Difference-in-Differences Estimation

PyFixest supports event study designs via the canonical two-way fixed effects design, the 2-Step imputation estimator, and local projections.

Setup

from importlib import resources

import pandas as pd

import pyfixest as pf
from pyfixest.report.utils import rename_event_study_coefs
from pyfixest.utils.dgps import get_sharkfin

%load_ext watermark
%watermark --iversions
%load_ext autoreload
%autoreload 2

pandas  : 2.2.3
pyfixest: 0.29.0

# one-shot adoption data - parallel trends is true
df_one_cohort = get_sharkfin()
df_one_cohort.head()

	year	Y
0	0	1.629307
1	1	0.825902
2	2	0.208988
3	3	-0.244739
4	4	0.804665

# multi-cohort adoption data
df_multi_cohort = pd.read_csv(
    resources.files("pyfixest.did.data").joinpath("df_het.csv")
)
df_multi_cohort.head()

	unit	state	group	unit_fe	g	year	year_fe	treat	rel_year	rel_year_binned	error	dep_var
0	1	33	Group 2	7.043016	2010	1990	0.066159	False	-20.0	-6	-0.086466	7.022709
1	1	33	Group 2	7.043016	2010	1991	-0.030980	False	-19.0	-6	0.766593	7.778628
2	1	33	Group 2	7.043016	2010	1992	-0.119607	False	-18.0	-6	1.512968	8.436377
3	1	33	Group 2	7.043016	2010	1993	0.126321	False	-17.0	-6	0.021870	7.191207
4	1	33	Group 2	7.043016	2010	1994	-0.106921	False	-16.0	-6	-0.017603	6.918492

Examining Treatment Timing

Before any DiD estimation, we need to examine the treatment timing, since it is crucial to our choice of estimator.

pf.panelview(
    df_one_cohort,
    unit="unit",
    time="year",
    treat="treat",
    collapse_to_cohort=True,
    sort_by_timing=True,
    ylab="Cohort",
    xlab="Year",
    title="Treatment Assignment Cohorts",
    figsize=(6, 5),
)

pf.panelview(
    df_multi_cohort,
    unit="unit",
    time="year",
    treat="treat",
    collapse_to_cohort=True,
    sort_by_timing=True,
    ylab="Cohort",
    xlab="Year",
    title="Treatment Assignment Cohorts",
    figsize=(6, 5),
)

We immediately see that we have staggered adoption of treatment in the second case, which implies that a naive application of 2WFE might yield biased estimates under substantial effect heterogeneity.

We can also plot treatment assignment in a disaggregated fashion, which gives us a sense of cohort sizes.

pf.panelview(
    df_multi_cohort,
    unit="unit",
    time="year",
    treat="treat",
    sort_by_timing=True,
    ylab="Unit",
    xlab="Year",
    title="Treatment Assignment (all units)",
    figsize=(6, 5),
)

Inspecting the Outcome Variable

pf.panelview() further allows us to inspect the “outcome” variable over time:

pf.panelview(
    df_multi_cohort,
    outcome="dep_var",
    unit="unit",
    time="year",
    treat="treat",
    collapse_to_cohort=True,
    title="Outcome Plot",
    legend=True,
    figsize=(7, 2.5),
)

We immediately see that the first cohort is switched into treatment in 2000, while the second cohort is switched into treatment by 2010. Before each cohort is switched into treatment, the trends are parallel.

We can additionally inspect individual units by dropping the collapse_to_cohort argument. Because we have a large sample, we might want to inspect only a subset of units.

pf.panelview(
    df_multi_cohort,
    outcome="dep_var",
    unit="unit",
    time="year",
    treat="treat",
    subsamp=100,
    title = "Outcome Plot",
    legend=True,
    figsize=(7, 2.5),
)

One-shot adoption: Static and Dynamic Specifications

After taking a first look at the data, let’s turn to estimation. We return to the df_one_cohort data set (without staggered treatment rollout).

fit_static_twfe = pf.feols(
    "Y ~ treat | unit + year",
    df_one_cohort,
    vcov={"CRV1": "unit"},
)
fit_static_twfe.summary()

###

Estimation:  OLS
Dep. var.: Y, Fixed effects: unit+year
Inference:  CRV1
Observations:  30000

| Coefficient   |   Estimate |   Std. Error |   t value |   Pr(>|t|) |   2.5% |   97.5% |
|:--------------|-----------:|-------------:|----------:|-----------:|-------:|--------:|
| treat         |      0.206 |        0.052 |     3.927 |      0.000 |  0.103 |   0.308 |
---
RMSE: 0.701 R2: 0.905 R2 Within: 0.003

Since this is a single-cohort dataset, this estimate is consistent for the ATT under parallel trends. We can estimate heterogeneous effects by time by interacting time with the treated group:

fit_dynamic_twfe = pf.feols(
    "Y ~ i(year, ever_treated,  ref = 14) | unit + year",
    df_one_cohort,
    vcov={"CRV1": "unit"},
)

fit_dynamic_twfe.iplot(
    coord_flip=False,
    title="Event Study",
    figsize=[1200, 400],
    yintercept=0,
    xintercept=13.5,
    labels=rename_event_study_coefs(fit_dynamic_twfe._coefnames),
)

Event study plots like this are very informative, as they allow us to visually inspect the parallel trends assumption and also the dynamic effects of the treatment.

Based on a cursory glance, one would conclude that parallel trends does not hold because one of the pre-treatment coefficient has a confidence interval that does not include zero. However, we know that parallel trends is true because the treatment is randomly assigned in the underlying DGP.

Pointwise vs Simultaneous Inference in Event Studies

This is an example of a false positive in testing for pre-trends produced by pointwise inference (where each element of the coefficient vector is tested separately).

As an alternative, we can use simultaneous confidence bands of the form \([a, b] = ([a_k, b_k])_{k=1}^K\) such that

\[ P(\beta \in [a, b]) = P(\beta_k \in [a_k, b_k] \forall k) \rightarrow 1 - \alpha \]

These bands can be constructed by using a carefully chosen critical value \(c\) that accounts for the covariance between coefficients using the multiplier bootstrap. In pointwise inference, the critical value is \(c = z_{1 - \alpha/2} = 1.96\) for \(\alpha = 0.05\); the corresponding critical value for simultaneous inference is typically larger. These are also known as sup-t bands in the literature (see lec 3 of the NBER SI methods lectures linked above).

This is implemented in the confint(joint=True) method in the feols class. If we pass the joint='both' argument to iplot, we get the simultaneous confidence bands (for all event study coefficients) in addition to the pointwise confidence intervals. Note that simultaneous inference for all event study coefficients may be overly conservative, especially when the number of coefficients is large; one may instead choose to perform joint inference for all pre-treatment coefficients and all post-treatment coefficients separately.

fit_dynamic_twfe.iplot(
    coord_flip=False,
    title="Event Study",
    figsize=[1200, 400],
    yintercept=0,
    xintercept=13.5,
    joint="both",
    labels=rename_event_study_coefs(fit_dynamic_twfe._coefnames),
)

The joint confidence bands are wider than the pointwise confidence intervals, and they include zero for all pre-treatment coefficients. This is consistent with the parallel trends assumption.

Event Study under Staggered Adoption via `feols()`, `event_study()`, `did2s()`, `lpdid()`

We now return to the data set with staggered treatment rollout, df_multi_cohort.

Two-Way Fixed Effects

As a baseline model, we can estimate a simple two-way fixed effects DiD regression via feols():

fit_twfe = pf.feols(
    "dep_var ~ i(rel_year, ref=-1.0) | state + year",
    df_multi_cohort,
    vcov={"CRV1": "state"},
)

You can also estimate a TWFE model via the event_study() function, which aims to provide a common interface to multiple difference-in-differences implementations:

fit_twfe_event = pf.event_study(
    data=df_multi_cohort,
    yname="dep_var",
    idname="unit",
    tname="year",
    gname="g",
    estimator="twfe",
)

Fully-Interacted / Saturated Event Study (Sun-Abraham)

In a similar spirit, you can fit a fully-interacted difference-in-differences model by selecting the estimator = "saturated":

fit_saturated = pf.event_study(
    data=df_multi_cohort,
    yname="dep_var",
    idname="unit",
    tname="year",
    gname="g",
    estimator="saturated",
)

fit_saturated.iplot()

/home/runner/work/pyfixest/pyfixest/pyfixest/did/saturated_twfe.py:68: UserWarning: The SaturatedEventStudyClass is currently in beta. Please report any issues you may encounter.
  warnings.warn(
/home/runner/work/pyfixest/pyfixest/pyfixest/estimation/feols_.py:2654: UserWarning: 
            22 variables dropped due to multicollinearity.
            The following variables are dropped: 
    C(rel_time, contr.treatment(base=-1.0))[-20.0]:cohort_dummy_2000
    C(rel_time, contr.treatment(base=-1.0))[-19.0]:cohort_dummy_2000
    C(rel_time, contr.treatment(base=-1.0))[-18.0]:cohort_dummy_2000
    C(rel_time, contr.treatment(base=-1.0))[-17.0]:cohort_dummy_2000
    C(rel_time, contr.treatment(base=-1.0))[-16.0]:cohort_dummy_2000
    ....
            
  warnings.warn(

We can obtain treatment effects by period via the aggregate() method

fit_saturated.aggregate(weighting = "shares")

	Estimate	Std. Error	t value	Pr(>\|t\|)	2.5%	97.5%
period
-20.0	-0.073055	0.083868	-0.871071	0.383715	-0.237434	0.091324
-19.0	-0.005748	0.087254	-0.065871	0.94748	-0.176763	0.165268
-18.0	0.019863	0.084817	0.234181	0.814844	-0.146376	0.186101
-17.0	0.075561	0.08836	0.85515	0.392468	-0.097622	0.248744
-16.0	-0.051278	0.088165	-0.581611	0.560829	-0.224077	0.121522
-15.0	0.047891	0.085371	0.560977	0.574813	-0.119433	0.215215
-14.0	0.030061	0.085232	0.352697	0.724316	-0.13699	0.197112
-13.0	0.03987	0.082118	0.485517	0.62731	-0.121079	0.200818
-12.0	0.048458	0.087172	0.555888	0.578287	-0.122396	0.219312
-11.0	-0.01282	0.091618	-0.13993	0.888715	-0.192388	0.166748
-10.0	0.001529	0.06211	0.02461	0.980366	-0.120205	0.123262
-9.0	0.017902	0.062642	0.285777	0.775049	-0.104874	0.140677
-8.0	0.02942	0.062148	0.473384	0.635939	-0.092389	0.151229
-7.0	0.06927	0.06336	1.093268	0.274276	-0.054914	0.193453
-6.0	-0.000168	0.06252	-0.002687	0.997856	-0.122705	0.122369
-5.0	-0.042447	0.060507	-0.701529	0.482973	-0.161039	0.076144
-4.0	0.025822	0.061739	0.418243	0.67577	-0.095185	0.146828
-3.0	0.062146	0.061508	1.010385	0.312311	-0.058406	0.182699
-2.0	0.031255	0.06339	0.493059	0.621971	-0.092988	0.155498
0.0	1.38732	0.061787	22.453107	0.0	1.266219	1.508421
1.0	1.628963	0.063322	25.724928	0.0	1.504854	1.753073
2.0	1.686253	0.062372	27.035432	0.0	1.564007	1.8085
3.0	1.796582	0.063137	28.455425	0.0	1.672837	1.920328
4.0	1.934649	0.063037	30.690833	0.0	1.8111	2.058199
5.0	2.065588	0.061803	33.422231	0.0	1.944456	2.186719
6.0	2.166252	0.064074	33.808421	0.0	2.040668	2.291835
7.0	2.263413	0.06352	35.632879	0.0	2.138916	2.387911
8.0	2.358293	0.06232	37.841747	0.0	2.236149	2.480438
9.0	2.354538	0.056481	41.687038	0.0	2.243837	2.465239
10.0	2.515993	0.060297	41.72651	0.0	2.397813	2.634174
11.0	2.552978	0.092825	27.503048	0.0	2.371044	2.734912
12.0	2.603939	0.090658	28.722574	0.0	2.426252	2.781625
13.0	2.578044	0.093739	27.502403	0.0	2.394319	2.761769
14.0	2.722969	0.092043	29.583642	0.0	2.542568	2.90337
15.0	2.909275	0.092438	31.472726	0.0	2.7281	3.09045
16.0	2.859325	0.092643	30.864025	0.0	2.677749	3.040901
17.0	2.907943	0.093763	31.013703	0.0	2.724171	3.091716
18.0	2.873678	0.093009	30.896721	0.0	2.691384	3.055973
19.0	2.882807	0.09253	31.155532	0.0	2.701453	3.064162
20.0	2.911501	0.091093	31.961938	0.0	2.732962	3.09004

and plot the effects

fit_saturated.iplot_aggregate(weighting = "shares")

/home/runner/work/pyfixest/pyfixest/pyfixest/did/saturated_twfe.py:271: UserWarning: No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.
  ax.legend()

When can we get away with using the two-way fixed effects regression?

We will motivate this section by lazily quoting the abstract of Lal (2025):

The use of the two-way fixed effects regression in empirical social science was historically motivated by folk wisdom that it uncovers the Average Treatment effect on the Treated (ATT) as in the canonical two-period two-group case. This belief has come under scrutiny recently due to recent results in applied econometrics showing that it fails to uncover meaningful averages of heterogeneous treatment effects in the presence of effect heterogeneity over time and across adoption cohorts, and several heterogeneity-robust alternatives have been proposed. However, these estimators often have higher variance and are therefore under-powered for many applications, which poses a bias-variance tradeoff that is challenging for researchers to navigate. In this paper, we propose simple tests of linear restrictions that can be used to test for differences in dynamic treatment effects over cohorts, which allows us to test for when the two-way fixed effects regression is likely to yield biased estimates of the ATT.

You can employ the proposed test after running a saturated event study by calling the test_treatment_heterogeneity() method:

fit_saturated.test_treatment_heterogeneity()

statistic    3256.497337
pvalue          0.000000
dtype: float64

In this case, we might be willing to rely on the simple TWFE model to produce unbiased estimates. If we’re not, two “new” difference-in-differences estimators are implemented (beyond the already-presented saturated Sun-Abraham approach) that produce unbiased estimates under staggered assignment and heterogeneous treatment effects: Gardner’s 2-Step Estimator and the Local Projections estimator from Dube et al.

Gardner’s 2-Step Estimator

To do the same via Gardners 2-stage estimator, we employ the the pf.did2s() function:

fit_did2s = pf.did2s(
    df_multi_cohort,
    yname="dep_var",
    first_stage="~ 0 | unit + year",
    second_stage="~i(rel_year,ref=-1.0)",
    treatment="treat",
    cluster="state",
)

Local Projections (Dube et al)

Last, we can estimate the ATT for each time period via local projections by using the lpdid() function:

fit_lpdid = pf.lpdid(
    data=df_multi_cohort,
    yname="dep_var",
    gname="g",
    tname="year",
    idname="unit",
    vcov={"CRV1": "state"},
    pre_window=-20,
    post_window=20,
    att=False,
)

Let’s look at some results:

figsize = [1200, 400]

fit_twfe.iplot(
    coord_flip=False,
    title="TWFE-Estimator",
    figsize=figsize,
    xintercept=18.5,
    yintercept=0,
    labels=rename_event_study_coefs(fit_twfe._coefnames),  # rename coefficients
).show()

fit_lpdid.iplot(
    coord_flip=False,
    title="Local-Projections-Estimator",
    figsize=figsize,
    yintercept=0,
    xintercept=18.5,
).show()

What if we are not interested in the ATT per treatment period, but in a pooled effects?

fit_twfe = pf.feols(
    "dep_var ~ i(treat) | unit + year",
    df_multi_cohort,
    vcov={"CRV1": "state"},
)

fit_did2s = pf.did2s(
    df_multi_cohort,
    yname="dep_var",
    first_stage="~ 0 | unit + year",
    second_stage="~i(treat)",
    treatment="treat",
    cluster="state",
)

fit_lpdid = pf.lpdid(
    data=df_multi_cohort,
    yname="dep_var",
    gname="g",
    tname="year",
    idname="unit",
    vcov={"CRV1": "state"},
    pre_window=-20,
    post_window=20,
    att=True,
)
pd.concat(
    [
        fit_twfe.tidy().assign(estimator="TWFE"),
        fit_did2s.tidy().assign(estimator="DID2s"),
        fit_lpdid.tidy().assign(estimator="LPDID").drop("N", axis=1),
    ],
    axis=0,
)

	Estimate	Std. Error	t value	2.5%	97.5%	estimator
C(treat)[T.True]	1.982540	0.019338	102.521988	1.943426	2.021655	TWFE
C(treat)[T.True]	2.230482	0.024709	90.271444	2.182052	2.278911	DID2s
treat_diff	2.506746	0.071420	35.098900	2.362287	2.651206	LPDID