ECON 730: Causal Inference with Panel Data

Lecture 5: Introduction to Difference-in-Differences

Pedro H. C. Sant’Anna

Emory University

Spring 2026

From Experiments to Observational Studies

Roadmap

• Lectures 3–4: Causal inference with randomized panel experiments

  • Design-based identification; Horvitz–Thompson estimation
  • Known treatment assignment mechanism \(\Rightarrow\) clear identification

• Today: What if treatment is not randomized?

• Our approach:

  • The canonical \(2 \times 2\) difference-in-differences (DiD) setup
  • Two groups (treated vs. untreated), two periods (pre vs. post)
  • Replace randomization with parallel trends assumption

• The full journey today:

  • Setup \(\to\) Identification \(\to\) Estimation \(\to\) Inference \(\to\) Application

The Challenge: Selection into Treatment

• In observational studies, units select into treatment
  • States choose to expand Medicaid; firms decide to adopt new technology
• Simple pre–post or treated-vs.-untreated comparisons are biased
  • Treated units may differ from comparison units even without treatment
• Difference-in-Differences (DiD): Exploit the time dimension
  • Use pre-treatment periods to account for selection concerns, as long as additional assumptions (Parallel Trends, No-Anticipation) are met
• The most widely used identification strategy in applied microeconomics
  • See Lecture 1; (Currie, Kleven, and Zwiers 2020; Goldsmith-Pinkham 2024)

The \(2 \times 2\) DiD Setup

Running Example: Medicaid Expansion and Mortality

• Baker et al. (2025): A Practitioner’s Guide to Difference-in-Differences

• Policy: Affordable Care Act (ACA) Medicaid expansion

  • 2014: Some states expand eligibility \(\to\) more residents gain health insurance
  • Other states never expand (through 2019)

• Outcome: County-level mortality rate (ages 20–64), deaths per 100,000

• Question: Did Medicaid expansion reduce mortality?

• \(2 \times 2\) Setup:

  • Two periods: 2013 (pre) and 2014 (post)
  • Two groups: States expanding in 2014 vs. never-expanding states
  • Unit of observation: Counties within states

Data Structure: Two Groups, Two Periods

• Time periods: \(t \in \{1, 2\}\) (e.g., 2013 and 2014)

• Treatment group indicator: \(G_i \in \{2, \infty\}\)

  • \(G_i = 2\): unit \(i\) is first treated at period 2 (expansion states)
  • \(G_i = \infty\): unit \(i\) is never treated (never-expansion states)

• Treatment indicator: \(D_{i,t} = \mathbf{1}\{G_i \leq t\}\)

  • \(D_{i,1} = 0\) for all units (no one is treated at \(t=1\))
  • \(D_{i,2} = \mathbf{1}\{G_i = 2\}\) (only the treated group at \(t=2\))

• Key features:

  • Treatment is binary and absorbing (once treated, stay treated)
  • No one is treated in the first period
  • This is the simplest DiD setup — building block for everything else

Potential Outcomes in the \(2 \times 2\) Setup

• Recall from Lecture 2: potential outcomes indexed by treatment sequence

• Specialize to \(2 \times 2\): Only two possible treatment paths

  • \(Y_{i,t}(\infty)\): outcome if unit \(i\) is never treated (untreated potential outcome)
  • \(Y_{i,t}(2)\): outcome if unit \(i\) is first treated at \(t=2\)

• SUTVA (Stable Unit Treatment Value Assumption):

  • No interference: unit \(i\)’s outcome depends only on \(i\)’s own treatment path
  • No hidden versions of treatment

• Observed outcome: \[ Y_{i,t} = \mathbf{1}\{G_i = 2\} \cdot Y_{i,t}(2) + \mathbf{1}\{G_i = \infty\} \cdot Y_{i,t}(\infty) \]

• This is the same potential outcomes framework from Lectures 2–4, just specialized to two groups and two periods

Target Parameter: The Average Treatment Effect on the Treated

• Parameter of interest: The ATT at period 2 \[ ATT \;=\; \mathbb{E}\big[Y_{i,t=2}(2) - Y_{i,t=2}(\infty) \;\big|\; G_i = 2\big] \]

• Interpretation:

  • Average causal effect of treatment for those who are actually treated
  • Not the ATE — we condition on \(G_i = 2\)

• In the Medicaid example:

  • Average effect of Medicaid expansion on mortality in counties that expanded
  • Not: what would happen if all counties expanded

• The fundamental problem: We observe \(Y_{i,t=2}(2)\) for the treated group, but we never observe \(Y_{i,t=2}(\infty)\) for the treated group

• How do we impute the missing counterfactual?

The DiD Estimand: A Preview

• The DiD estimand takes the form:

\[ \theta^{DiD} = \Big(\mathbb{E}[Y_{i,t=2} | G_i = 2] - \mathbb{E}[Y_{i,t=1} | G_i = 2]\Big) - \Big(\mathbb{E}[Y_{i,t=2} | G_i = \infty] - \mathbb{E}[Y_{i,t=1} | G_i = \infty]\Big) \]

• Four observable group-period means; treated group’s change minus comparison group’s change

Two fundamental questions for the rest of this lecture:

1. How do we arrive at this estimand? Why this particular form?
2. Under what assumptions does \(\boldsymbol{\theta^{DiD} = ATT}\)? Why?

Selection Bias and the Missing Data Problem

The Missing Data Problem: Under SUTVA

• SUTVA \(\Rightarrow\) each unit reveals outcomes from its own treatment path only
• What do we observe vs. what is missing?

	\(Y_{i,t=1}(\infty)\)	\(Y_{i,t=1}(2)\)	\(Y_{i,t=2}(\infty)\)	\(Y_{i,t=2}(2)\)
\(G_i = 2\) (Treated)	?	\(\checkmark\)	?	\(\checkmark\)
\(G_i = \infty\) (Comparison)	\(\checkmark\)	?	\(\checkmark\)	?

• Treated units reveal \(Y_{i,t}(2)\); comparison units reveal \(Y_{i,t}(\infty)\)
• Problem: The ATT requires \(\mathbb{E}[Y_{i,t=2}(\infty) | G_i = 2]\) — a missing cell!
• Can we learn more using No-Anticipation?

Filling In: Adding No-Anticipation

• No-Anticipation \(\Rightarrow\) \(Y_{i,t=1}(2) = Y_{i,t=1}(\infty)\) for all \(i\) (can be relaxed to hold on average among treated)
• So for treated units at \(t=1\): \(Y_{i,t=1} = Y_{i,t=1}(2) = Y_{i,t=1}(\infty)\) — both potential outcomes observed!

	\(Y_{i,t=1}(\infty)\)	\(Y_{i,t=1}(2)\)	\(Y_{i,t=2}(\infty)\)	\(Y_{i,t=2}(2)\)
\(G_i = 2\) (Treated)	\(\checkmark\)*	\(\checkmark\)	?	\(\checkmark\)
\(G_i = \infty\) (Comparison)	\(\checkmark\)	—	\(\checkmark\)	—

*Same as \(Y_{i,t=1}(2)\) under No-Anticipation

• The one remaining missing cell: \(\mathbb{E}[Y_{i,t=2}(\infty) | G_i = 2]\) — the counterfactual for the treated group at \(t=2\)

Selection Bias in the Post-Treatment Comparison

• Naive approach: Compare treated and comparison at \(t=2\)

\[ \begin{aligned} &\mathbb{E}[Y_{i,t=2} | G_i = 2] - \mathbb{E}[Y_{i,t=2} | G_i = \infty] \\[6pt] &= \mathbb{E}[Y_{i,t=2}(2) | G_i = 2] - \mathbb{E}[Y_{i,t=2}(\infty) | G_i = \infty] \\[6pt] &= \underbrace{\mathbb{E}[Y_{i,t=2}(2) - Y_{i,t=2}(\infty) | G_i = 2]}_{ATT} + \underbrace{\mathbb{E}[Y_{i,t=2}(\infty) | G_i = 2] - \mathbb{E}[Y_{i,t=2}(\infty) | G_i = \infty]}_{\color{#b91c1c}{\text{Selection Bias}}} \end{aligned} \]

• The selection bias reflects differences in untreated potential outcomes between groups at \(t=2\)

Worked Example: Job Training Program

• Setting: Job training for disadvantaged workers — workers with lower baseline wages select into the program

Unit	\(Y_{i,t=1}(\infty)\)	Wage	\(Y_{i,t=2}(\infty)\)	\(Y_{i,t=2}(2)\)
A (Trained)	20	20	22	27
B (Trained)	18	18	20	24
C (Not trained)	30	30	32	—
D (Not trained)	28	28	30	—

• True ATT $= = $ 4.5
• Naive: $ - = 25.5 - 31 = $ -5.5 — Selection bias \(= -10\)

Selection bias contaminates naive comparisons.

What assumptions let us recover the ATT?

Assumptions: No-Anticipation and Parallel Trends

Assumption: No Anticipation

Our first assumption ensures that future treatment does not contaminate pre-treatment outcomes:

No-Anticipation. For all units \(i\): \(Y_{i,t=1}(2) = Y_{i,t=1}(\infty)\). Treatment at \(t=2\) has no effect on outcomes at \(t=1\).

For ATT identification, we only need this on average across treated units: \(\mathbb{E}[Y_{i,t=1}(2) | G_i = 2] = \mathbb{E}[Y_{i,t=1}(\infty) | G_i = 2]\)

• Interpretation: Agents do not change behavior before treatment begins
• When it holds: Unexpected policy changes; units unaware of future treatment
• When it fails: Pre-announced policies \(\to\) behavioral adjustments before implementation (Malani and Reif 2015)
• Notational simplification: Under No-Anticipation, \(Y_{i,t=1} = Y_{i,t=1}(\infty)\) for all units

Assumption: Parallel Trends

Parallel Trends (PT). \[ \mathbb{E}[Y_{i,t=2}(\infty) - Y_{i,t=1}(\infty) \;|\; G_i = 2] = \mathbb{E}[Y_{i,t=2}(\infty) - Y_{i,t=1}(\infty) \;|\; G_i = \infty]. \]

• What PT says:
  • In the absence of treatment, both groups would have followed the same trend
  • Allows for permanent level differences between groups
• What PT does NOT require:
  • Same levels of outcomes: \(\mathbb{E}[Y_{i,t}(\infty) | G_i = 2] \neq \mathbb{E}[Y_{i,t}(\infty) | G_i = \infty]\) is fine!
  • Random treatment assignment or no selection into treatment
• What PT rules out: Differential time-varying selection — trends in untreated outcomes differ by group

This PT involves counterfactual outcomes \(\Rightarrow\) fundamentally untestable. We can assess plausibility using pre-treatment data (in a few lectures).

Parallel Trends: Graphical Intuition (1/4)

• Treated group has higher levels and rises more from \(t=1\) to \(t=2\)
• But how much of the rise is due to treatment?

Parallel Trends: Graphical Intuition (2/4)

• The naive comparison at \(t=2\) includes the level difference that existed before treatment
• This is selection bias \(+\) treatment effect, mixed together

Parallel Trends: Graphical Intuition (3/4)

• Under PT: the treated group’s counterfactual trajectory is parallel to the comparison group
• The hollow circle is the missing counterfactual \(\mathbb{E}[Y_{i,t=2}(\infty) | G_i = 2]\)

Parallel Trends: Graphical Intuition (4/4)

• ATT \(=\) (Treated group’s change) \(-\) (Comparison group’s change) \(=\) Difference-in-Differences
• Think: What would this diagram look like if Parallel Trends fails?

When Is Parallel Trends Plausible?

• PT is fundamentally untestable — it concerns counterfactual trends, not observed trends

• Suggestive evidence: Pre-treatment trends can provide support

  • If expansion and non-expansion states had similar mortality trends before 2014, PT is more credible
  • But parallel pre-trends \(\neq\) parallel trends (Roth 2022; Ghanem, Sant’Anna, and Wüthrich 2026)

• Potential violations in the Medicaid example:

  • Opioid crisis differentially affected states — could confound mortality trends
  • Expansion states may have had different health infrastructure investments

• Best practice: Argue for PT using institutional knowledge, not just pre-trend tests. We will revisit this in future lectures

Identification of the ATT

Constructive Imputation: Step by Step

Goal: Show that \(\theta^{DiD} = ATT\) under SUTVA + No-Anticipation + PT.

Start from the ATT:

\[ ATT = \underbrace{\mathbb{E}[Y_{i,t=2}(2) | G_i = 2]}_{\color{#15803d}{\text{observable}}} - \underbrace{\mathbb{E}[Y_{i,t=2}(\infty) | G_i = 2]}_{\color{#b91c1c}{\text{counterfactual}}} \]

Use PT to impute the counterfactual:

\[ \mathbb{E}[Y_{i,t=2}(\infty) | G_i = 2] = \mathbb{E}[Y_{i,t=1}(\infty) | G_i = 2] + \mathbb{E}[Y_{i,t=2}(\infty) | G_i = \infty] - \mathbb{E}[Y_{i,t=1}(\infty) | G_i = \infty] \]

Under No-Anticipation + SUTVA, all terms are observable:

\[ \mathbb{E}[Y_{i,t=2}(\infty) | G_i = 2] = \underbrace{\mathbb{E}[Y_{i,t=1} | G_i = 2]}_{\color{#15803d}{\checkmark\; \text{No-Anticipation}}} + \underbrace{\mathbb{E}[Y_{i,t=2} | G_i = \infty] - \mathbb{E}[Y_{i,t=1} | G_i = \infty]}_{\color{#15803d}{\checkmark\; \text{all observable}}} \]

The DiD Identification Result

Theorem (DiD Identification). Under SUTVA, No-Anticipation, and Parallel Trends: \[ ATT = \theta^{DiD} = \Big(\mathbb{E}[Y_{i,t=2} | G_i = 2] - \mathbb{E}[Y_{i,t=1} | G_i = 2]\Big) - \Big(\mathbb{E}[Y_{i,t=2} | G_i = \infty] - \mathbb{E}[Y_{i,t=1} | G_i = \infty]\Big) \]

• Three assumptions \(\Rightarrow\) observable formula with four population means
• Note: This is an identification result, not an estimation result
  • The formula involves population expectations, not sample averages
  • Estimation comes next

Weighted Expectations: Who Defines the Parameter?

Baker et al. (2025) emphasize: the weights define the ATT

Unweighted DiD:

\[ATT = \mathbb{E}[Y_{i,t=2}(2) - Y_{i,t=2}(\infty) | G_i = 2]\]

• Each unit gets equal weight
• “Average effect per county”

Population-weighted DiD:

\[ATT_\omega = \mathbb{E}_\omega[Y_{i,t=2}(2) - Y_{i,t=2}(\infty) | G_i = 2]\]

• Weight by population \(\omega_i\)
• “Average effect per person”

Key message: Both are valid ATTs, but they answer different questions

• The choice of weights is a substantive decision, not a statistical one
• The choice of weights also impacts the PT assumption!

We know what to estimate.

Now: how to estimate and do inference.

Estimation

The Analogy Principle

• Identification: \(ATT = f(\text{population means})\)
• Estimation: Replace population means with sample analogs
• Notation:
  • \(n\): total sample size; \(n_2\): treated units; \(n_\infty\): comparison units
  • \(\bar{Y}_{g,t} = \frac{1}{n_{g}} \sum_{i:\, G_i = g} Y_{i,t}\): sample mean for group \(g \in \{2, \infty\}\) at time \(t\)
• The DiD estimator (“DiD-by-hand”):

\[ \widehat{\theta}^{DiD} = \big(\bar{Y}_{g=2,t=2} - \bar{Y}_{g=2,t=1}\big) - \big(\bar{Y}_{g=\infty,t=2} - \bar{Y}_{g=\infty,t=1}\big) \]

• Simply the difference of two within-group time changes
• With panel data, this simplifies further…

Panel Data Simplification: First-Differencing

With panel data (same units in both periods), define \(\Delta Y_i = Y_{i,t=2} - Y_{i,t=1}\)

The DiD estimator as a two-sample difference in means: \[ \widehat{\theta}^{DiD} = \overline{\Delta Y}_2 - \overline{\Delta Y}_\infty = \frac{1}{n_2} \sum_{i:\, G_i = 2} \Delta Y_i - \frac{1}{n_\infty} \sum_{i:\, G_i = \infty} \Delta Y_i \]

Two steps:

1. First-difference \(\Rightarrow\) removes time-invariant unit effects
2. Compare treated vs. comparison changes

Requires panel data:

• Same units observed in both periods
• With repeated cross-sections, use the four-means formula

Notation: Treatment Indicator for Regressions

• We use \(G_i\) notation consistently for conditioning and parameters: \(ATT = \mathbb{E}[Y_{i,t=2}(2) - Y_{i,t=2}(\infty) \;|\; G_i = 2]\)

• For regression and influence function formulas, define a binary shorthand: \[ D_i = \mathbf{1}\{G_i = 2\} = \begin{cases} 1 & \text{if unit } i \text{ is in the treated group} \\ 0 & \text{if unit } i \text{ is in the comparison group} \end{cases} \]

• Why this convention?

  • \(G\)-notation generalizes naturally to staggered adoption
  • \(D_i\) is convenient in regression equations where we need 0/1 arithmetic

• Rule: All conditioning, parameters, and potential outcomes use \(G_i\); \(D_i\) appears only in regression specifications and influence function formulas

TWFE Regression

A common way to estimate DiD: Two-Way Fixed Effects (TWFE) regression

Pooled OLS form:

\[Y_{i,t} = \alpha_0 + \gamma_0 D_i + \lambda_0 T_t + \beta^{TWFE} (D_i \times T_t) + \varepsilon_{i,t}\]

where \(T_t = \mathbf{1}\{t = 2\}\), \(D_i = \mathbf{1}\{G_i = 2\}\)

Unit & time FE form:

\[Y_{i,t} = \alpha_i + \lambda_t + \beta^{TWFE} D_{i,t} + \varepsilon_{i,t}\]

where \(D_{i,t} = \mathbf{1}\{G_i \leq t\}\)

• Both are equivalent in the \(2 \times 2\) case with balanced panel data — same \(\hat{\beta}^{TWFE}\)
• Key question: Does \(\hat{\beta}^{TWFE} = \widehat{\theta}^{DiD}\)?
• Answer: Yes! In the \(2 \times 2\) case, they are numerically identical

TWFE = DiD-by-Hand: The Equivalence

• Claim: \(\hat{\beta}^{TWFE} = \widehat{\theta}^{DiD}\) (exact numerical equality)
• Proof sketch: OLS solves four moment conditions:

\[ \begin{aligned} \mathbb{E}_n[\varepsilon_{i,t}] &= 0 & &\Rightarrow \hat{\alpha}_0 = \bar{Y}_{g=\infty,t=1} \\ \mathbb{E}_n[D_i \cdot \varepsilon_{i,t}] &= 0 & &\Rightarrow \hat{\gamma}_0 = \bar{Y}_{g=2,t=1} - \bar{Y}_{g=\infty,t=1} \\ \mathbb{E}_n[T_t \cdot \varepsilon_{i,t}] &= 0 & &\Rightarrow \hat{\lambda}_0 = \bar{Y}_{g=\infty,t=2} - \bar{Y}_{g=\infty,t=1} \\ \mathbb{E}_n[(D_i \cdot T_t) \cdot \varepsilon_{i,t}] &= 0 & &\Rightarrow \hat{\beta}^{TWFE} = \widehat{\theta}^{DiD} \end{aligned} \]

• The four moment conditions uniquely pin down the four group-time means
• Bottom line: In the \(2 \times 2\) case, TWFE regression is just a convenient way to compute the DiD estimator. Nothing more, nothing less.
• Every student in this class should be able to derive these equivalence results!
• Think: Will this equivalence survive when we move to staggered adoption? Why or why not?

Three Equivalent Specifications

Baker et al. (2025): Three OLS specifications, same \(\hat{\beta}\)

Spec	Regression	Data Used
(1) Pooled OLS	\(Y_{i,t} = \alpha + \gamma D_i + \lambda T_t + \beta (D_i \cdot T_t) + \varepsilon_{i,t}\)	Panel (\(2n\) obs)
(2) First-diff	\(\Delta Y_i = \delta + \beta D_i + u_i\)	Panel, FD (\(n\) obs)
(3) Unit & time FE	\(Y_{i,t} = \alpha_i + \lambda_t + \beta D_{i,t} + \varepsilon_{i,t}\)	Panel (\(2n\) obs)

• All three yield identical point estimates for \(\hat{\beta}\) in the \(2 \times 2\) case with balanced panel
• BUT: Standard errors differ unless you cluster appropriately!
  • Spec (1) and (3) have \(2n\) observations; Spec (2) has \(n\) observations
  • Clustering at the unit level in Specs (1) and (3) reconciles the SEs

OLS: Regression as a Means to an End

• Important conceptual point: The regression is a computational device

• In the \(2 \times 2\) case, OLS does not add any statistical content

  • Same estimate as computing four means and subtracting
  • The interpretation comes from the DiD identification argument, not from the regression

• Why use regression then?

  • Convenience: standard software handles SEs and clustering
  • Reporting: tables with coefficients and SEs are standard in economics

• Warning for later: In more complex settings (staggered adoption, heterogeneous effects), TWFE \(\neq\) “the” DiD estimator

  • The \(2 \times 2\) equivalence is special and does not generalize
  • Recall Lecture 3: FE can be biased under carryover effects. Here, with no carryover in the \(2 \times 2\) case, TWFE is well-behaved

Empirical Application: Medicaid Expansion

Application: The ACA and County Mortality

• Data: County-level mortality rates (ages 20–64), deaths per 100,000

  • Source: Baker et al. (2025), replication data from pedrohcgs/JEL-DiD

• Treatment: State-level Medicaid expansion under the ACA

  • 24 states + DC expanded in January 2014
  • 19 states never expanded (through 2019)
  • Drop DC and pre-2014 adopters (DE, MA, NY, VT) for clean \(2 \times 2\)

• \(2 \times 2\) Setup:

  • Pre: 2013, Post: 2014
  • Treated: Counties in states expanding in 2014
  • Comparison: Counties in never-expanding states

• Sample: $$2,300 counties, roughly 900 treated and 1,400 comparison

• Key feature: Counties vary enormously in population size — weighting matters!

Simple \(2 \times 2\) Results: Unweighted vs. Weighted

Baker et al. (2025), Table 2: Simple means and DiD

	Pre (2013)	Post (2014)	Pre (2013)	Post (2014)
	Unweighted		Pop-Weighted
Treated (\(G_i=2\))	419.2	428.5	322.7	326.5
Comparison (\(G_i=\infty\))	474.0	483.1	376.4	382.7
\(\Delta\) Treated	+9.3		+3.7
\(\Delta\) Comparison	+9.1		+6.3
DiD	+0.1		$-$2.6

• Unweighted: DiD \(\approx +0.1\) (SE \(= 3.7\)) (essentially zero, “wrong sign”)
• Population-weighted: DiD \(\approx -2.6\) (SE \(= 1.5\)) (meaningful reduction in mortality)
• Why the difference? Different parameters! “Per county” vs. “per person” ATT

Regression Equivalence: Three Specifications

• All three specifications yield the same point estimate (county-clustered SEs)

	(1) Pooled OLS	(2) First-diff	(3) Unit & time FE
\(\hat{\beta}\) (unweighted)	0.1	0.1	0.1
SE (county-clustered)	(3.7)	(3.7)	(3.7)
\(\hat{\beta}\) (pop-weighted)	$-$2.6	$-$2.6	$-$2.6
SE (county-clustered)	(1.5)	(1.5)	(1.5)
Observations	\(2n\)	\(n\)	\(2n\)

• Point estimates are identical across all three specifications
• County-clustered SEs are also identical (as expected from theory)
• Neither result is statistically significant at conventional levels

R Code: DiD by Hand vs. Regression

DiD by hand:

# Four group-time means
means <- short_data %>%
  group_by(Treat, Post) %>%
  summarise(
    m = mean(crude_rate_20_64))

# DiD estimate
did_hat <- (means$m[4] - means$m[3]) -
           (means$m[2] - means$m[1])

TWFE regression:

library(fixest)

# TWFE with county + year FE
reg <- feols(
  crude_rate_20_64 ~ Treat:Post
  | county_code + year,
  data = short_data,
  cluster = ~stateID)

coef(reg)  # Same as did_hat!

• Both approaches give identical estimates
• short_data: balanced panel with 2013–2014 only (the \(2 \times 2\) subset)
• Full replication code: github.com/pedrohcgs/JEL-DiD

Key Takeaway: Weights Matter

Lesson from the Medicaid application:

• The choice of weights (unweighted vs. population-weighted) changes the target parameter
• Unweighted DiD: $ = $ +0.1 (“per county” ATT)
• Weighted DiD: $ = $ -2.6 (“per person” ATT)
• Both are valid, but they answer different questions

• Why this matters going forward:
  • When we add covariates, conditioning variables implicitly change weights
  • Different estimators (regression adjustment, IPW, doubly robust) target the same parameter but may use different implicit weights
• Researcher’s responsibility: Be explicit about what parameter you are estimating

We have seen DiD work in practice.

Now: what are its statistical properties?

Influence Functions and Asymptotic Theory

Why Influence Functions?

Having seen DiD work in practice, we now formalize the statistical properties of the estimator.

• We have an estimator \(\widehat{\theta}^{DiD}\). What are its large-sample properties?
• Goals: Consistency, asymptotic normality, variance estimation, and valid bootstrap inference
• Influence functions provide a unified approach:
  • Decompose the estimator as a sum of iid terms (+ remainder)
  • The influence function \(\psi_i\) captures unit \(i\)’s contribution to the estimator
  • Variance of the IF \(=\) asymptotic variance of \(\widehat{\theta}^{DiD}\)
• This framework generalizes naturally to more complex estimators (covariates, staggered designs) in later lectures

Panel Data Sampling Scheme

Panel Data Sampling. We observe an iid random sample \(\{Z_i\}_{i=1}^n\) where \(Z_i = (Y_{i,t=1}, Y_{i,t=2}, G_i)\) is drawn from the joint distribution of \((Y_{i,t=1}, Y_{i,t=2}, G_i)\).

• The same \(n\) units are observed in both periods (\(t=1\) and \(t=2\))
• Key: Units are iid across \(i\), but \(Y_{i,t=1}\) and \(Y_{i,t=2}\) are not independent within unit
• This allows us to compute \(\Delta Y_i = Y_{i,t=2} - Y_{i,t=1}\) for each unit
• Let \(D_i = \mathbf{1}\{G_i = 2\}\) and \(p = \Pr(G_i = 2) \in (0,1)\)
• Notation: \(\mathbb{E}_n[f(Z_i)] = \frac{1}{n}\sum_{i=1}^n f(Z_i)\) denotes the sample average

The DiD Estimator as a Function of Means

• Write the DiD estimator using empirical expectations: \[ \widehat{\theta}^{DiD} = \frac{\mathbb{E}_n[\Delta Y_i \cdot D_i]}{\mathbb{E}_n[D_i]} - \frac{\mathbb{E}_n[\Delta Y_i \cdot (1 - D_i)]}{\mathbb{E}_n[1 - D_i]} \]

• This is a smooth function of sample means: \(\widehat{\theta}^{DiD} = g(\mathbb{E}_n[\mathbf{m}(Z_i)])\)

• where \(\mathbf{m}(Z_i) = (\Delta Y_i \cdot D_i, \; D_i, \; \Delta Y_i \cdot (1-D_i), \; 1 - D_i)'\)

• By the Law of Large Numbers: \(\mathbb{E}_n[\mathbf{m}(Z_i)] \xrightarrow{p} \mathbb{E}[\mathbf{m}(Z_i)]\)

• By the Continuous Mapping Theorem: \(\widehat{\theta}^{DiD} = g(\mathbb{E}_n[\mathbf{m}(Z_i)]) \xrightarrow{p} g(\mathbb{E}[\mathbf{m}(Z_i)]) = \theta^{DiD}\)

• \(\Rightarrow\) Consistency follows from LLN + CMT

Asymptotic Normality: Setup

• Apply the delta method to \(g(\mathbb{E}_n[\mathbf{m}(Z_i)])\): \[ \sqrt{n}(\widehat{\theta}^{DiD} - \theta^{DiD}) = \sqrt{n} \cdot \nabla g(\boldsymbol{\mu})' \big(\mathbb{E}_n[\mathbf{m}(Z_i)] - \boldsymbol{\mu}\big) + o_p(1) \] where \(\boldsymbol{\mu} = \mathbb{E}[\mathbf{m}(Z_i)]\)

• By the CLT: \[ \sqrt{n}\big(\mathbb{E}_n[\mathbf{m}(Z_i)] - \boldsymbol{\mu}\big) \xrightarrow{d} N(\mathbf{0}, \text{Var}(\mathbf{m}(Z_i))) \]

• Combining (Slutsky): \[ \sqrt{n}(\widehat{\theta}^{DiD} - \theta^{DiD}) \xrightarrow{d} N\big(0, \;\nabla g(\boldsymbol{\mu})' \text{Var}(\mathbf{m}(Z_i)) \nabla g(\boldsymbol{\mu})\big) \]

• This can be written more compactly using the influence function…

The Influence Function: Panel Data Case

Panel data IF. Under the panel data sampling scheme: \[ \psi_i^{p} = \underbrace{\frac{D_i}{p}}_{w_1(D_i)} \big(\Delta Y_i - \mu_{\Delta,2}\big) \;-\; \underbrace{\frac{1 - D_i}{1 - p}}_{w_0(D_i)} \big(\Delta Y_i - \mu_{\Delta,\infty}\big) \] where \(p = \Pr(G_i = 2)\), \(\mu_{\Delta,g} = \mathbb{E}[\Delta Y_i | G_i = g]\) for \(g \in \{2,\infty\}\).

• Key property: \(\sqrt{n}(\widehat{\theta}^{DiD} - \theta^{DiD}) = \frac{1}{\sqrt{n}} \sum_{i=1}^n \psi_i^p + o_p(1)\)
• Asymptotic variance: \(V_p = \text{Var}(\psi_i^p)\)
• Full derivation in the appendix

Computing the IF: Medicaid Example

Suppose: \(p = 0.4\), \(\mu_{\Delta,2} = -4.0\), \(\mu_{\Delta,\infty} = -1.4\)

• Expansion county with \(\Delta Y_i = -6.5\): \[ \psi_i^p = \tfrac{1}{0.4}(-6.5 - (-4.0)) - 0 = \textcolor{#b91c1c}{-6.25} \] This county “pulls” the estimate toward a larger mortality reduction

• Non-expansion county with \(\Delta Y_i = -0.5\): \[ \psi_i^p = 0 - \tfrac{1}{0.6}(-0.5 - (-1.4)) = \textcolor{#15803d}{-1.5} \] Its mortality fell less than average \(\Rightarrow\) supports the treatment effect

• Takeaway: The IF assigns each unit a signed “credit” for the overall estimate

Understanding the Influence Function

• Two terms: treated group’s contribution and comparison group’s contribution

• Each term is a demeaned quantity, weighted by group proportion

• \(\mathbb{E}[\psi_i^p] = 0\) by construction — the IF is centered

• Intuition for the weights:

  • \(D_i/p\): up-weights treated units (rarer group gets more weight)
  • \((1-D_i)/(1-p)\): up-weights comparison units

• Why this matters: The IF gives us everything for inference — consistency, asymptotic normality, variance estimation, and bootstrap validity all follow from this representation

Asymptotic Normality Result

Theorem. Under the panel data sampling scheme, SUTVA, No-Anticipation, and \(\text{Var}(\Delta Y_i | G_i = g) < \infty\): \[ \sqrt{n}\big(\widehat{\theta}^{DiD} - \theta^{DiD}\big) \xrightarrow{d} N(0, V_p) \] where \(V_p = \text{Var}(\psi_i^p) = \frac{\sigma^2_{\Delta,2}}{p} + \frac{\sigma^2_{\Delta,\infty}}{1-p}\)

• \(\sigma^2_{\Delta,g} = \text{Var}(\Delta Y_i | G_i = g)\): within-group variance of the first-differenced outcome
• Intuition: Two components — treated group uncertainty (\(\sigma^2_{\Delta,2}/p\)) and comparison group uncertainty (\(\sigma^2_{\Delta,\infty}/(1-p)\))
• Design matters: \(p\) small \(\Rightarrow\) first term dominates \(\Rightarrow\) large variance. Minimized at \(p \approx 0.5\) (balanced design)
• Think: If only 5% of counties expanded, what does this formula say about power?

Repeated Cross-Sections: Brief Overview

• With repeated cross-sections (RCS), different units sampled at \(t=1\) and \(t=2\)

  • Let \(T_i = \mathbf{1}\{\text{unit } i \text{ sampled at } t=2\}\) (distinct from \(T_t\) in TWFE regression)

• Cannot first-difference \(\Rightarrow\) must estimate four means separately

• The IF has four components instead of two: \[ \psi_i^{rc} = w_1(D_i, T_i) \cdot (Y_i - \mu_{2,T_i}) - w_0(D_i, T_i) \cdot (Y_i - \mu_{\infty,T_i}) \] where each weight depends on both group and period membership

• Requires additional assumption: Stationarity of group composition

  • \(\Pr(G_i = 2 | T_i = 1) = \Pr(G_i = 2 | T_i = 2)\) (no compositional changes)

• Panel is strictly more efficient than RCS — panel exploits within-unit correlation

Inference: Standard Errors and Clustering

How to Conduct Inference

• Given: \(\sqrt{n}(\widehat{\theta}^{DiD} - \theta^{DiD}) \xrightarrow{d} N(0, V_p)\)

• Variance estimation (analogy principle): \[ \widehat{V}_p = \frac{1}{n} \sum_{i=1}^n \big(\widehat{\psi}_i^p\big)^2 \] where \(\widehat{\psi}_i^p\) replaces population quantities with sample analogs

• Standard error: \(\widehat{SE} = \sqrt{\widehat{V}_p / n}\)

• Confidence interval: \(\widehat{\theta}^{DiD} \pm z_{\alpha/2} \cdot \widehat{SE}\)

• \(t\)-statistic: \(t = \widehat{\theta}^{DiD} / \widehat{SE}\). Reject \(H_0: \theta^{DiD} = 0\) if \(|t| > z_{\alpha/2}\)

• But wait: Should we cluster the standard errors?

Why Cluster Standard Errors?

• DiD-by-hand operates on \(n\) units (first-differenced panel)
  • Natural degrees of freedom: \(n\) independent observations
• TWFE regression uses \(2n\) observations (\(n\) units \(\times\) 2 periods)
  • Without clustering: SE formula assumes \(2n\) independent observations
  • Artificially inflates sample size by a factor of 2!
• Clustering at the unit level accounts for within-unit correlation
  • Two observations from the same unit are not independent
  • Corrected SE matches the DiD-by-hand SE
• Bertrand, Duflo, and Mullainathan (2004): Ignoring clustering in DiD leads to massive over-rejection (up to 40% rejection at 5% nominal level)

Multiplier Bootstrap Using Influence Functions

• Key advantage of IF-based inference: The multiplier bootstrap

• Idea: Perturb the IF with random weights instead of resampling data \[ \widehat{\theta}^{*,b} = \widehat{\theta}^{DiD} + \frac{1}{n} \sum_{i=1}^n U_i^{(b)} \cdot \widehat{\psi}_i^p \] where \(U_i^{(b)} \sim N(0,1)\) are iid random weights

• No re-estimation needed: Each bootstrap draw is a simple weighted sum

  • Extremely fast compared to traditional bootstrap (which re-estimates \(\widehat{\theta}\) each time)

• For cluster-level inference: Draw \(U_s^{(b)}\) at the cluster level

  • All units in cluster \(s\) share the same weight \(U_s^{(b)}\)
  • This preserves the within-cluster correlation structure

Bootstrap Algorithm

Multiplier Bootstrap for DiD (Cluster-Robust)

1. Compute \(\widehat{\theta}^{DiD}\) and the influence function \(\widehat{\psi}_i^p\) for each unit \(i\)
2. For \(b = 1, \ldots, B\) (e.g., \(B = 999\)):
   1. Draw \(U_s^{(b)} \sim N(0,1)\) for each cluster \(s = 1, \ldots, S\)
   2. Assign \(U_i^{(b)} = U_{s(i)}^{(b)}\) for all units \(i\) in cluster \(s\)
   3. Compute: \(T^{*,b} = \frac{1}{\sqrt{n}} \sum_{i=1}^n U_i^{(b)} \cdot \widehat{\psi}_i^p\)
3. Compute bootstrap critical value: $c_= $ quantile\(_{1-\alpha}(|T^{*,1}|, \ldots, |T^{*,B}|)\)
4. Reject \(H_0: \theta^{DiD} = 0\) if \(|\sqrt{n} \cdot \widehat{\theta}^{DiD}| > c_\alpha\)

• Bootstrap CIs can also be constructed from quantiles of \(\widehat{\theta}^{*,b} = \widehat{\theta}^{DiD} + T^{*,b}/\sqrt{n}\)
• Rademacher weights (\(U_s^{(b)} \in \{-1, +1\}\) with equal probability) also valid and sometimes preferred for few clusters

The Few-Clusters Problem

• Standard cluster-robust inference relies on \(S \to \infty\) (number of clusters)

• Problem: In many DiD applications, \(S\) is small (e.g., 50 states, 10 provinces)

  • CLT approximation may be poor with few clusters
  • Cluster-robust SEs can be severely biased downward

• Approaches in the literature:

  • Donald and Lang (2007): \(t\)-distribution with \(S-2\) degrees of freedom (assumes homoskedasticity)
  • Conley and Taber (2011): Large untreated group to inform inference (fixed \(S_1\), growing \(S_0\))
  • Ferman and Pinto (2019): Allow for heteroskedasticity across clusters

• No silver bullet: Each approach requires additional assumptions; this is an active area of research (see Alvarez, Ferman, and Wüthrich 2025 for a recent survey)

Panel Data vs. Repeated Cross-Sections

We have assumed panel data. But what if we do not have it?

What changes with repeated cross-sections?

Repeated Cross-Section (RCS) Sampling Scheme

Repeated Cross-Section Sampling. Period \(t\) data \(\{(Y_{i,t}, G_i)\}\) is an iid sample from \(F_{Y,G|T=t}\). Observations across periods are independent.

• Different units sampled at \(t=1\) and \(t=2\)

• Requires additional assumption: Stationarity of group composition

  • \(\Pr(G_i = 2 | T_i = 1) = \Pr(G_i = 2 | T_i = 2)\)
  • Violation: compositional changes (Sant’Anna and Xu 2026)

• IF has four components: one per group-period cell

• Examples: CPS microdata, Census data, polling data before/after events

Comparison: Panel vs. Repeated Cross-Sections

Panel Data

• Same units both periods
• Can first-difference
• 2 IF components
• More efficient
• Risk: attrition, survivorship bias

Repeated Cross-Sections

• Different units each period
• Must estimate 4 means
• 4 IF components
• Less efficient
• Risk: compositional changes

• Key result: Panel data is strictly more efficient than RCS
• Intuition: Panel exploits within-unit correlation \(\Rightarrow\) \(\text{Var}(\Delta Y_i | G_i)\) can be much smaller than \(\text{Var}(Y_{i,t=2} | G_i) + \text{Var}(Y_{i,t=1} | G_i)\)
• Unbalanced panels: Some units observed once, some twice — use panel structure where available, but gains depend on strength of within-unit correlation

Taking Stock

What We Accomplished Today

Key takeaways from the \(2 \times 2\) DiD framework:

1. Identification: SUTVA + No-Anticipation + Parallel Trends \(\Rightarrow\) DiD identifies the ATT using four observable group-time means

2. Estimation: DiD-by-hand and TWFE regression are numerically identical in the \(2 \times 2\) case — regression is just a convenient computational device

3. Inference: Influence functions provide consistency, asymptotic normality, and variance estimation. Always cluster at least at the unit level; ideally at the treatment-assignment level

4. Weights matter: The choice of weights (unweighted vs. population-weighted) defines a different target parameter

References

Alvarez, Luis, Bruno Ferman, and Kaspar Wüthrich. 2025. “Inference with Few Treated Units.”

Baker, Andrew, Brantly Callaway, Scott Cunningham, Andrew Goodman-Bacon, and Pedro H. C. Sant’Anna. 2025. “Difference-in-Differences Designs: A Practitioner’s Guide.” Journal of Economic Literature Forthcoming.

Bertrand, Marianne, Esther Duflo, and Sendhil Mullainathan. 2004. “How Much Should We Trust Differences-in-Differences Estimates?” Quarterly Journal of Economics 119 (1): 249–75. https://doi.org/10.1162/003355304772839588.

Conley, Timothy G., and Christopher R. Taber. 2011. “Inference with ‘Difference in Differences’ with a Small Number of Policy Changes.” Review of Economics and Statistics 93 (1): 113–25. https://doi.org/10.1162/REST_a_00049.

Currie, Janet, Henrik Kleven, and Esmée Zwiers. 2020. “Technology and Big Data Are Changing Economics: Mining Text to Track Methods.” AEA Papers and Proceedings 110: 42–48.

Donald, Stephen G., and Kevin Lang. 2007. “Inference with Difference-in-Differences and Other Panel Data.” Review of Economics and Statistics 89 (2): 221–33. https://doi.org/10.1162/rest.89.2.221.

Ferman, Bruno, and Cristine Pinto. 2019. “Inference in Differences-in-Differences with Few Treated Groups and Heteroskedasticity.” The Review of Economics and Statistics 101 (3): 452–67.

Ghanem, Dalia, Pedro H. C. Sant’Anna, and Kaspar Wüthrich. 2026. “When Should Pre-Trends Be Parallel?” AEA Papers and Proceedings 116: forthcoming.

Goldsmith-Pinkham, Paul. 2024. “Tracking the Credibility Revolution across Fields.” arXiv:2405.20604.

Malani, Anup, and Julian Reif. 2015. “Interpreting Pre-Trends as Anticipation: Impact on Estimated Treatment Effects from Tort Reform.” Journal of Public Economics 124: 1–17. https://doi.org/10.1016/j.jpubeco.2015.01.001.

Roth, Jonathan. 2022. “Pre-Test with Caution: Event-Study Estimates After Testing for Parallel Trends.” American Economic Review: Insights 4 (3): 305–22.

Sant’Anna, Pedro H. C., and Qi Xu. 2026. “Difference-in-Differences with Compositional Changes.” Working Paper.

Appendix

Appendix

Detailed derivations and proofs

Appendix: Panel Data — Consistency (Detailed)

• Setup: \(Z_i = (Y_{i,t=1}, Y_{i,t=2}, G_i)\) iid, \(D_i = \mathbf{1}\{G_i = 2\}\), \(\Delta Y_i = Y_{i,t=2} - Y_{i,t=1}\)

• Define population quantities: \[ \begin{aligned} \mu_1 &= \mathbb{E}[\Delta Y_i \cdot D_i], \quad p = \mathbb{E}[D_i] \\ \mu_0 &= \mathbb{E}[\Delta Y_i \cdot (1-D_i)], \quad 1-p = \mathbb{E}[1-D_i] \end{aligned} \]

• The DiD estimand: \(\theta^{DiD} = \frac{\mu_1}{p} - \frac{\mu_0}{1-p}\)

• Sample analogs: \[ \widehat{\theta}^{DiD} = \frac{\mathbb{E}_n[\Delta Y_i \cdot D_i]}{\mathbb{E}_n[D_i]} - \frac{\mathbb{E}_n[\Delta Y_i \cdot (1-D_i)]}{\mathbb{E}_n[1-D_i]} \]

• By LLN: \(\mathbb{E}_n[\Delta Y_i \cdot D_i] \xrightarrow{p} \mu_1\), etc.

• By CMT (continuous mapping theorem): \(\widehat{\theta}^{DiD} \xrightarrow{p} \theta^{DiD}\) \(\square\)

Appendix: Panel Data — Asymptotic Linearity (1/3)

• Goal: Show \(\sqrt{n}(\widehat{\theta}^{DiD} - \theta^{DiD}) = \frac{1}{\sqrt{n}} \sum_{i=1}^n \psi_i^p + o_p(1)\)

• Step 1: Write the estimator as a function of sample means. Let \(\hat{\mu}_1 = \mathbb{E}_n[\Delta Y_i D_i]\), \(\hat{p} = \mathbb{E}_n[D_i]\), \(\hat{\mu}_0 = \mathbb{E}_n[\Delta Y_i(1-D_i)]\), \(\hat{q} = \mathbb{E}_n[1-D_i]\) \[ \widehat{\theta}^{DiD} = \frac{\hat{\mu}_1}{\hat{p}} - \frac{\hat{\mu}_0}{\hat{q}} \]

• Step 2: Linearize each ratio. For the first term: \[ \begin{aligned} \frac{\hat{\mu}_1}{\hat{p}} - \frac{\mu_1}{p} &= \frac{\hat{\mu}_1 p - \mu_1 \hat{p}}{\hat{p} \cdot p} \\ &= \frac{(\hat{\mu}_1 - \mu_1) p - \mu_1 (\hat{p} - p)}{\hat{p} \cdot p} \\ &= \frac{1}{p}\big(\hat{\mu}_1 - \mu_1\big) - \frac{\mu_1}{p^2}(\hat{p} - p) + o_p(n^{-1/2}) \end{aligned} \] where the last step uses \(\hat{p} \xrightarrow{p} p > 0\).

Appendix: Panel Data — Asymptotic Linearity (2/3)

• Step 3: Similarly for the second ratio: \[ \frac{\hat{\mu}_0}{\hat{q}} - \frac{\mu_0}{1-p} = \frac{1}{1-p}(\hat{\mu}_0 - \mu_0) + \frac{\mu_0}{(1-p)^2}(\hat{p} - p) + o_p(n^{-1/2}) \]

• Step 4: Combine: \[ \begin{aligned} \widehat{\theta}^{DiD} - \theta^{DiD} &= \frac{1}{p}(\hat{\mu}_1 - \mu_1) - \frac{\mu_{\Delta,2}}{p}(\hat{p} - p) \\ &\quad - \frac{1}{1-p}(\hat{\mu}_0 - \mu_0) - \frac{\mu_{\Delta,\infty}}{1-p}(\hat{p} - p) + o_p(n^{-1/2}) \end{aligned} \] where \(\mu_{\Delta,2} = \mu_1/p = \mathbb{E}[\Delta Y_i | G_i = 2]\) and \(\mu_{\Delta,\infty} = \mu_0/(1-p) = \mathbb{E}[\Delta Y_i | G_i = \infty]\).

• Note: \(\hat{p} - p = \mathbb{E}_n[D_i] - p\) and \(\hat{\mu}_1 - \mu_1 = \mathbb{E}_n[\Delta Y_i D_i] - \mu_1\)

Appendix: Panel Data — Asymptotic Linearity (3/3)

• Step 5: Express as average of iid terms. Each sample mean is an average: \[ \begin{aligned} \widehat{\theta}^{DiD} - \theta^{DiD} &= \frac{1}{n}\sum_{i=1}^n \bigg[ \frac{D_i \Delta Y_i - \mu_1}{p} - \frac{\mu_{\Delta,2}(D_i - p)}{p} \\ &\qquad\qquad - \frac{(1-D_i)\Delta Y_i - \mu_0}{1-p} - \frac{\mu_{\Delta,\infty}(D_i - p)}{1-p} \bigg] + o_p(n^{-1/2}) \end{aligned} \]

• Step 6: Simplify the treated group term: \[ \frac{D_i \Delta Y_i - \mu_1}{p} - \frac{\mu_{\Delta,2}(D_i - p)}{p} = \frac{D_i(\Delta Y_i - \mu_{\Delta,2})}{p} \]

• Similarly for the comparison term. Thus: \[ \widehat{\theta}^{DiD} - \theta^{DiD} = \frac{1}{n}\sum_{i=1}^n \underbrace{\left[\frac{D_i}{p}(\Delta Y_i - \mu_{\Delta,2}) - \frac{1-D_i}{1-p}(\Delta Y_i - \mu_{\Delta,\infty})\right]}_{\psi_i^p} + o_p(n^{-1/2}) \]

Appendix: Panel Data — Asymptotic Variance

• From the IF representation: \(\sqrt{n}(\widehat{\theta}^{DiD} - \theta^{DiD}) = \frac{1}{\sqrt{n}}\sum_{i=1}^n \psi_i^p + o_p(1)\)

• By CLT: \(\frac{1}{\sqrt{n}}\sum_{i=1}^n \psi_i^p \xrightarrow{d} N(0, \text{Var}(\psi_i^p))\)

• Computing \(\text{Var}(\psi_i^p)\): \[ \begin{aligned} V_p &= \text{Var}\left(\frac{D_i}{p}(\Delta Y_i - \mu_{\Delta,2}) - \frac{1-D_i}{1-p}(\Delta Y_i - \mu_{\Delta,\infty})\right) \\ &= \frac{1}{p^2}\text{Var}(D_i(\Delta Y_i - \mu_{\Delta,2})) + \frac{1}{(1-p)^2}\text{Var}((1-D_i)(\Delta Y_i - \mu_{\Delta,\infty})) \end{aligned} \] (cross-term is zero since \(D_i(1-D_i) = 0\))

• Simplifying: \(\text{Var}(D_i(\Delta Y_i - \mu_{\Delta,2})) = p \cdot \sigma^2_{\Delta,2}\) where \(\sigma^2_{\Delta,2} = \text{Var}(\Delta Y_i | G_i = 2)\)

\[ \boxed{V_p = \frac{\sigma^2_{\Delta,2}}{p} + \frac{\sigma^2_{\Delta,\infty}}{1-p}} \]

Appendix: Variance Estimation

• Plug-in estimator: Replace population quantities with sample analogs \[ \widehat{\psi}_i^p = \frac{D_i}{\hat{p}}(\Delta Y_i - \overline{\Delta Y}_2) - \frac{1-D_i}{1-\hat{p}}(\Delta Y_i - \overline{\Delta Y}_\infty) \]

• Variance estimate: \[ \widehat{V}_p = \frac{1}{n}\sum_{i=1}^n (\widehat{\psi}_i^p)^2 \]

• By LLN: \(\widehat{V}_p \xrightarrow{p} V_p\)

• Standard error: \(\widehat{SE}(\widehat{\theta}^{DiD}) = \sqrt{\widehat{V}_p / n}\)

• Alternative: Direct plug-in \[ \widehat{V}_p^{alt} = \frac{\hat{\sigma}^2_{\Delta,2}}{\hat{p}} + \frac{\hat{\sigma}^2_{\Delta,\infty}}{1 - \hat{p}} \] where \(\hat{\sigma}^2_{\Delta,g} = \frac{1}{n_g}\sum_{i:G_i=g}(\Delta Y_i - \overline{\Delta Y}_g)^2\)

Appendix: RCS — Influence Function Derivation (1/2)

• RCS data: \(Z_i = (Y_i, G_i, T_i)\), \(D_i = \mathbf{1}\{G_i = 2\}\), \(T_i = \mathbf{1}\{\text{sampled at } t=2\}\) (distinct from \(T_t\) in regression)

• Cannot first-difference. The DiD estimand uses four conditional means: \[ \theta^{DiD} = (\mu_{2,2} - \mu_{2,1}) - (\mu_{\infty,2} - \mu_{\infty,1}) \] where \(\mu_{g,t} = \mathbb{E}[Y_i | G_i = g, T_i = t]\)

• Under stationarity: \(\Pr(G_i = 2 | T_i = t)\) is constant across \(t\)

• Linearization: Apply the delta method to each of the four ratios

• The IF takes the form: \[ \begin{aligned} \psi_i^{rc} &= \frac{D_i T_i}{p \cdot \lambda}(Y_i - \mu_{2,2}) - \frac{D_i(1-T_i)}{p \cdot (1-\lambda)}(Y_i - \mu_{2,1}) \\ &\quad - \frac{(1-D_i)T_i}{(1-p)\cdot\lambda}(Y_i - \mu_{\infty,2}) + \frac{(1-D_i)(1-T_i)}{(1-p)(1-\lambda)}(Y_i - \mu_{\infty,1}) \end{aligned} \] where \(\lambda = \Pr(T_i = 1)\) and \(\mu_{g,t} = \mathbb{E}[Y_i | G_i = g, T_i = t]\)

Appendix: RCS — Influence Function Derivation (2/2)

• Asymptotic variance: \(V_{rc} = \text{Var}(\psi_i^{rc})\)

• Since \((D_i, T_i) \in \{0,1\}^2\) creates four disjoint groups, cross-products vanish: \[ V_{rc} = \frac{\sigma^2_{2,2}}{p \cdot \lambda} + \frac{\sigma^2_{2,1}}{p \cdot (1-\lambda)} + \frac{\sigma^2_{\infty,2}}{(1-p) \cdot \lambda} + \frac{\sigma^2_{\infty,1}}{(1-p)(1-\lambda)} \] where \(\sigma^2_{g,t} = \text{Var}(Y_i | G_i = g, T_i = t)\)

• Comparison with panel: \[ V_p = \frac{\sigma^2_{\Delta,2}}{p} + \frac{\sigma^2_{\Delta,\infty}}{1-p} \;\;<\;\; V_{rc} \]

  Two sources of RCS inefficiency:

  • 1. Cannot exploit within-unit correlation: \(\sigma^2_{\Delta,g} = \sigma^2_{g,2} + \sigma^2_{g,1} - 2\text{Cov}(Y_{i,2}, Y_{i,1}|G_i=g)\)
  • 2. Must split sample across periods: each cell has \(\leq n\) observations (factors \(1/\lambda\), \(1/(1-\lambda)\))

• Conclusion: Panel data is strictly more efficient than RCS for DiD (\(V_p < V_{rc}\) always)

Appendix: Optimal Sample Allocation for RCS

• Question: Given a total budget of \(n\) observations, how should we split between \(t=1\) and \(t=2\) in an RCS?

• Let \(\lambda = n_2/(n_1 + n_2)\) be the fraction sampled at \(t=2\)

• The asymptotic variance is: \[ V_{rc}(\lambda) = \frac{1}{\lambda}\left(\frac{\sigma^2_{2,2}}{p} + \frac{\sigma^2_{\infty,2}}{1-p}\right) + \frac{1}{1-\lambda}\left(\frac{\sigma^2_{2,1}}{p} + \frac{\sigma^2_{\infty,1}}{1-p}\right) \]

• Optimal allocation: \[ \lambda^* = \frac{\sqrt{A_2}}{\sqrt{A_1} + \sqrt{A_2}} \] where \(A_t = \frac{\sigma^2_{2,t}}{p} + \frac{\sigma^2_{\infty,t}}{1-p}\)

• Practical implication: If the outcome is more variable in the post-period (e.g., due to treatment effect heterogeneity), sample more observations in the post-period