Difference-in-Differences Models: Zero to Hero Tutorial

This comprehensive tutorial takes you from the foundational concepts of Difference-in-Differences (DiD) estimation all the way through advanced extensions, assumption testing, heterogeneity analysis, and practical usage within the DataStatPro application. Whether you are a complete beginner or an experienced analyst, this guide is structured to build your understanding step by step.

Table of Contents

1. Prerequisites and Background Concepts
2. What is the Difference-in-Differences Design?
3. The Mathematical Framework
4. The Parallel Trends Assumption
5. Identification and Causal Inference
6. Standard DiD Estimation
7. Hypothesis Testing and Inference
8. Effect Size Measures
9. Model Fit and Evaluation
10. Diagnostics and Assumption Testing
11. Extensions: Staggered DiD and Multiple Time Periods
12. Extensions: Heterogeneous Treatment Effects
13. Extensions: Continuous and Fuzzy Treatment
14. Covariates and Controls in DiD
15. Using the DiD Component
16. Computational and Formula Details
17. Worked Examples
18. Common Mistakes and How to Avoid Them
19. Troubleshooting
20. Quick Reference Cheat Sheet

1. Prerequisites and Background Concepts

Before diving into Difference-in-Differences, it is helpful to be familiar with the following foundational concepts. Do not worry if you are not — each concept is briefly explained here.

1.1 Counterfactuals and the Potential Outcomes Framework

The potential outcomes framework (Rubin Causal Model) is the conceptual foundation of causal inference. For each unit $i$ and time period $t$, define:

• $Y_{it}(1)$: The potential outcome that would occur if unit $i$ received treatment at time $t$.
• $Y_{it}(0)$: The potential outcome that would occur if unit $i$ did not receive treatment at time $t$.

The individual treatment effect for unit $i$ at time $t$ is:

$\tau_{it} = Y_{it}(1) - Y_{it}(0)$

The fundamental problem of causal inference: We can never observe both $Y_{it}(1)$ and $Y_{it}(0)$ for the same unit at the same time. We observe only one — the realised outcome. The unobserved outcome is called the counterfactual.

The Average Treatment Effect on the Treated (ATT) is:

$ATT = E[Y_{it}(1) - Y_{it}(0) \mid D_i = 1]$

DiD is one of the most widely used methods for estimating the ATT using a comparison group to approximate the unobserved counterfactual.

1.2 Selection Bias

Selection bias arises when the assignment to treatment is not random — treated and control units differ systematically in ways that also affect the outcome. A naïve comparison of treated vs. untreated units confounds the treatment effect with pre-existing differences:

$E[Y \mid D=1] - E[Y \mid D=0] = \underbrace{ATT}_{\text{causal effect}} + \underbrace{E[Y(0) \mid D=1] - E[Y(0) \mid D=0]}_{\text{selection bias}}$

DiD removes selection bias due to time-invariant unobserved differences between groups.

1.3 Panel Data

Panel data (also called longitudinal data) consists of observations on the same units (individuals, firms, regions, countries) over multiple time periods. It has two dimensions: a cross-sectional dimension ($n$ units) and a time dimension ($T$ periods).

Panel data are written as $\{Y_{it}, D_{it}, \mathbf{X}_{it}\}$ for unit $i = 1, \dots, n$ and period $t = 1, \dots, T$.

DiD most naturally arises in a panel data context, though it can also be implemented with repeated cross-sections.

1.4 Fixed Effects

A unit fixed effect $\alpha_i$ captures all time-invariant characteristics of unit $i$ — both observed and unobserved — that affect the outcome. By including fixed effects in a regression, we effectively compare each unit to itself over time, removing all time-invariant confounders.

A time fixed effect $\lambda_t$ captures factors that affect all units equally at time $t$ — common macroeconomic conditions, seasonal patterns, or universal policy changes.

1.5 Ordinary Least Squares (OLS) Regression

OLS regression finds the linear relationship between predictors $\mathbf{X}$ and outcome $Y$ by minimising the sum of squared residuals:

$\hat{\boldsymbol{\beta}} = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{y}$

DiD is typically implemented as an OLS regression with specific interaction terms and fixed effects, so familiarity with OLS is essential.

1.6 Treatment Assignment and Natural Experiments

A natural experiment is a situation in which the assignment of units to treatment and control conditions is determined by some external, exogenous factor — rather than by the researcher or by the units themselves. Natural experiments approximate the conditions of a randomised controlled trial. Common examples:

• A policy reform that applies to some regions but not others.
• A law change that takes effect at a specific date.
• Geographic boundaries that determine eligibility for a programme.
• Lotteries or other chance-based assignment mechanisms.

DiD is the workhorse estimator for natural experiments with a pre-period and a post-period.

2. What is the Difference-in-Differences Design?

2.1 The Core Idea

Difference-in-Differences (DiD) is a quasi-experimental research design that estimates the causal effect of a treatment or policy by comparing the change over time in the outcome for a treated group to the change over time in the outcome for an untreated (control) group.

The intuition is straightforward:

$\hat{\tau}_{DiD} = \underbrace{(\bar{Y}_{treated,post} - \bar{Y}_{treated,pre})}_{\text{before-after change in treated group}} - \underbrace{(\bar{Y}_{control,post} - \bar{Y}_{control,pre})}_{\text{before-after change in control group}}$

• The first difference (within the treated group, over time) removes time-invariant differences between treated units and the rest of the world.
• The second difference (between treated and control, within the same time period) removes common time trends that affect both groups equally.

By taking the difference of these two differences, DiD isolates the treatment effect from:

1. Pre-existing level differences between treated and control groups.
2. Common time trends affecting both groups equally.

2.2 The 2×2 DiD Setup

The simplest (canonical) DiD design has:

• Two groups: A treated group ($D_i = 1$) and a control group ($D_i = 0$).
• Two time periods: A pre-treatment period ($t = 0$) and a post-treatment period ($t = 1$).
• One treatment: Applied only to the treated group, only in the post-treatment period.

The canonical 2×2 DiD table of group-period means:

The DiD estimate:

$\hat{\tau}_{DiD} = (\bar{Y}_{1,1} - \bar{Y}_{1,0}) - (\bar{Y}_{0,1} - \bar{Y}_{0,0})$

2.3 Real-World Applications

DiD is one of the most widely applied methods in empirical social science, economics, public health, and policy evaluation:

• Labour Economics: Card & Krueger (1994) — Effect of New Jersey's minimum wage increase on fast-food employment, using Pennsylvania as the control group.
• Health Policy: Effect of the Affordable Care Act (ACA) Medicaid expansion on health insurance coverage and health outcomes, comparing expansion to non-expansion states.
• Education Policy: Effect of class size reductions (STAR experiment) or school voucher programmes on student achievement.
• Environmental Economics: Effect of environmental regulations (e.g., Clean Air Act) on air pollution and health outcomes.
• Finance: Effect of financial crises, banking regulations, or central bank interventions on lending and economic activity.
• Criminology: Effect of policing policies, incarceration changes, or gun laws on crime rates.
• Public Health: Effect of vaccination campaigns, smoking bans, or lockdown policies on health outcomes.
• Development Economics: Effect of microcredit programmes, cash transfers, or infrastructure investments on household welfare.

2.4 DiD vs. Other Quasi-Experimental Methods

3. The Mathematical Framework

3.1 The Canonical 2×2 DiD Model

The standard regression formulation of the 2×2 DiD model is:

$Y_{it} = \alpha + \beta \cdot \text{Treated}_i + \gamma \cdot \text{Post}_t + \delta \cdot (\text{Treated}_i \times \text{Post}_t) + \epsilon_{it}$

Where:

• $Y_{it}$ = outcome for unit $i$ at time $t$.
• $\text{Treated}_i \in \{0, 1\}$ = indicator for whether unit $i$ belongs to the treated group (time-invariant).
• $\text{Post}_t \in \{0, 1\}$ = indicator for whether time period $t$ is the post-treatment period (unit-invariant).
• $\text{Treated}_i \times \text{Post}_t$ = the DiD interaction term.
• $\alpha$ = baseline mean for the control group in the pre-period.
• $\beta$ = pre-treatment difference in levels between treated and control groups (selection bias term).
• $\gamma$ = common time trend from pre to post period (time effect for the control group).
• $\delta$ = the DiD estimator — the causal effect of the treatment.
• $\epsilon_{it}$ = idiosyncratic error term.

Predicted cell means from the regression:

3.2 The Two-Way Fixed Effects (TWFE) Model

The most general and widely used DiD regression extends the canonical model to panel data with unit fixed effects and time fixed effects:

$Y_{it} = \alpha_i + \lambda_t + \delta \cdot D_{it} + \mathbf{X}_{it}^T\boldsymbol{\beta} + \epsilon_{it}$

Where:

• $\alpha_i$ = unit (entity) fixed effect — absorbs all time-invariant unit characteristics.
• $\lambda_t$ = time fixed effect — absorbs all period-specific shocks common to all units.
• $D_{it} \in \{0, 1\}$ = treatment indicator (= 1 if unit $i$ is treated at time $t$).
• $\mathbf{X}_{it}$ = vector of time-varying controls.
• $\delta$ = the DiD estimator of the Average Treatment Effect on the Treated (ATT).

Key insight: The TWFE model is the natural extension of the 2×2 DiD to multiple units and multiple time periods. The coefficient $\delta$ on the treatment indicator $D_{it}$ is the DiD estimate once unit and time fixed effects are included.

3.3 The Within-Estimator Interpretation

The TWFE estimator is equivalent to the within estimator (demeaning). Define:

$\ddot{Y}_{it} = Y_{it} - \bar{Y}_i - \bar{Y}_t + \bar{Y}$

Where $\bar{Y}_i = T^{-1}\sum_t Y_{it}$ (unit mean), $\bar{Y}_t = n^{-1}\sum_i Y_{it}$ (time mean), and $\bar{Y} = (nT)^{-1}\sum_{it} Y_{it}$ (grand mean). Similarly define $\ddot{D}_{it}$ and $\ddot{\mathbf{X}}_{it}$.

The TWFE estimator is:

$\hat{\delta}_{TWFE} = \frac{\sum_{it} \ddot{D}_{it} \ddot{Y}_{it}}{\sum_{it} \ddot{D}_{it}^2}$

This identifies $\delta$ from within-unit, within-time variation in treatment status.

3.4 Potential Outcomes Representation

In the potential outcomes framework, the DiD estimand is:

$\delta_{DiD} = E[Y_{it}(1) - Y_{it}(0) \mid D_i = 1, t = 1]$

This is the ATT in the post-treatment period — the average effect on the treated units of receiving treatment.

The DiD identification strategy replaces the unobserved counterfactual $E[Y_{it}(0) \mid D_i = 1, t = 1]$ with the observable:

$E[Y_{it}(0) \mid D_i = 1, t = 1] = E[Y_{it}(0) \mid D_i = 1, t = 0] + \underbrace{\left(E[Y_{it}(0) \mid D_i = 0, t = 1] - E[Y_{it}(0) \mid D_i = 0, t = 0]\right)}_{\text{time trend from control group}}$

This is exactly the parallel trends assumption — the counterfactual trend for the treated group equals the observed trend for the control group.

4. The Parallel Trends Assumption

4.1 The Assumption Stated

The parallel trends assumption (also called the common trends or parallel paths assumption) is the key identifying assumption of DiD:

In the absence of treatment, the average outcome for the treated group would have followed the same trend as the average outcome for the control group.

Formally:

$E[Y_{it}(0) \mid D_i = 1, t = 1] - E[Y_{it}(0) \mid D_i = 1, t = 0] = E[Y_{it}(0) \mid D_i = 0, t = 1] - E[Y_{it}(0) \mid D_i = 0, t = 0]$

Crucially: This assumption is about the counterfactual — what would have happened to the treated group had it not been treated. It is fundamentally untestable with post-treatment data, but can be supported with:

• Pre-treatment trend evidence (parallel pre-trends test).
• Institutional knowledge about why the groups are similar in their trends.
• Placebo tests.

4.2 Visualising Parallel Trends

The canonical DiD diagram plots the outcome over time for both groups:

Outcome
  |
  |             ● Treated (actual)
  |           /
  |         /   ↑ δ = Treatment Effect
  |       /   /
  |     / ---/ ← Treated counterfactual (unobserved)
  |   /   /
  |  ●   /
  | / \ /
  |/   ● Control (observed, serves as counterfactual trend)
  |
  +--------+--------→ Time
        Pre       Post
           ↑
        Treatment
        begins

The treatment effect $\delta$ is the vertical distance between the actual treated outcome and the counterfactual treated outcome in the post-period. The control group's observed trajectory is the counterfactual trend.

4.3 When is Parallel Trends Plausible?

Parallel trends is more plausible when:

• Treatment and control groups are similar in observed characteristics and pre-treatment trends.
• Treatment is determined by a sharp, exogenous rule (geographic, legislative, administrative).
• The treatment and control groups come from the same broad population (e.g., neighbouring counties, similar industries, adjacent cohorts).
• There are no other contemporaneous changes that differentially affect treated and control groups.

Parallel trends is less plausible when:

• Groups are systematically different in ways related to the outcome trajectory (e.g., high-income vs. low-income countries).
• Treatment is self-selected based on anticipated trends (e.g., firms that chose to adopt a technology because they expected growth).
• There are anticipation effects — units change behaviour before the treatment officially starts.

4.4 Parallel Trends in Different Functional Forms

The parallel trends assumption is not scale-invariant. It may hold on the levels scale but not on the logarithmic scale (or vice versa):

• Levels scale: $\Delta E[Y(0) \mid D=1] = \Delta E[Y(0) \mid D=0]$ (additive parallel trends).
• Log scale: $\Delta E[\ln Y(0) \mid D=1] = \Delta E[\ln Y(0) \mid D=0]$ (multiplicative/proportional parallel trends).

The choice of outcome transformation (levels, logs, rates) should be guided by theory about the nature of the treatment effect and the plausibility of parallel trends.

4.5 Conditional Parallel Trends

The parallel trends assumption may only hold conditional on observable covariates $\mathbf{X}_i$:

$E[Y_{it}(0) \mid D_i = 1, \mathbf{X}_i, t = 1] - E[Y_{it}(0) \mid D_i = 1, \mathbf{X}_i, t = 0] = E[Y_{it}(0) \mid D_i = 0, \mathbf{X}_i, t = 1] - E[Y_{it}(0) \mid D_i = 0, \mathbf{X}_i, t = 0]$

When unconditional parallel trends is implausible, including covariates (Section 14) can restore the assumption by controlling for observable differences in time trends between groups.

5. Identification and Causal Inference

5.1 What DiD Identifies

Under the parallel trends assumption, the DiD regression coefficient $\delta$ identifies the Average Treatment Effect on the Treated (ATT) in the post-treatment period:

$\hat{\delta}_{DiD} \xrightarrow{p} ATT = E[Y_{it}(1) - Y_{it}(0) \mid D_i = 1, t \geq t_0]$

Where $t_0$ is the treatment onset period.

Not identified by DiD:

• The Average Treatment Effect (ATE) — the effect averaged over both treated and control units.
• The effect on the control group had it been treated.
• The long-run effect if treatment effects change over time (addressed in event study designs).

5.2 The No Anticipation Assumption

A supplementary assumption is no anticipation: treated units do not change their behaviour in the pre-treatment period in anticipation of receiving treatment.

Formally, for all $t < t_0$:

$Y_{it}(1) = Y_{it}(0) \quad \text{for all } i \text{ with } D_i = 1$

Why it matters: If treated units begin changing before the treatment officially starts (e.g., firms start investing as soon as a subsidy is announced), the pre-period outcome already reflects anticipatory responses. This violates the parallel trends assumption in the pre-period and biases the DiD estimator.

How to check: Pre-treatment placebo tests (event study coefficients for pre-period leads should be near zero).

5.3 The Stable Unit Treatment Value Assumption (SUTVA)

SUTVA has two components:

1. No interference: The treatment status of unit $i$ does not affect the potential outcomes of unit $j$ (no spillovers, general equilibrium effects, or cross-unit contamination).

2. No hidden versions of treatment: There is only one version of the treatment; all treated units receive the same treatment.

Violations: Spillovers arise when treatment of some units affects control units (e.g., a local employment policy in one area displaces workers to other areas, affecting those areas' outcomes). SUTVA violations bias the DiD estimator.

5.4 Exogeneity of Treatment Timing

In staggered DiD designs (Section 11), a key requirement is that the timing of treatment adoption is exogenous — not determined by pre-existing trends or anticipation of future outcomes. If units that were doing well adopt treatment earlier, the DiD estimator is biased.

5.5 DiD as a Special Case of the Fixed Effects Estimator

The 2×2 DiD estimator is numerically equivalent to the first-differences estimator in a two-period panel:

$\Delta Y_i = Y_{i,1} - Y_{i,0} = \delta \cdot \Delta D_i + \Delta \epsilon_i$

Where $\Delta D_i = D_{i,1} - D_{i,0} = 1$ for treated units and $0$ for control units. OLS on this first-differenced equation produces $\hat{\delta} = (\bar{\Delta Y}_{treated} - \bar{\Delta Y}_{control})$ — exactly the DiD formula.

6. Standard DiD Estimation

6.1 OLS Estimation of the Canonical DiD

The 2×2 DiD regression:

$Y_{it} = \alpha + \beta \cdot \text{Treated}_i + \gamma \cdot \text{Post}_t + \delta \cdot (\text{Treated}_i \times \text{Post}_t) + \epsilon_{it}$

is estimated by OLS. The DiD coefficient:

$\hat{\delta} = (\bar{Y}_{1,1} - \bar{Y}_{0,1}) - (\bar{Y}_{1,0} - \bar{Y}_{0,0})$

This can be written as:

$\hat{\delta} = \hat{\beta}_{OLS} = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{y}$

Where $\mathbf{X}$ contains the constant, $\text{Treated}_i$, $\text{Post}_t$, and their interaction $\text{Treated}_i \times \text{Post}_t$.

6.2 TWFE Estimation with Panel Data

The TWFE estimator is obtained by including unit and time dummies (or using the within-transformation):

Using dummy variables:

$Y_{it} = \sum_{i=1}^n \alpha_i \mathbf{1}[unit = i] + \sum_{t=1}^T \lambda_t \mathbf{1}[period = t] + \delta \cdot D_{it} + \mathbf{X}_{it}^T\boldsymbol{\beta} + \epsilon_{it}$

Using the within (demeaning) transformation:

$\ddot{Y}_{it} = \delta \cdot \ddot{D}_{it} + \ddot{\mathbf{X}}_{it}^T\boldsymbol{\beta} + \ddot{\epsilon}_{it}$

Where $\ddot{Y}_{it} = Y_{it} - \bar{Y}_i - \bar{Y}_t + \bar{\bar{Y}}$ and similarly for other variables.

The TWFE estimator $\hat{\delta}$ is consistent under the parallel trends assumption and strict exogeneity of treatment given fixed effects.

6.3 First Differences Estimator

An alternative to demeaning is first differencing, which subtracts the previous period's observation:

$\Delta Y_{it} = Y_{it} - Y_{i,t-1} = \lambda_t - \lambda_{t-1} + \delta(D_{it} - D_{i,t-1}) + \mathbf{X}_{it}^{*T}\boldsymbol{\beta} + \Delta\epsilon_{it}$

In a two-period model, first differences and within estimation are identical. For $T > 2$, they differ in efficiency — first differences is more efficient when $\epsilon_{it}$ follows a random walk; within estimation is more efficient when $\epsilon_{it}$ is serially uncorrelated.

6.4 Weighted DiD

When the groups have unequal sizes or when reweighting is needed to improve comparability, a weighted DiD uses weights $w_i$:

$\hat{\delta}_{WDiD} = \frac{\sum_i w_i D_i \Delta Y_i}{\sum_i w_i D_i} - \frac{\sum_i w_i (1-D_i) \Delta Y_i}{\sum_i w_i (1-D_i)}$

Common weighting schemes:

• Population weights: Weight by group size.
• Propensity score weights: Reweight control units to match the distribution of pre-treatment characteristics in the treated group (augmented inverse probability weighting — AIPW).
• Variance weights: Inverse of estimated error variance for each unit.

6.5 DiD with Repeated Cross-Sections

When panel data (the same units followed over time) are unavailable, DiD can be implemented with repeated cross-sections — independent samples drawn from the same population at each time period. The DiD regression:

$Y_{ig} = \alpha + \beta \cdot G_g + \gamma \cdot T_t + \delta \cdot (G_g \times T_t) + \epsilon_{ig}$

Where:

• $G_g \in \{0, 1\}$ = treated group indicator for individual $i$ in group $g$.
• $T_t \in \{0, 1\}$ = post-treatment period indicator.
• $\delta$ = DiD estimator (interpreted as a change in group-period means).

The DiD estimator is valid under the assumption that the cross-sectional samples are representative of the same underlying population in each period, even though different individuals are observed.

7. Hypothesis Testing and Inference

7.1 Standard Error Choices

The choice of standard errors is critical in DiD analyses. Several options are available, with different assumptions:

7.1.1 OLS Standard Errors

$SE_{OLS}(\hat{\delta}) = \sqrt{\hat{\sigma}^2 [(\mathbf{X}^T\mathbf{X})^{-1}]_{\delta\delta}}$

Valid only under homoscedasticity and no serial correlation. Almost never appropriate for DiD — treated group observations are typically serially correlated.

7.1.2 Heteroscedasticity-Robust (HC) Standard Errors

$SE_{HC}(\hat{\delta}) = \sqrt{[(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\hat{\Omega}\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}]_{\delta\delta}}$

Where $\hat{\Omega} = \text{diag}(\hat{\epsilon}_{it}^2)$. Accounts for heteroscedasticity but not serial correlation.

7.1.3 Cluster-Robust Standard Errors

The most recommended standard errors for DiD. Clustering at the group level (e.g., state, firm, country) allows for arbitrary heteroscedasticity and serial correlation within clusters:

$SE_{cluster}(\hat{\delta}) = \sqrt{\left[(\mathbf{X}^T\mathbf{X})^{-1}\left(\sum_{g=1}^G \mathbf{X}_g^T\hat{\epsilon}_g\hat{\epsilon}_g^T\mathbf{X}_g\right)(\mathbf{X}^T\mathbf{X})^{-1}\right]_{\delta\delta}}$

Where $g$ indexes clusters, $\mathbf{X}_g$ and $\hat{\epsilon}_g$ are the design matrix and residuals for cluster $g$.

Critical recommendation (Bertrand, Duflo & Mullainathan, 2004): Always cluster at the level of treatment assignment (e.g., state-level policy → cluster at state level). Failure to do so leads to severely underestimated standard errors and spurious significance.

7.1.4 Wild Cluster Bootstrap

When the number of clusters is small ($G < 30$), cluster-robust standard errors based on asymptotic approximations can be unreliable. The wild cluster bootstrap (Cameron, Gelbach & Miller) provides more reliable inference:

1. Estimate the model and obtain cluster-robust residuals $\hat{\epsilon}_g$.
2. For each bootstrap replication $b = 1, \dots, B$:
   • Draw $v_g^{(b)} \in \{-1, +1\}$ with equal probability for each cluster $g$.
   • Construct bootstrap residuals $\tilde{\epsilon}_{it}^{(b)} = v_g^{(b)} \hat{\epsilon}_{it}$.
   • Form the bootstrap outcome: $Y_{it}^{*(b)} = \hat{Y}_{it} + \tilde{\epsilon}_{it}^{(b)}$.
   • Re-estimate $\hat{\delta}^{(b)}$.
3. Use the distribution of $\hat{\delta}^{(b)}$ to compute p-values and confidence intervals.

7.2 The Wald Test for the DiD Coefficient

The Wald test for the DiD effect tests $H_0: \delta = 0$:

$t = \frac{\hat{\delta}}{SE(\hat{\delta})} \sim t_{n-k}$

Where $n - k$ is the residual degrees of freedom. With cluster-robust standard errors, use the $t$-distribution with $G - 1$ degrees of freedom (where $G$ is the number of clusters):

$t = \frac{\hat{\delta}}{SE_{cluster}(\hat{\delta})} \sim t_{G-1}$

A $(1-\alpha) \times 100\%$ confidence interval for $\delta$:

$\hat{\delta} \pm t_{\alpha/2,\, G-1} \times SE_{cluster}(\hat{\delta})$

7.3 F-Test for Joint Significance

To jointly test whether a vector of DiD coefficients is zero (e.g., in a model with multiple treatment indicators):

$H_0: \mathbf{R}\boldsymbol{\beta} = \mathbf{0}$

$F = \frac{(\mathbf{R}\hat{\boldsymbol{\beta}})^T[\mathbf{R}(\mathbf{X}^T\hat{\Omega}\mathbf{X})^{-1}\mathbf{R}^T]^{-1}(\mathbf{R}\hat{\boldsymbol{\beta}})}{q} \sim F_{q, G-q}$

Where $q$ is the number of restrictions.

7.4 Inference with Few Treated Units

A common challenge is when only a few units receive treatment (e.g., one state, two firms). In such cases:

• Cluster-robust standard errors have very few clusters → unreliable asymptotic approximations.
• Randomisation inference (permutation tests): Repeatedly re-assign treatment to randomly selected units and re-estimate $\hat{\delta}$. The p-value is the fraction of placebo estimates at least as large as the observed estimate.
• Synthetic control methods: Construct a weighted control unit that matches the treated unit's pre-period trajectory.

8. Effect Size Measures

8.1 The DiD Coefficient as Effect Size

The primary effect size in DiD is the DiD coefficient $\hat{\delta}$ itself. Its interpretation depends on the model specification:

• Levels regression ($Y$ in original units): $\hat{\delta}$ is the absolute change in the outcome caused by treatment (e.g., 3.2 percentage points, 500 USD, 2.1 hours).
• Log outcome ($\ln Y$): $\hat{\delta} \approx 100\hat{\delta}\%$ change for small values; more precisely, $(e^{\hat{\delta}} - 1) \times 100\%$ change.
• Standardised outcome (mean 0, SD 1): $\hat{\delta}$ is in standard deviation units — directly comparable to Cohen's $d$.

8.2 Percent Change Effect

When the outcome is in levels, the percent change caused by treatment is:

$\%\Delta = \frac{\hat{\delta}}{\bar{Y}_{treated,pre}} \times 100\%$

Where $\bar{Y}_{treated,pre}$ is the pre-treatment mean of the treated group. This contextualises the absolute effect size relative to the pre-treatment baseline.

8.3 Standardised Effect Size (Cohen's d Analogue)

Standardise the DiD estimate by the pre-treatment standard deviation of the outcome:

$d_{DiD} = \frac{\hat{\delta}}{s_{pre}}$

Where $s_{pre}$ is the pooled pre-treatment standard deviation across treated and control groups. Benchmarks follow Cohen (1988):

| $|d_{DiD}|$ | Effect Size | | :---------- | :---------- | | $0.20$ | Small | | $0.50$ | Medium | | $0.80$ | Large |

8.4 Relative Reduction/Increase

For outcomes where the baseline level matters (e.g., crime rates, disease incidence), report the relative effect:

$RR = \frac{\bar{Y}_{treated,post}^{observed}}{\bar{Y}_{treated,post}^{counterfactual}} = \frac{\bar{Y}_{1,1}}{\bar{Y}_{1,0} + (\bar{Y}_{0,1} - \bar{Y}_{0,0})}$

Or the relative DiD:

$\hat{\delta}_{rel} = \frac{\hat{\delta}}{\bar{Y}_{treated,pre}} \quad \text{(fractional change)}$

8.5 Number Needed to Treat (NNT)

For binary outcomes (e.g., employed/unemployed, insured/uninsured):

$NNT = \frac{1}{|\hat{\delta}|}$

The NNT represents the number of units that need to be treated to produce one additional success (or prevented failure), contextualising the policy significance of the effect.

8.6 $R^2$ and Explained Variance

While not a primary effect size for DiD, report the within-$R^2$ (after partialling out fixed effects) to convey how much treatment variation explains the residual variation in the outcome:

$R^2_{within} = 1 - \frac{SS_{residual}}{SS_{within-unit, within-time}}$

Report both the overall $R^2$ and the within $R^2$ for TWFE models.

9. Model Fit and Evaluation

9.1 Goodness-of-Fit Statistics

Standard regression fit statistics apply to the DiD regression:

9.2 Fit of the Counterfactual

A key model evaluation step is assessing how well the control group serves as a counterfactual for the treated group's pre-treatment trajectory. Visually:

• Plot the raw pre-treatment trends for treated and control groups.
• Assess whether the trends are parallel (or conditionally parallel after covariate adjustment).

Quantitatively: Compute the pre-treatment DiD — the difference in trends during the pre-period. If this is near zero and statistically insignificant, the parallel trends assumption is supported.

9.3 Information Criteria for Model Comparison

When comparing DiD specifications (e.g., different control sets, different functional forms, different clustering levels), use AIC and BIC:

$AIC = 2k - 2\ln(\hat{L})$

$BIC = k\ln(n) - 2\ln(\hat{L})$

Where $k$ is the number of parameters and $\hat{L}$ is the maximised likelihood.

Note: AIC and BIC comparisons are only valid for models fitted to the same sample using the same outcome variable (e.g., levels vs. logs are not comparable on these criteria).

9.4 Assessing Balance on Pre-Treatment Covariates

A critical validity check is whether treated and control groups have similar pre-treatment characteristics:

• Report a balance table of pre-treatment mean differences in key covariates between groups.
• Test statistical differences using two-sample $t$-tests or non-parametric tests.
• Report standardised mean differences (SMD = mean difference / pooled SD) as effect sizes for balance.
• SMD $< 0.10$ is commonly used as a threshold for acceptable balance.

10. Diagnostics and Assumption Testing

10.1 Pre-Trends Test (Event Study)

The event study (dynamic DiD) is the primary tool for testing the parallel trends assumption using pre-treatment data. It estimates a separate DiD coefficient for each time period:

$Y_{it} = \alpha_i + \lambda_t + \sum_{k \neq -1} \delta_k \cdot \mathbf{1}[t - T_i^* = k] \cdot D_i + \mathbf{X}_{it}^T\boldsymbol{\beta} + \epsilon_{it}$

Where:

• $T_i^*$ = treatment onset period for unit $i$.
• $k$ indexes periods relative to treatment onset (negative = pre-treatment, positive = post-treatment).
• Period $k = -1$ is the omitted base period (normalised to zero for identification).
• $D_i = 1$ for treated units.

Interpretation:

• Pre-treatment coefficients ($k < 0$): Should be statistically indistinguishable from zero. Significant pre-treatment coefficients indicate pre-existing differences in trends — a violation of parallel trends.
• Post-treatment coefficients ($k \geq 0$): Estimate the dynamic treatment effect at each horizon after treatment. Increasing post-treatment effects may indicate treatment ramp-up; decreasing effects may indicate decay.

Formal pre-trends test: Joint $F$-test (or $\chi^2$ test) that all pre-treatment coefficients $\delta_k$ ($k < 0$) are jointly zero:

$H_0: \delta_{-K} = \delta_{-K+1} = \dots = \delta_{-2} = 0$

A non-significant result supports the parallel trends assumption; a significant result casts doubt.

⚠️ Passing the pre-trends test does not prove parallel trends hold in the post-period — trends may diverge after treatment for reasons unrelated to treatment. The pre-trends test is a necessary but not sufficient condition for identification.

10.2 Placebo Tests

Placebo tests assess whether the estimated DiD effect could have arisen by chance or due to confounding:

10.2.1 Placebo Time Periods

Estimate the DiD using only pre-treatment data, using a false treatment date (e.g., 2 years before the true treatment):

$Y_{it} = \alpha_i + \lambda_t + \delta_{placebo} \cdot D_{it}^{placebo} + \epsilon_{it}$

A significant $\hat{\delta}_{placebo}$ when treatment has not yet occurred suggests confounding or a violation of parallel trends.

10.2.2 Placebo Treatment Groups

Assign treatment to groups that were not actually treated and estimate the DiD. If the "treatment effect" is significant for these falsely treated groups, the design has poor identification.

10.2.3 Outcome Placebo Tests

Estimate the DiD using outcomes that should not be affected by the treatment. A null result ($\hat{\delta} \approx 0$) for these placebo outcomes increases confidence that the design is not picking up spurious effects.

10.3 Sensitivity to Parallel Trends Violations

Rambachan and Roth's sensitivity analysis (2023) provides a formal framework for assessing how large a violation of parallel trends would need to be to reverse the conclusion. The key parameter $M$ captures the maximum allowable deviation from parallel trends:

$|\Delta_{post} - \Delta_{pre}| \leq M$

Report breakdown values of $M$ — the maximum deviation consistent with the estimated effect remaining statistically significant or of the correct sign.

10.4 Testing for Anticipation Effects

Estimate the DiD including period $k = -1$ (the period immediately before treatment) in the event study:

$Y_{it} = \alpha_i + \lambda_t + \sum_{k = -K}^{-1} \delta_k \cdot \mathbf{1}[t - T_i^* = k] \cdot D_i + \sum_{k=0}^{K} \delta_k \cdot \mathbf{1}[t - T_i^* = k] \cdot D_i + \epsilon_{it}$

If $\delta_{-1}$ is significantly different from zero, anticipation effects may be present.

10.5 Checking for Compositional Changes

In DiD with repeated cross-sections, the composition of the treated or control groups may change between periods. If the treatment induces sample selection (e.g., a health policy causes sick people to enter/exit the workforce), the DiD estimator may be biased.

How to check:

• Compare the distribution of pre-treatment characteristics across groups and periods.
• Test for treatment effects on selection-related outcomes (e.g., sample size, attrition rates).
• Use balanced panel data where possible to avoid compositional issues.

10.6 Residual Diagnostics

Standard regression diagnostics apply to the DiD residuals:

• Serial correlation test (Wooldridge, 2002): Tests whether the residuals from the first-differenced equation are serially correlated. Under H₀ (no serial correlation in levels), the first-differenced residuals have correlation $-0.5$. Significant deviation suggests serial correlation.
• Heteroscedasticity: Breusch-Pagan or White test; motivates cluster-robust standard errors.
• Normality of residuals: Jarque-Bera test; Q-Q plot (less critical with large $n$).
• Outlier detection: Cook's distance; leverage ($h_{ii}$); DFFITS.

11. Extensions: Staggered DiD and Multiple Time Periods

11.1 Staggered Treatment Adoption

In many real-world settings, different units adopt treatment at different points in time — this is called staggered (or differential timing) DiD. For example, different US states adopt a policy in different years.

The TWFE regression in this context:

$Y_{it} = \alpha_i + \lambda_t + \delta \cdot D_{it} + \epsilon_{it}$

The TWFE estimator $\hat{\delta}$ is a weighted average of all possible 2×2 DiD comparisons between groups that adopt treatment at different times — what Goodman-Bacon (2021) calls Bacon decomposition.

11.2 The Bacon Decomposition

Goodman-Bacon (2021) shows that the TWFE estimator in a staggered design decomposes as:

$\hat{\delta}_{TWFE} = \sum_{k \neq l} s_{kl} \hat{\delta}_{kl}$

Where $\hat{\delta}_{kl}$ is the 2×2 DiD comparing early adopters (treatment at time $k$) vs. late adopters (treatment at time $l > k$), and $s_{kl} > 0$ are weights summing to 1.

The problem of "forbidden comparisons": Some of these 2×2 DiDs compare a late adopter group in the post-period against an early adopter group that has already been treated — using already-treated units as a control group. If treatment effects are heterogeneous and dynamic (treatment effects change over time), this produces negative weights that can lead to:

• A significant $\hat{\delta}_{TWFE}$ of the wrong sign even when all individual group-time ATTs are positive.
• A misleading averaged effect that conceals substantial heterogeneity.

11.3 Robust Staggered DiD Estimators

Several robust estimators have been developed to address the staggered DiD problem:

11.3.1 Callaway & Sant'Anna (2021) — Cohort-Specific ATTs

Define a cohort as the set of units that first receive treatment at the same calendar time $g$. The cohort-average treatment effect on the treated is:

$ATT(g, t) = E[Y_t(g) - Y_t(\infty) \mid G = g]$

Where $Y_t(g)$ is the potential outcome at time $t$ if first treated at time $g$, and $Y_t(\infty)$ is the never-treated potential outcome.

Aggregation: Individual cohort-time ATTs are aggregated to form:

• Simple average: $\theta = \text{Average}_{g,t}[ATT(g,t)]$.
• Calendar time aggregation: $\theta(t)$ = average ATT at calendar time $t$ across all treated cohorts.
• Event time aggregation: $\theta(e)$ = average ATT at event time $e = t - g$ across all cohorts.

11.3.2 Sun & Abraham (2021) — Interaction-Weighted Estimator

Decompose the TWFE estimate using cohort × period interactions:

$Y_{it} = \alpha_i + \lambda_t + \sum_{g \neq \infty} \sum_{k \neq -1} \delta_{gk} \cdot \mathbf{1}[G_i = g] \cdot \mathbf{1}[t - g = k] + \epsilon_{it}$

The interaction-weighted (IW) estimator aggregates $\hat{\delta}_{gk}$ using the share of each cohort in each period as weights, producing a heterogeneity-robust estimate of the average effect.

11.3.3 de Chaisemartin & D'Haultfœuille (2020) — $\text{DID}_M$

The $\text{DID}_M$ estimator uses only "clean" comparisons — periods in which treatment status changes — to form the estimate:

$\text{DID}_M = \sum_{(i,t): S_{it}=1} w_{it} \Delta Y_{it} - \sum_{(i,t): S_{it}=0} w_{it} \Delta Y_{it}$

Where $S_{it} = 1$ if unit $i$ switches from untreated to treated between $t-1$ and $t$, and weights $w_{it}$ ensure comparability.

11.3.4 Borusyak, Jaravel & Spiess (2024) — Imputation Estimator

Imputes the counterfactual using a linear factor model:

1. Estimate $\alpha_i$ and $\lambda_t$ using untreated observations only.
2. Impute the counterfactual: $\hat{Y}_{it}(0) = \hat{\alpha}_i + \hat{\lambda}_t$ for treated observations.
3. Estimate ATTs: $\hat{\tau}_{it} = Y_{it} - \hat{Y}_{it}(0)$.

11.4 Choosing Among Staggered DiD Estimators

💡 For staggered designs, always report the Bacon decomposition to diagnose the extent of potentially problematic comparisons, and supplement TWFE with at least one robust estimator.

12. Extensions: Heterogeneous Treatment Effects

12.1 Why Treatment Effects May Be Heterogeneous

The standard DiD model estimates a single average treatment effect (ATT). In reality, treatment effects often vary across:

• Units: Different firms, regions, or individuals respond differently to the same treatment.
• Time: Treatment effects may grow, decay, or oscillate over time after treatment onset.
• Subgroups: Effects may differ by gender, income, size, geography, or other characteristics.
• Treatment intensity: Larger doses may produce larger effects (see Section 13).

12.2 Subgroup DiD Analysis

To examine how treatment effects vary across a categorical moderator $Z_i$:

$Y_{it} = \alpha_i + \lambda_t + \delta \cdot D_{it} + \gamma \cdot (D_{it} \times Z_i) + \mathbf{X}_{it}^T\boldsymbol{\beta} + \epsilon_{it}$

• $\delta$ = treatment effect for the reference subgroup ($Z_i = 0$).
• $\gamma$ = differential treatment effect — how much the effect differs for $Z_i = 1$ relative to $Z_i = 0$.
• Total effect for $Z_i = 1$: $\delta + \gamma$.

Test of effect heterogeneity: $H_0: \gamma = 0$ (no heterogeneity). Use cluster-robust standard errors.

12.3 Dynamic Treatment Effects (Event Study)

The event study design (Section 10.1) directly estimates dynamic treatment effects:

$\delta_k = E[Y_{it}(1) - Y_{it}(0) \mid D_i = 1, t - T_i^* = k]$

For $k = 0, 1, 2, \dots, K$ (periods since treatment onset). Plot $\hat{\delta}_k$ with confidence intervals to visualise:

• Immediate effects ($k = 0$): Effect in the year/period treatment begins.
• Ramp-up: Effects growing over time (learning, diffusion, cumulative investment).
• Decay: Effects diminishing over time (adaptation, spillovers, fading).
• Persistence: Effects stable over time (structural change).

12.4 Heterogeneity-Robust Aggregation

The robust staggered DiD estimators (Section 11.3) produce cohort-specific ATTs $ATT(g, t)$ that can be aggregated in various ways:

Average across all cohorts and post-periods: $\bar{\delta} = \frac{1}{|\mathcal{S}|}\sum_{(g,t) \in \mathcal{S}} ATT(g, t)$

Event-time average: ATT as a function of time since treatment: $\delta(k) = \frac{1}{|\{g: t \geq g + k\}|}\sum_{g: t \geq g+k} ATT(g, g + k)$

Calendar-time average: ATT in each calendar year: $\delta(t) = \frac{1}{|\{g: g \leq t\}|}\sum_{g \leq t} ATT(g, t)$

13. Extensions: Continuous and Fuzzy Treatment

13.1 Continuous Treatment Intensity (Dose-Response DiD)

When the treatment variable is continuous (e.g., amount of subsidy, level of minimum wage increase, exposure to a policy) rather than binary, the DiD model becomes:

$Y_{it} = \alpha_i + \lambda_t + \delta \cdot D_{it} + \mathbf{X}_{it}^T\boldsymbol{\beta} + \epsilon_{it}$

Where $D_{it}$ is now a continuous variable representing the intensity of treatment. The coefficient $\delta$ represents the effect of a one-unit increase in treatment intensity on the outcome.

Dose-response curve: Plot the predicted outcome as a function of treatment intensity at different time periods to visualise the dose-response relationship.

13.2 Fuzzy DiD (Instrumental Variables DiD)

In a fuzzy DiD design, the policy change shifts the probability of treatment but does not deterministically assign treatment. For example:

• A subsidy makes adoption cheaper but does not mandate it.
• An eligibility rule changes, but not all eligible units comply.

The binary treatment variable $D_{it}$ measures actual take-up; the policy indicator $Z_{it}$ is the instrument (discontinuous change in eligibility or incentive).

First stage: Treatment as a function of the instrument:

$D_{it} = \pi_0 + \pi_1 Z_{it} + \alpha_i^D + \lambda_t^D + \mathbf{X}_{it}^T\boldsymbol{\pi} + \nu_{it}$

Second stage: Outcome as a function of predicted treatment:

$Y_{it} = \alpha_i + \lambda_t + \delta \cdot \hat{D}_{it} + \mathbf{X}_{it}^T\boldsymbol{\beta} + \epsilon_{it}$

The IV-DiD estimator $\hat{\delta}_{IV}$ estimates the Local Average Treatment Effect (LATE) — the effect on compliers (units that switch treatment status in response to the policy change).

Fuzzy DiD estimator:

$\hat{\delta}_{FuzzyDiD} = \frac{\hat{\delta}_{reduced\, form}}{\hat{\pi}_1} = \frac{\Delta\bar{Y}_{treated} - \Delta\bar{Y}_{control}}{\Delta\bar{D}_{treated} - \Delta\bar{D}_{control}}$

13.3 Triple Differences (DDD)

Triple differences (DDD) adds a third source of variation to further control for confounders. The idea is to difference out group-specific time trends that are common to all individuals within a group:

$Y_{igjt} = \alpha + \sum \text{(all pairwise FE)} + \delta \cdot (\text{Treated}_g \times \text{Post}_t \times \text{Eligible}_j) + \epsilon_{igjt}$

Where $j$ is an additional dimension (e.g., age group, income group) that determines eligibility within the treated group.

DDD estimator:

$\hat{\delta}_{DDD} = \underbrace{[(\bar{Y}_{Tr,Post,El} - \bar{Y}_{Tr,Pre,El}) - (\bar{Y}_{Tr,Post,NoEl} - \bar{Y}_{Tr,Pre,NoEl})]}_{\text{DiD among Treated Group}} - \underbrace{[(\bar{Y}_{Ctrl,Post,El} - \bar{Y}_{Ctrl,Pre,El}) - (\bar{Y}_{Ctrl,Post,NoEl} - \bar{Y}_{Ctrl,Pre,NoEl})]}_{\text{DiD among Control Group}}$

DDD is valuable when the comparison across regions includes contamination from regional trends that differentially affect all groups in treated regions.

14. Covariates and Controls in DiD

14.1 Why Include Covariates?

Adding covariates to the DiD model serves two distinct purposes:

1. Improving efficiency (precision): Covariates that predict the outcome reduce residual variance, shrinking standard errors and widening the confidence interval.

2. Restoring conditional parallel trends: If unconditional parallel trends is implausible but trends are parallel after conditioning on observable characteristics, including covariates removes the confounding and restores identification.

14.2 Time-Invariant Covariates

In the TWFE model, time-invariant covariates $\mathbf{X}_i$ (e.g., gender, ethnicity, geographic characteristics) are absorbed by the unit fixed effect $\alpha_i$ and cannot be estimated separately. However, they can be included as interactions with the treatment or time variables to allow their effect to vary:

$Y_{it} = \alpha_i + \lambda_t + \delta \cdot D_{it} + \gamma \cdot (D_{it} \times X_i) + \epsilon_{it}$

14.3 Time-Varying Covariates

Time-varying covariates $X_{it}$ can be included directly in the TWFE model:

$Y_{it} = \alpha_i + \lambda_t + \delta \cdot D_{it} + \boldsymbol{\beta}^T \mathbf{X}_{it} + \epsilon_{it}$

⚠️ Including time-varying covariates that are themselves affected by the treatment (i.e., "bad controls" or "mediators") is a common mistake. Including such variables absorbs part of the treatment effect, leading to underestimation of $\delta$. Only include covariates that are determined before treatment or that are plausibly unaffected by treatment.

14.4 Regression Adjustment (Outcome Regression)

The regression-adjusted DiD uses the control group's pre-to-post relationship between covariates and the outcome to construct an improved counterfactual:

$\hat{\delta}_{RA} = (\bar{Y}_{1,1} - \bar{Y}_{1,0}) - \hat{m}(\mathbf{X}_{1,0})$

Where $\hat{m}(\mathbf{X}_{1,0})$ is the predicted counterfactual change based on the control group's covariate-outcome relationship. This improves efficiency and removes covariate-related bias.

14.5 Doubly Robust DiD (DR-DiD)

The doubly robust estimator combines propensity score weighting and outcome regression. It is consistent if either the propensity score model or the outcome regression model is correctly specified:

$\hat{\delta}_{DR} = \frac{1}{n_1}\sum_{D_i = 1}\left[\Delta Y_i - \hat{m}(\mathbf{X}_i)\right] - \frac{1}{n_0}\sum_{D_i = 0}\hat{w}(\mathbf{X}_i)\left[\Delta Y_i - \hat{m}(\mathbf{X}_i)\right]$

Where $\hat{w}(\mathbf{X}_i)$ are propensity score reweighting terms and $\hat{m}(\mathbf{X}_i)$ is the regression-adjusted counterfactual change.

14.6 Controlling for Pre-Treatment Trends (Linear Trend Adjustment)

When parallel trends is violated by unit-specific linear time trends, include unit-specific trend terms:

$Y_{it} = \alpha_i + \lambda_t + \rho_i \cdot t + \delta \cdot D_{it} + \epsilon_{it}$

Where $\rho_i \cdot t$ is a unit-specific linear time trend. This allows each unit to have its own pre-treatment trajectory, controlling for heterogeneous trends.

⚠️ Including unit-specific trends is a strong assumption (units would have continued on their pre-treatment trend indefinitely) and can overcontrol. Use only when unit-specific trends are well-established in the pre-period and when there are sufficient pre-period observations to estimate them.

15. Using the DiD Component

The Difference-in-Differences component in the DataStatPro application provides a comprehensive workflow for DiD estimation, testing, and visualisation.

Step-by-Step Guide

Step 1 — Select Dataset Choose the dataset from the "Dataset" dropdown. The dataset should have:

• A unit identifier column (individual, firm, region, country).
• A time period column (year, quarter, month).
• An outcome variable (continuous or binary).
• A treatment indicator (binary: 0 = control, 1 = treated).

Step 2 — Select DiD Design Choose the DiD specification:

• 2×2 DiD (two groups, two periods — canonical design)
• Two-Way Fixed Effects (TWFE) (panel data, multiple periods)
• Staggered DiD (multiple treatment timing groups)
• Event Study / Dynamic DiD (multiple pre- and post-periods)
• Triple Differences (DDD) (three sources of variation)
• Fuzzy DiD / IV-DiD (non-compliance with treatment)

Step 3 — Select Variables Map the required variables from your dataset:

• Unit ID: The unique identifier for each unit (individual, firm, state).
• Time ID: The time period variable.
• Outcome (Y): The continuous or binary dependent variable.
• Treatment Group ($D_i$): Binary indicator for treated group (time-invariant for 2×2).
• Treatment Indicator ($D_{it}$): Binary indicator for treatment status (may vary by time for TWFE/staggered).
• Covariates: Optional time-varying or time-invariant controls.

Step 4 — Specify Treatment Timing

• 2×2 DiD: Specify the pre- and post-period labels.
• TWFE/Staggered: The application detects treatment timing from the $D_{it}$ variable automatically. Review the detected cohort structure.
• Event Study: Specify the base period (omitted category, default: $k = -1$) and the number of leads and lags to include.

Step 5 — Configure Fixed Effects Select fixed effects to include:

• ✅ Unit Fixed Effects ($\alpha_i$) — strongly recommended for panel data.
• ✅ Time Fixed Effects ($\lambda_t$) — strongly recommended.
• Unit-Specific Linear Trends $\rho_i \cdot t$ — optional; use when pre-trend concerns exist.

Step 6 — Configure Standard Errors Select the standard error type:

• Cluster-Robust (default and recommended) — specify the clustering variable (typically the treatment assignment unit).
• HC (Heteroscedasticity-Robust) — use when clusters are very small or unavailable.
• Wild Cluster Bootstrap — specify for few clusters ($G < 30$).
• Block Bootstrap — for more general panel dependence.

Step 7 — Select Staggered DiD Estimator (if applicable) For staggered designs, choose the robust estimator:

• TWFE (standard but potentially biased with heterogeneous effects)
• Callaway-Sant'Anna
• Sun-Abraham
• Bacon Decomposition (diagnostic)
• de Chaisemartin-D'Haultfœuille

Step 8 — Configure Inference Options

• Confidence level: Default 95%.
• Bootstrap replications: For wild bootstrap (default: 999).
• Permutation replications: For randomisation inference (default: 999).

Step 9 — Select Display Options Choose which outputs to display:

• ✅ DiD Coefficient Table (estimate, SE, t, p, CI)
• ✅ Pre-Treatment Trends Plot
• ✅ Event Study Plot (with CIs)
• ✅ 2×2 DiD Table (group-period means)
• ✅ Parallel Trends Test (joint F-test on pre-period coefficients)
• ✅ Counterfactual Plot
• ✅ Placebo Test Results
• ✅ Bacon Decomposition Plot (staggered designs)
• ✅ Balance Table (pre-treatment covariate balance)
• ✅ Residual Diagnostics
• ✅ Coefficient Profile Plot across Subgroups
• ✅ Dynamic Effects Plot

Step 10 — Run the Analysis Click "Run DiD Model". The application will:

1. Construct the design matrix with appropriate interaction terms and fixed effects.
2. Estimate the DiD coefficient(s) via OLS with specified standard errors.
3. Compute the event study / dynamic effects coefficients.
4. Run the pre-trends test and parallel trends diagnostics.
5. Execute placebo tests (if requested).
6. Compute the Bacon decomposition (for staggered designs).
7. Generate all selected visualisations and tables.

16. Computational and Formula Details

16.1 The 2×2 DiD Estimator: Step-by-Step

Step 1: Compute group-period means

$\bar{Y}_{gt} = \frac{1}{n_{gt}}\sum_{i \in g}\sum_{t' = t} Y_{it}$

For groups $g \in \{0, 1\}$ (control, treated) and periods $t \in \{0, 1\}$ (pre, post).

Step 2: First differences

$\Delta\bar{Y}_{treated} = \bar{Y}_{1,1} - \bar{Y}_{1,0}$

$\Delta\bar{Y}_{control} = \bar{Y}_{0,1} - \bar{Y}_{0,0}$

Step 3: DiD estimate

$\hat{\delta} = \Delta\bar{Y}_{treated} - \Delta\bar{Y}_{control}$

Step 4: Standard error (homoscedastic OLS)

With $n_{gt}$ observations per cell, $n = n_{00} + n_{01} + n_{10} + n_{11}$:

$SE_{OLS}(\hat{\delta}) = \hat{\sigma}\sqrt{\frac{1}{n_{00}} + \frac{1}{n_{01}} + \frac{1}{n_{10}} + \frac{1}{n_{11}}}$

Where $\hat{\sigma}^2 = \sum_{igt}(Y_{it} - \hat{Y}_{it})^2 / (n - 4)$.

16.2 TWFE Estimation: The Demeaning Procedure

Step 1: Compute unit means, time means, and grand mean

$\bar{Y}_i = \frac{1}{T}\sum_{t=1}^T Y_{it}, \quad \bar{Y}_t = \frac{1}{n}\sum_{i=1}^n Y_{it}, \quad \bar{Y} = \frac{1}{nT}\sum_{i,t} Y_{it}$

Step 2: Demean all variables

$\ddot{Y}_{it} = Y_{it} - \bar{Y}_i - \bar{Y}_t + \bar{Y}$ $\ddot{D}_{it} = D_{it} - \bar{D}_i - \bar{D}_t + \bar{D}$

Step 3: Estimate TWFE by OLS on demeaned variables

$\hat{\delta}_{TWFE} = \frac{\sum_{it}\ddot{D}_{it}\ddot{Y}_{it}}{\sum_{it}\ddot{D}_{it}^2}$

16.3 Cluster-Robust Variance Estimator

With $G$ clusters and the TWFE estimator:

$\hat{V}_{cluster}(\hat{\delta}) = \frac{G}{G-1} \cdot \frac{n-1}{n-k} \cdot (\ddot{\mathbf{D}}^T\ddot{\mathbf{D}})^{-1} \left(\sum_{g=1}^G \ddot{\mathbf{D}}_g^T\hat{\boldsymbol{\epsilon}}_g\hat{\boldsymbol{\epsilon}}_g^T\ddot{\mathbf{D}}_g\right) (\ddot{\mathbf{D}}^T\ddot{\mathbf{D}})^{-1}$