Testing Monotonicity of Mean Potential Outcomes in a Continuous Treatment

Yu-Chin Hsu†

Institute of Economics, Academia Sinica

Department of Finance, National Central University

Department of Economics, National Chengchi University and

CRETA, National Taiwan University

Martin Huber∗

Department of Economics, University of Fribourg

Center for Econometrics and Business Analytics, St. Petersburg State
University

Ying-Ying Lee‡

Department of Economics, University of California, Irvine

Chu-An Liu§

Institute of Economics, Academia Sinica

This version: June 08, 2021

†<EMAIL_ADDRESS>∗<EMAIL_ADDRESS>‡<EMAIL_ADDRESS>§
<EMAIL_ADDRESS>
Acknowledgments: Yu-Chin Hsu gratefully acknowledges research support from the
Ministry of Science and Technology of Taiwan (MOST107-2410-H- 001-034-MY3),
the Academia Sinica Investigator Award of the Academia Sinica, Taiwan (AS-
IA-110-H01), and the Center for Research in Econometric Theory and
Applications (107L9002) from the Featured Areas Research Center Program within
the framework of the Higher Education Sprout Project by the Ministry of
Education of Taiwan. Chu-An Liu gratefully acknowledges research support from
the Academia Sinica Career Development Award (AS-CDA-110-H02).

Abstract

While most treatment evaluations focus on binary interventions, a growing
literature also considers continuously distributed treatments, e.g. hours
spent in a training program to assess its effect on labor market outcomes. In
this paper, we propose a Cramér-von Mises-type test for testing whether the
mean potential outcome given a specific treatment has a weakly monotonic
relationship with the treatment dose under a weak unconfoundedness assumption.
This appears interesting for testing shape restrictions, e.g. whether
increasing the treatment dose always has a non-negative effect, no matter what
the baseline level of treatment is. We formally show that the proposed test
controls asymptotic size and is consistent against any fixed alternative.
These theoretical findings are supported by the method’s finite sample
behavior in our Monte-Carlo simulations. As an empirical illustration, we
apply our test to the Job Corps study and reject a weakly monotonic
relationship between the treatment (hours in academic and vocational training)
and labor market outcomes like earnings or employment.

JEL classification: C01, C12, C21

Keywords: hypothesis testing, continuous treatment models, average dose
response functions, treatment monotonicity.

##  1 Introduction

Even though many studies on treatment or policy evaluation investigate the
effects of binary or discrete interventions, a growing literature also
considers the assessment of continuously distributed treatments, e.g. hours
spent in a training program whose effect on labor market performance is of
interest. Most contributions like Imbens (2000), Hirano and Imbens (2004),
Flores (2007), Flores et al. (2012), Galvao and Wang (2015), Lee (2018) and
Colangelo and Lee (2020) focus on the identification and estimation of the
average dose-response function (ADF), which corresponds to the conditional
mean outcome as a function of the treatment dose. This permits assessing the
average treatment effect (ATE) as the difference in the ADF assessed at two
distinct treatment doses of interest, while Hirano and Imbens (2004) and
Flores et al. (2012) also consider the marginal effect of slightly increasing
the treatment dose, which is the derivative of the ADF. Rather than
considering the total effect of the treatment, Huber et al. (2020) suggest a
causal mediation approach to disentangle the ATE into its direct effect and
indirect effect operating through intermediate variables or mediators to
assess the causal mechanisms of the treatment.

In this paper, we propose a method for testing whether the ADF has a weakly
monotonic relationship with (i.e. is weakly increasing or decreasing in) the
treatment dose under a weak unconfoundedness assumption, implying that
confounder of the treatment-outcome relation can be controlled for by observed
covariates. Such a test appears interesting for verifying shape restrictions,
e.g. whether increasing the treatment dose always has a non-negative effect,
no matter what the baseline level of treatment is. Moreover, the treatment
effect model is known to be equivalent to a nonseparable structural model of a
nonseparable outcome with a general disturbance, as for instance Imbens and
Newey (2009) and Lee (2018). In this case, the ADF corresponds to the average
structural function in Blundell and Powell (2003). Therefore our test can be
applied to testing monotonicity of the average structural function in a
nonseparable structural model under a conditional independence assumption.

To construct our test, we first transform the null hypothesis of a monotonic
relationship to countably many moment inequalities based on the generalized
instrumental function approach of Hsu et al. (2019) and Hsu and Shen (2020).
We construct a Cramér-von Mises-type test statistic based on the estimated
moments, which are shown to converge to a Gaussian process at the parametric
rate. Importantly, by making use of moment inequalities, our method does not
rely on the nonparametric estimation of the ADF or the marginal effects, which
would converge at slower nonparametric rates. To compute the critical value
for our test, we apply a multiplier bootstrap method and the generalized
moment selection (GMS) approach of Andrews and Shi (2013, 2014). We
demonstrate that our test controls asymptotic size and is consistent against
any fixed alternative.

Our paper is related to a growing literature on testing monotonicity in
regression problems such as Bowman et al. (1998), Ghosal et al. (2000),
Gijbels et al. (2000), Hall and Heckman (2000), Dümbgen and Spokoiny (2001),
Durot (2003), Baraud et al. (2005), Wang and Meyer (2011), Chetverikov (2019)
and Hsu et al. (2019). The main difference between our method and the
previously suggested tests is that we rely on a two-step estimation process
when computing the moments, with the first step consisting of
nonparametrically estimating the generalized propensity score, i.e. the
conditional density of a treatment dose given the covariates, by kernel
regression. For this reason, it is necessary to take into account the behavior
of the first step when we derive the limiting behavior of the estimated moment
inequalities underlying our test. In addition to the nonparametric method, we
also consider testing under parametrically estimated generalized treatment
propensity scores and investigate the finite sample behavior of either
approach in a simulation study. We also extend our method to testing
conditional monotonicity given observed covariates.

As an empirical illustration, we apply our test to data from an experimental
study on Job Corps, see Schochet et al. (2001) and Schochet et al. (2008), a
program aimed at increasing the human capital of youths from disadvantaged
backgrounds in the U.S. We consider hours in academic and vocational training
in the first year of the program as continuous treatment and investigate its
association with several labor market outcomes: weeks employed in the second
year after program assignment, weekly earnings in the fourth year, earnings
and hours worked per week in quarter 16, and a binary employment indicator
four years after assignment. For all outcomes, both weakly positive and
negative monotonicity in the treatment are clearly rejected. When, however,
considering lower treatment doses of up to 1000 hours, only weakly negative
monotonicity is rejected, while positive monotonicity is not. The opposite
holds true for larger treatment doses beyond 2000 hours, with weakly positive
monotonicity (but not negative monotonicity) being frequently refuted in the
data. In line with qualitative findings in Flores et al. (2012), our results
therefore point to a concave ADF, suggesting that the marginal effect of
training is positive for relatively low treatment doses but may even become
negative when hours in training surpass a certain amount. A potential
explanation could be that participants attending more training in the first
year might be induced to attain more education in the following years rather
than to participate in the labor market, resulting in a lock-in effect.

The paper is organized as follows. Section 2 formulates the hypothesis of weak
monotonicity to be tested. Sections 3 and 4 propose monotonicity tests under
nonparametric and parametric estimation of the generalized treatment
propensity score, respectively. Section 5 adapts the methods to testing
monotonicity with conditional (rather than unconditional) mean potential
outcomes given observed covariates. Section 6 presents a Monte-Carlo
simulation and discusses how to choose the tuning parameters of the test in
practice. Section 7 provides an empirical application to the Job Corps data.
Section 8 concludes. The technical proofs are relegated to the Appendix.

##  2 Hypothesis to be tested

Let $Y(t)$ denote the potential outcome corresponding to the level of
treatment intensity $t\in\mathcal{T}$, where we have $\mathcal{T}=[a,b]$ with
$-\infty<a<b<\infty$. $Y(t)$ is called the unit-level dose-response function
in Hirano and Imbens (2004). Let $\mu(t)=E[Y(t)]$ for $t\in\mathcal{T}$ denote
the average of the potential outcome function, also known as the average dose-
response function or the average structural function. In this paper, we are
interested in testing if the average dose-response function is weakly
increasing in the treatment intensity within a specific range and we define
the null hypothesis of our interest as

$\displaystyle H_{0}:~{}\mu(t_{1})\geq\mu(t_{2}),~{}~{}\text{for
all}~{}t_{1}\geq t_{2},~{}\text{for $t_{1},t_{2}\in[t_{\ell},t_{u}]$},$ (2.1)

where $a\leq t_{\ell}<t_{u}\leq b$ so that $[t_{\ell},t_{u}]$ is a convex and
compact subset of $[a,b]$. Without loss of generality, we assume that
$[t_{\ell},t_{u}]=[0,1]$.111If $[t_{\ell},t_{u}]$ is not $[0,1]$, we can
always apply an affine transformation $\phi$ on $t$ so that $\phi(t_{\ell})=0$
and $\phi(t_{u})=1$.

Note that the null hypothesis in (2.1) has a form that is similar to that in
the literature onregression monotonicity, see for instance Hsu et al. (2019).
However, the identification of $\mu(t)$ in our case is different from theirs.
We apply the generalized instrumental function approach of Hsu et al. (2019)
and Hsu and Shen (2020) to transform $H_{0}$ in (2.1) to countably many moment
inequalities without loss of information.222The generalized instrumental
function approach is a generalization of the instrumental function approach in
Andrews and Shi (2013, 2014). To be specific, suppose that $\mu(t)$ is a
continuous function on $t=[0,1]$ and $h(t)>0$ is a weighting function such
that $\int_{0}^{1}h(t)dt<\infty$. Then by Lemma 2.1 of Hsu and Shen (2020),
$H_{0}$ in (2.1) is equivalent to

$\displaystyle\frac{\int_{t_{2}}^{t_{2}+q^{-1}}\mu(\tilde{t})\cdot
h(\tilde{t})d\tilde{t}}{\int_{t_{2}}^{t_{2}+q^{-1}}h(\tilde{t})d\tilde{t}}-\frac{\int_{t_{1}}^{t_{1}+q^{-1}}\mu(\tilde{t})\cdot
h(\tilde{t})d\tilde{t}}{\int_{t_{1}}^{t_{1}+q^{-1}}h(\tilde{t})d\tilde{t}}\leq
0,~{}\text{or}$ (2.2)
$\displaystyle\int_{t_{2}}^{t_{2}+q^{-1}}\mu(\tilde{t})\cdot
h(\tilde{t})d\tilde{t}\cdot\int_{t_{1}}^{t_{1}+q^{-1}}h(\tilde{t})d\tilde{t}-\int_{t_{1}}^{t_{1}+q^{-1}}\mu(\tilde{t})\cdot
h(\tilde{t})d\tilde{t}\cdot\int_{t_{2}}^{t_{2}+q^{-1}}h(\tilde{t})d\tilde{t}\leq
0$ (2.3)

for any $q=2,\cdots,$ and for any $t_{1}\geq t_{2}$ such that $q\cdot
t_{1},q\cdot t_{2}\in\\{0,1,2,\cdots,q-1\\}$. Equations (2.2) and (2.3) hold
by the fact that if a function is non-decreasing, then its weighted average
over an interval will be non-decreasing as well when the interval moves to the
right. In addition, by Hsu et al. (2019), Equations (2.2) and (2.3) contain
the same information as the null hypothesis.

In the following, we discuss the identification of
$\int_{t}^{t+q^{-1}}\mu(\tilde{t})\cdot h(\tilde{t})d\tilde{t}$. Let
$p(t,x)=f_{T|X}(t|x)$ be the generalized propensity score, which corresponds
to the conditional density of the treatment given the covariates and
$p(t,x)>0$ for all $t$ and $x$.

######  Assumption 2.1.

(Weak Unconfoundedness): $Y(t)\perp T\mid X$ for all $t\in\mathcal{T}$.

Assumption 2.1 is a commonly invoked identifying assumption based on
observational data, also known as conditional independence and selection on
observables. It assumes that conditional on observables $X$, $T$ is as good as
randomly assigned, or conditionally exogenous. We then have the following
lemma concerning the identification of $\int_{t}^{t+q^{-1}}\mu(\tilde{t})\cdot
h(\tilde{t})d\tilde{t}$.

######  Lemma 2.1.

Suppose Assumption 2.1 holds. Let $h(t)>0$ for all $t$ be a known weight
function such that $\int_{0}^{1}h(t)dt<\infty$. Then for $r>0$,

$\displaystyle\int_{t}^{t+r}\mu(\tilde{t})h(\tilde{t})d\tilde{t}=\mathbb{E}\left[\frac{Y}{p(T,X)}\cdot
h(T)\cdot 1(T\in[t,t+r])\right].$

We now apply Lemma 2.1 of Hsu and Shen (2020) and the identification result in
Lemma 2.1 to transform (2.1) to countably many moment inequalities based on
which we will construct our test. For
$\ell=(t_{1},t_{2},q^{-1})\in[0,1]^{2}\times(0,1]$, define

$\displaystyle{\cal
L}=\Big{\\{}\ell=(t_{1},t_{2},q^{-1}):~{}q\cdot(t_{1},t_{2})\in\\{0,1,2,\cdots,q-1\\}^{2},~{}t_{1}>t_{2},\text{,
and }q=2,3,\cdots\Big{\\}}.$ (2.4)

For each $\ell$, we define

$\displaystyle\nu_{1}(\ell)=E\left[\frac{Y}{p(T,X)}1(T\in[t_{1},t_{1}+q^{-1}])\right],~{}~{}~{}\nu_{2}(\ell)=E\left[\frac{Y}{p(T,X)}1(T\in[t_{2},t_{2}+q^{-1}])\right].$

######  Lemma 2.2.

Suppose Assumption 2.1 holds. Assume that $\mu(t)$ is continuous in $t$. Then
$H_{0}$ in (2.1) is equivalent to

$\displaystyle H_{0}^{\prime}:$
$\displaystyle~{}\nu(\ell)=\nu_{2}(\ell)-\nu_{1}(\ell)\leq 0~{}\text{for any
$\ell=(t_{1},t_{2},q^{-1})\in\mathcal{L}$}.$ (2.5)

The proof of Lemma 2.2 is a direct implication of (2.3) and Lemma 2.1. To see
this, set $h(t)=1$ and note that $\int_{t}^{t+r}h(\tilde{t})d\tilde{t}=r$. By
(2.3) and Lemma 2.1, for any $\ell\in\mathcal{L}$,

$\displaystyle\int_{t_{2}}^{t_{2}+q^{-1}}\mu(\tilde{t})\cdot
h(\tilde{t})d\tilde{t}\cdot\int_{t_{1}}^{t_{1}+q^{-1}}h(\tilde{t})d\tilde{t}-\int_{t_{1}}^{t_{1}+q^{-1}}\mu(\tilde{t})\cdot
h(\tilde{t})d\tilde{t}\cdot\int_{t_{2}}^{t_{2}+q^{-1}}h(\tilde{t})d\tilde{t}\leq
0$ iff $\displaystyle~{}\nu_{2}(\ell)r-\nu_{1}(\ell)r\leq 0$ iff
$\displaystyle~{}\nu(\ell)=\nu_{2}(\ell)-\nu_{1}(\ell)\leq 0.$

In Lemma 2.2, we pick $h(t)=1$ for simplicity, but the result also holds for
any other known $h(t)>0$.

##  3 Nonparametric Monotonicity Test

In this section, we propose a testing procedure for $H_{0}$ defined in (2.1)
that is equivalent to $H_{0}^{\prime}$ in (2.5). The key for our test is the
estimation of $\nu_{1}(\ell)$ and $\nu_{2}(\ell)$, which consists of two
steps. In the first step, we estimate the generalized propensity score. In the
second step, we use a sample analog estimator of $\nu(\ell)$ based on plugging
in the estimate of the generalized propensity score. In this section, we
consider testing based on a nonparametric estimator for the generalized
propensity score, while the next section discusses the parametric case.

Our nonparametric conditional kernel density estimator for the generalized
propensity score is given by

$\displaystyle\hat{p}_{np}(T_{i},X_{i})=\frac{\sum_{j\neq
i}K_{h}(T_{j}-T_{i})K_{h}(X_{j}-X_{i})}{\sum_{j\neq i}K_{h}(X_{j}-X_{i})},$
(3.1)

where $K_{h}(u)=h^{-1}k(u/h)$, $k(\cdot)$ is an $r_{k}$-order kernel function,
and $h$ is a bandwidth that satisfies Assumption B.2 in the Appendix. For a
vector $X$ with dimension $d_{x}\geq 2$, we use a product kernel
$K_{h}(X_{j}-X_{i})=\Pi_{\ell=1}^{d_{x}}K_{h}(X_{\ell,j}-X_{\ell,i})=h^{-d_{x}}\Pi_{\ell=1}^{d_{x}}k\big{(}(X_{\ell,j}-X_{\ell,i})/h\big{)}$.

In the second step, $E\big{[}({Y}/{p(T,X)})\cdot 1(T\in[t,t+q^{-1}])\big{]}$
is estimated by the plug-in-based empirical analog
$n^{-1}\sum_{i=1}^{n}(Y_{i}/\hat{p}_{np}(T_{i},X_{i}))1(T_{i}\in[t,t+q^{-1}])$.
$\nu(\ell)$ with $\ell=(t_{1},t_{2},q^{-1})$ is then estimated by

$\displaystyle\hat{\nu}_{np}(\ell)=\hat{\nu}_{2,np}(\ell)-\hat{\nu}_{1,np}(\ell),~{}\text{where
for $j=1$ and 2},$ (3.2)
$\displaystyle\hat{\nu}_{j,np}(\ell)=\frac{1}{n}\sum_{i=1}^{n}\frac{Y_{i}}{\hat{p}_{np}(T_{i},X_{i})}1(T_{i}\in[t_{j},t_{j}+q^{-1}]).$

Lemma B.1 in the Appendix shows that uniformly over $\ell\in\mathcal{L}$ ,

$\displaystyle\sqrt{n}(\hat{\nu}_{np}(\ell)-\nu(\ell))$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\phi_{\ell,np}(Y_{i},T_{i},X_{i})+o_{p}(1),$
(3.3)

where

$\displaystyle\phi_{\ell,np}(Y_{i},T_{i},X_{i})=\phi_{2,\ell,np}(Y_{i},T_{i},X_{i})-\phi_{1,\ell,np}(Y_{i},T_{i},X_{i}),~{}\text{and
for $j=1$ and 2, }$
$\displaystyle\phi_{j,\ell,np}(Y,T,X)=\frac{1(T\in[t_{j},t_{j}+q^{-1}])}{p(T,X)}(Y-E[Y|T,X])+\rho_{j,\ell}(X)-\nu_{j}(\ell),$
$\displaystyle\rho_{j,\ell}(X)=E\left[\frac{Y1(T\in[t_{j},t_{j}+q^{-1}])}{p(T,X)}\Big{|}X\right].$

Note that $\phi_{\ell,np}(Y,T,X)$ is the influence function for
$\hat{\nu}_{np}(\ell)$. Given that $\phi_{\ell,np}(Y,T,X)$ is a
Vapnik–Chervonenkis (VC) type class of functions, it follows that
$\sqrt{n}(\hat{\nu}_{np}(\ell)-\nu(\ell))$ weakly converges to a Gaussian
process as the sample size increases.

Let $\hat{\phi}_{\ell,np}(Y,T,X)$ denote the estimated influence function with
$p(T,X)$, $E[Y|T,X]$, $\rho_{1,\ell}(X)$, $\rho_{2,\ell}(X)$, $\nu_{1}(\ell)$,
and $\nu_{2}(\ell)$ in ${\phi}_{\ell,np}(Y,T,X)$ replaced by correspondingly
uniformly consistent estimators $\hat{p}_{np}(T,X)$, $\widehat{E}[Y|T,X]$,
$\hat{\rho}_{1,\ell}(X)$, $\hat{\rho}_{2,\ell}(X)$, $\hat{\nu}_{1,np}(\ell)$,
and $\hat{\nu}_{2,np}(\ell)$, respectively. For example,

$\displaystyle\widehat{E}[Y|T_{i},X_{i}]=\frac{\sum_{j\neq
i}Y_{j}K_{h}(T_{j}-T_{i})K_{h}(X_{j}-X_{i})}{\sum_{j\neq
i}K_{h}(T_{j}-T_{i})K_{h}(X_{j}-X_{i})},$
$\displaystyle\hat{\rho}_{j,\ell}(X)=\frac{\sum_{l\neq
i}\frac{Y_{l}1(T_{l}\in[t_{j},t_{j}+r])}{\hat{p}_{np}(T_{l},X_{l})}K_{h}(X_{l}-X_{i})}{\sum_{l\neq
i}K_{h}(X_{l}-X_{i})}.$

Let
$\hat{\sigma}^{2}_{\nu,np}(\ell)=n^{-1}\sum_{i=1}^{n}\hat{\phi}^{2}_{\ell,np}(Y_{i},T_{i},X_{i})$,
which is a consistent estimator for the asymptotic variance of
$\sqrt{n}(\hat{\nu}_{np}(\ell)-{\nu}(\ell))$ under proper regularity
conditions. Let
$\hat{\sigma}_{\nu,\epsilon,np}(\ell)=\max\\{\hat{\sigma}_{\nu,np}(\ell),\epsilon\cdot\hat{\sigma}_{\nu,np}(0,1/2,1/2)\\}$,
by which we manually bound the variance estimator away from zero. To test the
null hypothesis $H_{0}^{\prime}$, we make use of a Cramér-von Mises test
statistic defined as

$\displaystyle\widehat{T}_{np}=\sum_{\ell\in\mathcal{L}}\max\Big{\\{}\sqrt{n}\frac{\hat{\nu}_{np}(\ell)}{\hat{\sigma}_{\nu,\epsilon,np}(\tau,\ell)},0\Big{\\}}^{2}Q(\ell),$
(3.4)

where $Q$ is a weighting function such that $Q(\ell)>0$ for all
$\ell\in\mathcal{L}$ and $\sum_{\ell\in\mathcal{L}}Q(\ell)<\infty$.

We next define the simulated critical value for our test. We first introduce a
multiplier bootstrap method that can simulate a process that converges to the
same limit as $\sqrt{n}(\hat{\nu}_{np}(\ell)-{\nu}(\ell))$. Let
$\\{U_{i}:~{}1\leq i\leq n\\}$ be a sequence of i.i.d. random variables that
satisfy Assumption B.4. We construct the simulated process as

$\displaystyle\widehat{\Phi}_{\nu,np}^{u}(\ell)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}U_{i}\cdot\hat{\phi}_{\ell,np}(Y_{i},T_{i},X_{i}),$
(3.5)

where $\hat{\phi}_{\ell,np}(Y_{i},T_{i},X_{i})$ is the estimated influence
function defined earlier. Under specific regularity conditions, we can show
that the simulated process weakly converges to a Gaussian process conditional
on the sample path with probability approaching one and that this limiting
Gaussian process corresponds to the limiting process of
$\sqrt{n}(\hat{\nu}_{np}(\ell)-\nu(\ell))$.

We adopt the GMS method to construct the simulated critical value as

$\displaystyle\hat{c}^{\eta}_{np}(\alpha)$
$\displaystyle=\sup\left\\{q\Big{|}P^{u}\left(\sum_{\ell\in\mathcal{L}}\max\Big{\\{}\frac{\widehat{\Phi}_{\nu,np}^{u}(\ell)}{\hat{\sigma}_{\nu,\epsilon,np}(\ell)}+\hat{\psi}_{\nu,np}(\ell),0\Big{\\}}Q(\ell)\leq
q\right)\leq 1-\alpha+\eta\right\\}+\eta,$
$\displaystyle\hat{\psi}_{\nu,np}(\ell)$ $\displaystyle=-B_{n}\cdot
1\left(\sqrt{n}\cdot\frac{\hat{\nu}_{np}(\ell)}{\hat{\sigma}_{\nu,\epsilon,np}(\ell)}<-a_{n}\right),$

in which $a_{n}$ and $B_{n}$ satisfy Assumption B.5 below.333The GMS approach
is similar to the recentering method of Hansen (2005) and Donald and Hsu
(2016), and the contact approach of Linton et al. (2010).

The decision rule is then given by

Reject $H^{\prime}_{0}$ if $\widehat{T}_{np}>\hat{c}^{\eta}_{np}(\alpha)$.
(3.6)

In Section B in the Appendix, we provide regularity conditions and show that
the proposed nonparametric test has asymptotic size control and is consistent
against any fixed alternative hypothesis.

##  4 Test Based on Parametric Estimation of $p(t,x)$

In this section, we consider a parametric test based on parametrically
specifying and estimating the generalized propensity score $p(t,x)$. More
specifically, let $\\{p(t,x;\gamma):x\in\mathcal{X},\gamma\in\Gamma\\}$ be a
parametric model of the conditional density, where $\Gamma$ is a compact set
of dimension $q$. To this end, we invoke the following assumption on the first
step estimation of the generalized propensity score.

######  Assumption 4.1.

(Parametric Generalized Propensity Scores):

  1. (i)

There exists $\gamma_{0}$ which is an interior point of $\Gamma$, such that
$p(t,x)=p(t,x;\gamma_{0})$ for all $(t,x)\in\mathcal{T}\times\mathcal{X}$.

  2. (ii)

There exists an estimator $\hat{\gamma}$ for $\gamma_{0}$ satisfying that

$\displaystyle\sqrt{n}(\hat{\gamma}-\gamma_{0})=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\phi_{\gamma}(T_{i},X_{i};\gamma_{0})+o_{p}(1)$
(4.1)

and $E[|\phi_{\gamma}(T,X;\gamma)|^{2+\delta}]<\infty$ for a neighborhood of
$\gamma_{0}$.

  3. (iii)

$p(t,x;\gamma)\geq\delta$ and $\|\bigtriangledown_{\gamma}p(t,x;\gamma)\|<M$
for some $\delta>0$ and $M>0$ for all $(t,x)\in\mathcal{T}\times\mathcal{X}$
and $\gamma\in\mathcal{N}_{\epsilon}(\gamma_{0})$ where
$\mathcal{N}_{\epsilon}(\gamma_{0})$ is an open ball with radius equal to
$\epsilon$ around $\gamma_{0}$.

Assumption 4.1 (i) requires that the generalized propensity score function is
correctly specified. Assumption 4.1 (ii) implies the existence of an estimator
with an asymptotically linear representation for the parameters in the
generalized propensity score specification and that the estimator is also
asymptotically normally distributed. In general, we would use a maximum
likelihood estimator (MLE) to estimate $\gamma_{0}$ and in this case,
$\phi_{\gamma}(T_{i},X_{i};\gamma_{0})$ is the score function of the log-
likelihood function multiplied by inverse matrix of the expected value of the
product of the score functions.444Given the unconditional information matrix
equality, the matrix of the expected value of the product of the score
functions is equal to the negative of the expected value of the Hessian matrix
of the log-likelihood function. To be specific, $p(t,x;\gamma)$ is the
parametric specification of the conditional density function of $T$ given $X$,
so the log-likelihood function will be $\log p(t,x;\gamma)$ and the MLE
estimator for $\gamma_{0}$ is given by

$\displaystyle\hat{\gamma}=\arg\min_{\gamma\in\Gamma}\frac{1}{n}\sum_{i=1}^{n}\log
p(T_{i},X_{i};\gamma).$

In this case, the score function is
$s(t,x;\gamma_{0})=\bigtriangledown_{\gamma}p(t,x;\gamma_{0})/p(t,x;\gamma_{0})$
and under suitable conditions, we have

$\displaystyle\sqrt{n}(\hat{\gamma}-\gamma_{0})$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}E\big{[}s(T,X;\gamma_{0})s(T,X;\gamma_{0})^{\prime}\big{]}^{-1}s(T_{i},X_{i};\gamma_{0})+o_{p}(1)$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\phi_{\gamma}(T_{i},X_{i};\gamma_{0})+o_{p}(1).$

See also Chapter 13 of Wooldridge (2010).

Let $p(t,x;\hat{\gamma})$ be the parametric estimator of the generalized
propensity score. Let
$\hat{\nu}_{pa}(\ell)=\hat{\nu}_{2,pa}(\ell)-\hat{\nu}_{1,pa}(\ell)$, where
for $j=1,2$, $\hat{\nu}_{j,pa}(\ell)$ is obtained by replacing those
$\hat{p}_{np}(T_{i},X_{i})$ in $\hat{\nu}_{j,np}(\ell)$ in (3.2) with
${p}(T_{i},X_{i};\hat{\gamma})$. Under specific regularity conditions, we can
show that uniformly over $\ell\in\mathcal{L}$,

$\displaystyle\sqrt{n}(\hat{\nu}_{pa}(\ell)-\nu(\ell))$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\phi_{\ell,pa}(Y_{i},T_{i},X_{i})+o_{p}(1),$
(4.2)

where

$\displaystyle\phi_{\ell,pa}(Y_{i},T_{i},X_{i})=\phi_{2,\ell,pa}(Y_{i},T_{i},X_{i})-\phi_{1,\ell,pa}(Y_{i},T_{i},X_{i}),~{}\text{and
for $j=1$ and 2, }$
$\displaystyle\phi_{j,\ell,pa}(Y_{i},T_{i},X_{i})=\frac{Y_{i}\cdot
1(T_{i}\in[t_{j},t_{j}+r])}{p(T_{i},X_{i};\gamma_{0})}-G_{j}(\ell)\cdot\phi_{\gamma}(T,X;\gamma_{0})-\nu_{j}(\ell),$
$\displaystyle G_{j}(\ell)=E\Big{[}\frac{Y\cdot
1(T\in[t_{j},t_{j}+r])\cdot\bigtriangledown_{\gamma}p(T,X;\gamma_{0})}{p^{2}(T,X,\gamma_{0})}\Big{]}.$

$\bigtriangledown_{\gamma}p(T_{i},X_{i},\gamma_{0})$ denotes the gradient of
$p(T_{i},X_{i};\gamma_{0})$ with respect to $\gamma$. Note that
$\phi_{\ell,pa}(Y,T,X)$ is the influence function for $\hat{\nu}_{pa}(\ell)$.
For $j=1$ and 2, we can estimate $G_{j}(\ell)$ by

$\displaystyle\widehat{G}_{j}(\ell)=\frac{1}{n}\sum_{i=1}^{n}\frac{Y_{i}\cdot
1(T_{i}\in[t_{j},t_{j}+r])\cdot\bigtriangledown_{\gamma}p(T_{i},X_{i};\hat{\gamma})}{p^{2}(T_{i},X_{i};\hat{\gamma})}$

and it is straightforward to show that $\widehat{G}_{j}(\ell)$ is uniformly
consistent for $G_{j}(\ell)$.

Let $\hat{\phi}_{\ell,pa}(Y,T,X)$ be the estimated influence function with
$\gamma_{0}$, $G_{1}(\ell)$, $G_{2}(\ell)$, $\nu_{1}(\ell)$, and
$\nu_{2}(\ell)$ in ${\phi}_{\ell,pa}(Y,T,X)$ replaced by correspondingly
uniformly consistent estimators $\hat{\gamma}$, $\widehat{G}_{1}(\ell)$,
$\widehat{G}_{2}(\ell)$, $\hat{\nu}_{1,pa}(\ell)$, and $\hat{\nu}_{2,pa}$,
respectively. Let
$\hat{\sigma}^{2}_{\nu,pa}(\ell)=n^{-1}\sum_{i=1}^{n}\hat{\phi}^{2}_{\ell,pa}(Y_{i},T_{i},X_{i})$,
which is a consistent estimator for the asymptotic variance of
$\sqrt{n}(\hat{\nu}_{pa}(\ell)-{\nu}(\ell))$ under proper regularity
conditions. Let
$\hat{\sigma}_{\nu,\epsilon,pa}(\ell)=\max\\{\hat{\sigma}_{\nu,pa}(\ell),\epsilon\cdot\hat{\sigma}_{\nu,pa}(0,1/2,1/2)\\}$.
Similar to the nonparametric case, we define the test statistic as

$\displaystyle\widehat{T}_{pa}=\sum_{\ell\in\mathcal{L}}\max\Big{\\{}\sqrt{n}\frac{\hat{\nu}_{pa}(\ell)}{\hat{\sigma}_{\nu,\epsilon,pa}(\tau,\ell)},0\Big{\\}}^{2}Q(\ell).$
(4.3)

We next define the simulated critical value for the parametric test. We
construct the simulated process as

$\displaystyle\widehat{\Phi}_{\nu,pa}^{u}(\ell)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}U_{i}\cdot\hat{\phi}_{\ell,pa}(Y_{i},T_{i},X_{i}),$

and the GMS simulated critical value as

$\displaystyle\hat{c}^{\eta}_{pa}(\alpha)$
$\displaystyle=\sup\left\\{q\Big{|}P^{u}\left(\sum_{\ell\in\mathcal{L}}\max\Big{\\{}\frac{\widehat{\Phi}_{\nu,pa}^{u}(\ell)}{\hat{\sigma}_{\nu,\epsilon,pa}(\ell)}+\hat{\psi}_{\nu,pa}(\ell),0\Big{\\}}Q(\ell)\leq
q\right)\leq 1-\alpha+\eta\right\\}+\eta,$
$\displaystyle\hat{\psi}_{\nu,pa}(\ell)$ $\displaystyle=-B_{n}\cdot
1\left(\sqrt{n}\cdot\frac{\hat{\nu}_{pa}(\ell)}{\hat{\sigma}_{\nu,\epsilon,pa}(\ell)}<-a_{n}\right).$

The decision rule is then given by

Reject $H^{\prime}_{0}$ if $\widehat{T}_{pa}>\hat{c}^{\eta}_{pa}(\alpha)$.
(4.4)

The size and power properties are similar to the nonparametric case and we
omit the details for this reason.

##  5 Testing Monotonicity Conditional on Covariates

In this section, we adapt our method to testing monotonicity with conditional
(rather than unconditional) mean potential outcomes given observed covariates
$X$. In this case, the null hypothesis considered corresponds to

$\displaystyle H_{0}:~{}\mu(t_{1},x)\geq\mu(t_{2},x),~{}~{}\text{for
all}~{}t_{1}\geq t_{2},~{}\text{for $t_{1},t_{2}\in[0,1]$ and
}~{}{x\in\mathcal{X}},$ (5.1)

where $\mu(t,x)=E[Y(t)|X=x]$ is the conditional average of the potential
outcome function or the average dose-response function. For simplicity and
without loss of generality, we henceforth assume that $X$ is a scalar with
$\mathcal{X}=[0,1]$. By Lemma 2.1 of Hsu and Shen (2020), $H_{0}$ in (5.1) is
equivalent to

$\displaystyle\int_{{x}}^{{x}+q^{-1}}\int_{t_{2}}^{t_{2}+q^{-1}}\mu(\tilde{t},\tilde{x})\cdot
h(\tilde{t},\tilde{x})dtd\tilde{x}\cdot\int_{{x}}^{{x}+q^{-1}}\int_{t_{1}}^{t_{1}+q^{-1}}h(\tilde{t},\tilde{x})d\tilde{t}d\tilde{x}-$
$\displaystyle~{}~{}~{}~{}~{}~{}\int_{{x}}^{{x}+q^{-1}}\int_{t_{1}}^{t_{1}+q^{-1}}\mu(\tilde{t},\tilde{x})\cdot
h(\tilde{t},\tilde{x})d\tilde{t}d\tilde{x}\cdot\int_{{x}}^{{x}+q^{-1}}\int_{t_{2}}^{t_{2}+q^{-1}}h(\tilde{t},\tilde{x})d\tilde{t}d\tilde{x}\leq
0$ (5.2)

for any $q=2,\cdots,$ and for any $t_{1}\geq t_{2}$ such that $q\cdot
t_{1},q\cdot t_{2},q\cdot x\in\\{0,1,2,\cdots,q-1\\}$. Following Lemma 2.1 and
defining $h(t,x)=f(x)$ to be the density function of $X$, we have for $r>0$,

$\displaystyle\int_{{x}}^{{x}+r}\int_{t}^{t+r}\mu(\tilde{t},\tilde{x})h(\tilde{t},\tilde{x})d\tilde{t}d\tilde{x}=\mathbb{E}\left[\frac{Y}{p(T,X)}\cdot
1(T\in[t,t+r])\cdot 1(X\in[x,x+r])\right],$
$\displaystyle\int_{{x}}^{{x}+r}\int_{t}^{t+r}h(\tilde{t},\tilde{x})d\tilde{t}d\tilde{x}=\mathbb{E}\left[1(X\in[x,x+r])\right].$

For $\ell_{x}=(t_{1},t_{2},x,q^{-1})\in[0,1]^{3}\times(0,1]$, we let

$\displaystyle{\cal
L}_{x}=\Big{\\{}\ell_{x}=(t_{1},t_{2},x,q^{-1}):~{}q\cdot(t_{1},t_{2},x)\in\\{0,1,2,\cdots,q-1\\}^{3},~{}t_{1}>t_{2},\text{,
and }q=2,3,\cdots\Big{\\}}.$ (5.3)

For each $\ell_{x}$, we define

$\displaystyle\nu_{1}(\ell_{x})=E\left[\frac{Y}{p(T,X)}1(T\in[t_{1},t_{1}+q^{-1}]1(X\in[x,x+q^{-1}]))\right],$
$\displaystyle\nu_{2}(\ell_{x})=E\left[\frac{Y}{p(T,X)}1(T\in[t_{2},t_{2}+q^{-1}])1(X\in[x,x+q^{-1}])\right].$

This permits establishing the following lemma.

######  Lemma 5.1.

Suppose Assumption 2.1 holds. Assume that $\mu(t,x)$ is continuous in $t$ for
all $x\in[0,1]$. Then $H_{0}$ in (5.1) is equivalent to

$\displaystyle H_{0}^{\prime}:$
$\displaystyle~{}\nu(\ell_{x})=\nu_{2}(\ell_{x})-\nu_{1}(\ell_{x})\leq
0~{}\text{for any $\ell_{x}=(t_{1},t_{2},x,q^{-1})\in\mathcal{L}_{x}$}.$ (5.4)

We subsequently only consider the nonparametric version of the test and note
that the parametric one can be constructed in a similar way. We estimate
$\nu(\ell_{x})$ with $\ell_{x}=(t_{1},t_{2},x,q^{-1})$ by

$\displaystyle\hat{\nu}_{np}(\ell_{x})=\hat{\nu}_{2,np}(\ell_{x})-\hat{\nu}_{1,np}(\ell_{x}),~{}\text{where
for $j=1$ and 2},$ (5.5)
$\displaystyle\hat{\nu}_{j,np}(\ell_{x})=\frac{1}{n}\sum_{i=1}^{n}\frac{Y_{i}}{\hat{p}_{np}(T_{i},X_{i})}1(T_{i}\in[t_{j},t_{j}+q^{-1}])1(X_{i}\in[x,x+q^{-1}]),$

where $\hat{p}_{np}(T_{i},X_{i})$ is given in (3.1). Similar to Lemma B.1 in
the Appendix, we can show that uniformly over $\ell_{x}\in\mathcal{L}_{x}$,

$\displaystyle\sqrt{n}(\hat{\nu}_{np}(\ell_{x})-\nu(\ell_{x}))$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\phi_{\ell_{x},np}(Y_{i},T_{i},X_{i})+o_{p}(1),$
(5.6)

where

$\displaystyle\phi_{\ell_{x},np}(Y_{i},T_{i},X_{i})=\phi_{2,\ell_{x},np}(Y_{i},T_{i},X_{i})-\phi_{1,\ell_{x},np}(Y_{i},T_{i},X_{i}),~{}\text{and
for $j=1$ and 2, }$
$\displaystyle\phi_{j,\ell_{x},np}(Y,T,X)=1(X\in[x,x+q^{-1}])\Big{(}\frac{1(T\in[t_{j},t_{j}+q^{-1}])}{p(T,X)}(Y-E[Y|T,X])+\rho_{j,\ell}(X)\Big{)}-\nu_{j}(\ell_{x}).$

Let $\hat{\phi}_{\ell_{x},np}(Y,T,X)$ be the estimated influence function with
$p(T,X)$, $E[Y|T,X]$, $\rho_{1,\ell}(X)$, $\rho_{2,\ell}(X)$,
$\nu_{1}(\ell_{x})$, and $\nu_{2}(\ell_{x})$ in ${\phi}_{\ell_{x},np}(Y,T,X)$
replaced by correspondingly uniformly consistent estimators. Let
$\hat{\sigma}^{2}_{\nu,np}(\ell_{x})=n^{-1}\sum_{i=1}^{n}\hat{\phi}^{2}_{\ell_{x},np}(Y_{i},T_{i},X_{i})$
which will be a consistent estimator for the asymptotic variance of
$\sqrt{n}(\hat{\nu}_{np}(\ell_{x})-{\nu}(\ell_{x}))$ under proper regularity
conditions. Furthermore, let
$\hat{\sigma}_{\nu,\epsilon,np}(\ell_{x})=\max\\{\hat{\sigma}_{\nu,np}(\ell_{x}),\epsilon\cdot\hat{\sigma}_{\nu,np}(0,1/2,0,1/2)\\}$.
The Cramér-von Mises test statistic is defined as

$\displaystyle\widehat{T}_{x,np}=\sum_{\ell_{x}\in\mathcal{L}_{x}}\max\Big{\\{}\sqrt{n}\frac{\hat{\nu}_{np}(\ell_{x})}{\hat{\sigma}_{\nu,\epsilon,np}(\tau,\ell_{x})},0\Big{\\}}^{2}Q(\ell_{x}),$
(5.7)

where $Q$ is a weighting function such that $Q(\ell_{x})>0$ for all
$\ell_{x}\in\mathcal{L}_{x}$ and
$\sum_{\ell_{x}\in\mathcal{L}_{x}}Q(\ell_{x})<\infty$. The simulated process
is constructed as

$\displaystyle\widehat{\Phi}_{\nu,x,np}^{u}(\ell_{x})=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}U_{i}\cdot\hat{\phi}_{\ell_{x},np}(Y_{i},T_{i},X_{i}).$
(5.8)

The GMS simulated critical value is given by

$\displaystyle\hat{c}^{\eta}_{x,np}(\alpha)$
$\displaystyle=\sup\left\\{q\Big{|}P^{u}\left(\sum_{\ell_{x}\in\mathcal{L}_{x}}\max\Big{\\{}\frac{\widehat{\Phi}_{\nu,x,np}^{u}(\ell_{x})}{\hat{\sigma}_{\nu,\epsilon,np}(\ell_{x})}+\hat{\psi}_{\nu,np}(\ell_{x}),0\Big{\\}}Q(\ell_{x})\leq
q\right)\leq 1-\alpha+\eta\right\\}+\eta,$
$\displaystyle\hat{\psi}_{\nu,np}(\ell_{x})$ $\displaystyle=-B_{n}\cdot
1\left(\sqrt{n}\cdot\frac{\hat{\nu}_{np}(\ell_{x})}{\hat{\sigma}_{\nu,\epsilon,np}(\ell_{x})}<-a_{n}\right).$

Finally, the decision rule is given by

Reject $H^{\prime}_{0}$ if $\widehat{T}_{np}>\hat{c}^{\eta}_{np}(\alpha)$.

The size and power properties are similar to the unconditional potential
outcome cases and the details are omitted for this reason.

##  6 Simulation

To implement our test in practice, one has to choose several tuning parameters
in advance. We make the following propositions concerning the choice of these
parameters and present related Monte Carlo simulation results further below.

  1. 1.

Instrumental functions: We opt for using a set of indicator functions of
countable hypercubes. For $\ell=(t_{1},t_{2},q^{-1})\in[0,1]^{2}\times(0,1]$,
define

$\displaystyle{\cal
L}=\Big{\\{}\ell=(t_{1},t_{2},q^{-1}):q\cdot(t_{1},t_{2})\in\\{0,1,2,\cdots,q-1\\}^{2},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}t_{1}>t_{2},\text{,
and }q=2,\cdots,q_{1}\Big{\\}},$ (6.1)

where $q_{1}$ is a natural number and is chosen such that the expected sample
size of the smallest cube is around 25. Our simulations show that the results
are robust to various expected sample sizes.

  2. 2.

$Q(\ell)$: The distribution $Q(\ell)$ assigns weight $\propto q^{-2}$ to each
$q$ and for each $q$, $Q(\ell)$ assigns an equal weight to each instrumental
function with last element of $\ell$ equal to $q^{-1}$. Recall that for each
$q$, there are $(q(q+1)/2)$ instrumental functions with the last element of
$\ell$ equal to$q^{-1}$.

  3. 3.

$a_{n}$, $B_{n}$, $\epsilon$, $\eta$: We set $a_{n}=0.15\cdot\ln(n)$,
$B_{n}=0.85\cdot\ln(n)/\ln\ln(n)$, $\epsilon=10^{-6}$, and $\eta=10^{-6}$ as
suggested by Hsu et al. (2019). These choices are used in all the simulations
that we report below and seem to perform well.

We consider samples of sizes $n=200$, $400$, and $800$. For $q_{1}$, we set
$q_{1}=5$ for $n=200$, $q_{1}=10$ for $n=400$, and $q_{1}=20$ for $n=800$. All
our Monte Carlo results are based on $1000$ simulations. In each simulation,
the critical value is approximated by $1000$ bootstrap replications. The
nominal size of the test is set at $10\%$.

Our first data generating process (DGP) considered in the simulation,
henceforth denoted by DGP 1, is given by

$\displaystyle Y=(-0.5+X)T+T^{2}+X+U_{y},$ $\displaystyle T=X+0.5U_{t},$
$\displaystyle X=U_{x},$

$U_{y}\sim\mathcal{N}(0,1)$, $U_{t}\sim\mathcal{N}(0,1)$ and $U_{x}\sim
Uni[0,1]$. $U_{y}$, $U_{t}$, and $U_{x}$ are mutually independent. In DGP 1,
$H_{0}$ holds.

The second DGP, denoted by DGP 2, corresponds to

$\displaystyle Y=(-0.5+X)T+\sin(\pi T)+X+U_{y},$ $\displaystyle T=X+0.5U_{t},$
$\displaystyle X=U_{x},$

$U_{y}\sim\mathcal{N}(0,1)$, $U_{t}\sim\mathcal{N}(0,1)$ and $U_{x}\sim
Uni[0,1]$. $U_{y}$, $U_{t}$, and $U_{x}$ are mutually independent. In DGP 2,
$H_{0}$ does not hold.

In the third scenario considered, denoted by DGP 3,

$\displaystyle Y=U_{y},$ $\displaystyle T=X+0.5U_{t},$ $\displaystyle
X=U_{x},$

$U_{y}\sim\mathcal{N}(0,1)$, $U_{t}\sim\mathcal{N}(0,1)$ and $U_{x}\sim
Uni[0,1]$. $U_{y}$, $U_{t}$, and $U_{x}$ are mutually independent. In DGP 3,
$H_{0}$ holds. In these DGPs, $1+d_{x}=2$. We use fourth-order Epanechnikov
kernels in all simulations. Concerning the bandwidth, we set $h=a\cdot
n^{-1/5}$, and the parameter $a$ is varied between $0.8$, $1$, and $1.2$.

Table 6 provides the rejection probabilities of our test with both
nonparametric and parametric generalized propensity scores. The results are
consistent with our theoretical findings. Both the nonparametric and
parametric tests control size well in DGP 1 and DGP 3. The rejection
probabilities increase with the sample size and are greater than the nominal
size of $0.1$ in DGP 2. As we can see, the choice of bandwidth parameter $a$
does not importantly affect the test performance. Furthermore, the parametric
test slightly outperforms the nonparametric test. Table 7 provides the
rejection probabilities of our tests based on the trimmed generalized
propensity score estimator, defined as
$\tilde{p}(T_{i},X_{i})=\max\\{\hat{p}(T_{i},X_{i}),0.05\\}$, implying that
conditional treatment densities below 5% are set to 5%.555Based on this
trimming rule, around 2% of the samples are trimmed. For the DGPs considered
in our simulations, the results of the trimmed estimators are quite similar to
those without trimming.

We now investigate the robustness of the performance of our tests to the
choice of $q_{1}$. Let $N$ denote the expected sample size of the smallest
cube. We consider three alternative choices of $q_{1}$, each resulting in
$N=33$, $50$, and $66$, respectively. Tables 8 and 9 shows the rejection
probabilities of our tests without and with trimming, respectively. For the
nonparametric test, we set $h=a\cdot n^{-1/5}$ and $a=1$. The results suggest
that the choice of $q_{1}$ does not affect the test performance much.
Therefore, the finite sample behavior of our tests appears to be reasonably
robust to different values of $q_{1}$.

##  7 Empirical application

As an empirical illustration, we apply our test to data from the Job Corps
study. The latter was conducted between November 1994 and February 1996 to
evaluate the publicly funded U.S. Job Corps program and used an experimental
design that randomly assigned access to the program. Job Corps targets youths
from low-income households who are between 16 and 24 years old and legally
reside in the U.S. Program participants obtained on average roughly 1200 hours
of vocational and/or academic classroom training as well as housing and board
over an average duration of 8 months. We refer to Schochet et al. (2001) and
Schochet et al. (2008) for a detailed discussion of the study design and the
average effects of program assignment on a range of different outcomes. Their
results suggest that Job Corps raises educational attainment, reduces criminal
activity, and increases labor market performance measured by employment and
earnings, at least for some years after the program.

Particularly relevant for our context is the study by Flores et al. (2012),
who consider the length of exposure to academic and/or vocational training as
continuously distributed treatment to assess its effect on earnings based on
regression and weighting estimators (using the inverse of the conditional
treatment density as weight). As the length of treatment exposure is (in
contrast to Job Corps assignment) not random, they impose a selection-on-
observables assumption and control for baseline characteristics at Job Corps
assignment. While the authors find overall positive average effects of
increasing hours in academic and vocational instruction, the effects appear to
diminish with length of exposure, pointing to a potential non-monotonicity in
the association of earnings and time of instruction. Relatedly, Lee (2018) and
Colangelo and Lee (2020) assess the effect of hours in training on the
proportion of weeks employed in the second year after program assignment based
on kernel regression and double machine learning,666The estimator of Colangelo
and Lee (2020) is based on the efficient influence function, whose plug-in
parameters (namely the conditional treatment densities and conditional mean
outcomes) are estimated by machine learning, see Chernozhukov et al. (2018)
for an introduction to the double machine learning framework. respectively.
Also for this outcome, the plotted regression lines in either study point to a
concave association with the treatment dose.777See also Huber et al. (2020),
who use a causal mediation approach to assess the direct effect of the
treatment dose on the number of arrests in the fourth year after program
assignment when controlling for employment behavior in the second year based
on inverse probability weighting and find a non-linear association.

However, in the light of estimation uncertainty, mere eye-balling of the
outcome-treatment associations in empirical applications does not tell us
whether specific shape restrictions can be refuted. For this reason, we use
our parametric method to formally test whether weak positive and negative
monotonicity can be rejected in the Job Corps data when considering several
labor market outcomes. To this end, we define the treatment variable $T$ as
the total hours spent in academic and vocational training in the 12 months
following the program assignment. Our outcomes $Y$ include weeks employed in
the second year after program assignment, weekly earnings in the fourth year,
earnings and hours worked per week in quarter 16, and a binary employment
indicator four years after assignment (i.e. in week 208).

For invoking weak unconfoundedness (Assumption 2.1), we consider the same set
of pre-treatment covariates $X$ as Lee (2018), Colangelo and Lee (2020), and
Huber et al. (2020), which overlaps with the control variables of Flores et
al. (2012).888A control variable in Flores et al. (2012) we do not have access
to is the local unemployment rate which was constructed by matching county-
level unemployment rates to individual postal codes of residence, which are
only available in a restricted-use data set. We condition on individual
characteristics like age, gender, ethnicity, language competency, education,
marital status, household size and income, previous receipt of social aid,
family background (e.g. parents’ education), criminal activity, as well as
health and health-related behavior (e.g. smoking, alcohol, or drug
consumption). Conditioning on such a rich set of socio-economic variables
appears important, as the satisfaction of weak unconfoundedness relies on
successfully controlling for all factors jointly affecting treatment duration
and labor market behavior. Furthermore, we include variables that might be
associated with the duration in training, namely expectations about Job Corps
and interaction with the recruiters, which might serve as proxies for
unobserved personality traits (like motivation) that could also affect the
outcomes. Finally, we control for pre-treatment outcomes, namely previous
labor market participation and earnings, to tackle any confounders that affect
the outcomes of interest through their respective pre-treatment values.

The original Job Corps data set consists of $15,386$ individuals prior to
program assignment, but a substantial share never enrolled in the program and
dropped out of the study, such that there are only $11,313$ individuals with
completed follow-up interviews four years after randomization. Among those,
$6,828$ had been randomized into Job Corps and had thus access to academic or
vocational training. To define our final evaluation sample, we follow Flores
et al. (2012), Lee (2018), Colangelo and Lee (2020), and Huber et al. (2020)
and consider observations with at least 40 hours (or one working week) of
training for our analysis, all in all $4,166$ individuals. Among these, there
are cases of item non-response in various elements of $X$ measured at the
baseline survey, for which we account by the inclusion of missing dummies as
additional regressors, while observations with missing values in the outcome
of interest need to be dropped when running the respective test. Table 1
provides descriptive statistics for selected covariates $X$ (see Huber et al.
(2020) for a full list of control variabes) as well as for the treatment $T$
and all outcomes $Y$, including the respective number of nonmissing
observations (nonmissing).

Table 1: Descriptives for selected covariates, treatment, and outcomes variable | mean | median | minimum | maximum | nonmissing
---|---|---|---|---|---
female | 0.432 | 0.495 | 0.000 | 1.000 | 4166
age | 18.325 | 2.142 | 16.000 | 24.000 | 4166
white | 0.249 | 0.433 | 0.000 | 1.000 | 4166
black | 0.502 | 0.500 | 0.000 | 1.000 | 4166
Hispanic | 0.172 | 0.378 | 0.000 | 1.000 | 4166
years of education | 10.045 | 1.535 | 0.000 | 20.000 | 4102
married | 0.016 | 0.126 | 0.000 | 1.000 | 4166
has children | 0.178 | 0.382 | 0.000 | 1.000 | 4166
ever worked | 0.145 | 0.352 | 0.000 | 1.000 | 4166
mean gross weekly earnings | 19.429 | 97.749 | 0.000 | 2000.000 | 4166
household size | 3.536 | 2.006 | 0.000 | 15.000 | 4101
mum’s years of education | 11.504 | 2.599 | 0.000 | 20.000 | 3397
dad’s years of education | 11.459 | 2.900 | 0.000 | 20.000 | 2604
welfare receipt during childhood | 2.064 | 1.189 | 1.000 | 4.000 | 3871
poor or fair general health | 0.124 | 0.330 | 0.000 | 1.000 | 4166
physical or emotional problems | 0.043 | 0.203 | 0.000 | 1.000 | 4166
extent of marijuana use | 2.540 | 1.549 | 0.000 | 4.000 | 1534
extent of smoking | 1.526 | 0.971 | 0.000 | 4.000 | 2171
extent of alcohol consumption | 3.140 | 1.210 | 0.000 | 4.000 | 2383
ever arrested | 0.241 | 0.428 | 0.000 | 1.000 | 4166
recruiter support | 1.592 | 1.059 | 1.000 | 5.000 | 4068
idea about desired training | 0.839 | 0.368 | 0.000 | 1.000 | 4166
expected months in Job Corps | 6.622 | 9.794 | 0.000 | 36.000 | 4166
hours in training ($T$) | 1192.130 | 966.945 | 0.857 | 6188.571 | 4166
weeks employed in second year ($Y$) | 43.862 | 37.900 | 0.000 | 100.000 | 4119
weekly earnings in fourth year ($Y$) | 215.521 | 202.619 | 0.000 | 1879.172 | 4024
weekly earnings in quarter 16 ($Y$) | 220.933 | 223.078 | 0.000 | 1970.445 | 4015
weekly hours worked quarter 16 ($Y$) | 28.187 | 22.746 | 0.000 | 84.000 | 4102
employed in week 208 ($Y$) | 0.627 | 0.484 | 0.000 | 1.000 | 4007

We verify whether the null hypotheses that the mean potential outcome is
weakly increasing in the treatment intensity, i.e.
$H_{0}:\mu(t_{1})\geq\mu(t_{2})$, or weakly decreasing in the treatment
intensity, i.e. $H_{0}:\mu(t_{1})\leq\mu(t_{2})$, are rejected in our
evaluation sample based on the parametric test statistic (4.3). To this end,
we follow Huber et al. (2020) and assume that treatment $T$ is conditionally
log-normally distributed given $X$ when parametrically estimating the
generalized treatment propensity score. The choices of nuisance parameters are
the same as in the simulations (see the previous section), and the expected
sample size of the smallest cube is $40$. The p-values of the tests for the
various outcomes are calculated based on 1000 bootstrap replications.

Table 2: Test statistic and p-value, $40\leq T\leq 3000$

| trim=0.0001 |  | trim=0.00007
---|---|---|---
$H_{0}:$ | $\mu(t_{1})\geq\mu(t_{2})$ | $\mu(t_{1})\leq\mu(t_{2})$ |  | $\mu(t_{1})\geq\mu(t_{2})$ | $\mu(t_{1})\leq\mu(t_{2})$
$Y$ | stat | p-value | stat | p-value |  | stat | p-value | stat | p-value
workwy2 | 4.438 | 0.000 | 6.584 | 0.000 |  | 2.050 | 0.000 | 9.114 | 0.000
earny4 | 2.749 | 0.000 | 9.750 | 0.000 |  | 1.101 | 0.000 | 13.367 | 0.000
earnq16 | 2.456 | 0.000 | 9.303 | 0.000 |  | 0.953 | 0.000 | 12.683 | 0.000
hrswq16 | 2.608 | 0.000 | 11.436 | 0.000 |  | 0.929 | 0.000 | 15.662 | 0.000
work208 | 3.800 | 0.000 | 11.920 | 0.000 |  | 1.695 | 0.000 | 15.724 | 0.000

Note: Outcomes ‘workwy2’, ‘earny4’, ‘earnq16’, ‘hrswq16’, and ‘work208’ are
weeks employed in the second year after program assignment, weekly earnings in
the fourth year, earnings and hours worked per week in quarter 16, and a
binary employment indicator four years after assignment (i.e. in week 208).
‘stat’ denotes the test statistic.

In a first step, we apply the test to a treatment interval of $T\in[40,3000]$,
where choosing 3000 hours of training as upper bound of the analysis is
motivated by the quickly decreasing number of observations beyond that point.
As in the simulations, we consider propensity score trimming and set the
trimming parameter to $0.0001$ such that the estimated generalized propensity
score corresponds to
$\tilde{p}(T_{i},X_{i})=\max\\{\hat{p}(T_{i},X_{i}),0.0001\\}$. As a result,
about $15\%$ of the propensity score estimates are set to $0.0001$. We also
consider $0.00007$, resulting in around $10\%$ of the estimates to be trimmed
to $0.00007$.999We follow Donald et al. (2014) and Hsu et al. (2020) and trim
the estimated generalized propensity scores to prevent them from being too
close zero, in order to obtain a more stable IPW estimator whose variance is
not affected by extremely low scores. Table 2 reports the results for all
outcomes under both null hypotheses of weakly increasing and decreasing mean
potential outcomes in the treatment, respectively, by either trimming rule.
Even though the test statistics (stat) vary considerably, all tests yield
p-values that are essentially zero and therefore clearly reject the null
hypotheses. Jointly refuting weakly positive and weakly negative monotonicity
therefore points to a non-monotonic impact of hours in training on the labor
market outcomes considered.

Table 3: Test statistic and p-value, $40\leq T\leq 1000$

| trim=0.0001 |  | trim=0.00007
---|---|---|---
$H_{0}:$ | $\mu(t_{1})\geq\mu(t_{2})$ | $\mu(t_{1})\leq\mu(t_{2})$ |  | $\mu(t_{1})\geq\mu(t_{2})$ | $\mu(t_{1})\leq\mu(t_{2})$
$Y$ | stat | p-value | stat | p-value |  | stat | p-value | stat | p-value
workwy2 | 0.053 | 0.493 | 3.742 | 0.000 |  | 0.054 | 0.452 | 3.596 | 0.001
earny4 | 0.148 | 0.143 | 5.051 | 0.000 |  | 0.149 | 0.135 | 4.992 | 0.000
earnq16 | 0.115 | 0.211 | 4.723 | 0.000 |  | 0.116 | 0.198 | 4.654 | 0.000
hrswq16 | 0.104 | 0.217 | 5.232 | 0.000 |  | 0.106 | 0.193 | 5.124 | 0.000
work208 | 0.101 | 0.225 | 5.706 | 0.000 |  | 0.102 | 0.215 | 5.633 | 0.000

Note: Outcomes ‘workwy2’, ‘earny4’, ‘earnq16’, ‘hrswq16’, and ‘work208’ are
weeks employed in the second year after program assignment, weekly earnings in
the fourth year, earnings and hours worked per week in quarter 16, and a
binary employment indicator four years after assignment (i.e. in week 208).
‘stat’ denotes the test statistic.

In a second step, we verify whether the null hypotheses are rejected for
subsets of the treatment support, motivated by the concavities in the outcome-
treatment associations spotted in the previously mentioned empirical
applications. We would therefore expect the null of weakly decreasing mean
potential outcomes to be rejected among relatively low treatment doses and
that of weakly increasing outcomes among rather high doses. To verify this
suspicion, we partition the treatment support into three sets of $[40,1000]$,
$[1000,2000]$, and $[2000,3000]$ and run the tests separately within each set.
Table 3 reports the test statistics and p-values for $T\in[40,1000]$.
Consistent with a concave outcome-treatment relation, none of the tests
rejects weakly positive monotonicity at any conventional level of
significance, while all tests strongly reject weakly negative monotonicity.
For the intermediate treatment range of $[1000,2000]$, the test results are
displayed in Table 4. The tests are again unanimous in that none of the tests
rejects weakly positive monotonicity at any conventional level of
significance, while all tests strongly reject weakly negative monotonicity.

Table 4: Test statistic and p-value, $1000\leq T\leq 2000$

| trim=0.0001 |  | trim=0.00007
---|---|---|---
$H_{0}:$ | $\mu(t_{1})\geq\mu(t_{2})$ | $\mu(t_{1})\leq\mu(t_{2})$ |  | $\mu(t_{1})\geq\mu(t_{2})$ | $\mu(t_{1})\leq\mu(t_{2})$
$Y$ | stat | p-value | stat | p-value |  | stat | p-value | stat | p-value
workwy2 | 0.022 | 0.830 | 4.616 | 0.000 |  | 0.020 | 0.841 | 5.360 | 0.000
earny4 | 0.014 | 0.880 | 5.990 | 0.000 |  | 0.012 | 0.917 | 6.555 | 0.000
earnq16 | 0.015 | 0.862 | 4.934 | 0.000 |  | 0.013 | 0.886 | 5.425 | 0.000
hrswq16 | 0.015 | 0.852 | 6.183 | 0.000 |  | 0.013 | 0.881 | 6.855 | 0.000
work208 | 0.005 | 0.993 | 8.238 | 0.000 |  | 0.005 | 0.998 | 8.536 | 0.000

Note: Outcomes ‘workwy2’, ‘earny4’, ‘earnq16’, ‘hrswq16’, and ‘work208’ are
weeks employed in the second year after program assignment, weekly earnings in
the fourth year, earnings and hours worked per week in quarter 16, and a
binary employment indicator four years after assignment (i.e. in week 208).
‘stat’ denotes the test statistic.

Table 5 provides the test results for the highest treatment bracket
$[2000,3000]$. Under the 0.0001 trimming rule, our test now rejects weakly
positive monotonicity for any outcome at the 1% level, but yields rather high
p-values when testing weakly negative monotonicity. when considering the
0.00007 trimming rule, weakly positive monotonicity is rejected at the 5%
level for the outcomes ‘workwy2’ and ‘work208’ as well as at the 10% level for
‘earny4’, while p-values are slightly above 10% for the remaining two
outcomes. Again, any p-values are far beyond conventional levels of
significance when considering weakly negative monotonicity. Summing up, our
empirical findings are consistent with a concave mean potential outcome-
treatment dependence under either trimming rule, albeit the shape of the
regression curves is not identical across the common support restrictions
considered. A potential explanation for the concavity of the ADF could be that
individuals attending more training in the first year might be induced to
attain more education also in the following years rather than to participate
in the labor market.

Table 5: Test statistic and p-value, $2000\leq T\leq 3000$

| trim=0.0001 |  | trim=0.00007
---|---|---|---
$H_{0}:$ | $\mu(t_{1})\geq\mu(t_{2})$ | $\mu(t_{1})\leq\mu(t_{2})$ |  | $\mu(t_{1})\geq\mu(t_{2})$ | $\mu(t_{1})\leq\mu(t_{2})$
$Y$ | stat | p-value | stat | p-value |  | stat | p-value | stat | p-value
workwy2 | 3.876 | 0.000 | 0.007 | 0.947 |  | 1.476 | 0.023 | 0.017 | 0.842
earny4 | 2.840 | 0.001 | 0.010 | 0.878 |  | 0.939 | 0.073 | 0.016 | 0.873
earnq16 | 2.182 | 0.007 | 0.007 | 0.964 |  | 0.609 | 0.134 | 0.012 | 0.923
hrswq16 | 2.474 | 0.005 | 0.007 | 0.966 |  | 0.681 | 0.126 | 0.011 | 0.955
work208 | 3.623 | 0.000 | 0.004 | 0.973 |  | 1.374 | 0.020 | 0.006 | 0.985

Note: Outcomes ‘workwy2’, ‘earny4’, ‘earnq16’, ‘hrswq16’, and ‘work208’ are
weeks employed in the second year after program assignment, weekly earnings in
the fourth year, earnings and hours worked per week in quarter 16, and a
binary employment indicator four years after assignment (i.e. in week 208).
‘stat’ denotes the test statistic.

##  8 Conclusion

In this paper, we proposed Cramér-von Mises-type tests for testing whether a
mean potential outcome is weakly monotonic in a continuously distributed
treatment under a weak unconfoundedness assumption. We considered versions of
the test for both nonparametrically and parametrically estimated generalized
treatment propensity scores and extended them to testing monotonicity
conditional on observed covariates. We also investigated the test’s finite
sample behavior in a simulation study and found it to perform decently under
our suggested choices of specific tuning parameters. Finally, we applied the
parametric version of our test to the Job Corps study, investigating the
associations of several labor market outcomes (earnings, employment, and hours
worked) with hours in training as treatment. We found that both weakly
increasing and decreasing treatment-outcome relations were generally rejected
over a treatment range of 40-3000 hours in training. However, when splitting
the treatment range into subsets, the test exclusively rejected weakly
negative associations among lower treatment values, while it only tended to
reject weakly positive associations among higher treatment values. These
findings were in line with a concave mean potential outcome-treatment
dependence.

APPENDIX

##  Appendix A Proof of Lemma 2.1

Under Assumption 2.1, Hirano and Imbens (2004) show that

$\displaystyle\mu(t)=\mathbb{E}\left[\mathbb{E}\left[Y|T=t,p(t,X)\right]\right]=\int_{\mathcal{X}}\mathbb{E}\left[Y|T=t,p(T,X)=p(t,X)\right]f(x)dx.$

Then

$\displaystyle\int_{t}^{t+r}\mu(t)h(\tilde{t})d\tilde{t}$
$\displaystyle=\int_{t}^{t+r}\int_{\mathcal{X}}\mathbb{E}\left[Y|T=\tilde{t},p(T,X)=p(\tilde{t},X)\right]f(x)h(\tilde{t})dxd\tilde{t}$
$\displaystyle=\mathbb{E}\left[\mathbb{E}\left[Y|T,p(T,X)\right]\frac{f(X)h(T)1(T\in[t,t+r])}{f_{TX}(T,X)}\right]$
$\displaystyle=\mathbb{E}\left[\mathbb{E}\left[Y|T,p(T,X)\right]\frac{h(T)1(T\in[t,t+r])}{p(T,X)}\right]$
$\displaystyle=\mathbb{E}\left[\frac{Y}{p(T,X)}h(T)1(T\in[t,t+r])\right].$

$\square$

##  Appendix B Appendix for Section 3

In this section, we provide conditions and proofs for the size and power
properties of the nonparametric tests. We first introduce some notation and
then several regularity conditions. Let $\|\cdot\|_{\infty}$ be the
$\sup$-norm for a function $f(x)$, i.e.,
$\|f\|_{\infty}=\sup_{x\in\mathcal{X}}|f(x)|$. $C$ denotes a generic constant.
Limits are taken as $n\rightarrow\infty$ unless stated otherwise.

######  Assumption B.1.

(Smoothness):

  1. (i)

The data $\left\\{(Y_{i},T_{i},X_{i})\right\\}$, $i=1,2,...,n$, are $i.i.d.$
random vectors taking values in
$\mathcal{Y}\times\mathcal{T}\times\mathcal{X}\subseteq\mathcal{R}^{2+d_{x}}$
with distribution $\mathbb{P}$.

  2. (ii)

$\mathbb{E}\left[Y^{2+\delta}\right]<\infty$ for some $\delta>0$.

  3. (iii)

The interior support of $(T,X)$, $\mathcal{T}_{0}\times\mathcal{X}_{0}$, is a
compact and convex subset of $\mathbb{R}^{1+d_{x}}$.

  4. (iv)

For $(t,x)\in\mathcal{T}_{0}\times\mathcal{X}_{0}$, $f_{TX}(t,x)$ and $f(x)$
are bounded.

  5. (v)

For some $\Delta\geq 0$, $f_{T|X}(t|x)$ and $\mathbb{E}[Y|T=t,X=x]$ are
$\Delta$-order continuously differentiable with respect to $(t,x)$, with
uniformly bounded derivatives.

Assumption B.1 is standard for kernel estimators to obtain uniform convergence
rates and control biases, e.g., Newey (1994). Condition (iv) implies that the
propensity score $f_{T|X}(t|x)$ is bounded away from zero.

######  Assumption B.2.

(Kernel): The kernel function $k(u):\mathbb{R}\rightarrow\mathbb{R}$ satisfies
the following conditions:
(i) ($r_{k}$-order) $\int k(u)du=1$, $\int u^{l}k(u)du=0$ for $0<l<r_{k}$, and
$\int|u^{r_{k}}k(u)|du<\infty$ for some $r_{k}\geq 2$.
(ii) (bounded support) For some $L<\infty$, $k(u)=0$ for $|u|>L$.
(iii) For an integer $\Delta_{k}$, the derivatives of the kernel up to order
$\Delta_{k}\geq 0$ exist and are Lipschitz.

######  Assumption B.3.

(Step 1 nonparametric tuning parameters): The bandwidth $h\propto n^{-\eta}$
for some positive $\eta$ satisfies $(2r_{k})^{-1}<\eta<(2d)^{-1}$ and
$r_{k}>d$.

######  Assumption B.4.

$\\{U_{i}:~{}1\leq i\leq n\\}$ is a sequence of i.i.d. random variables that
is independent of the sample path of $\\{(Y_{i},X_{i},Z_{i},T_{i}):~{}1\leq
i\leq n\\}$ such that $E[U_{i}]=0$, $E[U_{i}^{2}]=1$, and
$E[|U_{i}|^{2+\delta}]<M$ for some $\delta>0$ and $M>0$.

######  Assumption B.5.

(i) $a_{n}$ is a sequence of non-negative numbers satisfying
$\lim_{n\rightarrow\infty}a_{n}=\infty$ and
$\lim_{n\rightarrow\infty}a_{n}/\sqrt{n}=0$.
(ii) $B_{n}$ is a sequence of non-negative numbers satisfying that $B_{n}$ is
non-decreasing, $\lim_{n\rightarrow\infty}B_{n}=\infty$ and
$\lim_{n\rightarrow\infty}B_{n}/a_{n}=0$.

######  Assumption B.6.

There exists a uniformly consistent estimator $\widehat{E}[Y|T=t,X=x]$ for
$E[Y|T=t,X=x]$ over $t\in[0,1]$ and $x\in\mathcal{X}$, and uniformly
consistent estimators $\hat{\rho}_{1,\ell}(X=x)$ and
$\hat{\rho}_{2,\ell}(X=x)$ for ${\rho}_{1,\ell}(X=x)$ and
${\rho}_{2,\ell}(X=x)$ over $\ell\in\mathcal{L}$ and $x\in\mathcal{X}$.

######  Theorem B.1.

Suppose that Assumption 2.1, and Assumptions B.1-B.6 hold with $\Delta\geq
r_{k}$. Then the following statements are true:

  1. (a)

Under $H_{0}$,
$\lim_{n\rightarrow\infty}P(\widehat{T}_{np}>\hat{c}^{\eta}_{np}(\alpha))\leq\alpha$;

  2. (b)

Under $H_{1}$,
$\lim_{n\rightarrow\infty}P(\widehat{T}_{np}>\hat{c}^{\eta}_{np}(\alpha))=1$.

######  Lemma B.1.

Suppose the Assumptions 2.1, and B.1-B.3 hold. Then uniformly over
$\ell\in\mathcal{L}$,

$\displaystyle\sqrt{n}(\hat{\nu}_{np}(\ell)-\nu(\ell))$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\phi_{\ell,np}(Y_{i},T_{i},X_{i})+o_{p}(1),$
(B.1)

where

$\displaystyle\phi_{\ell,np}(Y_{i},T_{i},X_{i})=\phi_{2,\ell,np}(Y_{i},T_{i},X_{i})-\phi_{1,\ell,np}(Y_{i},T_{i},X_{i}),~{}\text{and
for $j=1$ and 2, }$
$\displaystyle\phi_{j,\ell,np}(Y,T,X)=\frac{1(T\in[t_{j},t_{j}+q^{-1}])}{p(T,X)}(Y-E[Y|T,X])+\rho_{j,\ell}(X)-\nu_{j}(\ell),$
$\displaystyle\rho_{j,\ell}(X)=E\left[\frac{Y1(T\in[t_{j},t_{j}+q^{-1}])}{p(T,X)}\Big{|}X\right].$

In addition,
$\sqrt{n}(\hat{\nu}_{np}(\cdot)-\nu(\cdot))\Rightarrow\Phi_{h_{np}}(\cdot)$
where $\Rightarrow$ denotes weak convergence and $\Phi_{h_{np}}$ denotes a
Gaussian process with variance-covariance kernel equal to $h_{np}$.
$h_{np}(\ell_{1},\ell_{2})=E[\phi_{\ell_{1},np}\cdot\phi_{\ell_{2},np}].$

Proof of Lemma B.1:
To show the first part, it is sufficient to demonstrate that uniformly over
$(t,r)$ such that $[t,t+r]\subseteq[0,1]$,

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\frac{Y_{i}1(T_{i}\in[t,t+r])}{\hat{p}(T_{i},X_{i})}-E\left[\frac{Y1(T\in[t,t+r])}{p(T,X)}\right]$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\frac{Y_{i}1(T_{i}\in[t,t+r])}{p(T_{i},X_{i})}-E[Y|T=T_{i},X=X_{i}]\frac{1(T_{i}\in[t,t+r])}{p(T_{i},X_{i})}$
$\displaystyle\ \ \
+E\left[\frac{Y1(T\in[t,t+r])}{p(T,X)}\Big{|}X=X_{i}\right]-E\left[\frac{Y1(T\in[t,t+r])}{p(T,X)}\right]+o_{p}(n^{-1/2}).$

We first derive the linear representation of the kernel estimator
$\hat{p}(T_{i},X_{i})$ of the generalized propensity score
$p(T_{i},X_{i})=f_{T|X}(T_{i},X_{i})$. Under Assumptions B.1 and B.2, Lemma
B.3 in Newey (1994) implies
$\sup_{(t,x)\in\mathcal{T}_{0}\times\mathcal{X}_{0}}\Big{|}\hat{f}_{T|X}(t|x)-f_{T|X}(t|x)\Big{|}=O_{p}\left(\big{(}\log
n\big{/}(nh^{d_{x}+1})\big{)}^{1/2}+h^{r_{k}}\right)$. Then, we obtain

$\displaystyle\hat{p}(t,x)-p(t,x)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\big{(}K_{h}(T_{i}-t)-f_{T|X}(t|x)\big{)}K_{h}(X_{i}-x)/f(x)+R_{n}^{v}(t,x),$
$\displaystyle\mbox{where}\ \|R_{n}^{v}\|_{\infty}$
$\displaystyle=O_{p}\Big{(}\|\hat{f}-f\|_{\infty}\|\hat{f}_{T|X}-f_{T|X}\|_{\infty}\Big{)}$
$\displaystyle=O_{p}\left(\left(\sqrt{\frac{\log
n}{nh^{d_{x}}}}+h^{r_{k}}\right)\left(\sqrt{\frac{\log
n}{nh^{d_{x}+1}}}+h^{r_{k}}\right)\right).$

Next, we derive the linear representation of the proposed estimator. Denote
$g_{i}=g(T_{i},t)\equiv 1(T_{i}\in(-\infty,t])$. By plugging in the above
result for $\hat{p}(T_{i},X_{i})$, the final estimator corresponds to

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\frac{Y_{i}g_{i}}{\hat{p}(T_{i},X_{i})}=$
$\displaystyle\ \frac{1}{n}\sum_{i=1}^{n}\frac{Y_{i}g_{i}}{p(T_{i},X_{i})}$
(B.2)
$\displaystyle-\frac{1}{n}\sum_{i=1}^{n}\frac{Y_{i}g_{i}}{p(T_{i},X_{i})^{2}}\frac{1}{n}\sum_{j=1}^{n}\big{(}K_{h}(T_{j}-T_{i})-p(T_{i},X_{i})\big{)}\frac{K_{h}(X_{j}-X_{i})}{f(X_{i})}$
$\displaystyle+o_{p}(n^{-1/2}),$

where the smaller order terms are controlled by Assumption B.3. The following
proof focuses on the second term from above that is a U-process indexed by
$\lambda\equiv(t,h)\in\Lambda\equiv\mathcal{T}\times\mathcal{R}_{+}$, denoted
by $U_{n}q\equiv{n\choose 2}^{-1}\sum_{i=1}^{n}\sum_{j\neq
i}q(Z_{i},Z_{j};\lambda)$, where
$q(Z_{i},Z_{j};\lambda)\equiv\frac{Y_{i}g_{i}}{p(T_{i},X_{i})^{2}}\big{(}K_{h}(T_{j}-T_{i})-p(T_{i},X_{i})\big{)}\frac{K_{h}(X_{j}-X_{i})}{f(X_{i})}$.

We apply the $U$-process theorems in Sherman (1994). Let
$(Z_{i},Z_{j})\in\mathcal{Z}^{2}\equiv\mathcal{Z}\times\mathcal{Z}$ from the
product measure $\mathbb{P}^{2}\equiv\mathbb{P}\times\mathbb{P}$. Let
$\mathcal{F}\equiv\\{q:(Z_{i},Z_{j})\mapsto
h^{d}q(Z_{i},Z_{j};\lambda):\lambda\in\Lambda\\}$ be a class of real-valued on
$\mathcal{Z}^{2}$. The collection $\big{\\{}U_{n}q:q\in\mathcal{F}\big{\\}}$
is a $U$-process of order two indexed by $\mathcal{F}$ where for each
$q\in\mathcal{F}$,

$\displaystyle U_{n}q$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[q(Z_{i},Z;\lambda)|Z_{i}]+\frac{1}{n}\sum_{j=1}^{n}\mathbb{E}[q(Z,Z_{j};\lambda)|Z_{j}]-\mathbb{P}^{2}[q(Z_{i},Z_{j};\lambda)]+U^{2}_{n},$
$\displaystyle\mbox{where}\ \ U^{2}_{n}$ $\displaystyle\equiv{n\choose
2}^{-1}\sum_{i=1}^{n}\sum_{j\neq i}r_{n}(Z_{i},Z_{j};\lambda)\ \ \mbox{and}$
$\displaystyle r_{n}(Z_{i},Z_{j};\lambda)$ $\displaystyle\equiv
q(Z_{i},Z_{j};\lambda)-\mathbb{E}[q(Z_{i},Z;\lambda)|Z_{i}]-\mathbb{E}[q(Z,Z_{j};\lambda)|Z_{j}]+\mathbb{P}^{2}[q(Z_{i},Z_{j};\lambda)]$

is $\mathbb{P}$-degenerate on $\mathcal{Z}^{2}$. The dominant term in the
$U$-statistic $U_{n}q$ is

$\displaystyle\mathbb{E}[q(Z,Z_{j};\lambda)|Z_{j}]$
$\displaystyle=\mathbb{E}\left[\frac{Yg}{p(T,X)^{2}}\big{(}K_{h}(T_{j}-T)-p(T,X)\big{)}\frac{K_{h}(X_{j}-X)}{f(X)}\right]$
$\displaystyle=\mathbb{E}\left[Y|T=T_{j},X=X_{j}\right]\frac{1(T_{j}\leq
t)}{p(T_{j},X_{j})}-\mathbb{E}\left[\frac{Yg}{p(T,X)}\Big{|}X=X_{j}\right]$
(B.3) $\displaystyle\ \ \ +O_{p}(h^{r_{k}}).$

(B.2) and (B.3) are the dominant terms in the influence function.

The smaller order terms are
$\mathbb{E}[q(Z,Z_{j};\lambda)|Z_{i}]=O_{p}(h^{r_{k}})$,
$\mathbb{P}^{2}[q(Z_{i},Z_{j};\lambda)]=O_{p}(h^{r_{k}})$, and $U_{n}^{2}$.
For $U_{n}^{2}$, we show $\sup_{\mathcal{F}}|U_{n}^{2}|=O_{p}(n^{-1}h^{-d})$
by Corollary 4 in Sherman (1994) in the following. First, $\\{T\mapsto 1(T\leq
t):t\in\mathcal{T}\\}$ and $\\{X\mapsto K((X-x)/h):h>0\\}$ are Euclidean by
Example 2.10 in Pakes and Pollard (1989). Then Lemma 2.14 in Pakes and Pollard
(1989) implies $\mathcal{F}$ is Euclidean. Hence $\mathcal{F}$ is Euclidean
for the envelope $F(Z_{i},Z_{j})=C\cdot\phi(Z_{i})$ satisfying
$\mathbb{P}^{2}F^{2}=C\cdot\mathbb{E}\left[Y^{2}g\big{/}\left(p(T,X)^{4}f(X)^{2}\right)\right]<\infty$
by Assumption B.1. By Lemma 6 and Corollary 4 in Sherman (1994),
$\sup_{\mathcal{F}}|U_{n}^{2}|=O_{p}(n^{-1}h^{-d})$. Assumption B.3 ensures
that $\sup_{\mathcal{F}}|U_{n}^{2}|=O_{p}(n^{-1}h^{-d})=o_{p}(n^{-1/2})$. This
is sufficient to show that uniformly over $\ell\in\mathcal{L}$,

$\displaystyle\sqrt{n}(\hat{\nu}_{np}(\ell)-\nu(\ell))=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\phi_{\ell,np}(Y_{i},T_{i},X_{i})+o_{p}(1)$

Let $\bar{\phi}_{np}$ be the envelope function of
$\\{\phi_{\ell,np}(Y,T,X):\ell\in\mathcal{L}\\}$ and note that
$E[\bar{\phi}^{2+\delta}_{np}]<\infty$. Then the second part follows from the
fact that $\\{\phi_{\ell,np}(Y,T,X):\ell\in\mathcal{L}\\}$ forms a VC type
class of functions. $\square$

######  Lemma B.2.

Suppose the Assumptions 2.1 and B.1-B.6 hold. Then,
$\sup_{\ell\in\mathcal{L}}|\hat{\sigma}_{\nu,np}(\ell)-{\sigma}_{\nu,np}(\ell)|\stackrel{{\scriptstyle
p}}{{\rightarrow}}0$ where
${\sigma}^{2}_{\nu,np}(\ell)=E[\phi^{2}_{\ell,np}]$, and
$\widehat{\Phi}^{u}_{\nu,np}\stackrel{{\scriptstyle
p}}{{\Rightarrow}}\Phi_{h_{np}}$.

Proof of Lemma B.2:
The fact that $\\{\phi_{\ell,np}:\ell\in\mathcal{L}\\}$ is a VC type class of
functions implies that $\\{\phi^{2}_{\ell,np}:\ell\in\mathcal{L}\\}$ is also a
VC type. In addition, given that $E[\bar{\phi}^{2+\delta}_{np}]<\infty$, we
have by the uniform weak law of large numbers that
$\sup_{\ell\in\mathcal{L}}|\tilde{\sigma}^{2}_{\nu,np}(\ell)-{\sigma}^{2}_{\nu,np}(\ell)|\stackrel{{\scriptstyle
p}}{{\rightarrow}}0$, where
$\tilde{\sigma}^{2}_{\nu,np}(\ell)=n^{-1}\sum_{i=1}^{n}\phi^{2}_{\ell,np}(Y_{i},T_{i},X_{i})$.
By Assumption B.6, we have that
$\sup_{\ell\in\mathcal{L}}|\tilde{\sigma}^{2}_{\nu,np}(\ell)-\hat{\sigma}^{2}_{\nu,np}(\ell)|\stackrel{{\scriptstyle
p}}{{\rightarrow}}0$. Then the first part follows. The proof of the second
part follows from the standard arguments for the multiplier bootstrap such as
Lemma 4.1 of Hsu (2017) and is omitted for the sake of brevity. $\square$

Proof of Theorem B.1:
The proof of Theorem B.1 follows from the same arguments as Theorem 5.1 of Hsu
(2017) once Lemmas B.1 and B.2 are established and is omitted for the sake of
brevity. $\square$

##  Appendix C Appendix for Section 4

######  Theorem C.1.

Suppose that Assumptions 2.1, 4.1, B.1(i), (ii), (iii), B.4, and B.5 are
satisfied. Then the following statements hold:

  1. (a)

Under $H_{0}$,
$\lim_{n\rightarrow\infty}P(\widehat{T}_{pa}>\hat{c}^{\eta}_{pa}(\alpha))\leq\alpha$;

  2. (b)

Under $H_{1}$,
$\lim_{n\rightarrow\infty}P(\widehat{T}_{pa}>\hat{c}^{\eta}_{pa}(\alpha))=1$.

Proof of Theorem C.1:
The proof is similar to that of Theorem B.1 once we establish the limiting
process result and simulated process result. Note that the influence function
result in (4.2) can be obtained by the delta method and the limiting process
result then follows from the arguments of the second part of Lemma B.1. The
uniform consistency result of $\hat{\sigma}_{\nu,pa}$ and the simulated
process result of $\widehat{\Phi}^{u}_{\nu,pa}$ follow from the same arguments
underlying Lemma B.2. Then Theorem C.1 follows. $\square$

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Table 6: Rejection probabilities of our test ($h=a\cdot n^{-1/5}$) DGP | n | $a=0.8$ | $a=1$ | $a=1.2$ | pa
---|---|---|---|---|---
1 | 200 | 0.015 | 0.017 | 0.012 | 0.003
1 | 400 | 0.014 | 0.012 | 0.011 | 0.000
1 | 800 | 0.012 | 0.010 | 0.008 | 0.000
2 | 200 | 0.689 | 0.908 | 0.962 | 1.000
2 | 400 | 0.974 | 0.998 | 1.000 | 1.000
2 | 800 | 1.000 | 1.000 | 1.000 | 1.000
3 | 200 | 0.037 | 0.055 | 0.073 | 0.095
3 | 400 | 0.041 | 0.059 | 0.076 | 0.100
3 | 800 | 0.068 | 0.082 | 0.090 | 0.100
Table 7: Rejection probabilities of our test with trimming ($h=a\cdot n^{-1/5}$) DGP | n | $a=0.8$ | $a=1$ | $a=1.2$ | pa
---|---|---|---|---|---
1 | 200 | 0.006 | 0.008 | 0.009 | 0.003
1 | 400 | 0.009 | 0.008 | 0.010 | 0.000
1 | 800 | 0.005 | 0.007 | 0.008 | 0.000
2 | 200 | 0.815 | 0.958 | 0.980 | 1.000
2 | 400 | 0.996 | 0.998 | 1.000 | 1.000
2 | 800 | 1.000 | 1.000 | 1.000 | 1.000
3 | 200 | 0.042 | 0.053 | 0.074 | 0.095
3 | 400 | 0.037 | 0.063 | 0.077 | 0.100
3 | 800 | 0.064 | 0.083 | 0.094 | 0.100
Table 8: Rejection probabilities of our test for different N |  | nonparametric |  | parametric
---|---|---|---|---
DGP | n | N=33 | N=40 | N=50 | N=66 |  | N=33 | N=40 | N=50 | N=66
1 | 200 | 0.019 | 0.017 | 0.014 | 0.016 |  | 0.003 | 0.003 | 0.002 | 0.005
1 | 400 | 0.012 | 0.012 | 0.009 | 0.009 |  | 0.000 | 0.000 | 0.000 | 0.000
1 | 800 | 0.010 | 0.010 | 0.009 | 0.009 |  | 0.000 | 0.000 | 0.000 | 0.000
2 | 200 | 0.927 | 0.908 | 0.886 | 0.841 |  | 1.000 | 1.000 | 1.000 | 1.000
2 | 400 | 0.998 | 0.998 | 0.997 | 0.996 |  | 1.000 | 1.000 | 1.000 | 1.000
2 | 800 | 1.000 | 1.000 | 1.000 | 1.000 |  | 1.000 | 1.000 | 1.000 | 1.000
3 | 200 | 0.055 | 0.055 | 0.057 | 0.058 |  | 0.098 | 0.095 | 0.095 | 0.091
3 | 400 | 0.058 | 0.059 | 0.059 | 0.058 |  | 0.099 | 0.100 | 0.099 | 0.097
3 | 800 | 0.082 | 0.082 | 0.082 | 0.082 |  | 0.100 | 0.100 | 0.100 | 0.099
Table 9: Rejection probabilities of our test with trimming for different N |  | nonparametric |  | parametric
---|---|---|---|---
DGP | n | N=33 | N=40 | N=50 | N=66 |  | N=33 | N=40 | N=50 | N=66
1 | 200 | 0.009 | 0.008 | 0.006 | 0.007 |  | 0.003 | 0.003 | 0.002 | 0.005
1 | 400 | 0.009 | 0.008 | 0.007 | 0.008 |  | 0.000 | 0.000 | 0.000 | 0.000
1 | 800 | 0.007 | 0.007 | 0.006 | 0.006 |  | 0.000 | 0.000 | 0.000 | 0.000
2 | 200 | 0.965 | 0.958 | 0.938 | 0.908 |  | 1.000 | 1.000 | 1.000 | 1.000
2 | 400 | 0.999 | 0.998 | 0.998 | 0.997 |  | 1.000 | 1.000 | 1.000 | 1.000
2 | 800 | 1.000 | 1.000 | 1.000 | 1.000 |  | 1.000 | 1.000 | 1.000 | 1.000
3 | 200 | 0.053 | 0.053 | 0.055 | 0.053 |  | 0.098 | 0.095 | 0.095 | 0.091
3 | 400 | 0.063 | 0.063 | 0.062 | 0.061 |  | 0.099 | 0.100 | 0.099 | 0.097
3 | 800 | 0.083 | 0.083 | 0.083 | 0.083 |  | 0.100 | 0.100 | 0.100 | 0.099