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Doubly Robust Instrumented Difference-in-Differences
\title{
Doubly Robust Instrumented Difference-in-Differences
}
\date{\today}
\author{
Jonas Skjold Raaschou-Pedersen\thanks{
Center for Social Data Science, University of Copenhagen.
Email: \texttt{[email removed]}.
} \thanks{
I thank Andreas Bjerre-Nielsen and Anders Munch for helpful comments.
}
}
\maketitle
\begin{abstract}
We study estimation of the local average treatment effect on the treated
($LATT$) in instrumented difference-in-differences (IDiD) designs with
covariates and staggered instrument exposure. We derive the efficient influence function
(EIF) of the target parameter in both panel and repeated cross-sections
settings, allowing for two classes of control groups: never-exposed and
not-yet-exposed.
Building on the EIF, we construct doubly robust estimands and corresponding
estimators from first principles. The resulting procedures are the IDiD
analogues of the difference-in-differences (DiD) procedures in \cite{csa},
targeting $LATT$ rather than $ATT$.
We further establish a Bloom-type result under one-sided compliance and
absorbing treatment, linking $LATT$ to a convex combination of
exposure-cohort-specific $ATT(g, t)$ parameters,
making the connection between IDiD and DiD explicit.
Asymptotic properties are established under conditions on the remainder
term and either Donsker conditions or via cross-fitting.
We also construct double machine learning (DML) estimators for the
$LATT$ in both data settings and show their equivalence to
cross-fitted estimators.
Simulations assess the double robustness and finite-sample performance of the
proposed methods.
An implementation is available in the Python package \texttt{idid}\footnote{
Replication code is available at \url{https://github.com/jsr-p/idid}.
}.
\end{abstract}
\textbf{Keywords:} Difference-in-differences, instrumental variables, causal
inference, semiparametric inference, instrumented difference-in-differences,
double machine learning
\newpage
\section{Introduction}
Difference-in-differences (DiD) is a central tool in applied econometrics for
estimating causal effects in non-experimental settings.
Recent work, often referred to as
the ``DiD renaissance'', has exposed important limitations of classical DiD
methods, particularly under treatment effect heterogeneity across cohorts and
over time. In such settings, coefficients from standard regression
specifications can lack a clear causal interpretation.
Two broad responses have emerged. One is to refine the underlying regression
model. The other is to begin with a well-defined, interpretable statistical
target, an \textit{estimand}, rather than starting from a regression model.
Under suitable identifying assumptions, this
estimand can be linked to observable quantities and estimated in a way that
admits a transparent causal interpretation.
A central object in this approach is the efficient influence function
(EIF), which is used to construct semiparametrically efficient estimators
and valid inference.
This perspective is adopted, for example, in the work of
\citet{sazhao} and \citet{csa}.
However, even modern DiD approaches can fail when the treatment is endogenous.
A classical remedy is the use of instrumental variables (IV). The integration
of IV with DiD remained underdeveloped until \citet{miyaji}, who introduced the
instrumented difference-in-differences (IDiD) framework with staggered
exposure to the instrument.
Adopting an estimand-based perspective, \citet{miyaji} defines the causal
parameter of interest, the cohort-specific time-varying local average treatment
effect on the treated ($LATT(e, t)$), and links it to an estimable quantity under
identifying assumptions.
However, the framework does not incorporate covariates, which are often
essential in empirical applications to support the plausibility of the
identifying assumptions.
In related work, \citet{csax} identify the $LATT$ parameter of
\citet{miyaji} with the inclusion of covariates as a special case of their
sequential conditional moment restriction framework for DiD.
They derive the efficient influence function of the parameter in the
panel data setting and construct a corresponding doubly robust estimator.
However, their analysis focuses on the case with a single exposure date to
the instrument.
Moreover, they leave the case of repeated cross-sections for future work.
This paper studies the general case with covariates and staggered exposure to
the instrument in both panel data and repeated cross-sections.
We derive the efficient influence function for the $LATT$ parameter in
both data settings.
The repeated cross-sections case is empirically relevant in settings where
balanced panels are unavailable, for example in ``trimmed panel'' data or
repeated survey samples.
A novel contribution is that the EIF is derived explicitly using the
approach developed by \citet{kennedy}.
Moreover, as done in \cite{miyaji}, the framework of \cite{csa} is extended to
the IDiD case allowing the use of not-yet-exposed units as controls, analogous
to the not-yet-treated control group in the staggered DiD setting.
Both control variables are handled generally by invoking either of the
corresponding identifying assumptions.
Using the derived EIFs, we construct doubly robust estimands and
corresponding estimators for the $LATT$ parameter in both data settings.
The construction follows the estimating equation approach:
the estimand solves the population moment condition implied by the EIF,
while the estimator solves its empirical counterpart.
This construction also underlies the doubly robust estimand and
estimators in \citet{sazhao} (and, by extension, \citet{csa}), although
implicitly.
The resulting doubly robust estimators have a structure closely related to
those proposed by \citet{sazhao} and \citet{csa}.
In particular, the estimator takes the form of a ratio of two
doubly robust estimators for ATT-type parameters.
The numerator corresponds to the outcome of interest and the denominator
to the treatment variable, with the instrument exposure variable playing
the role of the treatment indicator.
This structure arises naturally because the identified $LATT$ parameter
itself can be written as a ratio of two ATT-type parameters.
The estimators reduce to those of \cite{miyaji} in the case of no covariates.
The asymptotic behavior of the estimator is established via a decomposition
into an influence function term, an empirical process term, and a remainder
term (cf. \cite{kennedy}).
The remainder term is handled directly in the proofs.
The empirical process term is controlled either via Donsker class
assumptions or by employing cross-fitting, which permits the use of
flexible machine learning methods. We derive DML estimators, which we show are
equivalent to the cross-fitted estimator in \cite{kennedy}, for both data
settings.
A simulation study illustrates the finite-sample properties of the
proposed estimators.
A freely available implementation is provided in the Python package
\texttt{idid}.
\paragraph{Contributions}
This paper makes three contributions.
First, we derive the efficient influence function for the $LATT$ parameter
in the IDiD framework with covariates and
staggered instrument exposure.
The derivation covers both panel data and repeated cross-sections.
Second, using the EIF, we construct doubly robust estimands and estimators
for the $LATT$ parameter.
Third, the paper illustrates how the influence function derivation
strategy of \citet{kennedy} can be applied to econometric target
parameters.
This approach provides a practical alternative to classical tangent
space calculations \citep{newey1990,tsiatis} and ensures that the resulting
estimators target precisely the interpretable estimand specified
by the researcher.
We also construct
Neyman-orthogonal scores for DML estimation using the EIFs and relate IDiD with
staggered instrument exposure to DiD with staggered treatment via a
Bloom-type result.
\paragraph{Related literature}
This paper relates to several strands of the literature.
First, it contributes to the literature on IDiD.
\citet{miyaji} introduce the IDiD framework and define the $LATT$
parameter.
\citet{csax} extend this framework to allow for covariates and derive
a doubly robust estimator in the panel setting with a single exposure
date.
The present paper studies the general case with covariates and staggered
exposure to the instrument in both panel data and repeated cross-sections,
and derives corresponding doubly robust estimands and estimators.
Second, the paper relates to the literature on semiparametric
DiD estimation.
\citet{sazhao} and \citet{csa} develop doubly robust estimators for
ATT-type parameters in two-period and staggered adoption settings.
The estimators proposed here have a similar structure but tailored to the IDiD
framework.
Moreover, in deriving the influence function, doubly robust estimand, and
estimators for the LATT parameter, we simultaneously derive the corresponding
results for the ATT parameter, thereby recovering the results of \citet{sazhao}
and \citet{csa} from first principles.
Third, the paper connects to recent work at the intersection of modern
semiparametric methods and DiD/IDiD, as well as to recent work on doubly
robust IV estimators.
For example,
\cite{chang2020} develop DML estimators for DiD in the case of panel data and
repeated cross-sections,
\citet{deng} develop a TMLE estimator for the two-period
DiD parameter, while \citet{metalearner} propose a meta-learner
algorithm for the $LATT$ in two-period IDiD with panel data.
\citet{sloczynskiDoublyRobustEstimation2022} develop doubly robust
estimators for LATE and LATT in cross-sectional IV settings, whose
structure resembles the panel data estimator derived here.
More broadly, the paper relates to recent work emphasizing clearly defined
target parameters in causal inference and IV analysis,
including \citet{mogstad2024instrumental}, who distinguish between forward
and reverse engineering approaches to IV parameters.
In this paper, we take the forward engineering approach.
It also connects to the targeted learning literature
\cite{vdlrose,van2015statistics}, which emphasizes aligning the estimand
with the underlying scientific question and constructing estimators under
minimal modeling assumptions, rather than relying on potentially
misspecified parametric models. Finally, it relates to the double machine
learning literature \cite{chernozhukovDoubleDebiasedMachine2018a}, as we
derive two DML estimators.
\paragraph{Organization of the paper}
\Cref{sec:identification} introduces IDiD, the causal estimands, and the main
identification results.
\Cref{sec:weighting} studies aggregated effects.
\Cref{sec:estimation-and-inference} develops the doubly robust and DML estimators and
establishes their asymptotic properties. \Cref{sec:simulation} reports the
simulation results. Proofs and derivations are collected in the appendix.
\Cref{sec:influence-functions} provides a detailed account of how the influence
function calculations fit into the development of the paper.
\paragraph{Notation}
For a measurable function $f$, let $\Vert f \Vert_{q,P} = (\int |f|^q
dP)^{1/q}$ denote its $L^q(P)$ norm.
Write $P_n f = n^{-1}\sum_{i=1}^n f(X_i)$ for the empirical average and $Pf =
\int f\, dP$ for the expectation under $P$.
We also write $E[f(X)]$ for the expectation (under the relevant distribution) and
use the notation interchangeably where convenient.
If $f$ depends on a parameter $\tau$ and some nuisance $\eta$, we write $P_n
f(\cdot;\tau, \eta) =
n^{-1}\sum_{i=1}^n f(X_i;\tau, \eta)$ and $P f(\cdot;\tau, \eta) = \int
f(\cdot;\tau, \eta)\, dP$.
The empirical process is written $\sqrt{n}(P_n - P)[f]$.
Calligraphic letters denote supports of random variables, e.g.\ $\mathcal{E}$, $\mathcal{D}$, and $\mathcal{Z}$ for $E$, $D$, and $Z$.
For an event $A$, $\mathbf{1}\{A\}$ denotes its indicator.
We write $O$ for a generic tuple of random variables that is context-dependent.
See also \Cref{sec:eif-derivation} for details on the notation for the
influence function operator, $\mathbb{IF}$.
\section{Identification}
\label{sec:identification}
\subsection{Setup}
We first introduce the notation used throughout the article, building on \cite{miyaji}.
We consider the general case of $\mathcal{T}$ periods.
Let $D_{t} \in \{0, 1\}$ denote treatment status and
$Z_{t} \in \{0, 1\}$ the instrument status.
Moreover, let
$D = (D_{1}, D_{2}, \ldots, D_{\mathcal{T}})$
and
$Z = (Z_{1}, Z_{2}, \ldots, Z_{\mathcal{T}})$
denote the treatment and instrument paths.
We make the following assumption on the instrument:
\begin{assumption}[Staggered adoption]
\label{ass:staggered-instrument}
$Z_{1} = 0$\footnote{If not already stated explicitly, equalities involving random variables are understood to hold almost
surely.};
and for $t = 2, \ldots, \mathcal{T}$:
\begin{equation}
\label{eq:staggered-instrument}
Z_{t-1} = 1 \implies Z_{t} = 1.
\end{equation}
\label{i:ms0}
\end{assumption}
\Cref{ass:staggered-instrument} enforces that no units are exposed to the
instrument in the first period and all units that are exposed in some period
stay exposed\footnote{
This is analogous to Assumption 1 on the treatment variable assumed by
\cite{csa}.
}.
\Cref{ass:staggered-instrument}
implies that the time period where the instrument switches on characterizes
the instrument path $Z$ completely. Because of this we define the
\textit{cohort exposure variable} $E := \min \{t \mid Z_{t} = 1\}$ and $E_{e}
:= \mathbf{1}\{E = e\}$.
To identify the target parameter, valid control groups are needed.
In the DiD literature, commonly used control groups are never-treated and
not-yet-treated units.
Here, we construct similar control groups but based on the instrument instead
of the treatment variable.
Hence, we define
\begin{equation}
\label{eq:control-variables}
C^{nev} := \mathbf{1}\{E = \infty\},
\quad
C^{nye}_{e, s} := \mathbf{1}\{E_{e} = 0, Z_{s} = 0\},
\end{equation}
for the \textit{never-exposed} and \textit{not-yet-exposed} control groups,
respectively\footnote{
The control variable $C^{nye}_{e,s}$ is analogous to $(1-D_s)(1-G_g)$
in \cite{csa}, but here corresponds to units not in the cohort exposed
at $E = e$ and not yet exposed at time $s$, i.e., $Z_s = 0$.
In the appendix, we provide a table comparing the different objects in
the two-period DiD, staggered adoption DiD and our case of staggered
IDiD; see \Cref{table:comparison}.
}.
Let $\bar{e} := \max_{i}E_{i}$;
in the case of never-exposed units, $\bar{e} = \infty$,
and in the case of only not-yet-exposed units, $\bar{e} < \infty$.
Denote the support of the exposure variable excluding $\bar{e}$ as
$\mathcal{E} := \mathrm{supp}(E) \setminus \bar{e} \subseteq \{2, 3, \ldots,
\mathcal{T}\}$\footnote{
Analogous to \citet{csa}, when there is a never-exposed cohort, $E = \infty$,
$\mathcal{E}$ only excludes this.
In the case of not-yet-exposed control groups only, we exclude the
last exposed cohort because there are no available control groups for
this cohort.
}.
Denote the \textit{generalized propensity scores} corresponding to the control
variable in use as
\begin{align}
\label{eq:generalized-propensity-scores}
p_{e}(X) := P(E_{e} = 1 \mid X, E_{e} + C^{nev} = 1),
\quad
p_{e, s}(X) := P(E_{e} = 1 \mid X, E_{e} + C^{nye}_{e, s} = 1).
\end{align}
The conditioning on $E_{e} + C = 1$ restricts attention to the relevant
$2 \times 2$ comparison: units are either in the exposed cohort, $E_{e} = 1$,
or in the control group, $C = 1$. Within each such slice, the objects behave
as in the corresponding two-period setup.
\paragraph{Potential treatment and outcomes}
Let $D_{t}(\infty)$ denote a unit's unexposed potential treatment at time
$t$ if they remain untreated through time period $\mathcal{T}$,
i.e., if they were not to be exposed to the instrument across all available time
periods.
For $e = 2, 3, \ldots, \mathcal{T}$, let $D_{t}(e)$
denote the potential treatment
that a given unit would experience at time $t$ if they were to first become
exposed to the instrument in time period $e$.
The observed and potential treatment for a given unit are related through
\begin{align}
\label{eq:observed-treatment-status}
D_{t} = D_{t}(\infty) + \sum_{2 \leq e \leq \mathcal{T}}
[D_{t}(e) - D_{t}(\infty)] \cdot \mathbf{1}\{E = e\},
\end{align}
i.e., we only observe one potential treatment path for each unit.
Let $Y_{t}(d, z)$ be a given unit's potential outcome in period $t$ had they
been given treatment path $D = d$ and instrument path $Z = z$.
The analogy between the treatment and instrument for the DiD and IDiD
frameworks differs in the sense that the instrument does not affect the outcome
directly but only through the treatment. Specifically, the instrument creates
exogenous variation in the treatment that allows us to identify the effect of
the treatment on the outcome in the presence of hidden confounders.
This relation between the instrument and the potential outcomes is enforced in the
following assumption:
\begin{assumption}[No carryover and exclusion restriction in multiple time periods]
\label{ass:no-carryover}
\begin{align}
\label{eq:no-carryover}
\forall z \in \mathcal{Z},
\forall d \in \mathcal{D},
\forall t \in \{1, \ldots, \mathcal{T}\},
\,
Y_{t}(d, z) = Y_{t}(d_{t}),
\end{align}
\label{i:ms1}
\end{assumption}
\Cref{ass:no-carryover} means that the potential outcomes
at time $t$ only depend on the treatment variable at time $t$,
and that they do not depend directly on the instrument.
The latter is analogous to the exclusion restriction in the simple
cross-sectional IV design.
To arrive at an expression for the observed outcome,
let $Y_{t}(0)$ denote a given unit's potential outcome
at time $t$ if they are untreated at time period $t$,
and $Y_{t}(1)$ if they are treated at time period $t$.
With \eqref{eq:observed-treatment-status} we can write the observed
outcome in terms of the potential outcomes and treatments as:
\begin{align}
\label{eq:observed-outcome}
Y_{t} & = Y_{t}(0) + [Y_{t}(1) - Y_{t}(0)]D_{t}
\\&=
\label{eq:exposed-stag}
Y_{t}(0) + [Y_{t}(1) - Y_{t}(0)][
D_{t}(\infty) + \sum_{2 \leq e \leq \mathcal{T}}
[D_{t}(e) - D_{t}(\infty)] \cdot \mathbf{1}\{E = e\}
].
\end{align}
Using \eqref{eq:exposed-stag}, we define the \textit{exposed/unexposed
outcomes} as:
\begin{align}
\label{eq:exposed-unexposed-outcome}
Y_{t}(D_{t}(E)) := \begin{cases}
Y_{t}(D_{t}(e))
:= Y_{t}(0) + [Y_{t}(1) - Y_{t}(0)]D_{t}(e), &
E = e, \\
Y_{t}(D_{t}(\infty))
:= Y_{t}(0) + [Y_{t}(1) - Y_{t}(0)]D_{t}(\infty), &
E = \infty. \\
\end{cases}
\end{align}
The exposed/unexposed outcomes\footnote{
As noted by \cite{miyaji}, the concept of exposed and unexposed outcomes
is not new. In the standard cross-sectional binary IV setup with potential
outcomes $Y(0), Y(1)$, potential treatments $D(0), D(1)$, and instrument $Z$,
the observed treatment can be written as
$D = D(0) + [D(1) - D(0)]Z$.
Substituting this into the observed outcome equation
$Y = Y(0) + [Y(1) - Y(0)]D$
yields the exposed and unexposed outcomes
$Y(D(Z)) := Y(0) + [Y(1) - Y(0)]D(Z)$ for $Z \in \{0,1\}$.
This is analogous to the staggered exposure setting in
\cref{eq:exposed-stag}, which leads to the exposed and unexposed outcomes in
\cref{eq:exposed-unexposed-outcome}.
} play a key role in the identification results,
\Cref{prop:identification-panel,prop:identification-rc}.
\paragraph{Sampling assumption}
Our results apply to both panel data and repeated cross-sections,
which are covered by the following assumption.
Let $T \in \{1,\ldots,\mathcal{T}\}$ denote the period a unit is
observed in the repeated cross-sections case.
\begin{assumption}[Random Sampling or Repeated Cross-Sections]
\label{ass:sample}
Assume either:
\begin{enumerate}[label=(\Alph*), ref=\theassumption(\Alph*)]
\item \label{ass:sample:panel}
The data are i.i.d. from the distribution of
$(Y_{1},\ldots,Y_{\mathcal{T}},D_{1},\ldots,D_{\mathcal{T}},
E_{2},\ldots,E_{\mathcal{T}},C^{nev}, X)$.
\item \label{ass:sample:mixture}
Conditional on $T=t$, the data are i.i.d. from the distribution of
$(Y_t,D_t,E_2,\ldots,E_{\mathcal{T}},C^{nev},X)$
with $(E_{2}, E_{3}, \ldots, E_{\mathcal{T}}, C^{nev}, X)$ being
invariant to $T$.
\end{enumerate}
\end{assumption}
\Cref{ass:sample:panel} implies that the observed data consist of panel
data, whereas \cref{ass:sample:mixture} implies that the observed
data are i.i.d. draws from the mixture distribution
\begin{align}
\label{eq:mixture-csa}
F_{M}(y, d, e_{2}, \ldots, e_{\mathcal{T}}, c, t, x)
= \sum_{t=1}^{\mathcal{T}}
\lambda_{t} \cdot
F_{Y, D, E_{2}, E_{3}, \ldots, E_{\mathcal{T}}, C^{nev}, X \mid T}
(y, d, e_{2}, \ldots, e_{\mathcal{T}}, c, x \mid t),
\end{align}
where $\lambda_{t} := P(T_{t} = 1)$ and $T_{t} := \mathbf{1}\{T = t\}$.
This mixture distribution setup is analogous to
\cite{abadieSemiparametricDifferenceinDifferencesEstimators2005a,sazhao,csa}
but here for the IDiD design with staggered exposure.
\begin{remark}
All expectations in the repeated cross-sections case are taken
with respect to the mixture measure $P_{M}$, although we suppress this
subscript in the notation.
Moreover, in the repeated cross-section setting, for each unit in the pooled
sample we observe
$(Y, D, E_{2},\ldots,E_{\mathcal{T}}, C^{nev}, T, X)$.
\end{remark}
\subsection{The LATT parameter}
The target parameter in this paper is the cohort-specific time-varying local
average treatment effect on the treated of \cite{miyaji}
\begin{align}
\label{eq:latt}
LATT(e, t)
& := E[Y_{t}(1) - Y_{t}(0) \mid E_{e} = 1, D_{t}(e) > D_{t}(\infty)].
\end{align}
\Cref{eq:latt} is the treatment effect, $Y_{t}(1) - Y_{t}(0)$,
averaged over the subpopulation of compliers, $D_{t}(e) > D_{t}(\infty)$,
and the units exposed to the instrument in period $e$, $E_{e} = 1$.
The parameter varies across cohorts $E$ and time $t$;
hence it allows us to answer questions related to the heterogeneity across
cohorts and time.
In \Cref{sec:weighting} we show how to aggregate the parameters into
aggregated effects; similar to how \cite{csa} aggregates their $ATT(g, t)$
parameters.
\subsection{Identifying assumptions}
\begin{assumption}[Monotonicity assumption in multiple time periods]
\label{ass:monotonicity}
For all $e \in \mathcal{E}$ and $t \geq e$:
\begin{align*}
P(D_{t}(e) \geq D_{t}(\infty) \mid X) = 1
\ \text{a.s.}
\end{align*}
\end{assumption}
\Cref{ass:monotonicity} is analogous to the monotonicity assumption in
cross-sectional IV and requires that the instrument, here the exposure
cohort variable $E$, affects the treatment in only one direction.
\begin{assumption}[No anticipation in the first stage]
\label{ass:no-anticipation}
For all
$e \in \mathcal{E}$
and $t < e$:
\begin{align}
\label{eq:no-anticipation}
E[D_{t}(e) \mid X, E_{e} = 1]
= E[D_{t}(\infty) \mid X, E_{e} = 1]
\end{align}
\end{assumption}
\Cref{ass:no-anticipation} is analogous to the standard no-anticipation
assumption in DiD, with the exposure variable replacing the treatment,
and requires that exposure does not affect the treatment prior to the
exposure date.
\medskip\noindent
The following two assumptions use the never-exposed control group $C^{nev}$.
\begin{assumption}[Conditional parallel trends in the treatment based on "Never-Exposed" Group]
\label{ass:cpt-treat-ne}
For all $e, t$:
\begin{align}
\label{eq:cpt-treat-ne}
E[D_{t}(\infty)- D_{t-1}(\infty) \mid X, E_{e} = 1]
= E[D_{t}(\infty)- D_{t-1}(\infty) \mid X, C^{nev} = 1]
\; \text{a.s.}
\end{align}
\end{assumption}
\begin{assumption}[Conditional parallel trends in the unexposed outcome based
on "Never-Exposed" Groups]
\label{ass:cpt-unexpout-ne}
For all $e, t$:
\begin{align}
\label{eq:cpt-unexpout-ne}
E[Y_{t}(D_{t}(\infty))- Y_{t-1}(D_{t-1}(\infty)) \mid X, E_{e} = 1]
= E[Y_{t}(D_{t}(\infty))- Y_{t-1}(D_{t-1}(\infty)) \mid X, C^{nev} = 1]
\; \text{a.s.}
\end{align}
\end{assumption}
\medskip\noindent
The following two assumptions are analogous to the previous two, but use
the not-yet-exposed control group $C^{nye}_{e, s}$.
\begin{assumption}[Conditional parallel trends in the treatment based on "Not-Yet-Exposed" Groups]
\label{ass:cpt-treat-nye}
For all $e, t$:
\begin{align}
\label{eq:ptstagtreatmentX}
E[D_{t}(\infty)- D_{t-1}(\infty) \mid X, E_{e} = 1]
= E[D_{t}(\infty)- D_{t-1}(\infty) \mid X, C^{nye}_{e, s} = 1]
\; \text{a.s.}
\end{align}
\end{assumption}
\begin{assumption}[Conditional parallel trends in the unexposed outcome based on "Not-Yet-Exposed" Groups]
\label{ass:cpt-unexpout-nye}
For all $e, t$:
\begin{align}
\label{eq:ptstagoutcomeX}
E[Y_{t}(D_{t}(\infty))- Y_{t-1}(D_{t-1}(\infty)) \mid X, E_{e} = 1]
= E[Y_{t}(D_{t}(\infty))- Y_{t-1}(D_{t-1}(\infty)) \mid X, C^{nye}_{e, s} = 1]
\; \text{a.s.}
\end{align}
\end{assumption}
In the identification proofs we invoke either pair of the four assumptions above depending on
the control variable used.
If using the never-exposed control group $C^{nev}$, we assume
\Cref{ass:cpt-treat-ne,ass:cpt-unexpout-ne},
and if using the not-yet-exposed control group $ C^{nye}_{e,t}$, we assume
\Cref{ass:cpt-treat-nye,ass:cpt-unexpout-nye}.
Writing $C$ for a generic control variable allows us to encompass both control
variables in the identification arguments.
\begin{assumption}[Overlap]
\label{ass:overlap}
For each $t \in \{2, 3, \ldots, \mathcal{T}\}$, $e \in \mathcal{E}$,
there exist some $c > 0$ such that $P(E_{e} = 1) > c$
and $P(E_{e} = 1 \mid X) < 1 - c$ a.s.
\end{assumption}
\Cref{ass:overlap} is a standard overlap condition. It requires that, for
each exposure cohort, the probability of exposure is bounded away from
zero, and from one conditional on covariates. This ensures that all relevant
subpopulations have a non-negligible probability of being both exposed
and unexposed, which is necessary for identification and stable
estimation.
\subsection{Nonparametric identification of the group-time LATT parameter}
\label{sec:nonparametric-identification}
\subsubsection{Notation}
To derive the main identification result, we introduce the central objects and
notation below. Let $V_t$ be a generic random variable and define
\[
\Delta_{t-e+1} V_t := V_t - V_{e-1},
\]
i.e.\ the change of $V_{t}$ between period $t$ and the pre-exposure period
$e-1$.
\medskip
\noindent
\textit{Panel data.}
For panel data, define the outcome regression functions for treated,
never-exposed, and not-yet-exposed units as
\begin{align}
\label{eq:mean-functions-panel}
m_{e, t}^{trt, p}(X) & := E[\Delta_{t-e+1} Y_t \mid X, E_e = 1],
\\
\nonumber
m_{e, t}^{nev, p}(X) & := E[\Delta_{t-e+1} Y_t \mid X, C^{nev} = 1],
\quad
m_{e, t}^{nye, p}(X) := E[\Delta_{t-e+1} Y_t \mid X, C^{nye}_{e,t} = 1].
\end{align}
Analogously, define $g_{e, t}^{trt, p}(X)$, $g_{e, t}^{nev, p}(X)$, and
$g_{e, t}^{nye, p}(X)$ with outcome $\Delta_{t-e+1} D_t$.
\medskip
\noindent
\textit{Repeated cross-sections.}
For repeated cross-sections, the outcome regression functions are defined as
\begin{align}
\label{eq:mean-functions-rc}
m_{e, t}^{trt, rc}(X) & := E[Y \mid X, E_e = 1, T = t],
\\
\nonumber
m_{e, t}^{nev, rc}(X) & := E[Y \mid X, C^{nev} = 1, T = t],
\quad
m_{e, s, t}^{nye, rc}(X) := E[Y \mid X, C^{nye}_{e,s} = 1, T = t],
\end{align}
with corresponding definitions
$g_{e, t}^{trt, rc}(X)$, $g_{e, t}^{nev, rc}(X)$, and
$g_{e, s, t}^{nye, rc}(X)$ obtained by replacing $Y$ with $D$.
\paragraph{Encompassing both never exposed and not-yet-exposed}
A generic control variable is written as $C$ and the corresponding
propensity as $p(X)$.
Below we write $m_{e, t}^{c, p}(X), g_{e, t}^{c, p}(X)$
for a generic control variable outcome regression function in the case of panel
data,
and similarly
$m_{e, s, t}^{c, rc}(X), g_{e, s, t}^{c, rc}(X)$
in the case of repeated cross-sections.
The proofs for the not-yet-exposed and never-exposed cases are identical,
differing only in the control indicator and propensity score.
To avoid repetition, we present a unified argument covering both cases.
The results follow under \Cref{ass:cpt-treat-ne,ass:cpt-unexpout-ne}
for the never-exposed case and under
\Cref{ass:cpt-treat-nye,ass:cpt-unexpout-nye}
for the not-yet-exposed case.
This is done for both the panel-data and repeated-cross-sections settings.
\subsubsection{Panel Data}
\label{sec:identification-panel}
\begin{proposition}[Identification Panel Data]
\label{prop:identification-panel}
Fix $e \in \mathcal{E}$ and $t \ge e$.
Suppose \Cref{ass:sample:panel,ass:staggered-instrument,ass:no-carryover,ass:monotonicity,ass:no-anticipation,ass:overlap} hold.
Let $C$ denote a generic control-group indicator, where either $C = C^{nev}
\quad \text{or} \quad C = C^{nye}_{e,t}.$ If $C = C^{nev}$, additionally
assume
\Cref{ass:cpt-treat-ne,ass:cpt-unexpout-ne}.
If $C = C^{nye}_{e,t}$, additionally assume
\Cref{ass:cpt-treat-nye,ass:cpt-unexpout-nye}.
Let $m_{e, t}^{c, p}(X)$ and $g_{e, t}^{c, p}(X)$
be the mean functions \cref{eq:mean-functions-panel} for a generic control
variable.
Then the cohort-specific time-varying local average treatment effect on the
treated \cref{eq:latt} is identified by
\begin{align}
\label{eq:identification-panel}
LATT(e,t)
=
\frac{E[
m_{e, t}^{trt, p}(X) - m_{e, t}^{c, p}(X) \mid E_{e} = 1
]
}{E[
g_{e, t}^{trt, p}(X) - g_{e, t}^{c, p}(X) \mid E_{e} = 1
]} =: \tau^{p}_{e, t}.
\end{align}
\end{proposition}
\medskip\noindent
Proof in \Cref{sec:proofs-panel}.
\begin{remark}
The identification result in \cref{eq:identification-panel} is closely
related to \cite{csax}, but adapted here to also including the case of
not-yet-exposed controls. Its
general structure also mirrors the results of
\cite{frolichExploitingRegionalTreatment2010} and
\cite{sloczynskiDoublyRobustEstimation2022}, derived here for panel
data in the IDiD framework allowing for the different control group variables
in \cref{eq:control-variables}.
\end{remark}
\paragraph{Influence functions and construction of the DR Estimands}
Below we derive DR estimands and corresponding DR estimators for the $LATT(e,
t)$ in both data settings.
The steps are as follows:
\begin{enumerate}
\item Identification yields an initial estimand which is a ratio of two
ATT-type parameters, e.g.
\cref{eq:identification-panel,eq:identification-rc}.
\item We derive the efficient influence function (EIF) of the initial estimand\footnote{
Kennedy distinguishes between the influence function for a parameter
and that for an estimator, referring to the former as an
\textit{influence curve}. We do not adopt this distinction in terminology.
}.
This EIF involves weights that are normalized to Hájek-type weights
that are variance stabilizing cf. \citep{sazhao}.
\label{i:if-procedure-2}
\item Solving the population version of the estimating equation using the EIF
for the target parameter yields the doubly robust (DR) estimand.
\label{i:if-procedure-3}
\item We derive the EIF of the DR estimand\footnote{
The normalization in Step \ref{i:if-procedure-2} requires an adjusted
influence function that accounts for the normalization.
}.
\item The estimator is the plug-in estimator of the DR estimand, with
inference based on the estimated influence function from step 4.
\end{enumerate}
As the above procedure shows, the EIF of the target
parameter is central to deriving the DR estimands. A key
feature is that the identified LATT parameters are ratios of ATT
components and that the staggered exposure setting reduces to
two-period comparisons. This allows us to start from the canonical
two-period DiD case targeting the ATT parameter and derive the DR
estimands and EIFs in this setting. As part of this derivation, we
recover the DR DiD estimands of \cite{sazhao} from first principles. We
then exploit the ratio structure of the LATT parameters together with
the EIF machinery of \cite{kennedy} developed for our setting in
\Cref{sec:influence-functions} to combine the components. Constructing
the resulting estimators requires additional work and is taken up in
\Cref{sec:estimation-and-inference}.
\paragraph{Weights definition}
As noted in Step~\ref{i:if-procedure-3} of the procedure, the doubly robust
estimands and their corresponding EIFs rely on Hájek-type normalized weights.
The construction of these weights in both sampling settings follows the same
principle as in \cite{sazhao,csa}: the control weights are normalized to sum to one in
sample (in contrast to the unnormalized control weights appearing in the EIF
derived in Step~\ref{i:if-procedure-2}).
The weights in the panel data setting are:
\begin{align}
\label{eq:weights1-panel-norm}
w^{trt,p}_{e}
& := \frac{E_{e}}{\rho},
\quad
\rho := E[E_{e}]
\\
\label{eq:weights-nev-panel-norm}
w^{nev,p}_{e}
& :=
C^{nev}\frac{p_{e}(X)}{1 - p_{e}(X)}
/
E\left[C^{nev}\frac{p_{e}(X)}{1 - p_{e}(X)}\right]
\\
\label{eq:weights-nye-panel-norm}
w^{nye,p}_{e, t}
& :=
C^{nye}_{e, t}\frac{p_{e, t}(X)}{1 - p_{e, t}(X)}
/
E\left[C^{nye}_{e, t}\frac{p_{e, t}(X)}{1 - p_{e, t}(X)}\right]
\end{align}
The weights in the repeated cross-sections setting are:
\begin{align}
\label{eq:weights-trt-rc-norm}
w^{trt,rc}_{e, t}
& := \frac{E_{e} \cdot \mathbf{1}\{T = t\}}{ \rho^{trt, rc}_{e, t} },
\quad
\rho^{trt, rc}_{e, t} := E[E_{e} \cdot \mathbf{1}\{T = t\}]
\\
\label{eq:weights-nev-rc-norm}
w^{nev,rc}_{e, t}
& :=
\frac{C^{nev} \cdot \mathbf{1}\{T = t\}p_{e}(X)}{1 - p_{e}(X)}
/
E\left[
\frac{C^{nev} \cdot \mathbf{1}\{T = t\}p_{e}(X)}{1 - p_{e}(X)}
\right]
\\
\label{eq:weights-nye-rc-norm}
w^{nye,rc}_{e, s, t}
& :=
\frac{C^{nye}_{e, s} \cdot \mathbf{1}\{T = t\}p_{e, s}(X)}{1 - p_{e, s}(X)}
/
E\left[
\frac{C^{nye}_{e, s} \cdot \mathbf{1}\{T = t\}p_{e, s}(X)}{1 - p_{e, s}(X)}
\right].
\end{align}
In the repeated cross-sections setting, we further define:
\begin{align}
\label{eq:weights-trt-joint-rc}
w^{trt,rc}_{e}
& :=
w^{trt,rc}_{e, t} - w^{trt,rc}_{e, e - 1},
\\
\label{eq:weights-control-joint-rc}
w^{c,rc}_{e}
& :=
w^{c,rc}_{e, t, t}
-
w^{c,rc}_{e, t, e-1}
\end{align}
for $w^{c,rc}_{e, t, s}$ a generic control-weight.
\paragraph{Estimand and EIF}
\begin{proposition}
\label{prop:if-latt-panel}
Fix $e \in \mathcal{E}$ and $t \ge e$.
Suppose the conditions of \Cref{prop:identification-panel} hold.
Let $C$ denote a generic control-group indicator, where either
$C = C^{nev}$ or $C = C^{nye}_{e,t}$, and let $p(X)$ denote the corresponding
propensity score.
Write $m_{e, t}^{c, p}(X)$ and $g_{e, t}^{c, p}(X)$
for the mean functions in \cref{eq:mean-functions-panel}
corresponding to the chosen control group.
For \cref{eq:identification-panel}, define
$\tau^{p}_{e, t} =: \tau^{p, num}_{e, t}/\tau^{p, den}_{e, t}$.
Then the efficient influence function for $\tau^{p}_{e, t}$ is
\begin{align}
\label{eq:if-latt-panel}
& \varphi^{p}(O; \tau^{p}_{e, t}, \eta^{p}_{e, t})
\\
\nonumber
& =
\frac{1}{\tau^{p, den}_{e, t}}
[
\{ w^{trt,p}_{e} - w^{c,p}_{e, t} \}
\{\Delta_{t-e+1}Y_{t} - m_{e, t}^{c, p}(X)\}
- \tau^{p}_{e, t}
\{ w^{trt,p}_{e} - w^{c,p}_{e, t} \}
\{\Delta_{t-e+1}D_{t} - g_{e, t}^{c, p}(X)\}
],
\end{align}
and the nonparametric efficiency bound
equals
$E[\{\varphi^{p}(O; \tau^{p}_{e, t}, \eta^{p}_{e, t})\}^{2}]$,
where $\eta^{p}_{e, t} = (\rho, p, m_{e, t}^{c, p}, g_{e, t}^{c, p})$
is the nuisance-function tuple.
\end{proposition}
\medskip\noindent
Proof in \Cref{sec:latt-parameter-if-panel}.
\paragraph{DR estimands and estimator decomposition}
In the following, we denote a generic estimand by $\tau$
and its plug-in estimator by $\hat{\tau}$.
A generic decomposition of this plug-in estimator into a CLT, empirical process and
remainder term,
similar to \cite{kennedy}\footnote{
We depart from the notation in \cite{kennedy} by indexing the EIF explicitly
by the target parameter and nuisance functions, $\varphi(\cdot; \tau, \eta)$,
rather than by the distribution $P$ alone.
},
can be written as:
\begin{align}
\label{eq:decomposition}
& \sqrt{n}(\hat{\tau} - \tau)
\\
\nonumber
& =
\sqrt{n} (P_{n} - P) \varphi(\cdot; \tau, \eta)
+
\sqrt{n} (P_{n} - P)
[
\varphi(\cdot; \tau, \hat{\eta})
-
\varphi(\cdot; \tau, \eta)
]
+ \sqrt{n} P \varphi(\cdot; \tau, \hat{\eta})
+
o_{P}(1)
\end{align}
where $\varphi(\cdot; \tau, \eta)$ is the influence function
and $\eta$ is a tuple of nuisance parameters.
We derive our doubly robust estimands by using the EIF, $\varphi(\cdot; \tau,
\eta)$, corresponding to $\tau$ and solving
\begin{align}
\label{eq:dr-estimand-foc}
P\varphi(\cdot; \tau^{dr}, \eta) = 0
\end{align}
for the new target parameter $\tau^{dr}$, where the superscript "dr" means
doubly robust.
This is also how the estimands of \cite{sazhao} are derived (although not
explicitly shown in their paper).
In \Cref{sec:influence-functions} we show this from first principles, hence
demystifying where the estimands come from, building
up to our DR estimands for the $LATT$ parameter in both data settings.
The resulting estimators are of the "estimating equation form", i.e., for
$\hat{\eta}$ a generic estimator of the nuisance, the estimator, $\hat{\tau}$,
solves:
\begin{align}
\label{eq:estimating-equation}
P_{n}\varphi(\cdot; \hat{\tau}, \hat{\eta}) = 0.
\end{align}
Note that the DR estimands, e.g. the $\tau^{dr}$ estimand found through
\cref{eq:dr-estimand-foc}, have their own influence functions,
which differ from those of the original parameter, e.g. $\tau$.
Denote this influence function for the DR estimand as $\varphi^{dr}(\cdot;
\tau^{dr}, \eta^{dr})$.
We use this influence function to conduct inference for the DR estimators.
This is also what \cite{sazhao} does.
However, \cite{sazhao} also study the estimation effects arising from the
nuisance function estimators of $\eta^{dr}$.
The estimation effects arise from their linearization of their estimators and
entails Taylor expanding the components building up to their influence
function.
In the IDiD setting, the estimand and estimators are ratios and hence the
Taylor expansion becomes significantly more tedious to derive.
Hence, in this paper, we do not pursue this approach.
Instead, we take the more general approach of either assuming Donsker
conditions or using cross-fitting to tame the empirical process term in
\cref{eq:decomposition} cf. \cite{kennedy}.
The remainder term is handled explicitly in the proofs; see
\Cref{prop:remainder-term-panel,prop:remainder-term-rc}.
A drawback of taking the more general approach is that our estimator does not
inherit the DR-for-inference property as in \cite{sazhao}; we leave this for
future work (see also \cite{dukesvansteelandt}).
\paragraph{DR Estimand and EIF}
\begin{proposition}
\label{prop:dr-estimand-panel}
Fix $e \in \mathcal{E}$ and $t \ge e$.
Suppose the conditions of \Cref{prop:identification-panel} hold.
Then the doubly robust panel-data estimand for $LATT(e,t)$ is
\begin{align}
\label{eq:dr-estimand-panel}
\tau^{dr, p}_{e, t}
& = \frac{
E[
\{ w^{trt,p}_{e} - w^{c,p}_{e, t} \}
\{\Delta_{t-e+1}Y_{t} - m_{e, t}^{c, p}(X)\}
]
}{
E[
\{ w^{trt,p}_{e} - w^{c,p}_{e, t} \}
\{\Delta_{t-e+1}D_{t} - g_{e, t}^{c, p}(X)\}
]
}.
\end{align}
\end{proposition}
\medskip\noindent
Proof in \Cref{sec:dr-estimand-panel}.
\begin{remark}
\leavevmode
The estimand in \cref{eq:dr-estimand-panel} resembles a ratio of two
$ATT_{dr}(g,t;0)$ estimands from \cite{csa}, with the treatment replaced by
$E_e$ and $D_t$ used as the outcome in the denominator.
It is also related to the
cross-sectional LATT estimand in
\cite{sloczynskiDoublyRobustEstimation2022}. As in that work, the choice
of nuisance models follows the outcome structure: for count outcomes, a
Poisson model for $m^{c,p}(X)$ is natural, while for the denominator, a
binomial mean model for $g^{c,p}(X)$ is appropriate.
The resulting plug-in estimator, discussed in
\Cref{sec:estimation-and-inference}, is related to the estimator
of LATE in \cite{tan2006} in the cross-sectional IV case.
\end{remark}
\begin{proposition}
\label{prop:dr-estimand-weighted-panel}
Fix $e \in \mathcal{E}$ and $t \ge e$.
Let $C$ denote a generic control-group indicator, where either
$C = C^{nev}$ or $C = C^{nye}_{e,t}$.
Write $m_{e, t}^{c, p}(X)$ and $g_{e, t}^{c, p}(X)$
for the corresponding control-group mean functions.
Define the residuals
\begin{align*}
\varepsilon^{Y, p}_{e, t} := \Delta_{t-e+1}Y_{t} - m_{e, t}^{c, p}(X),
\quad
\varepsilon^{D, p}_{e, t} := \Delta_{t-e+1}D_{t} - g_{e, t}^{c, p}(X),
\end{align*}
and
$\tau^{dr, p}_{e, t}
=: \tau^{dr, p, num}_{e, t}/\tau^{dr, p, den}_{e, t}$.
Let $w^{c,p}_{e, t}$ be the control-weight for the generic control variable.
Then the influence function of the doubly robust estimand
\cref{eq:dr-estimand-panel} is
\begin{align}
\label{eq:if-dr-estimand-panel}
\varphi^{dr, p}(O; \tau^{dr, p}_{e, t}, \eta^{dr, p}_{e, t})
& =
\frac{1}{\tau^{dr, p, den}_{e, t}}
\Big[
w^{trt,p}_{e}
\{
\varepsilon^{Y, p}_{e, t}
-
E[ w^{trt,p}_{e} \varepsilon^{Y, p}_{e, t} ]
\}
-
w^{c,p}_{e, t}
\{
\varepsilon^{Y, p}_{e, t}
-
E[ w^{c,p}_{e, t} \varepsilon^{Y, p}_{e, t} ]
\}
\\ & \quad
\nonumber
- \tau^{dr, p}_{e, t}
(w^{trt,p}_{e}
\{
\varepsilon^{D, p}_{e, t}
-
E[ w^{trt,p}_{e} \varepsilon^{D, p}_{e, t} ]
\}
-
w^{c,p}_{e, t}
\{
\varepsilon^{D, p}_{e, t}
-
E[ w^{c,p}_{e, t} \varepsilon^{D, p}_{e, t} ]
\})
\Big]
\end{align}
where $\eta^{dr, p}_{e, t}
=
(\rho, p, m_{e, t}^{c, p}, g_{e, t}^{c, p})$
is the tuple of nuisance functions.
\end{proposition}
\medskip\noindent
Proof in \Cref{sec:eif-dr-estimand-panel}.
\subsubsection{Repeated Cross-Sections}
\begin{proposition}[Identification in repeated cross-sections]
\label{prop:identification-rc}
Fix $e \in \mathcal{E}$ and $t \ge e$.
Suppose
\Cref{ass:sample:mixture,ass:staggered-instrument,ass:no-carryover,ass:monotonicity,ass:no-anticipation,ass:overlap}
hold.
Let $C$ denote a generic control-group indicator, where either $C = C^{nev}
\quad \text{or} \quad
C = C^{nye}_{e,t}$, and let $p(X)$ denote the corresponding
propensity score.
Write $m_{e, s, t}^{c, rc}(X)$ and $g_{e, s, t}^{c, rc}(X)$
for the mean functions in \cref{eq:mean-functions-rc} corresponding to the
chosen control group (in the never-exposed case, the middle subscript $s$ is
redundant).
If $C = C^{nev}$, additionally assume
\Cref{ass:cpt-treat-ne,ass:cpt-unexpout-ne}.
If $C = C^{nye}_{e,t}$, additionally assume
\Cref{ass:cpt-treat-nye,ass:cpt-unexpout-nye}.
Then the cohort-time local average treatment effect on the treated is identified by
\begin{align}
\label{eq:identification-rc}
LATT(e,t)
=
\frac{
E[
m_{e,t}^{trt,rc}(X)
- m_{e,e-1}^{trt,rc}(X)
-
[
m_{e, t, t}^{c,rc}(X)
- m_{e, t, e-1}^{c,rc}(X)
]
\mid E_e = 1 ]
}{
E\left[
g_{e,t}^{trt,rc}(X)
- g_{e,e-1}^{trt,rc}(X)
-
[
g_{e, t, t}^{c,rc}(X)
- g_{e, t, e-1}^{c,rc}(X)
]
\mid E_e = 1
\right]
} =: \tau^{rc}_{e, t}.
\end{align}
\end{proposition}
\medskip\noindent
Proof in \Cref{sec:proofs-rc}.
\paragraph{Estimand and EIF}
In the repeated cross-sections case, we denote generic mean functions
for both control groups in \cref{eq:mean-functions-rc} by
$m^{c,rc}_{e,s,T}(X)$ and $g^{c,rc}_{e,s,T}(X)$ (again, in the never-exposed
case, the index $s$ is not used). This notation allows us to treat both
cases jointly.
Using this notation, we define the control mean functions unified across
periods $e-1$ and $t$ as:
\begin{align}
\label{eq:mean-function-m-period-rc-control}
m_{e, Y}^{c,rc}(X)
& :=
\mathbf{1}\{T = e-1\} \cdot m_{e, t, e-1}^{c, rc}(X)
+ \mathbf{1}\{T = t\} \cdot m_{e, t, t}^{c, rc}(X),
\\
\label{eq:mean-function-g-period-rc-control}
g_{e, D}^{c,rc}(X)
& :=
\mathbf{1}\{T = e-1\} \cdot g_{e, t, e-1}^{c, rc}(X)
+ \mathbf{1}\{T = t\} \cdot g_{e, t, t}^{c, rc}(X),
\end{align}
and
\begin{align}
\label{eq:mean-function-both-period-rc}
m_{e,Y}^{trt,rc}(X)
& :=
\mathbf{1}\{T = e-1\} \cdot m_{e, e - 1}^{trt, rc}(X)
+ \mathbf{1}\{T = t\} \cdot m_{e, t}^{trt, rc}(X),
\\
g_{e,D}^{trt,rc}(X)
& :=
\mathbf{1}\{T = e-1\} \cdot g_{e, e - 1}^{trt, rc}(X)
+ \mathbf{1}\{T = t\} \cdot g_{e, t}^{trt, rc}(X),
\end{align}
where implicitly, because we only consider $2 \times 2$ comparisons, $\mathbf{1}\{T = e
- 1\} + \mathbf{1}\{T = t\} = 1$.
\begin{proposition}
\label{prop:if-latt-rc}
Let $C$ be either of the control variables and $p(X)$ the corresponding
propensity.
Suppose the conditions of \Cref{prop:identification-rc} hold. For
\cref{eq:identification-rc},
define $\tau^{rc}_{e, t} =: \tau^{rc, num}_{e, t}/\tau^{rc, den}_{e, t}$.
Then the efficient influence function
for the LATT when repeated cross-sections are available
equals:
\begin{align}
\label{eq:if-latt-rc}
\varphi^{rc}(O; \tau^{rc}_{e, t}, \eta^{rc}_{e, t})
& =
\frac{1}{\tau^{rc, den}_{e, t}}
[
w^{trt,rc}_{e} \{Y - m_{e, Y}^{trt, rc}(X)\}
-
w^{c,rc}_{e}
\{Y - m_{e, Y}^{c,rc}(X)\}
\\
\nonumber
\nonumber & \quad
+ \frac{E_{e}}{\rho}
\{
m_{e,t}^{trt, rc}(X) - m_{e,e-1}^{trt, rc}(X)
-
[
m_{e, t, t}^{c, rc}(X) - m_{e, t, e-1}^{c, rc}(X)
]
\}
\\
\nonumber & \quad
\nonumber
- \tau^{rc}_{e, t}
(
w^{trt,rc}_{e} \{D - g_{e, Y}^{trt, rc}(X)\}
-
w^{c,rc}_{e}
\{D - g_{e, D}^{c,rc}(X)\}
\\
\nonumber
\nonumber & \quad
+ \frac{E_{e}}{\rho}
\{
g_{e,t}^{trt, rc}(X) - g_{e,e-1}^{trt, rc}(X)
-
[ g_{e, t, t}^{c, rc}(X) - g_{e, t, e-1}^{c, rc}(X) ]
\}
)
]
\end{align}
where the nonparametric efficiency bound equals
$E[\{\varphi^{rc}(O; \tau^{rc}_{e, t}, \eta^{rc}_{e, t})\}^{2}]$,
and where $\eta^{rc}_{e, t} = (
\rho,
\lambda_{e - 1},
\lambda_{t},
p,
m_{e, t}^{trt, rc},
m_{e, e - 1}^{trt, rc},
g_{e, t}^{trt, rc},
g_{e, e - 1}^{trt, rc}
m_{e, t, t}^{c, rc},
m_{e, t, e-1}^{c, rc},
g_{e, t, t}^{c, rc},
g_{e, t, e-1}^{c, rc}
)$
is the tuple of nuisance functions.
\end{proposition}
\medskip\noindent
Proof in \Cref{sec:latt-parameter-if-rc}.
\paragraph{DR Estimand and EIF}
\begin{proposition}
\label{prop:dr-estimand-rc}
Fix $e \in \mathcal{E}$ and $t \ge e$.
Suppose the conditions of \Cref{prop:identification-rc} hold.
Then the doubly robust repeated-cross-section estimand for $LATT(e,t)$ is
\begin{align}
\label{eq:dr-estimand-rc}
\tau^{dr, rc}_{e, t}
& =
\frac{
E [
\{w^{trt,rc}_{e} - w^{c,rc}_{e}\}
\{Y - m_{e, Y}^{c,rc}(X)\}
] + \kappa_{e, t}^{Y, rc}
}{
E [
\{w^{trt,rc}_{e} - w^{c,rc}_{e}\}
\{D - g_{e, Y}^{c,rc}(X)\}
] + \kappa_{e, t}^{D, rc}
},
\end{align}
where
\begin{align}
\label{eq:kappa-Y-rc}
& \kappa_{e, t}^{Y, rc}
\\
\nonumber
& :=
E [ m_{e, t}^{trt, rc}(X)
- m_{e, t, t}^{c, rc}(X) \mid E_{e} = 1 ]
-
E[ m_{e, t}^{trt, rc}(X)
- m_{e, t, t}^{c, rc}(X) \mid E_{e} = 1, T = t ]
\\ & \quad
\nonumber
-
\left(
E [ m_{e, e-1}^{trt, rc}(X)
- m_{e, t, e-1}^{c, rc}(X) \mid E_{e} = 1 ]
-
E[m_{e, e-1}^{trt, rc}(X)
- m_{e, t, e-1}^{c, rc}(X) \mid E_{e} = 1, T = e - 1]
\right),
\\
\label{eq:kappa-D-rc}
& \kappa_{e, t}^{D, rc}
\\
\nonumber
& :=
E [ g_{e, t}^{trt, rc}(X)
- g_{e, t, t}^{c, rc}(X) \mid E_{e} = 1 ]
-
E[ g_{e, t}^{trt, rc}(X)
- g_{e, t, t}^{c, rc}(X) \mid E_{e} = 1, T = t ]
\\ & \quad
\nonumber
-
\left(
E[ g_{e, e-1}^{trt, rc}(X)
- g_{e, t, e-1}^{c, rc}(X) \mid E_{e} = 1 ]
-
E[g_{e, e-1}^{trt, rc}(X)
- g_{e, t, e-1}^{c, rc}(X) \mid E_{e} = 1, T = e - 1]
\right).
\end{align}
\end{proposition}
\medskip\noindent
Proof in \Cref{sec:dr-estimand-rc}.
\begin{remark}
\leavevmode
The estimand in \cref{eq:dr-estimand-rc} again looks familiar; it
resembles a ratio of two $ATT_{dr,rc}(g,t;0)$ estimands from
\cite{csa}, with the treatment
replaced by the exposure variable $E_e$ and $D$ used as the outcome in
the denominator.
Also, \cref{eq:dr-estimand-rc} has a similar form as
\cref{eq:dr-estimand-panel},
a weighted residual including the control units mean function,
but in the repeated cross-sections setting also including the terms
$\kappa_{e, t}^{Y, rc},\kappa_{e, t}^{D, rc}$,
consisting of differences between the treated and control mean functions in
both periods.
\end{remark}
\begin{proposition}
\label{prop:dr-estimand-weighted-rc}
For the doubly robust estimand \cref{eq:dr-estimand-rc}, define
\begin{align}
\label{eq:resid-rc-case-individual}
\varepsilon^{Y,trt,rc}_{e,t} & := Y - m_{e, t}^{trt, rc}(X),
\quad
\varepsilon^{Y,c,rc}_{e,s,t} := Y - m_{e, s, t}^{c,rc}(X),
\\
\nonumber
\varepsilon^{D,trt,rc}_{e,t} & := D - g_{e, t}^{trt, rc}(X),
\quad
\varepsilon^{D,c,rc}_{e,s,t} := D - g_{e, s, t}^{c,rc}(X),
\end{align}
$\tau^{dr, rc}_{e, t}
=: \tau^{dr, rc, num}_{e, t}/\tau^{dr, rc, den}_{e, t}$,
and:
\begin{align}
\label{eq:mean-functions-rc-diffs}
m_{e, \Delta}^{trt, rc}(X)
:=
m_{e, t}^{trt, rc}(X) - m_{e, e - 1}^{trt, rc}(X),
\quad
m_{e, t, \Delta}^{c, rc}(X)
:= m_{e, t, t}^{c, rc}(X) - m_{e, t, e - 1}^{c, rc}(X).
\end{align}
Then the efficient influence function of \cref{eq:dr-estimand-rc} is
\begin{align}
\label{eq:if-dr-estimand-rc}
& \varphi^{dr,rc}(O; \tau^{dr, rc}_{e, t}; \eta^{dr, rc}_{e, t})
\\
\nonumber
& =
\frac{1}{\tau^{dr, rc, den}_{e, t}}
\Big[
\varphi^{dr,rc}_{num}(O; \tau^{dr, rc, num}_{e, t}; \eta^{dr, rc, num}_{e, t})
- \tau^{dr, rc}_{e, t}
\varphi^{dr,rc}_{den}(O; \tau^{dr, rc, den}_{e, t}; \eta^{dr, rc, den}_{e, t})
\Big],
\end{align}
where
\begin{align}
\label{eq:if-dr-num-estimand-rc}
& \varphi^{dr,rc}_{num}(O; \tau^{dr, rc, num}_{e, t}; \eta^{dr, rc, num}_{e, t})
\\
\nonumber
& :=
w^{trt,rc}_{e, t}
\left\{ \varepsilon^{Y,trt,rc}_{e,t}
- E[w^{trt,rc}_{e, t} \varepsilon^{Y,trt,rc}_{e,t} ] \right\}
+ w^{trt,p}_{e} \left\{ m_{e, \Delta}^{trt, rc}(X)
- E[w^{trt,p}_{e} m_{e, \Delta}^{trt, rc}(X) ] \right\}
\\ & \quad
\nonumber
-
\left[w^{trt,rc}_{e, e-1}
\left\{ \varepsilon^{Y,trt,rc}_{e,e-1}
- E[w^{trt,rc}_{e, e-1} \varepsilon^{Y,trt,rc}_{e,e-1} ] \right\}
+ w^{trt,p}_{e} \left\{ m_{e, t, \Delta}^{c, rc}(X)
- E[w^{trt,p}_{e} m_{e, t, \Delta}^{c, rc}(X) ] \right\}\right]
\\
\nonumber
& \quad
- [w^{c,rc}_{e, t, t}
\left\{ \varepsilon^{Y,c,rc}_{e,t,t}
- E[w^{c,rc}_{e, t, t} \varepsilon^{Y,c,rc}_{e,t,t} ] \right\}
- w^{c,rc}_{e, t, e - 1}
\left\{ \varepsilon^{Y,c,rc}_{e,t,e-1}
- E[w^{c,rc}_{e, t, e - 1} \varepsilon^{Y,c,rc}_{e,t,e-1} ] \right\}]
\\
\label{eq:if-dr-den-estimand-rc}
&
\varphi^{dr,rc}_{den}(O; \tau^{dr, rc, den}_{e, t}; \eta^{dr, rc, den}_{e, t})
\\
\nonumber
& :=
w^{trt,rc}_{e, t}
\left\{ \varepsilon^{D,trt,rc}_{e,t}
- E[w^{trt,rc}_{e, t} \varepsilon^{D,trt,rc}_{e,t} ] \right\}
+ w^{trt,p}_{e} \left\{ g_{e, \Delta}^{trt, rc}(X)
- E[w^{trt,p}_{e} g_{e, \Delta}^{trt, rc}(X) ] \right\}
\\ & \quad
\nonumber
-
\left[w^{trt,rc}_{e, e-1}
\left\{ \varepsilon^{D,trt,rc}_{e,e-1}
- E[w^{trt,rc}_{e, e-1} \varepsilon^{D,trt,rc}_{e,e-1} ] \right\}
+ w^{trt,p}_{e} \left\{ g_{e, t, \Delta}^{c, rc}(X)
- E[w^{trt,p}_{e} g_{e, t, \Delta}^{c, rc}(X) ] \right\}\right]
\\
\nonumber
& \quad
- [w^{c,rc}_{e, t, t}
\left\{ \varepsilon^{D,c,rc}_{e,t,t}
- E[w^{c,rc}_{e, t, t} \varepsilon^{D,c,rc}_{e,t,t} ] \right\}
- w^{c,rc}_{e, t, e - 1}
\left\{ \varepsilon^{D,c,rc}_{e,t,e-1}
- E[w^{c,rc}_{e, t, e - 1} \varepsilon^{D,c,rc}_{e,t,e-1} ] \right\}],
\end{align}
and
$\eta^{dr, rc}_{e, t} = (
\rho,
\lambda_{e - 1},
\lambda_{t},
p,
m_{e, t}^{trt, rc},
m_{e, e - 1}^{trt, rc},
g_{e, t}^{trt, rc},
g_{e, e - 1}^{trt, rc},
m_{e, t, t}^{c, rc},
m_{e, t, e-1}^{c, rc},
g_{e, t, t}^{c, rc},
g_{e, t, e-1}^{c, rc}
)$
is the nuisance-function tuple.
\end{proposition}
\medskip\noindent
Proof in \Cref{sec:eif-dr-estimand-rc}.
\subsubsection{Double Robustness}
\begin{proposition}
\label{prop:dr-both}
Fix $e \in \mathcal{E}$ and $t \ge e$, and let $C$ denote a generic
control-group indicator, where either $C = C^{nev}$ or
$C = C^{nye}_{e,t}$ and $p(X)$ the corresponding propensity.
Let $\pi(X)$ be a working model for the propensity $p(X)$.
In the panel data setting, let
$\mu^{c,p}_{e,t}(X)$ and $\eta^{c,p}_{e,t}(X)$ be working models of
$m^{c,p}_{e,t}(X)$ and $g^{c,p}_{e,t}(X)$, respectively.
Similarly, in the repeated cross-sections setting, let $\mu^{c,rc}_{e,Y}(X)$
and $\eta^{c,rc}_{e,Y}(X)$ be the working models
of $m_{e, Y}^{c,rc}(X)$ and $g_{e, D}^{c,rc}(X)$, respectively.
Assume the conditions of
\Cref{prop:identification-panel,prop:identification-rc} hold.
Then the following statements hold:
\begin{enumerate}[label=(\alph*), nosep]
\item In the panel-data setting,
\[
\tau^{dr,p}_{e,t} = LATT(e,t)
\]
if either (but not necessarily both) $\pi(X) = p(X)$ almost surely, or
\[
\mu^{c,p}_{e,t}(X) = m^{c,p}_{e,t}(X)
\quad \text{and} \quad
\eta^{c,p}_{e,t}(X) = g^{c,p}_{e,t}(X)
\]
almost surely.
\item In the repeated cross-section setting,
\[
\tau^{dr,rc}_{e,t} = LATT(e,t)
\]
if either (but not necessarily both) $\pi(X) = p(X)$ almost surely, or
\[
\mu_{e,Y}^{c,rc}(X) = m_{e,Y}^{c,rc}(X)
\quad \text{and} \quad
\eta_{e,Y}^{c,rc}(X) = g_{e,D}^{c,rc}(X)
\]
almost surely.
\end{enumerate}
\end{proposition}
\medskip\noindent
Proof in \Cref{sec:proof-other}.
\subsubsection{Bloom-type result}
In the special case of absorbing treatment (defined below) and
one-sided compliance (i.e., no units unexposed to the instrument are
treated), the $LATT(e,t)$ parameter can be related to the
instrument-exposure-cohort-specific $ATT(g,t)$ parameters of
\cite{csa}.
The following result shows this:
\begin{proposition}[Bloom-type result]
\label{prop:bloom}
Assume that panel data is available.
Let $C$ denote a generic control-group indicator, where either $C = C^{nev}$
or $C = C^{nye}_{e,t}$.
Assume that treatment is an absorbing state, i.e. $D_{1} = 0$
and
\begin{align}
\label{eq:treatment-absorbed}
D_{t-1} = 1 \implies D_{t} = 1,
\quad
t \in \{2, 3, \ldots, \mathcal{T}\},
\end{align}
and one-sided compliance:
\begin{align}
\label{eq:assumption-bloom}
C = 1 \implies D_{t} = 0.
\end{align}
Then:
\begin{align}
\label{eq:bloom}
LATT(e, t)
=
\sum_{g \leq t} ATT(g, t, e)P(G_{g} = 1 \mid D_{t} = 1, E_{e} = 1)
\end{align}
where
\begin{equation}
\label{eq:csa-objects}
ATT(g, t, e) := E[Y_t(g)-Y_t(0)\mid G_g=1, E_{e} = 1],
\end{equation}
is an exposure-cohort-specific $ATT(g, t)$-estimand of \cite{csa} and
$G := \min \{t \mid D_{t} = 1\}; \, G_{g} := \mathbf{1}\{G = g\}$
the corresponding treatment cohort dummies.
This result shows that the local average treatment effect on the treated in
period $t$ for exposure cohort $e$, $LATT(e, t)$, under absorbing treatment
and one-sided compliance, identifies an exposure-cohort-specific weighted
average of $ATT(g, t)$'s for all those treated by time $t$.
\end{proposition}
\medskip\noindent
Proof in \Cref{sec:proof-other}.
\begin{remark}
\leavevmode
\begin{enumerate}[nosep]
\item \Cref{prop:bloom}
is a Bloom-type result
\citep{bloom1984estimating,ap}
for the case of IDiD.
Moreover, \cref{eq:bloom} is a ``sensible causal estimand''
\citep{mogstad2024instrumental} in the sense that it is a convex
combination of
the causal $ATT(g, t, e)$ estimands with weights $P(G_{g} = 1 \mid
D_{t} = 1, E_{e} = 1)$, $g \leq t$.
\item \Cref{prop:bloom} is useful in randomized experiments with
one-sided \textit{non}-compliance, where treatment received $D_t$ may differ
from treatment assignment $Z_t$.
In this case, the random assignment $Z_t$ can be used as an instrument
for the received treatment $D_t$, allowing researchers to approximate the
treatment effect in the presence of hidden confounding, extending the
cross-sectional Bloom result, as described in \cite{ap}, to the IDiD
framework.
In the two-period case ($\mathcal{T}=2$), there is only a single exposed and
treatment cohort. By \cref{eq:bloom}, $LATT(2,2)=ATT(2,2,2)$, which
coincides with the DiD estimand $ATT(2,2)$, implying that IDiD exactly
recovers the average treatment effect on the treated.
\item The result is also related to the point that \textit{"IDiD is a natural
generalization of DiD, and the differences between the two designs
arise because IDiD allows the non-compliance of the assigned
treatment"} as stated in \cite{miyaji2025}.
Here, because of the mechanical one-sided compliance,
we can write the $LATT(e, t)$ as a convex average of
exposure-cohort-specific staggered DiD estimands, making the link
between IDiD with staggered exposure and DiD with staggered treatment
explicit.
\item An analogous result obtains under the one-sided compliance condition,
$E_{e} = 1 \implies D_{t} = 1$. In this case, $LATT(e,t)$ admits a
representation as a convex average of treatment effects for the
non-treated (cf. \cite{ap}); this result is omitted for brevity.
\item The result extends to repeated cross-sections with analogous
arguments; details are again omitted for brevity.
\end{enumerate}
\end{remark}
\subsubsection{Group-level difference estimand}
Researchers often compare the target parameter across subgroups. This
can be handled explicitly by expressing subgroup parameters as functionals of the
full population distribution. Differences are then represented by
embedding the corresponding influence functions in a common
full-sample framework, weighted by the group indicator and its
probability.
To see this, let $B \in \{m, f\}$ denote a group indicator.
We consider the difference in subgroup-specific LATTs,
$LATT^{\Delta}(e,t) := LATT^{m}(e,t) - LATT^{f}(e,t)$,
for example based on the DR estimand in the panel data setting:
\begin{align}
\label{eq:dr-estimand-group}
\tau^{dr, p, \Delta}_{e, t}
:= \tau^{dr, p, m}_{e, t} - \tau^{dr, p, f}_{e, t},
\end{align}
where each component admits an influence function under the corresponding
subgroup distribution,
$\varphi^{dr, p, m}(O; \tau^{dr, p, m}_{e, t}, \eta^{m})$
and
$\varphi^{dr, p, f}(O; \tau^{dr, p, f}_{e, t}, \eta^{f})$,
respectively.
These subgroup influence functions can be embedded into the full population as:
\begin{align}
\label{eq:dr-if-group}
& \varphi^{dr, p}(O; \tau^{dr, p, \Delta}_{e, t}, \eta^{dr, p, \Delta}_{e, t})
\\
\nonumber
& = \frac{\mathbf{1}\{B = m\}}{P(B = m)}
\varphi^{dr, p, m}(O; \tau^{dr, p, m}_{e, t}, \eta^{dr, p, m}_{e, t})
- \frac{\mathbf{1}\{B = f\}}{P(B = f)}
\varphi^{dr, p, f}(O; \tau^{dr, p, f}_{e, t}, \eta^{dr, p, f}_{e, t})
\end{align}
where $\eta^{dr, p, \Delta}_{e, t} = (\eta^{dr, p, m}_{e, t}, \eta^{dr, p,
f}_{e, t})$ collects the subgroup-specific nuisance functions.
Thus, inference on \cref{eq:dr-estimand-group} can be based on
\cref{eq:dr-if-group}.
The group-level difference estimand
in the case of repeated cross-sections is constructed in the exact same way and
is omitted for brevity.
\subsection{Aggregated effects}
\label{sec:weighting}
As shown in \cite{miyaji}, the $LATT(e,t)$ parameters can be aggregated into
estimands of interest, in a manner entirely analogous to the
aggregation of $ATT(g,t)$ parameters in \cite{csa}.
In this section, we apply the EIF machinery developed in
\Cref{sec:influence-functions} to derive the influence function for a
general weighted estimand $\theta$, defined below.
Consider an estimand that aggregates the $LATT(e,t)$ parameters:
\begin{align}
\label{eq:thetaiv}
\theta
:=
\sum_{e \in \mathcal{E}} \sum_{t = 2}^{\mathcal{T}}
w(e,t)\, LATT(e,t),
\end{align}
where $w(e,t)$ denotes a (possibly data-dependent) weighting scheme.
To ease notation, define $\tau_{e, t} := LATT(e, t)$.
Let $\mathcal{I} :=
\{(e, t) \in \mathcal{E} \times \{2, 3, \ldots, \mathcal{T}\} \mid
A_{e, t} \}$
be a set of indices that picks out the correct cohorts and time indices
related to the summary parameter $\theta$.
By the product rule \cref{eq:if-product-rule},
the EIF of the weighted estimand \cref{eq:thetaiv} equals:
\begin{align}
\label{eq:if-agg-estimand}
\varphi^{\theta}(O; \{\tau_{i}\}_{i \in \mathcal{I}};
\{\eta_{i}\}_{i \in \mathcal{I}})
:=
\sum_{(e, t) \in \mathcal{I}}
\left[
\mathbb{IF}(w(e, t))\tau_{e, t}+ w(e, t)\mathbb{IF}(\tau_{e, t})
\right].
\end{align}
\paragraph{Weighted estimands and weights}
Define the cohort-specific average exposed effect on the treated in the first
stage, i.e., the share of the compliers in cohort $e$ in period $t$:
\begin{align}
\label{eq:aet}
AET(e, t)
:= P(D_{t}(e) > D_{t}(\infty) \mid E = 1).
\end{align}
\Cref{tab:weighted-estimands} shows the weighted estimands and their
corresponding weights. For comparison, we also include the same aggregation
procedure as in \cite{csa}, but without the cohort-specific time-varying
complier weights. This allows researchers to choose whether to weight by
complier shares or not, although the complier-weighted version is the most
natural when the target parameter is the LATT.
The weights are themselves estimands, so the tools developed in
\Cref{sec:eif-derivation} can be applied to derive influence functions
for each weight in \Cref{tab:weighted-estimands}. With these in hand, we
apply \cref{eq:if-agg-estimand} to obtain the influence function of the
weighted estimand.
As is evident from \Cref{tab:weighted-estimands}, most of the weights
consist of sums of cohort probabilities conditional on different events, and the
time-varying complier share $AET(e, t)$.
For instance, for the weighted estimand $\theta^{o,IV}_{W}$, the main component
of the weight has influence function
(using the quotient rule \cref{eq:if-quotient-rule-estimand})
\begin{align}
\label{eq:if-component-event}
& \mathbb{IF}(P(E = e \mid E \leq \mathcal{T}))
\\
\nonumber
& =
\frac{1}{P(E \leq \mathcal{T})}
\left[
\mathbb{IF}\left(P(E = e)\right)
- P(E = e \mid E \leq \mathcal{T})
\mathbb{IF}\left(P(E \leq \mathcal{T})\right)
\right]
\end{align}
with
$\mathbb{IF}\left(P(E \leq \mathcal{T})\right) = \sum_{e \leq \mathcal{T}}
\mathbb{IF}\left(P(E = e)\right)$
and $\mathbb{IF}\left(P(E = e)\right) = \mathbf{1}\{E = e\} - P(E = e)$
cf. \cref{eq:if-block-prob}.
The influence function of the weight can now be derived using
\cref{eq:if-component-event} and the quotient-rule
\cref{eq:if-quotient-rule-estimand} summing the correct terms.
\medskip
\begin{threeparttable}
\centering
\small
\begin{tabular}{l l}
\hline
Weighted Estimand & $w(e,t)$ \\
\hline
\multicolumn{2}{l}{\textbf{IDiD}} \\
$\theta^{IV}_{es}(l)$
& $\mathbf{1}\{e+l \le \mathcal{T}\}\mathbf{1}\{t=e+l\}
P(E=e \mid E+l \le \mathcal{T})
\frac{AET(e,t)}{
\sum_{e \in \mathcal{E}} P(E=e \mid E+l \le \mathcal{T})AET(e,t)
}$ \\
$\theta^{IV}_{bal,es}(l,l')$
& $\mathbf{1}\{e+l' \le \mathcal{T}\}\mathbf{1}\{t=e+l\}
P(E=e \mid E+l' \le \mathcal{T})
\frac{AET(e,t)}{
\sum_{e \in \mathcal{E}} P(E=e \mid E+l' \le \mathcal{T})AET(e,t)
}$ \\
$\theta^{IV}_{sel}(\tilde{e})$
& $\mathbf{1}\{t \ge e\}\mathbf{1}\{e=\tilde{e}\}
\frac{AET(e,t)}{\sum_{t=\tilde{e}}^{\mathcal{T}} AET(e,t)}$ \\
$\theta^{IV}_{c}(\tilde{t})$
& $\mathbf{1}\{t \ge e\}\mathbf{1}\{t=\tilde{t}\}
P(E=e \mid E \le t)
\frac{AET(e,t)}{
\sum_{e \in \mathcal{E}} P(E=e \mid E \le t)AET(e,t)
}$ \\
$\theta^{cumm,IV}_{c}(\tilde{t})$
& $\mathbf{1}\{t \ge e\}\mathbf{1}\{t \le \tilde{t}\}
P(E=e \mid E \le t)
\frac{AET(e,t)}{
\sum_{e \in \mathcal{E}} P(E=e \mid E \le t)AET(e,t)
}$ \\
$\theta^{o,IV}_{W}$
& $\mathbf{1}\{t \ge e\} P(E=e \mid E \le \mathcal{T})
\Big/
\sum_{e \in \mathcal{E}} \sum_{t=2}^{\mathcal{T}}
\mathbf{1}\{t \ge e\} P(E=e \mid E \le \mathcal{T})$ \\
$\theta^{o,IV}_{sel}$
& $\mathbf{1}\{t \ge e\} P(E=e \mid E \le \mathcal{T})
\frac{AET(e,t)}{\sum_{t=e}^{\mathcal{T}} AET(e,t)}$ \\
\hline
\multicolumn{2}{l}{\textbf{CSA}} \\
$\theta_{es}(l)$
& $\mathbf{1}\{e+l \le \mathcal{T}\}\mathbf{1}\{t=e+l\}
P(E = e \mid E+l \le \mathcal{T})$ \\
$\theta_{bal,es}(l,l')$
& $\mathbf{1}\{e+l' \le \mathcal{T}\}\mathbf{1}\{t=e+l\}
P(E = e \mid E+l' \le \mathcal{T})$ \\
$\theta_{sel}(\tilde{e})$
& $\mathbf{1}\{t \ge e\}
\mathbf{1}\{e=\tilde{e}\}/(\mathcal{T}-e+1)$ \\
$\theta_{c}(\tilde{t})$
& $\mathbf{1}\{t \ge e\}\mathbf{1}\{t=\tilde{t}\}
P(E = e \mid E \le t)$ \\
$\theta_{c}^{cumm}(\tilde{t})$
& $\mathbf{1}\{t \ge e\}\mathbf{1}\{t \le \tilde{t}\}
P(E = e \mid E \le t)$ \\
$\theta^{o}_{sel}$
& $\mathbf{1}\{t \ge e\} P(E = e \mid E \le \mathcal{T})
/(\mathcal{T}-e+1)$ \\
\hline
\end{tabular}
\caption{Weighted estimands and their weights}
\begin{tablenotes}
\footnotesize
\item The table shows different versions of \cref{eq:thetaiv}
and the corresponding weights $w(e, t)$.
The upper IDiD part of the table is a modification of the table in
\cite{miyaji}.
The lower part is a modification of the one in \cite{csa} tailored to the
IDiD setting.
Note that the
$\theta^{IV}_{es}(l)$, $\theta^{bal,IV}_{es}(l,l')$,
$\theta^{IV}_{c}(\tilde{t})$,
have the group probability multiplied in the denominator compared to \cite{miyaji}
in order for the weights to sum to $1$ (and likewise for
$\theta^{cumm,IV}_{c}(\tilde{t})$ for it to sum to $\mathcal{T} - 1$).
The $\theta^{o,IV}_{W}$ parameter equals $\theta^{o}_{W}$, so the
latter is omitted. The indicators for cohort $e$ and time $t$ are
also implicitly included in the denominator notation.
\end{tablenotes}
\label{tab:weighted-estimands}
\end{threeparttable}
\paragraph{Event study at horizon $l$ parameter, $\theta_{es}^{IV}(l)$}
Here, we focus on the event study at horizon $l$ target parameter, which
aggregates the $LATT(e, t)$ parameter at time $t = e + l$, i.e., $l$ periods
after exposure to the instrument.
The target parameter is:
\begin{align}
\label{eq:esl-estimand}
\theta_{es}^{IV}(l)
& :=
\sum_{e \in \mathcal{E}} \sum_{t = 2}^{\mathcal{T}}
w^{IV}_{es(l)}(e, t) \tau_{e, t}
\end{align}
where the weight $w^{IV}_{es(l)}(e, t)$ is defined in the upper part of
\Cref{tab:weighted-estimands}.
\paragraph{Influence function of $\theta_{es}^{IV}(l)$}
To derive the influence function of the $\theta_{es}^{IV}(l)$ estimand,
we apply the product rule \cref{eq:if-product-rule} and derive the influence
function of each component separately.
First, we find the influence function of the weight $w^{IV}_{es(l)}(e, t)$.
We apply the product rule \cref{eq:if-product-rule} to get the
EIF of a generic term in the numerator and denominator, and then use the
quotient rule to get the EIF of the weight.
Towards this end, define
\begin{align*}
\varphi^{w^{IV}_{es(l)}, num}_{e, t}
& :=
\mathbb{IF}(
P(E = e \mid E + l \leq \mathcal{T}) AET(e, t)
)
\\
& =
\mathbb{IF}(P(E = e \mid E + l \leq \mathcal{T})) AET(e, t)
+ P(E = e \mid E + l \leq \mathcal{T})\mathbb{IF}(AET(e, t)).
\end{align*}
For the conditional probability we can use the block \cref{eq:if-block-cprob},
and the EIF for $AET(e, t)$ is the IF of the denominator for our (estimable) target
parameter at hand.
Hence, applying the quotient rule gives:
\begin{align}
\label{eq:if-weight-esl}
& \varphi^{w^{IV}_{es(l)}}_{e, t}(O; w^{IV}_{es(l)}(e, t), \eta)
\\
\nonumber
& :=
\frac{\mathbf{1}\{e + l \leq \mathcal{T}, t = e + l\}}{
\sum_{e \in \mathcal{E}}
P(E = e \mid E + l \leq \mathcal{T})AET(e, t)
}
\Big[
\varphi^{w^{IV}_{es(l)}, num}_{e, t}
- w^{IV}_{es(l)}(e, t)
\sum_{e \in \mathcal{E}}
\varphi^{w^{IV}_{es(l)}, num}_{e, t}
\Big].
\end{align}
Let $\mathcal{I}_{l} :=
\{(e, t) \in \mathcal{E} \times \{2, 3, \ldots, \mathcal{T}\} \mid
e + l \leq \mathcal{T} \wedge t = e + l\}$.
Thus, using \cref{eq:if-weight-esl},
and writing $\varphi_{e, t}(O; \tau_{e, t}, \eta)$
for the influence function of the generic $\tau_{e, t}$,
we get using \cref{eq:if-agg-estimand}:
\begin{align}
\label{eq:if-esl}
& \varphi^{\theta_{es}(l)}(O; \{\tau_{i}\}_{i \in \mathcal{I}_{l}};
\{\eta_{i}\}_{i \in \mathcal{I}})
\\
& :=
\nonumber
\sum_{(e, t) \in \mathcal{I}_{l}}
\left[
\varphi^{w^{IV}_{es(l)}}_{e, t}(O; w^{IV}_{es(l)}(e, t), \eta)\tau_{e, t}
+ w^{IV}_{es(l)}(e, t)\varphi_{e, t}(O; \tau_{e, t}, \eta)
\right].
\end{align}
\Cref{eq:if-esl} is the influence function we use to conduct inference with on
the horizon $l$ parameters, $\theta^{IV}_{es(l)}$.
Note that we chose $\theta^{IV}_{es(l)}$ over $\theta^{IV}_{bal,es}(l,l')$ for
simplicity.
The point on compositional changes making the interpretation of
$\theta^{IV}_{es(l)}$ harder, as pointed out in \cite{csa}, still applies,
and it is up to the econometrician to make the trade-off of interpretability vs
efficiency (more observations).
\paragraph{Multiplier Bootstrap and Simultaneous Confidence Bands}
Researchers often interpret multiple $\theta^{IV}_{es(l)}$ jointly.
In this case, simultaneous confidence bands are appropriate.
\cite{csa} show how to construct such bands via the multiplier bootstrap
and establish validity for the vector of $ATT(g,t)$ in their Theorem~3.
An analogous result applies to the vector of $LATT(e,t)$, allowing us to
use their Algorithm~1 for inference on our parameters of interest.
In particular, their Corollary~1 implies that the resulting bands
achieve correct asymptotic coverage, so simultaneous inference on
$\{\theta^{IV}_{es}(l) : l \in \{0,1,\ldots,h\}\}$ is valid.
The multiplier bootstrap is applied to the influence function
\cref{eq:if-esl} for each $l$.
We verify the procedure in \Cref{sec:simexp2}.
\section{Estimation and inference}
\label{sec:estimation-and-inference}
We consider plug-in estimators of $\tau^{dr,p}_{e,t}$ and
$\tau^{dr,rc}_{e,t}$ in
\cref{eq:dr-estimand-panel,eq:dr-estimand-rc}. We take two approaches.
First, we estimate the nuisance functions using simple parametric models
(e.g.\ linear or logistic regression) and invoke Donsker conditions to control
the empirical process term. Second, we allow for flexible machine learning
methods, avoid empirical process assumptions, and instead use DML.
We show below that the resulting DML estimators coincide with the cross-fitted estimator of \cite{kennedy}.
\paragraph{Asymptotically linear representation}
The plug-in estimators of $\tau^{dr,p}_{e,t}$ and $\tau^{dr,rc}_{e,t}$
admit the decomposition in \Cref{eq:decomposition}.
\cite{kennedy} shows that the general plug-in estimator has first-order bias
equal to $- \int \varphi(\cdot; \hat{\tau}, \hat{\eta}) \, dP$. This bias can
be removed by imposing the moment condition
$P_n[\varphi(\cdot; \hat{\tau}, \hat{\eta})] = 0$ and solving for the estimator,
yielding an estimating equation estimator\footnote{
We show in \Cref{sec:dml} that the cross-fitted estimating equation estimator
coincides with the DML estimator under the \texttt{DML2} algorithm;
see \Cref{remark:comparing-dml-cross-fit}.
}\footnote{
Alternative approaches include one-step estimators and targeted maximum
likelihood estimation (TMLE); see \cite{kennedy}.
}.
As discussed in \Cref{sec:identification-panel}, the estimators based on the DR
estimands in this paper are of this type and therefore eliminate the
first-order bias by construction\footnote{
Although the plug-in estimator of \cref{eq:dr-estimand-panel} solves the
estimating equation based on \cref{eq:if-latt-panel},
$P_n \varphi^{p}(O; \hat{\tau}^{dr,p}_{e,t}, \hat{\eta}^{p}_{e,t}) = 0$,
rather than the IF
$\varphi^{dr,p}(\cdot; \tau^{dr,p}_{e,t}, \eta^{dr,p}_{e,t})$ in
\cref{eq:if-dr-estimand-panel}, the plug-in bias is still zero since
$P_n \varphi^{dr,p}(O; \hat{\tau}^{dr,p}_{e,t}, \hat{\eta}^{dr,p}_{e,t}) = 0$
as well. The same holds for repeated cross-sections. This follows by direct
calculation; see \Cref{eq:plug-in-extra}.
}.
This justifies the decomposition in \cref{eq:decomposition} for our plug-in
estimators. To obtain the asymptotic linear representation
\begin{align}
\label{eq:decomposition-alin}
\sqrt{n}(\hat{\tau} - \tau)
=
\sqrt{n}(P_n - P)\varphi(\cdot; \tau, \eta)
+ o_P(1),
\end{align}
it remains to control the empirical process and remainder terms. Specifically,
to go from \cref{eq:decomposition} to \cref{eq:decomposition-alin}
we require
\begin{align}
\label{eq:conditions-alin}
\sqrt{n}(P_n - P)
\big[
\varphi(\cdot; \tau, \hat{\eta})
-
\varphi(\cdot; \tau, \eta)
\big]
= o_P(1),
\quad
\sqrt{n} P \varphi(\cdot; \tau, \hat{\eta}) = o_P(1).
\end{align}
The first condition is ensured by cross-fitting or Donsker assumptions, while
the second requires case-specific arguments \citep{kennedy}, formalized in
\Cref{prop:remainder-term-panel,prop:remainder-term-rc}.
\subsection{Remainder terms}
The following proposition provides conditions under which the remainder
term is asymptotically negligible, i.e., satisfies the right-hand side of
\cref{eq:conditions-alin}, in the panel data setting.
\begin{proposition}[Remainder term: panel data]
\label{prop:remainder-term-panel}
Suppose the identification in \Cref{prop:identification-panel} holds,
and that the second-order rate condition holds:
\begin{align*}
\Big\Vert
\frac{p}{1 - p}
-
\frac{\hat{p}}{1 - \hat{p}}
\Big\Vert_{2, P}
\Big\Vert
m_{e, t}^{c, p} - \hat{m}_{e, t}^{c, p}
\Big\Vert_{2, P}
& =
o_{P}(n^{-1/2}),
\\
\Big\Vert
\frac{p}{1 - p}
-
\frac{\hat{p}}{1 - \hat{p}}
\Big\Vert_{2, P}
\Big\Vert
g_{e, t}^{c, p} - \hat{g}_{e, t}^{c, p}
\Big\Vert_{2, P}
& =
o_{P}(n^{-1/2}).
\end{align*}
Then the remainder term for the influence function
\cref{eq:if-dr-estimand-panel} satisfies
\begin{align}
\label{eq:remainder-term-panel}
\sqrt{n}\,
P \varphi^{dr, p}(\cdot; \tau^{dr, p}_{e, t}, \hat{\eta}^{dr, p}_{e, t})
=
o_{P}(1).
\end{align}
\end{proposition}
\medskip\noindent
Proof in \Cref{sec:remainder-panel}.
\medskip\noindent
The following proposition provides conditions under which the remainder
term is asymptotically negligible, i.e., satisfies the right-hand side of
\cref{eq:conditions-alin}, in the repeated cross-sections setting.
\begin{proposition}[Remainder term: repeated cross-sections]
\label{prop:remainder-term-rc}
Suppose the identification in \Cref{prop:identification-rc} holds.
Moreover, assume:
\begin{enumerate}
\item (Treated nuisance consistency)
\begin{align}
\label{eq:treated-nuisance-rc-conditions}
\Vert m^{trt, rc}_{e,t} - \hat{m}^{trt, rc}_{e,t} \Vert_{2,P}
= o_P(1),
\quad
\Vert g^{trt, rc}_{e,t} - \hat{g}^{trt, rc}_{e,t} \Vert_{2,P}
= o_P(1),
\end{align}
and similarly for $t = e-1$.
\item (Weight fluctuations)
\begin{align*}
\left|
\frac{\lambda_t}{\hat{\rho}^{trt,rc}_{e,t}} - \frac{1}{\hat{\rho}}
\right|
= O_P(n^{-1/2}),
\quad
\left|
\frac{\lambda_{e-1}}{\hat{\rho}^{trt,rc}_{e,e-1}} - \frac{1}{\hat{\rho}}
\right|
= O_P(n^{-1/2}),
\end{align*}
where
$\hat{\rho}^{trt, rc}_{e, t} := P_{n}[E_{e} \cdot \mathbf{1}\{T = t\}]$
and $\hat{\rho} := P_{n}[E_{e}]$
are estimators of the denominators
of the weights in
\cref{eq:weights-trt-rc-norm,eq:weights1-panel-norm},
respectively.
\item (Second-order rate condition)
\begin{align*}
\Big\Vert
\frac{p}{1 - p}
-
\frac{\hat{p}}{1 - \hat{p}}
\Big\Vert_{2, P}
\Big\Vert
m^{c, rc}_{e,t} - \hat{m}^{c, rc}_{e,t}
\Big\Vert_{2, P}
& = o_P(n^{-1/2})
\\
\Big\Vert
\frac{p}{1 - p}
-
\frac{\hat{p}}{1 - \hat{p}}
\Big\Vert_{2, P}
\Big\Vert
g^{c, rc}_{e,t} - \hat{g}^{c, rc}_{e,t}
\Big\Vert_{2, P}
& = o_P(n^{-1/2}),
\end{align*}
and similarly for $t = e-1$.
\end{enumerate}
Then
\begin{align}
\label{eq:remainder-term-rc}
\sqrt{n}\,
P \varphi^{dr, rc}(\cdot; \tau^{dr, rc}_{e,t}, \hat{\eta}^{dr, rc}_{e,t})
= o_P(1).
\end{align}
\end{proposition}
\medskip\noindent
Proof in \Cref{sec:remainder-rc}.
\begin{remark}[Rate double robustness of the remainder term]
Compared to \Cref{prop:remainder-term-panel}, the repeated
cross-sections remainder does not exhibit the same rate double robustness,
in that the outcome and propensity models cannot offset each other’s
estimation error, as in the panel setting.
While double robustness for consistency still holds, only the
control-group nuisance components enter through the usual
propensity-outcome product structure; see also \cite[p.~26]{kennedy}.
\end{remark}
\subsection{Plug-in estimators}
In the following, we derive asymptotic linear representations for
plug-in estimators of $\tau^{dr,p}_{e,t}$ and
$\tau^{dr,rc}_{e,t}$ under Donsker class conditions, building on the
remainder results in
\Cref{prop:remainder-term-panel,prop:remainder-term-rc}.
Such conditions are often restrictive in high-dimensional settings.
Accordingly, \Cref{sec:dml} introduces a DML estimator that avoids these
assumptions by controlling the empirical process term via cross-fitting.
\paragraph{Panel Data}
\begin{proposition}
\label{prop:dr-estimator-alin-panel}
Consider the plug-in estimator of \cref{eq:dr-estimand-panel},
$\hat{\tau}^{dr,p}_{e, t}$.
Suppose \Cref{prop:remainder-term-panel} holds and that Donsker conditions
ensure the empirical process term is $o_{P}(1)$ as in
\cref{eq:conditions-alin}. Then
\begin{align}
\label{eq:dr-estimator-alin-panel}
\sqrt{n}(\hat{\tau}^{dr,p}_{e, t} - \tau^{dr,p}_{e, t})
=
\sqrt{n} (P_{n} - P)
\varphi^{dr, p}(\cdot; \tau^{dr, p}_{e, t}, \eta^{dr, p}_{e, t})
+ o_{P}(1),
\end{align}
and consequently,
\begin{align*}
\sqrt{n}(\hat{\tau}^{dr,p}_{e, t} - \tau^{dr,p}_{e, t})
\overset{d}{\to}
N\left(
0,
E[
\{\varphi^{dr, p}(O; \tau^{dr, p}_{e, t}, \eta^{dr, p}_{e, t})\}^{2}
]
\right).
\end{align*}
\end{proposition}
\paragraph{Repeated Cross-Sections}
\begin{proposition}
\label{prop:dr-estimator-alin-rc}
Consider the plug-in estimator of \cref{eq:dr-estimand-rc},
$\hat{\tau}^{dr,rc}_{e, t}$.
Suppose \Cref{prop:remainder-term-rc} holds and that Donsker conditions
ensure the empirical process term is $o_{P}(1)$ as in
\cref{eq:conditions-alin}.
Then
\begin{align}
\label{eq:dr-estimator-alin-rc}
\sqrt{n}(\hat{\tau}^{dr,rc}_{e, t} - \tau^{dr,rc}_{e, t})
=
\sqrt{n} (P_{n} - P)
\varphi^{dr, rc}(\cdot; \tau^{dr, rc}_{e, t}, \eta^{dr, rc}_{e, t})
+ o_{P}(1),
\end{align}
and consequently,
\begin{align*}
\sqrt{n}(\hat{\tau}^{dr, rc}_{e, t} - \tau^{dr, rc}_{e, t})
\overset{d}{\to}
N(
0,
E[
\{\varphi^{dr, rc}(O; \tau^{dr, rc}_{e, t}, \eta^{dr, rc}_{e, t})\}^{2}
]
).
\end{align*}
\end{proposition}
\subsection{Double Machine Learning / Cross-Fitting}
\label{sec:dml}
In this section, we present DML estimators for both data settings. In
each setting, the Neyman-orthogonal score is based on the corresponding
EIF in \cref{eq:if-latt-panel,eq:if-latt-rc}.
The resulting estimators coincide with the cross-fitted estimating
equation estimator of \cite{kennedy}. This follows because DML applies
cross-fitting to Neyman-orthogonal scores; when these scores are derived
from the EIFs in \cref{sec:nonparametric-identification}, the procedure
is equivalent to cross-fitting the corresponding estimating equation estimator.
\paragraph{Regularization and overfitting bias}
The Donsker assumptions made in the previous section are often unrealistic in
settings where the nuisances are high-dimensional and complex.
Here, flexible machine learning estimators are more appropriate.
Estimating nuisance functions with machine learning induces two forms of bias,
often referred to as regularization and overfitting bias in the DML literature
\citep{chernozhukovDoubleDebiasedMachine2018a}.
The remedy is to leverage a Neyman-orthogonal score for the DML estimator and
apply cross-fitting \citep{chernozhukovDoubleDebiasedMachine2018a}. This is
what we will do below.
\paragraph{DML Algorithm}
We construct our DML estimators using the \texttt{DML2} procedure of
\citep{chernozhukovDoubleDebiasedMachine2018a,bach2022doubleml}. The key components are:
\begin{enumerate}
\item a Neyman-orthogonal score identifying the target parameter
\item machine learning estimators of the nuisance functions
\item $K$-fold cross-fitting of the nuisance estimators
\item plugging the cross-fitted nuisances into the score and solving the
resulting moment condition yielding the DML estimator
\end{enumerate}
Hence, we first construct the Neyman-orthogonal scores.
For this we can use the influence functions
\cref{eq:if-latt-panel,eq:if-latt-rc}.
\paragraph{Panel Data}
Using the EIF for the panel data setting in
\cref{eq:if-latt-panel}, we obtain:
\begin{align}
\nonumber
& \varphi^{p}(O; \tau^{p}_{e, t}, \eta^{p}_{e, t})
\\
\nonumber
& \propto
\{ w^{trt,p}_{e} - w^{c,p}_{e, t} \}
\{\Delta_{t-e+1}Y_{t} - m_{e, t}^{c, p}(X)\}
- \tau^{p}_{e, t}
\{ w^{trt,p}_{e} - w^{c,p}_{e, t} \}
\{\Delta_{t-e+1}D_{t} - g_{e, t}^{c, p}(X)\}
\\
\nonumber
& =:
\psi^{p}_{b}(O; \eta^{p}_{e, t})
- \tau^{p}_{e, t}
\psi^{p}_{a}(O; \eta^{p}_{e, t})
\\
\label{eq:dml-score-panel}
& =: \psi^{p}(O; \tau^{p}_{e, t}, \eta^{p}_{e, t})
\end{align}
where we use DML notation for the score components, i.e. $\psi$, $\psi_{a}$ and
$\psi_{b}$.
Hence, we see that the influence function is proportional to a score (i.e.
function that identifies the target parameter) that is linear in the estimand
$\tau^{p}_{e, t}$.
Let $\{O_i\}_{i=1}^n$ be partitioned into $K$ folds of equal size
$n_k = n/K$, and denote the empirical measure on fold $k$ by $P_n^k$\footnote{
For equal-sized folds:
$K^{-1}P_{n}^{k}[f]
= K^{-1}n_{k}^{-1}\sum_{i=1}^{n} f(O_{i})
= n^{-1}\sum_{i=1}^{n}f(O_{i})$.
}.
Let $\hat{\eta}^{p,-k}_{e,t}$ be the nuisance estimator trained on all
observations outside fold $k$ (the superscript $-k$ indicates this).
Hats denote estimated quantities.
The DML estimator solves:
\begin{align*}
& K^{-1} \sum_{k=1}^{K}
P_{n}^{k}
[\psi^{p}(O; \hat{\tau}^{dml, p}_{e, t}, \hat{\eta}^{p, -k}_{e, t})]
= 0
\\
\iff
\hat{\tau}^{dml, p}_{e, t}
& =
\frac{
K^{-1} \sum_{k=1}^{K} P_{n}^{k}
[\psi^{p}_{b}(O; \hat{\eta}^{p, -k}_{e, t})]
}{
K^{-1} \sum_{k=1}^{K} P_{n}^{k}
[\psi^{p}_{a}(O_; \hat{\eta}^{p, -k}_{e, t})]
}
\\
& =
\frac{
K^{-1}\sum_{k=1}^{K} P_{n}^{k}
[
\{ \hat{w}^{trt, p, -k}_{e} - \hat{w}^{c, p, -k}_{e, t} \}
\{\Delta_{t-e+1}Y_{t} - \hat{m}_{e, t}^{c, p, -k}(X)\}
]
}{
K^{-1}\sum_{k=1}^{K} P_{n}^{k}
[\{ \hat{w}^{trt, p, -k}_{e} - \hat{w}^{c, p, -k}_{e, t} \}
\{\Delta_{t-e+1}D_{t} - \hat{g}_{e, t}^{c, p, -k}(X)\}]
}.
\end{align*}
Hence we have the result:
\begin{proposition}[DML Panel Data]
\label{prop:dml-panel}
Suppose the identification in
\Cref{prop:identification-panel} holds,
and also the condition on the remainder in
\cref{prop:remainder-term-panel}\footnote{
In \Cref{remark:identical-remainders} I show the remainder terms
using either \cref{eq:if-latt-panel}
or \cref{eq:if-dr-estimand-panel} are the same
and likewise for the repeated cross-sections setting
\cref{eq:if-latt-rc}
or \cref{eq:if-dr-estimand-rc}.
Hence, \Cref{prop:remainder-term-panel} can be used even though
the DML score was constructed based on \cref{eq:if-latt-panel}
and not \cref{eq:if-dr-estimand-panel}.
The same statements hold in the repeated cross-sections case for
\Cref{prop:dml-rc}.
}.
Then:
\begin{align*}
\sqrt{n}(\hat{\tau}^{dml, p}_{e, t} - \tau^{p}_{e, t, 0})
\overset{d}{\to} N(0, \sigma^{2, p}_{e, t}),
\end{align*}
where
\begin{align*}
\sigma^{2, p}_{e, t} :=
(J^{p}_{0})^{-2}
E[\{\psi^{p}(O; \tau^{p}_{e, t, 0}, \eta^{p}_{e, t, 0})\}^{2}],
\quad
J^{p}_{0} = E[\psi^{p}_{a}(O; \eta^{p}_{e, t, 0})]
\end{align*}
and we use the notation $\tau^{p}_{e, t, 0}, \eta^{p}_{e, t, 0}$
for the true value of the target parameter and nuisance function, respectively.
A variance estimator is:
\begin{align*}
\hat{\sigma}^{2, p}_{e, t}
=
(\hat{J}^{p}_{0})^{-2}
K^{-1} \sum_{k=1}^{K}
P_{n}^{k}
[\{\psi^{p}(O; \hat{\tau}^{dml, p}_{e, t}, \hat{\eta}^{p, -k}_{e, t})\}^{2}],
\quad
\hat{J}^{p}_{0}
=
K^{-1} \sum_{k=1}^{K}
P_{n}^{k}
[\psi^{p}_{a}(O; \hat{\eta}^{p, -k}_{e, t})]
\end{align*}
and an approximate $(1 - \alpha)\%$ confidence interval is constructed as:
\begin{align*}
[\hat{\tau}^{dml, p}_{e, t}
\pm \Phi^{-1}(1 - \alpha/2) \hat{\sigma}^{p}_{e, t}/\sqrt{n}],
\end{align*}
where $\Phi^{-1}(\cdot)$ is the quantile function of the standard normal
distribution.
\end{proposition}
\paragraph{Repeated Cross-Sections}
Using the EIF for the repeated cross-sections setting in
\cref{eq:if-latt-rc}, we obtain:
\begin{align}
\nonumber
\varphi^{rc}(O; \tau^{rc}_{e, t}, \eta^{rc}_{e, t})
& \propto
w^{trt,rc}_{e} \{Y - m_{e, Y}^{trt, rc}(X)\}
-
w^{c,rc}_{e}
\{Y - m_{e, Y}^{c,rc}(X)\}
\\
\nonumber
\nonumber & \quad
+ \frac{E_{e}}{\rho}
\{
m_{e,t}^{trt, rc}(X) - m_{e,e-1}^{trt, rc}(X)
-
[
m_{e, t, t}^{c, rc}(X) - m_{e, t, e-1}^{c, rc}(X)
]
\}
\\
\nonumber & \quad
\nonumber
- \tau^{rc}_{e, t}
(
w^{trt,rc}_{e} \{D - g_{e, Y}^{trt, rc}(X)\}
-
w^{c,rc}_{e}
\{D - g_{e, D}^{c,rc}(X)\}
\\
\nonumber
\nonumber & \quad
+ \frac{E_{e}}{\rho}
\{
g_{e,t}^{trt, rc}(X) - g_{e,e-1}^{trt, rc}(X)
-
[ g_{e, t, t}^{c, rc}(X) - g_{e, t, e-1}^{c, rc}(X) ]
\}
)
\\
\label{eq:dml-score-rc}
& =:
\psi^{rc}_{b}(O; \eta^{rc}_{e, t})
- \tau^{rc}_{e, t} \psi^{rc}_{a}(O; \eta^{rc}_{e, t})
\\
\nonumber
& =: \psi^{rc}(O; \tau^{rc}_{e, t}, \eta^{rc}_{e, t}).
\end{align}
Again, the influence function is proportional to a score that is
linear in the estimand $\tau^{rc}_{e, t}$.
Similarly to the panel data case, the DML estimator solves:
\begin{align*}
& K^{-1} \sum_{k=1}^{K}
P_{n}^{k}
[\psi^{rc}(O; \hat{\tau}^{dml, rc}_{e, t}, \hat{\eta}^{rc, -k}_{e, t})]
= 0
\\
\iff
\hat{\tau}^{dml, rc}_{e, t}
& =
\frac{
K^{-1} \sum_{k=1}^{K} P_{n}^{k}
[\psi^{rc}_{b}(O; \hat{\eta}^{rc, -k}_{e, t})]
}{
K^{-1} \sum_{k=1}^{K} P_{n}^{k}
[\psi^{rc}_{a}(O; \hat{\eta}^{rc, -k}_{e, t})]
},
\end{align*}
where the score components are defined in \cref{eq:dml-score-rc}.
Hence we have the result:
\begin{proposition}[DML Repeated Cross-Sections]
\label{prop:dml-rc}
Suppose the identification in
\Cref{prop:identification-rc} holds,
and also the condition on the remainder in \cref{prop:remainder-term-rc}.
Then:
\begin{align*}
\sqrt{n}(\hat{\tau}^{dml, rc}_{e, t} - \tau^{rc}_{e, t, 0})
\overset{d}{\to} N(0, \sigma^{2, rc}_{e, t}),
\end{align*}
where
\begin{align*}
\sigma^{2, rc}_{e, t} :=
(J^{rc}_{0})^{-2}
E[\{\psi^{rc}(O; \tau^{rc}_{e, t, 0}, \eta^{rc}_{e, t, 0})\}^{2}],
\quad
J^{rc}_{0} = E[\psi^{rc}_{a}(O; \eta^{rc}_{e, t, 0})]
\end{align*}
A variance estimator is:
\begin{align*}
\hat{\sigma}^{2, rc}_{e, t}
=
(\hat{J}^{rc}_{0})^{-2}
K^{-1} \sum_{k=1}^{K}
P_{n}^{k}
[\{\psi^{rc}(O; \hat{\tau}^{dml, rc}_{e, t}, \hat{\eta}^{rc, -k}_{e, t})\}^{2}],
\quad
\hat{J}^{rc}_{0}
=
K^{-1} \sum_{k=1}^{K}
P_{n}^{k}
[\psi^{rc}_{a}(O; \hat{\eta}^{rc, -k}_{e, t})]
\end{align*}
and an approximate $(1 - \alpha)\%$ confidence interval is constructed as:
\begin{align*}
[\hat{\tau}^{dml, rc}_{e, t}
\pm \Phi^{-1}(1 - \alpha/2) \hat{\sigma}^{rc}_{e, t}/\sqrt{n}].
\end{align*}
\end{proposition}
\begin{remark}[Comparing DML to cross-fitting for IDiD]
\label{remark:comparing-dml-cross-fit}
\leavevmode
DML with the Neyman-orthogonal scores
\cref{eq:dml-score-panel,eq:dml-score-rc}
is equivalent to a cross-fitted plug-in estimator based on
the double robust estimands \cref{eq:dr-estimand-panel,eq:dr-estimand-rc},
and conducting inference using the cross-fitted efficient influence
functions \cref{eq:if-latt-panel,eq:if-latt-rc}.
To see this, note that the cross-fitted estimator is obtained by solving
an equation of the form in \cref{eq:estimating-equation}. When
\Cref{eq:if-latt-panel,eq:if-latt-rc} are used, the resulting
estimating-equation estimator coincides with the plug-in estimators of
\Cref{eq:dr-estimand-panel,eq:dr-estimand-rc}. For instance, the
panel denominator is estimated by
\begin{align*}
P_{n}[
\{ \hat{w}^{trt,p}_{e} - \hat{w}^{c,p}_{e, t} \}
\{\Delta_{t-e+1}D_{t} - \hat{g}_{e, t}^{c, p}(X)\}
]
= P_{n}[\psi^{p}_{a}(O; \hat{\eta}^{p}_{e, t})]
\end{align*}
and in the repeated cross-sections case by
\begin{align*}
P_{n}
[
\{\hat{w}^{trt,rc}_{e} - \hat{w}^{c,rc}_{e}\}
\{D - \hat{g}_{e, Y}^{c,rc}(X)\}
] + \hat{\kappa}_{e, t}^{D, rc}
= P_{n}[\psi^{rc}_{a}(O; \hat{\eta}^{rc}_{e, t})],
\end{align*}
and similarly for the numerators.
Hence, the cross-fitted estimators are equivalent to the DML estimators.
Because of this equivalence, the remainder terms and nuisance-rate
conditions for the cross-fitted estimators are the same as those
needed for the DML estimators. The theorem for the cross-fitted
estimator in \cite{kennedy} therefore also applies to the DML estimators.
The DML estimators are also doubly robust, since they implicitly
estimate the doubly robust estimand, and cross-fitting handles
the empirical process term in the general decomposition automatically.
As a final point, the DML estimators are built from the EIF with normalized
weights. Hence
the DML estimators with $K=1$ correspond to the estimating-equation
estimators based on the normalized influence functions, which match
the DR plug-in estimators of \cref{eq:dr-estimand-panel,eq:dr-estimand-rc}.
However, the variance estimators of the DML estimators do not take into
account the normalization adjustments as done in the influence functions
\cref{eq:if-dr-estimand-panel,eq:if-dr-estimand-rc}
when conducting inference.
This does not seem to matter in the simulation experiments in
\Cref{sec:simexp1}.
\end{remark}
\paragraph{Estimators of aggregated effects}
For completeness, we state below a proposition for estimators of the weighted
estimands in \Cref{sec:weighting}.
\begin{proposition}
\label{prop:weighted-estimand-normality}
Suppose the assumptions of one of
\Cref{prop:dr-estimator-alin-panel,prop:dr-estimator-alin-rc,prop:dml-panel,prop:dml-rc}
hold for each cohort-time estimator entering the aggregated estimator
$\hat{\theta}$. Then
\begin{align*}
\sqrt{n}(\hat{\theta} - \theta)
\overset{d}{\to}
N(0, E[\{\varphi^{\theta}(O; \{\tau_{i}\}_{i \in \mathcal{I}};
\{\eta_{i}\}_{i \in \mathcal{I}})\}^{2}])
\end{align*}
where the estimand and influence function are defined in
\cref{eq:thetaiv,eq:if-agg-estimand}, respectively.
\end{proposition}
\Cref{prop:weighted-estimand-normality} is analogous to Corollary 1 in
\cite{miyaji} and Corollary 2 of \cite{csa}, and the proof is omitted.
\section{Simulation Experiments}
\label{sec:simulation}
In this section, we present three simulation experiments that assess the
finite-sample performance and double robustness of the estimators in
\Cref{sec:estimation-and-inference}.
The first experiment verifies double robustness in a simple two-period
DGP using the never-exposed control group. We compare the main estimators,
namely the DR estimators in
\Cref{prop:dr-estimator-alin-panel,prop:dr-estimator-alin-rc}
and the DML estimators in \Cref{prop:dml-panel,prop:dml-rc}, to three
non-doubly robust alternatives: inverse probability weighting (IPW),
standardized IPW (IPWS), and outcome regression (REG), each constructed by
exploiting the ratio-of-ATT-parameters structure in
\cref{eq:identification-panel,eq:identification-rc}. The standardization in
IPWS corresponds to the normalization of control weights as done in
\Cref{sec:nonparametric-identification}. This simulation setup follows the one in
\cite{sazhao}.
The second experiment considers staggered exposure to the instrument for
both control groups in \cref{eq:control-variables}, in both data
settings, and in the presence of an unobserved confounder affecting both
treatment and outcome. We focus on the $LATT(e,t)$ estimates varying
across cohorts and time, and aggregate these into estimates of the
weighted estimand in \cref{eq:esl-estimand}. Finally, we introduce a
group indicator under which $LATT$ differs across the two groups. We estimate
group-specific effects via \cref{eq:dr-estimand-group} and aggregate
these to horizon-specific effects as in \cref{eq:esl-estimand}, capturing
average dynamic differences across groups for each horizon $l$.
The third experiment follows the same setup as in the second experiment
(without the group indicator) but now imposes absorbing treatment
\cref{eq:treatment-absorbed}.
This setting allows us to verify the Bloom-type result in
\Cref{prop:bloom} and compare our estimators to those of \citet{csa}.
As before, a hidden confounder jointly determines treatment and outcome,
so the DiD estimator is biased, whereas the IDiD estimator is not.
\subsection{Simulation Experiment 1: Double Robustness}
\label{sec:simexp1}
We compare the DR and DML estimators to the IPW, IPWS and REG estimators
through a simulation experiment similar to \cite{sazhao}.
To assess the double robustness properties, we simulate four DGPs with varying
degrees of misspecification.
The first DGP has both the outcome regression and propensity score correctly
specified.
The second DGP has the outcome regression correctly specified and the propensity
score misspecified.
The third DGP has the outcome regression misspecified and the propensity
score correctly specified.
The fourth and last DGP has both the outcome regression and propensity score
misspecified.
We focus on a two-period setup with covariates using a never-exposed control
group with panel data.
Let
\begin{align*}
X = (X_{1}, X_{2})' \sim N(0, I_{2}),
\quad
\mathrm{supp}(E) = \{\infty,2\},
\end{align*}
and
\begin{align*}
h_{\ell}(X) = 0.8 X_{1} - 0.5 X_{2},
\quad
h_{n}(X) = 0.8\sin(X_{1}) - 0.5 \cdot \mathbf{1}\{X_{2}>0\} + 0.3 X_{1}X_{2},
\end{align*}
and define
\begin{align*}
g_{ps}(X) \in \{h_{\ell}(X), h_{n}(X)\},
\qquad
s(X) \in \{h_{\ell}(X), h_{n}(X)\}.
\end{align*}
Here, $h_{\ell}$ corresponds to correct specification and $h_{n}$ to
misspecification.
Exposure occurs in period $2$ with probability
\begin{align*}
P(E=2 \mid X) = \mathrm{expit}(g_{ps}(X)).
\end{align*}
Treatment propensities are
\begin{align*}
p_{t}(\infty) = \mathrm{expit}(0.2 + s(X) - 0.4t),
\quad
p_{t}(2)
= \mathrm{expit}(0.2 + s(X) - 0.4t + \kappa),
\end{align*}
and treatment states are generated as
\begin{align*}
D_{t}(\infty) = \mathbf{1}\{U \le p_{t}(\infty)\},
\quad
D_{t}(2) = \mathbf{1}\{U \le p_{t}(2)\},
\quad
U \sim \mathrm{Uniform}(0,1).
\end{align*}
Untreated potential outcomes follow
\begin{align*}
Y_{t}(0) & = [1 + 0.8 \cdot (t - 1)] s(X) + \varepsilon_{t},
\end{align*}
while treated outcomes are
\begin{align*}
Y_{t}(1) = Y_{t}(0) + \tau,
\end{align*}
where $\varepsilon_{1}, \varepsilon_{2} \sim N(0,0.2^{2})$.
The observed treatment is given by
\cref{eq:observed-treatment-status} and the observed outcome by
\cref{eq:observed-outcome}.
Note that the treatment and outcome evolutions depend on the same specification
$s(X)$.
Also, the true local average treatment effect on the treated for those exposed
in period $2$ equals $LATT(2, 2) = \tau$.
In the experiment, we set $\kappa, \tau = 1$,
$n = 5000$, $\mathcal{T} = 2$, and repeat the experiment $B=4999$ times.
For the repeated cross-sections case, we set $\lambda = 0.5$.
For the DML estimators, we use $K=5$ folds for cross-fitting. Outcome
regressions are estimated by linear regression, and propensity scores,
when required, by logistic regression.
The results are shown in \Cref{tab:sim-exp-1-panel,tab:sim-exp-1-rc} for the panel
and repeated cross-sections settings, respectively.
The panel results confirm the expected double robustness and efficiency
properties of the proposed estimators. When both nuisance components are
correctly specified (DGP1), all estimators are essentially
unbiased with coverage close to the nominal level.
The IPW estimators exhibit substantially higher variance than the other
estimators. When only the
outcome regression is correctly specified (DGP2), the DR, DML and regression estimators remain
approximately unbiased, whereas the IPW estimators become severely
biased and coverage deteriorates.
Conversely, when only the propensity score is correctly specified (DGP3), the
DR and DML estimators again remain unbiased, while the regression estimator is biased,
illustrating the double robustness property. When both nuisance components are
misspecified (DGP4), all estimators are biased, as expected.
The repeated cross-section results display the same qualitative double
robustness pattern but with larger dispersion. In DGP1-DGP3,
the DR and DML estimators remain approximately unbiased with reasonable coverage,
again verifying the double robustness property,
although their variance is higher than in the panel case. In contrast,
the IPW estimators, even in DGP1 under correct specification, exhibit extreme
variability, reflected in very large RMSE and asymptotic variance, driven by
instability in the weighting scheme.
In particular, the non-standardized IPW estimator explodes in variance, as was
also the case for the two-period DiD experiment in \cite{sazhao},
although here further amplified by the ratio structure of the estimators.
When the propensity score is misspecified (DGP2), the DR and DML estimators remain
relatively robust, while the regression estimator is biased. Finally, when both
nuisance components are misspecified (DGP4), all estimators are biased.
Overall, the results highlight the double robustness of the proposed
estimators. While this property holds in both data settings, the
repeated cross-section estimators are less stable in finite samples due
to noisier estimation of the underlying components; larger sample sizes
would mitigate this. The DR and DML estimators perform similarly in this
experiment, although the DML estimator exhibits slightly higher
variance, likely due to cross-fitting. In more complex settings, DML,
combined with more flexible nuisance models, is expected to outperform
the simpler DR estimator.
\begin{table}[H]
\centering
\caption{Simulation experiment 1: Panel data}
\label{tab:sim-exp-1-panel}
\begin{threeparttable}
\includegraphics[width=\textwidth]{tabs/dr_panel.pdf}
\begin{tablenotes}
\footnotesize
\item Notes: Simulations are based on $B=4999$ Monte Carlo repetitions
with sample size $n=5000$. The target parameter is $LATT(2, 2)$;
all subscripts $(e, t) = (2, 2)$.
$\hat{\tau}^{reg,s}_{e,t}$ denotes the outcome regression (OR)
estimator, $\hat{\tau}^{ipw,s}_{e,t}$ the inverse probability
weighting (IPW) estimator, $\hat{\tau}^{ipws,s}_{e,t}$ the
standardized IPW estimator, $\hat{\tau}^{dr,s}_{e,t}$ the doubly
robust (DR) estimator, and $\hat{\tau}^{dml,s}_{e,t}$ the DML estimator,
where $s \in \{p, rc\}$ indicates the panel and repeated cross-section
settings, respectively.
We use a linear working model for the outcome regression and a logistic
working model for the propensity score, with parameters estimated by OLS
and maximum likelihood. “Av. Bias”, “Med. Bias”, “RMSE”, “Asy. V”,
“Cover”, and “CIL” denote the average bias, median bias, root mean
squared error, average estimated asymptotic variance, 95\% coverage
probability, and average confidence interval length, respectively.
The semiparametric efficiency bounds are calculated using the true values
and the results from \Cref{prop:if-latt-panel,prop:if-latt-rc},
respectively.
The table follows the structure of \cite{sazhao}.
\end{tablenotes}
\end{threeparttable}
\end{table}
\begin{table}[H]
\centering
\caption{Simulation experiment 1: Repeated cross-sections}
\label{tab:sim-exp-1-rc}
\begin{threeparttable}
\includegraphics[width=\textwidth]{tabs/dr_rc.pdf}
\begin{tablenotes}
\footnotesize
\item Notes: See \Cref{tab:sim-exp-1-panel} for notes.
\end{tablenotes}
\end{threeparttable}
\end{table}
\subsection{Simulation Experiment 2: Staggered exposure and control groups}
\label{sec:simexp2}
In this simulation experiment we simulate a panel with staggered exposure and
introduce the time-varying hidden confounder $H_{t}$
determining both the treatment $D_{t}$ and outcome $Y_{t}$.
The setup for the simulation is as follows.
We simulate a time-invariant covariate and time-varying hidden confounder
as $X, H_{t} \sim N(0, 1),$
and exposure cohorts from
\begin{align}
\label{eq:propensity-and-support}
P(E = e \mid X) = \frac{\exp(\beta_{e} X,)}{
\sum_{e \in \mathcal{E}} \exp(\beta_{e} X,)
},
\quad
\mathrm{supp}(E) = \{\infty, 2, 3, \ldots, n_{E}\},
\end{align}
with $\beta_{e} \in \{0.1, \ldots, 0.3\}$, $e \in \mathrm{supp}(E)$,
equally spaced between $0.1$ and $0.3$\footnote{
For $P(E = e \mid X)$ simulated this way,
the generalized propensity \cref{eq:generalized-propensity-scores}
in the case of the never-exposed control group
equals
$p := P(E = e \mid X, E_{e} + C^{nev} = 1)
= \mathrm{expit}([\beta_{e} - \beta_{\infty}]X),
$
where $\beta_{\infty}$ is the parameter corresponding to the never-exposed control
group;
hence, $p / (1 - p) = \exp([\beta_{e} - \beta_{\infty}]X)
\sim \mathrm{LogNormal}(0, [\beta_{e} - \beta_{\infty}]^{2}).
$
Thus, in order to stabilize the propensity in the simulations,
we constrain the parameters $\beta_{e}$ to $\{0.1, \ldots, 0.3\}$ so the
maximum difference is $0.2$.
}.
The potential treatments and outcomes are simulated as:
\begin{align}
\label{eq:meat-treatment}
L_{t} & = \nu + 0.5 X + H_{t} + \frac{1}{8} t/\mathcal{T},
\quad
\nu \sim \mathrm{Uniform}(-1, -0.2),
\\
\nonumber
p_{t}(\infty) & = \mathrm{expit}(L_{t}),
\quad
p_{t}(e) = \mathrm{expit}(L_{t} + \delta),
\\
\nonumber
D_{t}(\infty) & = \mathbf{1}\{U_{t} \leq p_{t}(\infty)\},
\quad
D_{t}(e) = \mathbf{1}\{U_{t} \leq p_{t}(e)\},
\quad
U_{t} \sim \mathrm{Uniform}(0, 1),
\\
\label{eq:pos-simulation}
& Y_{t}(0) = \eta + X + H_{t} + t/\mathcal{T} + \varepsilon_{t},
\quad
Y_{t}(1) = Y_{t}(0) + \tau_{t} + \nu_{t},
\end{align}
where $\eta, \varepsilon_{t}, \nu_{t} \sim N(0, 1)$.
Note that $H_{t}$ determines both the treatment and outcome cf.
\cref{eq:meat-treatment,eq:pos-simulation}.
Again, the observed treatment is given by
\cref{eq:observed-treatment-status} and the observed outcome by
\cref{eq:observed-outcome}.
For simplicity, we focus on the DR estimators
\cref{eq:dr-estimator-alin-panel,eq:dr-estimator-alin-rc} and evaluate
their finite-sample performance under panel and repeated cross-section
sampling, respectively, using never-exposed and not-yet-exposed control
groups. We set $n = 10{,}000$, $\mathcal{T} = 5$, $\tau_t = 1$ for all $t$, $\delta = 1$,
$n_{E} = 5$
and perform $B = 1499$ simulation draws.
In the not-yet-exposed experiment the $E = \infty$ is removed from
\cref{eq:propensity-and-support}.
The average evolution of the
treatment $D_t$ and outcome $Y_t$, conditional on the exposure cohort
$E$, for a simulated dataset is shown in
\Cref{fig:evolution}.
\medskip
\noindent\textit{Cohort-time effects.}
\Cref{fig:simexp2-a} reports the distribution of $\hat{\tau}^{dr, p}_{e,t}$ and
$\hat{\tau}^{dr, rc}_{e,t}$ across all $(e, t)$ pairs. The estimates are
centered around the true value in all designs, showing the robustness of the
IDiD procedure to hidden confounding. The panel
case is most tightly concentrated, while the repeated cross-sections case exhibits
higher dispersion; both show occasional large realizations due to the ratio
structure.
In both designs, the not-yet-exposed estimates, compared to the never-exposed,
become increasingly dispersed at longer horizons, reflecting the corresponding
reduction in the size of the control group.
\begin{figure}[!htbp]
\centering
\begin{subfigure}{\textwidth}
\centering
\includegraphics[width=0.70\textwidth]{figs/evolution.pdf}
\caption{Average evolution of treatment $D_{t}$ and outcome $Y_{t}$
conditional on exposure cohort $E$ from the DGP of
Simulation Experiment 2. The upper row shows panel data and
the lower repeated cross-sections.}
\label{fig:evolution}
\end{subfigure}
\vspace{0.5em}
\begin{subfigure}{\textwidth}
\centering
\includegraphics[width=0.7\textwidth]{figs/sim2_dr_et.pdf}
\caption{Simulation experiment 2: distributions of the cohort-time
estimators $\hat{\tau}^{dr,p}_{e,t}$ and $\hat{\tau}^{dr,rc}_{e,t}$
across $(e,t)$ pairs, for panel data and repeated cross-sections under
never-exposed and not-yet-exposed controls.
The estimates for $(e, t) = (5, 5)$ using never-exposed controls are not
shown.
}
\label{fig:simexp2-a}
\end{subfigure}
\caption{Simulation experiment 2: design and estimator distributions.}
\label{fig:simexp2}
\end{figure}
\medskip
\noindent\textit{Event study aggregation.}
\Cref{fig:simexp2-b} shows the aggregated effects $\hat{\theta}^{IV}_{es(l)}$. The
same pattern emerges: estimates remain centered, but variability increases in
repeated cross-sections. Again, for the not-yet-exposed controls, dispersion
increases with the horizon due to the shrinking control group.
We conduct a multiplier bootstrap and construct simultaneous confidence
bands for $\{\theta^{IV}_{es}(l) \mid l = 0,1,2\}$ in
\cref{eq:esl-estimand}.
Let $\hat{C}^{IV}_{es}(l)$ denote the interval for the
horizon-$l$ effect. Then, as in \cite{csa}, the intervals are simultaneous in
the sense that\footnote{
The general simultaneous confidence band for
$LATT(e, t)$, $t \geq e$,
is simultaneous in the sense that
$
P (LATT(e, t) \in \hat{C}(e, t) \,
\forall (e,t) \in \mathcal{E} \times \{2, 3, \ldots,
\mathcal{T}\}
: t \geq e) \to 1 - \alpha
$; see \cite{csa} for details.
}
\begin{equation}
\label{eq:cov}
P (\theta^{IV}_{es}(l) \in \hat{C}^{IV}_{es}(l) \,
\forall l \in \{0,1,2\} )
\to 1 - \alpha.
\end{equation}
We verify this in \Cref{tab:sim-exp-2-tests} with $\alpha = 0.05$.
The block ``Pooled'' reports coverage rates \cref{eq:cov} computed from
simultaneous confidence bands (``Simultaneous'') obtained via the
multiplier bootstrap and from pointwise confidence intervals
(``Pointwise'').
As expected, the simultaneous coverage is close to $0.95$, whereas the
pointwise coverage is too low. The repeated cross-section estimates are
noisier, which propagates to the simultaneous bands, yielding slightly
conservative (too high) coverage.
\begin{figure}[H]
\centering
\includegraphics[width=0.7\textwidth]{figs/sim2_dr_agg.pdf}
\caption{Simulation experiment 2: aggregated effects using never-exposed
and not-yet-exposed controls, for panel data and repeated cross-sections.
The largest realizations are omitted for readability.
}
\label{fig:simexp2-b}
\end{figure}
\begin{table}[H]
\centering
\caption{Simulation experiment 2: aggregated effects coverage}
\label{tab:sim-exp-2-tests}
\begin{threeparttable}
\includegraphics[width=0.8\textwidth]{tabs/coverage_all.pdf}
\begin{tablenotes}
\footnotesize
\item Notes: The first two rows report joint coverage for the
horizon-aggregated effects in \Cref{fig:simexp2-b}. The last two rows
report joint coverage for the differenced group-specific
horizon-aggregated effects
in \Cref{fig:simexp2-c}.
\end{tablenotes}
\end{threeparttable}
\end{table}
\medskip
\noindent
\textit{Group-specific aggregation.}
Finally, we simulate a new dataset with a binary group indicator $F \sim
\mathrm{Bern}(0.5)$ splitting the units into two groups.
The groups have different $LATT$s with $F = 1$ being affected less.
We set $n=20,000$ so there are two groups of approximately $10,000$.
Our aim is to estimate the group-wise estimand $\tau^{dr,p,\Delta}_{e,t}$ as
defined in \cref{eq:dr-estimand-group}
and aggregate the effects into dynamic horizon-$l$ effects as in
\cref{eq:esl-estimand}.
The treated state in \cref{eq:pos-simulation} now becomes
\begin{align*}
Y_{t}(1) = Y_{t}(0) + \tau_{t}(1 - 0.5 F) + \nu_{t},
\end{align*}
where those with $F = 1$ have $LATT(e, t) = 0.5 \tau_{t}$.
\Cref{fig:simexp2-c} reports the aggregated group-specific horizon
effects and their difference, i.e., the aggregated estimates of
$\tau^{dr,p,\Delta}_{e,t}$.
The estimator tracks the targets
well in the panel setting, whereas repeated cross-sections exhibit
higher variability, particularly for the difference estimand.
Again, the not-yet-exposed case has higher variability because of the
shrinking control group.
We again verify the validity of the multiplier bootstrap to construct
simultaneous confidence bands for the aggregated effects, here for the
aggregated horizon-$l$ effects of $\tau^{dr,p,\Delta}_{e,t}$.
The results are reported in block ``Group'' of \Cref{tab:sim-exp-2-tests}.
For panel data, the
simultaneous coverage is close to $0.95$ for both control groups. In repeated
cross-sections, coverage is again slightly conservative (above $0.95$),
reflecting the higher variability of this design, compounded by the additional
noise from differencing. Pointwise coverage is too low across all simulations,
as expected.
\begin{figure}[H]
\centering
\includegraphics[width=0.7\textwidth]{figs/sim2_dr_group.pdf}
\caption{
Simulation experiment 2: group-specific aggregated effects and their difference
under never-exposed and not-yet-exposed controls.
The largest realizations are omitted for readability.
}
\label{fig:simexp2-c}
\end{figure}
\subsection{Simulation Experiment 3: One-sided compliance and staggered exposure}
We simulate a setting with staggered exposure to the instrument and
one-sided compliance in the presence of an unobserved confounder
affecting both treatment and outcomes to verify \Cref{prop:bloom}. The
setup follows Simulation Experiment 2, except that treatment is an
absorbing state, i.e. \cref{eq:treatment-absorbed}, and the treatment effects
are now heterogeneous across treatment cohorts and time.
The potential outcomes of \cref{eq:pos-simulation}
in the treated state are now given by
\begin{align}
\label{eq:att-effect-sim2}
Y_{t}(g) = Y_{t}(0) + \tau_{g, t} + \nu_{t},
\quad
\tau_{g,t}
=
\frac{t-g+1}{2},
\end{align}
where the dependence on $g$ reflects the absorbing treatment;
the $ATT(g,t,e)$ parameters increases linearly over post-treatment periods as
$1/2,\,1,\,3/2,\ldots$ and are invariant across exposure cohorts $e \in
\mathcal{E}$.
The potential outcomes in the untreated states are as in
\cref{eq:pos-simulation}.
In the simulation experiment, we set $n = 10,000$, $\mathcal{T}=5$,
$\mathrm{supp}(E)=\{\infty,2,3,4,5\}$, use as control group the
never-exposed units, $C^{nev}$,
and repeat the experiment $B=1499$ times.
The implied $ATT(g, t)$ parameters are
$
\{\tau_{2,2}, \tau_{2,3}, \tau_{2,4}, \tau_{2,5},
\tau_{3,3}, \tau_{3,4}, \tau_{3,5},
\tau_{4,4}, \tau_{4,5},
\tau_{5,5}\},
$
with true values given by \cref{eq:att-effect-sim2}.
For each draw, we estimate $LATT(e,t)$ using our DR
estimators in each data setting. We also construct the treatment cohort
variable $G = \min\{t \mid D_t = 1\}$, enabling two quantities:
(i) the true $ATT(g,t,e)$, computed from the simulated (unobserved)
potential outcomes, and (ii) $ATT(g,t)$ estimated via the
procedure of \cite{csa}.
By \Cref{prop:bloom}, for each $(e,t) \in \mathcal{E} \times \{2, 3, \ldots,
\mathcal{T}\}$ with $t \ge e$,
$\hat{LATT}(e,t)$ should approximately equal $\sum_{g \le t}
\hat{ATT}(g,t,e)\,\hat{P}(G = g \mid D_t = 1, E_e = 1),$ cf. \cref{eq:bloom}.
We compute this quantity using Oracle estimates for $\hat{ATT}(g,t,e)$ (the CSA
estimators won't work because of the hidden confounder) and empirical
conditional probabilities for the weights.
Results are reported in \Cref{tab:sim-exp-2}.
For horizon $l=t-e=0$, the $LATT(e, t)$ estimates coincide exactly with the
exposure-cohort-specific $ATT(g, t)$, so the aggregated $l=0$ effect equals the
cohort-weighted $ATT(g, t)$, which equals $1/2$ for each pair $(e, t) \in
\{(2, 2), (3, 3), \ldots, (5, 5)\}$.
For $l>0$, the estimates of $LATT(e,t)$ are a convex combination of the
underlying $ATT(g, t, e)$ parameters.
For instance,
\begin{align*}
\hat{LATT}(2, 4) = 1.25
& \approx
\hat{ATT}(2, 4, 2)\hat{P}(G_{2} = 1 \mid D_{4} = 1, E_{2} = 1)
\\ & \quad
+ \hat{ATT}(3, 4, 2)\hat{P}(G_{3} = 1 \mid D_{4} = 1, E_{2} = 1)
\\ & \quad
+ \hat{ATT}(4, 4, 2)\hat{P}(G_{4} = 1 \mid D_{4} = 1, E_{2} = 1)
\\
& = 1.501 \cdot 0.615 + 1.001 \cdot 0.266 + 0.493 \cdot 0.119
\\
& = 1.248
,
\end{align*}
use the first two entries in row $(e,t) = (2,4)$ of
\Cref{tab:sim-exp-2} (the oracle and weight estimates are not reported
in the table).
The weights $\hat{P}(G_g=1 \mid D_4=1, E_2=1)$, $g = 2, 3, 4$, are decreasing, reflecting that
treatment is absorbing. Among units first exposed at $t=2$, a large share is
treated at $t=2$, a smaller share at $t=3$, and only a small
fraction remains untreated until $t=4$. The heterogeneity in $\hat{ATT}(g,4,2)$
captures the different treatment horizons.
Comparing the Oracle and $LATT(e,t)$ columns across both data settings,
we see that the columns are approximately equal, verifying the result in
\Cref{prop:bloom}. In contrast, the DiD estimates of $ATT(g,t)$ in the CSA
column are biased because of the unobserved confounder $H_t$ affecting
both the treatment and outcome, cf. \cref{eq:meat-treatment,eq:pos-simulation}.
The instrumental variable component of the IDiD estimator addresses this source
of bias and remains unbiased, illustrating the robustness of the IDiD design to
hidden confounding,
and how the IDiD estimators can be leveraged in settings where DiD estimators
fail.
\begin{table}[H]
\centering
\caption{Simulation experiment 3}
\label{tab:sim-exp-2}
\begin{threeparttable}
\includegraphics[width=1\textwidth]{tabs/onesided_table.pdf}
\begin{tablenotes}
\footnotesize
\item Notes: The table shows the simulation results of simulation
experiment 3.
Each entry is an average over $B=1499$ replications.
The ``$LATT(e,t)$'' columns report the IDiD estimates, the
``Oracle'' columns report the right-hand side of \cref{eq:bloom} using
the simulated potential outcomes and estimated group probabilities, and
the ``$ATT(g,t)$'' columns report the DiD estimates based on
the estimator of \cite{csa}. The comparison between the ``$LATT(e,t)$''
and ``Oracle'' columns verifies \Cref{prop:bloom}.
The ''SD`` column reports the standard deviation of the $LATT(e, t)$
and $ATT(g, t)$ over all simulation draws.
\end{tablenotes}
\end{threeparttable}
\end{table}
\section{Conclusion}
This paper develops doubly robust estimands for the $LATT(e, t)$ parameter in
the IDiD setting with covariates, covering both panel data and repeated
cross-sections, and allowing for never-exposed and not-yet-exposed control
groups. We also construct corresponding DR and DML estimators.
Our approach is estimand-based: using the influence function machinery of
\cite{kennedy}, adapted to our setting in
\Cref{sec:influence-functions}, we derive the DR estimands from first
principles. The simulation results confirm the validity
of the corresponding estimators across both data settings and control groups, and verify
the group-difference estimands and the Bloom result under absorbing
treatment, linking IDiD with staggered instrument exposure to DiD with
staggered treatment.
At the same time, the analysis highlights an important distinction between the
two data settings: while both admit doubly robust identification and
estimation, the repeated-cross-sections case does not inherit the same
remainder-term behavior as the panel case because of the additional nuisance
components.
More broadly, the paper underscores the value of an estimand-based approach: it
replaces reverse-engineering of regression parameters with the specification of
interpretable estimands, and enables modular reuse of components across
settings, in particular when deriving the corresponding EIFs.
It also provides another example of constructing Neyman-orthogonal
scores from EIFs (cf.~\cite[Section 2.2.5]{chernozhukovDoubleDebiasedMachine2018a};
see also \cite{chen2026equivalence}) and of two cases in which the
cross-fitted estimator coincides with the DML estimator.
Practically, the paper delivers both the DR and DML estimators, together with
a software implementation, making the proposed methods directly usable in
applied work.
We hope that the approach and components developed here, together with the work
on which they build, serve as a useful guide and toolbox for future work.
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