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-2.5cmTwo Margins in Difference-in-Differences with a Continuous Treatment
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\begin{abstract}
This paper studies difference-in-differences with staggered adoption and a continuous, time-invariant dose. Each cohort-time comparison contains two margins. The level margin is the average treatment effect at realized doses. Under level parallel trends it equals the level contrast between the treated cohort and not-yet-treated controls. The response margin is the within-cohort slope of the outcome change on dose. It uses no controls, and its causal interpretation requires a response parallel trends assumption and a restriction on selection on gains. We show that the continuous-dose OLS coefficient in each cohort-time comparison is a convex combination of the response index and the level contrast per unit of mean dose, with a mixing weight that depends on the not-yet-treated share. We provide estimators of both margins, joint inference across cohort-time comparisons and event-time aggregates, and a covariate-adjusted extension. In an application to hydraulic fracturing, the level leads reject a joint zero restriction, whereas the response-index leads do not. The continuous-dose OLS coefficient draws primarily on the level margin. We report the level and response margin separately.
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\section{Introduction}\label{sec:intro}
Staggered difference-in-differences (DiD) with a binary treatment forms each cohort-time effect from a valid control group and aggregates only afterward \citep{dechaisemartin-dhaultfoeuille-2020,goodmanbacon-2021,sun-abraham-2021,callaway-santanna-2021,borusyak-jaravel-spiess-2024}. With a continuous dose that is fixed after adoption, each cohort-time comparison also contains variation in dose within the treated cohort. A regression of the outcome change on dose uses both the treated-control difference and this within-cohort variation. This paper separates the two comparisons and develops estimators and joint inference for their cohort-time and event-time parameters.
The level margin is the cohort's average treatment effect at its realized doses relative to zero treatment. Its observed-data counterpart, the level contrast, is the difference in mean outcome changes between the cohort and units not yet treated. The response index is the population OLS slope of the outcome change on dose within the cohort. The level contrast is measured in outcome units, and the response index in outcome units per unit of dose. Controls enter the level contrast, while the response index uses only the treated cohort. Neither determines the other.
We show that the OLS coefficient in each cohort-time comparison is an exact convex combination of the response index and the level contrast divided by the mean treated dose. Its mixing weight depends on the treated dose distribution and the not-yet-treated share. Even with the conditional outcome and dose distributions held fixed, changing this share changes the coefficient unless the two components coincide. Reporting the two objects separately shows which comparison accounts for the regression result.
We use the hydraulic fracturing application from \citet{bartik-et-al-2019} to illustrate this distinction. The continuous-dose OLS coefficient in each cohort-time comparison places most of its weight on the level-contrast component. Under county-level i.i.d. inference, the level leads reject a joint zero restriction, whereas the response-index leads do not. Removing counties whose missing prospectivity scores were coded as zero changes the level estimates and leaves the response estimates unchanged, because these counties enter only the controls. We report the two paths descriptively, distinguishing the employment contrast with not-yet-treated counties from the relation between employment and dose within cohorts.
The two comparisons require separate identifying restrictions. Level parallel trends (PT-L) equates mean untreated outcome changes for the treated cohort and its not-yet-treated controls and identifies the level margin. Response parallel trends (PT-R) removes the relation between dose and untreated outcome changes within the treated cohort. These restrictions are non-nested. PT-R removes dose-related untreated trends from the response index, but selection on gains may remain. Under the conditions for the derivative representation in Section~\ref{sec:why-two}, the index is the sum of a nonnegatively weighted average of causal derivatives and a selection term. A bound on the selection derivative gives a sensitivity bound for the causal component.
The paper builds on \citet{callaway-goodmanbacon-santanna-2026}, henceforth CGBS. They identify treatment-on-the-treated effects at realized doses under dose-specific parallel trends, show how comparisons across dose groups combine causal responses with selection on gains, and decompose continuous-treatment two-way fixed-effects (TWFE) regressions. Their staggered-adoption extension and event-study aggregations are developed in \citet{callaway-goodmanbacon-santanna-2024-event}. Our level margin is the average of their realized-dose effect over the cohort's dose distribution. We pair it with the within-cohort response index to separate the two sources of variation in each cohort-time regression and estimate the two objects jointly across the staggered design. In the two-period setting, CGBS's dose-specific parallel-trends condition combines PT-L and PT-R, while their aggregate condition coincides with PT-L. Stating the restrictions separately makes it possible to assess each comparison on its own terms.
The baseline estimators are the treated-control mean contrast and the within-cohort OLS slope. Their influence functions indexed by the same units retain dependence across cohort-time comparisons, margins, and estimated cohort-share weights. This yields joint Wald tests and simultaneous inference for fixed collections of comparisons and aggregates. The covariate-adjusted extension preserves the level margin, while residualizing dose within the cohort generally defines a different response index. Building on semiparametric DiD and orthogonal-score methods \citep{abadie-2005,santanna-zhao-2020,chang-2020,chernozhukov-et-al-2018}, we develop cross-fitted estimators and stacked limit theory for the adjusted targets. The response score has the residual-on-residual form used by \citet{clarke-polselli-2026} in their DML estimator for panel data. Both the baseline and adjusted procedures estimate scalar targets without a dose density, a derivative estimator, or a bandwidth.
Other continuous- or multivalued-treatment DiD methods use different designs or target different objects. \citet{fricke-2017} studies a two-period design with multiple treatments. \citet{dhaultfoeuille-hoderlein-sasaki-2023} use repeated cross-sections and a crossing change in the treatment distribution to identify nonparametric average and quantile effects under stationarity and rank restrictions. For treatments that change over time, \citet{dechaisemartin-et-al-2026-stayers} compare switchers with stayers, \citet{dechaisemartin-dhaultfoeuille-vazquezbare-2024} treat the case in which every unit switches, and \citet{haddad-et-al-2026} develop kernel DML for treatment histories. \citet{zhang-2026} develops DML for kernel-smoothed, dose-specific DiD effects with conditional-density nuisances and uniform-in-dose bands.
Section~\ref{sec:setup} defines the staggered design and the two cohort-time targets, and Section~\ref{sec:generic-assumptions} gives their identifying restrictions. Section~\ref{sec:why-two} derives the regression mixture and the causal and selection components of the response index. Section~\ref{sec:paths} defines cohort-specific and event-time parameters. Sections~\ref{sec:baseline-estimation} and~\ref{sec:covariates} develop baseline estimation and inference and the covariate-adjusted extension. Sections~\ref{sec:simulation} and~\ref{sec:application} report Monte Carlo evidence and the hydraulic fracturing application, and Section~\ref{sec:conclusion} concludes. The appendices collect proofs, the covariate-adjusted results and DML derivations, and further application results.
\section{Setup}\label{sec:setup}
This section formalizes the two-margin framework. The key step is to reduce the staggered-adoption design to cohort-time comparisons, each pairing one treated cohort with units that remain untreated at the outcome date. Within each comparison, the level margin summarizes treatment effects at realized doses relative to zero treatment, whereas the response margin summarizes the within-cohort slope of outcome changes on dose. The definitions below keep these two sources of variation separate from the beginning.
\subsection{Panel structure and cohort-time comparisons}
We consider a balanced panel observed over periods $1,\ldots,\mathcal T$, where $\mathcal T<\infty$. For one unit, write the observed vector as
\[
(Y_1,\ldots,Y_{\mathcal T},G,D).
\]
Here $G$ denotes the first treatment date, with $G=\infty$ for never-treated units. If $G=g<\infty$, the unit receives a scalar dose $D>0$ that remains fixed from period $g$ onward. For never-treated units, set $D=0$. The treatment-path setup follows CGBS.
Let $Y_s(g,d)$ denote the potential outcome at date $s$ under adoption at date $g$ with dose $d$, and let $Y_s(\infty,0)$ denote the never-treated potential outcome. Let $\mathcal C\subseteq\{(g,t):2\le g\le t\le\mathcal T\}$ index the cohort-time comparisons under consideration. For brevity, call each pair $c=(g,t)\in\mathcal C$ a cell. Define
\[
S_c=\mathbf 1\{G=g\text{ or }G>t\},
\qquad
T_c=\mathbf 1\{G=g\},
\qquad
Y_c=Y_t-Y_{g-1},
\]
and let $D_c=T_cD$. Whenever $P(S_c=1)>0$, let $P_c=P(\,\cdot\mid S_c=1)$ denote the corresponding conditional distribution. Thus $S_c$ selects cohort-$g$ units and units that remain untreated at date $t$, while $T_c$ distinguishes cohort-$g$ units within this sample. In particular, already-treated units never serve as controls.
For this cell, define the untreated outcome change and the date-$t$ treatment effect by
\[
U_c=Y_t(\infty,0)-Y_{g-1}(\infty,0),
\qquad
\tau_c(d)=Y_t(g,d)-Y_t(\infty,0).
\]
Consistency and no anticipation imply
\[
Y_c=U_c+\tau_c(D)\quad\text{when }G=g,
\qquad
Y_c=U_c\quad\text{when }G>t.
\]
Conditional on $S_c=1$, each cohort-time comparison is therefore a two-period continuous-treatment DiD problem under $P_c$.
\subsection{Cohort-time target parameters}
We study two parameters in each cohort-time cell. The first is the average treatment effect at the doses realized by cohort $g$, and the second is the within-cohort linear association between dose and the observed outcome change.
The level margin is
\begin{equation}\label{eq:cohort-level-def}
\tau_{L,g,t}=\mathbb E[\tau_c(D)\mid G=g].
\end{equation}
It is the average, within cohort $g$, of each unit's treatment effect at its realized dose relative to zero treatment at outcome date $t$. Whenever $P(G>t)>0$, its observed-data counterpart is the level contrast
\begin{equation}\label{eq:cohort-level-contrast-def}
\delta_{L,g,t}=\mathbb E[Y_c\mid G=g]-\mathbb E[Y_c\mid G>t],
\end{equation}
the difference in mean outcome changes between cohort $g$ and the units still untreated at date $t$. Section~\ref{sec:generic-assumptions} gives the condition under which the two coincide.
The level margin averages the $ATT(g,t,d\mid g,d)$ parameter of \citet{callaway-goodmanbacon-santanna-2026} over the realized-dose distribution $D\mid G=g$, where $g$, $t$, and $d$ index the timing group, outcome period, and dose group in their notation.
The response margin concerns variation across positive doses within cohort $g$. Let
\[
\mu_{D,g}=\mathbb E[D\mid G=g],
\qquad
V_g=\operatorname{Var}(D\mid G=g).
\]
When $V_g>0$, define the response index
\begin{equation}\label{eq:cohort-response-0-def}
\theta_{R,g,t}
=
\frac{\operatorname{Cov}(D,Y_c\mid G=g)}{V_g}
=
\mathbb E[\alpha_g(D)Y_c\mid G=g],
\qquad
\alpha_g(d)=\frac{d-\mu_{D,g}}{V_g}.
\end{equation}
The response index is the coefficient on $D$ in the population linear projection of $Y_c$ on a constant and $D$ within cohort $g$. Given finite second moments of $D$ and $Y_c$ within cohort $g$, $V_g>0$ is necessary and sufficient for it to be defined. Because the distribution of $D$ within cohort $g$ does not vary with the outcome date, $\mu_{D,g}$, $V_g$, $\alpha_g$, and the response weight introduced below carry no $t$ subscript.
Two naming conventions apply throughout. Objects defined by the observed outcome distribution carry the names "contrast" and "index": the level contrast $\delta_{L,g,t}$ and the response index $\theta_{R,g,t}$ require no causal assumption, and the response index in particular is a projection coefficient that we read as a statistical summary unless the conditions of Section~\ref{sec:why-two} are invoked explicitly. The name "margin" refers to the causal comparison: the level margin $\tau_{L,g,t}$, and the causal component of the response margin identified in Section~\ref{sec:why-two}.
We report the two margins separately throughout and never add them. The level margin is measured in outcome units and the response index in outcome units per unit of dose, so their sum has neither a common scale nor an interpretation as a total effect. The example at the end of the next subsection shows that neither margin determines the other.
\subsection{Generic two-period notation}
To state results that apply to every cohort-time cell, consider a generic two-period problem. Write
\[
O=(Y_1,Y_2,D),
\qquad
Y=Y_2-Y_1,
\qquad
T=\mathbf 1\{D>0\},
\]
\[
U=Y_2(0)-Y_1(0),
\qquad
\tau(d)=Y_2(d)-Y_2(0),
\]
so $Y=U+\tau(D)$ under consistency. Let $p=P(T=1)$, $\rho=P(T=0)=1-p$, and let $Q=P(\,\cdot\mid T=1)$, with expectation $\mathbb E_+[\cdot]$. Define
\[
\tau_L=\mathbb E_+[\tau(D)],
\]
and define the treated dose moments and, whenever $V_D>0$, the response index and representer by
\[
\begin{gathered}
\mu_D=\mathbb E_+[D],
\qquad
V_D=\operatorname{Var}_+(D),\\
\theta_R=\frac{\operatorname{Cov}_+(D,Y)}{V_D}
=\mathbb E_+[\alpha(D)Y],
\qquad
\alpha(d)=\frac{d-\mu_D}{V_D}.
\end{gathered}
\]
Under $P_c$, this notation applies cellwise by setting $(Y,T,D)=(Y_c,T_c,D_c)$. For example, if $D\in\{1,2\}$, a constant effect $\tau(d)=a$ gives $\tau_L=a$ and $\theta_R=0$, whereas a linear effect $\tau(d)=bd$ gives $\tau_L=b\mu_D$ and $\theta_R=b$.
\section{Identification}\label{sec:generic-assumptions}
This section states one parallel-trends restriction for each margin and shows what each identifies. Because the level margin compares treated with untreated units, level parallel trends (PT-L) equates their mean untreated outcome changes and identifies the level margin. Because the response margin uses variation in positive dose among treated units, response parallel trends (PT-R) requires the mean untreated outcome change to be invariant to dose within the treated group and removes dose-related untreated trends from the response index. Neither restriction implies the other, and the last subsection transfers both to cohort-time cells.
\subsection{Assumptions}
\begin{assumption}[Positivity and moments]\label{ass:sampling}
(i) $p>0$; (ii) $p<1$; (iii) $\mathbb E[Y^2+D^2]<\infty$.
\end{assumption}
\begin{assumption}[Positive treated-dose variation]\label{ass:variation}
$V_D>0$.
\end{assumption}
Part (i) of Assumption~\ref{ass:sampling} is needed whenever expectations under $Q$ enter, part (ii) whenever a control group enters, and Assumption~\ref{ass:variation} whenever the response index enters. Results that use only positive-dose variation do not invoke part (ii). Under part (i), part (iii) implies $\mathbb E_+[Y^2+D^2]<\infty$, so the treated-only projections and covariances below are well defined, and Assumption~\ref{ass:variation} then gives $0<V_D<\infty$.
\begin{assumption}[Consistency, no anticipation, and potential-outcome moments]\label{ass:consistency}
(i) $Y_1=Y_1(0)$ and $Y_2=Y_2(D)$; (ii) $\mathbb E[U^2+\tau(D)^2]<\infty$.
\end{assumption}
Part (i) combines consistency at the outcome date with no anticipation at the baseline date. Part (ii) is a potential-outcome moment condition for the causal covariance decompositions. It is not implied by Assumption~\ref{ass:sampling}(iii), because $U$ and $\tau(D)$ may offset in the observed change. Conversely, under part (i), part (ii) implies the $Y^2$ component of Assumption~\ref{ass:sampling}(iii), and the latter remains useful because it also controls $D^2$ and supports results stated entirely in terms of the observed distribution.
\begin{assumption}[Level parallel trends, PT-L]\label{ass:levelpt}
\[
\mathbb E[U\mid T=1]=\mathbb E[U\mid T=0].
\]
\end{assumption}
\begin{assumption}[Response parallel trends, PT-R]\label{ass:dosept}
\[
\mathbb E[U\mid D,T=1]=\mathbb E[U\mid T=1]
\quad\text{almost surely under }Q.
\]
\end{assumption}
PT-L is a between-group restriction, whereas PT-R is a within-treated restriction across positive doses.
\begin{proposition}[PT-L and PT-R are non-nested]\label{prop:pt-nonnested}
Neither Assumption~\ref{ass:levelpt} nor Assumption~\ref{ass:dosept} implies the other. If both hold, then
\begin{equation}\label{eq:all-dose-pt}
\mathbb E[U\mid D,T=1]=\mathbb E[U\mid T=0]
\quad\text{almost surely under }Q.
\end{equation}
\end{proposition}
In the generic two-period notation, equation~\eqref{eq:all-dose-pt} is the dose-specific parallel-trends condition in \citet{callaway-goodmanbacon-santanna-2026}. Their condition implies PT-R directly and PT-L after averaging over the positive-dose distribution, and Proposition~\ref{prop:pt-nonnested} gives the converse. Their aggregate parallel-trends condition coincides with PT-L in this two-period problem. The gain is therefore not a weaker parallel-trends assumption but a separate one for each margin.
\subsection{Identification of the two margins}
\begin{theorem}[Identification of the two margins]\label{thm:two-margins}
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item Suppose Assumptions~\ref{ass:sampling}(i), (ii) and~\ref{ass:consistency} hold. Under Assumption~\ref{ass:levelpt},
\[
\tau_L=\mathbb E[Y\mid T=1]-\mathbb E[Y\mid T=0].
\]
\item Under Assumptions~\ref{ass:sampling}(i), (iii) and~\ref{ass:variation},
\begin{equation}\label{eq:baseline-balancing}
\mathbb E_+[\alpha(D)]=0,
\qquad
\mathbb E_+[\alpha(D)D]=1,
\end{equation}
and $\theta_R$ is the unique population OLS slope of $Y$ on a constant and $D$ under $Q$.
\item Suppose Assumptions~\ref{ass:sampling}(i), (iii), \ref{ass:variation}, and~\ref{ass:consistency} hold. Under Assumption~\ref{ass:dosept},
\begin{equation}\label{eq:response-gain-id}
\theta_R
=
\frac{\operatorname{Cov}_+\{D,\tau(D)\}}{V_D}.
\end{equation}
\end{enumerate}
\end{theorem}
Part (ii) is a statement about the projection alone. PT-R removes the covariance between dose and untreated trends in part (iii), but selection on gains may remain. Since
\[
\theta_R
=
\frac{\operatorname{Cov}_+(D,U)}{V_D}
+
\frac{\operatorname{Cov}_+\{D,\tau(D)\}}{V_D},
\]
the condition $\operatorname{Cov}_+(D,U)=0$ is necessary and sufficient for \eqref{eq:response-gain-id} when $V_D>0$. We state PT-R because it is the within-cohort parallel-trends condition and, with PT-L, reproduces the dose-specific condition of CGBS. Section~\ref{sec:why-two} gives the additional conditions under which the response index admits a derivative interpretation.
\subsection{Cellwise assumptions and identification}
For $c=(g,t)\in\mathcal C$, extend the cell treatment effect to zero dose by setting $\widetilde\tau_c(0)=0$ and $\widetilde\tau_c(d)=\tau_c(d)$ for $d>0$. Under $P_c$, the generic two-period notation applies with
\[
(Y,T,D,U,\tau(\cdot))
=
(Y_c,T_c,D_c,U_c,\widetilde\tau_c(\cdot)).
\]
\begin{assumption}[Panel consistency and no anticipation]\label{ass:staggered}
Observed outcomes satisfy $Y_s=Y_s(G,D)$ when $G<\infty$ and $Y_s=Y_s(\infty,0)$ when $G=\infty$. For every feasible adoption date $h<\infty$, dose $d>0$, and period $s<h$, $Y_s(h,d)=Y_s(\infty,0)$.
\end{assumption}
Whenever $P(S_c=1)>0$, Assumption~\ref{ass:staggered} gives $Y_c=U_c+\widetilde\tau_c(D_c)$ under $P_c$, the cellwise counterpart of Assumption~\ref{ass:consistency}(i). The distribution $Q$ becomes $P(\,\cdot\mid G=g)$, the positivity conditions become $P(G=g)>0$ and $P(G>t)>0$, and $V_D$ becomes $V_g$. The two margin-specific restrictions become
\[
\mathrm{PT\text{-}L}_{g,t}:
\quad
\mathbb E[U_c\mid G=g]=\mathbb E[U_c\mid G>t],
\]
and
\[
\mathrm{PT\text{-}R}_{g,t}:
\quad
\mathbb E[U_c\mid D,G=g]=\mathbb E[U_c\mid G=g]
\quad\text{almost surely}.
\]
Thus the generic assumptions are a template for the cell conditions.
\begin{theorem}[Cohort-time identification]\label{thm:staggered}
Fix $c=(g,t)\in\mathcal C$ and suppose Assumption~\ref{ass:staggered} holds.
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item Suppose $P(G=g)>0$, $P(G>t)>0$,
\[
\mathbb E[U_c^2+\tau_c(D)^2\mid G=g]<\infty,
\qquad
\mathbb E[U_c^2\mid G>t]<\infty,
\]
and $\mathrm{PT\text{-}L}_{g,t}$ holds. Then
\[
\tau_{L,g,t}
=
\delta_{L,g,t}
=
\mathbb E[Y_c\mid G=g]-\mathbb E[Y_c\mid G>t].
\]
\item Suppose $P(G=g)>0$,
\[
\mathbb E[D^2+U_c^2+\tau_c(D)^2\mid G=g]<\infty,
\]
$V_g>0$, and $\mathrm{PT\text{-}R}_{g,t}$ holds. Then
\begin{equation}\label{eq:cohort-response-gain-id}
\theta_{R,g,t}
=
\frac{\operatorname{Cov}\{D,\tau_c(D)\mid G=g\}}{V_g}.
\end{equation}
\end{enumerate}
\end{theorem}
Figure~\ref{fig:pt-roles} in Appendix~\ref{app:illustrations} illustrates the separate roles of the two restrictions in a stylized cohort-time cell. In Panel~A, PT-L identifies the level margin, while failure of PT-R leaves a covariance between dose and untreated trends in the response index. In Panel~B, PT-R removes that covariance, while failure of PT-L prevents the level contrast from identifying the level margin.
\section{Why separate margins}\label{sec:why-two}
The level contrast and the response index summarize different comparisons. A regression using both zero and positive doses combines them in a single coefficient. This section derives the mixture and then examines the causal interpretation of the response index.
\subsection{The continuous-dose regression as a mixture}
Let
\[
\Delta_Y=\mathbb E[Y\mid T=1]-\mathbb E[Y\mid T=0],
\qquad
\beta_{\mathrm{TWFE}}=\frac{\operatorname{Cov}(D,Y)}{\operatorname{Var}(D)}.
\]
The latter is the OLS slope of the long-difference outcome on a constant and $D$. Equivalently, it is the coefficient on $D_i\mathbf 1\{s=2\}$ in the two-period panel regression with unit and time effects.
\begin{theorem}[Two-period TWFE mixture]\label{thm:mixture}
Suppose Assumptions~\ref{ass:sampling} and~\ref{ass:variation} hold and $D=0$ when $T=0$. Then
\begin{equation}\label{eq:share-formula}
\beta_{\mathrm{TWFE}}
=
\frac{V_D\theta_R+\rho\mu_D\Delta_Y}
{V_D+\rho\mu_D^2}
=
\lambda\theta_R
+(1-\lambda)\frac{\Delta_Y}{\mu_D},
\qquad
\lambda=\frac{V_D}{V_D+\rho\mu_D^2}.
\end{equation}
Under Assumptions~\ref{ass:consistency} and \ref{ass:levelpt}, $\Delta_Y=\tau_L$, and hence
\[
\beta_{\mathrm{TWFE}}
=
\lambda\theta_R
+(1-\lambda)\frac{\tau_L}{\mu_D}.
\]
Holding the conditional distributions of $(D,Y)$ given $T=1$ and of $Y$ given $T=0$ fixed while varying $\rho$ gives
\begin{equation}\label{eq:share-derivative}
\frac{\partial\beta_{\mathrm{TWFE}}}{\partial\rho}
=
\frac{\mu_DV_D(\Delta_Y-\mu_D\theta_R)}
{(V_D+\rho\mu_D^2)^2}.
\end{equation}
\end{theorem}
Equation~\eqref{eq:share-formula} makes the distinction explicit. The first component is the within-treated response index, and the second is the treated-control contrast $\Delta_Y$ per unit of mean treated dose. Dividing by $\mu_D$ puts both terms in units of outcome per dose. The mixture weight depends on the dispersion and mean of treated doses and on the untreated share. Holding the group-specific distributions fixed, an increase in the untreated share shifts weight from the response index toward the level contrast per unit of mean treated dose. As Equation~\eqref{eq:share-derivative} shows, the OLS coefficient can therefore change solely because the relative sizes of the two groups change, even though both components remain fixed. The coefficient is invariant to the untreated share only when $\Delta_Y/\mu_D=\theta_R$.
\begin{corollary}[Cellwise two-period TWFE mixture]\label{cor:share}
Fix $c=(g,t)$. Suppose $P(G=g)>0$, $P(G>t)>0$,
\[
\mathbb E[Y_c^2+D_c^2\mid S_c=1]<\infty,
\qquad
V_g>0.
\]
Let
\begin{equation}\label{eq:cell-twfe-components}
\rho_{g,t}=P_c(G>t)
=\frac{P(G>t)}{P(G=g)+P(G>t)},
\end{equation}
and let $\delta_{L,g,t}$ be the level contrast in \eqref{eq:cohort-level-contrast-def}. The OLS slope of $Y_c$ on a constant and $D_c$ under $P_c$, equivalently the two-period TWFE coefficient for the comparison between cohort $g$ and units not yet treated, is
\begin{equation}\label{eq:cell-twfe-mixture}
\beta_{\mathrm{TWFE},g,t}
=
\lambda_{g,t}\theta_{R,g,t}
+(1-\lambda_{g,t})\frac{\delta_{L,g,t}}{\mu_{D,g}},
\qquad
\lambda_{g,t}
=
\frac{V_g}{V_g+\rho_{g,t}\mu_{D,g}^2}.
\end{equation}
Under Assumption~\ref{ass:staggered} and the conditions of Theorem~\ref{thm:staggered}(i), $\delta_{L,g,t}=\tau_{L,g,t}$.
\end{corollary}
Theorem~\ref{thm:mixture} applies to a generic two-period regression, and Corollary~\ref{cor:share} applies it to a valid cohort-time comparison. As shown for binary treatment by \citet{goodmanbacon-2021}, pooled staggered TWFE can use already-treated cohorts as controls, so changes in their treatment effects enter the comparison. These comparisons are excluded from our cells comparing a treated cohort with units not yet treated, so characterizing the pooled continuous-dose coefficient would require a separate decomposition.
Figure~\ref{fig:two-margins-cell} in Appendix~\ref{app:illustrations} summarizes the two margins in outcome-level and long-difference views. Panel~A illustrates their distinct sources of variation and units. Panel~B visualizes the mixture in Corollary~\ref{cor:share}. The OLS slope using both treated and control units lies between the within-cohort dose slope and the level contrast divided by the mean treated dose.
\subsection{Causal interpretation of the response margin}
PT-R expresses the response index as a covariance between dose and treatment effects, but that covariance may reflect both causal responses and selection on gains. The representation below combines the OLS derivative-weight identity of \citet{yitzhaki-1996} with the decomposition in \citet{callaway-goodmanbacon-santanna-2026} of the realized-dose effect derivative into a causal response and a selection term. The mapping is made explicit after the theorem.
Define the response weight
\[
W(d)
=
-\mathbb E_+[\alpha(D)\mathbf 1\{D\le d\}].
\]
\begin{lemma}[Nonnegative response weight]\label{lem:weight-nonnegative}
Under Assumptions~\ref{ass:sampling}(i), (iii) and~\ref{ass:variation},
\[
W(d)\ge0\quad\text{for every }d.
\]
\end{lemma}
To obtain a derivative representation, impose the following support, smoothness, and compatibility conditions.
\begin{samepage}
\begin{assumption}[Connected support, smoothness, and compatibility]\label{ass:connected}
(i) The support of $D\mid T=1$ is a compact interval $[a,b]$, $a<b$; (ii) the effect surface
\[
\tau(u\mid d)=\mathbb E[\tau(u)\mid D=d,T=1]
\]
has a version that is continuously differentiable on an open neighborhood of the diagonal $\{(d,d):d\in[a,b]\}$; (iii) the chosen version, with $q(d)=\tau(d\mid d)$, satisfies
\begin{equation}\label{eq:diagonal-compatibility}
q(D)=\mathbb E[\tau(D)\mid D,T=1]
\qquad Q\text{-almost surely}.
\end{equation}
\end{assumption}
\end{samepage}
Define
\begin{equation}\label{eq:acrt-selection}
ACRT(d\mid d)
=
\left.\partial_u\tau(u\mid d)\right|_{u=d},
\qquad
S(d)
=
\left.\partial_{d'}\tau(d\mid d')\right|_{d'=d}.
\end{equation}
Here $ACRT(d\mid d)$ is the derivative of the conditional average treatment effect with respect to the assigned dose, evaluated at $u=d$, while $S(d)$ measures how average treatment effects vary across realized-dose groups. The diagonal $q(d)=\tau(d\mid d)$ of the effect surface is continuously differentiable on $[a,b]$ under Assumption~\ref{ass:connected}, with $q'(d)=ACRT(d\mid d)+S(d)$ by the chain rule.
\begin{theorem}[Response weights and selection]\label{thm:response-decomp}
Suppose Assumptions~\ref{ass:sampling}(i), (iii), \ref{ass:variation}, \ref{ass:consistency}, \ref{ass:dosept}, and \ref{ass:connected} hold. Then
\begin{equation}\label{eq:W-normalized}
W(d)\ge0,
\qquad
\int_a^bW(d)\,\mathrm d d=1,
\end{equation}
\begin{equation}\label{eq:qprime-representation}
\theta_R=\int_a^bW(d)q'(d)\,\mathrm d d,
\end{equation}
and
\begin{equation}\label{eq:selection-decomp}
\theta_R
=
\theta_R^{\mathrm{causal}}
+B_R^{\mathrm{selection}},
\end{equation}
where
\[
\theta_R^{\mathrm{causal}}
=
\int_a^bW(d)ACRT(d\mid d)\,\mathrm d d,
\]
\begin{equation}\label{eq:selection-part}
B_R^{\mathrm{selection}}
=
\int_a^bW(d)S(d)\,\mathrm d d.
\end{equation}
If $S(d)=0$ almost everywhere, then $\theta_R=\theta_R^{\mathrm{causal}}$. If $S(d)\ge0$ almost everywhere, then $\theta_R^{\mathrm{causal}}\le\theta_R$, with the inequality reversed when $S(d)\le0$. Finally, if $|S(d)|\le\bar s(d)$,
\begin{equation}\label{eq:sensitivity-bound}
|\theta_R-\theta_R^{\mathrm{causal}}|
\le
\int_a^bW(d)\bar s(d)\,\mathrm d d.
\end{equation}
\end{theorem}
For an absolutely continuous dose, $W$ is the nonnegative OLS derivative weight of \citet{yitzhaki-1996}. Once PT-R isolates $q$, the pointwise identity $q'(d)=ACRT(d\mid d)+S(d)$ is the cohort-specific counterpart of the decomposition in \citet{callaway-goodmanbacon-santanna-2026}, and Theorem~\ref{thm:response-decomp} integrates that identity against $W$.
The restriction $S(d)=0$ is a local no-selection condition that makes the response index a weighted average of causal derivatives without requiring treatment effects to coincide across all realized-dose groups. Equation~\eqref{eq:sensitivity-bound} allows departures from this condition to be expressed in economically interpretable units. A useful starting point is a constant envelope, $\bar s(d)=s_0$, where $s_0$ is measured in outcome units per unit of dose. It bounds the local variation in average treatment gains across neighboring realized-dose groups, holding the assigned dose fixed. For example, participants choosing more intensive training may have higher returns even when assigned the same training intensity.
Since the response weights integrate to one, the constant envelope implies $\theta_R^{\mathrm{causal}}\in[\theta_R-s_0,\theta_R+s_0]$. If $[L_R,U_R]$ is an asymptotically valid confidence interval for $\theta_R$, then $[L_R-s_0,U_R+s_0]$ is a conservative asymptotic confidence interval for $\theta_R^{\mathrm{causal}}$ under the selection bound and the assumptions of Theorem~\ref{thm:response-decomp}, and the criterion $L_R-s_0>r_{\min}$ accounts for both sampling uncertainty and the allowed selection when assessing whether the average causal response exceeds a prespecified threshold $r_{\min}$. Reporting this criterion over several values of $s_0$ shows how much selection on gains the conclusion tolerates, as \citet{rambachan-roth-2023} recommend for violations of parallel trends in binary-treatment DiD, and the additive bound itself parallels the bounded differential trends of \citet{manski-pepper-2018}. Here PT-R is maintained and $s_0$ bounds selection on gains across realized-dose groups rather than untreated trends, so pre-trend tests do not bound it without an additional assumption linking the two.
\begin{remark}[Relation to CGBS]\label{rem:cgbs-weights}
\citet[Theorem~3.4(a)]{callaway-goodmanbacon-santanna-2026} decompose $\beta_{\mathrm{TWFE}}$ in Theorem~\ref{thm:mixture} under their dose-specific parallel-trends condition~\eqref{eq:all-dose-pt} and an absolutely continuous treated dose with support $(a,b)$. With $q'(d)=ACRT(d\mid d)+S(d)$ as in Theorem~\ref{thm:response-decomp}, their decomposition is
\[
\begin{gathered}
\beta_{\mathrm{TWFE}}
=
\int_a^b w_1(d)\,q'(d)\,\mathrm d d
+w_0\,\frac{q(a)}{a},\\
w_1(d)=\frac{\mathbb E[(D-\mathbb E[D])\mathbf 1\{D\ge d\}]}{\operatorname{Var}(D)},
\qquad
w_0=\frac{(\mu_D-\mathbb E[D])\,p\,a}{\operatorname{Var}(D)},
\end{gathered}
\]
where $q(a)$ is the average effect of the lowest dose for the lowest-dose group. A direct calculation in Appendix~\ref{app:mainproofs} expresses their weights in the notation of Theorem~\ref{thm:mixture}:
\begin{equation}\label{eq:cgbs-weight-identity}
w_1(d)
=
\lambda W(d)
+(1-\lambda)\frac{Q(D>d)}{\mu_D}
\quad\text{for }d\in(a,b),
\qquad
w_0=(1-\lambda)\frac{a}{\mu_D}.
\end{equation}
Their causal-response weight at each dose is therefore a mixture of the within-treated response weight $W$ and a level component proportional to the treated-dose survival function. Under the hypotheses of Theorem~\ref{thm:response-decomp} and PT-L, integrating the second component against $q'$ gives $(1-\lambda)\{\tau_L-q(a)\}/\mu_D$, and the lowest-dose term contributes $(1-\lambda)q(a)/\mu_D$, and their sum is the second term of \eqref{eq:share-formula}. The level margin thus enters their decomposition through both terms, whereas \eqref{eq:share-formula} collects it in one term and leaves the response term with weights that do not depend on the untreated share. CGBS observe that $w_1$ depends on the size of the untreated group. In \eqref{eq:cgbs-weight-identity} that dependence runs only through $\lambda$, and $\rho\to0$ gives $\lambda\to1$ and $\beta_{\mathrm{TWFE}}\to\theta_R$. Their Remark~3.2 on designs without untreated units is the case $\lambda=1$, in which the coefficient equals $\theta_R$ and their decomposition reduces to Theorem~\ref{thm:response-decomp}, proved there under Assumption~\ref{ass:connected} without a dose density.
\end{remark}
\subsection{Balancing and local reweighting}
The response index is the linear functional $\theta_R=\mathbb E_+[\alpha(D)Y]$ of the treated outcome distribution. This subsection gives two characterizations of the representer $\alpha$ that use neither the derivative representation nor Assumption~\ref{ass:connected}.
The first concerns the two restrictions in \eqref{eq:baseline-balancing}. Define
\begin{equation}\label{eq:balancing-class-main}
\mathcal A_R
=
\left\{
a\in L_2(Q_D):
\mathbb E_+[a(D)]=0,
\quad
\mathbb E_+[a(D)D]=1
\right\}.
\end{equation}
For $a\in\mathcal A_R$, the functional $\mathbb E_+[a(D)Y]$ is invariant to adding a constant to $Y$ and equals one when $Y=D$. By Theorem~\ref{thm:two-margins}(ii), $\alpha\in\mathcal A_R$.
\begin{proposition}[Minimum-norm balancing weight]\label{prop:minnorm}
Under Assumptions~\ref{ass:sampling}(i), (iii) and~\ref{ass:variation}, $\alpha$ is the unique solution to
\[
\min_{a\in\mathcal A_R}\mathbb E_+[a(D)^2],
\]
and
\[
\min_{a\in\mathcal A_R}\mathbb E_+[a(D)^2]=\frac1{V_D}.
\]
If $Y=m(D)+\varepsilon$ under $Q$, with $\mathbb E_+[\varepsilon\mid D]=0$ and $\mathbb E_+[\varepsilon^2\mid D]=\sigma^2$ for some $0<\sigma^2<\infty$, the same representer uniquely minimizes $\operatorname{Var}_+\{a(D)\varepsilon\}$ over $\mathcal A_R$.
\end{proposition}
The least-squares estimands of \citet{hines-diazordaz-vansteelandt-2026} impose the same two restrictions. Appendix~\ref{app:response-characterizations} compares the two criteria.
The second characterization reweights the treated-dose distribution. Let $\mu(d)=\mathbb E_+[Y\mid D=d]$. For $\varepsilon$ near zero, define
\begin{equation}\label{eq:tilt-law}
\frac{\,\mathrm dQ_\varepsilon}{\,\mathrm dQ_D}(d)
=
\frac{\exp(\varepsilon d)}{M(\varepsilon)},
\qquad
M(\varepsilon)=\mathbb E_+[\exp(\varepsilon D)],
\end{equation}
and
\[
\mathcal Y(\varepsilon)=\int\mu(d)Q_\varepsilon(\,\mathrm d d),
\qquad
\mathcal D(\varepsilon)=\int d\,Q_\varepsilon(\,\mathrm d d),
\]
the mean outcome and mean dose under $Q_\varepsilon$ when the conditional distribution of $Y$ given $D$ is held fixed. A positive $\varepsilon$ shifts mass toward higher observed doses.
\begin{proposition}[Local tilt interpretation]\label{prop:tilt}
Suppose Assumptions~\ref{ass:sampling}(i), (iii) and~\ref{ass:variation} hold and $\mathbb E_+[\exp(\varepsilon_0|D|)]<\infty$ for some $\varepsilon_0>0$. Then $\mathcal Y$ and $\mathcal D$ are differentiable at $\varepsilon=0$, with
\[
\mathcal Y'(0)=\operatorname{Cov}_+(D,Y),
\qquad
\mathcal D'(0)=V_D,
\]
so
\begin{equation}\label{eq:tilt-interpretation}
\theta_R
=
\frac{\mathcal Y'(0)}{\mathcal D'(0)}
=
\left.\frac{\,\mathrm d\mathcal Y}{\,\mathrm d\mathcal D}\right|_{\varepsilon=0}.
\end{equation}
\end{proposition}
Along this path, $\theta_R$ is the derivative of the mean outcome with respect to the mean dose at the observed treated-dose distribution. The path changes the weight on units at each realized dose and does not change any unit's dose. Appendix~\ref{app:response-characterizations} contains the proofs, the comparison with the stochastic-policy tilts of \citet{jetsupphasuk-et-al-2025}, and the comparison with the global average derivative of \citet{callaway-goodmanbacon-santanna-2026}.
\section{Cohort-specific and event-time parameters}\label{sec:paths}
The preceding representations apply separately to every cohort-time cell. For a fixed cohort, however, the dose distribution is common across outcome dates.
For cohort $g$, define
\[
W_g(d)
=
-\mathbb E[\alpha_g(D)\mathbf 1\{D\le d\}\mid G=g].
\]
When a derivative interpretation is invoked, let the support of $D\mid G=g$ be $[a_g,b_g]$, define
\[
\tau_{g,t}(u\mid d)
=
\mathbb E[Y_t(g,u)-Y_t(\infty,0)\mid D=d,G=g],
\qquad
q_{g,t}(d)=\tau_{g,t}(d\mid d),
\]
and define $ACRT_{g,t}(d\mid d)$ and $S_{g,t}(d)$ as in \eqref{eq:acrt-selection}.
For the cohort tilt, let $\mu_{g,t}(d)=\mathbb E[Y_c\mid D=d,G=g]$ and define
\begin{equation*}
\frac{\,\mathrm d P_{g,\varepsilon}}{\,\mathrm d P(\,\cdot\mid G=g)}(d)
=
\frac{\exp(\varepsilon d)}
{\mathbb E[\exp(\varepsilon D)\mid G=g]}.
\end{equation*}
The induced outcome and dose paths are
\begin{equation}\label{eq:cohort-tilt-paths-0}
\mathcal Y_{g,t}(\varepsilon)
=
\int\mu_{g,t}(d)P_{g,\varepsilon}(\,\mathrm d d),
\qquad
\mathcal D_g(\varepsilon)
=
\int d\,P_{g,\varepsilon}(\,\mathrm d d).
\end{equation}
\begin{corollary}[Cohort balancing, tilt, and response weights]\label{cor:cohort-interpretations}
For a fixed cohort $g$ with $V_g>0$:
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item $\alpha_g$ uniquely minimizes
\[
\mathbb E[\alpha(D)^2\mid G=g]
\]
over square-integrable $\alpha$ satisfying $\mathbb E[\alpha(D)\mid G=g]=0$ and $\mathbb E[\alpha(D)D\mid G=g]=1$, with minimum $1/V_g$.
\item If $\mathbb E[Y_c^2\mid G=g]<\infty$ and $\mathbb E[\exp(\varepsilon_0|D|)\mid G=g]<\infty$ for some $\varepsilon_0>0$, then the paths in \eqref{eq:cohort-tilt-paths-0} are differentiable at zero and
\[
\theta_{R,g,t}
=
\frac{\mathcal Y_{g,t}'(0)}{\mathcal D_g'(0)}.
\]
\item $\alpha_g$ and $W_g$ are common to every $t\ge g$, $W_g(d)\ge0$, and, under connected support,
\[
\int_{a_g}^{b_g}W_g(d)\,\mathrm d d=1.
\]
If the conditions of Theorem~\ref{thm:staggered}(ii) hold and Assumption~\ref{ass:connected} holds for the cohort distribution $P(\,\cdot\mid G=g)$, with support $[a_g,b_g]$ and effect surface $\tau_{g,t}(\cdot\mid\cdot)$, then
\[
\theta_{R,g,t}
=
\int_{a_g}^{b_g}W_g(d)
\{ACRT_{g,t}(d\mid d)+S_{g,t}(d)\}\,\mathrm d d.
\]
The sign and sensitivity conclusions in Theorem~\ref{thm:response-decomp} apply cellwise.
\end{enumerate}
\end{corollary}
Because $W_g$ depends only on $D\mid G=g$, changes over $t$ in a cohort's response path reflect changes in the relation between outcome and dose under a fixed weighting rule.
For event time $e=t-g$, choose a nonempty cohort set $\mathcal G_e$, with $(g,g+e)\in\mathcal C$ for every included cohort, together with a weighting rule. Let $w_{g,e}$ denote the resulting population weights, which sum to one over $g\in\mathcal G_e$. The weights may be known constants or regular functionals of the treatment-timing distribution. Define
\begin{equation*}
\Theta_{L,e}
=
\sum_{g\in\mathcal G_e}w_{g,e}\tau_{L,g,g+e},
\qquad
\Theta_{R,e}
=
\sum_{g\in\mathcal G_e}w_{g,e}\theta_{R,g,g+e}.
\end{equation*}
Only the cohort set and the weighting rule need to be fixed in advance, because the population weights are part of the target. A common cohort set across event times holds cohort composition fixed, so the aggregates are comparable over $e$.
\section{Estimation and stacked inference}\label{sec:baseline-estimation}
These targets are functions of cohort means, dose moments, and within-cohort covariances, so they admit closed-form estimators. Stacking their unit-level influence functions then delivers joint inference across margins and cohort-time cells while allowing arbitrary serial dependence within a unit.\footnote{Stacking here refers to the joint treatment of unit-level influence functions across cells and margins for inference. It is unrelated to the stacked-regression estimator of \citet{cengiz-et-al-2019}, which appends cohort-specific event-study samples and estimates a single regression.}
\subsection{Direct estimators}
For each fixed cell $c=(g,t)$, let $n_g=\sum_i\mathbf 1\{G_i=g\}$ and $n_{0,g,t}=\sum_i\mathbf 1\{G_i>t\}$. Define the relevant sample means by
\[
\overline Y_{c,g}=\frac1{n_g}\sum_{i:G_i=g}Y_{c,i},
\qquad
\overline Y_{c,0}=\frac1{n_{0,g,t}}\sum_{i:G_i>t}Y_{c,i},
\qquad
\overline D_g=\frac1{n_g}\sum_{i:G_i=g}D_i.
\]
The estimators target the level contrast $\delta_{L,g,t}$ in \eqref{eq:cohort-level-contrast-def} and the response index $\theta_{R,g,t}$ in \eqref{eq:cohort-response-0-def}; under the conditions of Theorem~\ref{thm:staggered}(i) the former equals the level margin. The estimators are
\[
\widehat\delta_{L,g,t}
=
\overline Y_{c,g}-\overline Y_{c,0},
\]
and
\[
\widehat\theta_{R,g,t}
=
\frac{
\sum_{i:G_i=g}(D_i-\overline D_g)
(Y_{c,i}-\overline Y_{c,g})
}{
\sum_{i:G_i=g}(D_i-\overline D_g)^2
}.
\]
The response estimator is the cohort-$g$ OLS slope with an intercept. Compute
\[
\widehat V_g
=
\frac1{n_g}\sum_{i:G_i=g}(D_i-\overline D_g)^2
\]
once per cohort and reuse it for every $t$.
\subsection{Panel-level influence functions}
Let
\[
q_g=P(G=g),
\qquad
q_{0,g,t}=P(G>t),
\]
\[
\mu_{1,g,t}=\mathbb E[Y_c\mid G=g],
\qquad
\mu_{0,g,t}=\mathbb E[Y_c\mid G>t].
\]
The corresponding influence functions for the full panel population are
\begin{equation}\label{eq:phiL-0}
\phi_{L,g,t}^{P}(O)
=
\frac{\mathbf 1\{G=g\}}{q_g}(Y_c-\mu_{1,g,t})
-
\frac{\mathbf 1\{G>t\}}{q_{0,g,t}}(Y_c-\mu_{0,g,t}),
\end{equation}
and
\begin{equation}\label{eq:phiR-0}
\phi_{R,g,t}^{P}(O)
=
\frac{\mathbf 1\{G=g\}}{q_gV_g}
\left[
(D-\mu_{D,g})(Y_c-\mu_{1,g,t})
-\theta_{R,g,t}(D-\mu_{D,g})^2
\right].
\end{equation}
\begin{assumption}[Stacked-inference conditions]\label{ass:baseline-inference}
Full-path observations are i.i.d. across units. The cell collection $\mathcal C$ is finite, nonrandom, and not selected from the analysis sample. For every $c=(g,t)\in\mathcal C$, $q_g>0$, $q_{0,g,t}>0$, $V_g>0$,
\[
\mathbb E[|Y_c|^4+|D|^4\mid G=g]<\infty,
\qquad
\mathbb E[Y_c^2\mid G>t]<\infty.
\]
\end{assumption}
Let $\boldsymbol\vartheta$ stack $(\delta_{L,g,t},\theta_{R,g,t})$ in a fixed cell order, and let $\boldsymbol\phi_i^{P}$ stack the corresponding influence function columns on unit $i$. Plug-in columns replace the probabilities, means, variances, and slopes by their sample analogues and are zero outside the relevant groups. Every column is evaluated for all $n$ units in the same order, which preserves the cross-cell and cross-margin covariances.
\begin{theorem}[Stacked inference]\label{thm:baseline-stacked}
Under Assumption~\ref{ass:baseline-inference},
\begin{equation}\label{eq:baseline-vector-clt}
\sqrt n
(\widehat{\boldsymbol\vartheta}
-\boldsymbol\vartheta)
=
\frac1{\sqrt n}\sum_{i=1}^n
\boldsymbol\phi_i^{P}
+o_{\mathbb P}(1)
\xrightarrow{d}
N(0,\Sigma),
\end{equation}
where $\Sigma=\mathbb E[\boldsymbol\phi^{P} \boldsymbol\phi^{P\prime}]$. Moreover,
\begin{equation}\label{eq:baseline-sigma}
\widehat\Sigma
=
\mathbb P_n[
\widehat{\boldsymbol\phi}^{P}_i
\widehat{\boldsymbol\phi}^{P\prime}_i]
\xrightarrow{p}\Sigma.
\end{equation}
\end{theorem}
The covariance matrix of the estimator is estimated by $\widehat\Sigma/n$. Throughout the paper, Wald and max-$t$ statistics presuppose nonsingular limiting covariances for the coordinates they use.
The influence function stack can be augmented to test the cellwise restriction under which the regression coefficient in Corollary~\ref{cor:share} is invariant to the untreated share. The influence function of the cohort dose mean is
\begin{equation*}
\phi_{\mu,g}^{P}(O)
=
\frac{\mathbf 1\{G=g\}}{q_g}(D-\mu_{D,g}).
\end{equation*}
The delta method therefore gives the influence function of
$\delta_{L,g,t}-\mu_{D,g}\theta_{R,g,t}$ as
\begin{equation}\label{eq:phi-proportionality}
\phi_{L,g,t}^{P}
-\mu_{D,g}\phi_{R,g,t}^{P}
-\theta_{R,g,t}\phi_{\mu,g}^{P}.
\end{equation}
Thus a pre-specified finite collection of restrictions
$\delta_{L,g,t}=\mu_{D,g}\theta_{R,g,t}$ can be tested with the corresponding plug-in covariance matrix. Under the conditions of Theorem~\ref{thm:staggered}(i), the same restriction becomes $\tau_{L,g,t}=\mu_{D,g}\theta_{R,g,t}$.
The next result separates prespecification of an aggregate from knowledge of its numerical weights. It applies to either the stack in Theorem~\ref{thm:baseline-stacked} or any other fixed finite cell stack whose estimators have a joint asymptotic linear representation under the full-panel distribution $P$. Let $\mathcal J$ index the scalar coordinates of such a stack, with parameters $\vartheta_j$, estimators $\widehat\vartheta_j$, and full-panel influence functions $\phi_j^P$, stacked as $\boldsymbol\phi^P$ in the fixed cell-coordinate order. Let $r=1,\ldots,K_A$ index a finite family of aggregates, each defined by a coordinate set $\mathcal J_r\subseteq\mathcal J$ and a treatment-timing weighting rule with population weights $w_{rj}$, $\sum_{j\in\mathcal J_r}w_{rj}=1$, and sample weights $\widehat w_{rj}$ with influence columns $\xi^w_{rj}$, stacked as $\boldsymbol\xi^w$ in the fixed aggregate-coordinate order. We report only within-margin aggregates, so each $\mathcal J_r$ contains coordinates from one margin. Define
\[
\eta_r
=
\sum_{j\in\mathcal J_r}w_{rj}\vartheta_j,
\qquad
\widehat\eta_r
=
\sum_{j\in\mathcal J_r}\widehat w_{rj}\widehat\vartheta_j,
\]
and the combined influence function and its plug-in version
\begin{equation}\label{eq:event-psi}
\psi_{r,i}
=
\sum_{j\in\mathcal J_r}
\left(
w_{rj}\phi_{j,i}^{P}
+\vartheta_j\xi^w_{rj,i}
\right),
\qquad
\widehat\psi_{r,i}
=
\sum_{j\in\mathcal J_r}
\left(
\widehat w_{rj}\widehat\phi_{j,i}^{P}
+\widehat\vartheta_j\widehat\xi^w_{rj,i}
\right).
\end{equation}
\begin{theorem}[Inference for weighted aggregates]\label{thm:event-aggregation}
Let $K_A$ and the finite coordinate sets $\{\mathcal J_r\}_{r=1}^{K_A}$ be fixed as $n\to\infty$, with the coordinate sets and weighting rules nonrandom and not selected from the analysis sample. Let $Z_i=(\boldsymbol\phi_i^{P\prime},\boldsymbol\xi_i^{w\prime})'$ be a function of the panel observation $O_i$, i.i.d. across $i$, with $\mathbb E[Z]=0$ and $\mathbb E[\lVert Z\rVert^2]<\infty$. Suppose that, jointly over $j\in\mathcal J$,
\[
\sqrt n(\widehat\vartheta_j-\vartheta_j)
=
\frac1{\sqrt n}\sum_{i=1}^n\phi_{j,i}^{P}
+o_{\mathbb P}(1),
\]
and, jointly over $r=1,\ldots,K_A$ and $j\in\mathcal J_r$,
\begin{equation}\label{eq:event-weight-alr}
\sqrt n(\widehat w_{rj}-w_{rj})
=
\frac1{\sqrt n}\sum_{i=1}^n\xi^w_{rj,i}
+o_{\mathbb P}(1).
\end{equation}
Then, for $\boldsymbol\eta=(\eta_1,\ldots,\eta_{K_A})'$ and $\boldsymbol\psi_i=(\psi_{1,i},\ldots,\psi_{K_A,i})'$ from \eqref{eq:event-psi},
\begin{equation}\label{eq:event-if}
\sqrt n(\widehat{\boldsymbol\eta}-\boldsymbol\eta)
=
\frac1{\sqrt n}\sum_{i=1}^n\boldsymbol\psi_i
+o_{\mathbb P}(1)
\xrightarrow{d}
N(0,\Omega),
\qquad
\Omega=\mathbb E[\boldsymbol\psi\boldsymbol\psi'].
\end{equation}
If the plug-in columns $\widehat{\boldsymbol\psi}_i$ satisfy $\mathbb P_n[\lVert \widehat{\boldsymbol\psi}_i-\boldsymbol\psi_i\rVert^2]=o_{\mathbb P}(1)$, then
\begin{equation}\label{eq:event-sigma}
\widehat\Omega
=
\mathbb P_n[
\widehat{\boldsymbol\psi}_i
\widehat{\boldsymbol\psi}_i']
\xrightarrow{p}\Omega.
\end{equation}
\end{theorem}
The covariance matrix of $\widehat{\boldsymbol\eta}$ is estimated by $\widehat\Omega/n$.
For the event-time application and margin $m\in\{L,R\}$, take $r=(m,e)$ and $j=(m,g,g+e)$, with $\xi^w_{rj}=\xi^w_{g,e}$. Thus the same estimated cohort share enters the two separate margin aggregates, and the stacked covariance retains the induced cross-margin dependence.
An event-time aggregate of $\delta_{L,g,t}-\mu_{D,g}\theta_{R,g,t}$ uses the same formula, with the cell influence function in \eqref{eq:phi-proportionality}. When cohort shares are estimated, its weight term in \eqref{eq:event-psi} multiplies the cell value $\delta_{L,g,t}-\mu_{D,g}\theta_{R,g,t}$.
When the numerical weights are known and imposed in estimation, set $\widehat w_{rj}=w_{rj}$ and $\xi^w_{rj}=0$. If $L$ is the resulting fixed aggregation matrix, the influence function and covariance formulas in \eqref{eq:event-if} and \eqref{eq:event-sigma} reduce to $\boldsymbol\psi=L\boldsymbol\phi^{P}$ and $\widehat\Omega=L\widehat\Sigma L'$, the formulas for a fixed linear map.
For cohort-share weighting, let $q_g=P(G=g)$, $Q_e=P(G\in\mathcal G_e)$, and suppose $Q_e>0$. The population and sample weights are
\[
w_{g,e}=\frac{q_g}{Q_e},
\qquad
\widehat w_{g,e}
=
\frac{n_g}{\sum_{h\in\mathcal G_e}n_h},
\]
and the weight influence function is
\[
\xi^w_{g,e}(O)
=
\frac{
\mathbf 1\{G=g\}
-w_{g,e}\mathbf 1\{G\in\mathcal G_e\}
}{Q_e}.
\]
Its plug-in version replaces $(w_{g,e},Q_e)$ by $(\widehat w_{g,e},\widehat Q_e)$, where $\widehat Q_e=n^{-1}\sum_i\mathbf 1\{G_i\in\mathcal G_e\}$. These influence functions have mean zero and sum to zero over $g\in\mathcal G_e$. The cohort counts used in the weights come from the full panel, not from the units satisfying $S_c=1$.
\subsection{Pre-treatment comparisons}\label{sec:pretrend}
Let $\mathcal C_{\mathrm{pre}}\subseteq\{(g,t):3\le g\le\mathcal T,\ 1\le t<g-1\}$ be finite, nonrandom, and not selected from the analysis sample. For $c=(g,t)\in\mathcal C_{\mathrm{pre}}$, define
\[
C_c^{\mathrm{pre}}=\mathbf 1\{G>g\},
\qquad
S_c^{\mathrm{pre}}=\mathbf 1\{G=g\}+C_c^{\mathrm{pre}},
\qquad
Y_c^{\mathrm{pre}}=Y_t-Y_{g-1}.
\]
The control group excludes cohort $g$ and remains untreated at both dates. The pre-trend targets are the observed-data functionals
\begin{equation*}
\begin{aligned}
\delta_{L,g,t}^{\mathrm{pre}}
&=\mathbb E[Y_c^{\mathrm{pre}}\mid G=g]-\mathbb E[Y_c^{\mathrm{pre}}\mid G>g],\\
\theta_{R,g,t}^{\mathrm{pre}}
&=\frac{\operatorname{Cov}(D,Y_c^{\mathrm{pre}}\mid G=g)}{V_g}.
\end{aligned}
\end{equation*}
Estimate them by the corresponding sample mean contrast and within-cohort OLS slope. Their influence functions are \eqref{eq:phiL-0} and \eqref{eq:phiR-0} with $Y_c^{\mathrm{pre}}$ in place of $Y_c$, $G>g$ in place of $G>t$, and the pre-treatment means, probabilities, and response index in place of the post-treatment ones.
\begin{corollary}[Pre-trend inference]\label{cor:pretrend-inference}
Suppose observations are i.i.d. across units and, for every $c\in\mathcal C_{\mathrm{pre}}$, $P(G=g)>0$, $P(G>g)>0$, $V_g>0$, and
\[
\mathbb E[|Y_c^{\mathrm{pre}}|^4+|D|^4\mid G=g]<\infty,
\qquad
\mathbb E[(Y_c^{\mathrm{pre}})^2\mid G>g]<\infty.
\]
Then Theorem~\ref{thm:baseline-stacked} holds for the pre-trend stack, and under Assumption~\ref{ass:baseline-inference} also for the pre-trend and post-treatment stacks jointly; Theorem~\ref{thm:event-aggregation} applies to fixed pre-trend aggregates.
\end{corollary}
Corollary~\ref{cor:pretrend-inference} yields a pre-trend test for each margin. With $\mathcal C_{\mathrm{pre}}$ and the weighting rule fixed in advance, stack that margin's pre-trend aggregates with their influence columns from \eqref{eq:event-psi} and compare the Wald statistic $n\widehat{\boldsymbol\eta}'\widehat\Omega^{-1}\widehat{\boldsymbol\eta}$ with a $\chi^2$ distribution on as many degrees of freedom as there are aggregates. The binary-treatment literature cautions that such tests need careful interpretation. \citet{roth-2022} shows that they often have low power and that conditioning on passing them can enlarge bias and reduce coverage, so we report both margins regardless of the tests and do not read nonrejection as support for PT-L or PT-R. \citet{rambachan-roth-2023} instead bound the post-treatment violation by the pre-trends and obtain confidence sets valid under that bound; the jointly normal pre-treatment and post-treatment stacks supply their inputs. For the response index the bound covers dose-related trends, not selection on gains, which \eqref{eq:sensitivity-bound} bounds instead.
\section{Covariate adjustment}\label{sec:covariates}
Covariates matter in two ways: they provide a conditional identification route for the same level margin, while residualizing dose defines a different response index.
\subsection{Adjusted targets and conditional identification}
For cohort $g$, define
\[
e_g(x)=\mathbb E[D\mid G=g,X=x],
\qquad
V_g^{(X)}
=
\mathbb E[(D-e_g(X))^2\mid G=g],
\]
and, for cell $c=(g,t)$,
\[
g_{g,t}(x)=\mathbb E[Y_c\mid G=g,X=x].
\]
When $V_g^{(X)}>0$, the adjusted response index is
\begin{equation}\label{eq:cohort-response-X-def}
\theta_{R,g,t}^{(X)}
=
\frac{
\mathbb E[(D-e_g(X))\{Y_c-g_{g,t}(X)\}\mid G=g]
}{V_g^{(X)}}.
\end{equation}
Total covariance and total variance give the exact relation
\begin{equation}\label{eq:response-0-X-relation}
\theta_{R,g,t}
=
\frac{
V_g^{(X)}\theta_{R,g,t}^{(X)}
+\operatorname{Cov}\{e_g(X),g_{g,t}(X)\mid G=g\}
}{
V_g^{(X)}+\operatorname{Var}\{e_g(X)\mid G=g\}
}.
\end{equation}
Conditioning on $X$ therefore changes the residual dose variation and the index itself, and it may yield $V_g^{(X)}=0$ when $V_g>0$. When $X$ is constant, \eqref{eq:cohort-response-X-def} reduces to \eqref{eq:cohort-response-0-def}. Adjusted response objects carry the superscript $(X)$.
The level margin $\tau_{L,g,t}$ is still \eqref{eq:cohort-level-def}. Let $m_{0,g,t}(x)=\mathbb E[Y_c\mid G>t,X=x]$ and define the adjusted level contrast
\begin{equation}\label{eq:cohort-level-id-X}
\delta_{L,g,t}^{(X)}
=
\mathbb E[Y_c-m_{0,g,t}(X)\mid G=g].
\end{equation}
The identifying restrictions are conditional versions of those in Section~\ref{sec:generic-assumptions}.
\begin{assumption}[Conditional cohort-time conditions]\label{ass:covariate-identification}
For every $c=(g,t)\in\mathcal C$: (i) $X$ is a pre-treatment vector common to every outcome date and cell, and $P(G>t\mid X=x)>0$ for almost every $x$ in the support of $X\mid G=g$; (ii) conditional level parallel trends, $\mathrm{PT\text{-}L}^{(X)}_{g,t}$: $\mathbb E[U_c\mid G=g,X]=\mathbb E[U_c\mid G>t,X]$; (iii) conditional response parallel trends, $\mathrm{PT\text{-}R}^{(X)}_{g,t}$: $\mathbb E[U_c\mid D,G=g,X]=\mathbb E[U_c\mid G=g,X]$ almost surely; (iv) $V_g^{(X)}>0$.
\end{assumption}
Under Assumptions~\ref{ass:staggered} and~\ref{ass:covariate-identification}(i), (ii), $\delta_{L,g,t}^{(X)}=\tau_{L,g,t}$. Conditional PT-L does not imply PT-L, so $\delta_{L,g,t}^{(X)}$ and $\delta_{L,g,t}$ may differ. Conditional PT-L and conditional PT-R are non-nested, by the examples in the proof of Proposition~\ref{prop:pt-nonnested} applied conditionally on $X$. Part (i) fixes one covariate vector for every outcome date, so $e_g$ and $V_g^{(X)}$ are common over $t$ within a cohort, as $\mu_{D,g}$ and $V_g$ are. The conditions for a causal derivative reading of the adjusted index under part (iii) are the conditional analogues of those in Section~\ref{sec:why-two}, stated in Appendix~\ref{app:dmlproofs}.
Partialling out functions of $X$ from the cell regression does not isolate the adjusted response index: the dose coefficient retains a level component (Proposition~\ref{prop:adjusted-mixture}, Appendix~\ref{app:dmlproofs}).
\subsection{Cross-fitted DML and stacked inference}
Work under $P_c$, the distribution conditional on eligibility for cell $c$, and let $\pi_{g,t}(x)=P_c(T_c=1\mid X=x)$ denote the cell propensity. Together with $m_{0,g,t}$, $e_g$, and $g_{g,t}$, it forms the four nuisance functions. The propensity, control regression, and treated outcome regression vary by cell. The dose regression $e_g$ is defined conditional on $G=g$, so each fold-specific estimate uses only cohort-$g$ observations and is reused across all outcome dates for that cohort.
The corresponding orthogonal scores are given in equations~\eqref{eq:level-score} and \eqref{eq:response-score} of Appendix~\ref{app:dmlproofs}. The response score is a treated-cohort analogue of the residual-on-residual score in \citet{clarke-polselli-2026}, and the level score is an augmented DiD score. One fixed $K$-fold unit partition is reused across every cell and nuisance function, so that the stacked influence vectors are evaluated for the same units in the same order. Let $k(i)$ denote unit $i$'s fold and form all predictions out of fold. With $\mathbb P_{n,c}[h]=\mathbb P_n[S_ch]/\mathbb P_n[S_c]$, the resulting cross-fitted ratio estimators are
\begin{equation}\label{eq:deltahat-X}
\widehat\delta_{L,g,t}^{(X)}
=
\frac{
\mathbb P_{n,c}\left[
T_c\{Y_c-\widehat m_{0,-k}(X)\}
-(1-T_c)
\frac{\widehat\pi_{-k}(X)}{1-\widehat\pi_{-k}(X)}
\{Y_c-\widehat m_{0,-k}(X)\}
\right]
}{\mathbb P_{n,c}[T_c]},
\end{equation}
and
\begin{equation}\label{eq:thetahat-X}
\widehat\theta_{R,g,t}^{(X)}
=
\frac{
\mathbb P_{n,c}\left[
T_c\{D-\widehat e_{-k}(X)\}
\{Y_c-\widehat g_{-k}(X)\}
\right]
}{
\mathbb P_{n,c}\left[
T_c\{D-\widehat e_{-k}(X)\}^2
\right]
}.
\end{equation}
The product rates in the next assumption come from the exact nuisance remainders in Appendix~\ref{app:dmlproofs}. The asymptotic expansion uses no identifying restriction; conditional PT-L and PT-R govern only interpretation.
\begin{assumption}[Cross-fitted DML conditions]\label{ass:dml}
Full-path observations are i.i.d. across units. Outcomes, doses, and fitted nuisance functions are bounded by fixed constants; $p>0$ and $V^{(X)}>0$; and, for some $\epsilon\in(0,1/2)$, $\epsilon\le\pi(X)\le1-\epsilon$. Estimated propensities are truncated to $[\epsilon/2,1-\epsilon/2]$. The number of folds is fixed, and every prediction for unit $i$ is made by a model trained without unit $i$'s fold. Uniformly over folds, the four nuisances are $L_2$ consistent, with norms evaluated under the distribution of $X\mid S_c=1$ for $(m_0,\pi)$ and under the distribution of $X\mid G=g$ for $(e,g)$, and
\[
\lVert \widehat m_0-m_0\rVert_2
\lVert \widehat\pi-\pi\rVert_2
=o_{\mathbb P}(n^{-1/2}),
\]
\begin{equation}\label{eq:response-products}
\lVert \widehat e-e\rVert_2
\lVert \widehat g-g\rVert_2
+\lVert \widehat e-e\rVert_2^2
=o_{\mathbb P}(n^{-1/2}).
\end{equation}
\end{assumption}
The squared term in \eqref{eq:response-products} requires $\lVert \widehat e-e\rVert_2=o_{\mathbb P}(n^{-1/4})$. Appendix~\ref{app:dmlproofs} states the generic-cell asymptotic linear representation and efficiency result, Theorem~\ref{thm:joint-dml}.
\begin{assumption}[Cellwise DML conditions]\label{ass:staggered-dml}
The collection $\mathcal C$ is finite, nonrandom, and not selected from the analysis sample. For every $c\in\mathcal C$, $P(S_c=1)>0$ and Assumption~\ref{ass:dml} holds under $P_c$.
\end{assumption}
Let $\phi_{L,c}^{P_c}$ and $\phi_{R,c}^{(X),P_c}$ denote the gradients under $P_c$ derived in equations~\eqref{eq:phiL} and \eqref{eq:phiR} of Appendix~\ref{app:dmlproofs}. Because $P_c$ varies across cells, each gradient must first be transported to the common full-panel distribution $P$:
\begin{equation}\label{eq:cell-full-if}
\boldsymbol\phi_c^{(X),P}(O)
=
\frac{S_c}{P(S_c=1)}
\begin{pmatrix}
\phi_{L,c}^{P_c}(O) \\
\phi_{R,c}^{(X),P_c}(O)
\end{pmatrix}.
\end{equation}
\begin{theorem}[Stacked covariate-adjusted DML]\label{thm:staggered-dml}
Under Assumption~\ref{ass:staggered-dml}, let $\boldsymbol\vartheta^{(X)}$ stack $(\delta_{L,g,t}^{(X)},\theta_{R,g,t}^{(X)})$ over $c\in\mathcal C$ in a fixed order, let $\widehat{\boldsymbol\vartheta}^{(X)}$ stack the estimators \eqref{eq:deltahat-X} and \eqref{eq:thetahat-X}, computed with one unit-level fold partition for every cell and nuisance function, and let $\boldsymbol\phi^{(X),P}$ stack the rescaled columns in \eqref{eq:cell-full-if}, both in the same order. Then
\begin{equation}\label{eq:staggered-vector-clt}
\sqrt n
(\widehat{\boldsymbol\vartheta}^{(X)}
-\boldsymbol\vartheta^{(X)})
=
\frac1{\sqrt n}\sum_{i=1}^n
\boldsymbol\phi_i^{(X),P}
+o_{\mathbb P}(1)
\xrightarrow{d}
N(0,\Sigma^{(X)}),
\end{equation}
where $\Sigma^{(X)}=\mathbb E[\boldsymbol\phi^{(X),P}\boldsymbol\phi^{(X),P\prime}]$. Let $\widehat{\boldsymbol\phi}_i^{(X),P}$ stack the plug-in columns $\{S_{c,i}/\mathbb P_n[S_c]\}\widehat{\boldsymbol\phi}_c^{P_c}(O_i)$, with $\widehat{\boldsymbol\phi}_c^{P_c}$ the plug-in gradient of Theorem~\ref{thm:joint-dml}, evaluated for every unit $i$ in the same order. Then
\begin{equation}\label{eq:staggered-sigma}
\widehat\Sigma^{(X)}
=
\mathbb P_n[
\widehat{\boldsymbol\phi}_i^{(X),P}
\widehat{\boldsymbol\phi}_i^{(X),P\prime}]
\xrightarrow{p}\Sigma^{(X)}.
\end{equation}
\end{theorem}
Because \eqref{eq:staggered-vector-clt} is an asymptotic linear representation under $P$, Theorem~\ref{thm:event-aggregation} applies to the adjusted stack. Cohort-share weights and their influence functions are computed on the full panel without cross-fitting and enter \eqref{eq:event-psi} without the rescaling in \eqref{eq:cell-full-if}.
\section{Simulation}\label{sec:simulation}
We study the finite-sample performance of the baseline level and response estimators in a staggered-adoption design.
We generate a balanced panel over $t=1,\ldots,7$. Adoption dates satisfy
\[
\Pr(G_i=4)=\Pr(G_i=5)=\Pr(G_i=6)=0.2,
\qquad
\Pr(G_i=\infty)=0.4.
\]
The fixed post-treatment cell collection contains cohorts $g\in\{4,5,6\}$ at event times $e=t-g\in\{0,1\}$, giving six cohort-time cells. For treated units,
\[
D_i=\mu+h(2B_i-1),
\qquad
B_i\sim\operatorname{Beta}(3,3),
\qquad
(\mu,h)=(2,1.8),
\]
so the positive-dose support is $[0.2,3.8]$, and never-treated units have $D_i=0$. The untreated potential outcome is
\[
Y_{it}(\infty,0)
=A_i+m_t+\sum_{r=1}^t\varepsilon_{ir}
+\mathbf{1}\{G_i<\infty\}\,t\{b_L+b_R(D_i-\mu)\},
\qquad
m_t=0.1t+0.02t^2,
\]
where $A_i$ and the innovations $\varepsilon_{ir}$ are independent standard normal variables and are independent of $(G_i,D_i)$. The performance and benchmark designs set $b_L=b_R=0$, so PT-L and PT-R hold. The pre-trend experiment below varies these two trend parameters. For a unit in cohort $g$ with realized dose $d$, the treatment-effect surface at event time $e$ on the positive-dose support is
\begin{equation}\label{eq:simulation-effect-surface}
\tau_{g,e}(u\mid d)
=
3+a_g r_e\left\{0.5u+\kappa h
\tanh\!\left(3\frac{u-\mu}{h}\right)\right\}
+s(d-\mu),
\end{equation}
where $(a_4,a_5,a_6)=(0.8,1,1.2)$ and $(r_0,r_1)=(1,1.4)$. The selection derivative in Theorem~\ref{thm:response-decomp} is $s$. The centered $\tanh$ term averages to zero under the symmetric treated-dose distribution, so varying $\kappa$ changes the response index while leaving the level margin unchanged. The cubic specification of the \texttt{contdid} benchmark in Table~\ref{tab:simulation-benchmark} cannot represent this curve exactly, so the nonlinear design also exposes its approximation error.
With $b_L=b_R=0$, two designs differ only in $\kappa$, with $s=0$ in both: the nonlinear design $\kappa=0.4$ is the main design, and the linear design sets $\kappa=0$. Symmetry gives $\mu_D=2$, and the Beta variance gives $V_D=h^2/7$. At each event time, we aggregate the three cohort estimates using sample cohort shares computed from full-panel counts; their population values are one third. For each margin, the overall aggregate weights event times zero and one equally. In the main design, the aggregate level margin is $4.2$, and the aggregate response index is $1.4436$.
We use $n\in\{500,1{,}000,2{,}000\}$ and 5,000 replications at each sample size. The cell estimators are those of Section~\ref{sec:baseline-estimation}, and aggregate standard errors use the combined influence function in \eqref{eq:event-psi}.
\AddToHookNext{env/tabular/before}{}
\begin{table}[!htbp]
\centering
\caption{Finite-sample performance of the two-margin estimators}
\label{tab:simulation-staggered}
\small
\begin{tabular}{rcccc}
\toprule
$n$ & Bias & SD & Mean SE & Coverage \\
\midrule
\multicolumn{5}{l}{\textit{Panel A: Level margin}} \\
500 & 0.0001 & 0.102 & 0.101 & 0.9516 \\
1{,}000 & 0.0016 & 0.072 & 0.072 & 0.9472 \\
2{,}000 & 0.0001 & 0.051 & 0.051 & 0.9472 \\
\addlinespace
\multicolumn{5}{l}{\textit{Panel B: Response index}} \\
500 & 0.0044 & 0.099 & 0.097 & 0.9406 \\
1{,}000 & 0.0026 & 0.070 & 0.069 & 0.9480 \\
2{,}000 & -0.0003 & 0.048 & 0.049 & 0.9514 \\
\bottomrule
\end{tabular}
\begin{minipage}{0.98\textwidth}
\begin{singlespace}
\footnotesize
\textit{Notes:} 5{,}000 replications; $n$ is the number of units. The nonlinear design has no selection. The level and response targets are 4.2 and 1.4436, respectively. Both aggregates use full-panel sample cohort shares and equal weights on $e=0,1$. SD is the empirical standard deviation; Mean SE is the average influence-function standard error, including uncertainty in cohort shares and cross-cell covariance. Coverage refers to 95\% Wald intervals.
\end{singlespace}
\end{minipage}
\end{table}
Table~\ref{tab:simulation-staggered} reports results for both margins. Mean standard errors are close to the empirical standard deviations, and coverage of the 95 percent intervals is close to the nominal level. For the response aggregate, ignoring covariance between the two event-time estimates lowers coverage to 0.8568--0.8722.
Table~\ref{tab:simulation-benchmark} compares the within-cohort OLS estimator with the \texttt{contdid} estimator of the average causal response on the treated (ACRT), using the same cohort and event-time weights.\footnote{The benchmark is the ACRT estimator in \texttt{contdid}~0.1.1 with not-yet-treated controls, a varying base period, and the default cubic specification with no interior knots.} In the linear design the causal response is constant in dose within each cell, so the response index and the ACRT coincide at $0.6$ and the within-cohort regression is correctly specified. Both estimators have little bias, and within-cohort OLS has 22.1--25.3 percent lower empirical variance across the three sample sizes, the parsimony gain of a single correctly specified slope over a cubic dose curve. In the nonlinear design the two targets differ. The response index weights the causal response by $W$ and equals $1.4436$, whereas the ACRT averages it over the dose distribution and equals $1.3655$. Against its own target, within-cohort OLS again has smaller absolute bias and standard deviation and coverage closer to 95 percent, while the \texttt{contdid} bias includes an approximation error of $-0.0154$ from its fixed cubic specification.
\AddToHookNext{env/tabular/before}{}
\begin{table}[!htbp]
\centering
\caption{Comparison of response estimators}
\label{tab:simulation-benchmark}
\small
\setlength{\tabcolsep}{4pt}
\begin{tabular}{rcccccc}
\toprule
& \multicolumn{3}{c}{Within-cohort OLS} & \multicolumn{3}{c}{\texttt{contdid}} \\
\cmidrule(lr){2-4} \cmidrule(lr){5-7}
$n$ & Bias & SD & Coverage & Bias & SD & Coverage \\
\midrule
\multicolumn{7}{l}{\textit{Panel A: Linear design}} \\
500 & -0.003 & 0.095 & 0.946 & -0.003 & 0.110 & 0.938 \\
1{,}000 & -0.001 & 0.067 & 0.947 & -0.001 & 0.076 & 0.946 \\
2{,}000 & 0.001 & 0.047 & 0.951 & 0.001 & 0.053 & 0.947 \\
\addlinespace
\multicolumn{7}{l}{\textit{Panel B: Nonlinear design}} \\
500 & 0.004 & 0.099 & 0.941 & -0.022 & 0.118 & 0.930 \\
1{,}000 & 0.003 & 0.070 & 0.948 & -0.017 & 0.080 & 0.936 \\
2{,}000 & 0.000 & 0.048 & 0.951 & -0.018 & 0.055 & 0.937 \\
\bottomrule
\end{tabular}
\begin{minipage}{0.98\textwidth}
\begin{singlespace}
\footnotesize
\textit{Notes:} 5{,}000 replications; $n$ is the number of units. Aggregation uses full-panel sample cohort shares and equal weights on $e=0,1$. Both targets equal 0.6 in Panel A. In Panel B, the response index is 1.4436 and the ACRT is 1.3655; \texttt{contdid} bias includes cubic approximation error. SD is the empirical standard deviation. Coverage is for 95\% Wald intervals.
\end{singlespace}
\end{minipage}
\end{table}
\begingroup
\let\simulationtable\simulationtable[!htb]\AddToHookNext{env/tabular/before}{}
\begin{table}[!htbp]
\centering
\caption{Empirical rejection rates of pre-trend tests}
\label{tab:simulation-pretrend-tests}
\small
\begin{tabular}{lrcc}
\toprule
Design $(b_L,b_R)$ & $n$ & Level test & Response-index test \\
\midrule
Null $(0,0)$ & 500 & 0.049 & 0.056 \\
& 1{,}000 & 0.050 & 0.055 \\
& 2{,}000 & 0.054 & 0.053 \\
\addlinespace
Level trend $(0.10,0)$ & 500 & 0.264 & 0.067 \\
& 1{,}000 & 0.467 & 0.054 \\
& 2{,}000 & 0.778 & 0.047 \\
\addlinespace
Level trend $(0.20,0)$ & 500 & 0.771 & 0.059 \\
& 1{,}000 & 0.976 & 0.055 \\
& 2{,}000 & 1.000 & 0.054 \\
\addlinespace
Dose-related trend $(0,0.10)$ & 500 & 0.053 & 0.311 \\
& 1{,}000 & 0.050 & 0.563 \\
& 2{,}000 & 0.049 & 0.853 \\
\addlinespace
Dose-related trend $(0,0.20)$ & 500 & 0.055 & 0.861 \\
& 1{,}000 & 0.051 & 0.995 \\
& 2{,}000 & 0.049 & 1.000 \\
\bottomrule
\end{tabular}
\begin{minipage}{0.98\textwidth}
\begin{singlespace}
\footnotesize
\textit{Notes:} 5{,}000 replications per design and sample size; $n$ is the number of units. Entries are rejection rates of separate joint Wald tests of the level leads and response-index leads at $e=-3,-2$, relative to $e=-1$. Each test uses a 5\% nominal level and two degrees of freedom. Level comparisons use $G>g$ controls. Covariance estimates retain dependence across leads and account for full-panel sample cohort shares.
\end{singlespace}
\end{minipage}
\end{table}
\endgroup
\begin{samepage}
Table~\ref{tab:simulation-pretrend-tests} evaluates the separate pre-trend tests for the level and response-index leads. We hold $(\kappa,s)=(0.4,0)$ and vary $b_L$ and $b_R$ in the untreated outcome process. Each test assesses whether the two event-time leads at $e=-3,-2$, relative to $e=-1$, are jointly zero at the 5 percent nominal level. In this design, $b_L$ shifts the level leads, whereas the dose-related term $b_R(D-2)$ shifts the response-index leads.
\par
\end{samepage}
Under the null, level rejection rates range from 4.9 to 5.4 percent. Response rejection rates range from 5.3 to 5.6 percent. With $b_L=0.1$, level-test power rises from 26.4 to 77.8 percent as $n$ increases from 500 to 2,000, and with $b_R=0.1$, response-test power rises from 31.1 to 85.3 percent. Stronger trends increase power further, while the test for the unaffected margin generally remains near its nominal level, with a rejection rate of 6.7 percent for the response test when $b_L=0.1$ and $n=500$. Nonrejection establishes neither post-treatment PT-L or PT-R nor the absence of selection on gains.
\FloatBarrier
\section{Application}\label{sec:application}
\subsection{Research design and interpretation}
\citet{bartik-et-al-2019} study the local effects of hydraulic fracturing by comparing counties in the highest prospectivity quartile of a shale play with other counties in the same play. Using their data, \citet{callaway-goodmanbacon-santanna-2024-event} report employment effects for high- and low-dose groups and estimate level effects as a function of dose. We use the CGBS sample to estimate the average employment change relative to counties not yet treated and the association between employment changes and prospectivity within an adoption cohort. We then examine their pre-treatment diagnostics and their contributions to the cellwise continuous-dose regression.
The outcome is log total county employment. The dose is a time-invariant geological prospectivity score, and adoption occurs in the first year of fracking in the county's shale play. The processed CGBS sample is a balanced panel of 402 counties from 1990 through 2014, with 329 positive-dose counties and 73 zero-dose counties. Their preparation code assigns zero to 44 missing prospectivity scores; we retain this coding and examine its consequences below.
Prospectivity scores are constructed separately for each shale play and cannot be compared directly across plays \citep[p.~115]{bartik-et-al-2019}. We retain the CGBS score and read the response index as a statistical association in the recorded score units. Some adoption cohorts contain several plays, so their slopes may reflect differences across plays as well as differences among counties in the same play. Neither cohort centering nor play clustering makes a one-point score difference comparable across plays.
We use the baseline estimators in Section~\ref{sec:baseline-estimation} for seven cohorts, 2001 and 2005--2010, at event times $e=0,\ldots,4$. This gives 307 treated counties and 35 cohort-time cells. Excluding the 2012 cohort holds cohort composition fixed along the event-time path. Each cell uses the outcome change $Y_{g+e}-Y_{g-1}$. The level contrast uses all counties with $G>g+e$ as controls, between 73 and 368 depending on the cell, whereas the response index uses only cohort-$g$ counties. We aggregate with full-panel cohort shares and include the influence of the estimated shares. The common cohort set keeps these shares and the within-cohort response weights fixed over event time, as in Corollary~\ref{cor:cohort-interpretations}.
\subsection{Employment paths and separate diagnostics}
Figure~\ref{fig:fracking-margins} reports the event-time estimates. Both are close to zero at adoption and rise over the following four years. At event time four, the level contrast is 0.0474 with an i.i.d. standard error of 0.0116. The response index is 0.0284 log points per prospectivity point with an i.i.d. standard error of 0.0082. Under county-level i.i.d. sampling, joint 95 percent Gaussian max-$t$ intervals over the ten estimates exclude zero for both paths from event time two onward.
\begin{figure}[!htbp]
\centering
\caption{Event-time level contrast and response index}
\label{fig:fracking-margins}
\vspace{0.35em}
\includegraphics[width=0.97\textwidth]{figures/application_margins.pdf}
\vspace{0.35em}
\begin{minipage}{0.96\textwidth}
\begin{singlespace}
\footnotesize
\textit{Notes:} The figure reports aggregates with full-panel cohort shares for the 2001 and 2005--2010 cohorts. Vertical bars are joint 95 percent Gaussian max-$t$ intervals over both panels under county-level i.i.d. sampling. Panel~A is the level contrast, which equals the level margin under PT-L; Panel~B is the within-cohort statistical response index in the recorded prospectivity units.
\end{singlespace}
\end{minipage}
\end{figure}
For the pre-specified leads $e=-3,-2$, we use $g-1$ as the reference period and $G>g$ controls, as defined in Section~\ref{sec:pretrend}. The two level leads reject a joint zero restriction ($p<0.001$), whereas the response-index leads do not ($p=0.322$). These tests use Corollary~\ref{cor:pretrend-inference} and Theorem~\ref{thm:event-aggregation} under county-level i.i.d. sampling. We therefore report the level path as an observed-data contrast and retain the statistical interpretation of the response path. Nonrejection of its pre-trend restriction establishes neither PT-R nor the absence of selection on gains.
The difference between the diagnostics matters because PT-L concerns mean untreated changes across groups, while PT-R concerns their relation to dose within a cohort. Proposition~\ref{prop:pt-nonnested} shows that neither identifying restriction implies the other. The pre-trend tests do not determine which restriction holds after adoption. \citet[pp.~125, 130]{bartik-et-al-2019} also report positive employment pre-trends and allow for differential trends and a trend break in their richer specifications. CGBS interpret their dose-bin pre-treatment paths as supporting parallel trends. Those plots, the Bartik specifications, and our baseline contrasts use different samples and comparisons.
\subsection{What the cellwise dose coefficient measures}
Figure~\ref{fig:fracking-mixture} applies Corollary~\ref{cor:share} to the 35 cohort-time cells. The weight $\lambda_{g,t}$ on the response index ranges from 0.019 to 0.117. Within a cohort, the dose mean and variance are fixed, so this weight changes over event time only through the not-yet-treated share. Its mean, weighted by cohort shares, rises from 0.066 at event time zero to 0.084 at event time four, while the weighted number of eligible controls falls from 209.8 to 109.9.
Panel~B aggregates the exact cellwise components using the same cohort shares. At event time four, the aggregate of the cellwise OLS coefficients is 0.0111, with 0.0097 from the level-contrast component and 0.0013 from the response-index component. Thus the coefficient mainly reflects the treated-control contrast per unit of mean dose. Its units of log points per prospectivity point do not make it a within-cohort response. This is the comparison for which the level leads reject. The identity applies cell by cell and does not decompose a pooled multi-period staggered TWFE regression.
\begin{figure}[!htbp]
\centering
\caption{Cellwise continuous-dose regression weights and components}
\label{fig:fracking-mixture}
\vspace{0.35em}
\includegraphics[width=0.97\textwidth]{figures/application_mixture.pdf}
\vspace{0.35em}
\begin{minipage}{0.96\textwidth}
\begin{singlespace}
\footnotesize
\textit{Notes:} Panel~A reports $\lambda_{g,t}$ from \eqref{eq:cell-twfe-mixture} for each cohort-time cell. Panel~B reports cohort-share aggregates of the cellwise coefficient and its two exact components. The identity is applied cell by cell; the figure is not a decomposition of a pooled multi-period staggered TWFE coefficient.
\end{singlespace}
\end{minipage}
\end{figure}
We also test whether the five event-time aggregates of $\delta_{L,g,t}-\mu_{D,g}\theta_{R,g,t}$ are jointly zero. The i.i.d. Wald test does not reject ($p=0.228$). This is an aggregate restriction, not a test that the two endpoints coincide in every cell. Appendix~\ref{app:fracking-checks} gives the inference details.
Removing the 44 counties whose missing scores were coded as zero raises the long-window level contrast from 0.0432 to 0.0513. The response estimates are unchanged. These counties enter the control means but not the cohort-specific dose covariance, so their removal changes only the level comparison. This invariance follows from the definition of the response index; it does not establish a causal interpretation. Appendix~\ref{app:fracking-checks} reports all three windows.
\subsection{Inference and comparison with CGBS}
Table~\ref{tab:fracking-comparison} reports window averages. The short-run and long-run level contrasts are 0.0094 and 0.0432, and the corresponding response indices are 0.0088 and 0.0272 log points per prospectivity point. The larger long-run estimates agree with the employment pattern reported by \citet{bartik-et-al-2019} and \citet{callaway-goodmanbacon-santanna-2024-event}. The CGBS benchmark in Panel~B is a level effect at dose four. Its long-run value of 0.0277 has a different meaning and different units from the response index of 0.0272. Samples, controls, and aggregation also differ, and Appendix~\ref{app:fracking-checks} documents these choices.
\begin{table}[!htbp]
\centering
\caption{Level contrasts, response indices, and the CGBS benchmark}
\label{tab:fracking-comparison}
{ \begin{tabular}{llrrr}
\toprule
Object & Window & Estimate & iid SE & Play SE \\
\midrule
\multicolumn{5}{l}{\textit{A. Proposed level contrast and response index}} \\
$\delta_L$ & Short & 0.0094 & 0.0040 & 0.0069 \\
$\delta_L$ & Long & 0.0432 & 0.0101 & 0.0206 \\
$\delta_L$ & Overall & 0.0229 & 0.0063 & 0.0121 \\
$\theta_R$ & Short & 0.0088 & 0.0031 & 0.0063 \\
$\theta_R$ & Long & 0.0272 & 0.0075 & 0.0160 \\
$\theta_R$ & Overall & 0.0162 & 0.0046 & 0.0100 \\
\midrule
\multicolumn{5}{l}{\textit{B. CGBS level at $d=4$}} \\
CGBS $d=4$ & Short & 0.0059 & 0.0061 & 0.0077 \\
CGBS $d=4$ & Long & 0.0277 & 0.0131 & 0.0226 \\
CGBS $d=4$ & Overall & 0.0163 & 0.0089 & 0.0138 \\
\bottomrule
\end{tabular}
}
\vspace{0.5em}
\begin{minipage}{0.98\textwidth}
\begin{singlespace}
\footnotesize
\textit{Notes:} Short averages event times 0--2; long averages 3--4; overall averages 0--4. Level contrasts and CGBS estimates are in log points; response indices are in log points per recorded prospectivity point. Panel~A uses 307 treated counties in seven cohorts, with i.i.d. standard errors that include estimation of full-panel cohort shares. In Panel~B, the i.i.d. standard errors include estimation of the zero-dose control mean conditional on the empirical spline basis. Play SE is the CR1 standard error clustered by shale play. Appendix~\ref{app:fracking-checks} gives the CGBS sample details and archived standard errors.
\end{singlespace}
\end{minipage}
\end{table}
\looseness=-1 Adoption timing is shared within shale plays, so Table~\ref{tab:fracking-comparison} reports play-clustered CR1 standard errors alongside the standard errors under i.i.d. sampling across counties. The design has 14 shale plays, and several response cohorts contain only one play. Play-clustered intervals for all four short- and long-window estimates in Panel~A include zero. Inference under i.i.d. sampling across counties follows Theorems~\ref{thm:baseline-stacked} and \ref{thm:event-aggregation}, and the clustered results are a sensitivity analysis outside their sampling assumptions. CGBS inference also changes when uncertainty in the estimated control mean and dependence among counties are included, as shown in Appendix~\ref{app:fracking-checks}.
The positive response index describes the relation between employment and the prospectivity score within cohorts. A causal derivative interpretation would additionally require PT-R, Assumption~\ref{ass:connected}, and restrictions on selection on gains, as in Theorem~\ref{thm:response-decomp}. We do not impose those restrictions in this application. Reporting the index separately allows this dose association to be examined without incorporating the control-group differences that dominate the cellwise OLS coefficient.
\section{Conclusion}\label{sec:conclusion}
CGBS decompose the continuous-treatment TWFE coefficient, show that its weights on causal responses depend on the dose distribution and the size of the untreated group while selection on gains enters alongside, and recommend estimating treatment effects and causal responses at each dose instead. We take a different route through the same coefficient. In each cohort-time cell it is an exact convex combination of two comparisons that applied researchers already run: the difference in mean outcome changes between the cohort and its not-yet-treated controls, and the OLS slope of the outcome change on dose within the cohort. Our suggestion is to stop reporting their mixture and report the two comparisons separately.
Separating them pays in two ways. The first is identification. The level margin needs a control group and PT-L. The response index uses only the treated cohort and needs, for a causal reading, PT-R and a restriction on selection on gains. Because neither restriction implies the other, evidence against one leaves the other comparison intact, and each has its own pre-trend diagnostic. The second is estimation. A difference in means and an OLS slope involve no dose density, derivative estimator, bandwidth, or functional form for the dose-response curve, and their influence functions are explicit, so joint inference across cells, event times, and margins is closed form in the baseline case and carries over to the covariate-adjusted estimators. The price is that the response index is one number with fixed weights rather than a dose-response curve, and its causal content is only as good as the response restrictions behind it.
The hydraulic fracturing application illustrates the empirical relevance of the distinction. The level leads reject a joint zero restriction, whereas the response-index leads do not. The latter remain a diagnostic rather than evidence for a causal response interpretation. The exact cellwise mixture places most of its weight on the level-contrast component, and removing counties with missing prospectivity changes the level estimates but leaves the response estimates unchanged. These findings show what is lost when the two sources of variation are summarized by a single continuous-dose coefficient.
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