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.

See also NBER SI methods lectures on Linear Panel Event Studies.

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: 0.27.0

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

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

# 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	te	te_dynamic	dep_var
0	1	33	Group 2	7.043016	2010	1990	0.066159	False	-20.0	-6	-0.086466	0	0.0	7.022709
1	1	33	Group 2	7.043016	2010	1991	-0.030980	False	-19.0	-6	0.766593	0	0.0	7.778628
2	1	33	Group 2	7.043016	2010	1992	-0.119607	False	-18.0	-6	1.512968	0	0.0	8.436377
3	1	33	Group 2	7.043016	2010	1993	0.126321	False	-17.0	-6	0.021870	0	0.0	7.191207
4	1	33	Group 2	7.043016	2010	1994	-0.106921	False	-16.0	-6	-0.017603	0	0.0	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 |     3.929 |      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(), did2s() and 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"},
)

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 | state + 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	Pr(>|t|)	2.5%	97.5%	estimator
C(treat)[T.True]	1.982540	0.019331	102.556180	0.0	1.943439	2.021642	TWFE
C(treat)[T.True]	2.230482	0.024709	90.271444	0.0	2.180504	2.280459	DID2s
treat_diff	2.506746	0.071357	35.129648	0.0	2.362413	2.651080	LPDID