DiD | Carlos Mendez

Introduction to Difference-in-Differences (DiD) in Python

Mon, 27 Apr 2026 00:00:00 +0000

1. Overview

How much does an after-school tutoring program improve student performance? A school district implemented a new after-school tutoring program in 10 of its 35 high schools. After one year, the average GPA in tutored schools jumped from 60.17 to 96.37 — a staggering 36.20-point increase. Case closed?

Not quite. Over the same period, GPA also rose in the 25 schools that did not receive the program — from 71.22 to 82.10. Some of the improvement in tutored schools simply reflects a region-wide upward trend. Difference-in-Differences (DiD) strips away that common trend and reveals the tutoring program’s true causal effect: an ATT of approximately 25.32 GPA points.

This tutorial walks through DiD estimation in Python using PyFixest — a fast, Stata-flavored econometrics package — alongside Great Tables for publication-quality output. We use the simulated case study from Corral and Yang (2024), the same dataset used in the Stata companion tutorial.

1.1 Learning objectives

By the end of this tutorial, you will be able to:

Explain why naive before-after comparisons overstate treatment effects
Compute 2×2 DiD manually and via PyFixest’s feols() function
Estimate DiD using multiple equivalent approaches with a unified formula syntax
Compare inference under iid, HC1, CRV1, and CRV3 standard errors
Build publication-quality regression tables with etable() and Great Tables
Estimate and plot event study models with i() for dynamic treatment effects

1.2 Study design

graph LR
subgraph "Case Study Setting"
A["<b>35 High Schools</b><br/>in One Region"]
B["<b>10 Treated Schools</b><br/>(tutoring program)"]
C["<b>25 Comparison Schools</b><br/>(no program)"]
A --> B
A --> C
end
subgraph "DiD Design"
D["<b>Pre-Program</b><br/>GPA at baseline"]
E["<b>Post-Program</b><br/>GPA after intervention"]
F["<b>DiD Estimate</b><br/>ATT = 25.32"]
D --> E --> F
end
subgraph "Estimation Methods"
G["<b>Manual 2x2</b><br/>Subtraction"]
H["<b>TWFE Regression</b><br/>PyFixest feols()"]
I["<b>Event Study</b><br/>Dynamic effects"]
G --> H --> I
end
C --> D
F --> G
style A fill:#6a9bcc,stroke:#141413,color:#fff
style B fill:#d97757,stroke:#141413,color:#fff
style C fill:#6a9bcc,stroke:#141413,color:#fff
style F fill:#00d4c8,stroke:#141413,color:#fff
style I fill:#00d4c8,stroke:#141413,color:#fff

The data has a clean panel structure: each of the 35 schools is observed in two time periods (pre and post), giving us 70 observations for the 2×2 design. A second dataset extends this to 8 periods (280 observations) for the event study analysis.

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2. Setup and Imports

Install the required packages:

pip install pyfixest great_tables pandas matplotlib

Import the libraries:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import pyfixest as pf
from great_tables import GT, md, style, loc

Package	Purpose
`pyfixest`	Fast fixed-effects estimation with Stata-like formula syntax
`great_tables`	Publication-quality HTML/PNG tables from DataFrames
`pandas`	Data loading, manipulation, and summary statistics
`matplotlib`	Custom figure generation with dark theme styling

Dark theme figure styling (click to expand)

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NEAR_BLACK = "#141413"
TEAL = "#00d4c8"
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GRID_LINE = "#1f2b5e"
LIGHT_TEXT = "#c8d0e0"
WHITE_TEXT = "#e8ecf2"
plt.rcParams.update({
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"legend.frameon": False,
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3. Data Loading and Exploration

We load the 2×2 dataset directly from GitHub. This Stata .dta file contains 35 schools observed across 2 time periods:

url_did = "https://github.com/quarcs-lab/data-open/raw/master/isds/tutoring_did.dta"
df = pd.read_stata(url_did).astype(float)
print(df.shape)
print(df.dtypes)

(70, 7)
id float64
time float64
treated float64
post float64
txp float64
gpa float64
female_share float64
dtype: object

The dataset has 70 observations (35 schools × 2 periods) and 7 variables:

id — School identifier (1–35)
time — Time period (1 = pre, 2 = post)
treated — Treatment indicator (1 = received tutoring program)
post — Post-period indicator (1 = after program implementation)
txp — Interaction term (treated × post)
gpa — Outcome: average GPA of low-income students (0–100 scale)
female_share — Share of female students (covariate)

print(df.describe().round(2))

 id time treated post txp gpa female_share
count 70.00 70.0 70.00 70.0 70.00 70.00 70.00
mean 18.00 1.5 0.29 0.5 0.14 77.12 0.53
std 10.17 0.5 0.46 0.5 0.35 10.88 0.03
min 1.00 1.0 0.00 0.0 0.00 59.39 0.47
25% 9.25 1.0 0.00 0.0 0.00 70.68 0.51
50% 18.00 1.5 0.00 0.5 0.00 76.27 0.53
75% 26.75 2.0 1.00 1.0 0.00 82.66 0.55
max 35.00 2.0 1.00 1.0 1.00 99.15 0.57

A crosstab confirms the balanced 2×2 design:

ct = pd.crosstab(df["treated"], df["post"], margins=True)
print(ct)

 Pre (0) Post (1) Total
Comparison (0) 25 25 50
Treated (1) 10 10 20
Total 35 35 70

We have 10 treated schools observed in 2 periods (20 observations) and 25 comparison schools (50 observations). This is a perfectly balanced panel — every school appears exactly once in each period.

3.1 Panel structure visualization

The heatmap below shows the treatment assignment across schools and time. Steel blue cells represent the comparison group, while orange cells indicate treated schools in the post-program period.

This is a clean 2×2 design: treatment timing is simultaneous (all 10 schools receive the program at the same time), and no school switches treatment status.

4. The Problem with Naive Comparisons

The most intuitive approach to measuring the program’s effect is a simple before-after comparison for the treated schools:

treated_means = df[df["treated"] == 1].groupby("post")["gpa"].mean()
print(f"Pre-program: {treated_means[0]:.2f}")
print(f"Post-program: {treated_means[1]:.2f}")
print(f"Naive change: {treated_means[1] - treated_means[0]:.2f}")

Pre-program: 60.17
Post-program: 96.37
Naive change: 36.20

The naive estimate says the program boosted GPA by 36.20 points. But this ignores everything else that may have changed over the same period — curriculum reforms, new textbooks, regional economic shifts, or simply students maturing. Any of these factors could drive GPA upward in all schools, not just the treated ones.

The naive approach overstates the effect by conflating the treatment effect with time trends that would have occurred regardless of the program.

5. The DiD Design: Using a Comparison Group

The key insight of DiD is to use the comparison group as a mirror for what would have happened to the treated schools without the program. We compute all four group means:

means = df.groupby(["treated", "post"])["gpa"].mean()
pre_control = means[(0, 0)] # 71.22
post_control = means[(0, 1)] # 82.10
pre_treated = means[(1, 0)] # 60.17
post_treated = means[(1, 1)] # 96.37

Group means:
Comparison Pre: 71.22
Comparison Post: 82.10
Treated Pre: 60.17
Treated Post: 96.37

The comparison schools' GPA rose by 10.88 points (from 71.22 to 82.10) — this is the secular trend. We assume the treated schools would have experienced the same trend absent the program. This gives us the counterfactual:

counterfactual = pre_treated + (post_control - pre_control)
did_estimate = post_treated - counterfactual
print(f"Counterfactual: {pre_treated:.2f} + ({post_control:.2f} - {pre_control:.2f}) = {counterfactual:.2f}")
print(f"DiD estimate: {post_treated:.2f} - {counterfactual:.2f} = {did_estimate:.2f}")

Counterfactual: 60.17 + (82.10 - 71.22) = 71.05
DiD estimate: 96.37 - 71.05 = 25.32

The causal effect of the tutoring program is 25.32 GPA points — not 36.20. The naive approach overstated the effect by 43% because it attributed the 10.88-point common trend entirely to the program.

5.1 The parallel trends assumption

DiD rests on one critical assumption: parallel trends. In the absence of treatment, treated and comparison groups would have followed the same trajectory over time. Formally:

$$E[Y_{i,1}(0) - Y_{i,0}(0) \mid D=1] = E[Y_{i,1}(0) - Y_{i,0}(0) \mid D=0]$$

In words: the change in potential untreated outcomes is the same for both groups. Think of it like two runners on parallel tracks — they may start at different positions (treated schools have lower baseline GPA), but they run at the same pace. If one runner suddenly speeds up after receiving coaching, the difference between their new speed and the other runner’s speed measures the coaching effect.

Note what parallel trends does not require: the two groups do not need the same level of GPA, only the same trend. This is why DiD is powerful — it naturally handles time-invariant differences between groups (like school quality or student demographics).

5.2 SUTVA

The Stable Unit Treatment Value Assumption (SUTVA) requires that one school’s treatment does not affect another school’s outcome. If untreated schools lost students to tutored schools, or if tutored schools drew resources away from comparison schools, the DiD estimate would be biased. In this setting, schools serve distinct geographic catchments, making spillovers unlikely.

6. Manual DiD Calculation

We can organize the four group means into a 2×2 table and compute the DiD as a double difference:

means_table = df.groupby(["treated", "post"])["gpa"].mean().unstack()
means_table["Difference"] = means_table[1.0] - means_table[0.0]
print(means_table.round(2))

 Pre (0) Post (1) Difference
Comparison (0) 71.22 82.10 10.88
Treated (1) 60.17 96.37 36.20

The DiD formula takes the difference of differences:

$$DiD = \Big(E[Y_{i,1} \mid D=1] - E[Y_{i,0} \mid D=1]\Big) - \Big(E[Y_{i,1} \mid D=0] - E[Y_{i,0} \mid D=0]\Big)$$

Plugging in the numbers:

$$DiD = (96.37 - 60.17) - (82.10 - 71.22) = 36.20 - 10.88 = 25.32$$

Think of it this way: the treated schools improved by 36.20 points, but 10.88 of those points would have happened anyway (as evidenced by the comparison group). The remaining 25.32 points is the causal effect of the tutoring program.

Going back to the runner analogy: the treated runner sped up by 36.20 units while the comparison runner sped up by 10.88. The coaching effect is the extra 25.32 units of speed that only the coached runner gained.

7. DiD via Regression

7.1 Classical OLS with interaction

The manual calculation is equivalent to an OLS regression with the treatment indicator, time indicator, and their interaction:

$$Y_{it} = \alpha + \beta_1 \text{Treat}_i + \beta_2 \text{Post}_t + \beta_3 (\text{Treat}_i \times \text{Post}_t) + \varepsilon_{it}$$

Where:

$\alpha$ is the comparison group’s pre-period mean (intercept)
$\beta_1$ captures the baseline difference between groups
$\beta_2$ captures the common time trend
$\beta_3$ is the DiD estimate — the causal effect of treatment

In PyFixest, the feols() function handles this with a familiar formula syntax:

fit_ols = pf.feols("gpa ~ treated + post + txp", data=df, vcov="HC1")
print(fit_ols.summary())

Estimation: OLS
Dep. var.: gpa, Fixed effects: 0
Inference: HC1
Observations: 70
| Coefficient | Estimate | Std. Error | t value | Pr(>|t|) | 2.5% | 97.5% |
|:--------------|-----------:|-------------:|----------:|-----------:|--------:|--------:|
| Intercept | 71.215 | 0.218 | 326.123 | 0.000 | 70.779 | 71.651 |
| treated | -11.049 | 0.288 | -38.388 | 0.000 | -11.624 | -10.475 |
| post | 10.886 | 0.339 | 32.116 | 0.000 | 10.209 | 11.563 |
| txp | 25.315 | 0.615 | 41.164 | 0.000 | 24.087 | 26.543 |
---
RMSE: 1.15 R2: 0.989

Every coefficient maps directly to our group means:

Intercept (71.22) — Comparison group pre-period mean
treated (−11.05) — Treated schools start 11 points below comparison schools
post (10.89) — Common time trend (comparison group’s improvement)
txp (25.32) — The DiD estimate, matching our manual calculation

The vcov="HC1" option requests heteroskedasticity-robust (White) standard errors, the most common choice for cross-sectional data.

7.2 TWFE with fixed effects

A more flexible approach absorbs school-level and time-level heterogeneity using two-way fixed effects (TWFE). PyFixest uses the | pipe syntax to specify absorbed fixed effects:

$$Y_{it} = \beta_3 (\text{Treat}_i \times \text{Post}_t) + \gamma_i + \vartheta_t + \varepsilon_{it}$$

Here $\gamma_i$ are school fixed effects (absorbing all time-invariant school characteristics) and $\vartheta_t$ are time fixed effects (absorbing all common time shocks). Since treated is perfectly collinear with $\gamma_i$ and post is perfectly collinear with $\vartheta_t$, only the interaction term txp remains:

fit_twfe = pf.feols("gpa ~ txp | id + time", data=df, vcov={"CRV1": "id"})
print(fit_twfe.summary())

Estimation: OLS
Dep. var.: gpa, Fixed effects: id+time
Inference: CRV1
Observations: 70
| Coefficient | Estimate | Std. Error | t value | Pr(>|t|) | 2.5% | 97.5% |
|:--------------|-----------:|-------------:|----------:|-----------:|-------:|--------:|
| txp | 25.315 | 0.585 | 43.265 | 0.000 | 24.126 | 26.504 |
---
RMSE: 0.788 R2: 0.995 R2 Within: 0.981

The estimate is unchanged: 25.315. But the standard errors now use CRV1 (cluster-robust variance) clustered at the school level — the appropriate choice when treatment varies at the school level and observations within the same school are correlated.

The formula "gpa ~ txp | id + time" is one of PyFixest’s key strengths: everything to the left of | is estimated, everything to the right is absorbed. No need to manually create dummy variables.

7.3 TWFE with covariate

We can add female_share as a time-varying covariate to check robustness:

fit_cov = pf.feols("gpa ~ txp + female_share | id + time", data=df,
vcov={"CRV1": "id"})
print(fit_cov.summary())

Estimation: OLS
Dep. var.: gpa, Fixed effects: id+time
Inference: CRV1
Observations: 70
| Coefficient | Estimate | Std. Error | t value | Pr(>|t|) | 2.5% | 97.5% |
|:--------------|-----------:|-------------:|----------:|-----------:|--------:|--------:|
| txp | 25.328 | 0.605 | 41.881 | 0.000 | 24.099 | 26.557 |
| female_share | -3.216 | 8.700 | -0.370 | 0.714 | -20.898 | 14.465 |
---
RMSE: 0.785 R2: 0.995 R2 Within: 0.982

Adding female_share barely changes the DiD estimate (25.315 → 25.328, a shift of just 0.013). The covariate itself is statistically insignificant (p = 0.714), confirming that the two-way fixed effects already capture the relevant variation. This is reassuring — the treatment effect estimate is robust to the inclusion of observable covariates.

7.4 Programmatic access to results

PyFixest provides tidy methods for extracting specific quantities — useful for post-estimation workflows and building custom tables:

print(f"Coefficient: {fit_twfe.coef().values[0]:.4f}")
print(f"Std. Error: {fit_twfe.se().values[0]:.4f}")
print(f"t-statistic: {fit_twfe.tstat().values[0]:.4f}")
print(f"p-value: {fit_twfe.pvalue().values[0]:.4f}")
print(f"95% CI: [{fit_twfe.confint().values[0, 0]:.2f}, {fit_twfe.confint().values[0, 1]:.2f}]")

Coefficient: 25.3149
Std. Error: 0.5851
t-statistic: 43.2655
p-value: 0.0000
95% CI: [24.13, 26.50]

The .tidy() method returns a full DataFrame of results:

print(fit_twfe.tidy())

 Estimate Std. Error t value Pr(>|t|) 2.5% 97.5%
Coefficient
txp 25.314897 0.585106 43.265472 0.0 24.125818 26.503976

7.5 Comparison across specifications

All three specifications produce essentially the same DiD estimate:

Method	Estimate	Std. Error	95% CI	SE Type
OLS / HC1	25.315	0.615	[24.09, 26.54]	Heteroskedasticity-robust
TWFE / CRV1	25.315	0.585	[24.13, 26.50]	Cluster-robust (school)
TWFE + Cov / CRV1	25.328	0.605	[24.10, 26.56]	Cluster-robust (school)

The point estimates range from 25.315 to 25.328 — a difference of just 0.013 GPA points. The design (treatment assignment, fixed effects) does the heavy lifting; the choice of specification has negligible impact on the estimate.

8. Inference Comparison

One of PyFixest’s strengths is the ability to quickly compare different inference approaches on the same model. Here we estimate the TWFE model four times, each with a different variance-covariance estimator:

vcov_types = {
"iid": "iid",
"HC1": "HC1",
"CRV1": {"CRV1": "id"},
"CRV3": {"CRV3": "id"},
}
for label, vcov_spec in vcov_types.items():
fit_tmp = pf.feols("gpa ~ txp | id + time", data=df, vcov=vcov_spec)
tidy = fit_tmp.tidy()
txp_row = tidy[tidy.index == "txp"].iloc[0]
print(f" {label:5s}: SE = {txp_row['Std. Error']:.4f}, "
f"t = {txp_row['t value']:.2f}, "
f"p = {txp_row['Pr(>|t|)']:.4f}")

 iid : SE = 0.6071, t = 41.70, p = 0.0000
HC1 : SE = 0.5852, t = 43.26, p = 0.0000
CRV1 : SE = 0.5851, t = 43.27, p = 0.0000
CRV3 : SE = 0.6373, t = 39.72, p = 0.0000

SE Type	Description	SE	t-stat
iid	Classical (assumes homoskedasticity)	0.607	41.70
HC1	Heteroskedasticity-robust (White)	0.585	43.26
CRV1	Cluster-robust at school level	0.585	43.27
CRV3	Bias-corrected cluster-robust (Bell-McCaffrey)	0.637	39.72

iid assumes constant variance — a strong assumption rarely justified in practice
HC1 allows for heteroskedasticity but not within-cluster correlation
CRV1 is the workhorse for panel data: it accounts for arbitrary within-school correlation
CRV3 applies a small-sample bias correction, producing the most conservative SEs

The key takeaway: inference choice matters less than research design. Standard errors range from 0.585 to 0.637, but all four methods produce p-values that are essentially zero. When the treatment effect is 25 GPA points and the largest SE is 0.64, the t-statistic is still above 39. The signal-to-noise ratio is so strong that the choice of variance estimator is practically irrelevant here.

In applications with smaller effects or fewer clusters, the choice between CRV1 and CRV3 can make the difference between statistical significance and not. With only 35 clusters, CRV3 is the safer default.

9. Publication-Quality Tables with etable() and Great Tables

9.1 Stepwise specifications with csw0()

PyFixest’s csw0() operator lets you estimate multiple specifications in a single call. The csw0 (“cumulative stepwise from zero”) starts with a baseline model and progressively adds covariates:

fit_multi = pf.feols("gpa ~ txp + csw0(female_share) | id + time",
data=df, vcov={"CRV1": "id"})

This single line estimates two models: (1) gpa ~ txp | id + time and (2) gpa ~ txp + female_share | id + time. In Stata, you would need two separate regression commands; PyFixest handles it in one formula.

9.2 etable() output

The etable() method generates a publication-style regression table as a Great Tables object:

print(fit_multi.etable())

 (1) (2)
txp 25.315*** (0.585) 25.328*** (0.605)
female_share -3.216 (8.700)
---
FE: time x x
FE: id x x
Observations 70 70
S.E. type by: id by: id
R² 0.995 0.995
R² Within 0.981 0.982

The etable() output shows significance stars, standard errors in parentheses, fixed effects indicators, and model diagnostics — all formatted for immediate inclusion in a paper.

9.3 Custom Great Tables table

For full control over formatting, we can build a table from the .tidy() DataFrames:

rows = []
for name, fit in [("(1) OLS", fit_ols),
("(2) TWFE", fit_twfe),
("(3) TWFE + Cov", fit_cov)]:
tidy = fit.tidy()
txp_row = tidy[tidy.index == "txp"].iloc[0]
rows.append({
"Model": name,
"Estimate": txp_row["Estimate"],
"Std. Error": txp_row["Std. Error"],
"t value": txp_row["t value"],
"p-value": txp_row["Pr(>|t|)"],
"95% CI Lower": txp_row["2.5%"],
"95% CI Upper": txp_row["97.5%"],
"N": fit._N,
})
gt_df = pd.DataFrame(rows)
gt_table = (
GT(gt_df)
.tab_header(
title=md("**Table 2: DiD Estimates Across Specifications**"),
subtitle="Dependent variable: GPA"
)
.fmt_number(columns=["Estimate", "Std. Error", "t value",
"95% CI Lower", "95% CI Upper"], decimals=3)
.fmt_number(columns=["p-value"], decimals=4)
.fmt_integer(columns=["N"])
.tab_source_note(
"Notes: (1) OLS with HC1 robust SE. (2) TWFE with CRV1 "
"clustered at school level. (3) TWFE with female_share covariate and CRV1."
)
)
gt_table.save("did101_table2.png")

Great Tables provides fine-grained control over number formatting (.fmt_number()), column labels (.cols_label()), headers (.tab_header()), and styling (.tab_style()). The .save() method exports to PNG using a headless browser.

9.4 Exporting LaTeX tables for manuscripts

When submitting to academic journals, you need LaTeX-formatted tables rather than HTML or PNG. PyFixest’s etable() can generate publication-ready LaTeX directly by setting type="tex". The output uses booktabs for clean horizontal rules and threeparttable for properly aligned footnotes — the standard format expected by most economics and social science journals.

latex_output = pf.etable(
[fit_ols, fit_twfe, fit_cov],
type="tex",
labels={
"txp": "Treatment $\\times$ Post",
"treated": "Treatment",
"post": "Post",
"female_share": "Female Share",
"Intercept": "Constant",
},
notes="Standard errors in parentheses. * p<0.05, ** p<0.01, *** p<0.001.",
)
print(latex_output)

\begin{threeparttable}
\begin{tabular}{lcccc}
\toprule
& \multicolumn{3}{c}{gpa} \\
\cmidrule(lr){2-4}
& (1) & (2) & (3) \\
\midrule
Treatment & \makecell{-11.049*** \\ (0.288)} & & \\
Post & \makecell{10.886*** \\ (0.339)} & & \\
Treatment:Post & \makecell{25.315*** \\ (0.615)} & \makecell{25.315*** \\ (0.585)} & \makecell{25.328*** \\ (0.605)} \\
Female Share & & & \makecell{-3.216 \\ (8.700)} \\
Constant & \makecell{71.215*** \\ (0.218)} & & \\
\midrule
id & - & x & x \\
time & - & x & x \\
\midrule
Observations & 70 & 70 & 70 \\
S.E. type & hetero & by: id & by: id \\
$R^2$ & 0.989 & 0.995 & 0.995 \\
$R^2$ Within & - & 0.981 & 0.982 \\
\bottomrule
\end{tabular}
\footnotesize Standard errors in parentheses. * p<0.05, ** p<0.01, *** p<0.001.
\end{threeparttable}

To save the table directly to a .tex file that you can \input{} in your manuscript, use the file_name parameter:

pf.etable(
[fit_ols, fit_twfe, fit_cov],
type="tex",
labels={
"txp": "Treatment $\\times$ Post",
"treated": "Treatment",
"post": "Post",
"female_share": "Female Share",
"Intercept": "Constant",
},
notes="Standard errors in parentheses. * p<0.05, ** p<0.01, *** p<0.001.",
file_name="did101_table2.tex",
)

This saves the file to did101_table2.tex. In your LaTeX manuscript, include it with:

\begin{table}[htbp]
\centering
\caption{DiD Estimates Across Specifications}
\label{tab:did-results}
\input{did101_table2.tex}
\end{table}

The labels dictionary maps internal variable names to publication-friendly labels (e.g., "txp" becomes "Treatment $\times$ Post"). The notes parameter adds a footnote below the table. Your LaTeX document needs the booktabs, makecell, and threeparttable packages in the preamble:

\usepackage{booktabs}
\usepackage{makecell}
\usepackage{threeparttable}

10. Coefficient Comparison

A coefficient plot provides a visual comparison of the DiD estimate across specifications, including 95% confidence intervals:

fig, ax = plt.subplots(figsize=(9, 5))
model_names = ["(1) OLS\nHC1", "(2) TWFE\nCRV1", "(3) TWFE+Cov\nCRV1"]
estimates = [fit.tidy().loc["txp", "Estimate"] for fit in [fit_ols, fit_twfe, fit_cov]]
ci_lower = [fit.tidy().loc["txp", "2.5%"] for fit in [fit_ols, fit_twfe, fit_cov]]
ci_upper = [fit.tidy().loc["txp", "97.5%"] for fit in [fit_ols, fit_twfe, fit_cov]]
ax.errorbar(estimates, range(3), xerr=[[e-l for e,l in zip(estimates, ci_lower)],
[u-e for e,u in zip(estimates, ci_upper)]],
fmt="o", color=TEAL, markersize=10, capsize=6, elinewidth=2)
ax.set_xlabel("DiD Estimate (txp coefficient)")
ax.set_title("Coefficient Comparison Across Specifications")

The near-identical point estimates and overlapping confidence intervals across all three specifications reinforce that the DiD estimate is robust. The point estimates span a range of just 0.013 GPA points (25.315 to 25.328), demonstrating remarkable stability regardless of whether we include school fixed effects, time fixed effects, or time-varying covariates.

11. Event Study: Dynamic Treatment Effects

11.1 Loading the event study data

The 2×2 design tells us whether the program had an effect, but not when the effect kicked in or whether it grew or faded over time. The event study dataset extends the analysis to 8 time periods (4 pre-treatment and 4 post-treatment):

url_event = "https://github.com/quarcs-lab/data-open/raw/master/isds/tutoring_didevent.dta"
df_event = pd.read_stata(url_event).astype(float)
print(f"Shape: {df_event.shape}")

Shape: (280, 8)

The new variable timeToTreat measures periods relative to treatment onset for treated schools: −4 through −1 are pre-treatment periods, 0 through 3 are post-treatment. Untreated schools have NaN for this variable (they are never treated).

print(df_event["timeToTreat"].value_counts().sort_index())

timeToTreat
-4.0 10
-3.0 10
-2.0 10
-1.0 10
0.0 10
1.0 10
2.0 10
3.0 10

Each of the 10 treated schools contributes one observation per relative time period, giving 10 observations at each event time.

11.2 The event study specification

The event study model replaces the single txp interaction with a full set of event-time indicators, one for each period relative to treatment. We omit one period (the reference period, $t = -1$) to avoid perfect collinearity:

$$Y_{it} = \alpha + \sum_{j=-m}^{q} \theta_j \cdot \text{treat}_{it}(t = k + j) + \gamma_i + \vartheta_t + \varepsilon_{it}$$

Each $\theta_j$ measures the treatment effect at event time $j$:

Pre-treatment coefficients ($j < 0$): These should be near zero if parallel trends holds. Significant pre-treatment coefficients would indicate that treated schools were already diverging before the program — a red flag for the DiD design.
Post-treatment coefficients ($j \geq 0$): These capture the dynamic treatment effect at each lag after the program starts.

11.3 Estimation with i()

PyFixest’s i() function creates factor (indicator) variables with a specified reference level — perfect for event study designs:

df_event["timeToTreat"] = df_event["timeToTreat"].fillna(-99)
fit_event = pf.feols("gpa ~ i(timeToTreat, ref=-1) | id + time",
data=df_event, vcov={"CRV1": "id"})
print(fit_event.summary())

Estimation: OLS
Dep. var.: gpa, Fixed effects: id+time
Inference: CRV1
Observations: 280
| Coefficient | Estimate | Std. Error | t value | Pr(>|t|) | 2.5% | 97.5% |
|:--------------|-----------:|-------------:|----------:|-----------:|-------:|--------:|
| t = -4 | 0.342 | 0.401 | 0.852 | 0.400 | -0.474 | 1.157 |
| t = -3 | -0.322 | 0.441 | -0.730 | 0.471 | -1.219 | 0.575 |
| t = -2 | 0.593 | 0.423 | 1.401 | 0.170 | -0.267 | 1.454 |
| t = 0 | 25.028 | 0.445 | 56.232 | 0.000 | 24.123 | 25.932 |
| t = 1 | 24.705 | 0.559 | 44.174 | 0.000 | 23.569 | 25.842 |
| t = 2 | 24.768 | 0.739 | 33.534 | 0.000 | 23.267 | 26.270 |
| t = 3 | 25.701 | 0.797 | 32.268 | 0.000 | 24.083 | 27.320 |
---
RMSE: 1.134 R2: 0.991 R2 Within: 0.961

Note: Coefficient names simplified for readability. PyFixest outputs C(timeToTreat, contr.treatment(base=-1))[T.X.0] notation, where X is the relative time period.

The i(timeToTreat, ref=-1) syntax tells PyFixest to create indicator variables for each unique value of timeToTreat, using $t = -1$ as the reference period (coefficient normalized to zero). We fill NaN values with −99 for untreated schools — this creates a dummy that gets absorbed by the school fixed effects, keeping the remaining coefficients interpretable.

11.4 Event study plot

The event study plot is the signature visualization for DiD designs. Pre-treatment coefficients near zero validate the parallel trends assumption, while post-treatment coefficients reveal the dynamic treatment effect:

The plot shows a textbook event study pattern:

Pre-treatment (t = −4 to −2): Coefficients are small (0.34, −0.32, 0.59) and statistically insignificant (all p > 0.17). The confidence intervals comfortably include zero. This is strong evidence supporting the parallel trends assumption — treated and comparison schools were following similar trajectories before the program.
Post-treatment (t = 0 to 3): An immediate, sharp jump to ≈25 points at the moment of treatment, with the effect remaining remarkably stable across all four post-treatment periods (24.71 to 25.70). There is no evidence of fade-out or dynamic adjustment — the program’s effect is both immediate and sustained.

11.5 Event study coefficients table

Period	Estimate	95% CI	Significant?
t = −4	0.342	[−0.47, 1.16]	No
t = −3	−0.322	[−1.22, 0.57]	No
t = −2	0.593	[−0.27, 1.45]	No
t = −1	0.000	(reference)	—
t = 0	25.028	[24.12, 25.93]	Yes (p < 0.001)
t = 1	24.705	[23.57, 25.84]	Yes (p < 0.001)
t = 2	24.768	[23.27, 26.27]	Yes (p < 0.001)
t = 3	25.701	[24.08, 27.32]	Yes (p < 0.001)

The event study confirms three critical findings: (1) no pre-trends — the design is credible, (2) immediate effect — the program works from day one, and (3) sustained impact — no fade-out over four post-treatment periods.

12. Discussion

12.1 Four key findings

The naive before-after comparison overstates the effect by 43%. The raw change in treated schools is 36.20 GPA points, but 10.88 of these points reflect a common upward trend shared by all schools. DiD correctly attributes only 25.32 points to the program.
The event study confirms no differential pre-trends. All three pre-treatment coefficients (0.34, −0.32, 0.59) are small, close to zero, and statistically insignificant. The parallel trends assumption is well-supported by the data.
The effect is immediate and sustained. Post-treatment coefficients range from 24.71 to 25.70, showing no evidence of delayed onset, gradual ramp-up, or fade-out. The tutoring program’s impact is both immediate and remarkably stable.
Inference choice matters less than design. Standard errors ranged from 0.585 (CRV1) to 0.637 (CRV3) across four inference methods, but all produced t-statistics above 39. When the research design is clean and the signal is strong, the choice of variance estimator is practically irrelevant.

12.2 Caveats

This tutorial uses simulated data designed to illustrate DiD mechanics cleanly. In real applications, you should expect:

R-squared below 0.99: The R² of 0.995 in our TWFE models is unrealistically high. Real education data has far more noise.
Smaller treatment effects: A 25-point GPA increase is enormous. Real programs typically produce single-digit effects.
Imperfect parallel trends: Pre-treatment coefficients may not be exactly zero, requiring judgment about how much deviation is acceptable.
Staggered treatment timing: When different units receive treatment at different times, the standard TWFE estimator can be biased. Modern DiD estimators (Callaway & Sant’Anna, 2021; Gardner, 2022) address this.

13. Summary and Takeaways

DiD removes common time trends. The naive approach overstated the effect by 10.88 GPA points — exactly the secular trend captured by the comparison group.
Multiple approaches produce one answer. Three specifications (OLS, TWFE, TWFE with covariate) all yield a DiD estimate of 25.32–25.33, demonstrating the robustness of the design.
Event studies test parallel trends. Pre-treatment coefficients (0.34, −0.32, 0.59) are close to zero and insignificant, providing strong evidence that treated and comparison schools were on similar trajectories before the program.
The effect is immediate and sustained. Post-treatment coefficients (24.71–25.70) show no evidence of delayed onset or fade-out.
Inference flexibility is a PyFixest strength. Switching between iid, HC1, CRV1, and CRV3 requires only changing the vcov argument — no need for separate packages or commands.
etable() and Great Tables replace manual table construction. The csw0() operator estimates multiple specifications in one call, and etable() produces publication-ready output. For custom formatting, Great Tables provides full control via .tab_header(), .fmt_number(), .tab_style(), and .save().

14. Exercises

Robustness check: Load the event study dataset and collapse it to a 2×2 design by averaging GPA across all pre-treatment periods and all post-treatment periods for each school. Re-estimate the DiD. Does the estimate match the 2×2 result?
Inference sensitivity: Estimate the TWFE model with all four SE types (iid, HC1, CRV1, CRV3). At what significance level (α) would the choice of SE type change your conclusion about statistical significance? How few clusters would you need before the choice starts to matter?
Staggered DiD: Read about PyFixest’s did2s() function for the Gardner (2022) two-stage DiD estimator. How would you adapt this tutorial’s code to handle a setting where the 10 treated schools received the program at different times?

References

Corral, D. & Yang, M. (2024). An introduction to the difference-in-differences design in education policy research. Asia Pacific Education Review.
Callaway, B. & Sant’Anna, P.H. (2021). Difference-in-differences with multiple time periods. Journal of Econometrics, 225(2), 200–230.
Goodman-Bacon, A. (2021). Difference-in-differences with variation in treatment timing. Journal of Econometrics, 225(2), 254–277.
Sun, L. & Abraham, S. (2021). Estimating dynamic treatment effects in event studies with heterogeneous treatment effects. Journal of Econometrics, 225(2), 175–199.
Borusyak, K., Jaravel, X. & Spiess, J. (2024). Revisiting event-study designs: robust and efficient estimation. Review of Economic Studies, 91(6), 3253–3285.
Baker, A.C., Larcker, D.F. & Wang, C.C.Y. (2022). How much should we trust staggered difference-in-differences estimates? Journal of Financial Economics, 144(2), 370–395.
Correia, S. (2016). REGHDFE: Stata module to perform linear or instrumental-variable regression absorbing any number of high-dimensional fixed effects.
Fischer, A. & Schar, A. (2024). PyFixest: Fast high-dimensional fixed effects estimation in Python.
Great Tables: Presentation-ready display tables. Posit PBC.
Gardner, J. (2022). Two-stage differences in differences. Working paper.

Introduction to Difference-in-Differences (DiD) in Stata

Sat, 25 Apr 2026 00:00:00 +0000

Overview

How can we evaluate whether a government program actually works when a randomized controlled trial (RCT) is not feasible? Education researchers frequently face this challenge: a new policy is rolled out in some schools but not others, and we need to know whether it made a difference. Difference-in-Differences (DiD) is one of the most widely used quasi-experimental designs for answering this kind of causal question.

In this tutorial, we introduce the DiD method through a case study based on Corral and Yang (2024). A fictitious government implements an after-school tutoring program in 10 of 35 high schools to improve the GPA of low-income students. We compare these treated schools against 25 comparison schools that did not receive the program. Our goal is to estimate the Average Treatment Effect on the Treated (ATT) — by how many GPA points did the program improve academic performance?

We progress from a naive before-after comparison (which overstates the effect) to the full DiD regression framework, demonstrate five equivalent estimation approaches in Stata, and extend the analysis with an event study design that tests whether the parallel trends assumption holds. By the end, we find that the tutoring program increased GPA by approximately 25.32 points on a 0-100 scale — a large and statistically significant effect.

Learning objectives

Understand why naive before-after comparisons overstate treatment effects
Implement the 2x2 DiD design manually and via regression
Estimate the DiD using five equivalent Stata commands (diff, reg, didregress, xtreg, reghdfe)
Assess the parallel trends assumption using an event study design
Interpret event study coefficients as evidence for or against parallel pre-trends

Study design

The following diagram summarizes the case study setup and the analytical approach we follow throughout this tutorial.

graph LR
subgraph "Case Study Setting"
A["<b>35 High Schools</b><br/>in One Region"]
B["<b>10 Treated Schools</b><br/>(tutoring program)"]
C["<b>25 Comparison Schools</b><br/>(no program)"]
A --> B
A --> C
end
subgraph "DiD Design"
D["<b>Pre-Program</b><br/>GPA at baseline"]
E["<b>Post-Program</b><br/>GPA after intervention"]
F["<b>DiD Estimate</b><br/>ATT = 25.32"]
D --> E --> F
end
subgraph "Estimation Methods"
G["<b>Manual 2x2</b><br/>Subtraction"]
H["<b>TWFE Regression</b><br/>5 approaches"]
I["<b>Event Study</b><br/>Dynamic effects"]
G --> H --> I
end
C --> D
F --> G
style A fill:#6a9bcc,stroke:#141413,color:#fff
style B fill:#d97757,stroke:#141413,color:#fff
style C fill:#6a9bcc,stroke:#141413,color:#fff
style F fill:#00d4c8,stroke:#141413,color:#fff
style I fill:#00d4c8,stroke:#141413,color:#fff

The study uses panel data: the same 35 schools are observed at two time points (pre- and post-program), giving us 70 school-period observations. For the event study extension, we use an expanded dataset with 8 time periods (280 observations), allowing us to test for parallel pre-trends and examine dynamic treatment effects.

AI Podcast: Introduction to DiD in Stata

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AI Video: Introduction to DiD in Stata

Setup and packages

Before running the analysis, we install the required Stata packages. The capture prefix ensures the script does not fail if a package is already installed.

capture ssc install diff_plot, replace
capture ssc install diff, replace
capture net install ftools, from("https://raw.githubusercontent.com/sergiocorreia/ftools/master/src/") replace
capture ftools, compile
capture net install reghdfe, from("https://raw.githubusercontent.com/sergiocorreia/reghdfe/master/src/") replace
capture ssc install panelview, replace
capture ssc install eventdd, replace
capture ssc install matsort, replace
capture ssc install outreg2, replace

Package	Purpose
`diff`, `diff_plot`	Simple DiD estimation and visualization
`ftools`, `reghdfe`	High-dimensional fixed effects regression
`panelview`	Treatment timing visualization
`eventdd`	Event study estimation
`outreg2`	Formatted regression tables

Data loading and exploration

We load the 2x2 DiD dataset directly from GitHub. This simulated dataset contains school-level panel data with GPA outcomes for low-income students.

use "https://github.com/quarcs-lab/data-open/raw/master/isds/tutoring_did.dta", clear
describe
summarize
xtset id time
xtsum

Observations: 70
Variables: 7
Variable | Obs Mean Std. dev. Min Max
-------------+-------------------------------------------------
id | 70 18 10.17 1 35
time | 70 1.5 0.50 1 2
treated | 70 0.286 0.46 0 1
gpa | 70 77.12 10.88 59.39 99.15
female_share | 70 0.528 0.03 0.47 0.57
Panel variable: id (strongly balanced)
Time variable: time, 1 to 2

The dataset covers 35 schools observed at two time points (70 total observations). Ten schools (28.6%) are in the treated group and received the after-school tutoring program, while 25 schools serve as the comparison group. The panel is strongly balanced, meaning every school is observed in both periods with no missing data. GPA ranges from 59.4 to 99.2 on a 0-100 scale, with substantial variation (SD = 10.88). The xtsum output reveals that most GPA variation is within-school over time (within SD = 10.82) rather than between schools (between SD = 1.12), suggesting that a large treatment effect drives the time-series variation.

Treatment visualization

The panelview command provides a visual overview of the treatment timing. Each row is a school, and the shading indicates treatment status across time periods.

panelview gpa txp, i(id) t(time) type(treat) ///
prepost bytiming ///
xtitle("Time Period") ytitle("School ID") ///
legend(position(6))

The heatmap confirms a clean treatment design: all 10 treated schools (IDs 26-35) switch from pre-treatment (teal) to post-treatment (dark blue) simultaneously at time 2, while the 25 comparison schools (IDs 1-25) remain untreated throughout. There is no staggering — every treated school receives the program at the same time. This is the ideal setup for the standard 2x2 DiD design.

The problem with naive comparisons

Before introducing the DiD method, let us see what happens if we simply compare the treated group’s GPA before and after the program. This approach is called an Interrupted Time Series (ITS) — it tracks a single group over time and attributes any change to the intervention.

preserve
collapse (mean) gpa, by(time treated)
twoway (connected gpa time if treated==1, ///
msymbol(O) mcolor(gs1) lcolor(gs1) ///
ylab(0(10)100) xlab(1(1)2)), ///
ytitle("GPA") xtitle("Time") ///
xline(1.5, lcolor(red) lpattern(dash))
graph export "stata_did_its.png", replace width(2400)
restore

The treated group’s average GPA jumped from 60.17 (pre-program) to 96.37 (post-program), a raw increase of 36.20 GPA points. At first glance, this looks like a spectacular program effect. However, this naive comparison is misleading because it ignores secular time trends — students' GPA may naturally improve over time due to maturation, grade inflation, or other factors unrelated to the tutoring program. Without a comparison group, we cannot distinguish the program’s causal effect from these natural trends. This is precisely where the DiD design helps.

The DiD design: using a comparison group

The key insight of DiD is to use the comparison group’s change over time as a proxy for what would have happened to the treated group in the absence of the program. This unobserved scenario is called the counterfactual.

The counterfactual and parallel trends

preserve
collapse (mean) gpa, by(time treated)
* Add counterfactual observations
* Counterfactual = treated_pre + control_change
insobs 2
replace time = 1 in 5
replace time = 2 in 6
replace treated = 2 in 5
replace treated = 2 in 6
replace gpa = 60.17 in 5
replace gpa = 71.05 in 6
twoway (connected gpa time if treated==1, msymbol(O) mcolor(gs1) lcolor(gs1)) ///
(connected gpa time if treated==0, msymbol(+) mcolor(gs5) lcolor(gs5)) ///
(connected gpa time if treated==2, msymbol(O) mcolor(gs1) lcolor(gs1) lpattern(shortdash_dot)), ///
ylab(0(10)100) xlab(1(1)2) ///
legend(order(1 "Treated" 2 "Comparison" 3 "Counterfactual")) ///
ytitle("GPA") xtitle("Time")
graph export "stata_did_counterfactual.png", replace width(2400)
restore

Figure 2 shows three lines: the actual treated group (solid, rising sharply from 60.17 to 96.37), the comparison group (rising gently from 71.22 to 82.10), and the counterfactual (dashed line, showing where the treated group would have ended up without the program, at approximately 71.05). The gap between the actual treated outcome (96.37) and the counterfactual (71.05) is the DiD estimate of approximately 25.32 GPA points. The counterfactual is constructed by assuming the treated group would have experienced the same time trend as the comparison group — this is the parallel trends assumption, the fundamental assumption underlying DiD.

The parallel trends assumption

The parallel trends assumption states that in the absence of treatment, the difference between the treated and comparison groups would have remained constant over time. Formally:

$$E[Y_{i,1}(0) - Y_{i,0}(0) \mid D=1] = E[Y_{i,1}(0) - Y_{i,0}(0) \mid D=0]$$

In words, this says that the expected change in the untreated potential outcome over time is the same for both groups. Here, $Y_{i,t}(0)$ is the potential outcome for school $i$ at time $t$ without treatment, and $D$ is the treatment indicator. If this assumption holds, then the comparison group’s observed change serves as a valid estimate of what the treated group’s change would have been without the program. We cannot test this assumption directly (because we never observe the treated group’s outcome without treatment), but we can check whether the two groups followed parallel pre-trends before the intervention — a topic we address in the event study section.

The SUTVA assumption

A second assumption, the Stable Unit Treatment Value Assumption (SUTVA), requires two conditions: (1) one school’s treatment does not affect another school’s outcome (no spillovers — for example, students do not transfer between treated and untreated schools in response to the program), and (2) the treatment is applied consistently across all treated schools (no hidden variations in the tutoring program). SUTVA matters because if students transfer to treated schools or if the program varies in quality, our estimate could be biased.

Manual DiD calculation

The 2x2 DiD estimate is computed by subtracting the comparison group’s change from the treated group’s change. This “double difference” removes both baseline differences between groups and common time trends.

DiD means table (Table 1)

table treated post, stat(mean gpa) nformat(%12.2f)

 | Pre Post Diff
--------------------------+----------------------------
Control (25 schools) | 71.22 82.10 10.88
Treated (10 schools) | 60.17 96.37 36.20
--------------------------+----------------------------
DiD estimate | 25.32

Formally, the DiD estimator takes the following form:

$$DiD = \Big(E[Y_{i,1} \mid D=1] - E[Y_{i,0} \mid D=1]\Big) - \Big(E[Y_{i,1} \mid D=0] - E[Y_{i,0} \mid D=0]\Big)$$

In words, this says: take the treated group’s change over time (36.20) and subtract the comparison group’s change over time (10.88). The result (25.32) is the causal effect of the program, after removing the natural time trend. Think of it like measuring two runners' speed improvements between races: if both were expected to improve equally due to training, any extra improvement by the runner who received coaching can be attributed to the coaching itself. The comparison group’s 10.88-point improvement represents the natural “training effect,” and the remaining 25.32 points represent the “coaching effect” — the tutoring program.

DiD visualization

The diff_plot command produces a visual summary of the DiD, showing both groups' trajectories and the parallel trend line.

diff_plot gpa, group(treated) time(post)
graph export "stata_did_diff_plot.png", replace width(2400)

The plot labels each group’s mean GPA at both time points (60.17, 71.22, 96.37, 82.10) and displays the intervention effect of 25.31 GPA points. The dashed green line extending from the treated group’s pre-period mean shows the counterfactual trajectory under the parallel trends assumption. The vertical gap between the actual treated outcome and this counterfactual is the DiD estimate.

Formal DiD table

The diff command provides a formal DiD estimation with standard errors and significance tests.

diff gpa, treated(treated) period(post)

DIFFERENCE-IN-DIFFERENCES ESTIMATION RESULTS
Number of observations in the DIFF-IN-DIFF: 70
Outcome var. | gpa | S. Err. | |t| | P>|t|
----------------+---------+---------+---------+---------
Before
Diff (T-C) | -11.049 | 0.443 | -24.94 | 0.000***
After
Diff (T-C) | 14.266 | 0.443 | 32.20 | 0.000***
Diff-in-Diff | 25.315 | 0.627 | 40.40 | 0.000***
R-square: 0.99

The DiD estimate of 25.315 (SE = 0.627, t = 40.40, p < 0.001) is highly statistically significant and precisely estimated. Before the program, treated schools had GPAs 11.05 points lower than comparison schools (p < 0.001). After the program, treated schools had GPAs 14.27 points higher than comparison schools (p < 0.001). This reversal from a significant deficit to a significant advantage is one of the most compelling patterns in the data, and it is entirely attributable to the tutoring program under the DiD assumptions.

DiD via regression

While the manual subtraction approach is intuitive, researchers typically prefer regression-based methods because they allow for the inclusion of control variables, flexible standard error estimation, and extension to more complex designs. We demonstrate five equivalent approaches that all converge on the same DiD estimate.

Classical DiD regression

The simplest regression formulation explicitly includes the treatment indicator, the time indicator, and their interaction:

$$Y_{it} = \alpha + \beta_1 \text{Treat}_i + \beta_2 \text{Post}_t + \beta_3 (\text{Treat}_i \times \text{Post}_t) + \varepsilon_{it}$$

In words, this says: the outcome for school $i$ at time $t$ is a function of group membership ($\beta_1$), time period ($\beta_2$), and their interaction ($\beta_3$). The coefficient $\beta_3$ is the DiD estimate — the additional change in the treated group beyond what the comparison group experienced. Here, $\alpha$ is the comparison group’s pre-period mean, $\beta_1$ captures the baseline group difference, $\beta_2$ captures the common time trend, and $\varepsilon_{it}$ is the error term.

reg gpa treated post txp, robust

 gpa | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
treated | -11.04936 .2878309 -38.39 0.000 -11.62404 -10.47469
post | 10.88589 .3389564 32.12 0.000 10.20915 11.56264
txp | 25.3149 .6149733 41.16 0.000 24.08706 26.54273
_cons | 71.21514 .2183689 326.12 0.000 70.77915 71.65113

The regression decomposes the DiD into its building blocks. The constant (71.22) is the comparison group’s pre-period mean GPA. The treated coefficient (-11.05) tells us treated schools started with 11 fewer GPA points than comparison schools at baseline. The post coefficient (10.89) captures the natural time trend shared by both groups. The interaction txp (25.31, SE = 0.61, 95% CI: [24.09, 26.54]) is the DiD estimate, confirming the manual calculation. The tight 95% confidence interval (width of 2.46 points) indicates precise estimation.

Stata built-in DiD

Stata 17 introduced the didregress command, which estimates the DiD directly and labels the output as ATET (Average Treatment Effect on the Treated).

didregress (gpa) (txp), group(id) time(time)

ATET
txp (1 vs 0) | 25.3149 .8337103 30.36 0.000 23.62059 27.0092

The point estimate (25.31) is identical, but the standard error is larger (0.83 vs. 0.61) because didregress automatically clusters standard errors at the school level, accounting for within-school correlation of errors. The 95% CI [23.62, 27.01] is wider but still excludes zero by a large margin.

Two-Way Fixed Effects (TWFE)

The TWFE model replaces the explicit Treat and Post indicators with unit fixed effects ($\gamma_i$) and time fixed effects ($\vartheta_t$):

$$Y_{it} = \beta_3 (\text{Treat}_i \times \text{Post}_t) + \gamma_i + \vartheta_t + \varepsilon_{it}$$

In words, this says: after removing all time-invariant school characteristics (captured by $\gamma_i$) and all common time shocks (captured by $\vartheta_t$), the remaining variation in GPA attributable to the treatment interaction is the DiD estimate $\beta_3$. Think of fixed effects like a before-and-after photo filter: by comparing each school only to itself over time, the unit fixed effects automatically strip away all permanent differences between schools — whether they are rich or poor, urban or rural, large or small. The time fixed effects then remove any changes that hit all schools equally (like a nationwide curriculum reform). What remains is the treatment effect. This is equivalent to the classical regression but more flexible for larger panels.

xtreg gpa txp i.time, fe vce(cluster id)

Fixed-effects (within) regression Number of obs = 70
Group variable: id Number of groups = 35
R-squared:
Within = 0.9946
txp | 25.3149 .5851062 43.27 0.000 24.12582 26.50398

The xtreg command with fe estimates the within-school regression with clustered standard errors. The within R-squared of 0.9946 indicates that the treatment interaction alone explains 99.5% of the within-school GPA variation after removing fixed effects. The very high R-squared reflects the simulated nature of the data; real-world applications typically show lower values.

High-dimensional TWFE with reghdfe

The reghdfe command provides a computationally faster alternative for models with many fixed effects. It produces identical results to xtreg but scales better to large datasets.

reghdfe gpa txp, absorb(id time) cluster(id)

 txp | 25.3149 .5851062 43.27 0.000 24.12582 26.50398

The estimate is identical: 25.31 with clustered SE of 0.585 and a 95% CI of [24.13, 26.50].

Adding covariates

Researchers may include exogenous control variables to improve the precision of the DiD estimate. An important caveat is to never control for variables that are affected by the treatment (known as post-treatment bias). The share of female students (female_share) is a safe control because it is determined by school demographics, not by the tutoring program.

reghdfe gpa txp female_share, absorb(id time) cluster(id)

 txp | 25.32806 .6047651 41.88 0.000 24.09903 26.55709
female_share | -3.216239 8.700428 -0.37 0.714 -20.89764 14.46516

Adding the female share control has virtually no effect on the DiD estimate, which shifts from 25.31 to 25.33 (a change of ~0.01 points). The control itself is not statistically significant (p = 0.71), confirming it is unrelated to GPA in this dataset. This result demonstrates that in well-designed DiD settings with proper fixed effects, adding unrelated covariates does not change the estimate but may slightly increase standard errors.

Comparing all five approaches

All five estimation methods converge on the same DiD estimate, as summarized below:

Method	Estimate	SE	95% CI	Clustered
`diff` (manual)	25.315	0.627	–	No
`reg` (OLS interaction)	25.315	0.615	[24.09, 26.54]	No (robust)
`didregress` (Stata 17+)	25.315	0.834	[23.62, 27.01]	Yes
`xtreg` (TWFE)	25.315	0.585	[24.13, 26.50]	Yes
`reghdfe` (HD-TWFE)	25.315	0.585	[24.13, 26.50]	Yes
`reghdfe` + covariate	25.328	0.605	[24.10, 26.56]	Yes

The consistency across methods confirms the robustness of the 25.32-point DiD estimate. The differences in standard errors reflect whether and how clustering is applied. In this simulated dataset, clustering has minimal impact; in real-world applications, school-level clustering typically increases standard errors substantially.

Table 2: Three regression specifications

Following Corral and Yang (2024), we replicate their Table 2 with three specifications to show the stability of the estimate across modeling choices.

* (1) Baseline TWFE, no controls, no clustering
reghdfe gpa i.txp, absorb(id time)
outreg2 using table2.doc, replace keep(1.txp) ///
addtext(Controls, No, Clustered SEs, No) dec(2)
* (2) + Covariate (female_share), no clustering
reghdfe gpa i.txp c.female_share, absorb(id time)
outreg2 using table2.doc, append keep(1.txp) ///
addtext(Controls, Yes, Clustered SEs, No) dec(2)
* (3) No controls, + clustered SEs at school level
reghdfe gpa i.txp, absorb(id time) cluster(id)
outreg2 using table2.doc, append keep(1.txp) ///
addtext(Controls, No, Clustered SEs, Yes) dec(2)

Table 2: Difference-in-Differences Regression Coefficients
(1) (2) (3)
GPA GPA GPA
Treatment 25.31*** 25.33*** 25.31***
(0.607) (0.615) (0.585)
Observations 70 70 70
R-squared 0.99 0.99 0.99
Controls No Yes No
Clustered SEs No No Yes

The three specifications produce nearly identical estimates (25.31, 25.33, 25.31), all significant at the 1% level. This stability is encouraging: the result does not depend on whether we include covariates or cluster standard errors. In column (2), adding the female share control changes the estimate by only 0.02 points. In column (3), clustering at the school level slightly reduces the standard error (from 0.607 to 0.585), which is unusual — in practice, clustering almost always increases SEs because it accounts for within-school error correlation. The R-squared of 0.99 across all specifications reflects the strong treatment effect in the simulated data.

Event study: dynamic treatment effects

The 2x2 DiD assumes that the treatment effect is constant over time. But what if the program takes time to show results, or its effect fades out? An event study design addresses this by replacing the single treatment interaction with a set of time-specific treatment indicators — leads (pre-treatment periods) and lags (post-treatment periods). This serves two purposes: (1) it tests the parallel trends assumption by checking whether pre-treatment coefficients are near zero, and (2) it reveals the dynamic trajectory of the treatment effect.

Event study data

We load the expanded dataset with 8 time periods (4 pre-treatment, 4 post-treatment).

use "https://github.com/quarcs-lab/data-open/raw/master/isds/tutoring_didevent.dta", clear
describe
summarize
xtset id time

Observations: 280 (35 schools x 8 periods)
Variables: 8 (includes timeToTreat: relative time to treatment onset)
Panel variable: id (strongly balanced)
Time variable: time, 1 to 8

The event study dataset extends the case study to 8 time periods, with the tutoring program starting at period 5. The timeToTreat variable measures relative time to treatment onset, ranging from -4 (four periods before treatment) to +3 (three periods after treatment). This variable is defined only for the 10 treated schools (80 observations).

Treatment visualization

panelview gpa txp, i(id) t(time) type(treat) ///
prepost bytiming ///
xtitle("Time Period") ytitle("School ID")

The heatmap shows the same 10 treated schools now observed over 8 periods. The pre-treatment phase (periods 1-4, teal) allows us to assess whether treated and comparison schools followed similar GPA trajectories before the program, while the post-treatment phase (periods 5-8, dark blue) captures the dynamic treatment effects.

Event study model

The event study replaces the single treatment interaction from the TWFE model with a vector of lead and lag indicators:

$$Y_{it} = \alpha + \sum_{j=-m}^{q} \theta_j \cdot \text{treat}_{it}(t = k + j) + \gamma_i + \vartheta_t + \varepsilon_{it}$$

In words, this says: the outcome for school $i$ at time $t$ equals a constant, plus a separate coefficient ($\theta_j$) for each relative time period $j$ from the treatment onset at time $k$, plus school and time fixed effects. The leads ($j < 0$) capture pre-treatment differences, and the lags ($j \geq 0$) capture post-treatment effects. The reference period (typically $j = -1$, the period just before treatment) is omitted, so all coefficients are measured relative to this baseline.

eventdd gpa i.time, timevar(timeToTreat) ///
method(hdfe, absorb(id time) cluster(id)) ///
keepdummies ///
graph_op(ylab(-10(5)30) ///
ytitle("GPA Effect") ///
xtitle("Time to Treatment") ///
xlab(-4(1)4))
graph export "stata_did_event_study.png", replace width(2400)

The event study plot is the most informative figure in the analysis. The pre-treatment coefficients (periods -4 through -2) cluster around zero, with point estimates of 0.34, -0.32, and 0.59 — all statistically insignificant (p = 0.40, 0.47, 0.17). This provides compelling evidence that the parallel trends assumption holds: treated and control schools were following similar GPA trajectories in the four periods before the program started. At the moment of treatment (period 0), the effect jumps sharply to approximately 25 GPA points and remains stable through period +3. The tight confidence intervals (shown in blue) confirm that the effect is precisely estimated in every post-treatment period.

Event study coefficients (Table 4)

outreg2 using table4.doc, replace ///
keep(lead4 lead3 lead2 lag0 lag1 lag2 lag3) dec(2)

Table 4: Event Study Results
Pre-treatment (leads):
lead4 = 0.342 (SE = 0.401) p = 0.400
lead3 = -0.322 (SE = 0.441) p = 0.471
lead2 = 0.593 (SE = 0.423) p = 0.170
Post-treatment (lags):
lag0 = 25.028 (SE = 0.445) p = 0.000
lag1 = 24.705 (SE = 0.559) p = 0.000
lag2 = 24.768 (SE = 0.739) p = 0.000
lag3 = 25.701 (SE = 0.797) p = 0.000
N = 280, 35 schools, R-squared = 0.991

The event study coefficients tell a clear story. Before the program, none of the lead coefficients are statistically significant, and they range from -0.32 to 0.59 — fluctuations well within normal sampling variation. After the program begins, the treatment effect is immediate and persistent: lag coefficients range from 24.71 to 25.70, a span of less than 1 GPA point over four periods. There is no evidence of fade-out (declining effect over time) or ramp-up (gradually increasing effect). The program delivered its full benefit from the first period and maintained it consistently, suggesting a sustained structural change in academic support rather than a temporary boost.

Discussion

Returning to our case study question: Did the after-school tutoring program improve the GPA of low-income students? The evidence is clear. The DiD estimate of 25.32 GPA points is large, statistically significant (p < 0.001), and robust across five estimation methods, multiple regression specifications, and an event study design. The program transformed treated schools from having the lowest average GPA (60.17) to having the highest (96.37).

Three findings merit special attention for policymakers:

The naive before-after comparison overstates the effect by 43%. The ITS approach attributes the entire 36.20-point increase to the program, but 10.88 points (30% of the raw gain) are attributable to natural time trends. DiD corrects for this by netting out the comparison group’s change.
The event study confirms there were no differential pre-trends. All pre-treatment coefficients are near zero and insignificant, supporting the causal interpretation. If treated schools had been improving faster than comparison schools even before the program, our DiD estimate would be biased upward.
The effect is constant over time. The event study shows no fade-out, suggesting the program produces sustained benefits rather than temporary gains. This is important for cost-benefit analyses: policymakers can expect the GPA improvement to persist as long as the program continues.

Important caveats

This tutorial uses simulated data designed to illustrate DiD mechanics cleanly. Several features of this example would be unusual in a real-world application:

The R-squared of 0.99 reflects the simulated data’s low noise. Real educational interventions typically explain a much smaller share of outcome variation.
A 25-point GPA increase on a 100-point scale is unrealistically large. Real after-school programs typically produce effect sizes of 0.1-0.3 standard deviations.
The parallel pre-trends are nearly perfect by construction. In practice, researchers must carefully argue for the plausibility of this assumption using domain knowledge, pre-trend tests, and robustness checks.
This example uses simultaneous treatment timing (all schools treated at once). When treatment timing varies across units — called staggered DiD — the standard TWFE estimator can produce biased estimates. Modern estimators by Callaway and Sant’Anna (2021), Sun and Abraham (2021), and Borusyak et al. (2023) address this issue.

Summary and takeaways

DiD removes time trends: The naive ITS comparison overstated the program effect by 10.88 GPA points (43%). DiD corrects this by subtracting the comparison group’s change, yielding a causal estimate of 25.32 points.
Five methods, one answer: Classical OLS, didregress, xtreg, reghdfe, and reghdfe with covariates all produce the same DiD estimate (25.31-25.33), demonstrating the equivalence of these approaches in the standard 2x2 case.
Event studies test parallel trends: Pre-treatment coefficients (0.34, -0.32, 0.59, all p > 0.10) provide evidence that treated and comparison schools followed similar trajectories before the program, strengthening the causal claim.
The effect is immediate and sustained: Post-treatment coefficients range from 24.71 to 25.70 with no fade-out pattern, suggesting the program delivers lasting benefits.
Covariates matter less than design: Adding the female share control changed the estimate by only ~0.01 points. In a well-designed DiD with proper fixed effects, the research design does the heavy lifting.
Limitations: This tutorial covers the standard 2x2 DiD and event study. For staggered treatment timing (where units receive treatment at different times), modern estimators that avoid the negative-weights problem in TWFE are recommended.

Exercises

Robustness check: Re-estimate the DiD using only the event study dataset (280 observations) with a simple 2x2 specification (collapsing to pre/post). Does the estimate change compared to the 2-period dataset? Why or why not?
Placebo test: Using the event study dataset, restrict the sample to pre-treatment periods only (time 1-4) and assign a “fake” treatment at time 3. Run the DiD. If the parallel trends assumption holds, you should find no significant effect. What do you find?
Staggered DiD: Read about the Callaway and Sant’Anna (2021) estimator and the csdid Stata package. How would the analysis change if schools adopted the tutoring program at different times?