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Testing Partial Instrument Monotonicity
\title{Testing Partial Instrument Monotonicity}
\author{ Hongyi Jiang \\China Center for Economic Research\\National School of Development\\ Peking University\\ [email removed]\\
\and Zhenting Sun\thanks{This work was supported by the National Natural Science Foundation of China [Grant Number 72103004].}\\China Center for Economic Research\\National School of Development\\ Peking University\\
[email removed]}
\maketitle
\begin{abstract}
When multi-dimensional instruments are used to identify and estimate causal effects, the monotonicity condition may not hold due to heterogeneity in the population. Under a partial monotonicity condition, which only requires the monotonicity to hold for each instrument separately holding all the other instruments fixed, the 2SLS estimand can still be a positively weighted average of LATEs. In this paper, we provide a simple nonparametric test for partial instrument monotonicity. We demonstrate the good finite sample properties of the test through Monte Carlo simulations. We then apply the test to monetary incentives and distance from results centers as instruments for the knowledge of HIV status.
\end{abstract}
\textbf{Keywords:} Partial monotonicity, instrument validity, nonparametric test
\newpage
\section{Introduction}
Exclusion, random assignment, and monotonicity are three fundamental conditions for instrument variables (IVs) to be valid in the identification and estimation of causal effects. When the instrument variable is multi-dimensional, \citet{mogstad2021causal} show that the monotonicity condition only holds if choice behavior is effectively homogeneous. However, such homogeneity may break down in many empirical applications.
\citet{mogstad2021causal} then consider a weaker version of monotonicity, partial monotonicity, which permits heterogeneous
choice behavior. Under partial monotonicity, multi-dimensional instruments (multiple instruments) can be used for causal effects estimation even if the full monotonicity condition fails.
In the literature, the IV validity conditions can be tested by the methods of \citet{huber2015testing}, \citet{kitagawa2015test}, \citet{mourifie2016testing}, and \citet{sun2018ivvalidity}. In this paper, we extend the frameworks of \citet{kitagawa2015test} and \citet{sun2018ivvalidity} to the partial monotonicity of \citet{mogstad2021causal}. We show that the proposed test based on \citet{kitagawa2015test} and \citet{sun2018ivvalidity} performs well in practice. In the appendix, Monte Carlo studies demonstrate the finite sample properties of the test. We then apply the test to an empirical example from \citet{thornton2008demand} discussed in \citet{mogstad2021causal}.
\section{Setup and Test Formulation}
\citet{mogstad2021causal} introduce the concept of partial instrument monotonicity for multi-dimensional IVs. We follow \citet{mogstad2021causal} and mainly focus on the multivalued ordered treatment case.\footnote{It would be easy to extend the test to unordered treatments.} Let $( \Omega, \mathcal{A}, \mathbb{P} )$ be a probability space on which all the random elements are well defined. Suppose that the outcome variable $Y\in\mathbb{R}$, the treatment $D\in\mathcal{D}=\left\{
d_{1},\ldots,d_J\right\}$ for some $J\ge 2$, and the instrument $\boldsymbol{Z}\in\mathcal{Z}=\{\boldsymbol{z}_1,\ldots,\boldsymbol{z}_K\}$ for some $K\ge 2$, where every value $\boldsymbol{z}\in\mathcal{Z}$ is a vector $\boldsymbol{z}=(z_1,\ldots,z_L)$ for some $L\ge 2$. That is, $\boldsymbol{Z}$ is an $L$-dimensional instrument with $\boldsymbol{Z}=(Z_1,\ldots,Z_L)$, where $Z_l$ is a scalar variable for every $l$. Suppose the $l$th dimension of $\boldsymbol{Z}$ has $k_l$ possible values, i.e., $Z_l\in\{z_l^1,\ldots,z_l^{k_l}\}$. Following the rectangular support assumption of \citet{mogstad2021causal}, we suppose that $\mathcal{Z}=\mathrm{supp}(Z_1)\times\cdots\times\mathrm{supp}(Z_L)$, which implies $K=\prod_{l=1}^L k_l$. For every $\boldsymbol{z}=(z_1,\ldots,z_L)$, we define the vector $z_{-l}=(z_1,\ldots,z_{l-1},z_{l+1},\ldots,z_L)$, and $\boldsymbol{z}$ may be written as $\boldsymbol{z}=(z_{l},z_{-l})$.
Suppose that $Y_{d\boldsymbol{z}}$
for $d\in\mathcal{D}$ and $\boldsymbol{z}\in\mathcal{Z}$, and $D_{\boldsymbol{z}}$ for $\boldsymbol{z}\in
\mathcal{Z}$ are the potential random variables. The following assumption formalizes the IV validity assumption for multi-dimensional
instrument $\boldsymbol{Z}$ proposed by \citet{mogstad2021causal}.
\begin{assumption}
\label{ass.IV validity for multivalued Z}Partial IV validity for multi-dimensional $\boldsymbol{Z}$:
\begin{enumerate}[label=(\roman*)]
\item Instrument Exclusion: Almost surely, $Y_{d\boldsymbol{z}_{1}}=\cdots=Y_{d\boldsymbol{z}_{K}}$ for all $d\in\mathcal{D}$.
\item Random Assignment: The variable $\boldsymbol{Z}$ is jointly independent of $(
\tilde{Y},\tilde{D}) $, where
\begin{align*}
\tilde{Y} =\left( Y_{d_{1}\boldsymbol{z}_{1}},\ldots,Y_{d_{1}\boldsymbol{z}_{K}},\ldots, Y_{d_{J}\boldsymbol{z}_{1}
},\ldots,Y_{d_{J}\boldsymbol{z}_{K}}\right) \text{ and }\tilde{D} =\left( D_{\boldsymbol{z}_{1}},\ldots,D_{\boldsymbol{z}_{K}}\right) .
\end{align*}
\item Partial Instrument Monotonicity: For every $l\in\{1,\ldots,L\}$ and every given $z_{-l}$, the potential treatment response variables satisfy $D_{(z_{l}^{k+1},z_{-l})}\geq D_{(z_{l}^{k},z_{-l})}$ almost surely for all
$k\in \{1,\ldots, k_{l}-1\}$.
\end{enumerate}
\end{assumption}
\begin{remark}
Assumption \ref{ass.IV validity for multivalued Z}(iii) is a simplified version of Assumption PM (partial monotonicity) in \citet{mogstad2021causal}. \citet{mogstad2021causal} assume no direction in Assumption PM: For every $l\in\{1,\ldots,L\}$ and every given $z_{-l}$, the potential treatment response variables satisfy $D_{(z'_{l},z_{-l})}\geq D_{(z_{l},z_{-l})}$ almost surely or $D_{(z'_{l},z_{-l})}\leq D_{(z_{l},z_{-l})}$ almost surely for all
$z_{l},z'_{l}\in \{z_l^1,\ldots,z_l^{k_l}\}$.
In practice, we may usually assume a direction in Assumption PM: For every $l$ and every $z_{-l}$, there is a sequence $\{z_{l}^{1}(z_{-l}),\ldots,z_l^{k_l}(z_{-l})\}$ such that
\begin{align}\label{eq.generalized monotonicity}
D_{(z_{l}^{k+1}(z_{-l}),z_{-l})}\geq D_{(z_{l}^{k}(z_{-l}),z_{-l})}
\end{align}
almost surely for all
$k\in \{1,\ldots, k_{l}-1\}$.
For notational simplicity, we construct the test based on Assumption \ref{ass.IV validity for multivalued Z}(iii) which assumes that $z_l^k(z_{-l})=z_l^k$ for all $z_{-l}$. It would be straightforward to extend the test to \eqref{eq.generalized monotonicity}. We may also extend the test to Assumption PM in \citet{mogstad2021causal} following an idea similar to that in \citet[Section D]{sun2018ivvalidity} if there is no prior information about the direction in Assumption PM.
\end{remark}
Suppose that $D$ has maximum value $d_{\max}$ and minimum value $d_{\min}$. We provide a testable implication for Assumption \ref{ass.IV validity for multivalued Z} in the following lemma.
\begin{lemma}
\label{lemma.testable implication}
For all $1 \leq l \leq L$, all $1\le k \le k_{l}-1$, all possible values $z_{-l}$, all Borel sets $B$, and all $C=(-\infty,c]$ with $c\in\mathbb{R}$, it follows that
\begin{align}\label{eq.testable implication multivalue}
&\mathbb{P}\left( Y\in B,D=d_{\max}|\boldsymbol{Z}=(z_{l}^{k},z_{-l})\right) \leq \mathbb{P}\left( Y\in B,D=d_{\max}|\boldsymbol{Z}=(z_{l}^{k+1},z_{-l})\right)\nonumber\\
&\text{and }\mathbb{P}\left( Y\in B,D=d_{\min}|\boldsymbol{Z}=(z_{l}^{k},z_{-l})\right) \geq \mathbb{P}\left( Y\in B,D=d_{\min}|\boldsymbol{Z}=(z_{l}^{k+1},z_{-l})\right); \\
&\mathbb{P}\left( D\in C|\boldsymbol{Z}=(z_{l}^{k},z_{-l})\right) \geq \mathbb{P}\left( D\in C|\boldsymbol{Z}=(z_{l}^{k+1},z_{-l})\right). \label{eq.fosd multi}
\end{align}
\end{lemma}
Lemma \ref{lemma.testable implication} can be proved analogously to Lemma 2.1 of \citet{sun2018ivvalidity}.
In the following, we extend the tests of \citet{kitagawa2015test} and \citet{sun2018ivvalidity} to partial IV validity.
Without loss of generality, we assume that $d_{\min}=d_1\le\cdots\le d_J=d_{\max}$ with $d_{\min}=0$ and $d_{\max}=1$. Then the inequalities in \eqref{eq.testable implication multivalue} and \eqref{eq.fosd multi} are equivalent to
\begin{align}\label{eq.testable implication inequalities multi}
&(-1)^{d}\cdot\left\{\mathbb{P}\left( Y\in B,D=d|\boldsymbol{Z}=(z_{l}^{k+1},z_{-l})\right)-\mathbb{P}\left( Y\in B,D=d|\boldsymbol{Z}=(z_{l}^{k},z_{-l})\right)\right\} \leq 0 \notag\\
&\text{ and } \mathbb{P}\left( D\in C|\boldsymbol{Z}=(z_{l}^{k+1},z_{-l})\right)-\mathbb{P}\left( D\in C|\boldsymbol{Z}=(z_{l}^{k},z_{-l})\right)\le 0
\end{align}
for all $1 \leq l \leq L$, all $1\le k\le k_{l}-1$, all possible values $z_{-l}$, all closed intervals $B$ in $\mathbb{R}$, each $d\in\{0,1\}$, and all $C=(-\infty,c]$ with $c\in\mathbb{R}$. Following {Lemma B.7} of \citet{kitagawa2015test}, here we only consider all closed intervals $B$ instead of all Borel sets $B$ when constructing the test.
By definition, for all $B,C\in\mathcal{B}_{\mathbb{R}}$ and all possible values $\boldsymbol{z}$,
$
\mathbb{P}\left( Y\in B,D\in C|\boldsymbol{Z}=\boldsymbol{z}\right)={\mathbb{P}\left( Y\in B,D\in C,\boldsymbol{Z}=\boldsymbol{z}\right)
}/{\mathbb{P}\left( \boldsymbol{Z}=\boldsymbol{z}\right) }.
$
We then define function spaces
\begin{align}{\label{def.function spaces}}
&\mathcal{G}^l_{z_{-l}}=\left\{ \left( 1_{\mathbb{R}\times\mathbb{R} \times\{ (z_{l}^{k},z_{-l})\} },1_{\mathbb{R}\times\mathbb{R} \times\{ (z_{l}^{k+1},z_{-l})\} }\right)
:k=1,\ldots,k_l-1\right\}\text{ for every }l \text{ and every }z_{-l}, \notag \\
&{\mathcal{G}}=\cup_{l}\cup_{z_{-l}}\mathcal{G}^l_{z_{-l}},\notag\\
&\mathcal{H}_{1}=\left\{ \left( -1\right) ^{d}\cdot1_{B\times\left\{
d\right\} \times\mathbb{R}^L}:B\text{ is a closed interval in }\mathbb{R},
d\in\{0,1\}\right\},\notag\\
&\bar{\mathcal{H}}_1=\left\{ \left( -1\right) ^{d}\cdot1_{B\times\left\{
d\right\}\times\mathbb{R}^L }:B\text{ is a closed, open, or half-closed interval in }
\mathbb{R}
,d\in\left\{ 0,1\right\} \right\},\notag\\
&\mathcal{H}_{2}=\left\{ 1_{\mathbb{R}\times C \times\mathbb{R}^L}:C=(-\infty,c],c\in\mathbb{R}\right\},\notag\\
& \bar{\mathcal{H}}_2=\left\{ 1_{\mathbb{R}\times C \times\mathbb{R}^L }: C=(-\infty,c] \text{ or } C=(-\infty,c),c\in\mathbb{R} \right\},\notag\\
&\mathcal{H}=\mathcal{H}_{1}\cup\mathcal{H}_{2}, \text{ and } \bar{\mathcal{H}}=\bar{\mathcal{H}}_1\cup\bar{\mathcal{H}}_2.
\end{align}
Let $\{\left( Y_i,D_i,\boldsymbol{Z}_i \right)\}_{i=1}^{n} $ be an i.i.d.\ sample, which is distributed according to some probability distribution $P$, that is, the measure $P(G)=\mathbb{P}((Y_i,D_i,\boldsymbol{Z}_i)\in G)$ for all $G\in\mathcal{B}_{\mathbb{R}^{2+L}}$.
For every measurable function $v$, by an abuse of notation, we define
\begin{align}\label{eq.Q map}
P\left( v\right) =\int v\,\mathrm{d}P.
\end{align}
For every $\left( h,g\right) \in {\bar{\mathcal{H}}\times\mathcal{G}}$ with $g=(g_{1},g_{2})$, define
\begin{align}\label{eq.phi_Q}
\phi\left( h,g\right) =\frac{P\left( h\cdot g_{2}\right)
}{P\left( g_{2}\right) }-\frac{P\left( h\cdot g_{1}\right) }{P\left(
g_{1}\right) }.
\end{align}
The null hypothesis equivalent to \eqref{eq.testable implication inequalities multi} is
\begin{align}\label{eq.null order}
\mathrm H_0: \sup_{\left( h,g\right) \in
{{\mathcal{H}}\times\mathcal{ G}}}\phi\left( h,g\right)\le 0,
\end{align}
and the alternative hypothesis is
\begin{align*}
\mathrm H_1:\sup_{\left( h,g\right) \in
{{\mathcal{H}}\times\mathcal{ G}}}\phi\left( h,g\right)> 0.
\end{align*}
We then define the sample analogue of $\phi$ by
\begin{align*}
\hat{\phi}\left( h,g\right) =\frac{\hat{P}(h\cdot g_{2}) }{\hat{P}(g_{2})}-\frac{\hat{P}(h\cdot g_{1}) }{\hat{P}(g_{1}) },
\end{align*}
where $\hat{P}$ denotes the empirical probability measure of $P$ such that for every measurable function $v$,
\begin{align}\label{eq.defPn order}
\hat{P}\left( v\right) =\frac{1}{n}\sum_{i=1}^{n}v\left( Y_{i},D_{i},\boldsymbol{Z}_{i}\right),
\end{align}
and $\{( Y_{i},D_{i},\boldsymbol{Z}_{i})\}_{i=1}^n$ is the i.i.d.\ sample distributed according to $P$. Define
\begin{align}\label{eq.estimated stat variance multi order}
\hat{\sigma}^{2}\left( h,g\right) =\frac{T_n}{n}\cdot\left\{\frac{ \hat{P}\left( h^2\cdot g_{2}\right) }{\hat{P}^{2}\left(
g_{2}\right) } -\frac{ \hat{P}^2\left( h\cdot g_{2}\right) }{\hat{P}^{3}\left(
g_{2}\right) }
+\frac{ \hat{P}\left( h^2\cdot g_{1}\right) }{\hat{P}^{2}\left( g_{1}\right) }
-\frac{ \hat{P}^2\left( h\cdot g_{1}\right) }{\hat{P}^{3}\left( g_{1}\right) }\right\}
\end{align}
for all $(h,g)\in\bar{\mathcal{H}}\times\mathcal{G}$ with $g=(g_1,g_2)$, where $T_n=n\cdot\prod_{k=1}^{K}\hat{P}\left(1_{\mathbb{R}\times\mathbb{R}\times\{\boldsymbol{z}_k\}}\right)$.
We then specify a closed set $\Xi\subset(0,1]$ such that $\Xi$ contains all the values of $\xi$ used for constructing the test statistic in the following.
We also specify a positive measure $\nu$ on $\Xi$ that satisfies the following assumption.
\begin{assumption}\citep{sun2018ivvalidity}\label{ass.nu order}
The measure $\nu$ satisfies that $0<\nu(\Xi)<\infty$ and ${S}_n\in L^{1}(\nu)$ for all $\omega\in\Omega$ and all $n$ with
\begin{align*}
{S}_n(\xi)= \sup_{(h,g)\in{\mathcal{H}}\times\mathcal{G}} \frac{\hat{\phi}(h,g)}{\max\{\xi,\hat{\sigma}(h,g)\}}.
\end{align*}
\end{assumption}
We follow \citet{kitagawa2015test} and \citet{sun2018ivvalidity} and construct the test statistic as
\begin{equation}\label{eq.test stat expansion order}
TS_n=\int_{\Xi}\sup_{(h,g)\in{\mathcal{H}}\times\mathcal{G}} \frac{\sqrt{T_n}\hat{\phi}(h,g)}{\max\{\xi,\hat{\sigma}(h,g)\}}\,\mathrm{d}\nu(\xi).
\end{equation}
We may set the measure $\nu$ to be a Dirac measure centered at some fixed $\xi\in\Xi$, and this is equivalent to using a particular value of $\xi$ to construct the statistic.
We construct a random set $\widehat{\Psi_{{\mathcal{H}}\times\mathcal{G}}}$ by
\begin{align}\label{eq.Psi_hat order}
\widehat{\Psi_{{\mathcal{H}}\times\mathcal{G}}}=\left\{ \left( h,g\right)
\in{\mathcal{H}}\times\mathcal{G}:\sqrt{T_n}\left\vert \frac{\hat{\phi}(h,g)}{\max\{\xi_0,\hat{\sigma}(h,g)\}}\right\vert \leq\tau
_{n}\right\}
\end{align}
with $\tau_{n}\rightarrow\infty$ and $\tau_{n}/\sqrt{n}\rightarrow0$ as
$n\rightarrow\infty$, where $\xi_0$ is a fixed small positive number. In practice, we suggest setting $\xi_0=10^{-10}$ which is used in the simulations and the application in the paper.
This random set is an estimator for some contact set similar to those in \citet{Beare2015improved} and \citet{Beare2017improved} in different contexts.\footnote{See \citet{linton2010improved} and \citet{lee2013testing} for more discussions on estimation of contact sets.} Algorithm \ref{proc:test} illustrates the test procedure.
\begin{algorithm}
\caption{Test Procedure}\label{proc:test}
\begin{enumerate}[label=(\arabic*)]
\item Draw a bootstrap sample $\{ ( \hat{Y}_{i},\hat{D}_{i},\hat{\boldsymbol{Z}}_{i}) \} _{i=1}^{n}$ independently with replacement from the
sample $\left\{ \left( Y_{i},D_{i},\boldsymbol{Z}_{i}\right) \right\} _{i=1}^{n}$.
\item Let $\hat{P}^{*}\left( v\right) =n^{-1}\sum_{i=1}^{n}v( \hat{Y}_{i},\hat{D}_{i},\hat{\boldsymbol{Z}}_{i}) $ for all measurable function $v$, and $T_n^{*}=n\cdot\prod_{k=1}^{K}\hat{P}^{*}(1_{\mathbb{R}\times\mathbb{R}\times\{\boldsymbol{z}_k\}})$. Construct the bootstrap version of $\hat{\phi}$ by
\begin{align*}\label{eq.phi hat star order}
\hat{\phi}^{*}\left( h,g\right) =\frac{\hat{P}^{*}\left( h\cdot
g_{2}\right) }{\hat{P}^{*}\left( g_{2}\right) }-\frac{\hat{P}^{*}\left(
h\cdot g_{1}\right) }{\hat{P}^{*}\left( g_{1}\right) },
\end{align*}
and the bootstrap version of $\hat{\sigma}$ by
\begin{align*}
\hat{\sigma}^{*}\left( h,g\right) =\sqrt{\frac{T_n^{*}}{n}}\cdot\sqrt{\frac{ \hat{P}^{*}\left( h^2\cdot g_{2}\right) }{\hat{P}^{*}\left(g_{2}\right)^2 } -\frac{ \hat{P}^{*}\left( h\cdot g_{2}\right)^2 }{\hat{P}^{*}\left(
g_{2}\right)^3 }
+\frac{ \hat{P}^{*}\left( h^2\cdot g_{1}\right) }{\hat{P}^{*}\left( g_{1}\right)^2 }
-\frac{ \hat{P}^{*}\left( h\cdot g_{1}\right)^2 }{\hat{P}^{*}\left( g_{1}\right)^3 } }
\end{align*}
for all
$(h,g)\in\bar{\mathcal{H}}\times\mathcal{G}$ with $g=(g_1,g_2)$.
\item Construct the bootstrap version of the test statistic by
\begin{align*}
TS_n^{*}=\int_{\Xi}\sup_{(h,g)\in{\widehat{\Psi_{{\mathcal{H}}\times\mathcal{G}}}}} \frac{\sqrt{T_n^{*}}( \hat{\phi}^{*}(h,g)-\hat{\phi}(h,g))}{\max\{\xi,\hat{\sigma}^{*}(h,g)\}}\,\mathrm{d}\nu(\xi).
\end{align*}
\item Repeat steps (1), (2), and (3) $n_B$ times independently with $n_B$ chosen as large as is computationally convenient. Given
a prespecified nominal significance level $\alpha$, we construct the bootstrap critical value
$\hat{c}_{1-\alpha}$ by
\begin{align*}
\hat{c}_{1-\alpha}=\inf\left\{ c:\mathbb{P}\left( TS_n^{*} \leq c\bigg|\{(Y_{i},D_{i},Z_{i})\}_{i=1}^{n}\right) \geq
1-\alpha\right\} .
\end{align*}
In practice, we may approximate $\hat{c}_{1-\alpha}$ by the $1-\alpha$ quantile of the $n_B$ independently generated bootstrap statistics.
\item The decision rule for the test is: Reject $\mathrm H_{0}$ if $TS_n>\hat{c}_{1-\alpha}$.
\end{enumerate}
\end{algorithm}
\begin{proposition}
\label{prop.test multi order}Suppose Assumption \ref{ass.nu order} holds.
\begin{enumerate}[label=(\roman{*})]
\item If the $\mathrm H_0$ in \eqref{eq.null order} is true
and the CDF of the asymptotic limit of $TS_n$ is increasing and continuous at its $1-\alpha$ quantile, then
$\lim_{n\rightarrow\infty}\mathbb{P}(TS_n >\hat{c}_{1-\alpha}) =\alpha$.
\item If the $\mathrm H_0$ in \eqref{eq.null order} is false, then
$\lim_{n\rightarrow\infty}\mathbb{P}(
TS_n >\hat{c}_{1-\alpha}) =1$.
\end{enumerate}
\end{proposition}
Proposition \ref{prop.test multi order} may be proved analogously to Theorem 3.2 in \citet{sun2018ivvalidity}, so we omit the proof. The simulation study in Section \ref{sec.simulation} in the supplementary appendix demonstrates the good finite sample properties of the test. The numerical results show that the test is asymptotically size controlled and consistent.
Section \ref{sec.application} in the appendix provides an empirical application \citep{thornton2008demand} for the proposed test. We examine the partial validity of monetary incentives and distance from results centers as instruments for the knowledge of HIV status, and find that these instruments passed our test.
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