The exact contents of citations.db main_text.text for this paper — one flattened LaTeX string, title through conclusion, appendix excluded, unmodified except for removing email addresses. This is what our citation measures are computed over.
171,789 characters
Distribution Regression Difference-In-Differences
\title{Distribution Regression Difference-In-Differences}
\author{Iv\'an Fern\'andez-Val\thanks{Boston University} \ \quad Jonas Meier\thanks{Swiss National Bank.} \quad Aico van Vuuren\thanks{University of Groningen.}
\quad Francis Vella\thanks{Georgetown University.}}
\date{\today}
\maketitle
\begin{abstract}
We provide a simple distribution regression estimator for treatment effects in the difference-in-differences (DiD) design. Our procedure is particularly useful when the treatment effect differs across the distribution of the outcome variable. Our proposed estimator easily incorporates covariates and, importantly, can be extended to settings where the treatment potentially affects the joint distribution of multiple outcomes. Our key identifying restriction is that the counterfactual distribution of the treated in the untreated state has no interaction effect between treatment and time. This assumption results in a parallel trend assumption on a transformation of the distribution. We highlight the relationship between our procedure and assumptions with the changes-in-changes approach of Athey and Imbens (2006). We also reexamine the Card and Krueger (1994) study of the impact of minimum wages on employment to illustrate the utility of our approach.
\end{abstract}
\textbf{Keywords:} difference in differences, multiple outcomes, distributional and quantile treatment effects
\newpage
\section{Introduction}
The remarkable popularity of the difference-in-difference (DiD) estimator, inspired by an approach to evaluating the impact of policy interventions on economic outcomes introduced by David Card (see, for example, Card 1990, Card and Krueger, 1994), is one of the most striking features of empirical work on treatment and policy effects. While the methodological innovations in this literature (see Arkhangelsky and Imbens, 2024 for a recent review) include the use of constructed control groups, the staggered timing of treatments, and fuzzy rather than sharp designs, the vast majority of the associated empirical work has estimated the mean effect of the treatment on a single economic outcome. This seems somewhat limited and a fuller evaluation of a policy treatment would be based on an examination of the marginal and joint distributions of all outcomes it potentially influences. This paper provides a simple procedure for estimating distributional treatment effects in the presence of a single treatment when the outcomes of interest are potentially multivariate.
An initial methodological innovation focusing on distributional effects in DiD estimation is the changes-in-changes procedure of Athey and Imbens (2006), which estimates the counterfactual distribution of the treated group in the absence of treatment to compare with its observed distribution in the presence of treatment. Torous et al. (2024) extend the Athey and Imbens approach (2006) to the multivariate outcome setting.
Other work has adapted DiD estimation to examine the treatment effects at different quantiles of the outcome via the use of quantile regression. This includes, for example, Callaway and Li (2018, 2019). In contrast, Dube (2019), Goodman-Bacon (2021), and Goodman-Bacon and Schmidt (2020) employ conventional DiD estimation to explore the impact of the treatment at different points of the outcome distribution. Other distributional approaches include Kim and Wooldridge (2023) and Biewen, Fitzenberger, and Rümmele (2022). The former proposes an inverse probability weighting based procedure, while the latter employs a distribution regression (DR) approach. In this paper we also adopt a DR approach to constructing counterfactuals. In contrast to Biewen, Fitzenberger, and Rümmele (2022), who construct the counterfactual distributions via linear probability models, we employ non-linear link functions such as probit or logit models. This has a number of advantages, which we discuss below. In addition, we provide the associated identifying conditions required for this form of implementation of DR-DiD.
While DiD has typically been employed to evaluate the treatment effect on a specified economic outcome, there are many instances in which the treatment may affect multiple outcomes. For example, a change in tax rates on earnings of married couples may affect the hours of work of both husbands and wives. An analysis of such a tax change should include the impact on each of the outcomes. However, a richer analysis would not only examine the impact on the respective marginal hours distributions of husbands and wives but also the joint distribution of hours. Alternatively, while evaluations of minimum wage laws typically evaluate their impact on total employment, they may also affect the joint distribution of part-time and full-time employment. We illustrate how this joint effect can be evaluated via the bivariate distribution regression (BDR) approach of Fern\'andez-Val et al. (2024a). This requires that we first estimate the joint distribution by BDR and then construct the appropriate counterfactual. The treatment effects are then obtained via the appropriate comparisons. A recent alternative to this approach is extending the changes-in-changes procedure to multiple outcomes as is done in Torous et al. (2024).
The following section introduces the model and provides an analysis of the univariate case without covariates. We also extend our analysis to include covariates and contrast our approach with the Athey and Imbens (2006) changes-in-changes procedure. Section \ref{sec:multiple} extends our analysis to the multiple outcome case and Section \ref{sec:estimation} discusses estimation. Section \ref{sec:empirical} provides an empirical illustration via an application of our procedure to the data employed in the Card and Krueger (1994) study of the impact of increasing the minimum wage on employment. Section \ref{sec:conclusions} concludes.
\section{Econometric analysis of the univariate case}
Consider the standard DiD design with 2 periods, $T \in \{0,1\}$, and 2 groups, $G \in \{0,1\}$ in which a binary treatment, $D \in \{0,1\}$, is administered only to the treatment group with $G = 1$ in the second period $T=1$. Let $Y_0$ and $Y_1$ denote the potential outcomes under the non-treated and treated statuses. The observed outcome is $Y = Y_0 (1-D) + Y_1 D$, which corresponds to $Y_0$ for both groups at $T=0$, $Y_0$ for $G=0$ at $T=1$, and $Y_1$ for $G=1$ at $T=1$. Note that this implicitly imposes a non-anticipation assumption as we do not distinguish between the outcomes of the treated and non-treated state for $G=1$ in period $T=0$.
We are interested in the distributions of the potential outcomes of the treated at $T=1$, that is
$F_{Y_1 \,|\, G, T}(y \,|\, 1,1)$ and $F_{Y_0 \,|\, G, T}(y \,|\, 1,1)$. $F_{Y_1 \,|\, G, T}(y \,|\, 1,1)$ is identified from the observed outcome for $G=1$ at $T=1$,
$$
F_{Y_1 \,|\, G, T}(y \,|\, 1,1) = F_{Y \,|\, G, T}(y \,|\, 1,1);
$$
whereas $F_{Y_0 \,|\, G, T}(y \,|\, 1,1)$ is not identified without further assumptions.
The distribution of $Y_0$ conditional on $G$ and $T$ can be written as:
\begin{equation}\label{eq:dr}
F_{Y_0 \,|\, G, T}(y \,|\, g,t) = \Lambda(\alpha(y) + \beta(y) t + \gamma(y)g + \delta(y)gt), \quad y \in \mathbb{R},
\end{equation}
where $\Lambda$ is an invertible CDF such as the logistic, normal or uniform, and $y \mapsto $ $(\alpha(y), $ $ \beta(y), \gamma(y), \delta(y))$ is a vector of function-valued parameters.
The representation in \eqref{eq:dr} does not make any parametric assumption about the underlying distribution of $Y_0 \,|\, G, T$ since the dummy variable representation within the parentheses on the right-hand side is fully saturated. The parameters of the representation are local as they vary with $y$. To understand why \eqref{eq:dr} does not impose any restriction, note that $\alpha(y)$, $\beta(y)$, $\gamma(y)$ and $\delta(y)$ can be defined as:\footnote{See also Wooldridge (2023) equations (2.6) and (2.7).}
\[
\begin{split}
\alpha(y) & = \Lambda^{-1} \left( F_{Y_0 \,|\, G, T}(y \,|\, 0,0) \right) \\
\beta(y) & = \Lambda^{-1} \left( F_{Y_0 \,|\, G, T}(y \,|\, 0,1) \right) - \Lambda^{-1} \left( F_{Y_0 \,|\, G, T}(y \,|\, 0,0) \right) \\
\gamma(y) & = \Lambda^{-1} \left( F_{Y_0 \,|\, G, T}(y \,|\, 1,0) \right) - \Lambda^{-1} \left( F_{Y_0 \,|\, G, T}(y \,|\, 0,0) \right) \\
\delta(y) & = \Lambda^{-1} \left( F_{Y_0 \,|\, G, T}(y \,|\, 1,1) \right) - \Lambda^{-1} \left( F_{Y_0 \,|\, G, T}(y \,|\, 1,0) \right) \\
& - \left[ \Lambda^{-1} \left( F_{Y_0 \,|\, G, T}(y \,|\, 0,1) \right) - \Lambda^{-1} \left( F_{Y_0 \,|\, G, T}(y \,|\, 0,0) \right) \right]. \\
\end{split}
\]
We make the following identifying assumptions:
\begin{assumption}[No-interaction]\label{ass:no_interaction}
$$\delta(y)=0 \text{ for all } y \in \mathbb{R} \text{ in \eqref{eq:dr}}.$$
\end{assumption}
Let $\mathcal{Y}_d(G=g, T=t)$ denote the support of $Y_d \,|\, G=g,T=t$, for $d, g, t \in \{0,1\}$. We also assume:
\begin{assumption}[Support]\label{ass:support}
\[
\mathcal{Y}_0(G=1;T=1) \subseteq \mathcal{Y}_0(G=0;T=1) \cup \mathcal{Y}_0(G=1;T=0) \cup \mathcal{Y}_0(G=0;T=0).
\]
\end{assumption}
Assumption \ref{ass:no_interaction} implies that the distribution of the potential outcome $Y_0$ should not change differently in the second period for the treatment group compared to the control group. That is, we allow a difference between the distributions of the potential outcome $Y_0$ between the treatment and control group, but this difference should be identical in both periods. This is a parallel trend type assumption on a transformation of the distribution and can be written as:
\[
\begin{split}
\Lambda^{-1}\left(F_{Y_0 \,|\, G, T}(y \,|\, 1,1)\right) & - \Lambda^{-1}\left(F_{Y_0 \,|\, G, T}(y \,|\, 1,0)\right) = \\
& \Lambda^{-1}\left(F_{Y_0 \,|\, G, T}(y \,|\, 0,1)\right) - \Lambda^{-1}\left(F_{Y_0 \,|\, G, T}(y \,|\, 0,0) \right).
\end{split}
\]
This assumption is sensitive to the link function and imposes restrictions on the distribution $F_{Y_0 \,|\, G, T}$ for some link functions. For example, if $\Lambda$ is the identity link used in the linear probability model as in, for example, Almond et al. (2011), Dube (2019), Cengiz et al. (2019), Goodman-Bacon and Smith (2020), Goodman-Bacon (2021) and Biewen et al. (2022), one needs strong requirements in order to satisfy the parallel trends assumption (Blundell et al., 2004 and Wooldridge, 2023) That is, we need restrictions on the tails of the distribution of $F_{Y_0 \,|\, G, T}(y \,|\, 1,0)$, $F_{Y_0 \,|\, G, T}(y \,|\, 0,1)$ and $F_{Y_0 \,|\, G, T}(y \,|\, 0,0)$ to guarantee that $F_{Y_0 \,|\, G, T}(y \,|\, 1,1)$ is between $0$ and $1$. Thus, it requires that $F_{Y_0 \,|\, G, T}(y \,|\, 1,0) \leq 1 + F_{Y_0 \,|\, G, T}(y \,|\, 0,0) - F_{Y_0 \,|\, G, T}(y \,|\, 0,1)$, which might be restrictive at the top of the distribution, and $F_{Y_0 \,|\, G, T}(y \,|\, 1,0) $$\geq F_{Y_0 \,|\, G, T}(y \,|\, 0,0) - F_{Y_0 \,|\, G, T}(y \,|\, 0,1)$, which might be restrictive at the bottom of the distribution.\footnote{These requirements could be used to develop a specification test for the identity link. Roth and Sant'Anna (2023) proposed a test for the sharp hypothesis that $y \mapsto F_{Y_0 \,|\, G, T}(y \,|\, 1,0) + F_{Y_0 \,|\, G, T}(y \,|\, 0,1) - F_{Y_0 \,|\, G, T}(y \,|\, 0,0)$ be weakly increasing, which can be adapted to our setting. We do not pursue this route as we do not encourage the use of the linear probability model.}$^{,}$\footnote {For example, an increase in 0.2 in probability over time might be realistic for the control group when the initial probability was 0.5. However, if treatment group has a probability of, for example, 0.9, in the first period then it is not possible for the common trends assumption to hold.} Link functions such as the normal or logistic CDFs do not require such restrictions since the transformation expands the range of the distribution to the entire real line.
Assumption \ref{ass:no_interaction} cannot be empirically verified but when we have multiple observations in the pre-treatment period, it is possible to examine whether the ``parallel trends'' assumption holds pre-treatment. Assumption \ref{ass:support} is a restriction of the support of the counterfactual outcome of $Y_0$ for the treated group in the treated period.
These two assumptions identify $F_{Y_0 \,|\, G, T}(y \,|\, 1,1)$ since:
\begin{multline}\label{eq:id}
F_{Y_0 \,|\, G, T}(y \,|\, 1,1) = \Lambda(\alpha(y) + \beta(y) + \gamma(y) ) \\
= \Lambda\left[ \Lambda^{-1}\left(F_{Y_0 \,|\, G, T}(y \,|\, 1,0)\right) + \Lambda^{-1}\left(F_{Y_0 \,|\, G, T}(y \,|\, 0,1)\right) - \Lambda^{-1}\left(F_{Y_0 \,|\, G, T}(y \,|\, 0,0)\right) \right] \\
= \Lambda\left[ \Lambda^{-1}\left(F_{Y \,|\, G, T}(y \,|\, 1,0)\right) + \Lambda^{-1}\left(F_{Y \,|\, G, T}(y \,|\, 0,1)\right) - \Lambda^{-1}\left(F_{Y \,|\, G, T}(y \,|\, 0,0)\right) \right],
\end{multline}
under Assumption \ref{ass:no_interaction}. The support restrictions in Assumption \ref{ass:support} ensure that the term inside the squared brackets in \eqref{eq:id} is determined. Note that as $\lim_{x \rightarrow \infty}\Lambda(x) = 1$ and $\lim_{x \rightarrow -\infty}\Lambda(x) = 0$, our assumptions are sufficient but not necessary.
We present this identification result in the following lemma:
\begin{lemma}[Identification with Single Outcome]\label{lemma:did} $y \mapsto F_{Y_0 \,|\, G,T}(y \,|\, 1,1)$ is identified on $y \in \mathbb{R}$ under Assumptions \ref{ass:no_interaction} and \ref{ass:support}.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:did}] The results follow from equation \eqref{eq:id}.
\end{proof}
In empirical analysis, researchers typically would like to investigate objects that are related to the distributions of the potential outcome variables. One such object is the distributional treatment effect, defined as:
\[
\tau(y) := F_{Y_1 \,|\, G,T}(y \,|\, 1,1)(y) - F_{Y_0 \,|\, G,T}(y \,|\, 1,1), \quad y \in \mathbb{R}.
\]
The distributional treatment effect measures the change in the probability that the outcome is below $y$ as a result of the treatment. Another interesting object is the quantile treatment effect, defined as:
\[
\tau^*_q := F^{-1}_{Y_1 \,|\, G,T}(q \,|\, 1,1)(y) - F^{-1}_{Y_0 \,|\, G,T}(q \,|\, 1,1), \quad q \in (0,1).
\]
The quantile treatment effect measures the difference in the q-th quantile of the outcome variable as a result of the treatment.
\subsection{Inclusion of Covariates}
Including covariates is appealing as the assumption that $\delta(y) = 0$ may be harder to defend when there are differences in the trend between covariates and/or the composition of the treatment group changes over time in terms of observed characteristics; see also Melly and Santangelo (2015). Covariates are easily incorporated into the identification result by conditioning on them and adding an overlapping support assumption. Specifically, let $X$ be a vector of covariates such that the non-interaction assumption holds conditional on $X$; see Assumption \ref{ass:no_interaction_with_cov}. The distribution of $Y_0$ conditional on $G$, $T$ and $X$ can be written:
\begin{equation}\label{eq:dr_with_cov}
F_{Y_0 \,|\, G, T, X}(y \,|\, g,t,x) = \Lambda(\alpha(y,x) + \beta(y,x) t + \gamma(y,x)g + \delta(y,x)gt), \quad y \in \mathbb{R},
\end{equation}
where $(y,x) \mapsto (\alpha(y,x), \beta(y,x), \gamma(y,x), \delta(y,x))$ is a vector of unspecified functions.
Let $\mathcal{Y}_d(G=g, T=t; X=x)$ denote the support of $Y_d \,|\, G=g,T=t, X=x$. The identifying assumptions with covariates become:
\begin{assumption}[No-interaction with Covariates]\label{ass:no_interaction_with_cov}
$$\delta(y,X)=0 \text{ almost surely for all } y \in \mathbb{R} \text{ in \eqref{eq:dr_with_cov}.}$$
\end{assumption}
\begin{assumption}[Support conditions with Covariates]\label{ass:support_with_cov}
\[
\mathcal{Y}_0(G=1;T=1;X) \subseteq \mathcal{Y}_0(G=0;T=1;X) \cup \mathcal{Y}_0(G=1;T=0;X) \cup \mathcal{Y}_0(G=0;T=0;X),
\]
almost surely.
\end{assumption}
These two assumptions identify $F_{Y_0 \,|\, G, T, X}(y \,|\, 1,1,x)$ since
\begin{multline}\label{eq:id_with_cov}
F_{Y_0 \,|\, G, T,X}(y \,|\, 1,1,x) = \Lambda(\alpha(y,x) + \beta(y,x) + \gamma(y,x) ) \\
= \Lambda\left[ \Lambda^{-1}\left(F_{Y_0 \,|\, G, T,X}(y \,|\, 1,0,x)\right) + \Lambda^{-1}\left(F_{Y_0 \,|\, G, T,X}(y \,|\, 0,1,x)\right) - \Lambda^{-1}\left(F_{Y_0 \,|\, G, T,X}(y \,|\, 0,0,x)\right) \right] \\
= \Lambda\left[ \Lambda^{-1}\left(F_{Y \,|\, G, T,X}(y \,|\, 1,0,x)\right) + \Lambda^{-1}\left(F_{Y \,|\, G, T,X}(y \,|\, 0,1,x)\right) - \Lambda^{-1}\left(F_{Y \,|\, G, T,X}(y \,|\, 0,0,x)\right) \right],
\end{multline}
under the Assumption \ref{ass:no_interaction_with_cov}. The support restrictions in Assumption \ref{ass:support_with_cov} ensure that the term between parentheses in \eqref{eq:id_with_cov} is determined. Note that as $\lim_{x \rightarrow \infty}\Lambda(x) = 1$ and $\lim_{x \rightarrow -\infty}\Lambda(x) = 0$, our assumptions are sufficient but not necessary.
Let $\mathcal{X}_{11}$ denote the support of $X$ conditional on $G=1$ and $T=1$. The following Lemma states that $F_{Y_0,Z_0 \,|\, G,T,X}$ is identified under the previous assumptions.
\begin{lemma}[Identification with Covariates]\label{lemma:did_with_cov} Under Assumptions \ref{ass:no_interaction_with_cov} and \ref{ass:support_with_cov}, $(y,x) \mapsto $ \\ $F_{Y_0 \,|\, G,T,X}(y \,|\, 1,1,x)$ is identified on $(y,x) \in \mathbb{R}\times \mathcal{X}_{11}$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:did_with_cov}] The result follows from equations \eqref{eq:id_with_cov}.
\end{proof}
We can then identify $F_{Y_0 \,|\, G, T}(y \,|\, 1,1)$ as:
\begin{equation}\label{eq:id_with_cov2}
F_{Y_0 \,|\, G, T}(y \,|\, 1,1) = \int_{\mathcal{X}_{11}} F_{Y_0 \,|\, G, T,X}(y \,|\, 1,1,x) \mathrm{d}F_{X \,|\, G,T}(x \,|\, 1,1),
\end{equation}
where $F_{X \,|\, G,T}$ is the distribution of $X$ conditional on $G$ and $T$.
\subsection{Comparison with Changes-In-Changes}
\label{ss:cic}
As our proposals provide an alternative approach to the changes-in-changes (CiC) procedure of Athey and Imbens (2006), it is useful to contrast their setup and assumptions with ours. CiC assumes that the outcome of an individual without treatment satisfies the relationship $Y_0 = h(U,T)$ for the treatment and control groups, where $U$ is an unobserved and uniformly distributed random variable. It also assumes that $h$ is strictly increasing in the first term and that the distribution of $U$ is independent of time given the treatment outcome, i.e. $U \perp\!\!\!\perp T \,|\, G$. Finally, the support of $U$ for the treated population should be a subset of those of the untreated population. The final assumption implies in terms of the support of the potential outcomes that:
\[
\begin{split}
\mathcal{Y}_0(G=1,T=0) \subseteq \mathcal{Y}_0(G=0,T=0), \\
\mathcal{Y}_0(G=1,T=1) \subseteq \mathcal{Y}_0(G=0,T=1).
\end{split}
\]
Their second support restriction is less restrictive than ours but we do not need their first support restriction.
The previous assumptions identify the quantile function of $F_{Y_0 \,|\, G, T}(y \,|\, 1,1)$ as:
\[
\begin{split}
F^{-1}_{Y_0 \,|\, G, T}(u \,|\, 1,1) = &\phi\left( F^{-1}_{Y_0 \,|\, G, T}(u \,|\, 1,0) \right), \\
& \phi(y) := F^{-1}_{Y_0 \,|\, G, T}\left( F_{Y_0 \,|\, G, T}(y \,|\, 0,0) \,|\, 0,1\right), \quad u \in [0,1],
\end{split}
\]
where it is assumed that $Y_0$ is continuous with strictly increasing distribution function.
The transformation $\phi$ gives the second period outcome for an individual with an unobserved component $u$ such that $h(u,0) =y$, with $y$ the location at which the distribution function is evaluated (Athey and Imbens, 2006, page 441). Hence, their identification results follow
since $\phi$ evaluated in the first period observations of the treatment group is equally distributed as the distribution of the untreated outcome of the treatment group in the second period.
Their assumptions imply
the transformation $\phi$ that maps quantiles of $Y_0$ from period $0$ to period $1$ is the same for the treatment and control groups. This condition imposes the following restrictions on the coefficients of the representation of the conditional distribution in \eqref{eq:dr}:
$$
\alpha(y) = \alpha(\phi(y)) + \beta(\phi(y)), \quad \gamma(y) = \gamma(\phi(y)) + \delta(\phi(y)) .
$$
To see this, note that:
\begin{equation}\label{eq:cic1}
F_{Y_0 \,|\, G,T}(y \,|\, g,0) = F_{Y_0 \,|\, G,T}(h(h^{-1}(y,0),1) \,|\, g,1).
\end{equation}
Evaluating \eqref{eq:cic1} at $g=0$ and applying $F^{-1}_{Y_0 \,|\, G,T}(\cdot \,|\, 0,1)$ to both sides:
$$
h(h^{-1}(y,0),1) = F^{-1}_{Y_0 \,|\, G, T}\left( F_{Y_0 \,|\, G, T}(y \,|\, 0,0) \,|\, 0,1\right) =: \phi(y).
$$
Replacing $\phi(y)$ back in \eqref{eq:cic1} and using the representation \eqref{eq:dr}:
$$
\Lambda(\alpha(y) + \gamma(y)g) = \Lambda(\alpha(\phi(y)) + \beta(\phi(y)) + \gamma(\phi(y))g + \delta(\phi(y))g ).
$$
The restrictions then follow from equalizing the coefficients in both sides.\footnote{There is only a binding restriction because $\alpha(y) = \alpha(\phi(y)) + \beta(\phi(y))$ holds by definition of $\phi(y)$.}
They would complicate estimation in our framework as they involve two different levels of $Y$ and the transformation $\phi$ needs to be estimated.
\subsection{Comparison with Roth and Sant'Anna (2023)}
Roth and Sant'Anna (2023) derive the condition:
$$
F_{Y_0 \,|\, G, T}(y \,|\, 1,1) - F_{Y_0 \,|\, G, T}(y \,|\, 1,0) = F_{Y_0 \,|\, G, T}(y \,|\, 0,1) - F_{Y_0 \,|\, G, T}(y \,|\, 0,0), \quad y \in \mathbb{R},
$$
for the parallel trends assumption in expectations:
$$
{\mathbb{E}}(Y_0 \,|\, G = 1, T = 1) - {\mathbb{E}}(Y_0 \,|\, G = 1, T = 0) = {\mathbb{E}}(Y_0 \,|\, G = 0, T = 1) - {\mathbb{E}}(Y_0 \,|\, G = 0, T = 0),
$$
to be invariant to strictly monotone transformations of $Y_0$. This condition is different from our no-interaction assumption. Indeed, our DR model with no-interaction does not generally satisfy the parallel trends assumption in expectation as:
\[
\begin{split}
{\mathbb{E}}(Y_0 \,|\, G = g, T = 1) & - {\mathbb{E}}(Y_0 \,|\, G = g, T = 0) = \\
& \int_{-\infty}^{\infty} [ \Lambda(\alpha(y) + \gamma(y)g ) - \Lambda(\alpha(y) + \beta(y) + \gamma(y)g )] \mathrm{d} y
\end{split}
\]
depends on $g$ unless $\Lambda$ is the identity map, or $\beta(y)=0$ (no trend) or $\gamma(y)=0$ (random assignment) for $y \in \mathbb{R}$. Roth and Sant'Anna (2023) show that their condition holds if and only if there are no trends, random assignment or a mixture of both. Our model, however, generally satisfies a different invariance property with respect to strictly monotonic transformations that we specify in {subsection} \ref{rmk:invariance}.
\subsection{Invariance to Strictly Monotonic Transformations}
\label{rmk:invariance} The DR model in \eqref{eq:dr} with no-interaction is invariant to strictly monotonic transformations in the sense that we specify here. If $Y_0$ follows the DR model and satisfies the no-interaction assumption, then $\tilde Y_0 = h(Y_0)$ also follows the DR model and satisfies the no-interaction assumption for any strictly monotonic transformation $h$. To see this result note that if $h$ is strictly increasing:
$$
F_{\tilde Y_0 \,|\, G, T, X}(\tilde y \,|\, g,t,x) = \Lambda(\alpha(h^{-1}(\tilde y)) + \beta(h^{-1}(\tilde y)) t + \gamma(h^{-1}(\tilde y))g ) = \Lambda(\tilde \alpha(\tilde y) + \tilde \beta(\tilde y) t + \tilde \gamma(\tilde y)g ),
$$
where $\tilde y \mapsto h^{-1}(\tilde y)$ is the inverse function of $y \mapsto h(y)$, $\tilde \alpha = \alpha \circ h^{-1}$, $\tilde \beta = \beta \circ h^{-1}$ and $\tilde \gamma = \gamma \circ h^{-1}$. A similar argument applies when $h$ is strictly decreasing. Unlike the parallel trends in expectation, the no-interaction or parallel trends in distribution is invariant to strictly monotonic transformations.\footnote{The distributional approach of Kim and Wooldridge (2023) also satisfies this property.}
\section{Multiple Outcomes}
\label{sec:multiple}
Some settings may feature multiple outcomes that are potentially affected by the treatment. In these situations, we might be interested not only in how each of the outcomes is affected by the treatment, but also in how the relationship between the outcomes is affected by the treatment. For this, it is necessary to identify the joint distribution of the potential outcomes with and without treatment. We now consider a setting with two outcomes $Y$ and $Z$ and we focus on comparing features of the joint distribution of the potential outcomes with the treatment, $Y_1$ and $Z_1$, and the joint distribution of the potential outcomes without the treatment, $Y_0$ and $Z_0$, for the treated group $G=1$ in the post-treatment period $T=1$. For the sake of illustration we consider two measures of dependence. Namely, Spearman's and Kendall's rank correlation.
Let $F_{Y_d,Z_d \,|\, G,T}$ be the joint distribution of $Y_d$ and $Z_d$ conditional on $G$ and $T$, and $F_{Y_d \,|\, G,T}$ and $F_{Z_d \,|\, G,T}$ be the corresponding marginals. Spearman's rank correlation between $Y_d$ and $Z_d$, $d \in \{0,1\}$, can be expressed:
\begin{multline*}
\rho[Y_d,Z_d \,|\, G=1,T=1] = \text{Corr}[F_{Y_d \,|\, G,T}(Y_{d} \,|\, 1,1), F_{Z_d \,|\, G,T}(Z_{d} \,|\, 1,1) \,|\, G=1,T=1] = \\ 12 \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} [F_{Y_d \,|\, G,T}(y \,|\, 1,1) -1/2][F_{Z_d \,|\, G,T}(z \,|\, 1,1) - 1/2] F_{Y_d,Z_d \,|\, G,T}(\mathrm{d}y, \mathrm{d}z \,|\, 1,1);
\end{multline*}
and Kendall's rank correlation between $Y_d$ and $Z_d$, $d \in \{0,1\}$, can be expressed:
\begin{multline*}
\tau[Y_d,Z_d \,|\, G=1,T=1] = 4 \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} [F_{Y_d,Z_d \,|\, G,T}(y,z \,|\, 1,1) - 1/4] F_{Y_d,Z_d \,|\, G,T}(\mathrm{d}y, \mathrm{d}z \,|\, 1,1),
\end{multline*}
where we assume that $Y_d$ and $Z_d$ are continuous random variables to obtain the expressions on the right hand side.
As in the univariate case, $F_{Y_1,Z_1 \,|\, G,T}(y,z \,|\, 1,1)$ is identified by the joint distribution of the observed outcomes, $F_{Y,Z \,|\, G,T}(y,z \,|\, 1,1)$, whereas $F_{Y_0,Z_0 \,|\, G,T}(y,z \,|\, 1,1)$ is not identified from the data. To analyze identification, we use a variation of the local Gaussian representation (LGR) of a bivariate distribution from Chernozhukov, Fernand\'ez-Val and Luo (2018). Let $\Phi$ denote the Gaussian distribution function and $\Phi_2(\cdot,\cdot;\rho)$ denote the distribution of the bivariate standard normal with parameter $\rho$. Moreover, $\Lambda$ is, again, a strictly increasing cumulative distribution function. As we show in Section \ref{sec:estimation}, there is a benefit of using the logistic link function in our univariate analysis. Accordingly, we employ this in our empirical analysis for estimating both the univariate and bivariate effects.
\begin{lemma}[LGR with non-Normal Marginals]\label{lemma:lgr} The joint distribution of two random variables $Y$ and $Z$ conditional on $X$ can be represented by:
$$
F_{Y,Z \,|\, X}(y,z \,|\, x)(y,z \,|\, x) \equiv \Phi_2(\Phi^{-1}(\Lambda(\mu_{Y \,|\, X}(y \,|\, x))), \Phi^{-1}(\Lambda(\mu_{Z \,|\, X}(y \,|\, x))); \rho_{Y,Z \,|\, X}(y,z \,|\, x)),
$$
for all $y,z,x$, where $\mu_{Y \,|\, X}(y \,|\, x) = \Lambda^{-1}(F_{Y\,|\, X}(y \,|\, x))$, $\mu_{Z \,|\, X}(y \,|\, x) = \Lambda^{-1}(F_{Z\,|\, X}(z \,|\, x))$, and $\rho_{Y,Z \,|\, X}(y,z \,|\, x))$ is the unique solution in $\rho$ to the equation:
$$
F_{Y,Z \,|\, X}(y,z \,|\, x)(y,z \,|\, x) = \Phi_2(\Phi^{-1}(F_{Y\,|\, X}(y \,|\, x)(y,z \,|\, x)), \Phi^{-1}(F_{Z\,|\, X}(z \,|\, x)(y,z \,|\, x)); \rho).
$$
\end{lemma}
\begin{proof}
The proof is identical to the proof of Lemma 2.1 of Chernozhukov, Fernand\'ez-Val and Luo (2018) using:
$$
\Phi^{-1}(\Lambda(\mu_{Y \,|\, X}(y \,|\, x))) = \Phi^{-1}(F_{Y\,|\, X}(y \,|\, x))
$$
and
$$
\Phi^{-1}(\Lambda(\mu_{Z \,|\, X}(z \,|\, x))) = \Phi^{-1}(F_{Z\,|\, X}(z \,|\, x)).
$$
\end{proof}
The difference between Lemma \ref{lemma:lgr} and the LGR of Chernozhukov, Fernand\'ez-Val and Luo (2018) is that the marginals are represented by a general link rather than Gaussian links, that is:
$$
F_{Y\,|\, X}(y \,|\, x)(y \,|\, x) \equiv \Lambda(\mu_{Y \,|\, X}(y \,|\, x)), \quad F_{Z\,|\, X}(z \,|\, x) \equiv \Lambda(\mu_{Z \,|\, X}(z \,|\, x)).
$$
By the LGR, $F_{Y_0,Z_0 \,|\, G,T}$ can be expressed as:
\begin{multline}\label{eq:bdr}
F_{Y_0,Z_0 \,|\, G,T}(y,z \,|\, g,t) \equiv \\ \Phi_2(\Phi^{-1}(\Lambda(\mu_{Y_0 \,|\, G,T}(y \,|\, g,t))), \Phi^{-1}(\Lambda(\mu_{Z_0 \,|\, G,T}(y \,|\, g,t))); \rho_{Y_0,Z_0 \,|\, G,T}(y,z \,|\, g,t)),
\end{multline}
where $\mu_{Y_0 \,|\, G,T}(y \,|\, g,t) = \alpha_Y(y) + \beta_Y(y) t + \gamma_Y(y) g + \delta_Y(y) gt$, $\mu_{Z_0 \,|\, G,T}(y \,|\, g,t) = \alpha_Z(z) + \beta_Z(z) t + \gamma_Z(z) g + \delta_Z(z) gt$, and $\rho_{Y,Z \,|\, G,T}(y,z \,|\, g,t) = \alpha_{Y,Z}(y,z) + \beta_{Y,Z}(y,z) t + \gamma_{Y,Z}(y,z) g + \delta_{Y,Z}(y,z) gt$. In the LGR, the marginals are represented by:
$$
F_{Y_0 \,|\, G,T}(y \,|\, g,t) = \Lambda(\alpha_Y(y) + \beta_Y(y) t + \gamma_Y(y) g + \delta_Y(y) gt),
$$
and
$$
F_{Z_0 \,|\, G,T}(z \,|\, g,t) = \Lambda(\alpha_Z(z) + \beta_Z(z) t + \gamma_Z(z) g + \delta_Z(z) gt).
$$
We make the following identifying assumptions with respect to the distribution function in \eqref{eq:bdr}.
\begin{assumption}[Bivariate No-interaction]\label{ass:no_interaction_biv}
$$\delta_Y(y) = \delta_Z(z) = \delta_{Y,Z}(y,z) = 0 \text{ for all } (y,z) \in \mathbb{R}^2 \text{ in \eqref{eq:bdr}}.
$$
\end{assumption}
Let $\mathcal{YZ}_d(G=g, T=t)$ denote the support of $(Y_d,Z_d) \,|\, G=g,T=t$, for $d, g, t \in {0,1}$. We also assume:
\begin{assumption}[Bivariate Support]\label{ass:support_biv}
\[
\mathcal{YZ}_0(G=1;T=1) \subseteq \mathcal{YZ}_0(G=0;T=1) \cup \mathcal{YZ}_0(G=1;T=0) \cup \mathcal{YZ}_0(G=0;T=0).
\]
\end{assumption}
\begin{lemma}[Identification with Two Outcomes]\label{lemma:biv} $(y,z) \mapsto F_{Y_0,Z_0 \,|\, G,T}(y,z \,|\, 1,1)$ is identified on $\mathbb{R}^2$ under Assumptions \ref{ass:no_interaction_biv} and \ref{ass:support_biv}.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:biv}] Under the assumptions of the Lemma, $\mu_{Y_0 \,|\, G,T}(y \,|\, g,t) = \alpha_Y(y) + \beta_Y(y) t + \gamma_Y(y) g$, $\mu_{Z_0 \,|\, G,T}(y \,|\, g,t) = \alpha_Z(z) + \beta_Z(z) t + \gamma_Z(z) g$, and $\rho_{Y,Z \,|\, G,T}(y,z \,|\, g,t) = \alpha_{Y,Z}(y,z) + \beta_{Y,Z}(y,z) t + \gamma_{Y,Z}(y,z) g$.
The parameters $\alpha_Y(y)$, $\beta_Y(y)$, $\gamma_Y(y)$, $\alpha_Z(z)$, $\beta_Z(z)$, and $\gamma_Z(z)$ are identified from the marginals of $Y$ and $Z$, by Lemma \ref{lemma:did}. The parameter $\alpha_{Y,Z}(y,z)$ is identified as the solution in $\alpha$ to:
$$
F_{Y,Z \,|\, G,T}(y,z \,|\, 0,0) = \Phi_2(\alpha_Y(y) + \beta_Y(y) t + \gamma_Y(y) g, \alpha_Z(z) + \beta_Z(z) t + \gamma_Z(z) g ; \alpha).
$$
This solution exists and is unique because the RHS is strictly increasing in $\alpha$. The parameters $\beta_{Y,Z}(y,z)$ and $\gamma_{Y,Z}(y,z)$ are identified similarly as the solutions in $\beta$ and $\gamma$ of:
$$
F_{Y,Z \,|\, G,T}(y,z \,|\, 0,1) = \Phi_2(\alpha_Y(y) + \beta_Y(y) t + \gamma_Y(y) g, \alpha_Z(z) + \beta_Z(z) t + \gamma_Z(z) g ; \alpha_{Y,Z}(y,z) + \beta).
$$
and
$$
F_{Y,Z \,|\, G,T}(y,z \,|\, 1,0) = \Phi_2(\alpha_Y(y) + \beta_Y(y) t + \gamma_Y(y) g, \alpha_Z(z) + \beta_Z(z) t + \gamma_Z(z) g ; \alpha_{Y,Z}(y,z) + \gamma).
$$
Finally,
\[
\begin{split}
F_{Y_0,Z_0 \,|\, G,T}(y,z \,|\, 1,1) & = \Phi_2(\alpha_Y(y) + \beta_Y(y) + \gamma_Y(y), \alpha_Z(z) \beta_Z(z) + \gamma_Z(z); \alpha_{Y,Z}(y,z) + \\ & \beta_{Y,Z}(y,z) + \gamma_{Y,Z}(y,z)).
\end{split}
\]
\end{proof}
Covariates can be incorporated in a similar fashion as the univariate case. In particular, the LGR of $F_{Y_0,Z_0 \,|\, G,T,X}$ becomes:
\begin{multline}\label{eq:bdr_with_cov}
F_{Y_0,Z_0 \,|\, G,T,X}(y,z \,|\, g,t,x) \equiv \\ \Phi_2(\Phi^{-1}(\Lambda(\mu_{Y_0 \,|\, G,T,X}(y \,|\, g,t,x))), \Phi^{-1}(\Lambda(\mu_{Z_0 \,|\, G,T,X}(y \,|\, g,t,x))); \rho_{Y_0,Z_0 \,|\, G,T,X}(y,z \,|\, g,t,x)),
\end{multline}
where $\mu_{Y_0 \,|\, G,T,X}(y \,|\, g,t,x) = \alpha_Y(y,x) + \beta_Y(y,x) t + \gamma_Y(y,x) g + \delta_Y(y,x) gt$, $\mu_{Z_0 \,|\, G,T,X}(y \,|\, g,t,x) = \alpha_Z(z,x) + \beta_Z(z,x) t + \gamma_Z(z,x) g + \delta_Z(z,x) gt$, and $\rho_{Y,Z \,|\, G,T,X}(y,z \,|\, g,t,x) = \alpha_{Y,Z}(y,z,x) + \beta_{Y,Z}(y,z,x) t + \gamma_{Y,Z}(y,z,x) g + \delta_{Y,Z}(y,z,x) gt$.
Let $\mathcal{YZ}_d(G=g, T=t; X=x)$ denote the support of $(Y_d,Z_d) \,|\, G=g,T=t, X=x$. The identifying assumptions with covariates become:
\begin{assumption}[Bivariate No-interaction with Covariates]\label{ass:no_interaction_with_cov_biv}
$$\delta_Y(y,X) = \delta_Z(z,X) = \delta_{Y,Z}(y,z,X) = 0 \text{ almost surely for all } (y,z) \in \mathbb{R}^2 \text{ in \eqref{eq:bdr_with_cov}.}$$
\end{assumption}
\begin{assumption}[Bivariate Support with Covariates]\label{ass:support_with_cov_biv}
\begin{multline*}
\mathcal{YZ}_0(G=1;T=1;X) \subseteq \\ \mathcal{YZ}_0(G=0;T=1;X) \cup \mathcal{YZ}_0(G=1;T=0;X) \cup \mathcal{YZ}_0(G=0;T=0;X),
\end{multline*}
almost surely.
\end{assumption}
\begin{lemma}[Identification with Two Outcomes and Covariates]\label{lemma:biv_with_cov} Under Assumptions \ref{ass:no_interaction_with_cov_biv} and \ref{ass:support_with_cov_biv}, $(y,z) \mapsto F_{Y_0,Z_0 \,|\, G,T,X}(y,z \,|\, 1,1,x)$ is identified on $\mathbb{R}^2 \times \mathcal{X}_{11}$.
\end{lemma}
The marginalized distribution $F_{Y_0,Z_0 \,|\, G,T}(y \,|\, 1,1)$ is then identified by
$$
F_{Y_0,Z_0 \,|\, G,T}(y,z \,|\, 1,1) = \int_{\mathcal{X}_{11}} F_{Y_0,Z_0 \,|\, G,T,X}(y,z \,|\, 1,1,x) \mathrm{d} F_{X \,|\, G,T}(x \,|\, 1,1).
$$
\section{Estimation}
\label{sec:estimation}
\subsection{Univariate Case}
Assume we have a sample $\{(Y_{i}, X_{i}, G_{i}, T_{i}): 1\leq i \leq N\}$ of $(Y, X, G, T)$. For estimation, we replace the functions $(y,x) \mapsto (\alpha(y,x), \beta(y,x), \gamma(y,x))$ in \eqref{eq:dr_with_cov} by semiparametric linear indexes leading to the DR model for the conditional distribution:
\begin{equation} \label{eq:drx}
F_{Y_0 \,|\, G, T, X}(y \,|\, g,t,x) = \Lambda(p_{\alpha}(x)'\alpha(y) + p_{\beta}(x)'\beta(y) t + p_{\gamma}(x)'\gamma(y)g), \quad y \in \mathbb{R},
\end{equation}
where $p_{\alpha}(x)$, $p_{\beta}(x)$ and $p_{\gamma}(x)$ are vectors including the covariates and their transformations, and $y \mapsto (\alpha(y), \beta(y), \gamma(y))$ is a vector of function-valued parameters.
We implement the DR DiD estimator via the sequence of logit models at each point of the distribution of the outcome variable (Foresi and Peracchi, 1995, Chernozhukov, Fernandez-Val and Melly, 2013). We choose logit because it is the canonical link for binary outcomes allowing for pooled estimation of the distributions of the potential outcomes with and without the treatment (Wooldridge, 2023). Accordingly, we estimate the DR model for the observed outcomes on all observations including those with $D_i = 1$:
\begin{equation} \label{eq:uni}
F_{Y \,|\, G, T, X}(y \,|\, g,t,x) = \Lambda(p_{\alpha}(x)'\alpha(y) + p_{\beta}(x)'\beta(y) t + p_{\gamma}(x)'\gamma(y)g + p_{\theta}(x)'\theta(y)gt), \quad y \in \mathcal{Y},
\end{equation}
where $p_{\alpha}(x)$, $p_{\beta}(x)$, $p_{\gamma}(x)$ and $p_{\theta}(x)$ are vectors including a constant as the first component, and transformations of the covariates, and $\mathcal{Y}$ is a finite grid on $\mathbb{R}$. Let $I_i^y := 1(Y_i \leq y)$ and $\bar I_i^y = 1-I_i^y$.
\begin{algorithm}[Univariate Estimator] \label{algorithm_1}
\begin{enumerate}
\item Estimate the parameters of model \eqref{eq:uni} by distribution regression, that is, for $y \in \mathcal{Y}$,
\begin{footnotesize}{\begin{multline*}
(\hat \alpha(y), \hat \beta(y), \hat \gamma(y), \hat \theta(y)) \in \arg \max_{a,b,c,d} \sum_{i=1}^N \ell_i(a,b,c,d),\\
\ell_i(a,b,c,d) = I_i^y \log \Lambda(p_{\alpha}(X_i)'a + p_{\beta}(X_i)'b \ T_i + p_{\gamma}(X_i)'c \ G_i + p_{\theta}(X_i)'d \ G_i T_i) \\
+ \bar I_i^y \log \Lambda(-p_{\alpha}(X_i)'a - p_{\beta}(X_i)'b \ T_i - p_{\gamma}(X_i)'c \ G_i - p_{\theta}(X_i)'d \ G_i T_i).
\end{multline*}}\end{footnotesize}
\item Construct plug-in estimators of the distributions of the potential outcomes
\begin{footnotesize}{$$
\hat F_{Y_0 \,|\, G, T}(y \,|\, 1,1) = \frac{1}{N_{11}} \sum_{i=1}^N G_i T_i \ \Lambda(p_{\alpha}(X_i)'\hat \alpha(y) + p_{\beta}(X_i)'\hat \beta(y) + p_{\gamma}(X_i)'\hat \gamma(y)),
$$}\end{footnotesize}
and
\begin{footnotesize}{$$
\hat F_{Y_1 \,|\, G, T}(y \,|\, 1,1) = \frac{1}{N_{11}} \sum_{i=1}^N G_i T_i \ \Lambda(p_{\alpha}(X_i)'\hat \alpha(y) + p_{\beta}(X_i)'\hat \beta(y) + p_{\gamma}(X_i)'\hat \gamma(y) + p_{\theta}(X_i)'\hat \theta(y)),
$$}\end{footnotesize}
where $N_{11} = \sum_{i=1}^N G_i T_i$.
\item If needed, rearrange the estimates $y \mapsto \hat F_{Y_d \,|\, G, T}(y \,|\, 1,1)$ on $y \in \mathcal{Y}$, $d \in \{0,1\}$, to make them increasing.
\item Estimate functionals of the distributions of the potential outcome variables by using a plugin estimator, \emph{i.e.} the distributional treatment effect is estimated by
\[
\widehat \tau(y) = \widehat F_{Y_1 \,|\, G,T}(y \,|\, 1,1)(y) - \widehat F_{Y_0 \,|\, G,T}(y \,|\, 1,1),
\]
and the quantile treatment effect is estimated by:
\[
\widehat \tau ^*_q = \widehat F^{-1}_{Y_1 \,|\, G,T}(q \,|\, 1,1)(y) - \widehat F^{-1}_{Y_0 \,|\, G,T}(q \,|\, 1,1),
\]
where:
\[
\widehat F^{-1}_{Y_{j}|G,T}(q|1,1) = \inf \{ y: \widehat F_{Y_{j}|G,T}(y|1,1) \leq q\} \quad j = 0,1.
\]
\end{enumerate}
\end{algorithm}
By the properties of the logistic link, the estimator of $F_{Y_1 \,|\, G, T}(y \,|\, 1,1)$ is identical to the empirical distribution of $Y$ conditional on $G=1$ and $T=1$,
$$
\hat F_{Y_1 \,|\, G, T}(y \,|\, 1,1) \equiv \frac{1}{N_{11}} \sum_{i=1}^N G_i T_i \ 1(Y_i \leq y).
$$
Note that this estimator is therefore invariant to the specification of $p_{\theta}(x)$. We set $p_{\theta}(x)=1$ to speed up computation.
Note that the definition of the estimator for the inverse of the distribution function is appropriate as we rearranged the estimates $y \mapsto \hat F_{Y_d \,|\, G, T}(y \,|\, 1,1)$ on $y \in \mathcal{Y}$, $d \in \{0,1\}$.
Our algorithm has an advantage over the alternative of estimating $p_{\alpha}$, $p_{\beta}$, $p_{\gamma}$ using only those observations for which $D_i = 0$ via direct estimation of \eqref{eq:uni}. For example, the distributional treatment effect,
\[
\int_{\mathcal{X}_{11}} \left[ F_{Y_1|G,T,X}(y|1,1,X) - F_{Y_0|G,T,X}(y|1,1,X) \right] \mathrm{d}F_{X \,|\, G,T}(x \,|\, 1,1),
\]
equals the average partial effect of $D_i$ for the logit model used for distribution regression. Estimates and standard errors for these effects are routinely reported by many Statitical software packages (Wooldridge, 2023).
\subsection{Bivariate Case}
Assume we have a sample $\{(Y_{i}, Z_{i}, X_{i}, G_{i}, T_{i}): 1\leq i \leq N\}$ of $(Y, Z, X, G, T)$. For estimation, as in the univariate case, we replace the functions in $\mu_{Y_0 \,|\, G,T,X}$, $\mu_{Z_0 \,|\, G,T,X}$ and $\rho_{Y,Z \,|\, G,T,X}$ by semiparametric generalized linear indexes leading to a bivariate distribution regression (BDR) model:
\begin{equation} \label{eq:bdrx1}
\mu_{Y_0 \,|\, G,T,X}(y \,|\, g,t,x) = p_{\alpha}(x)'\alpha_Y(y) + p_{\beta}(x)'\beta_Y(y) t + p_{\gamma}(x)'\gamma_Y(y)g,
\end{equation}
\begin{equation} \label{eq:bdrx2}
\mu_{Z_0 \,|\, G,T,X}(y \,|\, g,t,x) = q_{\alpha}(x)'\alpha_Z(z) + q_{\beta}(x)'\beta_Z(z) t + q_{\gamma}(x)'\gamma_Z(z)g,
\end{equation}
and
\begin{equation} \label{eq:bdrx3}
\rho_{Y,Z \,|\, G,T,X}(y,z \,|\, g,t,x) = h(r_{\alpha}(x)'\alpha_{Y,Z}(y,z) + r_{\beta}(x)'\beta_{Y,Z}(y,z) t + r_{\gamma}(x)'\gamma_{Y,Z}(y,z)g),
\end{equation}
where $p_{\alpha}(x)$, $p_{\beta}(x)$, $p_{\gamma}(x)$, $q_{\alpha}(x)$, $q_{\beta}(x)$, $q_{\gamma}(x)$, $r_{\alpha}(x)$, $r_{\beta}(x)$ and $r_{\gamma}(x)$ are vectors including the covariates and their transformations, and $h(u) = \text{arctanh}(u)$ is the Fisher transformation that enforces $\rho_{Y,Z \,|\, G,T,X}$ to lie in $[-1,1]$.
We estimate all the parameters of $F_{Y_0,Z_0 \,|\, G,T}(y,z \,|\, 1,1)$ using the bivariate distribution regression estimator of Fernandez-Val et al. (2024a). We employ an imputation method that combines the parameter estimates from the sample of the first period for both groups and the sample of the second period for the untreated group, with the sample of the covariates in the second period for the treated group. The distribution $F_{Y_1,Z_1 \,|\, G,T}(y,z \,|\, 1,1)$ is estimated using the empirical distribution of $Y$ and $Z$ in the second period for the treated group. Algorithm \ref{algorithm_2} describes the estimation procedure. Let $\mathcal{Y}$ and $\mathcal{Z}$ be finite grids on $\mathbb{R}$, $I_i^y := 1(Y_i \leq y)$, $\bar I_i^y = 1-I_i^y$, $J_i^z := 1(Z_i \leq z)$, $\bar J_i^z = 1-J_i^z$.
\begin{algorithm}[Bivariate Estimator] \label{algorithm_2}
\begin{enumerate}
\item Estimate the parameters of \eqref{eq:bdrx1} and \eqref{eq:bdrx2} using Algorithm \ref{algorithm_1} on $y \in \mathcal{Y}$ and $z \in \mathcal{Z}$. Obtain
$$
\hat m_i^Y(y) = \Phi^{-1}(\Lambda(p_{\alpha}(X_i)'\hat \alpha_Y(y) + p_{\beta}(X_i)'\hat \beta_Y(y) \ T_i + p_{\gamma}(X_i)'\hat \gamma_Y(y) \ G_i)),
$$
and
$$
\hat m_i^Z(z) = \Phi^{-1}(\Lambda(q_{\alpha}(X_i)'\hat \alpha_Z(z) + q_{\beta}(X_i)'\hat \beta_Z(z) \ T_i + q_{\gamma}(X_i)'\hat \gamma_Z(z) \ G_i)),
$$
where $\hat \alpha_Y(y)$, $\hat \beta_Y(y)$, $\hat \gamma_Y(y)$, $\hat \alpha_Z(z)$, $\hat \beta_Z(z)$ and $\hat \gamma_Z(z)$ are the estimates of $\alpha_Y(y)$, $\beta_Y(y)$, $\gamma_Y(y)$, $\alpha_Z(z)$, $\beta_Z(z)$ and $\gamma_Z(z)$ obtained from Algorithm \ref{algorithm_1}.
\item Estimate the parameters of \eqref{eq:bdrx3} by BDR, that is, for $y \in \mathcal{Y}$ and $z \in \mathcal{Z}$,
\begin{footnotesize}{\begin{multline*}
(\hat \alpha_{Y,Z}(y,z), \hat \beta_{Y,Z}(y,z), \hat \gamma_{Y,Z}(y,z)) \in \arg \max_{a,b,c} \sum_{i=1}^N (1 - G_i T_i) \ \ell_i(a,b,c),\\
\ell_i(a,b,c) = I_i^y J_i^z \log \Phi_2(\hat m_i^Y(y), \hat m_i^Z(z); h(r_{\alpha}(X_i)'a + r_{\beta}(X_i)'b \ T_i + r_{\gamma}(X_i)'c \ G_i)) \\
+ I_i^y \bar J_i^z \log \Phi_2(\hat m_i^Y(y), - \hat m_i^Z(z); - h(r_{\alpha}(X_i)'a + r_{\beta}(X_i)'b \ T_i + r_{\gamma}(X_i)'c \ G_i)) \\
+ \bar I_i^y J_i^z \log \Phi_2(-\hat m_i^Y(y), \hat m_i^Z(z); - h(r_{\alpha}(X_i)'a + r_{\beta}(X_i)'b \ T_i + r_{\gamma}(X_i)'c \ G_i)) \\
+ \bar I_i^y \bar J_i^z \log \Phi_2(-\hat m_i^Y(y), -\hat m_i^Z(z); h(r_{\alpha}(X_i)'a + r_{\beta}(X_i)'b \ T_i + r_{\gamma}(X_i)'c \ G_i)).
\end{multline*}}\end{footnotesize}
\item Construct plug-in estimators of the distributions of the potential outcomes
\begin{footnotesize}{$$
\hat F_{Y_0 \,|\, G, T}(y \,|\, 1,1) = \frac{1}{N_{11}} \sum_{i=1}^N G_i T_i \ \Phi_2(\hat n_i^Y(y), \hat n_i^Z(z); \hat n_i^{Y,Z}(y,z)),
$$}\end{footnotesize}
and
$$
\hat F_{Y_1,Z_1 \,|\, G, T}(y,z \,|\, 1,1) = \frac{1}{N_{11}} \sum_{i=1}^N G_i T_i \ 1(Y_i \leq y, Z_i \leq z),
$$
where
$$
\hat n_i^Y(y) = \Phi^{-1}(\Lambda(p_{\alpha}(X_i)'\hat \alpha_Y(y) + p_{\beta}(X_i)'\hat \beta_Y(y) + p_{\gamma}(X_i)'\hat \gamma_Y(y))),
$$
$$
\hat n_i^Z(z) = \Phi^{-1}(\Lambda(q_{\alpha}(X_i)'\hat \alpha_Z(z) + q_{\beta}(X_i)'\hat \beta_Z(z) + q_{\gamma}(X_i)'\hat \gamma_Z(z) )),
$$
$$
\hat n_i^{Y,Z}(y,z) = h(r_{\alpha}(X_i)'\hat \alpha_{Y,Z}(y,z) + r_{\beta}(X_i)'\hat \beta_{Y,Z}(y,z) + r_{\gamma}(X_i)'\hat \gamma_{Y,Z}(y,z))
$$
and
$N_{11} = \sum_{i=1}^N G_i T_i$.
\end{enumerate}
\end{algorithm}
Estimators of the functionals of the joint distributions of potential outcomes such as Spearman's and Kendall's rank correlation coefficients can be constructed using the plug-in principle.
\subsection{Bootstrap Inference}
The estimators described in Algorithms \ref{algorithm_1} and \ref{algorithm_2} can be applied to panel and repeated cross-section data. Here we describe a weighted bootstrap algorithm to perform inference on functions of the distributions of potential outcomes designed for panel data. We focus on this case because it is relevant for our empirical application below.
To describe the procedure, we need to introduce an indicator $ID_i$, $i = 1, \ldots, N$, for the units in the panel. For example, if the sample is sorted by unit and time period, $ID = (1,1,2,2,\ldots,n,n)$, where $n=N/2$. The following algorithm describes the weighted bootstrap procedure to construct joint confidence bands for the distributions of the potential outcomes with and without the treatment in the univariate case. Inference for functionals of the distributions and for the bivariate case can be performed using similar algorithms.
\begin{algorithm}[Weighted Bootstrap Inference] \label{algorithm_3}
\begin{enumerate}
\item Choose the number of bootstrap repetitions $B$, e.g., $B=500$ or $B=1,000$.
\item Draw weights for each unit independent and identically from the standard exponential distribution, independently for the data. Construct a vector of weights $ \boldsymbol{\omega} = (\omega_1,\ldots,\omega_N)$, where $\omega_i = \omega_j$ if $ID_i = ID_j$, and normalize the components of $ \boldsymbol{\omega}$ to add up to one.
\item Estimate the parameters of model \eqref{eq:uni} by weighted distribution regression, that is, for $y \in \mathcal{Y}$,
\begin{footnotesize}{\begin{multline*}
(\tilde \alpha(y), \tilde \beta(y), \tilde \gamma(y), \tilde \theta(y)) \in \arg \max_{a,b,c,d} \sum_{i=1}^N \omega_i \ell_i(a,b,c,d),\\
\ell_i(a,b,c,d) = I_i^y \log \Lambda(p_{\alpha}(X_i)'a + p_{\beta}(X_i)'b \ T_i + p_{\gamma}(X_i)'c \ G_i + p_{\theta}(X_i)'d \ G_i T_i) \\
+ \bar I_i^y \log \Lambda(-p_{\alpha}(X_i)'a - p_{\beta}(X_i)'b \ T_i - p_{\gamma}(X_i)'c \ G_i - p_{\theta}(X_i)'d \ G_i T_i).
\end{multline*}}\end{footnotesize}
\item Construct plug-in weighted estimators of the distributions of the potential outcomes
\begin{footnotesize}{$$
\hat F_{Y_0 \,|\, G, T}^b(y \,|\, 1,1) = \frac{1}{N_{11}} \sum_{i=1}^N \omega_i \ G_i T_i \ \Lambda(p_{\alpha}(X_i)'\tilde \alpha(y) + p_{\beta}(X_i)'\tilde \beta(y) + p_{\gamma}(X_i)'\tilde \gamma(y)),
$$}\end{footnotesize}
and
\begin{footnotesize}{$$
\hat F_{Y_1 \,|\, G, T}^b(y \,|\, 1,1) = \frac{1}{N_{11}} \sum_{i=1}^N \omega_i \ G_i T_i \ \Lambda(p_{\alpha}(X_i)'\tilde \alpha(y) + p_{\beta}(X_i)'\tilde \beta(y) + p_{\gamma}(X_i)'\tilde \gamma(y) + p_{\theta}(X_i)'\tilde \theta(y)),
$$}\end{footnotesize}
where $N_{11} = \sum_{i=1}^N \omega_i G_i T_i$. If needed, rearrange the estimates $y \mapsto \hat F^b_{Y_d \,|\, G, T}(y \,|\, 1,1)$ on $y \in \mathcal{Y}$, $d \in \{0,1\}$, to make them increasing.
\item Repeat steps 1-3 $B$ times to obtain $$\left\{ \hat F_{Y_0 \,|\, G, T}^b(y \,|\, 1,1), \hat F_{Y_1 \,|\, G, T}^b(y \,|\, 1,1): y \in \mathcal{Y}, 1\leq b \leq B \right\}.$$
\item Construct an estimator of the $(1-\alpha)$-critical value of the maximal t-statistic, $\bar t_{\mathcal{Y}}(1-\alpha)$, as the $(1-\alpha)$-quantile of $\{\bar t_{\mathcal{Y}}^b: 1 \leq b \leq B\}$, where
$$
\bar t_{\mathcal{Y}}^b = \max_{y \in \mathcal{Y}} \left[\frac{|\hat F_{Y_0 \,|\, G, T}^b(y \,|\, 1,1) - \hat F_{Y_0 \,|\, G, T}(y \,|\, 1,1)|}{S_0(y)}, \frac{|\hat F_{Y_1 \,|\, G, T}^b(y \,|\, 1,1) - \hat F_{Y_1 \,|\, G, T}(y \,|\, 1,1)|}{S_1(y)} \right],
$$
and $S_d(y)$ is the interquartile range of $\left\{ \hat F_{Y_d \,|\, G, T}^b(y \,|\, 1,1): 1\leq b \leq B \right\}$ divided by $1.34896$, the interquartile range of the standard normal distribution, for $d \in \{0,1\}$.
\item Construct the $(1-\alpha)$-confidence bands as
$$
CB_{1-\alpha}[F_{Y_d \,|\, G, T}(\cdot \,|\, 1,1)] = \hat F_{Y_d \,|\, G, T}(y \,|\, 1,1) \pm \bar t_{\mathcal{Y}}(1-\alpha) S_d(y), \quad d \in \{0,1\}.
$$
\end{enumerate}
\end{algorithm}
\begin{remark}[Empirical Bootstrap] Empirical bootstrap can be implemented by drawing the weights in step 1 from a multinomial distribution with values $1,\ldots,n$ and equal probabilities $1/n$.
\end{remark}
\section{Empirical application}
\label{sec:empirical}
We illustrate our approach through a re-examination of data used in Card and Krueger (1994), hereafter CK, investigation of the impact on an increase in the minimum wage on the level of employment. In April 1992 New Jersey increased its minimum wage from the Federal level of 4.25 dollars per hour to
5.05 dollars per hour. CK investigated the impact of this increase on the change in the level of "full-time equivalent employment", measured as the sum of the number of full-time employees plus half the number of part-time employees, in New Jersey fast food restaurants. They use a DiD estimation strategy in which the control group comprises a group of comparable fast food restaurants in the region of Pennsylvania bordering New Jersey. CK concluded that this particular increase in the minimum wage led to a small increase in the level of full-time equivalent employment. This result produced a large and important related literature on the impact of minimum wages on employment.
We employ our procedure using the CK data to investigate the impact of this increase in the minimum wage on the level of full-time, part-time and full-time equivalent employment respectively.
We estimate the model using the 409 observations available in CK data set. 80.9 percent of these are observations are treated establishments in New Jersey. We acknowledge that the data set is relatively small and that this is likely to have implications for the level of statistical significance of the results. However, as our objective is to illustrate our approach in a well-known setting, we prefer to work with a data set which has been frequently used and is well understood (see, for a recent example, Torous et al. 2024) rather than providing our own original application. Note that for the models which include covariates, the additional variables are four dummy variables for franchise type and a dummy variable indicating that the establishment is company owned.
The respective actual and counterfactual distributions are reported in Figures \ref{fig:dist_total_employment}-\ref{fig:dist_fulltime_employment}. These figures are based on the models which include the additional covariates. While there are some differences in the distributions for full-time employment, indicating an increase in employment for establishment sizes above the 1st quartile, the larger differences appear in Figure \ref{fig:dist_fulltime_employment} which captures the increases in full-time employment. This figure suggests gains at all quantiles. Figure \ref{fig:dist_parttime_employment} is suggestive of some small reductions in part-time employment at some quantiles.
The results of the quantile treatment effects for the univariate analyses are reported in Tables \ref{tab:qte} and \ref{tab:qte_nox} noting that those in the former include the covariates while the latter does not. As the results are generally similar, we focus only on Table 1. The DiD estimate of the mean effect on full-time equivalent employment is 2.65 and this is statistically significant at the 10 percent level. An examination of the table reveals that this mean effect is driven by an increase in full-time employment as there is very weak statistical evidence of a small reduction in mean part-time employment. The level of statistical significance probably reflects the small sample size.
Tables \ref{tab:qte} and \ref{tab:qte_nox} also indicate that the effect of the minimum wage increase is different depending on the size of the establishment. For example, at smaller establishments the effect on full-time equivalent employment is negative and this reflects a reduction in these establishments' levels of part-time employment. However, there appears to be growth in full-time equivalent employment at the larger establishment sizes noting that the level of statistical significance is low. The point estimates capturing the changes in part-time employment are either zero or negative. The most striking feature of the table is that the increase in full-time equivalent employment is driven by gains in full-time employment. Moreover, the larger gains in employment are at the upper quantiles noting that the estimates with the higher degrees of statistical significance also occur at these quantiles.
To illustrate the applicability of our methodology to a bivariate analysis we consider the impact of the increase in the minimum wage on the joint distribution of the full-time and part-time employment levels. The counterfactual and actual distributions are shown in Figure \ref{fig:2dim}. The figure reveals that the joint distribution has changed due to the increase in the minimum wage and that the distribution of the treated population appears to have shifted downward and to the right. This appears to be a similar movement to that reported in Figure 5 of Torous et al. (2024). This suggests the treatment has changed the relationship between full-time and part-time employment. However, it is difficult to identify whether this merely reflecs the changes in the marginal distributions. It is also difficult to interpret the economic implications of the differences shown in this figure. Accordingly in Table 3 we also present estimates of Kendall's tau and Spearman's correlation index to capture the correlation between these two employment levels. Note that Kendall's tau is calculated as:
$$
\hat \tau_d = \frac{2}{n_{11}(n_{11}-1)} \sum_{i=1}^N \sum_{j=i+1}^N G_i G_j T_i T_j\text{sgn}(Y_{di} - Y_{dj}) \text{sgn}(Z_{di}-Z_{dj}), \ \ n_{11} = \sum_{i=1}^N G_i T_i, \ \ d \in \{0,1\}.
$$
Spearman's correlation index is calculated as:
\[
\hat \rho_{d} = 1 - \frac{6 \sum_{i=1}^{N} G_i T_i D_i^2}{n_{11} (n_{11}^2-1)}, \quad d \in \{0,1\},
\]
where $D_i$ is the difference between the ranks of $Y_d$ and $Z_d$ conditional on $G=1$ and $T=1$ for observation $i$.
The Kendall's tau and the Spearman's correlation index for the treated sample in the second period can be calculated from the observed data. For the counterfactual distribution of the treated sample in the second period when not treated, we first sample from the estimated distribution. That is, we sample a value of $Y_0$ using our estimate of its marginal distribution from above. We then sample $Z_0$ from the conditional distribution of $Z_0 \,|\, Y_0$ which can be obtained using our estimates for the bivariate model.
In the absence of treatment the estimates of Kendall's $\tau$ and Spearman's correlation index for these employment levels are -0.0095 and -0.0101 respectively. Following the increase in the minimum wage, this negative relationship becomes stronger, with the corresponding estimate values of -0.1709 and -0.2402. Moreover, despite the relatively small number of observations the difference in the Spearman's correlation is statistically significant at the 10 percent level. These two estimates of the change in the level of correlation both suggest that the increase in the minimum wage has changed the relationship between full-time and part-time employment. The statistically significant stronger negative correlation is consistent with a greater degree of substitutability between part-time and full-time employment in the presence of the higher minimum wage.
\begin{table}
\centering
\begin{turn}{-90}
\begin{footnotesize}
\begin{tabular}{lcccccc}
\hline
\hline
& Mean & 0.1 & 0.25 & 0.5 & 0.75 & 0.9 \\
\hline
\hline
Full-time equivalent employment & 2.6554 & -1.5 & 0.4383 & 2.5 & 1.5 & 1.5\\
95\% confidence intervals & (-0.331,5.642) & (-3.528,0.528) & (-0.921,1.797) & (-0.665,5.665) & (-0.94,3.94) & (-0.974,3.974)\\
90\% confidence intervals & (0.054,5.257) & (-3.48,0.48) & (-0.548,1.425) & (-0.487,5.487) & (-0.539,3.539) & (-0.925,3.925)\\
Part-time employment & -0.4618 & -1.0 & -2.0 & -0.5 & 0.0 & 0.0\\
95\% confidence intervals & (-3.596,2.672) & (-2.947,0.947) & (-4.972,0.972) & (-1.963,0.963) & (-0.99,0.99) & (-0.992,0.992)\\
90\% confidence intervals & (-3.022,2.098) & (-2.536,0.536) & (-4.946,0.946) & (-1.932,0.932) & (-0.978,0.978) & (-0.982,0.982)\\
Full-time employment & 3.0674 & $\cdot$ & 2.0 & 1.5 & 3.0 & 5.0\\
95\% confidence intervals & (-0.063,6.197) & $(\cdot, \cdot)$ & (-0.968,4.968) & (-0.956,3.956) & (-0.952,6.952) & (-0.981,10.981)\\
90\% confidence intervals & (0.49,5.645) & $(\cdot, \cdot)$ & (-0.923,4.923) & (-0.563,3.563) & (-0.553,6.553) & (-0.084,10.084)\\
\hline \hline
\end{tabular}
\end{footnotesize}
\end{turn}
\caption{Quantile treatment effects}
\label{tab:qte}
\end{table}
\begin{table}
\centering
\begin{turn}{-90}
\begin{footnotesize}
\begin{tabular}{lcccccc}
\hline
\hline
& Mean & 0.1 & 0.25 & 0.5 & 0.75 & 0.9 \\
\hline
\hline
Full-time equivalent employment & 2.565 & -1.5 & 0.5 & 2.5 & 1.5 & 1.5\\
95\% confidence intervals & (-0.785,5.915) & (-3.536,0.536) & (-0.928,1.928) & (-0.663,5.663) & (-0.726,3.726) & (-0.982,3.982)\\
90\% confidence intervals & (-0.371,5.501) & (-3.486,0.486) & (-0.557,1.557) & (-0.518,5.518) & (-0.51,3.51) & (-0.938,3.938)\\
Part-time employment & -0.5134 & -1.0 & -2.0 & 1.0 & 0.0 & 0.0\\
95\% confidence intervals & (-3.926,2.899) & (-2.511,0.511) & (-4.971,0.971) & (-0.977,2.977) & (-0.991,0.991) & (-0.992,0.992)\\
90\% confidence intervals & (-3.463,2.436) & (-2.447,0.447) & (-4.949,0.949) & (-0.944,2.944) & (-0.976,0.976) & (-0.982,0.982)\\
Full-time employment & 3.0512 & $\cdot$ & 2.0 & 2.0 & 3.0 & 5.0\\
95\% confidence intervals & (-0.254,6.357) & $(\cdot, \cdot)$ & (-0.965,4.965) & (-0.929,4.929) & (-0.968,6.968) & (-0.976,10.976)\\
90\% confidence intervals & (0.267,5.835) & $(\cdot, \cdot)$ & (-0.934,4.934) & (-0.461,4.461) & (-0.951,6.951) & (-0.956,10.956)\\
\hline\hline
\end{tabular}
\end{footnotesize}
\end{turn}
\caption{Quantile treatment effects -- No additional covariates included}
\label{tab:qte_nox}
\end{table}
\begin{figure}
\centering
\begin{tikzpicture}
\definecolor{darkgray176}{RGB}{176,176,176}
\definecolor{lightgray204}{RGB}{204,204,204}
\begin{axis}[
legend cell align={left},
legend style={
fill opacity=0.8,
draw opacity=1,
text opacity=1,
at={(0.97,0.03)},
anchor=south east,
draw=lightgray204
},
tick align=outside,
tick pos=left,
x grid style={darkgray176},
xlabel={Total employment},
xmin=-2.494, xmax=52.396,
xtick style={color=black},
y grid style={darkgray176},
ylabel={\(\displaystyle F_{Y_0|G,T}(\cdot|1,1), F_{Y_1|G,T}(\cdot|1,1)\)},
ymin=0, ymax=1,
ytick style={color=black}
]
\addplot [thick, black]
table {
0.001 0.00329331005915009
0.101 0.00329331005915009
0.201 0.00329331005915009
0.301 0.00329331005915009
0.401 0.00329331005915009
0.501 0.00329331005915009
0.601 0.00329331005915009
0.701 0.00329331005915009
0.801 0.00329331005915009
0.901 0.00329331005915009
1.001 0.00329331005915009
1.101 0.00329331005915009
1.201 0.00329331005915009
1.301 0.00329331005915009
1.401 0.00329331005915009
1.501 0.00329331005915009
1.601 0.00329331005915009
1.701 0.00329331005915009
1.801 0.00329331005915009
1.901 0.00329331005915009
2.001 0.00329331005915009
2.101 0.00329331005915009
2.201 0.00329331005915009
2.301 0.00329331005915009
2.401 0.00329331005915009
2.501 0.00329331005915009
2.601 0.00329331005915009
2.701 0.00329331005915009
2.801 0.00329331005915009
2.901 0.00329331005915009
3.001 0.00329331005915009
3.101 0.00329331005915009
3.201 0.00329331005915009
3.301 0.00329331005915009
3.401 0.00329331005915009
3.501 0.00329331005915009
3.601 0.00329331005915009
3.701 0.00329331005915009
3.801 0.00329331005915009
3.901 0.00329331005915009
4.001 0.00329331005915009
4.101 0.00329331005915009
4.201 0.00329331005915009
4.301 0.00329331005915009
4.401 0.00329331005915009
4.501 0.00329331005915009
4.601 0.00329331005915009
4.701 0.00329331005915009
4.801 0.00329331005915009
4.901 0.00329331005915009
5.001 0.00729601561162483
5.101 0.00729601561162483
5.201 0.00729601561162483
5.301 0.00729601561162483
5.401 0.00729601561162483
5.501 0.0113651450664049
5.601 0.0113651450664049
5.701 0.0113651450664049
5.801 0.0113651450664049
5.901 0.0113651450664049
6.001 0.0146487201852627
6.101 0.0146487201852627
6.201 0.0146487201852627
6.301 0.0146487201852627
6.401 0.0146487201852627
6.501 0.0233561968411153
6.601 0.0233561968411153
6.701 0.0233561968411153
6.801 0.0233561968411153
6.901 0.0233561968411153
7.001 0.0248009456619699
7.101 0.0248009456619699
7.201 0.0248009456619699
7.301 0.0248009456619699
7.401 0.0248009456619699
7.501 0.0265729203425682
7.601 0.0265729203425682
7.701 0.0265729203425682
7.801 0.0265729203425682
7.901 0.0265729203425682
8.001 0.029718121476418
8.101 0.029718121476418
8.201 0.029718121476418
8.301 0.0301264336821828
8.401 0.0301264336821828
8.501 0.0304406167461335
8.601 0.0304406167461335
8.701 0.0304406167461335
8.801 0.0304406167461335
8.901 0.0304406167461335
9.001 0.0312473559235803
9.101 0.0312473559235803
9.201 0.0312473559235803
9.301 0.0312473559235803
9.401 0.0312473559235803
9.501 0.0436775906458816
9.601 0.0436775906458816
9.701 0.0436775906458816
9.801 0.0440716196654573
9.901 0.0440716196654573
10.001 0.070727413075157
10.101 0.070727413075157
10.201 0.070727413075157
10.301 0.070727413075157
10.401 0.070727413075157
10.501 0.0812749229897064
10.601 0.0812749229897064
10.701 0.0812749229897064
10.801 0.0812749229897064
10.901 0.0812749229897064
11.001 0.0899397975756068
11.101 0.0899397975756068
11.201 0.0899397975756068
11.301 0.0905802680486096
11.401 0.0905802680486096
11.501 0.111300703489912
11.601 0.111300703489912
11.701 0.111300703489912
11.801 0.111300703489912
11.901 0.111300703489912
12.001 0.126220978742306
12.101 0.126220978742306
12.201 0.126220978742306
12.301 0.127177527539454
12.401 0.127177527539454
12.501 0.146341178846774
12.601 0.146341178846774
12.701 0.146341178846774
12.801 0.146341178846774
12.901 0.146341178846774
13.001 0.168191393895722
13.101 0.168191393895722
13.201 0.168191393895722
13.301 0.168191393895722
13.401 0.168191393895722
13.501 0.207043112028211
13.601 0.207043112028211
13.701 0.214864913915838
13.801 0.214864913915838
13.901 0.214864913915838
14.001 0.2499403283902
14.101 0.2499403283902
14.201 0.2499403283902
14.301 0.263249374458225
14.401 0.263249374458225
14.501 0.291667330275156
14.601 0.291667330275156
14.701 0.291667330275156
14.801 0.295436145154286
14.901 0.295436145154286
15.001 0.311049667639718
15.101 0.311049667639718
15.201 0.311049667639718
15.301 0.311049667639718
15.401 0.311049667639718
15.501 0.317630384441705
15.601 0.317630384441705
15.701 0.317630384441705
15.801 0.317630384441705
15.901 0.317630384441705
16.001 0.341853144559713
16.101 0.341853144559713
16.201 0.341853144559713
16.301 0.341853144559713
16.401 0.341853144559713
16.501 0.362628754019161
16.601 0.362628754019161
16.701 0.374465967921435
16.801 0.374465967921435
16.901 0.374465967921435
17.001 0.400448912654872
17.101 0.400448912654872
17.201 0.400448912654872
17.301 0.406792865778198
17.401 0.406792865778198
17.501 0.449908772587338
17.601 0.449908772587338
17.701 0.449908772587338
17.801 0.449908772587338
17.901 0.449908772587338
18.001 0.504743344732099
18.101 0.504743344732099
18.201 0.504743344732099
18.301 0.50608582251757
18.401 0.50608582251757
18.501 0.515025935464349
18.601 0.515025935464349
18.701 0.515025935464349
18.801 0.522924722684201
18.901 0.522924722684201
19.001 0.524239974911467
19.101 0.524239974911467
19.201 0.524239974911467
19.301 0.525863341182247
19.401 0.525863341182247
19.501 0.541541839944545
19.601 0.541541839944545
19.701 0.541541839944545
19.801 0.541541839944545
19.901 0.541541839944545
20.001 0.577731412260317
20.101 0.577731412260317
20.201 0.590039162699455
20.301 0.590039162699455
20.401 0.596419924062855
20.501 0.596419924062855
20.601 0.596419924062855
20.701 0.59994432942665
20.801 0.59994432942665
20.901 0.59994432942665
21.001 0.624569821580147
21.101 0.624569821580147
21.201 0.624569821580147
21.301 0.627325987280795
21.401 0.627325987280795
21.501 0.638607810160573
21.601 0.638607810160573
21.701 0.643574935235937
21.801 0.643574935235937
21.901 0.643574935235937
22.001 0.648471447541713
22.101 0.648471447541713
22.201 0.648471447541713
22.301 0.650298418711349
22.401 0.650298418711349
22.501 0.660586525738978
22.601 0.660586525738978
22.701 0.660586525738978
22.801 0.660586525738978
22.901 0.660586525738978
23.001 0.696026061378347
23.101 0.696026061378347
23.201 0.696026061378347
23.301 0.696026061378347
23.401 0.696026061378347
23.501 0.698275695610183
23.601 0.698275695610183
23.701 0.698275695610183
23.801 0.703234030726131
23.901 0.703234030726131
24.001 0.703234030726131
24.101 0.703234030726131
24.201 0.703234030726131
24.301 0.703523122756695
24.401 0.703523122756695
24.501 0.728129792273182
24.601 0.728129792273182
24.701 0.728433292460049
24.801 0.728433292460049
24.901 0.728433292460049
25.001 0.752452316672655
25.101 0.752452316672655
25.201 0.752452316672655
25.301 0.752452316672655
25.401 0.752452316672655
25.501 0.780140289888141
25.601 0.780140289888141
25.701 0.780140289888141
25.801 0.780140289888141
25.901 0.780140289888141
26.001 0.795077255559139
26.101 0.795077255559139
26.201 0.795077255559139
26.301 0.795077255559139
26.401 0.795077255559139
26.501 0.801306521297877
26.601 0.801306521297877
26.701 0.801306521297877
26.801 0.810744224212433
26.901 0.810744224212433
27.001 0.841119080762244
27.101 0.841119080762244
27.201 0.841119080762244
27.301 0.841119080762244
27.401 0.841119080762244
27.501 0.850423458640837
27.601 0.850423458640837
27.701 0.850423458640837
27.801 0.850423458640837
27.901 0.850423458640837
28.001 0.853896191891989
28.101 0.853896191891989
28.201 0.853896191891989
28.301 0.853896191891989
28.401 0.853896191891989
28.501 0.857967020626209
28.601 0.857967020626209
28.701 0.857967020626209
28.801 0.857967020626209
28.901 0.857967020626209
29.001 0.87850298689586
29.101 0.878926994401653
29.201 0.878926994401653
29.301 0.878926994401653
29.401 0.878926994401653
29.501 0.878926994401653
29.601 0.881192222239914
29.701 0.881192222239914
29.801 0.881192222239914
29.901 0.881192222239914
30.001 0.887420400245824
30.101 0.887420400245824
30.201 0.887420400245824
30.301 0.888927854615395
30.401 0.888927854615395
30.501 0.90721839043743
30.601 0.90721839043743
30.701 0.90721839043743
30.801 0.90721839043743
30.901 0.90721839043743
31.001 0.90721839043743
31.101 0.90721839043743
31.201 0.90721839043743
31.301 0.90721839043743
31.401 0.90721839043743
31.501 0.920927157610487
31.601 0.920927157610487
31.701 0.920927157610487
31.801 0.920927157610487
31.901 0.920927157610487
32.001 0.922326922833783
32.101 0.922326922833783
32.201 0.922326922833783
32.301 0.923395011231529
32.401 0.923395011231529
32.501 0.923395011231529
32.601 0.923395011231529
32.701 0.923395011231529
32.801 0.925613209549024
32.901 0.925613209549024
33.001 0.929611602115888
33.101 0.929611602115888
33.201 0.929611602115888
33.301 0.929611602115888
33.401 0.929611602115888
33.501 0.930926253359334
33.601 0.930926253359334
33.701 0.930926253359334
33.801 0.930926253359334
33.901 0.930926253359334
34.001 0.942113275665
34.101 0.942113275665
34.201 0.942113275665
34.301 0.942113275665
34.401 0.942113275665
34.501 0.950996384488654
34.601 0.950996384488654
34.701 0.950996384488654
34.801 0.950996384488654
34.901 0.950996384488654
35.001 0.959553045407306
35.101 0.959553045407306
35.201 0.959553045407306
35.301 0.959553045407306
35.401 0.959553045407306
35.501 0.969965242109941
35.601 0.969965242109941
35.701 0.969965242109941
35.801 0.969965242109941
35.901 0.969965242109941
36.001 0.972199147013985
36.101 0.972199147013985
36.201 0.972199147013985
36.301 0.972199147013985
36.401 0.972199147013985
36.501 0.977906328761506
36.601 0.977906328761506
36.701 0.977906328761506
36.801 0.977906328761506
36.901 0.977906328761506
37.001 0.977906328761506
37.101 0.977906328761506
37.201 0.977906328761506
37.301 0.977906328761506
37.401 0.977906328761506
37.501 0.978954725394519
37.601 0.978954725394519
37.701 0.978954725394519
37.801 0.978954725394519
37.901 0.978954725394519
38.001 0.978954725394519
38.101 0.978954725394519
38.201 0.978954725394519
38.301 0.978954725394519
38.401 0.978954725394519
38.501 0.978954725394519
38.601 0.978954725394519
38.701 0.978954725394519
38.801 0.978954725394519
38.901 0.978954725394519
39.001 0.978954725394519
39.101 0.978954725394519
39.201 0.978954725394519
39.301 0.978954725394519
39.401 0.978954725394519
39.501 0.980626636152806
39.601 0.980626636152806
39.701 0.980626636152806
39.801 0.981131556101651
39.901 0.981131556101651
40.001 0.981131556101651
40.101 0.981131556101651
40.201 0.981131556101651
40.301 0.981131556101651
40.401 0.981131556101651
40.501 0.981131556101651
40.601 0.981131556101651
40.701 0.981131556101651
40.801 0.982109958736266
40.901 0.982109958736266
41.001 0.982109958736266
41.101 0.982109958736266
41.201 0.982109958736266
41.301 0.985372379025165
41.401 0.985372379025165
41.501 0.98667341320962
41.601 0.98667341320962
41.701 0.98667341320962
41.801 0.98667341320962
41.901 0.98667341320962
42.001 0.98667341320962
42.101 0.98667341320962
42.201 0.98667341320962
42.301 0.98667341320962
42.401 0.98667341320962
42.501 0.98667341320962
42.601 0.98667341320962
42.701 0.98667341320962
42.801 0.98667341320962
42.901 0.98667341320962
43.001 0.98667341320962
43.101 0.98667341320962
43.201 0.98667341320962
43.301 0.98667341320962
43.401 0.98667341320962
43.501 0.990841933841338
43.601 0.990841933841338
43.701 0.990841933841338
43.801 0.990841933841338
43.901 0.990841933841338
44.001 0.991310090126993
44.101 0.991310090126993
44.201 0.991310090126993
44.301 0.991310090126993
44.401 0.991310090126993
44.501 0.992028166870941
44.601 0.992028166870941
44.701 0.992028166870941
44.801 0.992028166870941
44.901 0.992028166870941
45.001 0.992028166870941
45.101 0.992028166870941
45.201 0.992028166870941
45.301 0.992028166870941
45.401 0.992028166870941
45.501 0.992046959918012
45.601 0.992046959918012
45.701 0.992046959918012
45.801 0.992046959918012
45.901 0.992046959918012
46.001 0.992046959918012
46.101 0.992046959918012
46.201 0.992046959918012
46.301 0.992046959918012
46.401 0.992046959918012
46.501 0.993172257103547
46.601 0.993172257103547
46.701 0.993172257103547
46.801 0.993172257103547
46.901 0.993172257103547
47.001 0.993172257103547
47.101 0.993172257103547
47.201 0.993172257103547
47.301 0.993172257103547
47.401 0.993172257103547
47.501 0.993172257103547
47.601 0.993172257103547
47.701 0.993172257103547
47.801 0.993172257103547
47.901 0.993172257103547
48.001 0.99431108195146
48.101 0.99431108195146
48.201 0.99431108195146
48.301 0.99431108195146
48.401 0.99431108195146
48.501 0.99431108195146
48.601 0.99431108195146
48.701 0.99431108195146
48.801 0.99431108195146
48.901 0.99431108195146
49.001 0.99431108195146
49.101 0.99431108195146
49.201 0.99431108195146
49.301 0.99431108195146
49.401 0.99431108195146
49.501 0.994707260314526
49.601 0.994707260314526
49.701 0.994707260314526
49.801 0.994707260314526
49.901 0.994707260314526
};
\addlegendentry{$F_{Y_0|G,T}(\cdot|1,1)$}
\addplot [thick, black, dashed]
table {
0.001 0.0156739811912226
0.101 0.0156739811912226
0.201 0.0156739811912226
0.301 0.0156739811912226
0.401 0.0156739811912226
0.501 0.0156739811912226
0.601 0.0156739811912226
0.701 0.0156739811912226
0.801 0.0156739811912226
0.901 0.0156739811912226
1.001 0.0156739811912226
1.101 0.0156739811912226
1.201 0.0156739811912226
1.301 0.0156739811912226
1.401 0.0156739811912226
1.501 0.0156739811912226
1.601 0.0156739811912226
1.701 0.0156739811912226
1.801 0.0156739811912226
1.901 0.0156739811912226
2.001 0.0156739811912226
2.101 0.0156739811912226
2.201 0.0156739811912226
2.301 0.0156739811912226
2.401 0.0156739811912226
2.501 0.0156739811912226
2.601 0.0156739811912226
2.701 0.0156739811912226
2.801 0.0156739811912226
2.901 0.0156739811912226
3.001 0.0156739811912226
3.101 0.0156739811912226
3.201 0.0156739811912226
3.301 0.0156739811912226
3.401 0.0156739811912226
3.501 0.0156739811912226
3.601 0.0156739811912226
3.701 0.0156739811912226
3.801 0.0156739811912226
3.901 0.0156739811912226
4.001 0.0156739811912226
4.101 0.0156739811912226
4.201 0.0156739811912226
4.301 0.0156739811912226
4.401 0.0156739811912226
4.501 0.0156739811912226
4.601 0.0156739811912226
4.701 0.0156739811912226
4.801 0.0156739811912226
4.901 0.0156739811912226
5.001 0.0156739811912226
5.101 0.0156739811912226
5.201 0.0156739811912226
5.301 0.0156739811912226
5.401 0.0156739811912226
5.501 0.0156739811912226
5.601 0.0156739811912226
5.701 0.0156739811912226
5.801 0.0156739811912226
5.901 0.0156739811912226
6.001 0.0156739811912226
6.101 0.0156739811912226
6.201 0.0156739811912226
6.301 0.0156739811912226
6.401 0.0156739811912226
6.501 0.0188087774294671
6.601 0.0188087774294671
6.701 0.0188087774294671
6.801 0.0188087774294671
6.901 0.0188087774294671
7.001 0.0313479623824451
7.101 0.0313479623824451
7.201 0.0313479623824451
7.301 0.0313479623824451
7.401 0.0313479623824451
7.501 0.0407523510971787
7.601 0.0407523510971787
7.701 0.0407523510971787
7.801 0.0407523510971787
7.901 0.0407523510971787
8.001 0.0470219435736677
8.101 0.0470219435736677
8.201 0.0470219435736677
8.301 0.0470219435736677
8.401 0.0470219435736677
8.501 0.0532915360501567
8.601 0.0532915360501567
8.701 0.0532915360501567
8.801 0.0532915360501567
8.901 0.0532915360501567
9.001 0.0658307210031348
9.101 0.0658307210031348
9.201 0.0658307210031348
9.301 0.0658307210031348
9.401 0.0658307210031348
9.501 0.0846394984326019
9.601 0.0846394984326019
9.701 0.0846394984326019
9.801 0.0846394984326019
9.901 0.0846394984326019
10.001 0.103448275862069
10.101 0.103448275862069
10.201 0.103448275862069
10.301 0.103448275862069
10.401 0.103448275862069
10.501 0.125391849529781
10.601 0.125391849529781
10.701 0.125391849529781
10.801 0.125391849529781
10.901 0.125391849529781
11.001 0.134796238244514
11.101 0.134796238244514
11.201 0.134796238244514
11.301 0.134796238244514
11.401 0.134796238244514
11.501 0.15987460815047
11.601 0.15987460815047
11.701 0.15987460815047
11.801 0.15987460815047
11.901 0.15987460815047
12.001 0.172413793103448
12.101 0.172413793103448
12.201 0.172413793103448
12.301 0.172413793103448
12.401 0.172413793103448
12.501 0.200626959247649
12.601 0.200626959247649
12.701 0.200626959247649
12.801 0.203761755485893
12.901 0.203761755485893
13.001 0.216300940438871
13.101 0.216300940438871
13.201 0.216300940438871
13.301 0.216300940438871
13.401 0.216300940438871
13.501 0.22884012539185
13.601 0.22884012539185
13.701 0.22884012539185
13.801 0.22884012539185
13.901 0.22884012539185
14.001 0.244514106583072
14.101 0.244514106583072
14.201 0.244514106583072
14.301 0.247648902821317
14.401 0.247648902821317
14.501 0.260188087774295
14.601 0.260188087774295
14.701 0.260188087774295
14.801 0.260188087774295
14.901 0.260188087774295
15.001 0.269592476489028
15.101 0.269592476489028
15.201 0.269592476489028
15.301 0.269592476489028
15.401 0.269592476489028
15.501 0.294670846394984
15.601 0.294670846394984
15.701 0.294670846394984
15.801 0.294670846394984
15.901 0.294670846394984
16.001 0.322884012539185
16.101 0.322884012539185
16.201 0.322884012539185
16.301 0.322884012539185
16.401 0.322884012539185
16.501 0.35423197492163
16.601 0.35423197492163
16.701 0.35423197492163
16.801 0.357366771159875
16.901 0.357366771159875
17.001 0.366771159874608
17.101 0.366771159874608
17.201 0.366771159874608
17.301 0.366771159874608
17.401 0.366771159874608
17.501 0.38871473354232
17.601 0.38871473354232
17.701 0.38871473354232
17.801 0.391849529780564
17.901 0.391849529780564
18.001 0.41692789968652
18.101 0.41692789968652
18.201 0.41692789968652
18.301 0.41692789968652
18.401 0.41692789968652
18.501 0.426332288401254
18.601 0.426332288401254
18.701 0.426332288401254
18.801 0.426332288401254
18.901 0.426332288401254
19.001 0.448275862068966
19.101 0.448275862068966
19.201 0.448275862068966
19.301 0.448275862068966
19.401 0.448275862068966
19.501 0.45141065830721
19.601 0.45141065830721
19.701 0.45141065830721
19.801 0.45141065830721
19.901 0.45141065830721
20.001 0.482758620689655
20.101 0.482758620689655
20.201 0.482758620689655
20.301 0.482758620689655
20.401 0.482758620689655
20.501 0.507836990595611
20.601 0.507836990595611
20.701 0.507836990595611
20.801 0.507836990595611
20.901 0.507836990595611
21.001 0.520376175548589
21.101 0.520376175548589
21.201 0.520376175548589
21.301 0.523510971786834
21.401 0.523510971786834
21.501 0.557993730407524
21.601 0.557993730407524
21.701 0.557993730407524
21.801 0.561128526645768
21.901 0.561128526645768
22.001 0.576802507836991
22.101 0.576802507836991
22.201 0.576802507836991
22.301 0.576802507836991
22.401 0.576802507836991
22.501 0.598746081504702
22.601 0.598746081504702
22.701 0.598746081504702
22.801 0.601880877742947
22.901 0.601880877742947
23.001 0.623824451410658
23.101 0.623824451410658
23.201 0.623824451410658
23.301 0.626959247648903
23.401 0.626959247648903
23.501 0.642633228840125
23.601 0.642633228840125
23.701 0.642633228840125
23.801 0.648902821316614
23.901 0.648902821316614
24.001 0.67398119122257
24.101 0.67398119122257
24.201 0.67398119122257
24.301 0.67398119122257
24.401 0.67398119122257
24.501 0.68025078369906
24.601 0.68025078369906
24.701 0.68025078369906
24.801 0.683385579937304
24.901 0.683385579937304
25.001 0.711598746081505
25.101 0.711598746081505
25.201 0.711598746081505
25.301 0.711598746081505
25.401 0.711598746081505
25.501 0.733542319749216
25.601 0.733542319749216
25.701 0.733542319749216
25.801 0.733542319749216
25.901 0.733542319749216
26.001 0.74294670846395
26.101 0.74294670846395
26.201 0.74294670846395
26.301 0.74294670846395
26.401 0.74294670846395
26.501 0.77115987460815
26.601 0.77115987460815
26.701 0.77115987460815
26.801 0.77115987460815
26.901 0.77115987460815
27.001 0.793103448275862
27.101 0.793103448275862
27.201 0.793103448275862
27.301 0.793103448275862
27.401 0.793103448275862
27.501 0.799373040752351
27.601 0.799373040752351
27.701 0.799373040752351
27.801 0.799373040752351
27.901 0.799373040752351
28.001 0.818181818181818
28.101 0.818181818181818
28.201 0.818181818181818
28.301 0.818181818181818
28.401 0.818181818181818
28.501 0.830721003134796
28.601 0.830721003134796
28.701 0.830721003134796
28.801 0.830721003134796
28.901 0.830721003134796
29.001 0.84012539184953
29.101 0.84012539184953
29.201 0.84012539184953
29.301 0.84012539184953
29.401 0.84012539184953
29.501 0.843260188087774
29.601 0.843260188087774
29.701 0.843260188087774
29.801 0.843260188087774
29.901 0.843260188087774
30.001 0.855799373040752
30.101 0.855799373040752
30.201 0.855799373040752
30.301 0.858934169278997
30.401 0.858934169278997
30.501 0.871473354231975
30.601 0.871473354231975
30.701 0.871473354231975
30.801 0.871473354231975
30.901 0.871473354231975
31.001 0.884012539184953
31.101 0.884012539184953
31.201 0.884012539184953
31.301 0.884012539184953
31.401 0.884012539184953
31.501 0.890282131661442
31.601 0.890282131661442
31.701 0.890282131661442
31.801 0.890282131661442
31.901 0.890282131661442
32.001 0.905956112852665
32.101 0.905956112852665
32.201 0.905956112852665
32.301 0.905956112852665
32.401 0.905956112852665
32.501 0.909090909090909
32.601 0.909090909090909
32.701 0.909090909090909
32.801 0.909090909090909
32.901 0.909090909090909
33.001 0.918495297805643
33.101 0.918495297805643
33.201 0.918495297805643
33.301 0.918495297805643
33.401 0.918495297805643
33.501 0.921630094043887
33.601 0.921630094043887
33.701 0.921630094043887
33.801 0.921630094043887
33.901 0.921630094043887
34.001 0.931034482758621
34.101 0.931034482758621
34.201 0.931034482758621
34.301 0.931034482758621
34.401 0.931034482758621
34.501 0.934169278996865
34.601 0.934169278996865
34.701 0.934169278996865
34.801 0.934169278996865
34.901 0.934169278996865
35.001 0.93730407523511
35.101 0.93730407523511
35.201 0.93730407523511
35.301 0.93730407523511
35.401 0.93730407523511
35.501 0.940438871473354
35.601 0.940438871473354
35.701 0.940438871473354
35.801 0.940438871473354
35.901 0.940438871473354
36.001 0.946708463949843
36.101 0.946708463949843
36.201 0.946708463949843
36.301 0.946708463949843
36.401 0.946708463949843
36.501 0.949843260188088
36.601 0.949843260188088
36.701 0.949843260188088
36.801 0.949843260188088
36.901 0.949843260188088
37.001 0.952978056426332
37.101 0.952978056426332
37.201 0.952978056426332
37.301 0.952978056426332
37.401 0.952978056426332
37.501 0.956112852664577
37.601 0.956112852664577
37.701 0.956112852664577
37.801 0.959247648902821
37.901 0.959247648902821
38.001 0.962382445141066
38.101 0.962382445141066
38.201 0.962382445141066
38.301 0.962382445141066
38.401 0.962382445141066
38.501 0.96551724137931
38.601 0.96551724137931
38.701 0.96551724137931
38.801 0.96551724137931
38.901 0.96551724137931
39.001 0.968652037617555
39.101 0.968652037617555
39.201 0.968652037617555
39.301 0.968652037617555
39.401 0.968652037617555
39.501 0.968652037617555
39.601 0.968652037617555
39.701 0.968652037617555
39.801 0.968652037617555
39.901 0.968652037617555
40.001 0.968652037617555
40.101 0.968652037617555
40.201 0.968652037617555
40.301 0.968652037617555
40.401 0.968652037617555
40.501 0.968652037617555
40.601 0.968652037617555
40.701 0.968652037617555
40.801 0.968652037617555
40.901 0.968652037617555
41.001 0.971786833855799
41.101 0.971786833855799
41.201 0.971786833855799
41.301 0.971786833855799
41.401 0.971786833855799
41.501 0.971786833855799
41.601 0.971786833855799
41.701 0.971786833855799
41.801 0.971786833855799
41.901 0.971786833855799
42.001 0.971786833855799
42.101 0.971786833855799
42.201 0.971786833855799
42.301 0.971786833855799
42.401 0.971786833855799
42.501 0.971786833855799
42.601 0.971786833855799
42.701 0.971786833855799
42.801 0.971786833855799
42.901 0.971786833855799
43.001 0.971786833855799
43.101 0.971786833855799
43.201 0.971786833855799
43.301 0.971786833855799
43.401 0.971786833855799
43.501 0.974921630094044
43.601 0.974921630094044
43.701 0.974921630094044
43.801 0.974921630094044
43.901 0.974921630094044
44.001 0.981191222570533
44.101 0.981191222570533
44.201 0.981191222570533
44.301 0.981191222570533
44.401 0.981191222570533
44.501 0.981191222570533
44.601 0.981191222570533
44.701 0.981191222570533
44.801 0.981191222570533
44.901 0.981191222570533
45.001 0.984326018808777
45.101 0.984326018808777
45.201 0.984326018808777
45.301 0.984326018808777
45.401 0.984326018808777
45.501 0.984326018808777
45.601 0.984326018808777
45.701 0.984326018808777
45.801 0.984326018808777
45.901 0.984326018808777
46.001 0.984326018808777
46.101 0.984326018808777
46.201 0.984326018808777
46.301 0.984326018808777
46.401 0.984326018808777
46.501 0.987460815047022
46.601 0.987460815047022
46.701 0.987460815047022
46.801 0.987460815047022
46.901 0.987460815047022
47.001 0.987460815047022
47.101 0.987460815047022
47.201 0.987460815047022
47.301 0.987460815047022
47.401 0.987460815047022
47.501 0.990595611285266
47.601 0.990595611285266
47.701 0.990595611285266
47.801 0.990595611285266
47.901 0.990595611285266
48.001 0.990595611285266
48.101 0.990595611285266
48.201 0.990595611285266
48.301 0.990595611285266
48.401 0.990595611285266
48.501 0.990595611285266
48.601 0.990595611285266
48.701 0.990595611285266
48.801 0.990595611285266
48.901 0.990595611285266
49.001 0.990595611285266
49.101 0.990595611285266
49.201 0.990595611285266
49.301 0.990595611285266
49.401 0.990595611285266
49.501 0.990595611285266
49.601 0.990595611285266
49.701 0.990595611285266
49.801 0.990595611285266
49.901 0.990595611285266
};
\addlegendentry{$F_{Y_1|G,T}(\cdot|1,1)$}
\end{axis}
\end{tikzpicture}
\caption{Distribution of Total employment}
\label{fig:dist_total_employment}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\definecolor{darkgray176}{RGB}{176,176,176}
\definecolor{lightgray204}{RGB}{204,204,204}
\begin{axis}[
legend cell align={left},
legend style={
fill opacity=0.8,
draw opacity=1,
text opacity=1,
at={(0.97,0.03)},
anchor=south east,
draw=lightgray204
},
tick align=outside,
tick pos=left,
x grid style={darkgray176},
xlabel={Parttime employment},
xmin=-2.494, xmax=52.396,
xtick style={color=black},
y grid style={darkgray176},
ylabel={\(\displaystyle F_{Y_0|G,T}(\cdot|1,1), F_{Y_1|G,T}(\cdot|1,1)\)},
ymin=0, ymax=1,
ytick style={color=black}
]
\addplot [thick, black]
table {
0.001 0.00719937173478092
0.101 0.00719937173478092
0.201 0.00719937173478092
0.301 0.00719937173478092
0.401 0.00719937173478092
0.501 0.00719937173478092
0.601 0.00719937173478092
0.701 0.00719937173478092
0.801 0.00719937173478092
0.901 0.00719937173478092
1.001 0.00719937173478092
1.101 0.00719937173478092
1.201 0.00719937173478092
1.301 0.00719937173478092
1.401 0.00719937173478092
1.501 0.00719937173478092
1.601 0.00719937173478092
1.701 0.00719937173478092
1.801 0.00719937173478092
1.901 0.00719937173478092
2.001 0.00944299938793703
2.101 0.00944299938793703
2.201 0.00944299938793703
2.301 0.00944299938793703
2.401 0.00944299938793703
2.501 0.00944299938793703
2.601 0.00944299938793703
2.701 0.00944299938793703
2.801 0.00944299938793703
2.901 0.00944299938793703
3.001 0.0216731193642751
3.101 0.0216731193642751
3.201 0.0216731193642751
3.301 0.0216731193642751
3.401 0.0216731193642751
3.501 0.0216731193642751
3.601 0.0216731193642751
3.701 0.0216731193642751
3.801 0.0216731193642751
3.901 0.0216731193642751
4.001 0.0450021361750035
4.101 0.0450021361750035
4.201 0.0450021361750035
4.301 0.0450021361750035
4.401 0.0450021361750035
4.501 0.0450021361750035
4.601 0.0450021361750035
4.701 0.0450021361750035
4.801 0.0450021361750035
4.901 0.0450021361750035
5.001 0.0540648809648056
5.101 0.0540648809648056
5.201 0.0540648809648056
5.301 0.0540648809648056
5.401 0.0540648809648056
5.501 0.0620361109510146
5.601 0.0620361109510146
5.701 0.0620361109510146
5.801 0.0620361109510146
5.901 0.0620361109510146
6.001 0.0732575263881942
6.101 0.0732575263881942
6.201 0.0732575263881942
6.301 0.0732575263881942
6.401 0.0732575263881942
6.501 0.0868877263757681
6.601 0.0868877263757681
6.701 0.0868877263757681
6.801 0.0868877263757681
6.901 0.0868877263757681
7.001 0.117273734848147
7.101 0.117273734848147
7.201 0.117273734848147
7.301 0.117273734848147
7.401 0.117273734848147
7.501 0.117956170943056
7.601 0.117956170943056
7.701 0.117956170943056
7.801 0.117956170943056
7.901 0.117956170943056
8.001 0.126159751153983
8.101 0.126159751153983
8.201 0.126159751153983
8.301 0.126159751153983
8.401 0.126159751153983
8.501 0.12920693058692
8.601 0.12920693058692
8.701 0.12920693058692
8.801 0.12920693058692
8.901 0.12920693058692
9.001 0.133671131759373
9.101 0.133671131759373
9.201 0.133671131759373
9.301 0.133671131759373
9.401 0.133671131759373
9.501 0.133671131759373
9.601 0.133671131759373
9.701 0.133671131759373
9.801 0.133671131759373
9.901 0.133671131759373
10.001 0.189816359389361
10.101 0.189816359389361
10.201 0.189816359389361
10.301 0.189816359389361
10.401 0.189816359389361
10.501 0.19172625375139
10.601 0.19172625375139
10.701 0.19172625375139
10.801 0.19172625375139
10.901 0.19172625375139
11.001 0.205960031769117
11.101 0.205960031769117
11.201 0.205960031769117
11.301 0.205960031769117
11.401 0.205960031769117
11.501 0.208728949538517
11.601 0.208728949538517
11.701 0.208728949538517
11.801 0.208728949538517
11.901 0.208728949538517
12.001 0.299810531702508
12.101 0.299810531702508
12.201 0.299810531702508
12.301 0.299810531702508
12.401 0.299810531702508
12.501 0.312684694435202
12.601 0.312684694435202
12.701 0.312684694435202
12.801 0.312684694435202
12.901 0.312684694435202
13.001 0.350724449642121
13.101 0.350724449642121
13.201 0.350724449642121
13.301 0.350724449642121
13.401 0.350724449642121
13.501 0.350724449642121
13.601 0.350724449642121
13.701 0.350724449642121
13.801 0.350724449642121
13.901 0.350724449642121
14.001 0.377028440013055
14.101 0.377028440013055
14.201 0.377028440013055
14.301 0.377028440013055
14.401 0.377028440013055
14.501 0.382620944167319
14.601 0.382620944167319
14.701 0.382620944167319
14.801 0.382620944167319
14.901 0.382620944167319
15.001 0.491562957663961
15.101 0.491562957663961
15.201 0.491562957663961
15.301 0.491562957663961
15.401 0.491562957663961
15.501 0.491562957663961
15.601 0.491562957663961
15.701 0.491562957663961
15.801 0.491562957663961
15.901 0.491562957663961
16.001 0.49890392062993
16.101 0.49890392062993
16.201 0.49890392062993
16.301 0.49890392062993
16.401 0.49890392062993
16.501 0.50187616263305
16.601 0.50187616263305
16.701 0.50187616263305
16.801 0.50187616263305
16.901 0.50187616263305
17.001 0.507535283076455
17.101 0.507535283076455
17.201 0.507535283076455
17.301 0.507535283076455
17.401 0.507535283076455
17.501 0.513841850653868
17.601 0.513841850653868
17.701 0.513841850653868
17.801 0.513841850653868
17.901 0.513841850653868
18.001 0.544702736266161
18.101 0.544702736266161
18.201 0.544702736266161
18.301 0.544702736266161
18.401 0.544702736266161
18.501 0.544702736266161
18.601 0.544702736266161
18.701 0.544702736266161
18.801 0.544702736266161
18.901 0.544702736266161
19.001 0.572387921354244
19.101 0.572387921354244
19.201 0.572387921354244
19.301 0.572387921354244
19.401 0.572387921354244
19.501 0.572387921354244
19.601 0.572387921354244
19.701 0.572387921354244
19.801 0.572387921354244
19.901 0.572387921354244
20.001 0.683501747921443
20.101 0.683501747921443
20.201 0.683501747921443
20.301 0.683501747921443
20.401 0.683501747921443
20.501 0.684062117045212
20.601 0.684062117045212
20.701 0.684062117045212
20.801 0.684062117045212
20.901 0.684062117045212
21.001 0.684062117045212
21.101 0.684062117045212
21.201 0.684062117045212
21.301 0.684062117045212
21.401 0.684062117045212
21.501 0.690131071100096
21.601 0.690131071100096
21.701 0.690131071100096
21.801 0.690131071100096
21.901 0.690131071100096
22.001 0.705122070813485
22.101 0.705122070813485
22.201 0.705122070813485
22.301 0.705122070813485
22.401 0.705122070813485
22.501 0.710979396208898
22.601 0.710979396208898
22.701 0.710979396208898
22.801 0.710979396208898
22.901 0.710979396208898
23.001 0.724600497331596
23.101 0.724600497331596
23.201 0.724600497331596
23.301 0.724600497331596
23.401 0.724600497331596
23.501 0.727796749150171
23.601 0.727796749150171
23.701 0.727796749150171
23.801 0.727796749150171
23.901 0.727796749150171
24.001 0.734728421399329
24.101 0.734728421399329
24.201 0.734728421399329
24.301 0.734728421399329
24.401 0.734728421399329
24.501 0.734728421399329
24.601 0.734728421399329
24.701 0.734728421399329
24.801 0.734728421399329
24.901 0.734728421399329
25.001 0.765526141121988
25.101 0.765526141121988
25.201 0.765526141121988
25.301 0.765526141121988
25.401 0.765526141121988
25.501 0.770266473291406
25.601 0.770266473291406
25.701 0.770266473291406
25.801 0.770266473291406
25.901 0.770266473291406
26.001 0.77223779434644
26.101 0.77223779434644
26.201 0.77223779434644
26.301 0.77223779434644
26.401 0.77223779434644
26.501 0.777640306789369
26.601 0.777640306789369
26.701 0.777640306789369
26.801 0.777640306789369
26.901 0.777640306789369
27.001 0.781605350823748
27.101 0.781605350823748
27.201 0.781605350823748
27.301 0.781605350823748
27.401 0.781605350823748
27.501 0.782186794124187
27.601 0.782186794124187
27.701 0.782186794124187
27.801 0.782186794124187
27.901 0.782186794124187
28.001 0.78436960410118
28.101 0.78436960410118
28.201 0.78436960410118
28.301 0.78436960410118
28.401 0.78436960410118
28.501 0.78436960410118
28.601 0.78436960410118
28.701 0.78436960410118
28.801 0.78436960410118
28.901 0.78436960410118
29.001 0.787717932482892
29.101 0.787717932482892
29.201 0.787717932482892
29.301 0.787717932482892
29.401 0.787717932482892
29.501 0.792269620483456
29.601 0.792269620483456
29.701 0.792269620483456
29.801 0.792269620483456
29.901 0.792269620483456
30.001 0.831272288150006
30.101 0.831272288150006
30.201 0.831272288150006
30.301 0.831272288150006
30.401 0.831272288150006
30.501 0.831272288150006
30.601 0.831272288150006
30.701 0.831272288150006
30.801 0.831272288150006
30.901 0.831272288150006
31.001 0.833294800284595
31.101 0.833294800284595
31.201 0.833294800284595
31.301 0.833294800284595
31.401 0.833294800284595
31.501 0.833294800284595
31.601 0.833294800284595
31.701 0.833294800284595
31.801 0.833294800284595
31.901 0.833294800284595
32.001 0.856821874011011
32.101 0.856821874011011
32.201 0.856821874011011
32.301 0.856821874011011
32.401 0.856821874011011
32.501 0.856821874011011
32.601 0.856821874011011
32.701 0.856821874011011
32.801 0.856821874011011
32.901 0.856821874011011
33.001 0.866538812950489
33.101 0.866538812950489
33.201 0.866538812950489
33.301 0.866538812950489
33.401 0.866538812950489
33.501 0.866538812950489
33.601 0.866538812950489
33.701 0.866538812950489
33.801 0.866538812950489
33.901 0.866538812950489
34.001 0.870522522642777
34.101 0.870522522642777
34.201 0.870522522642777
34.301 0.870522522642777
34.401 0.870522522642777
34.501 0.870522522642777
34.601 0.870522522642777
34.701 0.870522522642777
34.801 0.870522522642777
34.901 0.870522522642777
35.001 0.904778089066988
35.101 0.904778089066988
35.201 0.904778089066988
35.301 0.904778089066988
35.401 0.904778089066988
35.501 0.904778089066988
35.601 0.904778089066988
35.701 0.904778089066988
35.801 0.904778089066988
35.901 0.904778089066988
36.001 0.915885065695571
36.101 0.915885065695571
36.201 0.915885065695571
36.301 0.915885065695571
36.401 0.915885065695571
36.501 0.920268269940605
36.601 0.920268269940605
36.701 0.920268269940605
36.801 0.920268269940605
36.901 0.920268269940605
37.001 0.920268269940605
37.101 0.920268269940605
37.201 0.920268269940605
37.301 0.920268269940605
37.401 0.920268269940605
37.501 0.926814607481556
37.601 0.926814607481556
37.701 0.926814607481556
37.801 0.926814607481556
37.901 0.926814607481556
38.001 0.926814607481556
38.101 0.926814607481556
38.201 0.926814607481556
38.301 0.926814607481556
38.401 0.926814607481556
38.501 0.926814607481556
38.601 0.926814607481556
38.701 0.926814607481556
38.801 0.926814607481556
38.901 0.926814607481556
39.001 0.932466137400706
39.101 0.932466137400706
39.201 0.932466137400706
39.301 0.932466137400706
39.401 0.932466137400706
39.501 0.932466137400706
39.601 0.932466137400706
39.701 0.932466137400706
39.801 0.932466137400706
39.901 0.932466137400706
40.001 0.961885553897417
40.101 0.961885553897417
40.201 0.961885553897417
40.301 0.961885553897417
40.401 0.961885553897417
40.501 0.961885553897417
40.601 0.961885553897417
40.701 0.961885553897417
40.801 0.961885553897417
40.901 0.961885553897417
41.001 0.961885553897417
41.101 0.961885553897417
41.201 0.961885553897417
41.301 0.961885553897417
41.401 0.961885553897417
41.501 0.961885553897417
41.601 0.961885553897417
41.701 0.961885553897417
41.801 0.961885553897417
41.901 0.961885553897417
42.001 0.977703180263244
42.101 0.977703180263244
42.201 0.977703180263244
42.301 0.977703180263244
42.401 0.977703180263244
42.501 0.977703180263244
42.601 0.977703180263244
42.701 0.977703180263244
42.801 0.977703180263244
42.901 0.977703180263244
43.001 0.977703180263244
43.101 0.977703180263244
43.201 0.977703180263244
43.301 0.977703180263244
43.401 0.977703180263244
43.501 0.977703180263244
43.601 0.977703180263244
43.701 0.977703180263244
43.801 0.977703180263244
43.901 0.977703180263244
44.001 0.977703180263244
44.101 0.977703180263244
44.201 0.977703180263244
44.301 0.977703180263244
44.401 0.977703180263244
44.501 0.977703180263244
44.601 0.977703180263244
44.701 0.977703180263244
44.801 0.977703180263244
44.901 0.977703180263244
45.001 0.986149316192802
45.101 0.986149316192802
45.201 0.986149316192802
45.301 0.986149316192802
45.401 0.986149316192802
45.501 0.986149316192802
45.601 0.986149316192802
45.701 0.986149316192802
45.801 0.986149316192802
45.901 0.986149316192802
46.001 0.989200996635358
46.101 0.989200996635358
46.201 0.989200996635358
46.301 0.989200996635358
46.401 0.989200996635358
46.501 0.989200996635358
46.601 0.989200996635358
46.701 0.989200996635358
46.801 0.989200996635358
46.901 0.989200996635358
47.001 0.989200996635358
47.101 0.989200996635358
47.201 0.989200996635358
47.301 0.989200996635358
47.401 0.989200996635358
47.501 0.989200996635358
47.601 0.989200996635358
47.701 0.989200996635358
47.801 0.989200996635358
47.901 0.989200996635358
48.001 0.993460554461942
48.101 0.993460554461942
48.201 0.993460554461942
48.301 0.993460554461942
48.401 0.993460554461942
48.501 0.993460554461942
48.601 0.993460554461942
48.701 0.993460554461942
48.801 0.993460554461942
48.901 0.993460554461942
49.001 0.993460554461942
49.101 0.993460554461942
49.201 0.993460554461942
49.301 0.993460554461942
49.401 0.993460554461942
49.501 0.993460554461942
49.601 0.993460554461942
49.701 0.993460554461942
49.801 0.993460554461942
49.901 0.993460554461942
};
\addlegendentry{$F_{Y_0|G,T}(\cdot|1,1)$}
\addplot [thick, black, dashed]
table {
0.001 0.0247678018575851
0.101 0.0247678018575851
0.201 0.0247678018575851
0.301 0.0247678018575851
0.401 0.0247678018575851
0.501 0.0247678018575851
0.601 0.0247678018575851
0.701 0.0247678018575851
0.801 0.0247678018575851
0.901 0.0247678018575851
1.001 0.0247678018575851
1.101 0.0247678018575851
1.201 0.0247678018575851
1.301 0.0247678018575851
1.401 0.0247678018575851
1.501 0.0247678018575851
1.601 0.0247678018575851
1.701 0.0247678018575851
1.801 0.0247678018575851
1.901 0.0247678018575851
2.001 0.0371517027863777
2.101 0.0371517027863777
2.201 0.0371517027863777
2.301 0.0371517027863777
2.401 0.0371517027863777
2.501 0.0371517027863777
2.601 0.0371517027863777
2.701 0.0371517027863777
2.801 0.0371517027863777
2.901 0.0371517027863777
3.001 0.0526315789473684
3.101 0.0526315789473684
3.201 0.0526315789473684
3.301 0.0526315789473684
3.401 0.0526315789473684
3.501 0.0557275541795666
3.601 0.0557275541795666
3.701 0.0557275541795666
3.801 0.0557275541795666
3.901 0.0557275541795666
4.001 0.0743034055727554
4.101 0.0743034055727554
4.201 0.0743034055727554
4.301 0.0743034055727554
4.401 0.0743034055727554
4.501 0.0743034055727554
4.601 0.0743034055727554
4.701 0.0743034055727554
4.801 0.0743034055727554
4.901 0.0743034055727554
5.001 0.0928792569659443
5.101 0.0928792569659443
5.201 0.0928792569659443
5.301 0.0928792569659443
5.401 0.0928792569659443
5.501 0.0928792569659443
5.601 0.0928792569659443
5.701 0.0928792569659443
5.801 0.0928792569659443
5.901 0.0928792569659443
6.001 0.126934984520124
6.101 0.126934984520124
6.201 0.126934984520124
6.301 0.126934984520124
6.401 0.126934984520124
6.501 0.126934984520124
6.601 0.126934984520124
6.701 0.126934984520124
6.801 0.126934984520124
6.901 0.126934984520124
7.001 0.164086687306502
7.101 0.164086687306502
7.201 0.164086687306502
7.301 0.164086687306502
7.401 0.164086687306502
7.501 0.164086687306502
7.601 0.164086687306502
7.701 0.164086687306502
7.801 0.164086687306502
7.901 0.164086687306502
8.001 0.179566563467492
8.101 0.179566563467492
8.201 0.179566563467492
8.301 0.179566563467492
8.401 0.179566563467492
8.501 0.179566563467492
8.601 0.179566563467492
8.701 0.179566563467492
8.801 0.179566563467492
8.901 0.179566563467492
9.001 0.201238390092879
9.101 0.201238390092879
9.201 0.201238390092879
9.301 0.201238390092879
9.401 0.201238390092879
9.501 0.201238390092879
9.601 0.201238390092879
9.701 0.201238390092879
9.801 0.201238390092879
9.901 0.201238390092879
10.001 0.253869969040248
10.101 0.253869969040248
10.201 0.253869969040248
10.301 0.253869969040248
10.401 0.253869969040248
10.501 0.253869969040248
10.601 0.253869969040248
10.701 0.253869969040248
10.801 0.253869969040248
10.901 0.253869969040248
11.001 0.287925696594427
11.101 0.287925696594427
11.201 0.287925696594427
11.301 0.287925696594427
11.401 0.287925696594427
11.501 0.287925696594427
11.601 0.287925696594427
11.701 0.287925696594427
11.801 0.287925696594427
11.901 0.287925696594427
12.001 0.334365325077399
12.101 0.334365325077399
12.201 0.334365325077399
12.301 0.334365325077399
12.401 0.334365325077399
12.501 0.337461300309598
12.601 0.337461300309598
12.701 0.337461300309598
12.801 0.337461300309598
12.901 0.337461300309598
13.001 0.356037151702786
13.101 0.356037151702786
13.201 0.356037151702786
13.301 0.356037151702786
13.401 0.356037151702786
13.501 0.362229102167183
13.601 0.362229102167183
13.701 0.362229102167183
13.801 0.362229102167183
13.901 0.362229102167183
14.001 0.396284829721362
14.101 0.396284829721362
14.201 0.396284829721362
14.301 0.396284829721362
14.401 0.396284829721362
14.501 0.39938080495356
14.601 0.39938080495356
14.701 0.39938080495356
14.801 0.39938080495356
14.901 0.39938080495356
15.001 0.495356037151703
15.101 0.495356037151703
15.201 0.495356037151703
15.301 0.495356037151703
15.401 0.495356037151703
15.501 0.495356037151703
15.601 0.495356037151703
15.701 0.495356037151703
15.801 0.495356037151703
15.901 0.495356037151703
16.001 0.501547987616099
16.101 0.501547987616099
16.201 0.501547987616099
16.301 0.501547987616099
16.401 0.501547987616099
16.501 0.501547987616099
16.601 0.501547987616099
16.701 0.501547987616099
16.801 0.501547987616099
16.901 0.501547987616099
17.001 0.532507739938081
17.101 0.532507739938081
17.201 0.532507739938081
17.301 0.532507739938081
17.401 0.532507739938081
17.501 0.544891640866873
17.601 0.544891640866873
17.701 0.544891640866873
17.801 0.544891640866873
17.901 0.544891640866873
18.001 0.569659442724458
18.101 0.569659442724458
18.201 0.569659442724458
18.301 0.569659442724458
18.401 0.569659442724458
18.501 0.569659442724458
18.601 0.569659442724458
18.701 0.569659442724458
18.801 0.569659442724458
18.901 0.569659442724458
19.001 0.582043343653251
19.101 0.582043343653251
19.201 0.582043343653251
19.301 0.582043343653251
19.401 0.582043343653251
19.501 0.582043343653251
19.601 0.582043343653251
19.701 0.582043343653251
19.801 0.582043343653251
19.901 0.582043343653251
20.001 0.653250773993808
20.101 0.653250773993808
20.201 0.653250773993808
20.301 0.653250773993808
20.401 0.653250773993808
20.501 0.653250773993808
20.601 0.653250773993808
20.701 0.653250773993808
20.801 0.653250773993808
20.901 0.653250773993808
21.001 0.659442724458204
21.101 0.659442724458204
21.201 0.659442724458204
21.301 0.659442724458204
21.401 0.659442724458204
21.501 0.659442724458204
21.601 0.659442724458204
21.701 0.659442724458204
21.801 0.659442724458204
21.901 0.659442724458204
22.001 0.681114551083591
22.101 0.681114551083591
22.201 0.681114551083591
22.301 0.681114551083591
22.401 0.681114551083591
22.501 0.684210526315789
22.601 0.684210526315789
22.701 0.684210526315789
22.801 0.684210526315789
22.901 0.684210526315789
23.001 0.702786377708978
23.101 0.702786377708978
23.201 0.702786377708978
23.301 0.702786377708978
23.401 0.702786377708978
23.501 0.702786377708978
23.601 0.702786377708978
23.701 0.702786377708978
23.801 0.702786377708978
23.901 0.702786377708978
24.001 0.718266253869969
24.101 0.718266253869969
24.201 0.718266253869969
24.301 0.718266253869969
24.401 0.718266253869969
24.501 0.718266253869969
24.601 0.718266253869969
24.701 0.718266253869969
24.801 0.718266253869969
24.901 0.718266253869969
25.001 0.770897832817337
25.101 0.770897832817337
25.201 0.770897832817337
25.301 0.770897832817337
25.401 0.770897832817337
25.501 0.770897832817337
25.601 0.770897832817337
25.701 0.770897832817337
25.801 0.770897832817337
25.901 0.770897832817337
26.001 0.780185758513932
26.101 0.780185758513932
26.201 0.780185758513932
26.301 0.780185758513932
26.401 0.780185758513932
26.501 0.780185758513932
26.601 0.780185758513932
26.701 0.780185758513932
26.801 0.780185758513932
26.901 0.780185758513932
27.001 0.789473684210526
27.101 0.789473684210526
27.201 0.789473684210526
27.301 0.789473684210526
27.401 0.789473684210526
27.501 0.795665634674923
27.601 0.795665634674923
27.701 0.795665634674923
27.801 0.795665634674923
27.901 0.795665634674923
28.001 0.801857585139319
28.101 0.801857585139319
28.201 0.801857585139319
28.301 0.801857585139319
28.401 0.801857585139319
28.501 0.801857585139319
28.601 0.801857585139319
28.701 0.801857585139319
28.801 0.801857585139319
28.901 0.801857585139319
29.001 0.801857585139319
29.101 0.801857585139319
29.201 0.801857585139319
29.301 0.801857585139319
29.401 0.801857585139319
29.501 0.801857585139319
29.601 0.801857585139319
29.701 0.801857585139319
29.801 0.801857585139319
29.901 0.801857585139319
30.001 0.863777089783282
30.101 0.863777089783282
30.201 0.863777089783282
30.301 0.863777089783282
30.401 0.863777089783282
30.501 0.863777089783282
30.601 0.863777089783282
30.701 0.863777089783282
30.801 0.863777089783282
30.901 0.863777089783282
31.001 0.869969040247678
31.101 0.869969040247678
31.201 0.869969040247678
31.301 0.869969040247678
31.401 0.869969040247678
31.501 0.869969040247678
31.601 0.869969040247678
31.701 0.869969040247678
31.801 0.869969040247678
31.901 0.869969040247678
32.001 0.888544891640867
32.101 0.888544891640867
32.201 0.888544891640867
32.301 0.888544891640867
32.401 0.888544891640867
32.501 0.888544891640867
32.601 0.888544891640867
32.701 0.888544891640867
32.801 0.888544891640867
32.901 0.888544891640867
33.001 0.894736842105263
33.101 0.894736842105263
33.201 0.894736842105263
33.301 0.894736842105263
33.401 0.894736842105263
33.501 0.894736842105263
33.601 0.894736842105263
33.701 0.894736842105263
33.801 0.894736842105263
33.901 0.894736842105263
34.001 0.897832817337461
34.101 0.897832817337461
34.201 0.897832817337461
34.301 0.897832817337461
34.401 0.897832817337461
34.501 0.897832817337461
34.601 0.897832817337461
34.701 0.897832817337461
34.801 0.897832817337461
34.901 0.897832817337461
35.001 0.934984520123839
35.101 0.934984520123839
35.201 0.934984520123839
35.301 0.934984520123839
35.401 0.934984520123839
35.501 0.934984520123839
35.601 0.934984520123839
35.701 0.934984520123839
35.801 0.934984520123839
35.901 0.934984520123839
36.001 0.938080495356037
36.101 0.938080495356037
36.201 0.938080495356037
36.301 0.938080495356037
36.401 0.938080495356037
36.501 0.938080495356037
36.601 0.938080495356037
36.701 0.938080495356037
36.801 0.938080495356037
36.901 0.938080495356037
37.001 0.947368421052632
37.101 0.947368421052632
37.201 0.947368421052632
37.301 0.947368421052632
37.401 0.947368421052632
37.501 0.947368421052632
37.601 0.947368421052632
37.701 0.947368421052632
37.801 0.947368421052632
37.901 0.947368421052632
38.001 0.956656346749226
38.101 0.956656346749226
38.201 0.956656346749226
38.301 0.956656346749226
38.401 0.956656346749226
38.501 0.956656346749226
38.601 0.956656346749226
38.701 0.956656346749226
38.801 0.956656346749226
38.901 0.956656346749226
39.001 0.959752321981424
39.101 0.959752321981424
39.201 0.959752321981424
39.301 0.959752321981424
39.401 0.959752321981424
39.501 0.959752321981424
39.601 0.959752321981424
39.701 0.959752321981424
39.801 0.959752321981424
39.901 0.959752321981424
40.001 0.975232198142415
40.101 0.975232198142415
40.201 0.975232198142415
40.301 0.975232198142415
40.401 0.975232198142415
40.501 0.975232198142415
40.601 0.975232198142415
40.701 0.975232198142415
40.801 0.975232198142415
40.901 0.975232198142415
41.001 0.975232198142415
41.101 0.975232198142415
41.201 0.975232198142415
41.301 0.975232198142415
41.401 0.975232198142415
41.501 0.975232198142415
41.601 0.975232198142415
41.701 0.975232198142415
41.801 0.975232198142415
41.901 0.975232198142415
42.001 0.975232198142415
42.101 0.975232198142415
42.201 0.975232198142415
42.301 0.975232198142415
42.401 0.975232198142415
42.501 0.978328173374613
42.601 0.978328173374613
42.701 0.978328173374613
42.801 0.978328173374613
42.901 0.978328173374613
43.001 0.978328173374613
43.101 0.978328173374613
43.201 0.978328173374613
43.301 0.978328173374613
43.401 0.978328173374613
43.501 0.978328173374613
43.601 0.978328173374613
43.701 0.978328173374613
43.801 0.978328173374613
43.901 0.978328173374613
44.001 0.981424148606811
44.101 0.981424148606811
44.201 0.981424148606811
44.301 0.981424148606811
44.401 0.981424148606811
44.501 0.981424148606811
44.601 0.981424148606811
44.701 0.981424148606811
44.801 0.981424148606811
44.901 0.981424148606811
45.001 0.990712074303406
45.101 0.990712074303406
45.201 0.990712074303406
45.301 0.990712074303406
45.401 0.990712074303406
45.501 0.990712074303406
45.601 0.990712074303406
45.701 0.990712074303406
45.801 0.990712074303406
45.901 0.990712074303406
46.001 0.990712074303406
46.101 0.990712074303406
46.201 0.990712074303406
46.301 0.990712074303406
46.401 0.990712074303406
46.501 0.990712074303406
46.601 0.990712074303406
46.701 0.990712074303406
46.801 0.990712074303406
46.901 0.990712074303406
47.001 0.990712074303406
47.101 0.990712074303406
47.201 0.990712074303406
47.301 0.990712074303406
47.401 0.990712074303406
47.501 0.990712074303406
47.601 0.990712074303406
47.701 0.990712074303406
47.801 0.990712074303406
47.901 0.990712074303406
48.001 0.990712074303406
48.101 0.990712074303406
48.201 0.990712074303406
48.301 0.990712074303406
48.401 0.990712074303406
48.501 0.990712074303406
48.601 0.990712074303406
48.701 0.990712074303406
48.801 0.990712074303406
48.901 0.990712074303406
49.001 0.990712074303406
49.101 0.990712074303406
49.201 0.990712074303406
49.301 0.990712074303406
49.401 0.990712074303406
49.501 0.990712074303406
49.601 0.990712074303406
49.701 0.990712074303406
49.801 0.990712074303406
49.901 0.990712074303406
};
\addlegendentry{$F_{Y_1|G,T}(\cdot|1,1)$}
\end{axis}
\end{tikzpicture}
\caption{Part-time employment}
\label{fig:dist_parttime_employment}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\definecolor{darkgray176}{RGB}{176,176,176}
\definecolor{lightgray204}{RGB}{204,204,204}
\begin{axis}[
legend cell align={left},
legend style={
fill opacity=0.8,
draw opacity=1,
text opacity=1,
at={(0.97,0.03)},
anchor=south east,
draw=lightgray204
},
tick align=outside,
tick pos=left,
x grid style={darkgray176},
xlabel={Fulltime employment},
xmin=-2.494, xmax=52.396,
xtick style={color=black},
y grid style={darkgray176},
ylabel={\(\displaystyle F_{Y_0|G,T}(\cdot|1,1), F_{Y_1|G,T}(\cdot|1,1)\)},
ymin=0, ymax=1,
ytick style={color=black}
]
\addplot [thick, black]
table {
0.001 0.310283666913142
0.101 0.310283666913142
0.201 0.310283666913142
0.301 0.310283666913142
0.401 0.310283666913142
0.501 0.310283666913142
0.601 0.310283666913142
0.701 0.310283666913142
0.801 0.310283666913142
0.901 0.310283666913142
1.001 0.333545370023105
1.101 0.333545370023105
1.201 0.333545370023105
1.301 0.333545370023105
1.401 0.333545370023105
1.501 0.345436819589401
1.601 0.345436819589401
1.701 0.345436819589401
1.801 0.345436819589401
1.901 0.345436819589401
2.001 0.351849652907392
2.101 0.351849652907392
2.201 0.351849652907392
2.301 0.351849652907392
2.401 0.351849652907392
2.501 0.351849652907392
2.601 0.351849652907392
2.701 0.351849652907392
2.801 0.351849652907392
2.901 0.351849652907392
3.001 0.416572049516381
3.101 0.416572049516381
3.201 0.416572049516381
3.301 0.416572049516381
3.401 0.416572049516381
3.501 0.422275906726586
3.601 0.422275906726586
3.701 0.422275906726586
3.801 0.422275906726586
3.901 0.422275906726586
4.001 0.484595287841694
4.101 0.484595287841694
4.201 0.484595287841694
4.301 0.484595287841694
4.401 0.484595287841694
4.501 0.502542182786913
4.601 0.502542182786913
4.701 0.502542182786913
4.801 0.502542182786913
4.901 0.502542182786913
5.001 0.565606894735175
5.101 0.565606894735175
5.201 0.565606894735175
5.301 0.565606894735175
5.401 0.565606894735175
5.501 0.581513303074874
5.601 0.581513303074874
5.701 0.581513303074874
5.801 0.581513303074874
5.901 0.581513303074874
6.001 0.609354864822778
6.101 0.609354864822778
6.201 0.609354864822778
6.301 0.609354864822778
6.401 0.609354864822778
6.501 0.611435158435049
6.601 0.611435158435049
6.701 0.611435158435049
6.801 0.611435158435049
6.901 0.611435158435049
7.001 0.647635070530288
7.101 0.647635070530288
7.201 0.647635070530288
7.301 0.647635070530288
7.401 0.647635070530288
7.501 0.651200259905936
7.601 0.651200259905936
7.701 0.651200259905936
7.801 0.651200259905936
7.901 0.651200259905936
8.001 0.706941664590729
8.101 0.706941664590729
8.201 0.706941664590729
8.301 0.706941664590729
8.401 0.706941664590729
8.501 0.709916556047761
8.601 0.709916556047761
8.701 0.709916556047761
8.801 0.709916556047761
8.901 0.709916556047761
9.001 0.723018947513683
9.101 0.723018947513683
9.201 0.723018947513683
9.301 0.723018947513683
9.401 0.723018947513683
9.501 0.723018947513683
9.601 0.723018947513683
9.701 0.723018947513683
9.801 0.723018947513683
9.901 0.723018947513683
10.001 0.761475974953671
10.101 0.761475974953671
10.201 0.761475974953671
10.301 0.761475974953671
10.401 0.761475974953671
10.501 0.770336162135448
10.601 0.770336162135448
10.701 0.770336162135448
10.801 0.770336162135448
10.901 0.770336162135448
11.001 0.786333075892174
11.101 0.786333075892174
11.201 0.786333075892174
11.301 0.786333075892174
11.401 0.786333075892174
11.501 0.790029000761231
11.601 0.790029000761231
11.701 0.790029000761231
11.801 0.790029000761231
11.901 0.790029000761231
12.001 0.815397516213478
12.101 0.815397516213478
12.201 0.815397516213478
12.301 0.815397516213478
12.401 0.815397516213478
12.501 0.822507876699984
12.601 0.822507876699984
12.701 0.822507876699984
12.801 0.822507876699984
12.901 0.822507876699984
13.001 0.831035779731273
13.101 0.831035779731273
13.201 0.831035779731273
13.301 0.831035779731273
13.401 0.831035779731273
13.501 0.83398395619812
13.601 0.83398395619812
13.701 0.83398395619812
13.801 0.83398395619812
13.901 0.83398395619812
14.001 0.841640998232687
14.101 0.841640998232687
14.201 0.841640998232687
14.301 0.841640998232687
14.401 0.841640998232687
14.501 0.843315393468916
14.601 0.843315393468916
14.701 0.843315393468916
14.801 0.843315393468916
14.901 0.843315393468916
15.001 0.90919871175383
15.101 0.90919871175383
15.201 0.90919871175383
15.301 0.90919871175383
15.401 0.90919871175383
15.501 0.90919871175383
15.601 0.90919871175383
15.701 0.90919871175383
15.801 0.90919871175383
15.901 0.90919871175383
16.001 0.914269800143712
16.101 0.914269800143712
16.201 0.914269800143712
16.301 0.914269800143712
16.401 0.914269800143712
16.501 0.915524989215635
16.601 0.915524989215635
16.701 0.915524989215635
16.801 0.915524989215635
16.901 0.915524989215635
17.001 0.915524989215635
17.101 0.915524989215635
17.201 0.915524989215635
17.301 0.915524989215635
17.401 0.915524989215635
17.501 0.915816437098505
17.601 0.915816437098505
17.701 0.915816437098505
17.801 0.915816437098505
17.901 0.915816437098505
18.001 0.930013707790841
18.101 0.930013707790841
18.201 0.930013707790841
18.301 0.930013707790841
18.401 0.930013707790841
18.501 0.930013707790841
18.601 0.930013707790841
18.701 0.930013707790841
18.801 0.930013707790841
18.901 0.930013707790841
19.001 0.937948913783768
19.101 0.937948913783768
19.201 0.937948913783768
19.301 0.937948913783768
19.401 0.937948913783768
19.501 0.937948913783768
19.601 0.937948913783768
19.701 0.937948913783768
19.801 0.937948913783768
19.901 0.937948913783768
20.001 0.947930370372704
20.101 0.947930370372704
20.201 0.947930370372704
20.301 0.947930370372704
20.401 0.947930370372704
20.501 0.947930370372704
20.601 0.947930370372704
20.701 0.947930370372704
20.801 0.947930370372704
20.901 0.947930370372704
21.001 0.948247654029025
21.101 0.948247654029025
21.201 0.948247654029025
21.301 0.948247654029025
21.401 0.948247654029025
21.501 0.948247654029025
21.601 0.948247654029025
21.701 0.948247654029025
21.801 0.948247654029025
21.901 0.948247654029025
22.001 0.948247654029025
22.101 0.948247654029025
22.201 0.948247654029025
22.301 0.948247654029025
22.401 0.948247654029025
22.501 0.948247654029025
22.601 0.948247654029025
22.701 0.948247654029025
22.801 0.948247654029025
22.901 0.948247654029025
23.001 0.957404666584611
23.101 0.957404666584611
23.201 0.957404666584611
23.301 0.957404666584611
23.401 0.957404666584611
23.501 0.957404666584611
23.601 0.957404666584611
23.701 0.957404666584611
23.801 0.957404666584611
23.901 0.957404666584611
24.001 0.958893303260747
24.101 0.958893303260747
24.201 0.958893303260747
24.301 0.958893303260747
24.401 0.958893303260747
24.501 0.958893303260747
24.601 0.958893303260747
24.701 0.958893303260747
24.801 0.958893303260747
24.901 0.958893303260747
25.001 0.964208243455312
25.101 0.964208243455312
25.201 0.964208243455312
25.301 0.964208243455312
25.401 0.964208243455312
25.501 0.964208243455312
25.601 0.964208243455312
25.701 0.964208243455312
25.801 0.964208243455312
25.901 0.964208243455312
26.001 0.971172896502793
26.101 0.971172896502793
26.201 0.971172896502793
26.301 0.971172896502793
26.401 0.971172896502793
26.501 0.974465562323628
26.601 0.974465562323628
26.701 0.974465562323628
26.801 0.974465562323628
26.901 0.974465562323628
27.001 0.974465562323628
27.101 0.974465562323628
27.201 0.974465562323628
27.301 0.974465562323628
27.401 0.974465562323628
27.501 0.976171479128888
27.601 0.976171479128888
27.701 0.976171479128888
27.801 0.976171479128888
27.901 0.976171479128888
28.001 0.981677212784048
28.101 0.981677212784048
28.201 0.981677212784048
28.301 0.981677212784048
28.401 0.981677212784048
28.501 0.981677212784048
28.601 0.981677212784048
28.701 0.981677212784048
28.801 0.981677212784048
28.901 0.981677212784048
29.001 0.981677212784048
29.101 0.981677212784048
29.201 0.981677212784048
29.301 0.981677212784048
29.401 0.981677212784048
29.501 0.981677212784048
29.601 0.981677212784048
29.701 0.981677212784048
29.801 0.981677212784048
29.901 0.981677212784048
30.001 0.989307931264235
30.101 0.989307931264235
30.201 0.989307931264235
30.301 0.989307931264235
30.401 0.989307931264235
30.501 0.989307931264235
30.601 0.989307931264235
30.701 0.989307931264235
30.801 0.989307931264235
30.901 0.989307931264235
31.001 0.989307931264235
31.101 0.989307931264235
31.201 0.989307931264235
31.301 0.989307931264235
31.401 0.989307931264235
31.501 0.989307931264235
31.601 0.989307931264235
31.701 0.989307931264235
31.801 0.989307931264235
31.901 0.989307931264235
32.001 0.989307931264235
32.101 0.989307931264235
32.201 0.989307931264235
32.301 0.989307931264235
32.401 0.989307931264235
32.501 0.989307931264235
32.601 0.989307931264235
32.701 0.989307931264235
32.801 0.989307931264235
32.901 0.989307931264235
33.001 0.990376540232749
33.101 0.990376540232749
33.201 0.990376540232749
33.301 0.990376540232749
33.401 0.990376540232749
33.501 0.990376540232749
33.601 0.990376540232749
33.701 0.990376540232749
33.801 0.990376540232749
33.901 0.990376540232749
34.001 0.990376540232749
34.101 0.990376540232749
34.201 0.990376540232749
34.301 0.990376540232749
34.401 0.990376540232749
34.501 0.990376540232749
34.601 0.990376540232749
34.701 0.990376540232749
34.801 0.990376540232749
34.901 0.990376540232749
35.001 0.995403553988656
35.101 0.995403553988656
35.201 0.995403553988656
35.301 0.995403553988656
35.401 0.995403553988656
35.501 0.995403553988656
35.601 0.995403553988656
35.701 0.995403553988656
35.801 0.995403553988656
35.901 0.995403553988656
36.001 0.995403553988656
36.101 0.995403553988656
36.201 0.995403553988656
36.301 0.995403553988656
36.401 0.995403553988656
36.501 0.995403553988656
36.601 0.995403553988656
36.701 0.995403553988656
36.801 0.995403553988656
36.901 0.995403553988656
37.001 0.995403553988656
37.101 0.995403553988656
37.201 0.995403553988656
37.301 0.995403553988656
37.401 0.995403553988656
37.501 0.995548887892671
37.601 0.995548887892671
37.701 0.995548887892671
37.801 0.995548887892671
37.901 0.995548887892671
38.001 0.995548887892671
38.101 0.995548887892671
38.201 0.995548887892671
38.301 0.995548887892671
38.401 0.995548887892671
38.501 0.995548887892671
38.601 0.995548887892671
38.701 0.995548887892671
38.801 0.995548887892671
38.901 0.995548887892671
39.001 0.995548887892671
39.101 0.995548887892671
39.201 0.995548887892671
39.301 0.995548887892671
39.401 0.995548887892671
39.501 0.995548887892671
39.601 0.995548887892671
39.701 0.995548887892671
39.801 0.995548887892671
39.901 0.995548887892671
40.001 0.997300319099687
40.101 0.997300319099687
40.201 0.997300319099687
40.301 0.997300319099687
40.401 0.997300319099687
40.501 0.997300319099687
40.601 0.997300319099687
40.701 0.997300319099687
40.801 0.997300319099687
40.901 0.997300319099687
41.001 0.997300319099687
41.101 0.997300319099687
41.201 0.997300319099687
41.301 0.997300319099687
41.401 0.997300319099687
41.501 0.997300319099687
41.601 0.997300319099687
41.701 0.997300319099687
41.801 0.997300319099687
41.901 0.997300319099687
42.001 0.997300319099687
42.101 0.997300319099687
42.201 0.997300319099687
42.301 0.997300319099687
42.401 0.997300319099687
42.501 0.997300319099687
42.601 0.997300319099687
42.701 0.997300319099687
42.801 0.997300319099687
42.901 0.997300319099687
43.001 0.997300319099687
43.101 0.997300319099687
43.201 0.997300319099687
43.301 0.997300319099687
43.401 0.997300319099687
43.501 0.997300319099687
43.601 0.997300319099687
43.701 0.997300319099687
43.801 0.997300319099687
43.901 0.997300319099687
44.001 0.997300319099687
44.101 0.997300319099687
44.201 0.997300319099687
44.301 0.997300319099687
44.401 0.997300319099687
44.501 0.997300319099687
44.601 0.997300319099687
44.701 0.997300319099687
44.801 0.997300319099687
44.901 0.997300319099687
45.001 0.997300319099687
45.101 0.997300319099687
45.201 0.997300319099687
45.301 0.997300319099687
45.401 0.997300319099687
45.501 0.997300319099687
45.601 0.997300319099687
45.701 0.997300319099687
45.801 0.997300319099687
45.901 0.997300319099687
46.001 0.997300319099687
46.101 0.997300319099687
46.201 0.997300319099687
46.301 0.997300319099687
46.401 0.997300319099687
46.501 0.997300319099687
46.601 0.997300319099687
46.701 0.997300319099687
46.801 0.997300319099687
46.901 0.997300319099687
47.001 0.997300319099687
47.101 0.997300319099687
47.201 0.997300319099687
47.301 0.997300319099687
47.401 0.997300319099687
47.501 0.997300319099687
47.601 0.997300319099687
47.701 0.997300319099687
47.801 0.997300319099687
47.901 0.997300319099687
48.001 0.997300319099687
48.101 0.997300319099687
48.201 0.997300319099687
48.301 0.997300319099687
48.401 0.997300319099687
48.501 0.997300319099687
48.601 0.997300319099687
48.701 0.997300319099687
48.801 0.997300319099687
48.901 0.997300319099687
49.001 0.997300319099687
49.101 0.997300319099687
49.201 0.997300319099687
49.301 0.997300319099687
49.401 0.997300319099687
49.501 0.997300319099687
49.601 0.997300319099687
49.701 0.997300319099687
49.801 0.997300319099687
49.901 0.997300319099687
};
\addlegendentry{$F_{Y_0|G,T}(\cdot|1,1)$}
\addplot [thick, black, dashed]
table {
0.001 0.180685358255452
0.101 0.180685358255452
0.201 0.180685358255452
0.301 0.180685358255452
0.401 0.180685358255452
0.501 0.180685358255452
0.601 0.180685358255452
0.701 0.180685358255452
0.801 0.180685358255452
0.901 0.180685358255452
1.001 0.205607476635514
1.101 0.205607476635514
1.201 0.205607476635514
1.301 0.205607476635514
1.401 0.205607476635514
1.501 0.21183800623053
1.601 0.21183800623053
1.701 0.21183800623053
1.801 0.21183800623053
1.901 0.21183800623053
2.001 0.271028037383178
2.101 0.271028037383178
2.201 0.271028037383178
2.301 0.271028037383178
2.401 0.271028037383178
2.501 0.274143302180685
2.601 0.274143302180685
2.701 0.274143302180685
2.801 0.274143302180685
2.901 0.274143302180685
3.001 0.330218068535826
3.101 0.330218068535826
3.201 0.330218068535826
3.301 0.330218068535826
3.401 0.330218068535826
3.501 0.330218068535826
3.601 0.330218068535826
3.701 0.330218068535826
3.801 0.330218068535826
3.901 0.330218068535826
4.001 0.392523364485981
4.101 0.392523364485981
4.201 0.392523364485981
4.301 0.392523364485981
4.401 0.392523364485981
4.501 0.398753894080997
4.601 0.398753894080997
4.701 0.398753894080997
4.801 0.398753894080997
4.901 0.398753894080997
5.001 0.473520249221184
5.101 0.473520249221184
5.201 0.473520249221184
5.301 0.473520249221184
5.401 0.473520249221184
5.501 0.473520249221184
5.601 0.473520249221184
5.701 0.473520249221184
5.801 0.473520249221184
5.901 0.473520249221184
6.001 0.514018691588785
6.101 0.514018691588785
6.201 0.514018691588785
6.301 0.514018691588785
6.401 0.514018691588785
6.501 0.514018691588785
6.601 0.514018691588785
6.701 0.514018691588785
6.801 0.514018691588785
6.901 0.514018691588785
7.001 0.538940809968847
7.101 0.538940809968847
7.201 0.538940809968847
7.301 0.538940809968847
7.401 0.538940809968847
7.501 0.545171339563863
7.601 0.545171339563863
7.701 0.545171339563863
7.801 0.545171339563863
7.901 0.545171339563863
8.001 0.579439252336449
8.101 0.579439252336449
8.201 0.579439252336449
8.301 0.579439252336449
8.401 0.579439252336449
8.501 0.582554517133956
8.601 0.582554517133956
8.701 0.582554517133956
8.801 0.582554517133956
8.901 0.582554517133956
9.001 0.595015576323987
9.101 0.595015576323987
9.201 0.595015576323987
9.301 0.595015576323987
9.401 0.595015576323987
9.501 0.595015576323987
9.601 0.595015576323987
9.701 0.595015576323987
9.801 0.595015576323987
9.901 0.595015576323987
10.001 0.663551401869159
10.101 0.663551401869159
10.201 0.663551401869159
10.301 0.663551401869159
10.401 0.663551401869159
10.501 0.663551401869159
10.601 0.663551401869159
10.701 0.663551401869159
10.801 0.663551401869159
10.901 0.663551401869159
11.001 0.697819314641745
11.101 0.697819314641745
11.201 0.697819314641745
11.301 0.697819314641745
11.401 0.697819314641745
11.501 0.697819314641745
11.601 0.697819314641745
11.701 0.697819314641745
11.801 0.697819314641745
11.901 0.697819314641745
12.001 0.747663551401869
12.101 0.747663551401869
12.201 0.747663551401869
12.301 0.747663551401869
12.401 0.747663551401869
12.501 0.747663551401869
12.601 0.747663551401869
12.701 0.747663551401869
12.801 0.747663551401869
12.901 0.747663551401869
13.001 0.766355140186916
13.101 0.766355140186916
13.201 0.766355140186916
13.301 0.766355140186916
13.401 0.766355140186916
13.501 0.769470404984424
13.601 0.769470404984424
13.701 0.769470404984424
13.801 0.769470404984424
13.901 0.769470404984424
14.001 0.785046728971963
14.101 0.785046728971963
14.201 0.785046728971963
14.301 0.785046728971963
14.401 0.785046728971963
14.501 0.785046728971963
14.601 0.785046728971963
14.701 0.785046728971963
14.801 0.785046728971963
14.901 0.785046728971963
15.001 0.844236760124611
15.101 0.844236760124611
15.201 0.844236760124611
15.301 0.844236760124611
15.401 0.844236760124611
15.501 0.844236760124611
15.601 0.844236760124611
15.701 0.844236760124611
15.801 0.844236760124611
15.901 0.844236760124611
16.001 0.853582554517134
16.101 0.853582554517134
16.201 0.853582554517134
16.301 0.853582554517134
16.401 0.853582554517134
16.501 0.853582554517134
16.601 0.853582554517134
16.701 0.853582554517134
16.801 0.853582554517134
16.901 0.853582554517134
17.001 0.85981308411215
17.101 0.85981308411215
17.201 0.85981308411215
17.301 0.85981308411215
17.401 0.85981308411215
17.501 0.85981308411215
17.601 0.85981308411215
17.701 0.85981308411215
17.801 0.85981308411215
17.901 0.85981308411215
18.001 0.878504672897196
18.101 0.878504672897196
18.201 0.878504672897196
18.301 0.878504672897196
18.401 0.878504672897196
18.501 0.878504672897196
18.601 0.878504672897196
18.701 0.878504672897196
18.801 0.878504672897196
18.901 0.878504672897196
19.001 0.881619937694704
19.101 0.881619937694704
19.201 0.881619937694704
19.301 0.881619937694704
19.401 0.881619937694704
19.501 0.881619937694704
19.601 0.881619937694704
19.701 0.881619937694704
19.801 0.881619937694704
19.901 0.881619937694704
20.001 0.91588785046729
20.101 0.91588785046729
20.201 0.91588785046729
20.301 0.91588785046729
20.401 0.91588785046729
20.501 0.91588785046729
20.601 0.91588785046729
20.701 0.91588785046729
20.801 0.91588785046729
20.901 0.91588785046729
21.001 0.922118380062305
21.101 0.922118380062305
21.201 0.922118380062305
21.301 0.922118380062305
21.401 0.922118380062305
21.501 0.922118380062305
21.601 0.922118380062305
21.701 0.922118380062305
21.801 0.922118380062305
21.901 0.922118380062305
22.001 0.934579439252336
22.101 0.934579439252336
22.201 0.934579439252336
22.301 0.934579439252336
22.401 0.934579439252336
22.501 0.934579439252336
22.601 0.934579439252336
22.701 0.934579439252336
22.801 0.934579439252336
22.901 0.934579439252336
23.001 0.937694704049844
23.101 0.937694704049844
23.201 0.937694704049844
23.301 0.937694704049844
23.401 0.937694704049844
23.501 0.937694704049844
23.601 0.937694704049844
23.701 0.937694704049844
23.801 0.937694704049844
23.901 0.937694704049844
24.001 0.940809968847352
24.101 0.940809968847352
24.201 0.940809968847352
24.301 0.940809968847352
24.401 0.940809968847352
24.501 0.940809968847352
24.601 0.940809968847352
24.701 0.940809968847352
24.801 0.940809968847352
24.901 0.940809968847352
25.001 0.962616822429907
25.101 0.962616822429907
25.201 0.962616822429907
25.301 0.962616822429907
25.401 0.962616822429907
25.501 0.962616822429907
25.601 0.962616822429907
25.701 0.962616822429907
25.801 0.962616822429907
25.901 0.962616822429907
26.001 0.97196261682243
26.101 0.97196261682243
26.201 0.97196261682243
26.301 0.97196261682243
26.401 0.97196261682243
26.501 0.975077881619938
26.601 0.975077881619938
26.701 0.975077881619938
26.801 0.975077881619938
26.901 0.975077881619938
27.001 0.975077881619938
27.101 0.975077881619938
27.201 0.975077881619938
27.301 0.975077881619938
27.401 0.975077881619938
27.501 0.975077881619938
27.601 0.975077881619938
27.701 0.975077881619938
27.801 0.975077881619938
27.901 0.975077881619938
28.001 0.981308411214953
28.101 0.981308411214953
28.201 0.981308411214953
28.301 0.981308411214953
28.401 0.981308411214953
28.501 0.981308411214953
28.601 0.981308411214953
28.701 0.981308411214953
28.801 0.981308411214953
28.901 0.981308411214953
29.001 0.984423676012461
29.101 0.984423676012461
29.201 0.984423676012461
29.301 0.984423676012461
29.401 0.984423676012461
29.501 0.984423676012461
29.601 0.984423676012461
29.701 0.984423676012461
29.801 0.984423676012461
29.901 0.984423676012461
30.001 0.993769470404984
30.101 0.993769470404984
30.201 0.993769470404984
30.301 0.993769470404984
30.401 0.993769470404984
30.501 0.993769470404984
30.601 0.993769470404984
30.701 0.993769470404984
30.801 0.993769470404984
30.901 0.993769470404984
31.001 0.993769470404984
31.101 0.993769470404984
31.201 0.993769470404984
31.301 0.993769470404984
31.401 0.993769470404984
31.501 0.993769470404984
31.601 0.993769470404984
31.701 0.993769470404984
31.801 0.993769470404984
31.901 0.993769470404984
32.001 0.993769470404984
32.101 0.993769470404984
32.201 0.993769470404984
32.301 0.993769470404984
32.401 0.993769470404984
32.501 0.993769470404984
32.601 0.993769470404984
32.701 0.993769470404984
32.801 0.993769470404984
32.901 0.993769470404984
33.001 0.993769470404984
33.101 0.993769470404984
33.201 0.993769470404984
33.301 0.993769470404984
33.401 0.993769470404984
33.501 0.993769470404984
33.601 0.993769470404984
33.701 0.993769470404984
33.801 0.993769470404984
33.901 0.993769470404984
34.001 0.993769470404984
34.101 0.993769470404984
34.201 0.993769470404984
34.301 0.993769470404984
34.401 0.993769470404984
34.501 0.993769470404984
34.601 0.993769470404984
34.701 0.993769470404984
34.801 0.993769470404984
34.901 0.993769470404984
35.001 0.996884735202492
35.101 0.996884735202492
35.201 0.996884735202492
35.301 0.996884735202492
35.401 0.996884735202492
35.501 0.996884735202492
35.601 0.996884735202492
35.701 0.996884735202492
35.801 0.996884735202492
35.901 0.996884735202492
36.001 0.996884735202492
36.101 0.996884735202492
36.201 0.996884735202492
36.301 0.996884735202492
36.401 0.996884735202492
36.501 0.996884735202492
36.601 0.996884735202492
36.701 0.996884735202492
36.801 0.996884735202492
36.901 0.996884735202492
37.001 0.996884735202492
37.101 0.996884735202492
37.201 0.996884735202492
37.301 0.996884735202492
37.401 0.996884735202492
37.501 0.996884735202492
37.601 0.996884735202492
37.701 0.996884735202492
37.801 0.996884735202492
37.901 0.996884735202492
38.001 0.996884735202492
38.101 0.996884735202492
38.201 0.996884735202492
38.301 0.996884735202492
38.401 0.996884735202492
38.501 0.996884735202492
38.601 0.996884735202492
38.701 0.996884735202492
38.801 0.996884735202492
38.901 0.996884735202492
39.001 0.996884735202492
39.101 0.996884735202492
39.201 0.996884735202492
39.301 0.996884735202492
39.401 0.996884735202492
39.501 0.996884735202492
39.601 0.996884735202492
39.701 0.996884735202492
39.801 0.996884735202492
39.901 0.996884735202492
40.001 1
40.101 1
40.201 1
40.301 1
40.401 1
40.501 1
40.601 1
40.701 1
40.801 1
40.901 1
41.001 1
41.101 1
41.201 1
41.301 1
41.401 1
41.501 1
41.601 1
41.701 1
41.801 1
41.901 1
42.001 1
42.101 1
42.201 1
42.301 1
42.401 1
42.501 1
42.601 1
42.701 1
42.801 1
42.901 1
43.001 1
43.101 1
43.201 1
43.301 1
43.401 1
43.501 1
43.601 1
43.701 1
43.801 1
43.901 1
44.001 1
44.101 1
44.201 1
44.301 1
44.401 1
44.501 1
44.601 1
44.701 1
44.801 1
44.901 1
45.001 1
45.101 1
45.201 1
45.301 1
45.401 1
45.501 1
45.601 1
45.701 1
45.801 1
45.901 1
46.001 1
46.101 1
46.201 1
46.301 1
46.401 1
46.501 1
46.601 1
46.701 1
46.801 1
46.901 1
47.001 1
47.101 1
47.201 1
47.301 1
47.401 1
47.501 1
47.601 1
47.701 1
47.801 1
47.901 1
48.001 1
48.101 1
48.201 1
48.301 1
48.401 1
48.501 1
48.601 1
48.701 1
48.801 1
48.901 1
49.001 1
49.101 1
49.201 1
49.301 1
49.401 1
49.501 1
49.601 1
49.701 1
49.801 1
49.901 1
};
\addlegendentry{$F_{Y_1|G,T}(\cdot|1,1)$}
\end{axis}
\end{tikzpicture}
\caption{Full-time employment}
\label{fig:dist_fulltime_employment}
\end{figure}
\begin{figure}
\caption{Joint Distribution of Full-time and Part-time Employment with and without Treatment}
\label{fig:2dim}
\begin{tikzpicture}
\definecolor{darkgrey176}{RGB}{176,176,176}
\begin{groupplot}[group style={group size=2 by 1}]
\nextgroupplot[
tick align=outside,
tick pos=left,
title={\(\displaystyle F_{Y_0, Z_0|G,T}(y,z|1,1)\)},
x grid style={darkgrey176},
xlabel={Part-time employment},
xmin=0, xmax=50,
xtick style={color=black},
y grid style={darkgrey176},
ylabel={Full-time employment},
ymin=0, ymax=50,
ytick style={color=black}
]
\path [draw=black, semithick]
(axis cs:0.001,18.001)
--(axis cs:0.501,18.001)
--(axis cs:1.001,18.001)
--(axis cs:1.501,18.001)
--(axis cs:1.501,18.001)
--(axis cs:1.501,17.501)
--(axis cs:1.501,17.001)
--(axis cs:1.501,16.501)
--(axis cs:1.501,16.001)
--(axis cs:1.501,15.501)
--(axis cs:1.501,15.001)
--(axis cs:1.501,14.501)
--(axis cs:1.501,14.001)
--(axis cs:1.501,13.501)
--(axis cs:1.501,13.001)
--(axis cs:1.501,12.501)
--(axis cs:1.501,12.001)
--(axis cs:2.001,11.501)
--(axis cs:2.001,11.501)
--(axis cs:2.501,11.001)
--(axis cs:2.501,11.001)
--(axis cs:2.501,10.501)
--(axis cs:2.501,10.001)
--(axis cs:2.501,9.501)
--(axis cs:3.001,9.001)
--(axis cs:3.001,9.001)
--(axis cs:3.501,8.501)
--(axis cs:3.501,8.501)
--(axis cs:3.501,8.001)
--(axis cs:3.501,7.501)
--(axis cs:3.501,7.001)
--(axis cs:3.501,6.501)
--(axis cs:4.001,6.001)
--(axis cs:4.001,6.001)
--(axis cs:4.001,5.80061831702536);
\path [draw=black, semithick]
(axis cs:11.7354789982839,49.501)
--(axis cs:11.745474753912,47.100000226867)
(axis cs:11.7479425410311,38.901974025974)
--(axis cs:11.7877582453586,30.001)
--(axis cs:11.8638291320599,29.001)
--(axis cs:11.9677421042724,24.001)
--(axis cs:12.0503470895026,20.001)
--(axis cs:12.501,19.9553885775751)
--(axis cs:13.001,19.6714456169725)
--(axis cs:13.501,19.6714456169725)
--(axis cs:13.9818964317202,19.501)
--(axis cs:14.001,18.9225798956606)
--(axis cs:14.1563834402011,18.501)
--(axis cs:14.1563834402011,18.001)
--(axis cs:14.5082214473137,17.501)
--(axis cs:14.5972362050034,15.001)
--(axis cs:14.7583203164465,14.501)
--(axis cs:14.7836772837853,14.001)
--(axis cs:15.001,13.5153303458844)
--(axis cs:15.5293119713747,13.501)
--(axis cs:15.5766356617584,13.001)
--(axis cs:16.001,12.1055574839509)
--(axis cs:16.5994089449796,12.001)
--(axis cs:16.7770892517638,11.501)
--(axis cs:17.001,11.2250446105336)
--(axis cs:17.501,10.9469272848419)
--(axis cs:17.668595058825,10.501)
--(axis cs:17.6934155219768,10.001)
--(axis cs:17.9344679258385,9.501)
--(axis cs:18.001,8.84221069317035)
--(axis cs:18.501,8.84221069317035)
--(axis cs:19.001,5.14244364433432)
--(axis cs:19.501,5.14244364433432)
--(axis cs:19.5329654330597,5.001)
--(axis cs:20.001,4.69668565577148)
--(axis cs:21.501,4.61932372736111)
--(axis cs:22.501,4.50360306894046)
--(axis cs:23.501,4.47888824600677)
--(axis cs:24.001,4.23805496149208)
--(axis cs:25.001,4.10649474831655)
--(axis cs:27.001,3.96223293507315)
--(axis cs:27.501,3.96223293507315)
--(axis cs:28.119256871669,3.501)
--(axis cs:29.501,2.99879709094098)
--(axis cs:30.001,2.62253162130889)
--(axis cs:32.501,2.56648305319843)
--(axis cs:32.8793563546719,2.501)
--(axis cs:33.001,2.09519996648178)
--(axis cs:33.5447225957998,2.001)
--(axis cs:35.501,1.82556994362535)
--(axis cs:36.001,1.54530152986287)
--(axis cs:37.0604088805878,1.501)
--(axis cs:37.0604088805878,1.001)
--(axis cs:37.501,0.84428194370151)
--(axis cs:39.501,0.777832146199693)
--(axis cs:39.7759008570467,0.501)
--(axis cs:39.7759008570467,0.001)
--(axis cs:39.7759008570467,0.001);
\path [draw=black, semithick]
(axis cs:14.6545913812005,49.501)
--(axis cs:14.6642344928489,33.5990556125871)
(axis cs:14.8846857693059,25.4129632701695)
--(axis cs:14.9601061378415,23.001)
--(axis cs:15.001,22.7811611785363)
--(axis cs:15.501,22.7811611785363)
--(axis cs:15.7007666185802,22.501)
--(axis cs:15.7479103220809,21.001)
--(axis cs:16.001,20.6714790083301)
--(axis cs:16.7782306958758,20.501)
--(axis cs:16.7782306958758,20.001)
--(axis cs:17.3013537230468,19.501)
--(axis cs:17.3013537230468,19.001)
--(axis cs:17.501,18.9237438886532)
--(axis cs:17.6395384564422,18.501)
--(axis cs:17.7416785997617,15.001)
--(axis cs:18.001,14.7349628453784)
--(axis cs:18.501,14.7349628453784)
--(axis cs:18.7868167932093,14.501)
--(axis cs:18.8321859076559,14.001)
--(axis cs:19.001,13.7167265004927)
--(axis cs:19.501,13.7167265004927)
--(axis cs:19.537597510975,13.001)
--(axis cs:19.6624650707066,10.001)
--(axis cs:19.747754107667,9.001)
--(axis cs:19.7690786728399,8.001)
--(axis cs:19.955171339579,7.501)
--(axis cs:20.001,6.99807812553784)
--(axis cs:20.501,6.99807812553784)
--(axis cs:21.001,6.84134744676945)
--(axis cs:21.501,6.82282635306409)
--(axis cs:21.9283662239974,6.501)
--(axis cs:22.001,5.99508715553676)
--(axis cs:23.501,5.90262735362329)
--(axis cs:26.910421983485,5.501)
--(axis cs:27.001,5.43831110407035)
--(axis cs:27.501,5.43831110407035)
--(axis cs:27.796474944692,5.001)
--(axis cs:28.501,4.89767564521403)
--(axis cs:29.501,4.87764585177842)
--(axis cs:29.9101939676475,4.501)
--(axis cs:30.001,4.22212965656222)
--(axis cs:30.501,4.22212965656222)
--(axis cs:31.001,3.97036185752409)
--(axis cs:34.501,3.88620442117179)
--(axis cs:37.501,3.60105625765074)
--(axis cs:41.932875736177,3.501)
--(axis cs:42.001,3.31306436987524)
--(axis cs:44.501,3.31306436987524)
--(axis cs:44.5726145746546,3.001)
--(axis cs:45.001,2.87180230142732)
--(axis cs:49.501,2.82141821703937)
--(axis cs:49.501,2.82141821703937);
\path [draw=black, semithick]
(axis cs:17.5726735113478,49.501)
--(axis cs:17.6445846757715,30.001)
--(axis cs:17.6874536600321,28.501)
--(axis cs:17.7780776992187,26.001)
--(axis cs:17.9250783596998,25.501)
--(axis cs:18.001,24.816923731754)
--(axis cs:18.501,24.816923731754)
--(axis cs:18.654700069316,24.501)
--(axis cs:18.6919177105908,23.001)
--(axis cs:18.9080469408882,22.501)
--(axis cs:18.9080469408882,21.6265567575173)
(axis cs:19.802704592908,13.9153012312372)
--(axis cs:19.8678372985863,12.501)
--(axis cs:20.001,9.950653403559)
--(axis cs:21.501,9.89841272111731)
--(axis cs:22.001,9.68531287072809)
--(axis cs:24.5266759586955,9.501)
--(axis cs:24.5372776262144,8.001)
--(axis cs:25.001,7.62779254903359)
--(axis cs:27.7712198529695,7.501)
--(axis cs:28.001,7.14928172134862)
--(axis cs:28.501,6.9798693512298)
--(axis cs:29.501,6.93696622701335)
--(axis cs:29.7623850541017,6.501)
--(axis cs:29.7725425923925,6.001)
--(axis cs:30.001,5.51668769312969)
--(axis cs:30.5548355293321,5.501)
--(axis cs:31.001,5.02147505935818)
--(axis cs:35.501,4.94687494478896)
--(axis cs:36.501,4.83461273114841)
--(axis cs:44.501,4.76574008223594)
--(axis cs:45.001,4.6172628927102)
--(axis cs:49.501,4.60404271081753)
--(axis cs:49.501,4.60404271081753);
\path [draw=black, semithick]
(axis cs:19.671324334887,49.501)
--(axis cs:19.6818240293889,47.0996004752852)
(axis cs:19.7327037485879,38.9020871413443)
--(axis cs:19.8234173337376,28.001)
--(axis cs:19.9836486301852,27.001)
--(axis cs:20.001,23.8451364121187)
--(axis cs:20.501,23.8451364121187)
--(axis cs:21.5364013540689,23.501)
--(axis cs:21.5393162031827,23.001)
--(axis cs:22.001,22.8210014351504)
--(axis cs:22.501,22.7655362712213)
--(axis cs:23.001,22.5383625112362)
--(axis cs:23.5253170757577,22.501)
--(axis cs:23.7455863600358,22.001)
--(axis cs:23.7630511579555,20.501)
--(axis cs:23.953884293435,20.001)
--(axis cs:24.001,17.8544409240615)
--(axis cs:24.501,17.8544409240615)
--(axis cs:24.595528657859,17.501)
--(axis cs:24.6525765597172,15.501)
--(axis cs:24.7069759919068,15.001)
--(axis cs:25.001,14.7897022352779)
--(axis cs:25.501,14.7787970948571)
--(axis cs:25.9359145131115,14.501)
--(axis cs:26.001,13.976904392952)
--(axis cs:26.501,13.8678291778853)
--(axis cs:27.5336721737114,13.001)
--(axis cs:27.7380358140119,12.501)
--(axis cs:28.001,12.3281598596702)
--(axis cs:28.501,12.3281598596702)
--(axis cs:28.7892389352055,12.001)
--(axis cs:28.9946956985834,11.501)
--(axis cs:29.001,9.9985898328005)
--(axis cs:30.501,9.99078177290102)
--(axis cs:31.001,9.72695802189891)
--(axis cs:32.501,9.6584902805884)
--(axis cs:33.501,9.28764854691362)
--(axis cs:34.001,9.06161196479524)
--(axis cs:34.5348541805017,9.001)
--(axis cs:34.6204578507484,8.001)
--(axis cs:36.001,7.66850576492525)
--(axis cs:38.501,7.57792549845442)
--(axis cs:40.001,7.5105526342707)
--(axis cs:41.501,7.4043045642647)
--(axis cs:41.6817402795608,7.001)
--(axis cs:42.001,6.8736340694701)
--(axis cs:44.501,6.84288100539544)
--(axis cs:45.001,6.72587784668593)
--(axis cs:49.501,6.62264636312956)
--(axis cs:49.501,6.62264636312956);
\path [draw=black, semithick]
(axis cs:23.6554178829319,49.501)
--(axis cs:23.6554178829319,49.001)
--(axis cs:23.675716475564,48.501)
--(axis cs:23.675716475564,48.001)
--(axis cs:23.7988890644622,47.501)
--(axis cs:23.8442226563072,47.001)
--(axis cs:23.9061569253455,46.501)
--(axis cs:23.9061569253455,46.001)
--(axis cs:24.001,45.5564702208611)
--(axis cs:24.501,45.5316399929736)
--(axis cs:24.5033732376967,45.501)
--(axis cs:24.5189675326725,45.001)
--(axis cs:24.6022586641669,44.501)
--(axis cs:24.6022586641669,44.001)
--(axis cs:24.7061869187146,43.501)
--(axis cs:24.7061869187146,43.001)
--(axis cs:24.9817469974649,42.501)
--(axis cs:24.9932699990946,42.001)
--(axis cs:25.001,41.8795243192259)
--(axis cs:25.2178423163612,41.501)
--(axis cs:25.2178423163612,41.001)
--(axis cs:25.429839276253,40.501)
--(axis cs:25.429839276253,40.001)
--(axis cs:25.501,39.9878789069069)
--(axis cs:25.8569526750983,39.501)
--(axis cs:25.8569526750983,39.001)
--(axis cs:25.9114845746884,38.501)
--(axis cs:25.9114845746884,38.001)
--(axis cs:26.001,37.8944797052113)
--(axis cs:26.501,37.8944797052113)
--(axis cs:26.5726649519419,37.501)
--(axis cs:26.5745563531265,37.001)
--(axis cs:26.5877338529196,36.501)
--(axis cs:26.5877338529196,36.001)
--(axis cs:26.5983982818353,35.501)
--(axis cs:26.5983982818353,35.001)
--(axis cs:26.6788860336478,34.501)
--(axis cs:26.6808141177539,34.001)
--(axis cs:26.8078475895918,33.501)
--(axis cs:26.8078475895918,33.001)
--(axis cs:27.001,32.7863109049365)
--(axis cs:27.501,32.7158998632327)
--(axis cs:28.001,32.5259865073608)
--(axis cs:28.501,32.5259865073608)
--(axis cs:29.001,32.5196639728185)
--(axis cs:29.3861125383523,32.501)
--(axis cs:29.501,32.4876625908685)
--(axis cs:29.5474307631084,32.001)
--(axis cs:29.5698941482043,31.501)
--(axis cs:29.5875055834764,31.001)
--(axis cs:29.6519959408296,30.501)
--(axis cs:29.6871361958737,30.001)
--(axis cs:30.001,29.7938911322843)
--(axis cs:30.501,29.7802798054606)
--(axis cs:31.001,29.5443613644506)
--(axis cs:31.501,29.5411076322553)
--(axis cs:32.001,29.5202324757299)
--(axis cs:32.4152008019821,29.501)
--(axis cs:32.4152008019821,29.001)
--(axis cs:32.4152008019821,28.501)
--(axis cs:32.4152008019821,28.001)
--(axis cs:32.501,27.9543365566032)
--(axis cs:33.001,27.5862398211809)
--(axis cs:33.501,27.5862398211809)
--(axis cs:34.001,27.5862398211809)
--(axis cs:34.501,27.5862398211809)
--(axis cs:35.001,27.5862398211809)
--(axis cs:35.501,27.5862398211809)
--(axis cs:35.5567323153596,27.501)
--(axis cs:35.5604070499618,27.001)
--(axis cs:35.5604070499618,26.501)
--(axis cs:35.5604070499618,26.001)
--(axis cs:35.5604070499618,25.501)
--(axis cs:35.613372358257,25.001)
--(axis cs:36.001,24.8999642807935)
--(axis cs:36.501,24.8869398470464)
--(axis cs:37.001,24.8313323189994)
--(axis cs:37.501,24.8194423407472)
--(axis cs:38.001,24.7907980568165)
--(axis cs:38.501,24.7907980568165)
--(axis cs:39.001,24.7542119421908)
--(axis cs:39.501,24.7542119421908)
--(axis cs:40.001,24.6651383876281)
--(axis cs:40.501,24.664483078281)
--(axis cs:41.001,24.5767294508635)
--(axis cs:41.501,24.5557229659295)
--(axis cs:42.001,24.5096793356584)
--(axis cs:42.501,24.5096793356584)
--(axis cs:42.5681676529954,24.501)
--(axis cs:42.5681676529954,24.001)
--(axis cs:42.9128531057214,23.501)
--(axis cs:42.9197668184463,23.001)
--(axis cs:43.001,22.8942667449182)
--(axis cs:43.501,22.8267369973937)
--(axis cs:43.6260768037522,22.501)
--(axis cs:43.6715899169428,22.001)
--(axis cs:43.6715899169428,21.501)
--(axis cs:43.6715899169428,21.001)
--(axis cs:43.7620178729658,20.501)
--(axis cs:43.7620178729658,20.001)
--(axis cs:43.8301966711429,19.501)
--(axis cs:43.8301966711429,19.001)
--(axis cs:43.8672160394557,18.501)
--(axis cs:43.8672160394557,18.001)
--(axis cs:43.8684490859629,17.501)
--(axis cs:43.8684490859629,17.001)
--(axis cs:43.872439020872,16.501)
--(axis cs:43.872439020872,16.001)
--(axis cs:43.8781135747431,15.501)
--(axis cs:43.8804789582602,15.001)
--(axis cs:44.001,14.8021416783908)
--(axis cs:44.501,14.788231311634)
--(axis cs:44.8546512579286,14.501)
--(axis cs:44.8772329294363,14.001)
--(axis cs:45.001,13.8583860876403)
--(axis cs:45.501,13.8168662598941)
--(axis cs:45.8335458744296,13.7868957070458);
\path [draw=black, semithick]
(axis cs:33.6434522410648,49.501)
--(axis cs:33.781956561387,49.001)
--(axis cs:34.001,48.7910715850286)
--(axis cs:34.501,48.7910715850286)
--(axis cs:34.5118238929948,48.7765691150644)
(axis cs:39.2732176771809,43.8057143680788)
--(axis cs:39.501,43.8057143680788)
--(axis cs:39.6935972495456,43.501)
--(axis cs:39.7120812616504,43.001)
--(axis cs:39.7742541494698,42.501)
--(axis cs:39.7742541494698,42.001)
--(axis cs:39.92452079632,41.501)
--(axis cs:39.9277775687693,41.001)
--(axis cs:39.9647286342424,40.501)
--(axis cs:39.9647286342424,40.001)
--(axis cs:40.001,39.9382148957221)
--(axis cs:40.501,39.9380535074739)
--(axis cs:41.001,39.9323547627518)
--(axis cs:41.501,39.8407157440272)
--(axis cs:42.001,39.7378829490742)
--(axis cs:42.501,39.7345249134962)
--(axis cs:43.001,39.7013923128498)
--(axis cs:43.501,39.6689656232137)
--(axis cs:43.7869381770915,39.501)
--(axis cs:43.8370096926787,39.001)
--(axis cs:43.9633150286958,38.501)
--(axis cs:43.9639834855802,38.001)
--(axis cs:44.001,37.9663449061313)
--(axis cs:44.501,37.9663449061313)
--(axis cs:45.001,37.7959199484048)
--(axis cs:45.501,37.7959199484048)
--(axis cs:46.001,37.6296999157028)
--(axis cs:46.501,37.6295258448213)
--(axis cs:46.9605354107323,37.501)
--(axis cs:47.001,37.4345862150766)
--(axis cs:47.501,37.4345862150766)
--(axis cs:48.001,37.3017828478628)
--(axis cs:48.501,37.3017828478628)
--(axis cs:48.6282282304553,37.001)
--(axis cs:49.001,36.8212809080811)
--(axis cs:49.501,36.8212809080811);
\path [draw=black, semithick]
;
\draw (axis cs:5.501,2.501) node[
scale=0.45,
text=black,
rotate=283.9
]{0.00};
\draw (axis cs:11.7479425410311,43.001) node[
scale=0.45,
text=black,
rotate=270.0
]{0.15};
\draw (axis cs:14.7088054289751,29.501) node[
scale=0.45,
text=black,
rotate=270.8
]{0.30};
\draw (axis cs:19.6137436260328,18.001) node[
scale=0.45,
text=black,
rotate=274.7
]{0.45};
\draw (axis cs:19.7143661693659,43.001) node[
scale=0.45,
text=black,
rotate=270.4
]{0.60};
\draw (axis cs:47.8469660167278,10.501) node[
scale=0.45,
text=black,
rotate=308.3
]{0.75};
\draw (axis cs:36.501,45.9158696318381) node[
scale=0.45,
text=black,
rotate=307.3
]{0.90};
\end{groupplot}
\end{tikzpicture}
\begin{tikzpicture}
\definecolor{darkgrey176}{RGB}{176,176,176}
\begin{groupplot}[group style={group size=2 by 1}]
\nextgroupplot[
tick align=outside,
tick pos=left,
title={\(\displaystyle F_{Y_1, Z_1|G,T}(y,z|1,1)\)},
x grid style={darkgrey176},
xlabel={Part-time employment},
xmin=0, xmax=50,
xtick style={color=black},
ymin=0, ymax=50
]
\path [draw=black, semithick]
;
\path [draw=black, semithick]
(axis cs:8.63671428571429,49.501)
--(axis cs:8.63671428571429,47.100025974026)
(axis cs:8.63671428571429,38.901974025974)
--(axis cs:8.73195238095238,27.501)
--(axis cs:9.001,20.9635)
--(axis cs:9.501,20.9635)
--(axis cs:9.56451351351351,19.501)
--(axis cs:9.65385714285714,17.001)
--(axis cs:10.001,11.8367142857143)
--(axis cs:10.501,11.6938571428571)
--(axis cs:10.6048461538462,11.501)
--(axis cs:10.6433076923077,11.001)
--(axis cs:10.8635,10.501)
--(axis cs:10.8635,10.001)
--(axis cs:11.001,9.891)
--(axis cs:11.501,9.891)
--(axis cs:11.735,9.501)
--(axis cs:11.8205652173913,8.501)
--(axis cs:11.8640434782609,8.001)
--(axis cs:12.001,7.776)
--(axis cs:12.501,7.776)
--(axis cs:12.776,7.501)
--(axis cs:12.8117142857143,7.001)
--(axis cs:13.001,6.62242857142857)
--(axis cs:13.501,6.62242857142857)
--(axis cs:13.586,6.501)
--(axis cs:13.686,6.001)
--(axis cs:14.001,5.791)
--(axis cs:14.501,5.791)
--(axis cs:14.6154736842105,5.501)
--(axis cs:14.6286315789474,5.001)
--(axis cs:14.8244375,4.501)
--(axis cs:14.8348709677419,4.001)
--(axis cs:15.001,3.75576190476191)
--(axis cs:15.501,3.75576190476191)
--(axis cs:17.001,3.515)
--(axis cs:17.176,3.501)
--(axis cs:17.501,2.99544444444444)
--(axis cs:19.001,2.77168965517241)
--(axis cs:19.501,2.77168965517241)
--(axis cs:20.001,2.4635)
--(axis cs:20.501,2.4635)
--(axis cs:21.001,2.3385)
--(axis cs:21.501,2.3385)
--(axis cs:21.771,2.001)
--(axis cs:23.001,1.93475)
--(axis cs:24.501,1.89978048780488)
--(axis cs:25.001,1.7416976744186)
--(axis cs:27.501,1.65668181818182)
--(axis cs:28.501,1.59986363636364)
--(axis cs:29.8356153846154,1.501)
--(axis cs:29.8740769230769,1.001)
--(axis cs:30.001,0.909333333333334)
--(axis cs:31.501,0.853777777777778)
--(axis cs:32.001,0.703631578947369)
--(axis cs:34.501,0.624684210526316)
--(axis cs:34.6968333333333,0.501)
--(axis cs:34.6968333333333,0.001)
--(axis cs:34.6968333333333,0.001);
\path [draw=black, semithick]
(axis cs:11.9083170731707,49.501)
--(axis cs:12.001,27.676)
--(axis cs:12.351,27.501)
--(axis cs:12.5105238095238,27.001)
--(axis cs:12.6771904761905,25.001)
--(axis cs:12.886,24.501)
--(axis cs:12.9720088200383,21.7609411997445)
(axis cs:14.6884389992448,14.2667330400282)
--(axis cs:14.7990392156863,13.001)
--(axis cs:15.001,11.5163846153846)
--(axis cs:15.5163846153846,11.501)
--(axis cs:15.5548461538462,11.001)
--(axis cs:15.901,10.501)
--(axis cs:16.001,9.9561724137931)
--(axis cs:16.501,9.93893103448276)
--(axis cs:17.501,9.63648387096774)
--(axis cs:17.8828181818182,9.501)
--(axis cs:17.8828181818182,9.001)
--(axis cs:18.001,8.81528571428572)
--(axis cs:18.501,8.81528571428572)
--(axis cs:18.701,8.501)
--(axis cs:18.7919090909091,8.001)
--(axis cs:19.001,7.89645454545455)
--(axis cs:19.501,7.89645454545455)
--(axis cs:19.7131951219512,7.501)
--(axis cs:19.7375853658537,7.001)
--(axis cs:20.001,5.869)
--(axis cs:21.501,5.829)
--(axis cs:22.501,5.63803703703704)
--(axis cs:23.001,5.50840740740741)
--(axis cs:23.5676666666667,5.501)
--(axis cs:23.7343333333333,5.001)
--(axis cs:24.501,4.97877777777778)
--(axis cs:25.001,4.706)
--(axis cs:27.501,4.54147619047619)
--(axis cs:27.6898888888889,4.501)
--(axis cs:28.001,4.03433333333334)
--(axis cs:28.501,3.9935)
--(axis cs:29.501,3.9935)
--(axis cs:30.001,3.76766666666667)
--(axis cs:32.501,3.65681395348837)
--(axis cs:34.001,3.58509090909091)
--(axis cs:34.7652857142857,3.501)
--(axis cs:34.8367142857143,3.001)
--(axis cs:35.001,2.94872727272727)
--(axis cs:39.501,2.85781818181818)
--(axis cs:40.501,2.78322222222222)
--(axis cs:49.501,2.68322222222222)
--(axis cs:49.501,2.68322222222222);
\path [draw=black, semithick]
(axis cs:14.9358684210526,49.501)
--(axis cs:15.001,29.5081428571429)
--(axis cs:15.5043333333333,29.501)
--(axis cs:15.5376666666667,28.001)
--(axis cs:15.7043333333333,27.001)
--(axis cs:15.771,26.001)
--(axis cs:15.971,25.501)
--(axis cs:16.001,24.9663846153846)
--(axis cs:16.501,24.9279230769231)
--(axis cs:16.8274705882353,24.501)
--(axis cs:16.8862941176471,23.001)
--(axis cs:17.2843333333333,20.001)
--(axis cs:17.501,19.871)
--(axis cs:17.8093333333333,19.501)
--(axis cs:17.8648888888889,18.001)
--(axis cs:18.001,17.7287777777778)
--(axis cs:18.501,17.7287777777778)
--(axis cs:18.6586923076923,17.501)
--(axis cs:18.8125384615385,16.001)
--(axis cs:18.9663846153846,15.501)
--(axis cs:19.001,14.983)
--(axis cs:19.501,14.983)
--(axis cs:19.6985409836066,14.501)
--(axis cs:19.8050983606557,13.001)
--(axis cs:20.001,11.7035)
--(axis cs:21.501,11.6285)
--(axis cs:21.7135,11.501)
--(axis cs:21.7551666666667,11.001)
--(axis cs:22.001,10.7740769230769)
--(axis cs:22.69475,10.501)
--(axis cs:22.69475,10.001)
--(axis cs:24.001,9.86376595744681)
--(axis cs:24.501,9.86376595744681)
--(axis cs:25.026,9.501)
--(axis cs:25.026,9.001)
--(axis cs:25.501,8.86528571428572)
--(axis cs:26.001,8.50814285714286)
--(axis cs:26.506,8.501)
--(axis cs:26.606,8.001)
--(axis cs:28.001,7.70242857142857)
--(axis cs:28.5327879765383,7.68814285714286)
(axis cs:36.3271026258267,5.59070588235294)
--(axis cs:39.50725,5.501)
--(axis cs:39.63225,5.001)
--(axis cs:40.501,4.942)
--(axis cs:49.501,4.84511764705882)
--(axis cs:49.501,4.84511764705882);
\path [draw=black, semithick]
(axis cs:19.7562631578947,49.501)
--(axis cs:19.7562631578947,47.100025974026)
(axis cs:19.7596666666667,38.901983724062)
--(axis cs:19.8468333333333,28.001)
--(axis cs:20.001,24.6724285714286)
--(axis cs:21.5276666666667,24.501)
--(axis cs:21.5943333333333,23.001)
--(axis cs:21.7795714285714,21.501)
--(axis cs:21.7795714285714,21.001)
--(axis cs:21.9224285714286,20.501)
--(axis cs:22.001,19.9486190476191)
--(axis cs:22.501,19.853380952381)
--(axis cs:23.001,19.5914761904762)
--(axis cs:23.6565555555556,19.501)
--(axis cs:23.6565555555556,19.001)
--(axis cs:24.001,17.791)
--(axis cs:24.501,17.791)
--(axis cs:24.5819523809524,17.001)
--(axis cs:24.6890952380952,15.001)
--(axis cs:25.001,14.6267142857143)
--(axis cs:26.591,14.501)
--(axis cs:26.641,14.001)
--(axis cs:27.001,13.641)
--(axis cs:27.501,13.451)
--(axis cs:27.591,13.001)
--(axis cs:28.001,12.591)
--(axis cs:29.501,12.451)
--(axis cs:29.5235,12.001)
--(axis cs:29.8496486486487,11.501)
--(axis cs:29.8631621621622,11.001)
--(axis cs:30.001,10.701)
--(axis cs:31.501,10.6127647058824)
--(axis cs:31.6367142857143,10.501)
--(axis cs:31.6724285714286,10.001)
--(axis cs:32.001,9.91581481481481)
--(axis cs:34.501,9.851)
--(axis cs:35.001,9.69222807017544)
--(axis cs:37.501,9.60450877192982)
--(axis cs:39.8787777777778,9.501)
--(axis cs:39.8787777777778,9.001)
--(axis cs:40.001,8.8635)
--(axis cs:41.501,8.8635)
--(axis cs:42.001,8.676)
--(axis cs:43.001,8.6135)
--(axis cs:44.501,8.551)
--(axis cs:44.581,8.501)
--(axis cs:44.781,8.001)
--(axis cs:45.501,7.97417073170732)
--(axis cs:49.501,7.9619756097561)
--(axis cs:49.501,7.9619756097561);
\path [draw=black, semithick]
(axis cs:24.9424893617021,49.501)
--(axis cs:24.9424893617021,49.001)
--(axis cs:24.9424893617021,48.501)
--(axis cs:24.9424893617021,48.001)
--(axis cs:24.9424893617021,47.501)
--(axis cs:24.9424893617021,47.001)
--(axis cs:24.9424893617021,46.501)
--(axis cs:24.9424893617021,46.001)
--(axis cs:24.9424893617021,45.501)
--(axis cs:24.9424893617021,45.001)
--(axis cs:24.9424893617021,44.501)
--(axis cs:24.9424893617021,44.001)
--(axis cs:24.9424893617021,43.501)
--(axis cs:24.9424893617021,43.001)
--(axis cs:24.9424893617021,42.501)
--(axis cs:24.9424893617021,42.001)
--(axis cs:24.9424893617021,41.501)
--(axis cs:24.9424893617021,41.001)
--(axis cs:24.9424893617021,40.501)
--(axis cs:24.9424893617021,40.001)
--(axis cs:24.9531276595745,39.501)
--(axis cs:24.9531276595745,39.001)
--(axis cs:24.9531276595745,38.501)
--(axis cs:24.9531276595745,38.001)
--(axis cs:24.9531276595745,37.501)
--(axis cs:24.9637659574468,37.001)
--(axis cs:24.9637659574468,36.501)
--(axis cs:24.9637659574468,36.001)
--(axis cs:24.9637659574468,35.501)
--(axis cs:24.9637659574468,35.001)
--(axis cs:24.9956808510638,34.501)
--(axis cs:24.9956808510638,34.001)
--(axis cs:24.9956808510638,33.501)
--(axis cs:24.9956808510638,33.001)
--(axis cs:24.9956808510638,32.501)
--(axis cs:24.9956808510638,32.001)
--(axis cs:25.001,31.751)
--(axis cs:25.126,31.501)
--(axis cs:25.126,31.001)
--(axis cs:25.126,30.501)
--(axis cs:25.126,30.001)
--(axis cs:25.501,29.926)
--(axis cs:26.001,29.676)
--(axis cs:26.501,29.676)
--(axis cs:26.676,29.501)
--(axis cs:26.676,29.001)
--(axis cs:26.726,28.501)
--(axis cs:26.726,28.001)
--(axis cs:26.926,27.501)
--(axis cs:26.976,27.001)
--(axis cs:27.001,26.751)
--(axis cs:27.0635,26.501)
--(axis cs:27.1885,26.001)
--(axis cs:27.501,25.7926666666667)
--(axis cs:27.676,25.501)
--(axis cs:27.676,25.001)
--(axis cs:28.001,24.7843333333333)
--(axis cs:28.501,24.751)
--(axis cs:29.001,24.7176666666667)
--(axis cs:29.501,24.6843333333333)
--(axis cs:29.5664761904762,24.501)
--(axis cs:29.5664761904762,24.001)
--(axis cs:29.5902857142857,23.501)
--(axis cs:29.5902857142857,23.001)
--(axis cs:29.626,22.501)
--(axis cs:29.626,22.001)
--(axis cs:29.6617142857143,21.501)
--(axis cs:29.6617142857143,21.001)
--(axis cs:29.7093333333333,20.501)
--(axis cs:29.7093333333333,20.001)
--(axis cs:29.9950476190476,19.501)
--(axis cs:29.9950476190476,19.001)
--(axis cs:30.001,18.9176666666667)
--(axis cs:30.501,18.9176666666667)
--(axis cs:30.8135,18.501)
--(axis cs:30.8135,18.001)
--(axis cs:31.001,17.926)
--(axis cs:31.501,17.926)
--(axis cs:31.7843333333333,17.501)
--(axis cs:31.8176666666667,17.001)
--(axis cs:31.951,16.501)
--(axis cs:31.951,16.001)
--(axis cs:32.001,15.876)
--(axis cs:32.501,15.876)
--(axis cs:32.876,15.501)
--(axis cs:32.876,15.001)
--(axis cs:33.001,14.9801666666667)
--(axis cs:33.501,14.9801666666667)
--(axis cs:34.001,14.9523888888889)
--(axis cs:34.0677235653324,14.9523888888889)
(axis cs:41.8319778400699,12.8880060436173)
--(axis cs:42.001,12.8419090909091)
--(axis cs:42.501,12.7964545454546)
--(axis cs:43.001,12.7964545454546)
--(axis cs:43.501,12.7964545454546)
--(axis cs:44.001,12.751)
--(axis cs:44.501,12.751)
--(axis cs:45.001,12.5237272727273)
--(axis cs:45.501,12.5237272727273)
--(axis cs:46.001,12.5237272727273)
--(axis cs:46.501,12.5237272727273)
--(axis cs:47.001,12.5237272727273)
--(axis cs:47.501,12.5237272727273)
--(axis cs:47.751,12.501)
--(axis cs:48.001,12.376)
--(axis cs:48.501,12.376)
--(axis cs:49.001,12.376)
--(axis cs:49.501,12.376);
\path [draw=black, semithick]
(axis cs:34.7444782608696,49.501)
--(axis cs:34.7444782608696,49.001)
--(axis cs:34.7444782608696,48.501)
--(axis cs:34.7444782608696,48.001)
--(axis cs:34.7444782608696,47.501)
--(axis cs:34.7444782608696,47.100025974026)
(axis cs:34.8096956521739,38.905522365213)
--(axis cs:34.8096956521739,38.501)
--(axis cs:34.8096956521739,38.001)
--(axis cs:34.8096956521739,37.501)
--(axis cs:34.8314347826087,37.001)
--(axis cs:34.8314347826087,36.501)
--(axis cs:34.8314347826087,36.001)
--(axis cs:34.8314347826087,35.501)
--(axis cs:34.8314347826087,35.001)
--(axis cs:34.9183913043478,34.501)
--(axis cs:34.9183913043478,34.001)
--(axis cs:34.9183913043478,33.501)
--(axis cs:34.9183913043478,33.001)
--(axis cs:34.9183913043478,32.501)
--(axis cs:34.9183913043478,32.001)
--(axis cs:34.9401304347826,31.501)
--(axis cs:34.9401304347826,31.001)
--(axis cs:34.9401304347826,30.501)
--(axis cs:34.9401304347826,30.001)
--(axis cs:35.001,29.861)
--(axis cs:35.501,29.861)
--(axis cs:36.001,29.561)
--(axis cs:36.501,29.561)
--(axis cs:36.701,29.501)
--(axis cs:36.701,29.001)
--(axis cs:36.8676666666667,28.501)
--(axis cs:36.8676666666667,28.001)
--(axis cs:37.001,27.901)
--(axis cs:37.501,27.776)
--(axis cs:37.776,27.501)
--(axis cs:37.901,27.001)
--(axis cs:38.001,26.601)
--(axis cs:38.501,26.601)
--(axis cs:38.551,26.501)
--(axis cs:38.801,26.001)
--(axis cs:39.001,25.9343333333333)
--(axis cs:39.501,25.9343333333333)
--(axis cs:39.7176666666667,25.501)
--(axis cs:39.7176666666667,25.001)
--(axis cs:40.001,24.7743333333333)
--(axis cs:40.501,24.7743333333333)
--(axis cs:41.001,24.7743333333333)
--(axis cs:41.501,24.7743333333333)
--(axis cs:42.001,24.6743333333333)
--(axis cs:42.501,24.641)
--(axis cs:43.001,24.641)
--(axis cs:43.501,24.641)
--(axis cs:44.001,24.6076666666667)
--(axis cs:44.501,24.6076666666667)
--(axis cs:44.7676666666667,24.501)
--(axis cs:44.7676666666667,24.001)
--(axis cs:44.9343333333333,23.501)
--(axis cs:44.9343333333333,23.001)
--(axis cs:45.001,22.8676666666667)
--(axis cs:45.501,22.8676666666667)
--(axis cs:46.001,22.701)
--(axis cs:46.501,22.701)
--(axis cs:47.001,22.701)
--(axis cs:47.501,22.701)
--(axis cs:48.001,22.5343333333334)
--(axis cs:48.501,22.5343333333334)
--(axis cs:49.001,22.5343333333334)
--(axis cs:49.501,22.5343333333334);
\path [draw=black, semithick]
;
\draw (axis cs:8.63671428571428,43.001) node[
scale=0.45,
text=black,
rotate=270.0
]{0.15};
\draw (axis cs:13.9062631578947,18.001) node[
scale=0.45,
text=black,
rotate=281.2
]{0.30};
\draw (axis cs:32.001,6.13850000000001) node[
scale=0.45,
text=black,
rotate=339.6
]{0.45};
\draw (axis cs:19.7562631578947,43.001) node[
scale=0.45,
text=black,
rotate=270.0
]{0.60};
\draw (axis cs:38.001,13.926) node[
scale=0.45,
text=black,
rotate=341.6
]{0.75};
\draw (axis cs:34.7444782608696,43.001) node[
scale=0.45,
text=black,
rotate=270.0
]{0.90};
\end{groupplot}
\end{tikzpicture}
\end{figure}
\begin{table}
\caption{Estimates of the Kendall's $\tau$ and Spearman's correlation index.}
\centering
\begin{turn}{-90}
\begin{tabular}{lccc}
\hline
\hline
& \multicolumn{1}{c}{Treatment} & \multicolumn{1}{c}{Without treatment} & \multicolumn{1}{c}{Difference}\\
\hline
\hline
\multicolumn{4}{l}{Kendall's $\tau$}\\
\hline
Part-time and full-time employment &-0.1709 & -0.0095 & -0.1613\\
95\% confidence intervals & (-0.2407,-0.1011) & (-0.2401,0.221) & (-0.3911,0.0684)\\
95\% confidence intervals & (-0.2305,-0.1113) & (-0.1848,0.1657) & (-0.339,0.0164)\\
\hline
\multicolumn{4}{l}{Spearman's correlation index}\\
\hline
Part-time and full-time employment &-0.2402 & -0.0101 & -0.23\\
95\% confidence intervals & (-0.3383,-0.142) & (-0.2596,0.2393) & (-0.5027,0.0325)\\
90\% confidence intervals & (-0.3216,-0.1588) & (-0.1923,0.1721) & (-0.4192,-0.0409)\\
\hline
\hline
\end{tabular}
\end{turn}
\end{table}
\begin{table}
\caption{Results of the Kendall's $\tau$ and Spearman's correlation index -- No additional covariates.}
\centering
\begin{turn}{-90}
\begin{tabular}{lccc}
\hline
\hline
& \multicolumn{1}{c}{Treatment} & \multicolumn{1}{c}{Without treatment} & \multicolumn{1}{c}{Difference}\\
\hline
\hline
\multicolumn{4}{l}{Kendall's $\tau$}\\
\hline
Part-time and full-time employment &-0.1709 & -0.0136 & -0.1573\\
95\% confidence intervals & (-0.23,-0.1117) & (-0.2127,0.1855)& (-0.371,0.0565)\\
95\% confidence intervals & (-0.2268,-0.1149) & (-0.2035,0.1764) & (-0.3374,0.0228)\\
\hline
\multicolumn{4}{l}{Spearman's correlation index}\\
\hline
Part-time and full-time employment &-0.2402 & -0.0143 & -0.2259\\
95\% confidence intervals & (-0.3237,-0.1566) & (-0.2352,0.2066) & (-0.4751,0.0091)\\
90\% confidence intervals & (-0.3172,-0.1632) & (-0.2148,0.1862) & (-0.4316,-0.0202)\\
\hline
\hline
\end{tabular}
\end{turn}
\end{table}
\section{Conclusion}
\label{sec:conclusions}
We provide a simple distribution regression based estimator to implement the evaluation of treatment effects in a difference-in-difference setting. As our approach provides counterfactual distributions we are able to explore the impact of the treatment at different quantiles of the distribution of the outcome variable. For both the univariate and multivariate cases we provide the identifying assumption and the associated estimation algorithms. A re-examination of the Card and Krueger (1994) study highlights the utility of various aspects of our approach.
Our analysis can easily be extended to the case of multiple time periods and more than two outcomes. We can also extend our distributional regression framework to use time and unit weights as in the synthetic difference-in-difference estimation method of Arkhangelsky et al. (2021). We leave each of these extensions to future research (e.g. Fern\'andez-Val et al., 2024b).
\section*{\thinspace\ References}
\begin{description}
\item \textsc{Almond, D., H.W. Hoynes, and D.W. Schanzenbach} (2011), \textquotedblleft Inside the war on poverty: the impact of food stamps
on birth outcomes\textquotedblright, \emph{Review of Economics and Statistics} \textbf{93}, 387--403.
\item \textsc{Arkhangelsky, D. and G. Imbens} (2024), \textquotedblleft Causal Models for longitudinal and
panel Data: a survey\textquotedblright, \emph{The Econometrics Journal} \textbf{27}, C1-61.
\item \textsc{Arkhangelsky, D. S. Athey, D.A. Hirshberg, G.W. Imbens, and S. Wager} (2021), \textquotedblleft Synthetic difference-in-differences, \emph{American Economic Review} \textbf{111}, 4088–18.
\item \textsc{Athey, S. and G.D. Imbens} (2006),
\textquotedblleft Identification and inference in nonlinear
difference-in-differences models\textquotedblright, \emph{Econometrica}
\textbf{74} 431--97.
\item \textsc{Biewen, M., M. R\"ummele, and B. Fitzenberger} (2022), \textquotedblleft Using distribution regression Difference-in-Differences to evaluate the effects of a minimum wage introduction on the distribution of hourly wages and hours worked, working paper, N\"urnberg.
\item \textsc{Blundell, R., C. Meghir, M. Costa Dias and J. Van Reenen} (2004), \textquotedblleft Evaluating the employment impact of a mandatory job search program\textquotedblright, \emph{Journal of the European Economic Association} \textbf{2},569--606.
\item \textsc{Callaway, B. and T. Li} (2019), \textquotedblleft Quantile treatment effects in difference in differences models with panel data\textquotedblright \emph{Quantitative Economics} \textbf{10}, 1579--1618.
\item \textsc{Card, D.} (1990), \textquotedblleft The Impact of the Mariel Boatlift on the Miami Labor Market\textquotedblright \emph{Industrial and Labor Relations Review,} \textbf{43}, 245–-57.
\item \textsc{Card, D. and A.B. Krueger} (1994), \textquotedblleft Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania\textquotedblright \emph{American Economic Review} \textbf{84}, 772–-93.
\item \textsc{Cengiz, D., A. Dube, A. Lindner, and B. Zipperer} (2019), \textquotedblleft The effect of minimum wages on low-wage jobs \textquotedblright, \emph{Quarterly Journal of Economics} \textbf{134}, 1405--54.
\item \textsc{Chernozhukov, V., I. Fern\'{a}ndez-Val, and Melly B.}
(2013), \textquotedblleft Inference on counterfactual
distributions\textquotedblright , \emph{Econometrica}, \textbf{81}, 2205--68.
\item \textsc{Chernozhukov, V., I. Fern\'{a}ndez-Val, and S. Luo}{\small \ }
(2019), \textquotedblleft Distribution regression with sample selection,
with an application to wage decompositions in the UK", working paper, MIT,
Cambridge (MA).
\item \textsc{Chernozhukov, V. I. Fern\'andez-Val, B. Melly, and K. Wüthrich} (2020), \textquotedblleft Generic inference on quantile and quantile effect functions for discrete outcomes\textquotedblright, \emph{Journal of the American Statistical Association} \textbf{115}, 123--37.
\item \textsc{Dube, A.} (2019), \textquotedblleft Minimum wages and the distribution of family incomes\textquotedblright, \emph{American Economic Journal: Applied Economics} \textbf{11}, 268–-304.
\item \textsc{Fern\'{a}ndez-Val I., J. Meier, A. van Vuuren and F. Vella}
(2024a) \textquotedblleft Bivariate distribution regression with an application to
intergenerational mobility\textquotedblright, working paper, Boston
University.
\item \textsc{Fern\'{a}ndez-Val I., J. Meier, A. van Vuuren and F. Vella}
(2024b) \textquotedblleft Distributional synthetic difference-in-differences\textquotedblright, working paper, Boston University.
\item \textsc{Foresi, S. and F. Peracchi} (1995), ``The conditional
distribution of excess returns: an empirical analysis'', \emph{Journal of
the American Statistical Association}, \textbf{90}, 451--66.
\item \textsc{Goodman-Bacon, A.} (2021), \textquotedblleft The long-run effects of childhood insurance coverage: medicaid implementation, adult health, and labor market outcomes, \emph{American Economic Review} \textbf{111}, 2550-93.
\item \textsc{Goodman-Bacon, A. and L. Schmidt} (2020), \textquotedblleft Federalizing benefits: The introduction of supplemental security income and the size of the safety net.\textquotedblright, \emph{Journal of Public Economics} \textbf{185}, 104174.
\item \textsc{Kim, D. and J.M. Wooldridge} (2024), \textquotedblleft Difference-in-differences estimator of quantile
treatment effect on the treated \textquotedblright, \emph{Journal of Business and Economic Statistics}, forthcoming.
\item \textsc{MaCurdy, T.} (2015), \textquotedblleft How effective is the minimum wage at supporting the poor?\textquotedblright, \emph{Journal of Political Economy} \textbf{123}, 497--545.
\item \textsc{Malesky, E.J., C.V. Nguyen, and A. Trahn} (2014),
\textquotedblleft The Impact of recentralization on public services:
A difference-in-differences analysis of the abolition
of elected councils in Vietnam\textquotedblright, \emph{American Political Science Review}
\textbf{108} 144--68.
\item \textsc{Melly, B. and Santangelo} (2015), \textquotedblleft The changes-in-changes model with covariates\textquotedblright, working paper, Bern University.
\item \textsc{Roth, J. and P.H.C. Sant'Anna} (2023),
\textquotedblleft When Is parallel trends sensitive to functional form?\textquotedblright, \emph{Econometrica}
\textbf{91} 737--47.
\item \textsc{Torous, W., F. Gunsilius, and P. Rigollet} (2024), \textquotedblleft An optimal transport approach to estimating
causal effects via nonlinear difference-in-differences, working paper\textquotedblright, University of California, Berkeley.
\item \textsc{Wooldridge, J.M.} (2023),
\textquotedblleft Simple approaches to nonlinear difference-in-differences with panel data\textquotedblright, \emph{Econometric Journal}
\textbf{26} C31--66.
\item \textsc{Williams, O. D. and J.E. Grizzle} (1972), \textquotedblleft Analysis of contingency tables having ordered response categories\textquotedblright, \emph{Journal of the American Statistical Association} \textbf{67}, 55–63.
\end{description}