Testing Partial Instrument Monotonicity

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abstractWhen 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.

Keywords: Partial monotonicity, instrument validity, nonparametric test

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, mogstad2021causal show that the monotonicity condition only holds if choice behavior is effectively homogeneous. However, such homogeneity may break down in many empirical applications. 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 huber2015testing, kitagawa2015test, mourifie2016testing, and sun2018ivvalidity. In this paper, we extend the frameworks of kitagawa2015test and sun2018ivvalidity to the partial monotonicity of mogstad2021causal. We show that the proposed test based on kitagawa2015test and 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 thornton2008demand discussed in mogstad2021causal.

Setup and Test Formulation

mogstad2021causal introduce the concept of partial instrument monotonicity for multi-dimensional IVs. We follow 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 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 mogstad2021causal.

assumptionPartial IV validity for multi-dimensional $\boldsymbol{Z}$: \begin{enumerate}[label=(\roman*)] • Instrument Exclusion: Almost surely, $Y_{d\boldsymbol{z}_{1}}=\cdots=Y_{d\boldsymbol{z}_{K}}$ for all $d\in\mathcal{D}$. • 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) and \tilde{D} =\left( D_{\boldsymbol{z}_{1}},\ldots,D_{\boldsymbol{z}_{K}}\right) . \end{align*} • 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}

remarkAssumption (ref)(iii) is a simplified version of Assumption PM (partial monotonicity) in mogstad2021causal. 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} 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)(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 (ref). We may also extend the test to Assumption PM in mogstad2021causal following an idea similar to that in sun2018ivvalidity if there is no prior information about the direction in Assumption PM.

Suppose that $D$ has maximum value $d_{\max}$ and minimum value $d_{\min}$. We provide a testable implication for Assumption (ref) in the following lemma.

lemmaFor 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} &\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\\ &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). \end{align}

Lemma (ref) can be proved analogously to Lemma 2.1 of sun2018ivvalidity. In the following, we extend the tests of kitagawa2015test and 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 (ref) and (ref) are equivalent to

align[align omitted — 407 chars of source]

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 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

align[align omitted — 1,188 chars of source]

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

align[align omitted — 72 chars of source]

For every $\left( h,g\right) \in {\bar{\mathcal{H}}\times\mathcal{G}}$ with $g=(g_{1},g_{2})$, define

align[align omitted — 188 chars of source]

The null hypothesis equivalent to (ref) is

align[align omitted — 143 chars of source]

and the alternative hypothesis is

align*[align* omitted — 120 chars of source]

We then define the sample analogue of $\phi$ by

align*[align* omitted — 142 chars of source]

where $\hat{P}$ denotes the empirical probability measure of $P$ such that for every measurable function $v$,

align[align omitted — 136 chars of source]

and $\{( Y_{i},D_{i},\boldsymbol{Z}_{i})\}_{i=1}^n$ is the i.i.d.\ sample distributed according to $P$. Define

align[align omitted — 473 chars of source]

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.

assumptionsun2018ivvalidity 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*}

We follow kitagawa2015test and sun2018ivvalidity and construct the test statistic as

equation[equation omitted — 200 chars of source]

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

align[align omitted — 264 chars of source]

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 Beare2015improved and Beare2017improved in different contexts.\footnote{See linton2010improved and lee2013testing for more discussions on estimation of contact sets.} Algorithm (ref) illustrates the test procedure.

algorithm[algorithm omitted — 2,513 chars of source]

propositionSuppose Assumption (ref) holds. \begin{enumerate}[label=(\roman{*})] • If the $\mathrm H_0$ in (ref) is true and the CDF of the asymptotic limit of $TS_n$ is increasing and continuous at its $1-\alpha$ quantile, then $\lim_{n\rightarrow\infty}\mathbb{P}(TS_n >\hat{c}_{1-\alpha}) =\alpha$. • If the $\mathrm H_0$ in (ref) is false, then $\lim_{n\rightarrow\infty}\mathbb{P}( TS_n >\hat{c}_{1-\alpha}) =1$. \end{enumerate}

Proposition (ref) may be proved analogously to Theorem 3.2 in sun2018ivvalidity, so we omit the proof. The simulation study in Section (ref) 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) in the appendix provides an empirical application 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.