> [1] 0.08181162
Twoway Fixed effect estimate
library(broom)
fit_tw < lm(l_homicide ~ post + factor(state) + factor(year),
data = bacondecomp::castle)
head(tidy(fit_tw))
#> # A tibble: 6 × 5
#> term estimate std.error statistic p.value
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 (Intercept) 1.95 0.0624 31.2 2.84e118
#> 2 post 0.0818 0.0317 2.58 1.02e 2
#> 3 factor(state)Alaska 0.373 0.0797 4.68 3.77e 6
#> 4 factor(state)Arizona 0.0158 0.0797 0.198 8.43e 1
#> 5 factor(state)Arkansas 0.118 0.0810 1.46 1.44e 1
#> 6 factor(state)California 0.108 0.0810 1.34 1.82e 1
Hence, naive TWFE fixed effect equals the weighted average of the Bacon decomposition (= 0.08).
library(ggplot2)
ggplot(df_bacon) +
aes(
x = weight,
y = estimate,
# shape = factor(type),
color = type
) +
labs(x = "Weight", y = "Estimate", shape = "Type") +
geom_point() +
causalverse::ama_theme()
With timevarying controls that can identify variation withintreatment timing group, the”early vs. late” and “late vs. early” estimates collapse to just one estimate (i.e., both treated).
26.9.4 DID with in and out treatment condition
26.9.4.1 Panel Match
Imai and Kim (2021)
This case generalizes the staggered adoption setting, allowing units to vary in treatment over time. For \(N\) units across \(T\) time periods (with potentially unbalanced panels), let \(X_{it}\) represent treatment and \(Y_{it}\) the outcome for unit \(i\) at time \(t\). We use the twoway linear fixed effects model:
\[ Y_{it} = \alpha_i + \gamma_t + \beta X_{it} + \epsilon_{it} \]
for \(i = 1, \dots, N\) and \(t = 1, \dots, T\). Here, \(\alpha_i\) and \(\gamma_t\) are unit and time fixed effects. They capture timeinvariant unitspecific and unitinvariant timespecific unobserved confounders, respectively. We can express these as \(\alpha_i = h(\mathbf{U}_i)\) and \(\gamma_t = f(\mathbf{V}_t)\), with \(\mathbf{U}_i\) and \(\mathbf{V}_t\) being the confounders. The model doesn’t assume a specific form for \(h(.)\) and \(f(.)\), but that they’re additive and separable given binary treatment.
The least squares estimate of \(\beta\) leverages the covariance in outcome and treatment (Imai and Kim 2021, 406). Specifically, it uses the withinunit and withintime variations. Many researchers prefer the two fixed effects (2FE) estimator because it adjusts for both types of unobserved confounders without specific functionalform assumptions, but this is wrong (Imai and Kim 2019). We do need functionalform assumption (i.e., linearity assumption) for the 2FE to work (Imai and Kim 2021, 406)

TwoWay Matching Estimator:
It can lead to mismatches; units with the same treatment status get matched when estimating counterfactual outcomes.
Observations need to be matched with opposite treatment status for correct causal effects estimation.
Mismatches can cause attenuation bias.
The 2FE estimator adjusts for this bias using the factor \(K\), which represents the net proportion of proper matches between observations with opposite treatment status.

Weighting in 2FE:
Observation \((i,t)\) is weighted based on how often it acts as a control unit.
The weighted 2FE estimator still has mismatches, but fewer than the standard 2FE estimator.
Adjustments are made based on observations that neither belong to the same unit nor the same time period as the matched observation.
This means there are challenges in adjusting for unitspecific and timespecific unobserved confounders under the twoway fixed effect framework.

Equivalence & Assumptions:
Equivalence between the 2FE estimator and the DID estimator is dependent on the linearity assumption.
The multiperiod DiD estimator is described as an average of twotimeperiod, twogroup DiD estimators applied during changes from control to treatment.

Comparison with DiD:
In simple settings (two time periods, treatment given to one group in the second period), the standard nonparametric DiD estimator equals the 2FE estimator.
This doesn’t hold in multiperiod DiD designs where units change treatment status multiple times at different intervals.
Contrary to popular belief, the unweighted 2FE estimator isn’t generally equivalent to the multiperiod DiD estimator.
While the multiperiod DiD can be equivalent to the weighted 2FE, some control observations may have negative regression weights.

Conclusion:
 Justifying the 2FE estimator as the DID estimator isn’t warranted without imposing the linearity assumption.
Application (Imai, Kim, and Wang 2021)

Matching Methods:
Enhance the validity of causal inference.
Reduce model dependence and provide intuitive diagnostics (Ho et al. 2007)
Rarely utilized in analyzing time series crosssectional data.
The proposed matching estimators are more robust than the standard twoway fixed effects estimator, which can be biased if misspecified
Better than synthetic controls (e.g., (Xu 2017)) because it needs less data to achieve good performance and and adapt the the context of unit switching treatment status multiple times.

Notes:
 Potential carryover effects (treatment may have a longterm effect), leading to posttreatment bias.

Proposed Approach:
Treated observations are matched with control observations from other units in the same time period with the same treatment history up to a specified number of lags.
Standard matching and weighting techniques are employed to further refine the matched set.
Apply a DiD estimator to adjust for time trend.
The goal is to have treated and matched control observations with similar covariate values.

Assessment:
 The quality of matches is evaluated through covariate balancing.

Estimation:
 Both shortterm and longterm average treatment effects on the treated (ATT) are estimated.
library(PanelMatch)
Treatment Variation plot
Visualize the variation of the treatment across space and time
Aids in discerning whether the treatment fluctuates adequately over time and units or if the variation is primarily clustered in a subset of data.
DisplayTreatment(
unit.id = "wbcode2",
time.id = "year",
legend.position = "none",
xlab = "year",
ylab = "Country Code",
treatment = "dem",
hide.x.tick.label = TRUE, hide.y.tick.label = TRUE,
# dense.plot = TRUE,
data = dem
)
 Select \(F\) (i.e., the number of leads  time periods after treatment). Driven by what authors are interested in estimating:
\(F = 0\) is the contemporaneous effect (shortterm effect)
\(F = n\) is the the treatment effect on the outcome two time periods after the treatment. (cumulative or longterm effect)
 Select \(L\) (number of lags to adjust).
Driven by the identification assumption.
Balances biasvariance tradeoff.
Higher \(L\) values increase credibility but reduce efficiency by limiting potential matches.
Model assumption:
No spillover effect assumed.
Carryover effect allowed up to \(L\) periods.
Potential outcome for a unit depends neither on others’ treatment status nor on its past treatment after \(L\) periods.
After defining causal quantity with parameters \(L\) and \(F\).
 Focus on the average treatment effect of treatment status change.
 \(\delta(F,L)\) is the average causal effect of treatment change (ATT), \(F\) periods posttreatment, considering treatment history up to \(L\) periods.
 Causal quantity considers potential future treatment reversals, meaning treatment could revert to control before outcome measurement.
Also possible to estimate the average treatment effect of treatment reversal on the reversed (ART).
Choose \(L,F\) based on specific needs.

A large \(L\) value:
Increases the credibility of the limited carryover effect assumption.
Allows more past treatments (up to \(t−L\)) to influence the outcome \(Y_{i,t+F}\).
Might reduce the number of matches and lead to less precise estimates.

Selecting an appropriate number of lags
Researchers should base this choice on substantive knowledge.
Sensitivity of empirical results to this choice should be examined.

The choice of \(F\) should be:
Substantively motivated.
Decides whether the interest lies in shortterm or longterm causal effects.
A large \(F\) value can complicate causal effect interpretation, especially if many units switch treatment status during the \(F\) lead time period.
Identification Assumption
Parallel trend assumption conditioned on treatment, outcome (excluding immediate lag), and covariate histories.
Doesn’t require strong unconfoundedness assumption.
Cannot account for unobserved timevarying confounders.

Essential to examine outcome time trends.
 Check if they’re parallel between treated and matched control units using pretreatment data

Constructing the Matched Sets:
For each treated observation, create matched control units with identical treatment history from \(t−L\) to \(t−1\).
Matching based on treatment history helps control for carryover effects.
Past treatments often act as major confounders, but this method can correct for it.
Exact matching on time period adjusts for timespecific unobserved confounders.
Unlike staggered adoption methods, units can change treatment status multiple times.
Matched set allows treatment switching in and out of treatment

Refining the Matched Sets:
Initially, matched sets adjust only for treatment history.
Parallel trend assumption requires adjustments for other confounders like past outcomes and covariates.

Matching methods:
Match each treated observation with up to \(J\) control units.
Distance measures like Mahalanobis distance or propensity score can be used.
Match based on estimated propensity score, considering pretreatment covariates.
Refined matched set selects most similar control units based on observed confounders.

Weighting methods:
Assign weight to each control unit in a matched set.
Weights prioritize more similar units.
Inverse propensity score weighting method can be applied.
Weighting is a more generalized method than matching.
The DifferenceinDifferences Estimator:
Using refined matched sets, the ATT (Average Treatment Effect on the Treated) of policy change is estimated.
For each treated observation, estimate the counterfactual outcome using the weighted average of control units in the refined set.
The DiD estimate of the ATT is computed for each treated observation, then averaged across all such observations.

For noncontemporaneous treatment effects where \(F > 0\):
The ATT doesn’t specify future treatment sequence.
Matched control units might have units receiving treatment between time \(t\) and \(t + F\).
Some treated units could return to control conditions between these times.
Checking Covariate Balance:
The proposed methodology offers the advantage of checking covariate balance between treated and matched control observations.
This check helps to see if treated and matched control observations are comparable with respect to observed confounders.
Once matched sets are refined, covariate balance examination becomes straightforward.
Examine the mean difference of each covariate between a treated observation and its matched controls for each pretreatment time period.
Standardize this difference using the standard deviation of each covariate across all treated observations in the dataset.
Aggregate this covariate balance measure across all treated observations for each covariate and pretreatment time period.

Examine balance for lagged outcome variables over multiple pretreatment periods and timevarying covariates.
 This helps evaluate the validity of the parallel trend assumption underlying the proposed DiD estimator.
Relations with Linear Fixed Effects Regression Estimators:

The standard DiD estimator is equivalent to the linear twoway fixed effects regression estimator when:
Only two time periods exist.
Treatment is given to some units exclusively in the second period.

This equivalence doesn’t extend to multiperiod DiD designs, where:
More than two time periods are considered.
Units might receive treatment multiple times.
Despite this, many researchers relate the use of the twoway fixed effects estimator to the DiD design.
Standard Error Calculation:

Approach:
Condition on the weights implied by the matching process.
These weights denote how often an observation is utilized in matching (G. W. Imbens and Rubin 2015)

Context:
Analogous to the conditional variance seen in regression models.
Resulting standard errors don’t factor in uncertainties around the matching procedure.
They can be viewed as a measure of uncertainty conditional upon the matching process (Ho et al. 2007).
Key Findings:
Even in conditions favoring OLS, the proposed matching estimator displayed higher robustness to omitted relevant lags than the linear regression model with fixed effects.
The robustness offered by matching came at a cost  reduced statistical power.
This emphasizes the classic statistical tradeoff between bias (where matching has an advantage) and variance (where regression models might be more efficient).
Data Requirements

The treatment variable is binary:
0 signifies “assignment” to control.
1 signifies assignment to treatment.
Variables identifying units in the data must be: Numeric or integer.
Variables identifying time periods should be: Consecutive numeric/integer data.
Data format requirement: Must be provided as a standard
data.frame
object.
Basic functions:
Utilize treatment histories to create matching sets of treated and control units.

Refine these matched sets by determining weights for each control unit in the set.
 Units with higher weights have a larger influence during estimations.
Matching on Treatment History:
Goal is to match units transitioning from untreated to treated status with control units that have similar past treatment histories.

Setting the Quantity of Interest (
qoi =
)att
average treatment effect on treated unitsatc
average treatment effect of treatment on the control unitsart
average effect of treatment reversal for units that experience treatment reversalate
average treatment effect
library(PanelMatch)
# All examples follow the package's vignette
# Create the matched sets
PM.results.none <
PanelMatch(
lag = 4,
time.id = "year",
unit.id = "wbcode2",
treatment = "dem",
refinement.method = "none",
data = dem,
match.missing = TRUE,
size.match = 5,
qoi = "att",
outcome.var = "y",
lead = 0:4,
forbid.treatment.reversal = FALSE,
use.diagonal.variance.matrix = TRUE
)
# visualize the treated unit and matched controls
DisplayTreatment(
unit.id = "wbcode2",
time.id = "year",
legend.position = "none",
xlab = "year",
ylab = "Country Code",
treatment = "dem",
data = dem,
matched.set = PM.results.none$att[1],
# highlight the particular set
show.set.only = TRUE
)
Control units and the treated unit have identical treatment histories over the lag window (19881991)
DisplayTreatment(
unit.id = "wbcode2",
time.id = "year",
legend.position = "none",
xlab = "year",
ylab = "Country Code",
treatment = "dem",
data = dem,
matched.set = PM.results.none$att[2],
# highlight the particular set
show.set.only = TRUE
)
This set is more limited than the first one, but we can still see that we have exact past histories.

Refining Matched Sets
Refinement involves assigning weights to control units.

Users must:
Specify a method for calculating unit similarity/distance.
Choose variables for similarity/distance calculations.

Select a Refinement Method
Users determine the refinement method via the
refinement.method
argument.
Options include:
mahalanobis
ps.match
CBPS.match
ps.weight
CBPS.weight
ps.msm.weight
CBPS.msm.weight
none
Methods with “match” in the name and Mahalanobis will assign equal weights to similar control units.
“Weighting” methods give higher weights to control units more similar to treated units.

Variable Selection
Users need to define which covariates will be used through the
covs.formula
argument, a onesided formula object.Variables on the right side of the formula are used for calculations.
“Lagged” versions of variables can be included using the format:
I(lag(name.of.var, 0:n))
.

Understanding
PanelMatch
andmatched.set
objectsThe
PanelMatch
function returns aPanelMatch
object.The most crucial element within the
PanelMatch
object is the matched.set object.Within the
PanelMatch
object, the matched.set object will have names like att, art, or atc.If
qoi = ate
, there will be two matched.set objects: att and atc.

Matched.set Object Details
matched.set is a named list with added attributes.

Attributes include:
Lag
Names of treatment
Unit and time variables
Each list entry represents a matched set of treated and control units.
Naming follows a structure:
[id variable].[time variable]
.Each list element is a vector of control unit ids that match the treated unit mentioned in the element name.
Since it’s a matching method, weights are only given to the
size.match
most similar control units based on distance calculations.
# PanelMatch without any refinement
PM.results.none <
PanelMatch(
lag = 4,
time.id = "year",
unit.id = "wbcode2",
treatment = "dem",
refinement.method = "none",
data = dem,
match.missing = TRUE,
size.match = 5,
qoi = "att",
outcome.var = "y",
lead = 0:4,
forbid.treatment.reversal = FALSE,
use.diagonal.variance.matrix = TRUE
)
# Extract the matched.set object
msets.none < PM.results.none$att
# PanelMatch with refinement
PM.results.maha <
PanelMatch(
lag = 4,
time.id = "year",
unit.id = "wbcode2",
treatment = "dem",
refinement.method = "mahalanobis", # use Mahalanobis distance
data = dem,
match.missing = TRUE,
covs.formula = ~ tradewb,
size.match = 5,
qoi = "att" ,
outcome.var = "y",
lead = 0:4,
forbid.treatment.reversal = FALSE,
use.diagonal.variance.matrix = TRUE
)
msets.maha < PM.results.maha$att
# these 2 should be identical because weights are not shown
msets.none > head()
#> wbcode2 year matched.set.size
#> 1 4 1992 74
#> 2 4 1997 2
#> 3 6 1973 63
#> 4 6 1983 73
#> 5 7 1991 81
#> 6 7 1998 1
msets.maha > head()
#> wbcode2 year matched.set.size
#> 1 4 1992 74
#> 2 4 1997 2
#> 3 6 1973 63
#> 4 6 1983 73
#> 5 7 1991 81
#> 6 7 1998 1
# summary(msets.none)
# summary(msets.maha)
Visualizing Matched Sets with the plot method
Users can visualize the distribution of the matched set sizes.
A red line, by default, indicates the count of matched sets where treated units had no matching control units (i.e., empty matched sets).
Plot adjustments can be made using
graphics::plot
.
plot(msets.none)
Comparing Methods of Refinement

Users are encouraged to:
Use substantive knowledge for experimentation and evaluation.

Consider the following when configuring
PanelMatch
:The number of matched sets.
The number of controls matched to each treated unit.
Achieving covariate balance.
Note: Large numbers of small matched sets can lead to larger standard errors during the estimation stage.
Covariates that aren’t well balanced can lead to undesirable comparisons between treated and control units.

Aspects to consider include:
Refinement method.
Variables for weight calculation.
Size of the lag window.
Procedures for addressing missing data (refer to
match.missing
andlistwise.delete
arguments).Maximum size of matched sets (for matching methods).

Supportive Features:
print
,plot
, andsummary
methods assist in understanding matched sets and their sizes.
get_covariate_balance
helps evaluate covariate balance: Lower values in the covariate balance calculations are preferred.
PM.results.none <
PanelMatch(
lag = 4,
time.id = "year",
unit.id = "wbcode2",
treatment = "dem",
refinement.method = "none",
data = dem,
match.missing = TRUE,
size.match = 5,
qoi = "att",
outcome.var = "y",
lead = 0:4,
forbid.treatment.reversal = FALSE,
use.diagonal.variance.matrix = TRUE
)
PM.results.maha <
PanelMatch(
lag = 4,
time.id = "year",
unit.id = "wbcode2",
treatment = "dem",
refinement.method = "mahalanobis",
data = dem,
match.missing = TRUE,
covs.formula = ~ I(lag(tradewb, 1:4)) + I(lag(y, 1:4)),
size.match = 5,
qoi = "att",
outcome.var = "y",
lead = 0:4,
forbid.treatment.reversal = FALSE,
use.diagonal.variance.matrix = TRUE
)
# listwise deletion used for missing data
PM.results.listwise <
PanelMatch(
lag = 4,
time.id = "year",
unit.id = "wbcode2",
treatment = "dem",
refinement.method = "mahalanobis",
data = dem,
match.missing = FALSE,
listwise.delete = TRUE,
covs.formula = ~ I(lag(tradewb, 1:4)) + I(lag(y, 1:4)),
size.match = 5,
qoi = "att",
outcome.var = "y",
lead = 0:4,
forbid.treatment.reversal = FALSE,
use.diagonal.variance.matrix = TRUE
)
# propensity score based weighting method
PM.results.ps.weight <
PanelMatch(
lag = 4,
time.id = "year",
unit.id = "wbcode2",
treatment = "dem",
refinement.method = "ps.weight",
data = dem,
match.missing = FALSE,
listwise.delete = TRUE,
covs.formula = ~ I(lag(tradewb, 1:4)) + I(lag(y, 1:4)),
size.match = 5,
qoi = "att",
outcome.var = "y",
lead = 0:4,
forbid.treatment.reversal = FALSE
)
get_covariate_balance(
PM.results.none$att,
data = dem,
covariates = c("tradewb", "y"),
plot = FALSE
)
#> tradewb y
#> t_4 0.07245466 0.291871990
#> t_3 0.20930129 0.208654876
#> t_2 0.24425207 0.107736647
#> t_1 0.10806125 0.004950238
get_covariate_balance(
PM.results.maha$att,
data = dem,
covariates = c("tradewb", "y"),
plot = FALSE
)
#> tradewb y
#> t_4 0.04558637 0.09701606
#> t_3 0.03312750 0.10844046
#> t_2 0.01396793 0.08890753
#> t_1 0.10474894 0.06618865
get_covariate_balance(
PM.results.listwise$att,
data = dem,
covariates = c("tradewb", "y"),
plot = FALSE
)
#> tradewb y
#> t_4 0.05634922 0.05223623
#> t_3 0.01104797 0.05217896
#> t_2 0.01411473 0.03094133
#> t_1 0.06850180 0.02092209
get_covariate_balance(
PM.results.ps.weight$att,
data = dem,
covariates = c("tradewb", "y"),
plot = FALSE
)
#> tradewb y
#> t_4 0.014362590 0.04035905
#> t_3 0.005529734 0.04188731
#> t_2 0.009410044 0.04195008
#> t_1 0.027907540 0.03975173
get_covariate_balance Function Options:
Allows for the generation of plots displaying covariate balance using
plot = TRUE
.Plots can be customized using arguments typically used with the base R
plot
method.
Option to set
use.equal.weights = TRUE
for:Obtaining the balance of unrefined sets.
Facilitating understanding of the refinement’s impact.
# Use equal weights
get_covariate_balance(
PM.results.ps.weight$att,
data = dem,
use.equal.weights = TRUE,
covariates = c("tradewb", "y"),
plot = TRUE,
# visualize by setting plot to TRUE
ylim = c(1, 1)
)
# Compare covariate balance to refined sets
# See large improvement in balance
get_covariate_balance(
PM.results.ps.weight$att,
data = dem,
covariates = c("tradewb", "y"),
plot = TRUE,
# visualize by setting plot to TRUE
ylim = c(1, 1)
)
balance_scatter(
matched_set_list = list(PM.results.maha$att,
PM.results.ps.weight$att),
data = dem,
covariates = c("y", "tradewb")
)
PanelEstimate

Standard Error Calculation Methods

There are different methods available:
Bootstrap (default method with 1000 iterations).
Conditional: Assumes independence across units, but not time.
Unconditional: Doesn’t make assumptions of independence across units or time.

For
qoi
values set toatt
,art
, oratc
(Imai, Kim, and Wang 2021): You can use analytical methods for calculating standard errors, which include both “conditional” and “unconditional” methods.

PE.results < PanelEstimate(
sets = PM.results.ps.weight,
data = dem,
se.method = "bootstrap",
number.iterations = 1000,
confidence.level = .95
)
# point estimates
PE.results[["estimates"]]
#> t+0 t+1 t+2 t+3 t+4
#> 0.2609565 0.9630847 1.2851017 1.7370930 1.4871846
# standard errors
PE.results[["standard.error"]]
#> t+0 t+1 t+2 t+3 t+4
#> 0.6089145 1.0465362 1.4004193 1.7646397 2.2089610
# use conditional method
PE.results < PanelEstimate(
sets = PM.results.ps.weight,
data = dem,
se.method = "conditional",
confidence.level = .95
)
# point estimates
PE.results[["estimates"]]
#> t+0 t+1 t+2 t+3 t+4
#> 0.2609565 0.9630847 1.2851017 1.7370930 1.4871846
# standard errors
PE.results[["standard.error"]]
#> t+0 t+1 t+2 t+3 t+4
#> 0.4844805 0.8170604 1.1171942 1.4116879 1.7172143
summary(PE.results)
#> Weighted DifferenceinDifferences with Propensity Score
#> Matches created with 4 lags
#>
#> Standard errors computed with conditional method
#>
#> Estimate of Average Treatment Effect on the Treated (ATT) by Period:
#> $summary
#> estimate std.error 2.5% 97.5%
#> t+0 0.2609565 0.4844805 0.6886078 1.210521
#> t+1 0.9630847 0.8170604 0.6383243 2.564494
#> t+2 1.2851017 1.1171942 0.9045586 3.474762
#> t+3 1.7370930 1.4116879 1.0297644 4.503950
#> t+4 1.4871846 1.7172143 1.8784937 4.852863
#>
#> $lag
#> [1] 4
#>
#> $qoi
#> [1] "att"
plot(PE.results)
Moderating Variables
# moderating variable
dem$moderator < 0
dem$moderator < ifelse(dem$wbcode2 > 100, 1, 2)
PM.results <
PanelMatch(
lag = 4,
time.id = "year",
unit.id = "wbcode2",
treatment = "dem",
refinement.method = "mahalanobis",
data = dem,
match.missing = TRUE,
covs.formula = ~ I(lag(tradewb, 1:4)) + I(lag(y, 1:4)),
size.match = 5,
qoi = "att",
outcome.var = "y",
lead = 0:4,
forbid.treatment.reversal = FALSE,
use.diagonal.variance.matrix = TRUE
)
PE.results <
PanelEstimate(sets = PM.results,
data = dem,
moderator = "moderator")
# Each element in the list corresponds to a level in the moderator
plot(PE.results[[1]])
plot(PE.results[[2]])
To write up for journal submission, you can follow the following report:
In this study, closely aligned with the research by (Acemoglu et al. 2019), two key effects of democracy on economic growth are estimated: the impact of democratization and that of authoritarian reversal. The treatment variable, \(X_{it}\), is defined to be one if country \(i\) is democratic in year \(t\), and zero otherwise.
The Average Treatment Effect for the Treated (ATT) under democratization is formulated as follows:
\[ \begin{aligned} \delta(F, L) &= \mathbb{E} \left\{ Y_{i, t + F} (X_{it} = 1, X_{i, t  1} = 0, \{X_{i,tl}\}_{l=2}^L) \right. \\ &\left.  Y_{i, t + F} (X_{it} = 0, X_{i, t  1} = 0, \{X_{i,tl}\}_{l=2}^L)  X_{it} = 1, X_{i, t  1} = 0 \right\} \end{aligned} \]
In this framework, the treated observations are countries that transition from an authoritarian regime \(X_{it1} = 0\) to a democratic one \(X_{it} = 1\). The variable \(F\) represents the number of leads, denoting the time periods following the treatment, and \(L\) signifies the number of lags, indicating the time periods preceding the treatment.
The ATT under authoritarian reversal is given by:
\[ \begin{aligned} &\mathbb{E} \left[ Y_{i, t + F} (X_{it} = 0, X_{i, t  1} = 1, \{ X_{i, t  l}\}_{l=2}^L ) \right. \\ &\left.  Y_{i, t + F} (X_{it} = 1, X_{it1} = 1, \{X_{i, t  l} \}_{l=2}^L )  X_{it} = 0, X_{i, t  1} = 1 \right] \end{aligned} \]
The ATT is calculated conditioning on 4 years of lags (\(L = 4\)) and up to 4 years following the policy change \(F = 1, 2, 3, 4\). Matched sets for each treated observation are constructed based on its treatment history, with the number of matched control units generally decreasing when considering a 4year treatment history as compared to a 1year history.
To enhance the quality of matched sets, methods such as Mahalanobis distance matching, propensity score matching, and propensity score weighting are utilized. These approaches enable us to evaluate the effectiveness of each refinement method. In the process of matching, we employ both uptofive and uptoten matching to investigate how sensitive our empirical results are to the maximum number of allowed matches. For more information on the refinement process, please see the Web Appendix
The Mahalanobis distance is expressed through a specific formula. We aim to pair each treated unit with a maximum of \(J\) control units, permitting replacement, denoted as \( \mathcal{M}_{it} \le J\). The average Mahalanobis distance between a treated and each control unit over time is computed as:
\[ S_{it} (i') = \frac{1}{L} \sum_{l = 1}^L \sqrt{(\mathbf{V}_{i, t  l}  \mathbf{V}_{i', t l})^T \mathbf{\Sigma}_{i, t  l}^{1} (\mathbf{V}_{i, t  l}  \mathbf{V}_{i', t l})} \]
For a matched control unit \(i' \in \mathcal{M}_{it}\), \(\mathbf{V}_{it'}\) represents the timevarying covariates to adjust for, and \(\mathbf{\Sigma}_{it'}\) is the sample covariance matrix for \(\mathbf{V}_{it'}\). Essentially, we calculate a standardized distance using timevarying covariates and average this across different time intervals.
In the context of propensity score matching, we employ a logistic regression model with balanced covariates to derive the propensity score. Defined as the conditional likelihood of treatment given pretreatment covariates (Rosenbaum and Rubin 1983), the propensity score is estimated by first creating a data subset comprised of all treated and their matched control units from the same year. This logistic regression model is then fitted as follows:
\[ \begin{aligned} & e_{it} (\{\mathbf{U}_{i, t  l} \}^L_{l = 1}) \\ &= Pr(X_{it} = 1 \mathbf{U}_{i, t 1}, \ldots, \mathbf{U}_{i, t  L}) \\ &= \frac{1}{1 = \exp( \sum_{l = 1}^L \beta_l^T \mathbf{U}_{i, t  l})} \end{aligned} \]
where \(\mathbf{U}_{it'} = (X_{it'}, \mathbf{V}_{it'}^T)^T\). Given this model, the estimated propensity score for all treated and matched control units is then computed. This enables the adjustment for lagged covariates via matching on the calculated propensity score, resulting in the following distance measure:
\[ S_{it} (i') =  \text{logit} \{ \hat{e}_{it} (\{ \mathbf{U}_{i, t  l}\}^L_{l = 1})\}  \text{logit} \{ \hat{e}_{i't}( \{ \mathbf{U}_{i', t  l} \}^L_{l = 1})\}  \]
Here, \(\hat{e}_{i't} (\{ \mathbf{U}_{i, t  l}\}^L_{l = 1})\) represents the estimated propensity score for each matched control unit \(i' \in \mathcal{M}_{it}\).
Once the distance measure \(S_{it} (i')\) has been determined for all control units in the original matched set, we finetune this set by selecting up to \(J\) closest control units, which meet a researcherdefined caliper constraint \(C\). All other control units receive zero weight. This results in a refined matched set for each treated unit \((i, t)\):
\[ \mathcal{M}_{it}^* = \{i' : i' \in \mathcal{M}_{it}, S_{it} (i') < C, S_{it} \le S_{it}^{(J)}\} \]
\(S_{it}^{(J)}\) is the \(J\)th smallest distance among the control units in the original set \(\mathcal{M}_{it}\).
For further refinement using weighting, a weight is assigned to each control unit \(i'\) in a matched set corresponding to a treated unit \((i, t)\), with greater weight accorded to more similar units. We utilize inverse propensity score weighting, based on the propensity score model mentioned earlier:
\[ w_{it}^{i'} \propto \frac{\hat{e}_{i't} (\{ \mathbf{U}_{i, tl} \}^L_{l = 1} )}{1  \hat{e}_{i't} (\{ \mathbf{U}_{i, tl} \}^L_{l = 1} )} \]
In this model, \(\sum_{i' \in \mathcal{M}_{it}} w_{it}^{i'} = 1\) and \(w_{it}^{i'} = 0\) for \(i' \notin \mathcal{M}_{it}\). The model is fitted to the complete sample of treated and matched control units.
Checking Covariate Balance A distinct advantage of the proposed methodology over regression methods is the ability it offers researchers to inspect the covariate balance between treated and matched control observations. This facilitates the evaluation of whether treated and matched control observations are comparable regarding observed confounders. To investigate the mean difference of each covariate (e.g., \(V_{it'j}\), representing the \(j\)th variable in \(\mathbf{V}_{it'}\)) between the treated observation and its matched control observation at each pretreatment time period (i.e., \(t' < t\)), we further standardize this difference. For any given pretreatment time period, we adjust by the standard deviation of each covariate across all treated observations in the dataset. Thus, the mean difference is quantified in terms of standard deviation units. Formally, for each treated observation \((i,t)\) where \(D_{it} = 1\), we define the covariate balance for variable \(j\) at the pretreatment time period \(t  l\) as: \[\begin{equation} B_{it}(j, l) = \frac{V_{i, t l,j} \sum_{i' \in \mathcal{M}_{it}}w_{it}^{i'}V_{i', tl,j}}{\sqrt{\frac{1}{N_1  1} \sum_{i'=1}^N \sum_{t' = L+1}^{TF}D_{i't'}(V_{i', t'l, j}  \bar{V}_{t'  l, j})^2}} \label{eq:covbalance} \end{equation}\] where \(N_1 = \sum_{i'= 1}^N \sum_{t' = L+1}^{TF} D_{i't'}\) denotes the total number of treated observations and \(\bar{V}_{tl,j} = \sum_{i=1}^N D_{i,tl,j}/N\). We then aggregate this covariate balance measure across all treated observations for each covariate and pretreatment time period: \[\begin{equation} \bar{B}(j, l) = \frac{1}{N_1} \sum_{i=1}^N \sum_{t = L+ 1}^{TF}D_{it} B_{it}(j,l) \label{eq:aggbalance} \end{equation}\] Lastly, we evaluate the balance of lagged outcome variables over several pretreatment periods and that of timevarying covariates. This examination aids in assessing the validity of the parallel trend assumption integral to the DiD estimator justification.
In Figure ??, we demonstrate the enhancement of covariate balance thank to the refinement of matched sets. Each scatter plot contrasts the absolute standardized mean difference, as detailed in Equation (??), before (horizontal axis) and after (vertical axis) this refinement. Points below the 45degree line indicate an improved standardized mean balance for certain timevarying covariates postrefinement. The majority of variables benefit from this refinement process. Notably, the propensity score weighting (bottom panel) shows the most significant improvement, whereas Mahalanobis matching (top panel) yields a more modest improvement.
library(PanelMatch)
library(causalverse)
runPanelMatch < function(method, lag, size.match=NULL, qoi="att") {
# Default parameters for PanelMatch
common.args < list(
lag = lag,
time.id = "year",
unit.id = "wbcode2",
treatment = "dem",
data = dem,
covs.formula = ~ I(lag(tradewb, 1:4)) + I(lag(y, 1:4)),
qoi = qoi,
outcome.var = "y",
lead = 0:4,
forbid.treatment.reversal = FALSE,
size.match = size.match # setting size.match here for all methods
)
if(method == "mahalanobis") {
common.args$refinement.method < "mahalanobis"
common.args$match.missing < TRUE
common.args$use.diagonal.variance.matrix < TRUE
} else if(method == "ps.match") {
common.args$refinement.method < "ps.match"
common.args$match.missing < FALSE
common.args$listwise.delete < TRUE
} else if(method == "ps.weight") {
common.args$refinement.method < "ps.weight"
common.args$match.missing < FALSE
common.args$listwise.delete < TRUE
}
return(do.call(PanelMatch, common.args))
}
methods < c("mahalanobis", "ps.match", "ps.weight")
lags < c(1, 4)
sizes < c(5, 10)
You can either do it sequentailly
res_pm < list()
for(method in methods) {
for(lag in lags) {
for(size in sizes) {
name < paste0(method, ".", lag, "lag.", size, "m")
res_pm[[name]] < runPanelMatch(method, lag, size)
}
}
}
# Now, you can access res_pm using res_pm[["mahalanobis.1lag.5m"]] etc.
# for treatment reversal
res_pm_rev < list()
for(method in methods) {
for(lag in lags) {
for(size in sizes) {
name < paste0(method, ".", lag, "lag.", size, "m")
res_pm_rev[[name]] < runPanelMatch(method, lag, size, qoi = "art")
}
}
}
or in parallel
library(foreach)
library(doParallel)
registerDoParallel(cores = 4)
# Initialize an empty list to store results
res_pm < list()
# Replace nested forloops with foreach
results <
foreach(
method = methods,
.combine = 'c',
.multicombine = TRUE,
.packages = c("PanelMatch", "causalverse")
) %dopar% {
tmp < list()
for (lag in lags) {
for (size in sizes) {
name < paste0(method, ".", lag, "lag.", size, "m")
tmp[[name]] < runPanelMatch(method, lag, size)
}
}
tmp
}
# Collate results
for (name in names(results)) {
res_pm[[name]] < results[[name]]
}
# Treatment reversal
# Initialize an empty list to store results
res_pm_rev < list()
# Replace nested forloops with foreach
results_rev <
foreach(
method = methods,
.combine = 'c',
.multicombine = TRUE,
.packages = c("PanelMatch", "causalverse")
) %dopar% {
tmp < list()
for (lag in lags) {
for (size in sizes) {
name < paste0(method, ".", lag, "lag.", size, "m")
tmp[[name]] <
runPanelMatch(method, lag, size, qoi = "art")
}
}
tmp
}
# Collate results
for (name in names(results_rev)) {
res_pm_rev[[name]] < results_rev[[name]]
}
stopImplicitCluster()
library(gridExtra)
# Updated plotting function
create_balance_plot < function(method, lag, sizes, res_pm, dem) {
matched_set_lists < lapply(sizes, function(size) {
res_pm[[paste0(method, ".", lag, "lag.", size, "m")]]$att
})
return(
balance_scatter_custom(
matched_set_list = matched_set_lists,
legend.title = "Possible Matches",
set.names = as.character(sizes),
legend.position = c(0.2, 0.8),
# for compiled plot, you don't need x,y, or main labs
x.axis.label = "",
y.axis.label = "",
main = "",
data = dem,
dot.size = 5,
# show.legend = F,
them_use = causalverse::ama_theme(base_size = 32),
covariates = c("y", "tradewb")
)
)
}
plots < list()
for (method in methods) {
for (lag in lags) {
plots[[paste0(method, ".", lag, "lag")]] <
create_balance_plot(method, lag, sizes, res_pm, dem)
}
}
# # Arranging plots in a 3x2 grid
# grid.arrange(plots[["mahalanobis.1lag"]],
# plots[["mahalanobis.4lag"]],
# plots[["ps.match.1lag"]],
# plots[["ps.match.4lag"]],
# plots[["ps.weight.1lag"]],
# plots[["ps.weight.4lag"]],
# ncol=2, nrow=3)
# Standardized Mean Difference of Covariates
library(gridExtra)
library(grid)
# Create column and row labels using textGrob
col_labels < c("1year Lag", "4year Lag")
row_labels < c("Maha Matching", "PS Matching", "PS Weigthing")
major.axes.fontsize = 40
minor.axes.fontsize = 30
png(
file.path(getwd(), "images", "did_balance_scatter.png"),
width = 1200,
height = 1000
)
# Create a listoflists, where each inner list represents a row
grid_list < list(
list(
nullGrob(),
textGrob(col_labels[1], gp = gpar(fontsize = minor.axes.fontsize)),
textGrob(col_labels[2], gp = gpar(fontsize = minor.axes.fontsize))
),
list(textGrob(
row_labels[1],
gp = gpar(fontsize = minor.axes.fontsize),
rot = 90
), plots[["mahalanobis.1lag"]], plots[["mahalanobis.4lag"]]),
list(textGrob(
row_labels[2],
gp = gpar(fontsize = minor.axes.fontsize),
rot = 90
), plots[["ps.match.1lag"]], plots[["ps.match.4lag"]]),
list(textGrob(
row_labels[3],
gp = gpar(fontsize = minor.axes.fontsize),
rot = 90
), plots[["ps.weight.1lag"]], plots[["ps.weight.4lag"]])
)
# "Flatten" the listoflists into a single list of grobs
grobs < do.call(c, grid_list)
grid.arrange(
grobs = grobs,
ncol = 3,
nrow = 4,
widths = c(0.15, 0.42, 0.42),
heights = c(0.15, 0.28, 0.28, 0.28)
)
grid.text(
"Before Refinement",
x = 0.5,
y = 0.03,
gp = gpar(fontsize = major.axes.fontsize)
)
grid.text(
"After Refinement",
x = 0.03,
y = 0.5,
rot = 90,
gp = gpar(fontsize = major.axes.fontsize)
)
dev.off()
#> png
#> 2
Note: Scatter plots display the standardized mean difference of each covariate \(j\) and lag year \(l\) as defined in Equation (??) before (xaxis) and after (yaxis) matched set refinement. Each plot includes varying numbers of possible matches for each matching method. Rows represent different matching/weighting methods, while columns indicate adjustments for various lag lengths.
# Step 1: Define configurations
configurations < list(
list(refinement.method = "none", qoi = "att"),
list(refinement.method = "none", qoi = "art"),
list(refinement.method = "mahalanobis", qoi = "att"),
list(refinement.method = "mahalanobis", qoi = "art"),
list(refinement.method = "ps.match", qoi = "att"),
list(refinement.method = "ps.match", qoi = "art"),
list(refinement.method = "ps.weight", qoi = "att"),
list(refinement.method = "ps.weight", qoi = "art")
)
# Step 2: Use lapply or loop to generate results
results < lapply(configurations, function(config) {
PanelMatch(
lag = 4,
time.id = "year",
unit.id = "wbcode2",
treatment = "dem",
data = dem,
match.missing = FALSE,
listwise.delete = TRUE,
size.match = 5,
outcome.var = "y",
lead = 0:4,
forbid.treatment.reversal = FALSE,
refinement.method = config$refinement.method,
covs.formula = ~ I(lag(tradewb, 1:4)) + I(lag(y, 1:4)),
qoi = config$qoi
)
})
# Step 3: Get covariate balance and plot
plots < mapply(function(result, config) {
df < get_covariate_balance(
if (config$qoi == "att")
result$att
else
result$art,
data = dem,
covariates = c("tradewb", "y"),
plot = F
)
causalverse::plot_covariate_balance_pretrend(df, main = "", show_legend = F)
}, results, configurations, SIMPLIFY = FALSE)
# Set names for plots
names(plots) < sapply(configurations, function(config) {
paste(config$qoi, config$refinement.method, sep = ".")
})
To export
library(gridExtra)
library(grid)
# Column and row labels
col_labels <
c("None",
"Mahalanobis",
"Propensity Score Matching",
"Propensity Score Weighting")
row_labels < c("ATT", "ART")
# Specify your desired fontsize for labels
minor.axes.fontsize < 16
major.axes.fontsize < 20
png(file.path(getwd(), "images", "p_covariate_balance.png"), width=1200, height=1000)
# Create a listoflists, where each inner list represents a row
grid_list < list(
list(
nullGrob(),
textGrob(col_labels[1], gp = gpar(fontsize = minor.axes.fontsize)),
textGrob(col_labels[2], gp = gpar(fontsize = minor.axes.fontsize)),
textGrob(col_labels[3], gp = gpar(fontsize = minor.axes.fontsize)),
textGrob(col_labels[4], gp = gpar(fontsize = minor.axes.fontsize))
),
list(
textGrob(
row_labels[1],
gp = gpar(fontsize = minor.axes.fontsize),
rot = 90
),
plots$att.none,
plots$att.mahalanobis,
plots$att.ps.match,
plots$att.ps.weight
),
list(
textGrob(
row_labels[2],
gp = gpar(fontsize = minor.axes.fontsize),
rot = 90
),
plots$art.none,
plots$art.mahalanobis,
plots$art.ps.match,
plots$art.ps.weight
)
)
# "Flatten" the listoflists into a single list of grobs
grobs < do.call(c, grid_list)
# Arrange your plots with text labels
grid.arrange(
grobs = grobs,
ncol = 5,
nrow = 3,
widths = c(0.1, 0.225, 0.225, 0.225, 0.225),
heights = c(0.1, 0.45, 0.45)
)
# Add main x and y axis titles
grid.text(
"Refinement Methods",
x = 0.5,
y = 0.01,
gp = gpar(fontsize = major.axes.fontsize)
)
grid.text(
"Quantities of Interest",
x = 0.02,
y = 0.5,
rot = 90,
gp = gpar(fontsize = major.axes.fontsize)
)
dev.off()
library(knitr)
include_graphics(file.path(getwd(), "images", "p_covariate_balance.png"))
Note: Each graph displays the standardized mean difference, as outlined in Equation (??), plotted on the vertical axis across a pretreatment duration of four years represented on the horizontal axis. The leftmost column illustrates the balance prior to refinement, while the subsequent three columns depict the covariate balance post the application of distinct refinement techniques. Each individual line signifies the balance of a specific variable during the pretreatment phase.The red line is tradewb and blue line is the lagged outcome variable.
In Figure ??, we observe a marked improvement in covariate balance due to the implemented matching procedures during the pretreatment period. Our analysis prioritizes methods that adjust for timevarying covariates over a span of four years preceding the treatment initiation. The two rows delineate the standardized mean balance for both treatment modalities, with individual lines representing the balance for each covariate.
Across all scenarios, the refinement attributed to matched sets significantly enhances balance. Notably, using propensity score weighting considerably mitigates imbalances in confounders. While some degree of imbalance remains evident in the Mahalanobis distance and propensity score matching techniques, the standardized mean difference for the lagged outcome remains stable throughout the pretreatment phase. This consistency lends credence to the validity of the proposed DiD estimator.
Estimation Results
We now detail the estimated ATTs derived from the matching techniques. Figure below offers visual representations of the impacts of treatment initiation (upper panel) and treatment reversal (lower panel) on the outcome variable for a duration of 5 years posttransition, specifically, (F = 0, 1, …, 4). Across the five methods (columns), it becomes evident that the point estimates of effects associated with treatment initiation consistently approximate zero over the 5year window. In contrast, the estimated outcomes of treatment reversal are notably negative and maintain statistical significance through all refinement techniques during the initial year of transition and the 1 to 4 years that follow, provided treatment reversal is permissible. These effects are notably pronounced, pointing to an estimated reduction of roughly X% in the outcome variable.
Collectively, these findings indicate that the transition into the treated state from its absence doesn’t invariably lead to a heightened outcome. Instead, the transition from the treated state back to its absence exerts a considerable negative effect on the outcome variable in both the short and intermediate terms. Hence, the positive effect of the treatment (if we were to use traditional DiD) is actually driven by the negative effect of treatment reversal.
# sequential
# Step 1: Apply PanelEstimate function
# Initialize an empty list to store results
res_est < vector("list", length(res_pm))
# Iterate over each element in res_pm
for (i in 1:length(res_pm)) {
res_est[[i]] < PanelEstimate(
res_pm[[i]],
data = dem,
se.method = "bootstrap",
number.iterations = 1000,
confidence.level = .95
)
# Transfer the name of the current element to the res_est list
names(res_est)[i] < names(res_pm)[i]
}
# Step 2: Apply plot_PanelEstimate function
# Initialize an empty list to store plot results
res_est_plot < vector("list", length(res_est))
# Iterate over each element in res_est
for (i in 1:length(res_est)) {
res_est_plot[[i]] <
plot_PanelEstimate(res_est[[i]],
main = "",
theme_use = causalverse::ama_theme(base_size = 14))
# Transfer the name of the current element to the res_est_plot list
names(res_est_plot)[i] < names(res_est)[i]
}
# check results
# res_est_plot$mahalanobis.1lag.5m
# Step 1: Apply PanelEstimate function for res_pm_rev
# Initialize an empty list to store results
res_est_rev < vector("list", length(res_pm_rev))
# Iterate over each element in res_pm_rev
for (i in 1:length(res_pm_rev)) {
res_est_rev[[i]] < PanelEstimate(
res_pm_rev[[i]],
data = dem,
se.method = "bootstrap",
number.iterations = 1000,
confidence.level = .95
)
# Transfer the name of the current element to the res_est_rev list
names(res_est_rev)[i] < names(res_pm_rev)[i]
}
# Step 2: Apply plot_PanelEstimate function for res_est_rev
# Initialize an empty list to store plot results
res_est_plot_rev < vector("list", length(res_est_rev))
# Iterate over each element in res_est_rev
for (i in 1:length(res_est_rev)) {
res_est_plot_rev[[i]] <
plot_PanelEstimate(res_est_rev[[i]],
main = "",
theme_use = causalverse::ama_theme(base_size = 14))
# Transfer the name of the current element to the res_est_plot_rev list
names(res_est_plot_rev)[i] < names(res_est_rev)[i]
}
# parallel
library(doParallel)
library(foreach)
# Detect the number of cores to use for parallel processing
num_cores < 4
# Register the parallel backend
cl < makeCluster(num_cores)
registerDoParallel(cl)
# Step 1: Apply PanelEstimate function in parallel
res_est <
foreach(i = 1:length(res_pm), .packages = "PanelMatch") %dopar% {
PanelEstimate(
res_pm[[i]],
data = dem,
se.method = "bootstrap",
number.iterations = 1000,
confidence.level = .95
)
}
# Transfer names from res_pm to res_est
names(res_est) < names(res_pm)
# Step 2: Apply plot_PanelEstimate function in parallel
res_est_plot <
foreach(
i = 1:length(res_est),
.packages = c("PanelMatch", "causalverse", "ggplot2")
) %dopar% {
plot_PanelEstimate(res_est[[i]],
main = "",
theme_use = causalverse::ama_theme(base_size = 10))
}
# Transfer names from res_est to res_est_plot
names(res_est_plot) < names(res_est)
# Step 1: Apply PanelEstimate function for res_pm_rev in parallel
res_est_rev <
foreach(i = 1:length(res_pm_rev), .packages = "PanelMatch") %dopar% {
PanelEstimate(
res_pm_rev[[i]],
data = dem,
se.method = "bootstrap",
number.iterations = 1000,
confidence.level = .95
)
}
# Transfer names from res_pm_rev to res_est_rev
names(res_est_rev) < names(res_pm_rev)
# Step 2: Apply plot_PanelEstimate function for res_est_rev in parallel
res_est_plot_rev <
foreach(
i = 1:length(res_est_rev),
.packages = c("PanelMatch", "causalverse", "ggplot2")
) %dopar% {
plot_PanelEstimate(res_est_rev[[i]],
main = "",
theme_use = causalverse::ama_theme(base_size = 10))
}
# Transfer names from res_est_rev to res_est_plot_rev
names(res_est_plot_rev) < names(res_est_rev)
# Stop the cluster
stopCluster(cl)
To export
library(gridExtra)
library(grid)
# Column and row labels
col_labels < c("Mahalanobis 5m",
"Mahalanobis 10m",
"PS Matching 5m",
"PS Matching 10m",
"PS Weighting 5m")
row_labels < c("ATT", "ART")
# Specify your desired fontsize for labels
minor.axes.fontsize < 16
major.axes.fontsize < 20
png(file.path(getwd(), "images", "p_did_est_in_n_out.png"), width=1200, height=1000)
# Create a listoflists, where each inner list represents a row
grid_list < list(
list(
nullGrob(),
textGrob(col_labels[1], gp = gpar(fontsize = minor.axes.fontsize)),
textGrob(col_labels[2], gp = gpar(fontsize = minor.axes.fontsize)),
textGrob(col_labels[3], gp = gpar(fontsize = minor.axes.fontsize)),
textGrob(col_labels[4], gp = gpar(fontsize = minor.axes.fontsize)),
textGrob(col_labels[5], gp = gpar(fontsize = minor.axes.fontsize))
),
list(
textGrob(row_labels[1], gp = gpar(fontsize = minor.axes.fontsize), rot = 90),
res_est_plot$mahalanobis.1lag.5m,
res_est_plot$mahalanobis.1lag.10m,
res_est_plot$ps.match.1lag.5m,
res_est_plot$ps.match.1lag.10m,
res_est_plot$ps.weight.1lag.5m
),
list(
textGrob(row_labels[2], gp = gpar(fontsize = minor.axes.fontsize), rot = 90),
res_est_plot_rev$mahalanobis.1lag.5m,
res_est_plot_rev$mahalanobis.1lag.10m,
res_est_plot_rev$ps.match.1lag.5m,
res_est_plot_rev$ps.match.1lag.10m,
res_est_plot_rev$ps.weight.1lag.5m
)
)
# "Flatten" the listoflists into a single list of grobs
grobs < do.call(c, grid_list)
# Arrange your plots with text labels
grid.arrange(
grobs = grobs,
ncol = 6,
nrow = 3,
widths = c(0.1, 0.18, 0.18, 0.18, 0.18, 0.18),
heights = c(0.1, 0.45, 0.45)
)
# Add main x and y axis titles
grid.text(
"Methods",
x = 0.5,
y = 0.02,
gp = gpar(fontsize = major.axes.fontsize)
)
grid.text(
"",
x = 0.02,
y = 0.5,
rot = 90,
gp = gpar(fontsize = major.axes.fontsize)
)
dev.off()
library(knitr)
include_graphics(file.path(getwd(), "images", "p_did_est_in_n_out.png"))
26.9.4.2 Counterfactual Estimators
 Also known as imputation approach
 This class of estimator consider observation treatment as missing data. Models are built using data from the control units to impute conterfactuals for the treated observations.
 It’s called counterfactual estimators because they predict outcomes as if the treated observations had not received the treatment.
 Advantages:
 Avoids negative weights and biases by not using treated observations for modeling and applying uniform weights.
 Supports various models, including those that may relax strict exogeneity assumptions.
 Methods including
 Fixedeffects conterfactual estimator (FEct) (DiD is a special case):
 Based on the Twoway Fixedeffects, where assumes linear additive functional form of unobservables based on unit and time FEs. But FEct fixes the improper weighting of TWFE by comparing within each matched pair (where each pair is the treated observation and its predicted counterfactual that is the weighted sum of all untreated observations).
 Interactive Fixed Effects conterfactual estimator (IFEct) [Gobillon and Magnac (2016)](Xu 2017):
 When we suspect unobserved timevarying confounder, FEct fails. Instead, IFEct uses the factoraugmented models to relax the strict exogeneity assumption where the effects of unobservables can be decomposed to unit FE + time FE + unit x time FE.
 Generalized Synthetic Controls are a subset of IFEct when treatments don’t revert.
 Matrix completion (MC) (Athey et al. 2021):
 Generalization of factoraugmented models. Different from IFEct which uses hard impute, MC uses soft impute to regularize the singular values when decomposing the residual matrix.
 Only when latent factors (of unobservables) are strong and sparse, IFEct outperforms MC.
 [Synthetic Controls] (case studies)
 Fixedeffects conterfactual estimator (FEct) (DiD is a special case):
Identifying Assumptions:

Function Form: Additive separability of observables, unobservables, and idiosyncratic error term.
 Hence, these models are scale dependent (Athey and Imbens 2006) (e.g., logtransform outcome can invadiate this assumption).

Strict Exogeneity: Conditional on observables and unobservables, potential outcomes are independent of treatment assignment (i.e., baseline quasirandomization)
 In DiD, where unobservables = unit + time FEs, this assumption is the parallel trends assumption

Lowdimensional Decomposition (Feasibility Assumption): Unobservable effects can be decomposed in lowdimension.
 For the case that \(U_{it} = f_t \times \lambda_i\) where \(f_t\) = common time trend (time FE), and \(\lambda_i\) = unit heterogeneity (unit FE). If \(U_{it} = f_t \times \lambda_i\) , DiD can satisfy this assumption. But this assumption is weaker than that of DID, and allows us to control for unobservables based on data.
Estimation Procedure:
 Using all control observations, estimate the functions of both observable and unobservable variables (relying on Assumptions 1 and 3).
 Predict the counterfactual outcomes for each treated unit using the obtained functions.
 Calculate the difference in treatment effect for each treated individual.
 By averaging over all treated individuals, you can obtain the Average Treatment Effect on the Treated (ATT).
Notes:
 Use jackknife when number of treated units is small (Liu, Wang, and Xu 2022, 166).
26.9.4.2.1 Imputation Method
Liu, Wang, and Xu (2022) can also account for treatment reversals and heterogeneous treatment effects.
Other imputation estimators include
library(fect)
PanelMatch::dem
model.fect <
fect(
Y = "y",
D = "dem",
X = "tradewb",
data = na.omit(PanelMatch::dem),
method = "fe",
index = c("wbcode2", "year"),
se = TRUE,
parallel = TRUE,
seed = 1234,
# twfe
force = "twoway"
)
print(model.fect$est.avg)
plot(model.fect)
plot(model.fect, stats = "F.p")
Ftest \(H_0\): residual averages in the pretreatment periods = 0
To see treatment reversal effects
plot(model.fect, stats = "F.p", type = 'exit')
26.9.4.2.2 Placebo Test
By selecting a part of the data and excluding observations within a specified range to improve the model fitting, we then evaluate whether the estimated Average Treatment Effect (ATT) within this range significantly differs from zero. This approach helps us analyze the periods before treatment.
If this test fails, either the functional form or strict exogeneity assumption is problematic.
26.9.4.2.3 (No) Carryover Effects Test
The placebo test can be adapted to assess carryover effects by masking several posttreatment periods instead of pretreatment ones. If no carryover effects are present, the average prediction error should approximate zero. For the carryover test, set carryoverTest = TRUE
. Specify a posttreatment period range in carryover.period to exclude observations for model fitting, then evaluate if the estimated ATT significantly deviates from zero.
Even if we have carryover effects, in most cases of the staggered adoption setting, researchers are interested in the cumulative effects, or aggregated treatment effects, so it’s okay.
out.fect.c <
fect(
Y = "y",
D = "dem",
X = "tradewb",
data = na.omit(PanelMatch::dem),
method = "fe",
index = c("wbcode2", "year"),
se = TRUE,
carryoverTest = TRUE,
# how many periods of carryover
carryover.period = c(1, 3)
)
plot(out.fect.c, stats = "carryover.p")
We have evidence of carryover effects.
26.9.5 Gardner (2022) and Borusyak, Jaravel, and Spiess (2021)
Estimate the time and unit fixed effects separately
Known as the imputation method (Borusyak, Jaravel, and Spiess 2021) or twostage DiD (Gardner 2022)
# remotes::install_github("kylebutts/did2s")
library(did2s)
library(ggplot2)
library(fixest)
library(tidyverse)
data(base_stagg)
est < did2s(
data = base_stagg > mutate(treat = if_else(time_to_treatment >= 0, 1, 0)),
yname = "y",
first_stage = ~x1  id + year,
second_stage = ~i(time_to_treatment, ref = c(1, 1000)),
treatment = "treat" ,
cluster_var = "id"
)
fixest::esttable(est)
#> est
#> Dependent Var.: y
#>
#> time_to_treatment = 9 0.3518** (0.1332)
#> time_to_treatment = 8 0.3130* (0.1213)
#> time_to_treatment = 7 0.0894 (0.2367)
#> time_to_treatment = 6 0.0312 (0.2176)
#> time_to_treatment = 5 0.2079 (0.1519)
#> time_to_treatment = 4 0.1152 (0.1438)
#> time_to_treatment = 3 0.0127 (0.1483)
#> time_to_treatment = 2 0.1503 (0.1440)
#> time_to_treatment = 0 5.139*** (0.3680)
#> time_to_treatment = 1 3.480*** (0.3784)
#> time_to_treatment = 2 2.021*** (0.3055)
#> time_to_treatment = 3 0.6965. (0.3947)
#> time_to_treatment = 4 1.070** (0.3501)
#> time_to_treatment = 5 2.173*** (0.4456)
#> time_to_treatment = 6 4.449*** (0.3680)
#> time_to_treatment = 7 4.864*** (0.3698)
#> time_to_treatment = 8 6.187*** (0.2702)
#> ______________________ __________________
#> S.E. type Custom
#> Observations 950
#> R2 0.62486
#> Adj. R2 0.61843
#> 
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
fixest::iplot(
est,
main = "Event study",
xlab = "Time to treatment",
ref.line = 1
)
coefplot(est)
mult_est < did2s::event_study(
data = fixest::base_stagg >
dplyr::mutate(year_treated = dplyr::if_else(year_treated == 10000, 0, year_treated)),
gname = "year_treated",
idname = "id",
tname = "year",
yname = "y",
estimator = "all"
)
#> Error in purrr::map(., function(y) { : ℹ In index: 1.
#> ℹ With name: y.
#> Caused by error in `.subset2()`:
#> ! no such index at level 1
did2s::plot_event_study(mult_est)
26.9.6 Chaisemartind’Haultfoeuille
use twowayfeweights
from GitHub (Clément De Chaisemartin and d’Haultfoeuille 2020)

Average instant treatment effect of changes in the treatment
 This relaxes the nocarryovereffect assumption.

Drawbacks:
 Cannot observe treatment effects that manifest over time.
26.9.7 didimputation
use didimputation
from GitHub
26.9.8 Callaway and Sant’Anna (2021)
staggered
packageGrouptime average treatment effect
library(staggered)
library(fixest)
data("base_stagg")
# simple weighted average
staggered(
df = base_stagg,
i = "id",
t = "year",
g = "year_treated",
y = "y",
estimand = "simple"
)
#> estimate se se_neyman
#> 1 0.7110941 0.2211943 0.2214245
# cohort weighted average
staggered(
df = base_stagg,
i = "id",
t = "year",
g = "year_treated",
y = "y",
estimand = "cohort"
)
#> estimate se se_neyman
#> 1 2.724242 0.2701093 0.2701745
# calendar weighted average
staggered(
df = base_stagg,
i = "id",
t = "year",
g = "year_treated",
y = "y",
estimand = "calendar"
)
#> estimate se se_neyman
#> 1 0.5861831 0.1768297 0.1770729
res < staggered(
df = base_stagg,
i = "id",
t = "year",
g = "year_treated",
y = "y",
estimand = "eventstudy",
eventTime = 9:8
)
head(res)
#> estimate se se_neyman eventTime
#> 1 0.20418779 0.1045821 0.1045821 9
#> 2 0.06215104 0.1669703 0.1670886 8
#> 3 0.02744671 0.1413273 0.1420377 7
#> 4 0.02131747 0.2203695 0.2206338 6
#> 5 0.30690897 0.2015697 0.2036412 5
#> 6 0.05594029 0.1908101 0.1921745 4
ggplot(
res > mutate(
ymin_ptwise = estimate + 1.96 * se,
ymax_ptwise = estimate  1.96 * se
),
aes(x = eventTime, y = estimate)
) +
geom_pointrange(aes(ymin = ymin_ptwise, ymax = ymax_ptwise)) +
geom_hline(yintercept = 0) +
xlab("Event Time") +
ylab("Estimate") +
causalverse::ama_theme()
# Callaway and Sant'Anna estimator for the simple weighted average
staggered_cs(
df = base_stagg,
i = "id",
t = "year",
g = "year_treated",
y = "y",
estimand = "simple"
)
#> estimate se se_neyman
#> 1 0.7994889 0.4484987 0.4486122
# Sun and Abraham estimator for the simple weighted average
staggered_sa(
df = base_stagg,
i = "id",
t = "year",
g = "year_treated",
y = "y",
estimand = "simple"
)
#> estimate se se_neyman
#> 1 0.7551901 0.4407818 0.4409525
Fisher’s Randomization Test (i.e., permutation test)
\(H_0\): \(TE = 0\)
staggered(
df = base_stagg,
i = "id",
t = "year",
g = "year_treated",
y = "y",
estimand = "simple",
compute_fisher = T,
num_fisher_permutations = 100
)
#> estimate se se_neyman fisher_pval fisher_pval_se_neyman
#> 1 0.7110941 0.2211943 0.2214245 0 0
#> num_fisher_permutations
#> 1 100
26.9.9 L. Sun and Abraham (2021)
This paper utilizes the Cohort Average Treatment Effects on the Treated (CATT), which measures the cohortspecific average difference in outcomes relative to those never treated, offering a more detailed analysis than GoodmanBacon (2021). In scenarios lacking a nevertreated group, this method designates the last cohort to be treated as the control group.
Parameter of interest is the cohortspecific ATT \(l\) periods from intital treatment period \(e\)
\[ CATT = E[Y_{i, e + I}  Y_{i, e + I}^\inftyE_i = e] \]
This paper uses an interactionweighted estimator in a panel data setting, where the original paper Gibbons, Suárez Serrato, and Urbancic (2018) used the same idea in a crosssectional setting.
Callaway and Sant’Anna (2021) explores grouptime average treatment effects, employing cohorts that have not yet been treated as controls, and permits conditioning on timevarying covariates.
Athey and Imbens (2022) examines the treatment effect in relation to the counterfactual outcome of the alwaystreated group, diverging from the conventional focus on the nevertreated.
Borusyak, Jaravel, and Spiess (2021) presumes a uniform treatment effect across cohorts, effectively simplifying CATT to ATT.
Identifying Assumptions for dynamic TWFE:

Parallel Trends: Baseline outcomes follow parallel trends across cohorts before treatment.
 This gives us all CATT (including own, included bins, and excluded bins)
No Anticipatory Behavior: There is no effect of the treatment during pretreatment periods, indicating that outcomes are not influenced by the anticipation of treatment.

Treatment Effect Homogeneity: The treatment effect is consistent across cohorts for each relative period. Each adoption cohort should have the same path of treatment effects. In other words, the trajectory of each treatment cohort is similar. Compare to other designs:
Athey and Imbens (2022) assume heterogeneity of treatment effects vary over adoption cohorts, but not over time.
Borusyak, Jaravel, and Spiess (2021) assume heterogeneity of treatment effects vary over time, but not over adoption cohorts.
Callaway and Sant’Anna (2021) assume heterogeneity of treatment effects vary over time and across cohorts.
Clement De Chaisemartin and D’haultfœuille (2023) assume heterogeneity of treatment effects vary across groups and over time.
GoodmanBacon (2021) assume heterogeneity either “vary across units but not over time” or “vary over time but not across units”.
L. Sun and Abraham (2021) allows for treatment effect heterogeneity across units and time.
Sources of Heterogeneous Treatment Effects
Adoption cohorts can differ based on certain covariates. Similarly, composition of units within each adoption cohort is different.
The response to treatment varies among cohorts if units selfselect their initial treatment timing based on anticipated treatment effects. However, this selfselection is still compatible with the parallel trends assumption. This is true if units choose based on an evaluation of baseline outcomes  that is, if baseline outcomes are similar (following parallel trends), then we might not see selection into treatment based on the evaluation of the baseline outcome.
Treatment effects can vary across cohorts due to calendar timevarying effects, such as changes in economic conditions.
Notes:

If you do TWFE, you actually have to drop 2 terms to avoid multicollinearity:
Period right before treatment (this one was known before this paper)
Drop or bin or trim a distant lag period (this one was clarified by the paper). The reason is before of the multicollinearity in the linear relationship between TWFE and the relative period indicators.

Contamination of the treatment effect estimates from excluded periods is a type of “normalization”. To avoid this, we have to assume that all pretreatment periods have the same CATT.
 L. Sun and Abraham (2021) estimation method gives reasonable weights to CATT (i..e, weights that sum to 1, and are non negative). They estimate the weighted average of CATT where the weights are shares of cohorts that experience at least \(l\) periods after to treatment, normalized by the size of total periods \(g\).
Aggregation of CATT is similar to that of Callaway and Sant’Anna (2021)
Application
can use fixest
in r with sunab
function
library(fixest)
data("base_stagg")
res_sa20 = feols(y ~ x1 + sunab(year_treated, year)  id + year, base_stagg)
iplot(res_sa20)
Using the same syntax as fixest
# devtools::install_github("kylebutts/fwlplot")
library(fwlplot)
fwl_plot(y ~ x1, data = base_stagg)
fwl_plot(y ~ x1  id + year, data = base_stagg, n_sample = 100)
fwl_plot(y ~ x1  id + year, data = base_stagg, n_sample = 100, fsplit = ~ treated)
26.9.10 Wooldridge (2022)
use etwfe(Extended twoway Fixed Effects) (Wooldridge 2022)
26.9.11 Doubly Robust DiD
Also known as the locally efficient doubly robust DiD (Sant’Anna and Zhao 2020)
The package (not method) is rather limited application:
Use OLS (cannot handle
glm
)Canonical DiD only (cannot handle DDD).
library(DRDID)
data("nsw_long")
eval_lalonde_cps <
subset(nsw_long, nsw_long$treated == 0  nsw_long$sample == 2)
head(eval_lalonde_cps)
#> id year treated age educ black married nodegree dwincl re74 hisp
#> 1 1 1975 NA 42 16 0 1 0 NA 0.000 0
#> 2 1 1978 NA 42 16 0 1 0 NA 0.000 0
#> 3 2 1975 NA 20 13 0 0 0 NA 2366.794 0
#> 4 2 1978 NA 20 13 0 0 0 NA 2366.794 0
#> 5 3 1975 NA 37 12 0 1 0 NA 25862.322 0
#> 6 3 1978 NA 37 12 0 1 0 NA 25862.322 0
#> early_ra sample experimental re
#> 1 NA 2 0 0.0000
#> 2 NA 2 0 100.4854
#> 3 NA 2 0 3317.4678
#> 4 NA 2 0 4793.7451
#> 5 NA 2 0 22781.8555
#> 6 NA 2 0 25564.6699
# locally efficient doubly robust DiD Estimators for the ATT
out <
drdid(
yname = "re",
tname = "year",
idname = "id",
dname = "experimental",
xformla = ~ age + educ + black + married + nodegree + hisp + re74,
data = eval_lalonde_cps,
panel = TRUE
)
summary(out)
#> Call:
#> drdid(yname = "re", tname = "year", idname = "id", dname = "experimental",
#> xformla = ~age + educ + black + married + nodegree + hisp +
#> re74, data = eval_lalonde_cps, panel = TRUE)
#> 
#> Further improved locally efficient DR DID estimator for the ATT:
#>
#> ATT Std. Error t value Pr(>t) [95% Conf. Interval]
#> 901.2703 393.6247 2.2897 0.022 1672.7747 129.766
#> 
#> Estimator based on panel data.
#> Outcome regression est. method: weighted least squares.
#> Propensity score est. method: inverse prob. tilting.
#> Analytical standard error.
#> 
#> See Sant'Anna and Zhao (2020) for details.
# Improved locally efficient doubly robust DiD estimator
# for the ATT, with panel data
# drdid_imp_panel()
# Locally efficient doubly robust DiD estimator for the ATT,
# with panel data
# drdid_panel()
# Locally efficient doubly robust DiD estimator for the ATT,
# with repeated crosssection data
# drdid_rc()
# Improved locally efficient doubly robust DiD estimator for the ATT,
# with repeated crosssection data
# drdid_imp_rc()
26.10 Multiple Treatment
When you have 2 treatments in a setting, you should always try to model both of them under one regression to see whether they are significantly different.
 Never use one treated groups as control for the other, and run separate regression.
 Could check this answer
\[ \begin{aligned} Y_{it} &= \alpha + \gamma_1 Treat1_{i} + \gamma_2 Treat2_{i} + \lambda Post_t \\ &+ \delta_1(Treat1_i \times Post_t) + \delta_2(Treat2_i \times Post_t) + \epsilon_{it} \end{aligned} \]
(Clement De Chaisemartin and D’haultfœuille 2023) video code
26.11 Assumption Violation
26.11.1 Endogenous Timing
If the timing of units can be influenced by strategic decisions in a DID analysis, an instrumental variable approach with a control function can be used to control for endogeneity in timing.
26.11.2 Questionable Counterfactuals
In situations where the control units may not serve as a reliable counterfactual for the treated units, matching methods such as propensity score matching or generalized random forest can be utilized. Additional methods can be found in Matching Methods.
26.12 Mediation Under DiD
Check this post
26.13 Assumptions

Parallel Trends: Difference between the treatment and control groups remain constant if there were no treatment.

should be used in cases where
you observe before and after an event
you have treatment and control groups

not in cases where
treatment is not random
confounders.

To support we use


Linear additive effects (of group/unit specific and timespecific):
If they are not additively interact, we have to use the weighted 2FE estimator (Imai and Kim 2021)
Typically seen in the Staggered Difndif
No anticipation: There is no causal effect of the treatment before its implementation.
Possible issues

Estimate dependent on functional form:
 When the size of the response depends (nonlinearly) on the size of the intervention, we might want to look at the the difference in the group with high intensity vs. low.

Selection on (time–varying) unobservables
 Can use the overall sensitivity of coefficient estimates to hidden bias using Rosenbaum Bounds

Longterm effects
 Parallel trends are more likely to be observed over shorter period (window of observation)

Heterogeneous effects
 Different intensity (e.g., doses) for different groups.

Ashenfelter dip (Ashenfelter and Card 1985) (job training program participant are more likely to experience an earning drop prior enrolling in these programs)
 Participants are systemically different from nonparticipants before the treatment, leading to the question of permanent or transitory changes.
 A fix to this transient endogeneity is to calculate longrun differences (exclude a number of periods symmetrically around the adoption/ implementation date). If we see a sustained impact, then we have strong evidence for the causal impact of a policy. (Proserpio and Zervas 2017b) (James J. Heckman and Smith 1999) (Jepsen, Troske, and Coomes 2014) (X. Li, Gan, and Hu 2011)

Response to event might not be immediate (can’t be observed right away in the dependent variable)
 Using lagged dependent variable \(Y_{it1}\) might be more appropriate (Blundell and Bond 1998)
Other factors that affect the difference in trends between the two groups (i.e., treatment and control) will bias your estimation.
Correlated observations within a group or time
Incidental parameters problems (Lancaster 2000): it’s always better to use individual and time fixed effect.
When examining the effects of variation in treatment timing, we have to be careful because negative weights (per group) can be negative if there is a heterogeneity in the treatment effects over time. Example: [Athey and Imbens (2022)](Borusyak, Jaravel, and Spiess 2021)(GoodmanBacon 2021). In this case you should use new estimands proposed by @callaway2021difference(Clément De Chaisemartin and d’Haultfoeuille 2020), in the
did
package. If you expect lags and leads, see (L. Sun and Abraham 2021)(Gibbons, Suárez Serrato, and Urbancic 2018) caution when we suspect the treatment effect and treatment variance vary across groups
26.13.1 Prior Parallel Trends Test
 Plot the average outcomes over time for both treatment and control group before and after the treatment in time.
 Statistical test for difference in trends (using data from before the treatment period)
\[ Y = \alpha_g + \beta_1 T + \beta_2 T\times G + \epsilon \]
where
\(Y\) = the outcome variable
\(\alpha_g\) = group fixed effects
\(T\) = time (e.g., specific year, or month)
\(\beta_2\) = different time trends for each group
Hence, if \(\beta_2 =0\) provides evidence that there are no differences in the trend for the two groups prior the time treatment.
You can also use different functional forms (e..g, polynomial or nonlinear).
If \(\beta_2 \neq 0\) statistically, possible reasons can be:
Statistical significance can be driven by large sample
Or the trends are so consistent, and just one period deviation can throw off the trends. Hence, statistical statistical significance.
Technically, we can still salvage the research by including time fixed effects, instead of just the beforeandafter time fixed effect (actually, most researchers do this mechanically anyway nowadays). However, a side effect can be that the time fixed effects can also absorb some part your treatment effect as well, especially in cases where the treatment effects vary with time (i.e., stronger or weaker over time) (Wolfers 2003).
Debate:

(KahnLang and Lang 2020) argue that DiD will be more plausible when the treatment and control groups are similar not only in trends, but also in levels. Because when we observe dissimilar in levels prior to the treatment, why is it okay to think that this will not affect future trends?
Show a plot of the dependent variable’s time series for treated and control groups and also a similar plot with matched sample. (Ryan et al. 2019) show evidence of matched DiD did well in the setting of nonparallel trends (at least in health care setting).
In the case that we don’t have similar levels ex ante between treatment and control groups, functional form assumptions matter and we need justification for our choice.

Pretrend statistical tests: (Roth 2022) provides evidence that these test are usually under powered.
 See PretrendsPower and pretrends packages for correcting this.
library(tidyverse)
library(fixest)
od < causaldata::organ_donations %>%
# Use only pretreatment data
filter(Quarter_Num <= 3) %>%
# Treatment variable
dplyr::mutate(California = State == 'California')
# use my package
causalverse::plot_par_trends(
data = od,
metrics_and_names = list("Rate" = "Rate"),
treatment_status_var = "California",
time_var = list(Quarter_Num = "Time"),
display_CI = F
)
#> [[1]]
# do it manually
# always good but plot the dependent out
od >
# group by treatment status and time
dplyr::group_by(California, Quarter) >
dplyr::summarize_all(mean) >
dplyr::ungroup() >
# view()
ggplot2::ggplot(aes(x = Quarter_Num, y = Rate, color = California)) +
ggplot2::geom_line() +
causalverse::ama_theme()
# but it's also important to use statistical test
prior_trend < fixest::feols(Rate ~ i(Quarter_Num, California)  State + Quarter,
data = od)
fixest::coefplot(prior_trend, grid = F)
fixest::iplot(prior_trend, grid = F)
This is alarming since one of the periods is significantly different from 0, which means that our parallel trends assumption is not plausible.
In cases where you might have violations of parallel trends assumption, check (Rambachan and Roth 2023)
Impose restrictions on how different the posttreatment violations of parallel trends can be from the pretrends.
Partial identification of causal parameter
Sensitivity analysis
# https://github.com/asheshrambachan/HonestDiD
# remotes::install_github("asheshrambachan/HonestDiD")
# library(HonestDiD)
Alternatively, Ban and Kedagni (2022) propose a method that with an information set (i.e., pretreatment covariates), and an assumption on the selection bias in the posttreatment period (i.e., lies within the convex hull of all selection biases), they can still identify a set of ATT, and with stricter assumption on selection bias from the policymakers perspective, they can also have a point estimate.
Alternatively, we can use the pretrends
package to examine our assumptions (Roth 2022)
26.13.2 Placebo Test
Procedure:
 Sample data only in the period before the treatment in time.
 Consider different fake cutoff in time, either
Try the whole sequence in time
Generate random treatment period, and use randomization inference to account for sampling distribution of the fake effect.
 Estimate the DiD model but with the posttime = 1 with the fake cutoff
 A significant DiD coefficient means that you violate the parallel trends! You have a big problem.
Alternatively,
 When data have multiple control groups, drop the treated group, and assign another control group as a “fake” treated group. But even if it fails (i.e., you find a significant DiD effect) among the control groups, it can still be fine. However, this method is used under Synthetic Control
library(tidyverse)
library(fixest)
od < causaldata::organ_donations %>%
# Use only pretreatment data
dplyr::filter(Quarter_Num <= 3) %>%
# Create fake treatment variables
dplyr::mutate(
FakeTreat1 = State == 'California' &
Quarter %in% c('Q12011', 'Q22011'),
FakeTreat2 = State == 'California' &
Quarter == 'Q22011'
)
clfe1 < fixest::feols(Rate ~ FakeTreat1  State + Quarter,
data = od)
clfe2 < fixest::feols(Rate ~ FakeTreat2  State + Quarter,
data = od)
fixest::etable(clfe1,clfe2)
#> clfe1 clfe2
#> Dependent Var.: Rate Rate
#>
#> FakeTreat1TRUE 0.0061 (0.0051)
#> FakeTreat2TRUE 0.0017 (0.0028)
#> FixedEffects:  
#> State Yes Yes
#> Quarter Yes Yes
#> _______________ _______________ ________________
#> S.E.: Clustered by: State by: State
#> Observations 81 81
#> R2 0.99377 0.99376
#> Within R2 0.00192 0.00015
#> 
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
We would like the “supposed” DiD to be insignificant.
Robustness Check

Placebo DiD (if the DiD estimate \(\neq 0\), parallel trend is violated, and original DiD is biased):
Group: Use fake treatment groups: A population that was not affect by the treatment
Time: Redo the DiD analysis for period before the treatment (expected treatment effect is 0) (e..g, for previous year or period).
Possible alternative control group: Expected results should be similar
Try different windows (further away from the treatment point, other factors can creep in and nullify your effect).
Treatment Reversal (what if we don’t see the treatment event)
Higherorder polynomial time trend (to relax linearity assumption)

Test whether other dependent variables that should not be affected by the event are indeed unaffected.
 Use the same control and treatment period (DiD \(\neq0\), there is a problem)
26.13.3 Rosenbaum Bounds
Rosenbaum Bounds assess the overall sensitivity of coefficient estimates to hidden bias (Rosenbaum and Rosenbaum 2002) without having knowledge (e.g., direction) of the bias. This method is also known as worst case analyses (DiPrete and Gangl 2004).
Consider the treatment assignment is based in a way that the odds of treatment of a unit and its control is different by a multiplier \(\Gamma\) (where \(\Gamma = 1\) mean that the odds of assignment is identical, which mean random treatment assignment).
 This bias is the product of an unobservable that influences both treatment selection and outcome by a factor \(\Gamma\) (omitted variable bias)
Using this technique, we may estimate the upper limit of the pvalue for the treatment effect while assuming selection on unobservables of magnitude \(\Gamma\).
Usually, we would create a table of different levels of \(\Gamma\) to assess how the magnitude of biases can affect our evidence of the treatment effect (estimate).
If we have treatment assignment is clustered (e.g., within school, within state) we need to adjust the bounds for clustered treatment assignment (Hansen, Rosenbaum, and Small 2014) (similar to clustered standard errors)
Then, we can report the minimum value of \(\Gamma\) at which the treatment treat is nullified (i.e., become insignificant). And the literature’s rules of thumb is that if \(\Gamma > 2\), then we have strong evidence for our treatment effect is robust to large biases (Proserpio and Zervas 2017a)
Packages
rbounds
(Keele 2010)sensitivitymv
(Rosenbaum 2015)