## 1. Conceptual Framework

### 1.1 Notation and Key Concepts

- \(i\): Index for individual unit.
- \(t\): Time period.
- \(D_{i,t}\): Binary indicator for treatment. We assume throughout that treatment is received permanently once it has been received for the first time. In other words, \(D_{i,t}=1 \implies D_{i,t+1}=1\).
- \(G_i\): Treatment cohort, i.e., the time at which treatment is first received by \(i\). That is, \(G_i = g \implies D_{i,t}=1, \forall t\geq g\). Note: If treatment is not received, \(G_i = \infty\).
- \(Y_{i,t}\): Observed outcome of interest.
- \(Y_{i,t}(g)\): Counterfactual outcome if treatment cohort were \(G_i=g\).

### 1.2 Goal

Our goal is to identify the average treatment effect on the treated (ATT), for cohort \(g\) at event time \(e \equiv t-g\), which is defined by:

\[ \text{ATT}_{g,e} \equiv \mathbb{E}[Y_{i,g+e}(g) - Y_{i,g+e}(\infty) | G_i = g] \]

We may also be interested in the average ATT across treated cohorts for a given event time:

\[ \text{ATT}_{e} \equiv \sum_g \omega_{g,e} \text{ATT}_{g,e}, \quad \omega_{g,e} \equiv \frac{\sum_i 1\{G_i=g\}}{\sum_i 1\{G_i < \infty\}} \] Lastly, we may be interested in the average across certain event times of the average ATT across cohorts:

\[ \text{ATT}_{E} \equiv \frac{1}{|E|} \sum_{e \in E} \text{ATT}_{e} \] where \(E\) is a set of event times, e.g., \(E = \{1,2,3\}\).

### 1.3 Difference-in-differences

**Control group:** For the treated cohort \(G_i = g\), let \(C_{g,e}\) denote the corresponding set of
units \(i\) that belong to a control
group.

- At a minimum, the control group must satisfy \(i \in C_{g,e} \implies G_i > \max\{g, g+e\}\). This says that the control group must belong to a later cohort than the treated group of interest, and the control group must not have been treated yet by the event time of interest.

**Base event time:** We consider a reference event time
from before treatment \(b\), which
satisfies \(b<0\).

**Difference-in-differences:** The
difference-in-differences estimand is defined by, \[
\text{DiD}_{g,e} \equiv \mathbb{E}[Y_{i,g+e} - Y_{i,g+b} | G_i = g]
- \mathbb{E}[Y_{i,g+e} - Y_{i,g+b} | i \in C_{g,e}]
\]

## 2. Identification

Throughout this section, our goal is to identify \(\text{ATT}_{g,e}\) for some treated cohort \(g\) and some event time \(e\). We take the base event time \(b<0\) as given.

### 2.1 Identifying Assumptions

**Parallel Trends:**

\[ \mathbb{E}[Y_{i,g+e}(\infty) - Y_{i,g+b}(\infty) | G_i = g] = \mathbb{E}[Y_{i,g+e}(\infty) - Y_{i,g+b}(\infty) | i \in C_{g,e}] \] This says that, in the absence of treatment, the treatment and control groups would have experienced the same average change in their outcomes between event time \(b\) and event time \(e\).

**No Anticipation:**

\[ \mathbb{E}[ Y_{i,g+b}(g) | G_i = g] = \mathbb{E}[ Y_{i,g+b}(\infty) | G_i = g] \] This says that, at base event time \(b\), the observed outcome for the treated cohort would have been the same if it had instead been assigned to never receive treatment.

### 2.2 Proof of Identification by DiD

We prove that \(\text{DiD}_{g,e}\) identifies \(\text{ATT}_{g,e}\) in three steps:

**Step 1:** Add and subtract \(Y_{i,g+b}(\infty)\) from the ATT
definition:

\[ \text{ATT}_{g,e} \equiv \mathbb{E}[Y_{i,g+e}(g) - Y_{i,g+e}(\infty) | G_i = g] \] \[ = \mathbb{E}[Y_{i,g+e}(g) - Y_{i,g+b}(\infty) | G_i = g] - \mathbb{E}[Y_{i,g+e}(\infty) - Y_{i,g+b}(\infty) | G_i = g] \]

**Step 2:** Assume that Parallel Trends holds. Then, we
can replace the conditioning set \(G_i=g\) with the conditioning set \(i \in C_{g,e}\) in the second term:

\[ \text{ATT}_{g,e} = \mathbb{E}[Y_{i,g+e}(g) - Y_{i,g+b}(\infty) | G_i = g] - \mathbb{E}[Y_{i,g+e}(\infty) - Y_{i,g+b}(\infty) | G_i = g] \] \[ = \mathbb{E}[Y_{i,g+e}(g) - Y_{i,g+b}(\infty) | G_i = g] - \mathbb{E}[Y_{i,g+e}(\infty) - Y_{i,g+b}(\infty) | i \in C_{g,e}] \]

**Step 3:** Assume that No Anticipation holds. Then, we
can replace \(Y_{i,g+b}(\infty)\) with
\(Y_{i,g+b}(g)\) if the conditioning
set is \(G_i = g\):

\[ \text{ATT}_{g,e} = \mathbb{E}[Y_{i,g+e}(g) - Y_{i,g+b}(\infty) | G_i = g] - \mathbb{E}[Y_{i,g+e}(\infty) - Y_{i,g+b}(\infty) | i \in C_{g,e}] \] \[ = \mathbb{E}[Y_{i,g+e}(g) - Y_{i,g+b}(g) | G_i = g] - \mathbb{E}[Y_{i,g+e}(\infty) - Y_{i,g+b}(\infty) | i \in C_{g,e}] \] where the final expression is \(\text{DiD}_{g,e}\).

Thus, we have shown that \(\text{DiD}_{g,e} = \text{ATT}_{g,e}\) if Parallel Trends and No Anticipation hold.

## 3. The `DiDge(...)`

Command

\(\text{DiD}_{g,e}\) is estimated in
`DiDforBigData`

by the `DiDge(...)`

command, which
is documented here.

### 3.1 Automatic Control Group Selection

**All:** The largest valid control group is \(C_{g,e} \equiv \{ i : G_i > \max\{g,
g+e\}\}\). To use this control group, specify
`control_group = "all"`

in the `DiDge(...)`

command. This option is selected by default.

Two alternatives can be specified.

**Never-treated:** The never-treated control group is
defined by \(C_{g,e} \equiv \{ i : G_i =
\infty \}\). To use this control group, specify
`control_group = "never-treated"`

in the
`DiDge(...)`

command.

**Future-treated:** The future-treated control group is
defined by \(C_{g,e} \equiv \{ i : G_i >
\max\{g, g+e\} \text{ and } G_i < \infty\}\). To use this
control group, specify `control_group = "future-treated"`

in
the `DiDge(...)`

command.

**Base event time:** The base event time can be
specified using the `base_event`

argument in
`DiDge(...)`

, where `base_event = -1`

by
default.

### 3.2 DiD Estimation for a Single \((g,e)\) Combination

The `DiDge()`

command performs the following sequence of
steps:

**Step 1.** Define the \((g,e)\)-specific sample of treated and
control units, \(S_{g,e} \equiv \{G_i=g\} \cup
\{i \in C_{g,e}\}\). Drop any observations that do not satisfy
\(i \in S_{g,e}\).

**Step 2.** Construct the within-\(i\) differences \(\Delta Y_{i,g+e} \equiv Y_{i,g+e} -
Y_{i,g+b}\) for each \(i \in
S_{g,e}\).

**Step 3.** Estimate the simple linear regression \(\Delta Y_{i,g+e} = \alpha_{g,e} + \beta_{g,e}
1\{G_i =g\} + \epsilon_{i,g+e}\) by OLS for \(i \in S_{g,e}\).

The OLS estimate of \(\beta_{g,e}\) is equivalent to \(\text{DiD}_{g,e}\). The standard error provided by OLS for \(\beta_{g,e}\) is equivalent to the standard error from a two-sample test of equal means for the null hypothesis \[\mathbb{E}[\Delta Y_{i,g+e} | G_i = g] = \mathbb{E}[\Delta Y_{i,g+e} | i \in C_{g,e}] \] which is equivalent to testing that \(\text{ATT}_{g,e}=0\).

## 4. The `DiD(...)`

Command

`DiDforBigData`

uses the `DiD(...)`

command to
estimate \(\text{DiD}_{g,e}\) for all
available cohorts \(g\) across a range
of possible event times \(e\);
`DiD(...)`

is documented here.

### 4.1 DiD Estimation for All Possible \((g,e)\) Combinations

`DiD(...)`

uses the `control_group`

and
`base_event`

arguments the same way as
`DiDge(...)`

.

`DiD(...)`

also uses the `min_event`

and
`max_event`

arguments to choose the minimum and maximum event
times \(e\) of interest. If these
arguments are not specified, it assumes all possible event times are of
interest.

In practice, `DiD(...)`

completes the following steps:

**Step 1.** Determine all possible combinations of \((g,e)\) available in the data. The
`min_event`

and `max_event`

arguments allow the
user to restrict the minimum and maximum event times \(e\) of interest.

**Step 2.** In parallel, for each \((g,e)\) combination, construct the
corresponding control group \(C_{g,e}\)
the same way as `DiDge(...)`

. Drop any \((g,e)\) combination for which the control
group is empty.

**Step 3.** Within each \((g,e)\)-specific process, define the \((g,e)\)-specific sample of treated and
control units, \(S_{g,e} \equiv \{G_i=g\} \cup
\{i \in C_{g,e}\}\). Drop any observations that do not satisfy
\(i \in S_{g,e}\).

**Step 4.** Within each \((g,e)\)-specific process, construct the
within-\(i\) differences \(\Delta Y_{i,g+e} \equiv Y_{i,g+e} -
Y_{i,g+b}\) for each \(i\) that
remains in the sample.

**Step 5.** Within each \((g,e)\)-specific process, estimate \(\Delta Y_{i,g+e} = \alpha_{g,e} + \beta_{g,e}
1\{G_i =g\} + \epsilon_{i,g+e}\) by OLS.

The OLS estimate of \(\beta_{g,e}\) is equivalent to \(\text{DiD}_{g,e}\). The standard error provided by OLS for \(\beta_{g,e}\) is equivalent to the standard error from a two-sample test of equal means for the null hypothesis \[\mathbb{E}[\Delta Y_{i,g+e} | G_i = g] = \mathbb{E}[\Delta Y_{i,g+e} | i \in C_{g,e}] \] which is equivalent to testing that \(\text{ATT}_{g,e}=0\). Note that \(\text{ATT}_{g,e}=0\) is tested as a single hypothesis for each \((g,e)\) combination; no adjustment for multiple hypothesis testing is applied.

### 4.2 Estimate the Average DiD across Cohorts and Event Times

Aside from estimating each \(\text{DiD}_{g,e}\), `DiD(...)`

also estimates \(\text{DiD}_{e}\) for
each \(e\) included in the event times
of interest.

To do so, `DiD(...)`

completes the following steps:

**Step 1.** At the end of the \((g,e)\)-specific estimation in parallel
described above, it returns the various \((g,e)\)-specific samples of the form \(S_{g,e} \equiv \{G_i=g\} \cup \{i \in
C_{g,e}\}\).

**Step 2.** It defines an indicator for corresponding to
cohort \(g\), then stacks all of the
samples \(S_{g,e}\) that have the same
\(e\). Note that the same \(i\) can appear multiple times due to
membership in both \(S_{g_1,e}\) and
\(S_{g_2,e}\), so the distinct
observations are distinguished by the indicators for \(g\).

**Step 3.** It estimates \(\Delta Y_{i,g+e} = \sum_g \alpha_{g,e} + \sum_g
\beta_{g,e} 1\{G_i =g\} + \epsilon_{i,g+e}\) by OLS for the
stacked sample across \(g\).

**Step 4.** It constructs \(\text{DiD}_e = \sum_g \omega_{g,e}
\beta_{g,e}\), where \(\omega_{g,e}
\equiv \frac{\sum_i 1\{G_i=g\}}{\sum_i 1\{G_i < \infty\}}\).
Since each \(\beta_{g,e}\) is an
estimate of the corresponding \(\text{ATT}_{g,e}\), it follows that \(\text{DiD}_e\) is an estimate of the
weighted average \(\text{ATT}_{e} \equiv
\sum_g \omega_{g,e} \text{ATT}_{g,e}\).

**Step 5.** To test the null hypothesis that \(\text{ATT}_{e} = 0\), it defines \(\bar\beta_e = (\beta_{g,e})_g\) and \(\bar\omega_e = (\omega_{g,e})_g\). Note
that \(\text{DiD}_e = \bar\omega_e'
\bar\beta_e\). To get the standard error, for \(\text{DiD}_e\), it uses that \(\text{Var}(\text{DiD}_e) = \bar\omega_e'
\text{Var}(\bar\beta_e) \bar\omega_e\), where \(\text{Var}(\bar\beta_e)\) is the usual
(heteroskedasticity-robust) variance-covariance matrix of the OLS
coefficients. Since the same unit \(i\)
appears on multiple rows of the sample, we must cluster on \(i\) when estimating \(\text{Var}(\bar\beta_e)\). Finally, the
standard error corresponding to the null hypothesis of \(\text{ATT}_{e} = 0\) is \(\sqrt{\text{Var}(\text{DiD}_e)}\).

A similar approach is used to estimate \(\text{DiD}_{E}\), the average \(\text{DiD}_{e}\) across a set of event times \(E\). It again uses that these average DiD parameters can be represented as a linear combination of OLS coefficients \(\beta_{g,e}\) with appropriate weights to construct the standard error for \(\text{ATT}_{E}\).