Quantile and Distribution Treatment Effects on the Treated with Possibly Non-Continuous Outcomes

refsection\section{Introduction} Treatment effects may be heterogeneous, which could be important for policymakers. The quantile treatment effect on the treated (QTT) and the distribution treatment effect on the treated (DTT) are useful tools that enable the researcher to explore the effects of treatment at different points on the distribution separately or jointly. This is sometimes in contrast to average treatment effects (on the treated) parameters that can mask substantial heterogeneity in treatment effects. Heterogeneity in the effects of treatment is often encountered in practice. For instance, legal minimum wage policies are anticipated to impact low-wage workers rather than those earning significantly above the minimum wage ghanem2023evaluating,cengiz-Arindrajit-2019effect. almond-hoynes-schanzenbach-2011-inside reports that the largest birth-weight gains from the Food Stamp Program in the United States occur among infants with the lowest birth weights. Moreover, callaway2021bounds finds that the average effects of job displacement on workers' earnings are a mix of large negative effects and more modest impacts. In another study, callaway-li-oka-2018 demonstrates statistically significant effects of minimum wage increases on earnings, particularly at the lower end of the distribution. The non-continuity of outcomes, which is also prevalent in empirical work, arises for several reasons. For example, non-continuity of the outcome may be intrinsic or an artefact of the way the outcome is measured, e.g., the count of car theft incidents diTella-2004, the attitude to gun control on a three-point ordinal scale, e.g., newman-hartman-2019-mass,yamauchi2020difference, and the frequency of marijuana consumption on a three-point cardinal scale gutknecht2024generalized. Also, discontinuities in observed outcomes can result from policy interventions, e.g., minimum wage laws ghanem2023evaluating,cengiz-Arindrajit-2019effect and price controls ruback1982effect. In spite of the prevalence of non-continuous outcomes in empirical work, existing QTT/DTT methods cannot provide a pathway from point identification to inference when the outcome is non-continuous. For example, with only two periods of data and under the distributional difference-in-differences assumption of callaway-li-2019, QTT is only partially identified (fan-yu-2012). Bounds identified in, e.g., fan-yu-2012 tend to be too wide to be useful in most applications (callaway-li-2019). Typical asymptotic arguments of existing QTT methods require continuously distributed outcome variables, e.g., callaway-li-2019. It is not clear how to accompany the partial identification results in ghanem2023evaluating with inference using uniform inference techniques that typically involve Hadamard differentiability. At the heart of this problem for existing methods is the Hadamard non-differentiability of the quantile function (QF) of non-continuous variables. Moreover, simulations in the supplement of chernozhukov2019generic (see Section C therein) reveal that jittering, i.e., smoothing out discontinuities in the DF of non-continuous outcomes, can be an unsatisfactory remedy. To get around the aforementioned challenge, this paper (1) models the counterfactual distribution function (DF) (instead of the QF) of observed untreated potential outcomes, (2) point identifies the DF of the untreated potential outcome of the treated group in the post-treatment period, (3) obtains estimates and uniform confidence bands of the DF, the counterfactual DF, and DTT, and (4) conducts inference on the QF, the counterfactual QF, and QTT thanks to the geometric arguments of chernozhukov2019generic. This pathway to conducting inference has clear advantages vis-\`a-vis methods that model the counterfactual QF. Valid uniform confidence bands on DFs, unlike QFs, are easier to obtain even for discrete outcomes. DFs, as commonly modelled using a linear index, tend to be easier to fit to the data relative to QFs rothe-2012. This paper provides point identification of the counterfactual DF in the same setting as a standard difference-in-differences (DiD) parallel trends (applicable to non-continuous outcomes) -- see puhani2012treatment,wooldridge-2023-simple -- and no-anticipation assumptions extended to the support $\mathcal{Y}$ of outcomes $Y$ that may be discrete, continuous, or mixed. Previous work highlights the drawbacks of identification via the kind of functional index parallel trends (FIPT) used in this paper. The FIPT rules out models with shifts in the location and/or scale that occur across groups and over time athey-imbens-2006 and gutknecht2024generalized. Thus, the proposed method requires treatment assignment or some distributional stability through time. As ghanem2023evaluating observes, such conditions may not always hold in interesting applications. In its general form as presented in this paper, the FIPT, valid under a non-linear transformation of the outcome, is not guaranteed to be preserved under other non-linear transformations ghanem2023evaluating,yamauchi2020difference. Going down this road, nonetheless, has its merits. There do exist empirical applications where treatment can be deemed “as (if) randomly assigned" unconditionally or conditional on some pre-treatment covariates. For example, the empirical example considered in his paper has treatment determined by an exogenous shock. Treatment status, out of the given population, can be considered as (if) randomly assigned and concerns about significant heterogeneity between the treated and untreated groups can be laid to rest. Moreover, the proposed method provides a pathway to inference not only on DTT but also on QTT even when outcomes are non-continuous. The distributional DiD of havnes-mogstad-2015-universal,roth-santanna-2023parallel is a special case of the proposed method. There are other appealing features of the proposed method. The method does not require panel data; repeated cross-sections are usable. Results in this paper cover fairly general sampling schemes: (1) balanced panels, (2) unbalanced panels, and (3) repeated cross-sections under both the canonical two-group and staggered treatment designs. Like the Changes-in-Changes (CIC) and partial identification approaches in athey-imbens-2006,ghanem2023evaluating, respectively, only two periods are required. The proposed method applies to discrete, mixed, and continuous outcomes, and it naturally extends to the staggered treatment setting unlike, e.g., the CIC of athey-imbens-2006 -- see discussion on page C33 of wooldridge-2023-simple. The key identifying assumption of the method is a DF-generalisation of the index parallel trends assumption of wooldridge-2023-simple -- see equations 2.6 and 2.7 therein. By including covariates, an extension of the baseline model shows that weaker conditional forms of the functional index parallel and functional no-anticipation assumptions are sufficient for identification. The rest of the paper is laid out as follows. (ref) spells out the identification assumptions and the identification result in a two-period baseline model. (ref) provide the estimator and the asymptotic theory, respectively, for the baseline model. (ref) extends the baseline model and its asymptotic theory to multiple periods pre-treatment, post-treatment, or both under non-staggered (two-group) and staggered treatment designs. (ref) extends the two-period baseline model to accommodate covariates. Simulations in (ref) examine the small sample performance of the proposed method, the empirical application in (ref) demonstrates its empirical usefulness, and (ref) concludes. (ref) collects the proofs of all theoretical results and (ref) provides extra simulation results. \section{The Baseline Two-Period Model} Let $Y$ denote the outcome variable on a non-binary support $\mathcal{Y}\subseteq \mathbb{R}$.\footnote{The binary case is to the non-linear DiD studied in puhani2012treatment,wooldridge-2023-simple.} The baseline setting considered in this paper involves two groups, namely a treated group $(d=1)$ and an untreated group $(d=0)$ on the one hand and a pre-treatment $(t=0)$ and post-treatment $(t=1)$ period on the other. Let $Y_{1t}(d)$ and $Y_{0t}(d)$, respectively, denote the treated and untreated potential outcomes of a unit in group $d\in\{0,1\} $ at period $t\in \{0,1\} $. \subsection{Preliminaries} Denote by $F_{Y_{d't}|D=d}(y)$, the DF of $Y_{d't}(d)$, i.e., the treated $d'=1$ (untreated $d'=0$) potential outcome of a unit in group $d$ at period $t$. For example, $F_{Y_{01}|D=1}(y)$ would have been the DF of a treated unit had it not been treated. The Distributional Treatment Effect on the Treated (DTT) is defined as \begin{equation} DTT(y) = F_{Y_{11}|D=1}(y) - F_{Y_{01}|D=1}(y), \quad y\in\mathcal{Y} \end{equation} where $\mathcal{Y}$ denotes the support of the outcome. While QTT is already well-known and easily interpretable, e.g., athey-imbens-2006,callaway-li-2019, DTT does not appear to have received similar emphasis in the literature. This may be due to the easier interpretability of the QF and QTT in terms of units of the outcome, while the DF and DTT, which are in terms of probabilities, are less intuitive. That notwithstanding, DTT in certain empirical contexts can be interesting in its own right. The interpretation of $DTT(y)$ applies a natural functional generalisation of the Average treatment effect on the treated (ATT) for binary outcomes -- wooldridge-2023-simple,puhani2012treatment. Although labelled DTT in both papers, the parameter of interest in gutknecht2024generalized applies to the probability mass function of an ordered discrete outcome, whereas (ref) applies to the Cumulative Distribution Function (CDF). Thus, the framework of gutknecht2024generalized rules out continuous or mixed outcomes, unlike the current paper. $DTT(y)$ in (ref) is the change in the probability that the outcome of the treated group is no larger than $y$ thanks to treatment. To make the interpretation more concrete, consider the following empirical example. \begin{example}[$DTT$ -- Police and car theft] diTella-2004 considers the impact of police on crime. The treatment $D$ is increased police presence, the outcome $Y$ is the count of car theft, and the unit of observation is the block. $F_{Y_{11}|D=1}(y):= \mathbb{P}(Y_{11}\leq y|D=1) $ is the probability/likelihood that at most $y$ vehicles are stolen in a block with increased police presence. The counterfactual DF, namely $F_{Y_{01}|D=1}(y):= \mathbb{P}(Y_{01}\leq y|D=1) $ would be the probability/likelihood that at most $y$ vehicles are stolen in a block with increased police presence were there no increased police presence. $-DTT(y)$ is thus the reduction in the likelihood of at most $y$ incidents of car theft on a block with increased police presence thanks to increased police presence. \end{example} The QF, as a function of a generic DF $F(y)$, is a crucial function used in defining the $QTT$. \begin{definition}[Quantile Function] Given a DF $y \mapsto F(y),\ y\in\mathcal{Y}$, $\tau \mapsto F^{-1}(\tau)$ is the corresponding QF defined by the left-inverse function $F^{-1}(\tau) := \inf\{y \in \mathcal{Y} : F(y) \geq \tau\}$ if $\displaystyle\sup_{y\in \mathcal{Y}}\{F(y) \}\geq \tau $ and $F^{-1}(\tau) := \sup\{y\in\mathcal{Y}\}$ otherwise for $\tau \in [0,1]$. \end{definition} The Quantile Treatment Effect on the Treated is defined as \begin{equation} QTT(\tau) = F_{Y_{11}|D=1}^{-1}(\tau) - F_{Y_{01}|D=1}^{-1}(\tau). \end{equation} The following example provides a QTT counterpart of the interpretation in (ref). \begin{example}[QTT -- Police and car theft] $F_{Y_{11}|D=1}^{-1}(\tau)$ is the $\tau^{th}$ quantile of the count of car theft on a block with increased police presence, and $F_{Y_{01}|D=1}^{-1}(\tau)$ would be the $\tau^{th}$ quantile on a block with increased police presence were there no increased police presence. Thus, $QTT(\tau)$ is the decrease in the $\tau^{th}$ quantile of the count of car theft on a block with increased police presence, thanks to increased police presence. \end{example} \subsection{Identification} The fundamental problem of identifying DTT and QTT is that, unlike $F_{Y_{11}|D=1}(y)$ which is identified from the data sampling process, the distribution of the untreated potential outcome of treated units in the post-treatment period, i.e., $F_{Y_{0t}|D=1}(y)$, is not identified without assumptions. Thus, identification assumptions are needed much like in the wider DiD and QTT literature, e.g., athey-imbens-2006,callaway-li-2019,bonhomme-sauder-2011,ghanem2023evaluating,gutknecht2024generalized,wooldridge-2023-simple. To this end, a functional index parallel trends (FIPT) and a functional no-anticipation (FNA) are leveraged to identify $F_{Y_{0t}|D=1}(y)$. Let $\widetilde{Y}_{d't}(y)$ be an underlying continuously distributed latent variable corresponding to untreated potential outcomes of group $d$ such that $ \text{\usefont{U}{bbm}{m}{n}1}\{Y_{0t}(d)\leq y\} = \text{\usefont{U}{bbm}{m}{n}1}\{\widetilde{Y}_{0t}(d;y)\leq 0\}$. Then, assume the latent untreated potential outcome follows \[ \widetilde{Y}_{0t}(D;y) = U_{Dt}(0) - \alpha(y) - \beta(y)D - t\gamma(y), \ t\in \{0,1\}. \] Observe that the above restriction is imposed only on the untreated potential outcome and the treated potential outcome is left unrestricted. From the foregoing, \begin{align} F_{Y_{0t}|D=d}(y) &= \mathbb{P}(\widetilde{Y}_{0t}(D;y) \leq 0|D=d) \nonumber \\ & = \mathbb{P}\big(U_{dt}(0) \leq \alpha(y) + d\beta(y) + t\gamma(y)\big)\nonumber\\ & = \theta \Phi_d\big(\alpha(y) + d\beta(y) + t\gamma(y)\big) + (1-\theta)\widetilde{\Phi}_t\big(\widetilde{\alpha}(y) + d\widetilde{\beta}(y) + t\widetilde{\gamma}(y)\big) \end{align} for a strictly increasing functions $\Phi_d,\widetilde{\Phi}_t: \mathbb{R} \rightarrow (0,1), \ (d,t,\theta) \in \{0,1\}^3 $. (ref) allows the CDF of untreated potential outcomes to vary by period or group. The approach to imposing identifying restrictions on latent untreated potential outcomes has antecedents in the literature -- e.g., wooldridge-2023-simple,yamauchi2020difference,gutknecht2024generalized. The threshold-crossing framework used in gutknecht2024generalized,yamauchi2020difference, for example, is not practical in the current setting as outcomes in this paper are not only discrete, they can also mixed or continuous. Moreover, the flexibility afforded by allowing the function valued $\big(\alpha(y),\beta(y),\gamma(y)\big)', y\in\mathcal{Y}$, unlike wooldridge-2023-simple, allows the researcher to locally approximate the DFs of untreated potential outcomes and avoid global distributional assumptions on $Y_{01}(1)$. It remains to identify $F_{Y_{01}|D=1}(y)$ from (ref). Consider the following parallel trends and no-anticipation assumptions. \begin{assumption}[Functional Index Parallel Trends] For each $y\in\mathcal{Y}$, \begin{align*} F_{Y_{0t}|D=d}(y) &= \theta \Phi_d\big(\alpha(y) + d\beta(y) + t\gamma(y)\big) + (1-\theta)\widetilde{\Phi}_t\big(\widetilde{\alpha}(y) + d\widetilde{\beta}(y) + t\widetilde{\gamma}(y)\big) \end{align*}where $\theta\in \{0,1\}$ is a known constant, $\Phi_d,\widetilde{\Phi}_t:\mathbb{R} \mapsto [0,1] $ and the inverses $\Phi_d^{-1},\widetilde{\Phi}_t^{-1}: [0,1] \mapsto \mathbb{R} $ are known strictly increasing and continuously differentiable functions for $d \in \{0,1\}$ or $t \in \{0,1\}$. \end{assumption} (ref) is imposed on the untreated potential outcome only -- the DF of the treated potential outcome in the post-treatment period is left unrestricted. Also, (ref) requires that $\Phi_d(\cdot)$ or $ \widetilde{\Phi}_t $ be known/specified. In practice, one can use known cumulative distribution functions such as the standard normal, the logistic, or the Cauchy. (ref) with $\theta=1$ allows the functional form of the DF of untreated potential outcomes to vary across groups but not across time; this bears a striking similarity to distributional stability assumptions in the literature -- cf. callaway-li-2019,callaway-li-oka-2018,ghanem2023evaluating. The smoothness condition imposed on $\Phi_d(\cdot)$ is crucial for the asymptotic results -- it is an essential ingredient in circumventing the failure of existing inference techniques that require Hadamard differentiability of the QF. By flexibly modelling each point $y$ on $\mathcal{Y}$, the choice of $\Phi_d(\cdot)$ is a working CDF, i.e., it ought not hold stricto sensu as the approximation of the DF is local, and not global distributional assumptions. This property is borne out in the simulation results in (ref). No or limited anticipation assumptions often encountered in the DiD literature have their analogue in the current setting. \begin{assumption}[Distributional No Anticipation] For each $y\in\mathcal{Y}$, \[F_{Y_{10}|D=1}(y) - F_{Y_{00}|D=1}(y)=0.\] \end{assumption} (ref) says the DF of both treated and untreated potential outcomes of the treated group in the pre-treatment period is the same -- this rules out anticipatory effects of treatment at any point of the distribution of the outcome. Per the running example, (ref) holds as treatment -- increased police presence -- is driven by an unanticipated event. Potential perpetrators of car theft could not have (1) foreseen the terrorist attack and (2) subsequently adjusted their tendency to commit car theft before treatment actually began.\footnote{Implicit in this reasoning is an almost zero time lag between the terrorist attack and the treatment, i.e., increased deployment of police officers -- see diTella-2004.} With (ref) in hand, one can proceed with the identification of the counterfactual DF $F_{Y_{01}|D=1}(y)$. \begin{theorem}[Identification] Suppose (ref) hold, then, $F_{Y_{01}|D=1}(y)$ is identified for $\theta\in \{0,1\}$ and each $y\in\mathcal{Y}$: \begin{equation} \begin{split} F_{Y_{01}|D=1}(y) &= \theta\Phi_1\Big(\Phi_1^{-1}\big(F_{10}(y)\big) + \Phi_0^{-1}\big(F_{01}(y)\big) - \Phi_0^{-1}\big(F_{00}(y)\big) \Big)\\ &+ (1-\theta)\widetilde{\Phi}_1\Big(\widetilde{\Phi}_0^{-1}(F_{10}(y)) + \widetilde{\Phi}_1^{-1}(F_{01}(y)) - \widetilde{\Phi}_0^{-1}(F_{00}(y)) \Big). \end{split} \end{equation} \end{theorem} Consider the case $\theta=1$. It follows from (ref) that $F_{Y_{00}|D=d}(y) = \Phi_d(\alpha(y)+d\beta(y)) $ and $F_{Y_{01}|D=d}(y) = \Phi_d(\alpha(y)+d\beta(y) +\gamma(y)) $. Then combining the above, and thanks to the invertibility of $\Phi_d(\cdot)$, $\gamma(y) = \Phi_d^{-1}\big(F_{Y_{01}|D=d}(y)\big) - \Phi_d^{-1}\big(F_{Y_{00}|D=d}(y)\big)$ for each $d\in\{0,1\}$. The latter implies a parallel trends assumption in a non-linear transformation of $F_{Y_{0t}|D=d}(y), \ t\in\{0,1\} $ for each $y\in\mathcal{Y}$ -- cf. wooldridge-2023-simple. Under (ref), (ref) has the equivalent expression for all $y\in\mathcal{Y}$ \begin{align*} \Phi_1^{-1}(F_{Y_{01}|D=1}(y)) - \Phi_1^{-1}(F_{Y_{00}|D=1}(y)) &= \Phi_0^{-1}(F_{Y_{01}|D=0}(y)) - \Phi_0^{-1}(F_{Y_{00}|D=0}(y)) if \theta=1 and\\ \widetilde{\Phi}_1^{-1}(F_{Y_{01}|D=1}(y)) -\widetilde{\Phi}_0^{-1}(F_{Y_{00}|D=1}(y)) &= \widetilde{\Phi}_1^{-1}(F_{Y_{01}|D=0}(y)) - \widetilde{\Phi}_0^{-1}(F_{Y_{00}|D=0}(y)) \text{ if } \theta=0. \end{align*} The above equations are parallel trends conditions on strictly monotonic transformations of the DF of untreated potential outcomes $F_{Y_{0t}|D=d}(y)$, $(d,t) \in \{0,1\}^2 $ for all $y\in\mathcal{Y}$. If $\theta=1$, (ref) entails a distributional stability condition across time, i.e., the distribution of $U_{Dt}(0)$ does not vary over time although it can depend on the treatment status $D$. A symmetric argument applies with distributional stability in treatment $D$ while allowing the distribution to vary with time $t$ if $\theta=0$. (ref) only considers the boundary points of the unit interval $[0,1]$. The following result explores the generalisation of the condition $\theta\in\{0,1\}$ in (ref) to $\theta\in [0,1]$. \begin{proposition}[Identification] (a) Suppose (ref) hold with $\theta\in [0,1]$, then, $F_{Y_{01}|D=1}(y)$ is not identified for each $y\in\mathcal{Y}$ if $\theta\in (0,1)$. (b) Under the restrictions $\Phi(x)=\widetilde{\Phi}(x)=x$ and $\alpha(y)=\widetilde{\alpha}(y)$, $\beta(y)=\widetilde{\beta}(y)$, and $\gamma(y)=\widetilde{\gamma}(y)$, the identification of $F_{Y_{01}|D=1}(y)$ is invariant to $\theta\in [0,1]$: \[ F_{Y_{01}|D=1}(y)=F_{10}(y) + F_{01}(y) - F_{00}(y). \] \end{proposition} (ref) presents an impossibility result for generic strictly increasing $\Phi_d(\cdot)$ and $\widetilde{\Phi}_t(\cdot)$ when $\theta\in (0,1)$. For the specific choices of the identity functions $\Phi(x)=\widetilde{\Phi}(x)=x$, identification is invariant to $\theta\in[0,1]$. This latter case coincides with the distributional DiD of havnes-mogstad-2015-universal,roth-santanna-2023parallel. \subsection{Examples} The following examples illustrate some data-generating processes of untreated potential outcomes. The case $\theta=1$ is considered for ease of exposition. \begin{example} (ref) results in a special case of the counterfactual when the working CDF $\Phi_d(\cdot), \ d\in\{0,1\} $ is the standard uniform, namely $\Phi_d(x)=\widetilde{\Phi}_t(x)= x\text{\usefont{U}{bbm}{m}{n}1}\{x\geq 0\} + (1-x)\text{\usefont{U}{bbm}{m}{n}1}\{x>1\} $. (ref) becomes \begin{equation} F_{Y_{01}|D=1}(y) = \Pi_{01}(y)\text{\usefont{U}{bbm}{m}{n}1}\{ \Pi_{01}(y) \geq 0 \} + (1-\Pi_{01}(y))\text{\usefont{U}{bbm}{m}{n}1}\{\Pi_{01}(y) > 1\} \end{equation} where $\Pi_{01}(y): = F_{11}(y) + F_{01}(y) - F_{00}(y), \ y\in\mathcal{Y}$. \end{example} The distributional DiD -- see havnes-mogstad-2015-universal and roth-santanna-2023parallel -- is a special case of (ref) without support restrictions. It has the virtue of a “non-parametrically" identified counterfactual DF as all three summands on the right-hand side of (ref) are non-parametrically identified from the sampling process of the data and $\Phi_d(\cdot)$ “disappears". For this special case, an important insight that can be drawn from roth-santanna-2023parallel concerning the invariance of identification to strictly monotonic transformations of the outcome; it holds if and only if (ref) holds without support restrictions under the restriction $\Phi_0(\cdot)=\Phi_1(\cdot)$. The characterisation of roth-santanna-2023parallel shows that the invariance to strictly monotonic transformations holds if the DF of untreated potential outcomes for each group and period is a mixture of a time-varying and group-varying DF. Without the support restrictions of the standard uniform in (ref), $F_{Y_{01}|D=1}(y)$ is not guaranteed to be a DF -- cf. roth-santanna-2023parallel. Thus, without support restrictions, the test proposed in roth-santanna-2023parallel is applicable to detect violations of (ref) in the special case of (ref). Thus, $\Phi_1(\cdot)$ in the general formulation (ref) plays the role of enforcing the $[0,1]$ bound on the counterfactual DF. This, however, has the drawback of eliminating the testability of the implications of (ref). \begin{example} Consider a standard two-way fixed effect model: $ Y_{0t}(d) = U_{0t}(d) - \alpha - d\beta - t\gamma, \ t\in \{0,1\} $ where $U_{0t}(d)$ is continuously distributed for each $d\in\{0,1\}$. Then, $F_{Y_{0t}|D=d}(y) = \Phi_d\big(\alpha(y) + d\beta + t\gamma\big), $ where $\alpha(y) = y+\alpha $ and $ \Phi_d(y)=\mathbb{P}(U_{0t}(d)\leq y)$. Thus only $\alpha(y)$ varies in $y$. From the proof of (ref), the following restrictions hold: (1) $\alpha(y) = \alpha + y = \Phi_0^{-1}\big(F_{00}(y)\big) $; (2) $ \Phi_1^{-1}\big(F_{10}(y)\big) = \alpha + \beta +y $; and (3) $\Phi_0^{-1}\big(F_{01}(y)\big) = \alpha + \gamma + y $. \end{example} (ref) involves a \emph{pure location shift} in the DF at $y\in \mathcal{Y}$. Varying $y$ and fixing $D=d$ results in the variation of neither slope parameter. Further, the “latent" variable corresponding to the untreated group and pre-treatment of the treated group coincides with the observed continuous outcome. However, depending on whether $y$ is in the tail or the middle of the distribution of ${Y_{0t}|D=d}$, the effects of the treatment can differ. In (ref), the latent variable coincides with the observed outcome. This is no longer the case when the observed outcome is limited, e.g, discrete, or censored. Nonetheless, thanks to the latent variable representation of discrete outcomes, the approach adopted in this paper continues to be feasible even with discrete outcomes. This is further explored in (ref) and simulations in (ref). \begin{example} Consider a count untreated potential outcome $Y_{01}(d)$ defined on the support $\{y_0, y_1,\ldots,y_K\}$. Assume an underlying latent variable: $\widetilde{Y}_{0t}(d) = U_{0t}(d) - \alpha - d\beta - t\gamma, \ t\in \{0,1\}$. Then, for ordered thresholds $\{\tilde{y}_0, \tilde{y}_1, \ldots, \tilde{y}_K\} $ such that $ \mathbb{P}( \widetilde{Y}_{0t}(d) < \tilde{y}_0) = \mathbb{P}( \widetilde{Y}_{0t}(d) > \tilde{y}_K) = 0 $, $$ \mathbb{P}(Y_{0t}(d) \leq y) = \mathbb{P}( U_{0t}(d) \leq \alpha_k + d\beta + t\gamma) = \Phi_d(\alpha_k + d\beta + t\gamma) $$ where $\alpha_k:= \alpha+\tilde{y}_k$ and $y\in (y_{k-1},y_k]$ if $k\geq 1$ else $y=y_0$. \end{example} In (ref), the latent $\widetilde{Y}_{0t}(d)$ no longer coincides with the untreated potential outcomes. The observed outcome is discrete, but the latent variable follows known distributions. (ref) still holds at different levels of $y$ with discrete outcomes with only the location parameter $\alpha_k$ varying on the discrete support $\{\alpha+\tilde{y}_0, \alpha+\tilde{y}_1, \ldots, \alpha+\tilde{y}_K \}$. Finally, the following example concerns discrete outcomes censored from below. \begin{example} Let $\widetilde{Y}_{0t}(d) = U_{0t}(d)- \alpha - \beta d - t\gamma $, where $U_{0t}(d)$ is continuously distributed for each $d\in \{0,1\}$ and the (censored) potential outcome is defined such that $Y_{0t}(d) = \max\{\lceil \widetilde{Y}_{0t}(d)\rceil,0\} $. $Y_{0t}(d)$ is discrete and $\widetilde{Y}_{0t}(d)$ is the latent generating variable. For $y \in \mathcal{Y}:= \{0,1,\ldots\} $: \begin{align} F_{Y_{0t}|D=d}(y) &= \mathbb{P}(Y_{0t}(d) \leq y) \nonumber \\ &= \mathbb{P}(\max\{\lceil \widetilde{Y}_{0t}(d)\rceil,0\} \leq y) \nonumber \\ &= \mathbb{P}(\lceil \widetilde{Y}_{0t}(d)\rceil \leq y) \nonumber \text{ since $y\geq 0$ } \\ &= \mathbb{P}( \widetilde{Y}_{0t}(d) \leq y) \nonumber \text{ since } \lceil Y\rceil >y \text{ iff } Y>y \text{ for discrete } y \\ & = \Phi_d(\alpha(y) + \beta d + t\gamma) \nonumber \end{align} where $\alpha(y) = \alpha+y $ and $\Phi_d(y) = \mathbb{P}( U_{0t}(d)\leq y) $. \end{example} Like in (ref), the latent variable does not coincide with untreated potential outcomes. Moreover, the outcome variable is censored from below and is discrete. The above representation involves a pure location shift in $y$. \section{Estimation} The parameters of interest are DFs, QFs, $DTT(y), \ y\in\mathcal{Y} $ and $QTT(\tau), \ \tau \in (0,1) $. The first terms in (ref) and (ref) are estimable from data using observed outcomes for the treated group in the post-treatment period. It remains to estimate the counterfactual DF $F_{Y_{01}|D=1}(y)$ and counterfactual QF $F_{Y_{01}|D=1}^{-1}(\tau)$, which are identified for each $(y,\tau) \in \mathcal{Y}\times (0,1) $ under the conditions of (ref). Let data $W_{it}:=(Y_{it},D_{it})$ denote the observation corresponding to a unit $i$ at period $t$. Also let $n_t$ denote the sample size at period $t$, and $\hat{p}_{dt}:= n_t^{-1}\sum_{i=1}^{n_t}\text{\usefont{U}{bbm}{m}{n}1}\{D_{it}=d\} $ denote the fraction of observations in group $d \in \{0,1\} $ at period $t \in \{0,1\} $. Let $\widehat{F}_{dt}(y):= n_t^{-1}\sum_{i=1}^{n_t} \text{\usefont{U}{bbm}{m}{n}1}\{D_{it}=d\}\text{\usefont{U}{bbm}{m}{n}1}\{Y_{it}\leq y\}/\hat{p}_{dt}$ be the empirical CDF of group $d$ at period $t$. Estimation and theoretical results under the case $\theta=0$ follow that of $\theta=1$ analogously, so it is suppressed throughout the rest of the paper for ease of exposition. From the proof of (ref), $\big(\alpha(y),\beta(y),\gamma(y)\big)' = \Big(\Phi_0^{-1}\big(F_{00}(y)\big), \Phi_1^{-1}\big(F_{11}(y)\big) - \alpha(y), \Phi_0^{-1}\big(F_{01}(y)\big) - \alpha(y) \Big)'$. Applying the analogue and plug-in principles provide the consistent estimators: $\big(\hat{\alpha}(y), \big)' = \Big(\Phi_0^{-1}\big(\widehat{F}_{00}(y)\big), \Phi_1^{-1} \big(\widehat{F}_{10}(y)\big)-\hat{\alpha}(y), \Phi_0^{-1} \big(\widehat{F}_{01}(y)\big)-\hat{\alpha}(y) \Big)' $. An estimator of the counterfactual distribution is thus given by \begin{align*} \widehat{F}_{Y_{01}|D=1}(y) &= \Phi_1\big(\hat{\alpha}(y) + \hat{\beta}(y) + \hat{\gamma}(y)\big)\\ & = \Phi_1\Big(\Phi_1^{-1} \big(\widehat{F}_{10}(y)\big) + \Phi_0^{-1} \big(\widehat{F}_{01}(y)\big) - \Phi_0^{-1} \big(\widehat{F}_{00}(y)\big) \Big). \end{align*} $DTT$ and $QTT$ can then be estimated using \[ \widehat{DTT} (y) = \widehat{F}_{Y_{11}|D=1}(y) - \widehat{F}_{Y_{01}|D=1}(y) \] and \[ \widehat{QTT}(\tau) = \widehat{F}_{Y_{11}|D=1}^{-1}(\tau) - \widehat{F}_{Y_{01}|D=1}^{-1}(\tau) \] where $\widehat{F}_{Y_{11}|D=1}(y) = \widehat{F}_{11}(y)$. \section{Asymptotic Theory} This section concerns the asymptotic theory of the baseline two-period model. The sampling scheme is, however, specified in a generic form with $\mathcal{T}$ pre-treatment periods where no unit is treated, and $T$ post-treatment periods when some units are treated. It is maintained throughout this text that $\mathcal{T}$ and $T$ are fixed. This baseline entails varying degrees of temporal dependence via special cases: balanced panels, unbalanced panels, and repeated cross-sections. \subsection{Sampling} Let $N$ denote the number of unique cross-sectional units observed in the entire sample across periods $[-\mathcal{T}:T]\setminus \{0\}$ where $ [-\mathcal{T}:T]:=\{- \mathcal{T}, \ldots,T\} $, and $S_{jt}$ be a dummy which equals one if unit $j \in \{1,\ldots,N\} $ is observed at period $t$. Thus $n_t:= \sum_{j=1}^{N} S_{jt} $ is the number of observations at period $t$, and $n:=\sum_{t=-\mathcal{T}}^T n_t $ is the size of the pooled sample. Let $W_{jt}:= (Y_{jt},D_{jt})$ denote observed data, and $\widetilde{W}_{jt}:= (Y_{jt},D_{jt})$ denote a latent version of it such that $ W_{jt} = \text{\usefont{U}{bbm}{m}{n}1}\{S_{jt}=1\}\widetilde{W}_{jt} $. This sampling framework is particularly useful as it spans the spectrum of sampling schemes of interest: balanced panels, unbalanced panels, and repeated cross-sections. Define $\widetilde{W}_j:=\{(\widetilde{W}_{jt},S_{jt}), t\in [-\mathcal{T} : T]\}$. \begin{assumption} $ \{\widetilde{W}_j: 1\leq j \leq N \} $ are independent and identically ($i.i.d.$) distributed. \end{assumption} By imposing the sampling assumption on $\widetilde{W}_j$, (ref) allows for several interesting sampling schemes as special cases: (1) arbitrary dependence across time for the same cross-sectional unit (2) balanced panels, (3) unbalanced panels, and (4) repeated cross sections. This makes the results presented in this paper generally applicable whenever the posited identification conditions viz. (ref) hold. Further, jointly imposing (ref) precludes non-random missingness that may jeopardise valid inference on the DFs. (ref) says data for each group at each period are $i.i.d.$. It does not impose stationarity or restrict dependence across time. However, dependence must be observable, i.e., cross-sectional units through time (if they occur) are observable to the researcher. The following regularity assumption is imposed on $\displaystyle \rho_t: = \lim_{n\rightarrow\infty}n_t/n$ and $p_{dt}:=\mathbb{E}[D_t=d]$. \begin{assumption} $(\rho_t,p_{dt})' \in (0,1)^2 $ for each $(d,t) \in \{0,1\}\times [-\mathcal{T}:T] $. \end{assumption} (ref) ensures the number of observations in any period is not asymptotically negligible relative to the sample size in other periods. This is important in repeated cross-sections or unbalanced panels. In balanced panels, however, (ref) is not necessary as $n_t=N$ for any $t$ by construction. \subsection{Weak Convergence of DFs and DTT} For ease of exposition, the case with $\theta=1$ in (ref) is maintained throughout the rest of the paper as the other possible cases follow analogously. Define the empirical process $\widetilde{H}_{dt}(y):=\sqrt{n}(\widehat{F}_{dt}(y)-F_{dt}(y))$, the space $\mathbb{S}:=\ell^\infty(\mathcal{Y})$, and $p_{dt}:= \mathbb{E}[D_t=d] $. Empirical processes of the form $\widetilde{H}_{dt}(y)$ constitute the building blocks of the asymptotic theory in this paper. The following is a standard weak convergence result of such processes. \begin{lemma} Under (ref), $\widetilde{H}_{dt}(y) \rightsquigarrow \mathbb{G}_{dt}^H(y) $ in the the space $\mathbb{S}:=\ell^\infty(\mathcal{Y})$ for each pair $(d,t) $, where $\mathbb{G}_{dt}^H$ is a tight Gaussian process with mean $0$ and covariance function $\Omega_{dt}^H(y_1,y_2)$ defined on $\mathbb{S}$. The expression of the covariance function is provided in the appendix. \end{lemma} The next result studies the asymptotic behaviour of the $\widehat{F}_{Y_{01}|D=1}(y)$ and $\widehat{DTT}(y)$. \begin{theorem} Suppose (ref) hold, then for any $y\in \mathcal{Y} $, (a) \[ \sqrt{n}(\widehat{F}_{Y_{01}|D=1}(y) - F_{Y_{01}|D=1}(y)) \rightsquigarrow \mathbb{G}_{01}^C(y) \] and (b) \[ \sqrt{n}(\widehat{DTT}(y) - DTT(y)) \rightsquigarrow \mathbb{G}_{DTT}(y) \] where $\mathbb{G}_{01}^C(y)$ and $\mathbb{G}_{DTT}(y)$ are tight Gaussian processes with mean $0$ and covariance functions $\Omega_{01}^C(y_1,y_2)$ and $\Omega_{DTT}(y_1,y_2)$ defined on $\mathbb{S}$. The expressions of the covariance functions are provided in the appendix. \end{theorem} A crucial component to the result in (ref) is the application of the functional delta method. This is where the differentiability of $\Phi_d(\cdot)$ and $\Phi_d^{-1}(\cdot) $ for each $d\in \{0,1\}$ plays a central role in ensuring the use of standard uniform inference techniques even if the distribution of potential outcomes is non-continuous. \section{Extension - Multiple Periods} The baseline setting considered so far in the paper concerns a two-group-two-period setting. In several empirical contexts, (ref) in this paper, for example, panel data (balanced or unbalanced) may be available. Moreover, treatment in some settings need not occur simultaneously for all treated units. Also, in such settings, treatment effects may be heterogeneous in time and group. This section extends results in the baseline model, namely (ref) to multiple periods under non-staggered (two-group) and staggered (multiple-group) designs. The maintained assumption in the presence of multiple post-treatment periods is that treatment is an absorbing state, i.e., a unit remains treated after it receives treatment in a post-treatment period. The following notation are useful: (1) $[-\mathcal{T}]:= \{-\mathcal{T},\ldots,-1\} $ and (2) $[T]:= \{1,\ldots,T\} $. \subsection{Non-staggered design} This section considers the extension of the baseline model to the non-staggered multiple-period (NSMP) setting with common treatment timing. In this setting, there are $\mathcal{T}$ pre-treatment periods and $T$ post-treatment periods. In terms of identification, the building block involves one pre-treatment period $t'\in [-\mathcal{T}]$ and one post treatment period $t\in[T]$ \`a la callaway-santanna-2020. Define $t_t:= \text{\usefont{U}{bbm}{m}{n}1}\{t\geq 1\} $. \begin{namedassumption}{(ref)-NSMP}[Functional Index Parallel Trends] For each $y\in\mathcal{Y}$, pre-treatment period $t' \in \{ - \mathcal{T}, \ldots, -1\}$, and post-treatment period $t \in \{ 1, \ldots,T\}$, \begin{align*} F_{Y_{0t}|D=d}(y) &= \Phi_d(\alpha_{t'}(y) + d\beta_{t'}(y) + t_t\gamma_{tt'}(y)). \end{align*} \end{namedassumption} The suitability of each pre-treatment period as stated in (ref) implies the following restriction: \[ \Phi_d(\alpha_{t_a'}(y) + d\beta_{t_a'}(y) + t_t\gamma_{tt_a'}(y)) = \Phi_d(\alpha_{t_b'}(y) + d\beta_{t_b'}(y) + t_t\gamma_{tt_b'}(y)) \]for all $(t_a',t_b')\in \{ - \mathcal{T}, \ldots, -1\}^2$ and each fixed $t \in [T] $ and $y\in\mathcal{Y}$.\footnote{A hypothesis test of such “functional over-identifying restrictions" lies beyond the scope of the current paper.} The following provides the form of the no-anticipation identification condition suited to the non-staggered multiple-period setting. \begin{namedassumption}{(ref)-NSMP}[Functional No Anticipation] For each $y\in\mathcal{Y}$ and $t'\in[-\mathcal{T}]$, \begin{align*} F_{Y_{1t'}|D=1}(y) = F_{Y_{0 t'}|D=1}(y). \end{align*} \end{namedassumption} As there are possibly multiple pre-treatment periods, the notation $F_{Y_{0 t}^{t'}|D=1}(y)$ is used to emphasise a counterfactual distribution function constructed using the pre-treatment period $t'$. The following result extends the baseline identification result in (ref) to the NSMP setting. Although identification of the counterfactual DF $F_{Y_{0 t}|D=1}(y)$ ought to be invariant to the pre-treatment period $t'$ under (ref), the notation $F_{Y_{0 t}^{t'}|D=1}(y)$ is used for emphasis and clarity. \begin{corollary}[Identification -- NSMP Setting] Suppose (ref) hold, then, $F_{Y_{0 t}^{t'}|D=1}(y)$ is identified for each $y\in\mathcal{Y}$, $t\in [T]$, and $t'\in[-\mathcal{T}]$: \begin{equation*} F_{Y_{0 t}^{t'}|D=1}(y) = \Phi_1\big(\Phi_1^{-1}(F_{Y_{1t'}}(y)) + \Phi_0^{-1}(F_{Y_{0t}}(y)) - \Phi_0^{-1}(F_{Y_{0t'}}(y)) \big), \ y\in\mathcal{Y}. \end{equation*} \end{corollary} Next, the following result extends (ref) to the NSMP setting. \begin{corollary} Suppose (ref) hold, then for any $y\in \mathcal{Y} $, (a) \[ \sqrt{n}(\widehat{F}_{Y_{0 t}^{t'}|D=1}(y) - F_{Y_{0 t}^{t'}|D=1}(y)) \rightsquigarrow \mathbb{G}_{0t}^{t'}(y) \] and (b) \[ \sqrt{n}(\widehat{DTT}_t^{t'}(y) - DTT_t^{t'}(y)) \rightsquigarrow \mathbb{G}_{DTT_t^{t'}}(y) \] where $\mathbb{G}_{0t}^{t'}(y)$ and $\mathbb{G}_{DTT_t^{t'}}(y)$ are tight Gaussian processes with mean $0$ and covariance functions $\Omega_{0t}^{t'}(y_1,y_2)$ and $\Omega_{DTT_t^{t'}}(y_1,y_2)$ defined on $\mathbb{S}$. The expressions of the covariance functions are provided in the appendix. \end{corollary} \subsection{Staggered design} In addition to having multiple post-treatment periods, one sometimes encounters staggered treatment designs where not all units are treated simultaneously. Further, the effect of treatment may depend on when a unit first receives treatment, i.e., its group, thus introducing another dimension to heterogeneity in the effects of treatment on the treated. Thus, this section extends the inferential tools developed in (ref) to explore heterogeneity in the effects of treatment in (1) $(y,\tau) \in \mathcal{Y}\times (0,1) $, i.e., in the DF or QF, (2) in post-treatment periods $t=1,\ldots,T$, and in groups $g\in\mathcal{G}\setminus \infty $. Define the set of groups $\mathcal{G}:=\{1,\ldots,\mathcal{T},\infty\}$ where $\mathcal{T}$ is the number of post-treatment periods. A unit belongs to group $g\in\mathcal{G}$ if it first receives treatment at period $g \geq 1 $. $g=\infty$ denotes a never-treated group which serves as a control group for treated groups. Then, adapting notation from the preceding sections, let $F_{Y_{gt}|G=g}(y)$ and $F_{Y_{\infty t}|G=g}(y)$ respectively, denote the DF and the counterfactual DF of group $g$ at post-treatment period $t$. To adapt the baseline two-period model to staggered treatment adoption designs, consider for example, a treated group $G=g$ and a never-treated control group $G=\infty$, a group $g$ pre-treatment period $t' < g$, and a group $g$ post-treatment period $t\geq g$. Next, define the following dummies: $d_g:= \text{\usefont{U}{bbm}{m}{n}1}\{G=g\} $ and $t_g:= \text{\usefont{U}{bbm}{m}{n}1}\{t\geq g\} $. \begin{namedassumption}{(ref)-Staggered}[Functional Index Parallel Trends] For each $y\in\mathcal{Y}$, $g\in\mathcal{G}\setminus \infty $, $t\in[g:T]$, and $t' \in[-\mathcal{T}:g-1]\setminus \{0\} $, \begin{align*} F_{Y_{\infty t}^{t'}|G=g}(y) &= \Phi_g\big(\alpha_{t'}^{\infty}(y) + d_g\beta_{t'}^{g\infty}(y) + t_g \gamma_{tt'}^{g\infty}(y)\big). \end{align*} \end{namedassumption} The parameters are indexed by $g\in\mathcal{G}$ for emphasis as there are multiple treated groups under a staggered treatment adoption design. \begin{remark} At a time when units in group $g\in\mathcal{G}$ are treated, i.e., time $g$, groups $ \mathcal{G} \ni g' > g $ are not yet treated. This effectively makes such groups in addition to the never-treated group $G=\infty$ suitable controls for group $g$ at such periods -- see, e.g., callaway-tsyawo-2023treatment. Although interesting from an efficiency perspective, such an extension of (ref) is not explicitly handled in this paper for considerations of space. \end{remark} \begin{namedassumption}{(ref)-Staggered}[Functional No Anticipation] For each $y\in\mathcal{Y}$, $g\in\mathcal{G}\setminus \infty $, and $t'\in [-\mathcal{T}:g-1] \setminus \{0\} $, \begin{align*} F_{Y_{gt'}|G=g}(y) = F_{Y_{\infty t'}|G=g}(y). \end{align*} \end{namedassumption} (ref), respectively, extend the parallel trends and no anticipation assumptions of the baseline case to the staggered treatment adoption setting, using the never treated group as control group. In the special case of panel data or repeated cross-sections with multiple post-treatment periods, there is a single group observed over multiple post-treatment periods. This is a non-staggered design. Let $F_{Y_{\infty t}^{t'}|G=g}(y)$ denote the counterfactual DF based on the pre-treatment period $t'$. \begin{corollary}[Identification - Staggered Treatment Timing] Under (ref), $F_{Y_{\infty t}^{t'}|G=g}(y)$ is identified for each $y\in\mathcal{Y}$, $g\in\mathcal{G} $, and $t\in[g:T]$ given any $t'\in[-\mathcal{T}:g-1]\setminus \{0\} $: \begin{equation} F_{Y_{\infty t}^{t'}|G=g}(y) = \Phi_g\big(\Phi_g^{-1}(F_{Y_{gt'}}(y)) + \Phi_\infty^{-1}(F_{Y_{\infty t}}(y)) - \Phi_\infty^{-1}(F_{Y_{\infty t'}}(y)) \big), \ y\in\mathcal{Y}. \end{equation} \end{corollary} The multiplicity of groups under staggered treatment designs calls for a modification of (ref). Define $p_{gt}:=\mathbb{P}[G_t=g]$. \begin{namedassumption}{(ref)-Staggered} $p_{gt} \in (0,1) $ for any group $g\in\mathcal{G}$ and period $t\in\{-\mathcal{T},\ldots,T\} \setminus \{0\} $. \end{namedassumption} The following result extends (ref) to the staggered treatment design settings. \begin{corollary} Suppose (ref) hold, then for any $y\in \mathcal{Y} $, (a) \[ \sqrt{n}(\widehat{F}_{Y_{\infty t}^{t'}|G=g}(y) - F_{Y_{\infty t}^{t'}|G=g}(y)) \rightsquigarrow \mathbb{G}_{gt}^{\infty t'}(y) \] and (b) \[ \sqrt{n}(\widehat{DTT}_{gt}^{t'}(y) - DTT_{gt}^{t'}(y)) \rightsquigarrow \mathbb{G}_{DTT_{gt}^{t'}}(y) \] where $\mathbb{G}_{gt}^{\infty t'}(y)$ and $\mathbb{G}_{DTT_{gt}^{t'}}(y)$ are tight Gaussian processes with mean $0$ and covariance functions $\Omega_{gt}^{\infty t'}(y_1,y_2)$ and $\Omega_{DTT_{gt}^{t'}}(y_1,y_2)$ defined on $\mathbb{S}$. The expressions of the covariance functions are provided in the appendix. \end{corollary} \subsection{Weighting Schemes} While multiple period settings with both non-staggered and staggered treatment adoption designs generate heterogeneity in treatment effects that may be interesting from a policy perspective, it may be challenging to interpret many group-time average treatment effects if the number of groups and/or periods is relatively large. Therefore, it may be useful to report interesting convex-weighted effects. In what follows, define the convex weighting scheme by $ \{\omega(g,t',t), \ g\in\mathcal{G}\setminus \infty, \ -\mathcal{T},\ldots,g-1 \leq t' \leq g-1, \ g \leq t \leq T \} $. Weighted DFs and DTT are of the general form \begin{align*} F_{Y^\omega}(y) &= \sum_{g\in\mathcal{G}}\sum_{\substack{-t'=g-1 \\ t'\neq 0}}^{\mathcal{T}} \sum_{t=g}^{T} \omega(g,t',t) F_{Y_{gt}|G=g}(y);\\ F_{Y_{\infty }^\omega}(y) &= \sum_{g\in\mathcal{G}}\sum_{\substack{-t'=g-1 \\ t'\neq 0}}^{\mathcal{T}} \sum_{t=g}^{T} \omega(g,t',t) F_{Y_{\infty t}^{t'}|G=g}(y); \text{ and}\\ DTT_\omega(y) &= \sum_{g\in\mathcal{G}}\sum_{\substack{-t'=g-1 \\ t'\neq 0}}^{\mathcal{T}} \sum_{t=g}^{T} \omega(g,t',t) \big(F_{Y_{gt}|G=g}(y) - F_{Y_{\infty t}^{t'}|G=g}(y)\big) \end{align*} where the weighting scheme, which satisfies $\displaystyle \sum_{g\in\mathcal{G}}\sum_{\substack{-t'=g-1 \\ t'\neq 0}}^{\mathcal{T}} \sum_{t=g}^{T} \omega(g,t',t) = 1$, is known and/or estimable from data. Notice that $F_{Y_{gt}|G=g}(y)$, which is identified from the sampling process, does not depend on pre-treatment periods. For example, $\displaystyle \omega(g,t',t) = \frac{\text{\usefont{U}{bbm}{m}{n}1}\{G=g\}}{(\mathcal{T}+g-1) (T-g+1)} $ produces equal weights for all periods and a fixed group $G=g$. The weighting scheme can also be specialised to obtain average functional parameters of a specific group $g'\in\mathcal{G}$ or period $t$ where $ \omega(g,t',t) = 0 $ for any $g\neq g'$ while satisfying the convexity constraint. Assume a researcher is interested in the average DTT of participating in the treatment exactly $e$ periods after the treatment was adopted across all groups that are ever observed to have participated in the treatment for exactly at least $e$ periods. Then the following weighting scheme can be used: \begin{align*} \omega(g,t',t):&= \frac{\omega^e(g,t',t)}{\sum_{\tilde{g}\in\mathcal{G}}\sum_{\substack{-\tilde{t}'=\tilde{g}-1 \\ \tilde{t}'\neq 0}}^{\mathcal{T}} \sum_{\tilde{t}=\tilde{g}}^{T} \omega^e(\tilde{g},\tilde{t}',\tilde{t})}\\ &\text{ where } \omega^e(g,t',t) = \frac{\text{\usefont{U}{bbm}{m}{n}1}\{g+e\leq T \}\text{\usefont{U}{bbm}{m}{n}1}\{t-g = e\}\mathbb{P}(G=g|G+e\leq T)}{\mathcal{T}+g-1} \end{align*} averages over treated groups $e$ periods after being first treated. The quantile treatment on the treated based on weighted DFs can then be defined as \begin{align*} QTT_\omega(\tau) = F_{Y^\omega}^{-1}(\tau) - F_{Y_\infty^\omega}^{-1}(\tau). \end{align*} \begin{remark} Observe that $QTT_\omega(\tau)$ is not convex-weighted QTT averaged over groups and post-treatment periods but rather QTT obtained from convex-weighted DFs averaged over groups and periods. Thus for interpretability, the weighting schemes applied to the DFs ought to be sufficiently intuitive. \end{remark} In practice, however, a weighting scheme may depend on the data, say $\widehat{\omega}(g,t',t)$, i.e, $\omega(g,t',t)$ may need to be estimated. To this end, we impose the following regularity condition on the weights -- see, e.g., callaway-santanna-2020,callaway-tsyawo-2023treatment. The following regularity condition is imposed on the estimator $\widehat{\omega} $. \begin{assumption} \begin{align*} \sqrt{n}\big(\widehat{\omega}(g,t',t) - \omega(g,t',t)\big) = \frac{1}{\sqrt{N}}\sum_{j=1}^{N} \mathcal{W}_j(g,t',t) + o_p(1) \end{align*} \end{assumption}with $\mathbb{E}[\mathcal{W}_j(g,t',t)]=0$ and $\Omega(g,t',t)=\mathbb{E}[\mathcal{W}_j(g,t',t)^2]<\infty$ for each $(g,t',t) \in \mathcal{G} \times [-T:g-1]\setminus \{0\} \times [g:T] $. The following builds on (ref) in extending (ref) to the convex-weighted counterfactual DF and DTT. \begin{corollary} Suppose (ref) hold for each $(g,t',t) \in \mathcal{G} \times [-T:g-1]\setminus \{0\} \times [g:T] $, then for any $y\in \mathcal{Y} $, (a) \[ \sqrt{n}(\widehat{F}_{Y_{\infty}^{\widehat{\omega}}}(y) - F_{Y_{\infty}^\omega}(y)) \rightsquigarrow \mathbb{G}_{\infty}^{\omega C}(y) \] and (b) \[ \sqrt{n}(\widehat{DTT}_{\widehat{\omega}}(y) - DTT_\omega(y)) \rightsquigarrow \mathbb{G}_{DTT}^\omega(y) \] where $\mathbb{G}_{\infty}^{\omega C}(y)$ and $\mathbb{G}_{DTT}^{\omega}(y)$ are tight Gaussian processes with mean $0$ and covariance functions $\Omega_{\infty}^{\omega C}(y_1,y_2)$ and $\Omega_{DTT}^{\omega}(y_1,y_2)$ defined on $\mathbb{S}$. The expressions of the covariance functions are provided in the appendix. \end{corollary} \section{Extension - Covariates} Let $p(X) \in \mathbb{R}^{p_x}$ denote a dictionary of transformations of some elementary set of characteristics $X$, including the constant term $1$. The following state conditional (on pre-treatment covariates $X$) versions of (ref) in the two-period baseline model. Conditioning on covariates is limited to the two-period baseline case for ease of exposition. \begin{namedassumption}{(ref)-X}[Functional Index Parallel Trends] For each $y\in\mathcal{Y}$ and $t\in\{0, 1\}$, \begin{align*} F_{Y_{0t}|D=d,X}(y) = \Phi_d\Big(p(X)\big(\alpha(y) + \beta(y)d + \gamma(y)t\big)\Big) \ almost\ surely. \end{align*} \end{namedassumption} Observe in this case that $ (\alpha(y)', \beta(y)', \gamma(y)')' \in \mathbb{R}^{3p_x} $. \begin{namedassumption}{(ref)-X}[Functional No Anticipation] \begin{align*} F_{Y_{10}|D=1,X}(y) = F_{Y_{00}|D=1,X}(y) \ almost\ surely. \end{align*} \end{namedassumption} (ref) constitute weaker forms of (ref) as they only require that the FIPT and FNA hold in sub-populations characterised by $X$ but not necessarily globally. Further, by using dictionaries of transformations of $X$, namely $p(X)$, (ref) are more likely to hold in given applications. \begin{assumption}[No Perfect Collinearity] For each group and period $(d,t) \in \{0,1\}^2 $, $p(X)$ is not perfectly collinear. \end{assumption} Let $F_X(\cdot)$ denote the (joint) cumulative distribution function of $X$ on its support $\mathcal{X}$. The following extends (ref) to the covariate setting. \begin{theorem}[Identification] Suppose (ref) hold then for each $y\in\mathcal{Y}$, (a) $\big(\alpha(y)', \beta(y)', \gamma(y)'\big)'$ is identified, (b) $F_{Y_{01}|D=1,X}(y)$ is identified : \begin{align*} F_{Y_{01}|D=1,X}(y) = \Phi_1\Big(p(X)\big(\alpha(y) + \beta(y) + \gamma(y)\big)\Big) \end{align*}and (c) $F_{Y_{01}|D=1}(y)$ is identified: \begin{align*} F_{Y_{01}|D=1}(y) = \int_{\mathcal{X}}\Phi_1\Big(p(x)\big(\alpha(y) + \beta(y) + \gamma(y)\big)\Big)dF_X(x) = \mathbb{E}\Big[\Phi_1\Big(p(X)\big(\alpha(y) + \beta(y) + \gamma(y)\big)\Big)\Big]. \end{align*} \end{theorem} Unlike the preceding estimators, which have closed-form expressions, the introduction of covariates $X$ does not lead to closed-form expressions. Thus, the analogue and plug-in principles cannot be applied, and estimation is required. Thus, the quasi-maximum likelihood cum distribution regression, e.g., wooldridge-2023-simple,chernozhukov-val-melly-2013, becomes useful. However, given that a user may choose to allow $\Phi_d(\cdot)$ to differ by $d$ (or $\widetilde{\Phi}_t(\cdot)$ to differ by $t$ as the case may be), one may consider the following three-step estimation approach. Regressions are done using the quasi-maximum likelihood of a binary response model with link functions $\Phi_1(\cdot)$ and $\Phi_0(\cdot)$. \begin{algorithm} Fix $y\in\mathcal{Y}$, $\Phi_1(\cdot)$, and $\Phi_0(\cdot)$. \begin{enumerate}[(i)] • On sub-sample $(d,t)=(0,0)$, estimate $ \widehat{\eta}_{00}(y)=:\widehat{\alpha}(y) $ by regressing $\text{\usefont{U}{bbm}{m}{n}1}\{Y\leq y\} $ on $p(X)$ using $\Phi_0(p(X)\alpha(y))$; • (a) On sub-sample $(d,t)=(1,0)$, estimate $ \widehat{\eta}_{10}(y)$ by regressing $\text{\usefont{U}{bbm}{m}{n}1}\{Y\leq y\} $ on $p(X)$ using $\Phi_1(p(X)\eta_{10}(y))$. (b) Compute $\widehat{\beta}(y) = \widehat{\eta}_{10}(y) - \widehat{\alpha}(y) $; and • (a) On sub-sample $(d,t)=(0,1)$, estimate $ \widehat{\eta}_{01}(y)$ by regressing $\text{\usefont{U}{bbm}{m}{n}1}\{Y\leq y\} $ on $p(X)$ using $\Phi_0(p(X)\eta_{01}(y))$. (b) Compute $\widehat{\gamma}(y) = \widehat{\eta}_{01}(y) - \widehat{\alpha}(y) $. \end{enumerate} \end{algorithm} Where the researcher sets $\Phi_1(\cdot) = \Phi_0(\cdot)$, then the distribution regression of chernozhukov-val-melly-2013 can be applied to estimate \begin{align*} F_{Y_{0t}|D=d,X}(y) = \Phi\big(p(X)\alpha(y) + dp(X)\beta(y) + tp(X)\gamma(y)\big). \end{align*}on the pooled sample $(d,t) \in \{0,1\}^2\setminus (1,1)$. By the Law of Iterated Expectations (LIE), \begin{align*} F_{Y_{0t}|D=1}(y) &= \mathbb{E}[\text{\usefont{U}{bbm}{m}{n}1}\{Y_{0t}(D) \leq y\}|D=1] = \mathbb{E}\big[\mathbb{E}[\text{\usefont{U}{bbm}{m}{n}1}\{Y_{0t}(D) \leq y\}|D=1,X]\big| D=1\big]\\ &= \mathbb{E}[F_{Y_{0t}|D=1,X}(y)|D=1]\\ &= \frac{1}{p_{1t}}\mathbb{E}\Big[ \text{\usefont{U}{bbm}{m}{n}1}\{D_t=1\} \Phi_1\Big(p(X)\big(\alpha(y) + \beta(y) + \gamma(y)\big)\Big) \Big]. \end{align*} The last equality shows that in practice, $F_{Y_{0 1}|D=1}(y)$ is computed by averaging $F_{Y_{01}|D=1,X}(y)$ over $X$ of the treated group. The estimator of $F_{Y_{01}|D=1}(y)$ when covariates are involved is given by \begin{align*} \widehat{F}_{Y_{01}^X|D=1}(y) = \frac{1}{n_1} \sum_{i=1}^{n_1} \text{\usefont{U}{bbm}{m}{n}1}\{D_{i1}=1\} \Phi_1\Big(p(X_i)\big(\widehat{\alpha}(y) + \widehat{\beta}(y) + \widehat{\gamma}(y)\big)\Big)/\hat{p}_{11} \end{align*}where $\big(\widehat{\alpha}(y),\widehat{\beta}(y),\widehat{\gamma}(y)\big)'$ are estimated using group-period pairs $(d,t) \in \{0,1\}^2\setminus (1,1)$ and $DTT(y)$ is estimated using $\widehat{DTT}^X(y) = \widehat{F}_{Y_{11}|D=1}(y) - \widehat{F}_{Y_{01}^X|D=1}(y) = \widehat{F}_{11}(y) - \widehat{F}_{Y_{01}^X|D=1}(y) $. Consider the following high-level uniform (in $\mathcal{Y}$) asymptotic linearity condition on the estimators: $ \big(\widehat{\eta}_{00}(y)', \widehat{\eta}_{10}(y)', \widehat{\eta}_{01}' \big)' $. This is an important step in extending (ref) to the covariate setting. \begin{condition} $\displaystyle \sqrt{n}\big( \widehat{\eta}_{dt}(y) - \eta_{dt}(y) \big) = \frac{1}{\sqrt{n_t}} \sum_{i=1}^{n_t} \mathcal{S}_{dt}(W_{it};y) + o_p(1) $ where $\mathcal{S}_{dt}(W_{it};y), \ i \in [n_t], \ (d,t) \in \{0,1\}^2\setminus (1,1) $ are zero-mean square-integrable random variables at each $y\in\mathcal{Y}$. \end{condition} The following extends (ref) to the covariate setting. \begin{theorem} Suppose (ref) and (ref) hold, then for any $y\in \mathcal{Y} $, (a) \[ \sqrt{n}(\widehat{F}_{Y_{01}^X|D=1}(y) - F_{Y_{01}|D=1}(y)) \rightsquigarrow \mathbb{G}_{01}^X(y) \] and (b) \[ \sqrt{n}(\widehat{DTT}^X(y) - DTT(y)) \rightsquigarrow \mathbb{G}_{DTT^X}(y) \] where $\mathbb{G}_{01}^X(y)$ and $\mathbb{G}_{DTT^X}(y)$ are tight Gaussian processes with mean $0$ and covariance functions $\Omega_{01}^X(y_1,y_2)$ and $\Omega_{DTT^X}(y_1,y_2)$ defined on $\mathbb{S}$. The expressions of the covariance functions are provided in the appendix. \end{theorem} \section{Uniform Inference} (ref), (ref), (ref), (ref), (ref), and (ref) provide functional Central Limit Theorem results that are an important step in conducting inference on DFs and DTT, and consequently, QFs and QTT. In practice, however, it is more convenient to perform inference using the bootstrap. \subsection{DFs and DTT} Let $\widehat{\mathscr{F}}$ denote the estimator of a functional $\mathscr{F}$ on $\mathcal{Y}$. The functionals of interest are: $F_{dt}$, $F_{Y_{01}|D=1}$, $DTT$, $F_{Y_{0t}^{t'}|D=1}$, $DTT_t^{t'}$, $F_{Y_{\infty t}^{t'}|G=g}$, $DTT_{gt}^{t'}$, $F_{Y_{\infty}^{\omega}}$, and $DTT_{\omega} $. From the uniform asymptotically linear expressions in the proofs of (ref), (ref), (ref), (ref), (ref), and (ref), a generic expression is as follows: \[ \sqrt{n}\big( \widehat{\mathscr{F}}(y) - \mathscr{F}(y)\big) = \frac{1}{\sqrt{N}}\sum_{j=1}^N \Psi_{\mathscr{F}}\big(\widetilde{W}_j;y\big) + o_p(1) \]where $\Psi_{\mathscr{F}}\big(\widetilde{W};y\big)$ is Donsker, zero-mean, and square-integrable. Then the bootstrapped version is given by \[ \sqrt{n}\big( \widehat{\mathscr{F}}^*(y) - \mathscr{F}(y)\big) = \frac{1}{\sqrt{N}}\sum_{j=1}^N \xi_j\Psi_{\mathscr{F}}\big(\widetilde{W}_j;y\big) + o_p(1) \]where $\{\xi_j, 1\leq j \leq N\} $ are $i.i.d.$ random variables with mean zero, variance $1$, and $||\xi||_{2,1}:= \int_0^{\infty}\sqrt{\mathbb{P}[|\xi|>x]} dx < \infty $ independent of the sample data $ \{\widetilde{W}_j, 1\leq j \leq N \} $. The above includes commonly used forms of the multiplier bootstrap and the non-parametric bootstrap. The following result shows the validity of the non-parametric bootstrap procedure for inference on DFs and $DTT$. \begin{theorem}[Bootstrap Validity] Under (ref) in addition to (ref), (a) $\sqrt{n}(\widehat{F}_{dt}^*(y)-F_{dt}(y)) \rightsquigarrow \mathbb{G}_{dt}^H(y) $, (b) $ \sqrt{n}(\widehat{F}_{Y_{01}|D=1}^*(y) - F_{Y_{01}|D=1}(y)) \rightsquigarrow \mathbb{G}_{01}^C(y) $, (c) $\sqrt{n}(\widehat{DTT}^*(y) - DTT(y)) \rightsquigarrow \mathbb{G}_{DTT}(y)$, (d) $\sqrt{n}(\widehat{F}_{Y_{0 t}^{t'}|D=1}^*(y) - F_{Y_{0 t}^{t'}|D=1}(y)) \rightsquigarrow \mathbb{G}_{0t}^{t'}(y)$, (e) $\sqrt{n}(\widehat{DTT}_t^{t'*}(y) - DTT_t^{t'}(y)) \rightsquigarrow \mathbb{G}_{DTT_t^{t'}}(y)$, (f) $\sqrt{n}(\widehat{F}_{Y_{\infty t}^{t'}|G=g}(y) - F_{Y_{\infty t}^{t'}|G=g}^*(y)) \rightsquigarrow \mathbb{G}_{gt}^{\infty t'}(y)$, (g) $\sqrt{n}(\widehat{DTT}_{gt}^{t'*}(y) - DTT_{gt}^{t'}(y)) \rightsquigarrow \mathbb{G}_{DTT_{gt}^{t'}}(y)$, (h) $\sqrt{n}(\widehat{F}_{Y_{\infty}^{\widehat{\omega}}}^*(y) - F_{Y_{\infty}^\omega}(y)) \rightsquigarrow \mathbb{G}_{\infty}^{\omega C}(y)$, (i) $\sqrt{n}(\widehat{DTT}_{\widehat{\omega}}^*(y) - DTT_\omega(y)) \rightsquigarrow \mathbb{G}_{DTT}^\omega(y)$, (j) $\sqrt{n}(\widehat{F}_{Y_{01}^X|D=1}^*(y) - F_{Y_{01}|D=1}(y)) \rightsquigarrow \mathbb{G}_{01}^X(y)$, and (k) $\sqrt{n}(\widehat{DTT}^{X*}(y) - DTT(y)) \rightsquigarrow \mathbb{G}_{DTT^X}(y)$. \end{theorem} (ref), whose proof is provided in (ref), guarantees the validity of the bootstrap procedure on DFs, the counterfactual DF, and DTT. Thus, DF and DTT confidence bands can be computed using, for example, Algorithm 3 of chernozhukov-val-melly-2013. Details are omitted for concerns of space. Furthermore, these bands are valid -- see Remark 3.2 of chernozhukov-val-melly-2013. For ease of exposition, the rest of the discussion in this section focuses on the two-period model and (ref). Consider the confidence bands $\widehat{I}_1(y) = [\widehat{L}_1(y), \widehat{U}_1(y)]$ and $\widehat{I}_0(y) = [\widehat{L}_0(y), \widehat{U}_0(y)]$ of DFs $F_{Y_{11}|D=1}(y)$ and $ F_{Y_{01}|D=1}(y) $, respectively, for $y\in\mathcal{Y}$ with joint coverage probability $p$. Also, define the confidence bands $\widehat{I}_{DTT}(y) = [\widehat{L}_{DTT}(y), \widehat{U}_{DTT}(y)] $. In view of the foregoing, (ref), and (ref), the following result states the validity of the uniform confidence bands for completeness. \begin{theorem}[Coverage of Uniform Confidence Bands] Let (ref) hold, then \begin{align*} &\lim_{n\rightarrow\infty} \mathbb{P}\Big( \{F_{Y_{11}|D=1}(y) \in \widehat{I}_1(y)\} \bigcap \{F_{Y_{01}|D=1}(y) \in \widehat{I}_0(y)\} \Big) \geq p, \text{ and}\\ &\lim_{n\rightarrow\infty} \mathbb{P}\Big( DTT(y) \in \widehat{I}_{DTT}(y) \Big) \geq p. \end{align*} \end{theorem} \begin{proof} See Theorem SA.1 in the Supplemental material of chernozhukov-val-melly-2013 for proof. \end{proof} \subsection{QFs and QTT} Establishing the validity of confidence bands of QFs and QTT typically requires the smoothness (Hadamard differentiability) of the left-inverse (quantile) function. As the current paper explicitly allows discrete outcomes, the standard approach to conducting inference on QFs, QTT among other quantile treatment effects e.g., chernozhukov-hansen-2013,callaway-li-2020 does not apply as Hadamard differentiability breaks down. Instead, this paper uses the approach of chernozhukov2019generic in conducting inference on QFs and QTT. A crucial ingredient for using the authors' results is valid confidence bands on DFs ((ref)). For completeness, theoretical results from chernozhukov2019generic are adapted to the QFs and QTT in the current paper. The QF bands are given by $\widehat{I}_1^{-1}(\tau) = [\widehat{U}_1^{-1}(\tau), \widehat{L}_1^{-1}(\tau)]$ and $\widehat{I}_0^{-1}(\tau) = [\widehat{U}_0^{-1}(\tau), \widehat{L}_0^{-1}(\tau)],\ \tau\in[0,1]$, respectively, where $\widehat{L}_j^{-1}$ and $\widehat{U}_j^{-1}$ are left inverses of $\widehat{L}_j$ and $\widehat{U}_j, \ j\in\{0,1\}$ -- see (ref). The following definition is essential in deriving the confidence bands of QTT. \begin{definition}[Minkowski Difference] The Minkowski difference between two subsets $I$ and $J$ of a vector space is $I\ominus J := \{i - j : i \in I, j\in J\}$. If $I$ and $J$ are intervals $[i_1, i_2]$ and $[j_1, j_2]$, then $I\ominus J = [i_1, i_2] \ominus [j_1, j_2] = [i_1 - j_2, i_2 - j_1]$. \end{definition} From (ref), the confidence bands for QTT are given by $\widehat{I}_{QTT}^{-1}(\tau) = \widehat{I}_1^{-1}(\tau) \ominus \widehat{I}_0^{-1}(\tau)$. \begin{theorem} Under (ref), \begin{align*} &\lim_{n\rightarrow\infty} \mathbb{P}\Big( \{F_{Y_{11}|D=1}^{-1}(\tau) \in \widehat{I}_1^{-1}(\tau)\} \bigcap \{F_{Y_{01}|D=1}^{-1}(\tau) \in \widehat{I}_0(y)\} \Big) \geq p, \text{ and}\\ &\lim_{n\rightarrow\infty} \mathbb{P}\Big( QTT(\tau) \in \widehat{I}_{QTT}^{-1}(\tau)\} \Big) \geq p. \end{align*} \end{theorem} \begin{proof} (ref) follows from (ref) and Theorem 2 of chernozhukov2019generic. \end{proof} \section{Simulations} This section conducts a limited simulation exercise to examine the finite sample performance of the DiD procedure proposed in this paper. To that end, the latent outcome is generated as $\widetilde{Y}_i = \alpha + D_i\beta + t_i\gamma + D_it_i\delta + U_i$. $D_i\overset{i.i.d.}{\sim} Ber(0.5) $, $t_i = \text{\usefont{U}{bbm}{m}{n}1}\{i > N/2\} $, $U_i \overset{i.i.d.}{\sim} \mathcal{N}(0,1)$, $\alpha=0.1, \beta=0.2, \gamma=-0.1$, and $\delta=0.0$. The following sets of data-generating processes are considered: \[ Y_i = \begin{cases} \max\{\lceil \widetilde{Y}_i+1\rceil,0\} & \text{ DGP1} \\ \widetilde{Y}_i & \text{ DGP2} \end{cases} \] DGP2 is a simple model applicable to repeated cross-sections where the latent outcome coincides with the observed outcome, and DGP1 is a censored and discretised version of DGP2. For considerations of space, only results on DGP1 are presented in the main text. See (ref) for results on DGP2. Observe that both DGP1 and DGP2 satisfy (ref). Further, $DTT(y)=QTT(\tau)=0$ for all $(y,\tau)\in\mathcal{Y}\times (0,1)$. This simple setup enables one to gauge the proposed method's performance and evaluate the empirical performance of the uniform confidence bands on $DTT$. (ref) presents simulation results on DGP1. Each table is characterised by the distribution of $U$, namely the standard normal, the Asymmetric Laplace Distribution $ADL(0,1,\kappa), \kappa \in \{0.1,0.25,0.5\} $, and the working CDF, namely the Normal and the Uniform. The inclusion of the ADL distribution with $\kappa$ varied from zero is useful in examining the performance of the procedure when $U$ does not follow a symmetric distribution although the working CDF is symmetric. Using the Cauchy working CDF does not produce significantly different results -- see (ref). For each table, sample sizes $n \in \{200,400,600,800,1000\} $ are considered. Three parameters are of interest: DTT, mean DTT $ADTT =: \int DTT(y) dy $, and the counterfactual DF $F_{Y_{0 1}|D=1}(y)$. Results are evaluated via the following criteria: (1) $\mathcal{L}_2(\widehat{DTT})$, (2) 10% Rej.(DTT), (3) MB(ADTT), (4) MAD(ADTT), (5) 10% Rej.(ADTT), (6) $\mathcal{L}_2(\widehat{F})$ where $ \displaystyle \mathcal{L}_p(\widehat{F}):= \Big(\frac{1}{L}\sum_{l=1}^L |\widehat{F}(y_l) - F(y_l)|^p\Big)^{1/p} $ for the function $F(\cdot)$ DTT and $F_{Y_{0 1}|D=1}(y)$ and grid points $\{y_l,\ l \in [L]\}$ on $\mathcal{Y}$, Rej. denotes empirical rejection rates, MB denotes the mean bias, MAD denotes the median absolute deviation. For each set of results, $499$ non-parametric bootstrap samples are used with each of the $500$ Monte Carlo samples. For DGP1, the grid points $\{y_l,\ l \in [L]\}$ comprise the unique values of the outcome excluding the largest two realisations and points exceeding the 90th percentile of the outcome. \begin{table}[!htbp] \caption{ DGP1} \begin{tabular}{@llllllll@} \toprule $n$ & $\mathcal{L}_2(\widehat{DTT})$ & 10% Rej.(DTT) & MB(ADTT) & MAD(ADTT) & 10% Rej.(ADTT) & $\mathcal{L}_2(\widehat{cDF})$ \\ \midrule & \multicolumn{6}{l}{$U\sim\mathcal{N}(0,1),\ G=$ Normal}\\ \cmidrule{2-4} 200 & 0.108 & 0.114 & 0.001 & 0.058 & 0.090 & 0.095 \\ 400 & 0.079 & 0.090 & 0.010 & 0.039 & 0.086 & 0.069 \\ 600 & 0.057 & 0.092 & 0.002 & 0.029 & 0.060 & 0.050 \\ 800 & 0.050 & 0.076 & 0.002 & 0.028 & 0.060 & 0.043 \\ 1000 & 0.043 & 0.084 & 0.001 & 0.022 & 0.078 & 0.038 \\ \midrule & \multicolumn{6}{l}{$U\sim ALD(0, 1, 0.5),\ G=$ Normal}\\ \cmidrule{2-4} 200 & 0.107 & 0.108 & 0.009 & 0.059 & 0.860 & 0.158 \\ 400 & 0.076 & 0.100 & 0.005 & 0.041 & 0.094 & 0.141 \\ 600 & 0.063 & 0.104 & 0.004 & 0.035 & 0.130 & 0.136 \\ 800 & 0.054 & 0.104 & 0.002 & 0.029 & 0.106 & 0.132 \\ 1000 & 0.048 & 0.096 & 0.002 & 0.025 & 0.106 & 0.131 \\ \midrule & \multicolumn{6}{l}{$U\sim ALD(0, 1, 0.25),\ G=$ Normal}\\ \cmidrule{2-4} 200 & 0.104 & 0.110 & 0.005 & 0.055 & 0.860 & 0.276 \\ 400 & 0.074 & 0.098 & 0.004 & 0.041 & 0.100 & 0.273 \\ 600 & 0.063 & 0.128 & 0.004 & 0.035 & 0.130 & 0.267 \\ 800 & 0.053 & 0.110 & 0.001 & 0.028 & 0.094 & 0.266 \\ 1000 & 0.047 & 0.106 & 0.001 & 0.026 & 0.116 & 0.264 \\ \midrule & \multicolumn{6}{l}{$U\sim ALD(0, 1, 0.1),\ G=$ Normal}\\ \cmidrule{2-4} 200 & 0.105 & 0.074 & -0.001 & 0.055 & 0.106 & 0.387 \\ 400 & 0.074 & 0.090 & 0.003 & 0.040 & 0.088 & 0.386 \\ 600 & 0.063 & 0.104 & 0.003 & 0.036 & 0.134 & 0.382 \\ 800 & 0.053 & 0.082 & 0.001 & 0.028 & 0.104 & 0.382 \\ 1000 & 0.047 & 0.082 & 0.001 & 0.025 & 0.110 & 0.380 \\ \midrule & \multicolumn{6}{l}{$U\sim\mathcal{N}(0,1),\ G=$ Uniform}\\ \cmidrule{2-4} 200 & 0.100 &0.146 &-0.007 &0.053 &0.112 &0.086 \\ 400 & 0.071 &0.118 &0.002 &0.037 &0.106 &0.061 \\ 600 & 0.056 &0.140 &0.001 &0.030 &0.098 &0.049 \\ 800 & 0.050 &0.104 &0.002 &0.028 &0.092 &0.042 \\ 1000 & 0.043 &0.094 &0.001 &0.022 &0.086 &0.037 \\ \bottomrule \end{tabular} \end{table} Certain clear patterns emerge from (ref). The bias on DTT and the counterfactual DF is decreasing in the sample size. As the misspecification appears to increase in the skewness parameter $\kappa$ of the ALD distribution, the speed at which the bias shrinks to zero appears slower. The empirical rejection rates are decent even for the highly skewed case with $\kappa=0.1$, and they appear quite insensitive to the choice of the standard normal working CDF. In sum, the proposed method performs satisfactorily. \section{Empirical Application} This section applies the proposed method to estimate the DTT and QTT of increased police presence on crime, using data from diTella-2004. The data cover car theft incidents over nine months, from April 1, 1994, to December 31, 1994. The incidents of car theft are reported monthly in 876 blocks across three neighbourhoods in Buenos Aires, Argentina. In July 1994, a “treatment" was implemented, which involved assigning more police to specific blocks in Buenos Aires following a terrorist attack on the main Jewish Center. Thus, July is not used in the analyses, April through June constitute the pre-treatment months and August through December constitute the post-treatment months. This intervention affected 37 blocks, and diTella-2004 also considered blocks located one and two blocks away from the nearest Jewish institution.\footnote{A treated block contains a Jewish institution.} The outcome variable is the weekly average number of car theft incidents per block. diTella-2004 employed a standard linear Difference-in-Differences (DiD) model with time and block fixed effects -- see the paper for more details. This paper is interested in the DTT and QTT of police on crime. There are 15 unique pre- and post-treatment period pairs. Following (ref), pair-specific DFs and DTTs are equally averaged to obtain the DTT and QTT shown in (ref). The working CDF is set to the standard normal.\footnote{The standard uniform working CDF produces the same results and conclusions -- see (ref) in the appendix.} 999 empirical bootstrap samples at the block level are used on the grid $\mathcal{Y}:=\{0.00,0.25,0.50,0.75,1.0\}$.\footnote{Only $0.1\%$ of outcomes in the sample exceed $1.0$.} \begin{figure}[!htbp] \caption{QTT/DTT with uniform confidence bands} \begin{subfigure}{0.49\textwidth} \caption \end{subfigure} \begin{subfigure}{0.49\textwidth} \caption \end{subfigure} { \textit{Notes:} DTT and QTT are computed from weighted DFs. Weighted DFs are obtained from the average of all pre-post combinations of periods in the data. 999 cluster empirical bootstrap (at the block level) are used. The shaded areas correspond to 90% uniform confidence bands using \texttt{R} code from chernozhukov2019generic. QTT confidence bands are constructed from DF bands with joint $90\%$ coverage. The standard normal is the working CDF. } \end{figure} (ref) presents the DTT and QTT graphs with the shaded regions corresponding to the 90% uniform confidence bands. The dashed horizontal black line corresponds to the null effect. As can be observed, there is heterogeneity in both DTT and QTT in $(y,\tau) \in \mathcal{Y} \times (0,1) $. By increasing the probability that counts of car theft are capped at $y$, as can be observed from the largely positive DTT effects, one deduces that increased police presence decreases the probability that more than $y$ crimes are committed on a block with increased police presence. It can be concluded from the DTT graph that the effect of police on crime is not uniformly null on $\mathcal{Y}$. Moreover, the DTT effect appears to be decreasing in $y$. Thus, blocks experiencing relatively smaller numbers of crime incidents benefit more from the intervention. The QTT estimate lies below the horizontal zero line. QTT bands tell a story similar to that of the DTT; effects are not uniformly null. The effect of police on crime is most statistically significant at low quantile indices and less at higher quantiles. This is partly explained by the fewer occurrences of car theft incidents in the upper tail of the distribution. In sum, significant effects occur in some parts of the distribution. Effects are heterogeneous and statistically not uniformly null at the $10\%$-level. \section{Conclusion} This paper proposes a simple method to estimate DTT and QTT in the presence of possibly non-continuous outcomes viz. discrete, mixed, or continuous outcomes. Identification of DTT and QTT rests on intuitive functional extensions of the index parallel trends and no anticipation assumptions of wooldridge-2023-simple. By modelling the counterfactual DF and not the QF, this paper charts the path to valid inference under possibly non-continuous outcomes by (1) providing valid (joint) uniform confidence bands on DFs and (2) using the geometric arguments of chernozhukov2019generic to generate valid confidence bands on QFs and QTT. Thus, this paper circumvents the Hadamard non-differentiability problem that hampers uniform inference on QF and QTT parameters when the outcome is non-continuous. The sampling settings considered in this paper are quite general: (1) balanced panels, (2) unbalanced panels, and (3) repeated cross-sections under both non-staggered and staggered treatment designs. As existing literature points to the validity of the proposed method holding under “as (if) randomly assigned" treatment types of settings, the extension in the paper to conditional PT and NA weakens the identification restrictions required. Simulation results point to a reasonably robust performance of the method to the choice of working CDF even under cases where it is “misspecified". Also, by flexibly modelling the DFs before estimating QTT/DTT, the empirical example serves to illustrate the practical usefulness of the proposed method. \printbibliography