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Bounds for within-household encouragement designs with interference
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We obtain partial identification of direct and spillover effects in settings with strategic interaction and discrete treatments, outcome and independent instruments. We consider a framework with two decision-makers who play pure-strategy Nash equilibria in treatment take-up, whose outcomes are determined by their joint take-up decisions. We obtain a latent-type representation at the pair level. We enumerate all types that are consistent with pure-strategy Nash equilibria and exclusion restrictions, and then impose conditions such as symmetry, strategic complementarity/substitution, several notions of monotonicity, and homogeneity. Under any combination of the above restrictions, we provide sharp bounds for our parameters of interest via a simple Python optimization routine. Our framework allows the empirical researcher to tailor the above menu of assumptions to their empirical application and to assess their individual and joint identifying power.\\
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Keywords: Heterogeneity, instrumental variables, spillover effects, partial identification.
JEL Codes: C31, C36
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Policy experiments within households, in which either the household head or their spouse receive a treatment offer, are very widespread in the social sciences. For example, there have been many studies on the impact of gender-targeted conditional cash transfers to parents armand2020effect,benhassine2015turning,haushofer2016short, gender-targeted family planning d2022women,mccarthy2019intimate, the impact of banking vouchers dupas2019effect, or voter turnout encouragement foos2017all,nickerson2008voting,sinclair2012detecting. Analogously, there are many household experiments with siblings, particularly in educational contexts dizon2014parents,rogers2017intervening.
Household experiments are likely to exhibit two forms of network interference: $(i)$ treating one member of the household can impact the other; and $(ii)$ there may be strategic bargaining to decide who should take up the treatment, if voluntarily offered. This “two-stage” interference phenomenon opens up an avenue for treatment effects to be mediated by strategic take-up decisions. In turn, this presents a unique set of identification issues in experiments with non-compliance, which have not been the focus of the most of the existing literature. One strand has focused on identifying complier-type effects that are only valid if agents are $(i)$ not best-responding to others' take-up, and $(ii)$ weakly encouraged to enroll via offers Blackwell2017,DITRAGLIA20231589,Vazquez-Bare2020,ryu2024local. Another strand has focused on interpreting intention-to-treat or instrumental variable estimates as weighted averages of subgroup effects KangImbens2016,ImaiJiangMalani2021.
We propose an approach to partially identify average treatment effects that is robust to arbitrary outcome and treatment interference. The benefit of our approach is that we can bound average effects for the entire population--not just local effects on compliers--, and under cases of low non-compliance, obtain at least sign preservation. The key to our approach is to define a large type space that captures relevant strategic behaviors. While the exact proportion of people from each type may be unknown, it is constrained by support restrictions and by the joint distribution of observables (outcome, take-up, and instruments) across all members of the household. Computationally, we show that partial bounds on treatment effects can be obtained from a tractable linear program in the spirit of balke1997bounds, and illustrate our approach with two empirical applications.
For us to be able to model assumptions on strategic behavior in a causal way, we introduce a novel notion of take-up best response function. In potential outcomes language, this means that an individual's treatment can depend on the vector of joint household offers and the treatment decision of others. As is common in game-theoretic models, there may be multiple equilibria, and we complete the model by introducing a class of potential equilibrium selection rules, where households randomize across Nash equilibria.\footnote{Lazzati2025 develop a procedure to test Nash equilibirum play in a game with monotone best responses.} Modeling best-responses allows us to define counterfactual estimands for $(i)$ “fixed allocation” effects where treatment assignment is enforced by the policy maker, e.g. both treated vs both control, and $(ii)$ “policy targeting” estimands where some individuals are forced into treatment while others best respond.
We contribute theoretically by assessing the identifying power (or lack thereof) attributed to different restrictions on the household type space. To state our results, we define an identified set $\Theta(\mathcal{T})$ associated with a type space $\mathcal{T}$. We say that a restriction $\mathcal{T}'\subseteq \mathcal{T}$ is irrelevant if $\Theta(\mathcal{T}') = \Theta(\mathcal{T})$. We show that restriction irrelevance depends on whether a set is “closed” with respect to a particular set of restrictions. Our findings allow researchers to verify whether their assumptions lead to such closed type spaces.
Our first main theoretical result is that researchers can restrict attention to types with deterministic (as opposed to stochastic) equilibrium selection rules without loss of generality. Absent further structure, there are infinitely many stochastic equilibrium selection rules even if outcomes, take-up, and instruments are all discrete. Fortunately, the type space of deterministic rules is finite, and this allows us to frame the partial identification problem as a tractable linear program in matrix form. Building on this basic structure, we provide guidance on how researchers can layer-in further assumptions with the aim of obtaining tighter bounds.
Our second theoretical result is to analyze the identifying power of two commonly discussed restrictions: best-response dominance (other player's actions irrelevant for take-up) and best-response symmetry (identical types within households). Surprisingly, whether these assumptions are relevant or not depends on the type of estimand. On one hand, neither assumption can narrow the identified set for “fixed allocation” estimands, under mild closure conditions.\footnote{Closure conditions can be violated when too many restrictions are stacked together. In fact, we give a simple counter-example where joint assumptions of dominance and symmetry can be falsified by the data. Under certain conditions, dominance and symmetry can be viewed as mutually exclusive modeling strategies.} On the other hand, we provide a simple example where imposing either dominance or symmetry can strictly narrow the set for “policy targeting” estimands. We find similar results for supermodularity and submodularity restrictions.
Our theoretical findings have meaningful implications for empirical practice in terms of how we analyze “fixed allocation” or “policy targeting effects”. For “fixed allocation” estimands, the combination of deterministic rules and best-response dominance (which we show are irrelevant for the identified set), leads to a mathematically equivalent setup to a canonical partial identification problem with exogenous instruments and endogenous treatment Manski1990,balke1994counterfactual,balke1997bounds,mourifie2015sharp,chesher2017generalized, Laffer2019, Gunsilius2020, kitagawa2021identification, HanYang2024, bai2024identifying.\footnote{For a review aimed at the biostatistics literature see Swanson2018.} That means that researchers can bring familiar tools to tackle situations with interference, and these would still be robust to the presence of strategic agents and arbitrary equilibrium selection. For “policy targeting” estimands, dominance may constrain the identified set, and if researchers want to be agnostic, they could use our more general setup that allows for arbitrary equilibrium selection rules.
Our procedure is also well equipped to encode the researcher's application-specific restrictions in a comprehensive, yet tractable way, and assess their individual and joint identification power. We can incorporate a wide range of restrictions on the features of the data-generating process, capturing the researchers' institutional knowledge and the specifics of their application. For example, the researcher may have credible knowledge about the pairs assortment, the relative timing of the offer, treatment and outcome, the information structure, the strength of the incentives or the strategic interaction of the players. This information could thus be exploited via our procedure to provide identification of relevant parameters. In addition, researchers may achieve results by combining non-nested assumptions. Our procedure assists in categorizing such scenarios, as demonstrated in cases involving no strategic interaction alongside either monotonicity assumptions or a form of irrelevance of the instrumental variable for other members of the group.
Related Literature Our computational approach is heavily inspired by balke1994counterfactual,balke1997bounds who introduce a probabilistic response function framework where the structural relationship between observable variables are encoded as additional latent variables. In the original balke1994counterfactual,balke1997bounds exercise there is no interference of any kind, and take-up behavior is completely determined by the classic four principal strata: always-takers, compliers, never-takers, and defiers. Within this framework, a latent type encodes both the strata that an individual belongs to and their vector of potential outcomes under different treatment statuses. In our case, because strategic interaction is allowed, an individual latent type must specify the treatment response for any hypothetical value of their partner treatment, for all possible instrument values.\footnote{An individual latent type coincides with the collection of the individuals best response function (BRF), one for each of the games induced by varying the value of the instruments. The joint distribution of BRFs (for all individuals in a group can be the target for (partial) identification in its own right, like in the setting of kline2012bounds, where no instruments are present.} Moreover, because the unit of analysis is the pair, each latent type concatenates the two individual latent types (potential treatment response and potential outcomes) for each of the two individuals. While this representation is agnostic about the network formation process, it makes it easy to transparently incorporate further assumptions on matching on latent pair types. These type of assumptions can be justified by the empirical design and, as we show below, may carry identifying power.
Extending the balke1994counterfactual,balke1997bounds formulation to a setting with strategic response behavior is not straightforward due to the presence of multiple equilibria. Crucially, with multiple equilibria, the mapping from the latent types to the observable probabilities is no longer unique, a feature deeply linked to model incompleteness Jovanovic1989,tamer2003incomplete,honore2006bounds,Chesher2010,hahn2010bounds,chen2012bounds,kitagawa2021identification,HanBalat2023. When multiple equilibria do arise, being agnostic about the equilibrium selection mechanisms (which we represent as a distribution over deterministic equilibria) means that we can suitably extend the set of types and encode it as an additional optimization variable in the linear program formulation. Further, Theorem (ref) shows that when certain conditions are satisfied, encoding the equilibrium section rules is not even necessary. We describe this situation in detail in Section (ref).
Our paper complements the results in bai2024inference, who propose an approach to bound treatment effect parameters with multi-valued outcomes, treatments and instruments. Both their paper and ours are generalizations of balke1997bounds in that we recast the problem of partial identification as a linear program where the criterion is a particular estimand, equality restrictions map latent types to observed behavior, and inequality constraints encode bounds on proportions and certain sub-restrictions of types. While bai2024inference assume SUTVA, our approach explicitly focuses on interference and encodes a rich enough set of types to capture other features that uniquely arise in this type of setting, e.g., strategic response, multiple equilibria, and group symmetry. These are not present under SUTVA. Our contribution is related to proposing and analyzing the properties of these specific types of restrictions and their empirical content in applied examples.
While we focus on “global” parameters, recent work has focused on whether any effects can be point-identified for a meaningful subgroup of people under different assumptions on their take-up behavior. This is the avenue taken, among others, by Blackwell2017, Vazquez-Bare2017, Vazquez-Bare2020, DITRAGLIA20231589, and kormos2023. In order to obtain meaningful local effects from instrumental variables one requires strong monotonicity assumptions blandhol2022tsls, that are harder to satisfy in settings with interference. This is because when multiple instruments and strategic interactions are present, the number of possible latent types grows exponentially, absent further restrictions.\footnote{In the original LATE framework of Imbens1994 there are only 4 latent types --always-takers, compliers, never-takers, defiers-- and the uniform monotonicity assumption rules out only one of them, usually the defiers.} As a result, the point-identified estimand might represent a vanishingly small percentage of the underlying population, making extrapolation even harder or less credible.
The possibility of strategic take-up relates our paper to the econometric analysis of games. An excellent review is provided in DePaula2013. The focus of most of this literature corresponds to what in our setting are the first stage parameters, e.g., entry games as treatment take-up. For example, Lazzati2015 focuses on the first stage and is a combination of IOR with strategic interactions assuming super modularity, and treatment monotonicity. Instead, our ultimate goal are various ATEs, i.e., the second stage. Since the relationship between choice behaviour and outcome is left unspecified, many of the usual game-theoretical restrictions such as supermodularity, submodularity, dominance, and equilibrium selection do not yield any identifying power. A notable exception, and closely related to our paper is HanBalat2023, where the focus is actually on different ATEs and LATEs of endogenous treatments which result from the strategic interaction of the agents. The main difference is that, contrary to our paper, there are no spillovers at the instrument level in their setting. This means they assume a particular best-response function that depends on own offer and other's take-up, but not others' offers. Moreover, HanBalat2023 model take-up behaviour under the assumptions of strategic substitution/complementarity and symmetry of the other players' take-up behavior. This is one of the many restrictions we can impose on the latent space.\footnote{See Section (ref) for a list.}
The rest of the paper is organized as follows, in Section (ref) we introduce notation and the econometric framework, the latent type representation and in Section (ref) we describe its mapping to observable distribution and set up the linear program. Sections (ref) and (ref) present the menu of restrictions we can impose, and their impact on the identified set. Section (ref) illustrates our methods in an empirical application. Finally, Section (ref) concludes. Additional results are collected in several online appendices.
The sample contains $G$ groups, indexed by $g=1,\ldots,G$, each containing 2 individuals $i=1,2$. A subset of individuals and their partners receive an offer to participate in a program. We denote the offer status of a pair by $(Z_{ig},Z_{-ig})$, which can take values $ (z,z') \in \{0,1\}^2$. There may be non-compliance in the sense that eventual treatment take-up, denoted by $(D_{ig},D_{-ig})$, may differ from the offer status. We define a potential outcome as $Y_{ig}(d,d',z,z')$ and assume that it is not directly affected by offers, only by treatment status.
We model strategic take-up decisions by defining an individual best-response function $D_{ig}(z,z',d')$, to represent the take-up decision of $\{ig\}$ when the pair is offered $(z,z')$ and their partner chooses $d'$. We define an individual's type $S_{ig}$\footnote{$S_{ig}$ is supported on $\{0,1\}^{12}$.} as their set of contingent behaviors, given different treatment status and offers made.
Table (ref) illustrates how different values of $(z,z')$ induce different games which in turn can encourage different take-up responses for a given realization $(s,s')$ of $(S_{1g},S_{2g})$. In general, for each pair of individual types $(s,s')$ and offers $(z,z')$, we can define the set of pure-strategy Nash equilibria in take-up decisions $(D_{1g},D_{2g})$, which we denote by
In strategic settings such as ours, a single pair type $(s,s')$ can give rise to multiple equilibria: the pair shown in Table (ref) is one such example. This feature can result in model incompleteness: some latent distribution of pair types $(S_{1g},S_{2g})$ may be compatible with more than one distribution of the observables. To complete the model, we associate to each type an equilibrium selection rule $E_g(z,z',d,d')$ which gives the probability of choosing $(d,d')$ given a choice of instruments $(z,z')$. Crucially, the equilibrium selection rule itself can be shifted by the instruments, i.e., the equilibrium selection rule is game-dependent. We stack all the equilibrium selection rules across $(z,z',d,d')$ into a single vector $E_g$. For any fixed $(z,z')$ the equilibrium selection rule belongs to the collection of probability distributions on $\{0,1\}^2$. The extreme points of such a collection are deterministic equilibrium selection rules, which we take as the building block to construct any equilibrium selection via convex combination.
For this reason, it is convenient to encode a complete latent type as $(s,s',e) =: T_g \in \mathcal{T}$ where $e$ is vector of deterministic equilibria, one for each value of $(z,z')$. Then, a distribution $\mu \in \Delta(\mathcal{T})$ over the types triples is in fact a distribution over the latent type pairs $(s,s')$ together with a pair-specific equilibrium selection mechanism.\footnote{This is because the convex hull and the Cartesian product operations commute:
which shows that we can interpret $\mu \in \Delta(\mathcal{T})$ as specifying jointly a probability distribution on the types in $\Delta(\mathcal{S})$ together with a type-specific equilibrium selection in $\Delta(\mathcal{E})$. Moreover this latter term takes the form:
.} In Section (ref), distributions $\mu$ in a suitable subset $\Delta(\mathcal{T})$ will play a central role in our definition of the identified set and in its characterization via linear programming.
The collection of types $\mathcal{T}$ is very rich. We would like to impose some minimal conditions on this collection. We start by requiring that a type is consistent with Nash equilibrium play:
With Assumption (ref) we require that the equilibrium selection rule only randomizes over the set of Nash equilibria that $(s,s')$ could produce. If such a set is empty, then we rule out the type $t$. Therefore, this assumption guarantees that there exist a Nash equilibrium with probability 1. Analogous assumptions about the existence of a pure-strategy Nash equilibrium, appear also in Tamer2010, kline2012bounds and HanBalat2023. Consider the latent type as a random variable $T_g$, whose distribution is given by $\mu$. We introduce a standard exogenous assumptions for the treatment assignments:
This says that instruments are assigned to a group $g$ without systematic knowledge of $i)$ any individual's potential treatment choice, $ii)$ any individual's potential outcome, (both encoded by the $(S_{1g},S_{2g})$ entries of $T_g$, and $iii)$ the equilibrium selection encoded by the $E_g$ entry of $T_g$. Random assignment of $(Z_{1g}, Z_{2g})$ is a sufficient condition for Assumption (ref) to hold automatically, but the instruments could also come from a quasi-experiment as long as (ref) is deemed plausible. Here we focus on unconditional independence but the analysis may be extended to the case where Assumption (ref) holds conditional on a set of covariates $(X_{1g}, X_{2g})$.
The main parameter of interest is the average effect comparing two different treatment allocations $(d_1,d_1')$ and $(d_2,d_2')$ as $$ \theta(\mu) := \int [Y_s(d_1,d_1') - Y_s(d_2,d_2')] \ \text{d}\mu(s,s',e). $$ We also consider another type of “policy targeting” estimand, where one individual in the couple is assigned to the program without the option to refuse, the partner best responds optimally, and offers are fixed at $(z,z')$. For example, one might fix an individual's treatment to $d_1 = 1$, and only make an optional offer to their partner, i.e. $(z_1,z_2) = (0,1)$. This takes the form: $$ \gamma(\mu) := \int Y_s(d_1,D_{s'}(z_2,z_1,d_1)) \ \text{d}\mu(s,s',e). $$ The estimand $\gamma$ depends on the types of both individuals in the couple. We can define variants of this estimand that switch which members is targeted, or whether we obtain the difference between two counterfactual estimands. Different contrasts offer alternative ways to decompose the role of peer effects within a household. \\
Some special cases of the parameter $\theta$ include average direct effect (ADE), for $(d_1,d_2) = (1,0), (d'_1,d'_2)= (0,0)$, and average spillover effects (ASE), for $(d_1,d_2) = (0,1), (d'_1,d'_2)= (0,0)$. Unlike Vazquez-Bare2020 and kormos2023, we do not restrict attention to any particular latent subpopulation for which a LATE-type of parameter may be point-identified. There is a trade-off between targeting a subpopulation LATE which may be point-identified under some of the assumptions in Section 3, and a more aggregate parameters like the ADE we consider above, which is usually only partially identified. As the appropriate target may depend on the policy question that the empirical researcher is considering, we see our approach as complementary to Vazquez-Bare2020 and kormos2023. \\
The researcher only observes data on outcomes and take-up, conditional on the offers (instrument assignments) made to the group. The observed data is fully characterized by the conditional probability mass function
The observed probabilities aggregate the behavior across many different pair types $t = (s,s',e)$. Our representation guarantees that a pair $t$ will randomize over Nash equilibria over $(d,d')$, and thereafter have a predetermined set of potential outcomes:
The identified set will be determined by restrictions over the pair type space, which we denote by $\mathcal{T}$. A parameter of interest satisfies the definition $\theta(\mu)$ and the observed the probabilities aggregate over types.
Our baseline assumptions allow us to encode how the type distribution maps into observed probabilities like we describe in Section (ref). Because all restrictions are linear in $\mu$, the sets $\Theta(\mathcal{T})$ and $\Gamma(\mathcal{T})$ are convex, and hence each identified set is entirely determined by the end points of the respective intervals. In the next section, after introducing the constraint that the observable data puts on $\mu$, we provide the characterization of the identified set via a linear program.
Definition (ref) is challenging to operationalize directly because the set of stochastic equilibirum selection rules is a continuum. To tackle this problem, we show that there is no loss in focusing on deterministic equilibrium selection rules, which are more tractable for estimation purposes.
To ensure a fair comparison between restrictions on stochastic and deterministic types, we only focus on sets $\mathcal{T}$ that nest equivalent deterministic types as special cases.
The closure condition $\mathcal{T}$ asks where ther type space is rich enough. For example, suppose that $\mathcal{T}$ includes a pair type $(s,s',e)$ which randomizes between two Nash equilibria $(d,d') \in \{(0,0),(1,1)\}$ with probablity of 50% when $(z,z') = (0,0)$. The closure condition implies that $\mathcal{T}$ is large enough to also include two more types with the same $s$ and $s'$, but where the equilibria $(d,d') = (0,0)$ and $(d,d')=(1,1)$ are each selected with 100% probability, respectively.
The following theorem shows that we can focus on deterministic equilibrium selection rules without loss of generality.
Theorem (ref) proves that it is possible to reconstruct any identified set with only deterministic selection rules. The statement is written with generality in mind: the super-set $\mathcal{T}^*$ can include any number of additional restrictions on potential outcomes, best-responses, and selection rules, as long as they satisfy a minimal closure condition. The statement also applies to general types of parameters belonging to two main classes: $(i)$ parameters evaluating fixed $(d,d')$ allocations in $\Theta(\mathcal{T})$, and $(ii)$ treatment targeting parameters $\Gamma(\mathcal{T})$.
The proof is constructive. For each pair $(s,s')$, we find an equivalence between two alternate ways of modeling multiplicity. On one hand, individuals might choose to randomize across equilibria. On the other hand, there might heterogeneity in coordination mechanisms/signals that lead to unique equilibria for a particular pair, that are known to agents but unknown to the researcher. Since choices are only observed in the aggregate, in the absence of additional assumptions, both modeling approach are observationally equivalent. This construction applies to both $\Theta(\mathcal{T})$ and $\Gamma(\mathcal{T})$, because the counterfactual parameters in either case only depend on the distribution of types $(s,s')$ and not on their equilibrium selection rules.
In this section, after briefly outlining the relationship between latent types and observable probabilities, we provide a simple linear programming characterization of the identified set.
The researcher only observes data on outcomes and take-up, conditional on the offers (instrument assignments) made to the group. The observed data is fully characterized by the conditional probability mass function $p(y,y',d,d'|z,z')$ defined in (ref). Considering the conditioning set $(Z_{ig},Z_{jg},S_{ig},S_{jg}, E_g) = (z,z',s,s',e)$ we can also define a type-specific conditional probability $p(y,y',d,d'|z,z',s,s',e)$. Because we defined a latent $t$ as a triplet $(s,s',e)$, instead of just $(s,s')$, this conditional probability simply checks whether $(d,d')$ coincides with the deterministic equilibrium selected by $e$. As a result, $p(y,y',d,d'|z,z',s,s',e)$ takes the special form: $$ p(y,y',d,d'\mid z,z',s,s',e) = \mathbbm{1}\{ e(z,z') = (d,d') \} \times \mathbbm{1}\{(Y_{s}(d,d'),Y_{s'}(d,d')) = (y,y')\} $$ and its values are in $\{ 0,1 \}$. By the law of total expectations and Assumption (ref) the observed conditional probability and the latent one are related by:
We can stack the set of observed conditional probabilities in a vector $p$ of size $K$ (for the binary case $K=2^6 = 64$) and the probabilities of latent types in a vector $\mu$ of size $|\mathcal{T}|$. We can also represent the set of type-specific probabilities in the $(K \times |\mathcal{T}|)$ matrix $A$, whose generic element is $p(y,y',d,d'\mid z,z',s,s',e)$. This leads to a convenient matrix representation
Without imposing further restrictions, $|\mathcal{T}| > 64$, and hence the linear system is under-identified, similar to balke1997bounds. This means that multiple latent distributions $q$ are observationally equivalent in the sense that they induce the same $p$.
The choice of including the equilibrium selection in our definition of a latent type guarantees that, under Assumption (ref) the matrix $A$ is unique. Suppose we did not specify $e$ in a latent type, so a type is fully described by $(s,s')$. When $Nash(s,s',z,z')$ is a singleton, $p(y,y',d,d'|z,z',s,s'):= \sum\limits _{e} p(y,y',d,d'|z,z',s,s',e)$ is 0 for all except one combination of $(y,y',d,d')$, for which it is 1. In this case, $p(y,y',d,d'|z,z',s,s',e)$ is a degenerate probability distribution on $(d_i,d_j,y_i,y_j)$. When $Nash(s,s',z,z')$ contains more than one element (i.e. whenever the game features multiple equilibria) $p(y,y',d,d'|z,z',s,s')$ is an unknown, non-degenerate distribution on $(d_i,d_j,y_i,y_j)$. This is precisely because the equilibrium selection has not been specified. This means that a latent pair $(S_{ig}, S_{jg})$ could be mapped to more than one observable outcome.
Given a parameter $\theta$ and a set of latent types $\mathcal{T}$ we can write the bounds as value functions for the pair of linear programs below:
with \\
\vskip 0.5cm with $\mu \in \Delta(|\mathcal{T}|), c_{\theta} \in \mathbb{R}^{|\mathcal{T}|}, p \in{\Delta(K)}, A \in \textrm{Mat}(K, |\mathcal{T}|)$. The characterization of $\Gamma$ is analogous, for appropriate choices of the linear representer $c_{\gamma}$. The high dimensionality of our program prevents us from exploiting the duality to obtain closed form expressions of the bounds as in balke1997bounds.
The procedure described in (ref) is unfeasible because it depends on $p$. To estimate $\underline{\theta}(\mathcal{T})$ and $\overline{\theta}(\mathcal{T})$, we replace $p$ by its empirical counterpart, denoted by $\hat p$. Each entry of $p$ is of the form $p(y,y',d,d'|z,z')$, so that $\hat p$ consists of the empirical counterparts
for all the possible combinations of $(y,y',d,d',z,z')$. The feasible version of (ref) is
with $\hat{\underline{\theta}}(\mathcal{T}):= c'\hat{\underline{\mu}}$ and $\hat{\overline{\theta}}(\mathcal{T}):= c'\hat{\overline{\mu}}$ where\\
As pointed out by Hsieh2022Inference, the solution to a linear program is, in general, not differentiable as a function of the inputs. This implies that the sampling variation in $\hat p$ might not lead to asymptotic normality of $(\hat{\underline{\mu}} , \hat{\overline{\mu}})'$. A consequence of Theorem 3.1 in FangSantos2018 is that the canonical bootstrap does not work if the limiting distribution is not Gaussian. There are several potential approaches to statistical inference involving linear programs.\footnote{Among the potential inferential procedures, there is bai2022testing which exploits subsampling and two-step method for testing moment inequalities. fang2023inference which provides a novel geometric characterization of the feasible set to produce a valid testing procedure. syrgkanis2018inferenceauctionsweakassumptions which give finite sample inference methods exploiting concentration inequalities. FREYBERGER201541 which provides two bootstrap procedures that estimate the asymptotic distributions of the value function of a linear program, one requiring a user specified tunning parameter and another one free of tunning parameters but under the restriction of uniqueness of the optimal solution. For another approach on linear programming with partially identified econometric models see voronin2024.} We focus on providing pointwise confidence intervals for the upper and lower bounds of the parameter of interest, leveraging existing inferential methods for linear programs from horowitz2023inference. Their procedure gives finite-sample lower bounds on the coverage probabilities of the confidence intervals. We briefly describe their procedure in our setting. Recall that $\hat{p}$ contains 4 sub-vectors, denoted $p_z$, one for each instrument. Each of them has size 16 with entries equal to the empirical frequencies $\hat{p}(y,y',d,d'|z,z')$. Following horowitz2023inference we consider replacing $\hat{p}$ with an additional choice variable $p$, and solve the relaxed program:
Here, instead of imposing the linear constraint $A \mu = \hat{p}$, we only require that $p$, the image of $A\mu$ is inside a region $\mathcal{P}(\hat{p})$, which always includes $\hat{p}$. From this is immediate that the above program is a relaxed version of the original LP problem. $\mathcal{P}(\hat{p})$ is a high probability region, for example a 95% confidence region for $\hat{p}$. There are many options for this region, including a box or an ellipsoid. Because we want our procedure to adapt to cases where the randomized offers do not cover the all possible values of the instruments with positive probability, we construct the region as the product of up to 4 ellipses, one for each conditional distribution. That is, we take $\mathcal{P}(\hat{p}) = \Pi_{z} \mathcal{P}(\hat{p}_z)$ where, for each instrument:
with critical value $\kappa_{z,(1-\alpha)}$, and a positive definite, deterministic, finite matrix $\Upsilon$. Recall that because $A\mu=p$ we can write each of these regions as a quadratic constraint in $\mu$.\footnote{Defining the $A_k$ as the $(16 \times \mathcal{T})$ sub-matrix corresponding to $p_k$ we can write the regions as:
} The collection of values $\kappa_{z,(1-\alpha)}$ is chosen so that $\mathbb{P}(\mathcal{P}(\hat{p})) \geq 1-\alpha$. The critical value can be obtained from theorem 2.3 from horowitz2023inference. Regarding the choice of matrix $\Upsilon$, because $\hat{p}_k$ is a draw of the multinomial distribution, it satisfies $1^T \hat{p}_k=1$. As a consequence, there are at most $15$ linearly independent entries in $\hat{p}$ and $\hat{\Sigma} = \textrm{diag}(\hat{p}) - \hat{p} \hat{p}^T$ is rank deficient. Because the choice of $\Upsilon = \hat{\Sigma}^{-1}$ is not feasible, we consider the robust choice $\Upsilon = I$. One could also use regularization to choose $\Upsilon = (\hat{\Sigma} + \epsilon I)^{-1}$ interpolating $\hat{\Sigma}$ and the identity matrix with same dimension. Note that $\hat{\Sigma}$ is positive semi-definite. Moreover, because $\epsilon I \hat{\Sigma} = \hat{\Sigma} \epsilon I$ we can simultaneously eigen-decompose $\epsilon I$ and $\hat{\Sigma}$. If $\hat{\Sigma} = Q^T \Lambda Q$ then:
so $(\epsilon I + \hat{\Sigma})$'s $j$-th eigenvalue is $\epsilon + \lambda_j$. All entries in $\Lambda$ are non-negative so now the minimal eigenvalue of $(\epsilon I + \hat{\Sigma})$ is bounded below by $\epsilon$ making $\Upsilon=(\epsilon I + \hat{\Sigma})^{-1}$ a valid choice for the procedure of horowitz2023inference.
We consider two choices of $\kappa_{z,(1-\alpha)}$. The first is obtained by bootstrapping the quadratic form. The second one is as suggested by method 2 of horowitz2023inference. When $\hat{p}_z$ is in general position, this amounts to $\kappa_{z,(1-\alpha)}=\sigma^2 \times 2\times 15\times log(2\times \frac{15}{\alpha})$ where $\sigma^2$ is the maximum of the variance of each element of $\hat{p}$ and 15 is the number of linearly independent elements in $\hat{p}_z$.
In addition to the restrictions imposed by the observable distribution discussed in the previous section, there are a number of restrictions that researchers might like to impose with the goal of shrinking the identified set. In this section, we prove a set of negative results showing that some of these choices do not reduce the size of the identified set. Nevertheless, these results turn out to be advantageous from a computational side. The reason being that they narrow the scope of what the researcher needs to focus on, and enables us to formulate more tractable versions of the problem.
There a number of restrictions on $\mathcal{T}$ that can capture strategic complementarity or substitution. A supermodular game is one where individual payoffs for an action are weakly higher if other players also take that action. For household experiments, this might emerge when taking up treatment is mutually beneficial to both members. Differing payoffs translate into specific restrictions on the best response functions.
By contrast, in a submodular game payoffs are weakly lower, encouraging substitution.
Dominant best-response functions arise when the take-up decision does not depend on the partner's actions, which occurs when best response are both supermodular and submodular.
Recent work on causal inference, interference, and non-compliance relies on the dominance assumption to identify local effects DITRAGLIA20231589,Vazquez-Bare2020.
Assumption (ref) holds when the type space is sufficiently rich to include agents with dominant strategies that can $(i)$ mimic the same equilibrium as strategic types, and $(ii)$ they have the same potential outcomes. Considering sets of types $\mathcal{T}$ that are closed under dominant strategies makes sense if the researcher wishes to be agnostic about strategic behavior.
Theorem (ref) states that the identified set cannot be made smaller by restricting attention to pair types with dominant strategies, as long as the closure condition in Assumption (ref) is satisfied.
Since all dominant strategies are simultaneously supermodular and submodular, then by a “sandwich” argument we can show that these other restrictions have no identifying power either.
The counterfactuals estimands captured by $\Gamma(\mathcal{T})$ are very different than those in $\Theta(\mathcal{T})$ because the parameters explicitly depend on the best response functions. One might wonder whether dominance, supermodularity, and submodularity restrictions are relevant for $\Gamma(\mathcal{T})$. We show they are, by introducing a simple counter-example that satisfies Assumptions (ref), (ref), (ref), (ref), and (ref), in which imposing assumptions on best-responses can strictly shrink the identified set.
In this example, we observe full compliance in the sense that take-up exactly matches the offers made to pair. Best-responses are always dominant for all offer pairs except for $(z_1,z_2) = (0,0)$. In this case, the first individual has a dominant strategy, but their partner has a “mirror” best response, which exactly matches their behavior. However, their “mirror” behavior only matters off-equilibrium. Pairs with these characteristics display unique, deterministic, Nash equilibria, and by inspection we can verify that $\mathcal{T}^{counter}$ satisfies (ref), (ref), and (ref). Assumption (ref) is satisfied as long as offers are made at random. Because $\mathcal{T}^{counter}$ includes non-dominant best-responses, it also satisfies Assumption (ref).
Type symmetry is an important restriction for discrete choice games with peer effects, previously used in Tamer2010 and kline2012bounds, among others. To analyze best-response symmetry and its connection to best-response dominance, it is useful to represent the best response function as a vector, as follows: $$ \delta_s(z,z') = (D_s(z,z',0), D_s(z,z',1)).$$ We say that a type pair is symmetric in best responses if they both members have the same response function.
Table (ref) illustrates the set of possible equilibria for a pair $(s,s')$ depending on the best-response of each member. It facilitates the process of visualizing which type pairs and which sets of equilibria are being restricted by imposing symmetry, submodularity, and dominance, respectively.
Table (ref), shows that symmetric types can generate any $(d,d') \in \{(0,0),(0,1),(1,0),(1,1)\}$, similar to the dominance assumption, though in practice the type pairs that achieve this may differ. Inspired by this finding, we propose a particular type of closure assumption, based on the ability to map any type pair to an equivalent symmetric pair that leads to the same take-up equilibria.
Theorem (ref) shows that best response symmetry does not reduce the identified set, when the type space is closed over symmetric best responses. Since the superset of all types $\mathcal{T}^*$ that satisfies the core assumptions (ref), (ref), (ref) satisfies this closure condition, then we can conclude that symmetry on its own does not reduce the identified set for average treatment effect type parameters $\theta$.
Theorems (ref), (ref), and (ref) convey a similar message to the bounds derived for the joint Best Response Function (BRF) of two players in kline2012bounds. However, their framework does not include instruments or outcomes, and the joint BRF, rather than policy parameters, is the primary object of interest. We elaborate on this connection in Appendix (ref).
In the literature, point or partial identification of different parameters of interest is often obtained under additional assumptions each of which imposes restrictions on the latent types. Through our representation of the latent types, it is easy to see that restrictions are imposed via either equality or inequality constraints, on the space of types. Equality constraints typically rule out latent types altogether. This is the case for the Imbens1994 monotonicity assumption, where the probability mass on defier-types is equal to 0. Inequality constraints impose a bound on the total probability of certain types. This is the case for various forms of stochastic monotonicity, probabilities of strategic types and probability of assortative matching. In this paper, we collect a large menu of restrictions each of which may be motivated by economic theory, by knowledge of the institutional context, or by the information structure that the individuals face.
Because here a latent type completely characterizes the complete contingent plan of a pair of individuals (i.e. both the best response function for all induced games as well as all potential outcomes), some restrictions may be enforced separately on each individual of the pair. This may be useful when the individuals in the pair are not anonymous or when they belong to potentially different populations. For example, instrument monotonicity may be imposed separately for each individual in the pair. Other assumptions about strategic interaction may not directly restrict any distribution of the marginal types, but rule out certain types occurring jointly. For example strategic complementarity, and even the weaker Nash equilibrium in pure strategy assumptions rule out certain joint strategic responses.
Below we collect (in the notation of our setting) a (non-exhaustive) list of the assumptions that appear in the literature and can be motivated by the institutional context at hand.
dupas2019effect studied the impact of expanding access to bank accounts on several outcomes in rural Kenya. To this end, individuals who were randomly selected for the treatment received a non-transferable voucher for a free savings account. One of the treatment arms focused on dual headed households, and since the randomization was done at the individual level, this is precisely the setting studied in this paper. While dupas2019effect study a variety of outcomes, we focus on the effect that having a bank account has on owning a business as the main source of income.
This causal question suffers from the issues described in our paper since potential social interaction occurs between spouses in a non-exogenous way. This interaction, is potentially affecting choices of income sources as well as savings choices. Additionally, exogenous variation in the cost of opening bank accounts of an individual and his partner can affect the individual choice of savings if the household takes joint choices or shares the same budget. In this context, hoping to point identify average effects seems complex since it would require significant restrictions on the non-exogenous behavior of households. We thus in-turn, provide a partial identification alternative with less restrictive assumptions on the data generating process.
In terms of the notation of our paper, $Z_{men}$, $Z_{women}$ are, respectively, the indicators for the male and female spouse receiving the voucher. Similarly, $D_{men}$, $D_{women}$ are, respectively, the indicators for the male and female spouse opening a bank account. Finally $Y_{men},Y_{women}$ range over a small collection of outcomes of interest. They include indicators for male and female spouse owning a business, but also indicators for whether the spouses send or receive financial transfers from friends or next of kin. While presenting each set of bounds we specify which outcome of interest we are considering.
The complicated pattern of non-compliance that can arise in this empirical setting calls for an ITT analysis. While we go beyond by offering an analysis focused on average treatment effects, we consider two ITT style regressions:
for the subsample of men and women separately. The results can be seen in Table (ref).
Note that in both cases the coefficient on the spouses' instrument, is not significant. Moreover, for the case of women, neither coefficient is significantly different from 0. For the case of men, the interaction term coefficient is negative and statistically significant. When both are assigned to treatment, this reduces the likelihood of men owning a business.
Now we go beyond the ITT analysis, to try to bound the effects of the treatment on the outcome. We focus on four estimands, distinguishing between men and women. Let $\text{women}_i$ be equal to 1 if the individual is female, and 0 otherwise. Let $1$ denote males while $2$ denotes females. The first two estimands are
which are average direct and spillover effects for the subpopulations of men. The other two estimands are the analogous for the subpopulation of women:
First, we compute the bounds under three types of latent restrictions we previously discussed: strategic neutrality, strategic neutrality and IOR, strategic neutrality and Vazquez-Bare monotonicity (VB).
The results are in Table (ref). Notably, both IOR and VB restrictions result in empty bounds. A possible explanation for this is that monotonicity is violated because the probability of opening an account for the female partner is weakly lower when both individuals receive the offer, $0.65$ vs. $0.64$. This can be seen in Table (ref). Perhaps this may be due to the husband receiving slight preference for opening the account.
The bounds are non-empty if we allow a small number of non-monotonic types (2%). As it can be seen in the third column of Table (ref), further imposing a higher degree of monotonicity does not tighten the bounds for aggregate affects like the ADE and ASE.
For completeness, we note that an interesting feature of the setting in dupas2019effect is that the vouchers were also offered to single headed households, consisting of women. Thus, we have a no spillovers setting, akin to the one studied in the seminal paper by balke1997bounds. Applying their procedure, the no-assumption bounds on the ATE are $[-0.078,0.277]$. The range of values for the ATE are mostly positive, and is somewhat consistent with the ADE for women with a high degree of VB monotonicity. However, we note that there are possible confounders that simultaneously affect the outcome and the composition of the household.
We propose a linear programming formulation to obtain bounds on a variety of treatment effect parameters in the presence of spillovers and strategic interactions. The possibility of strategic interactions results in an incomplete model in the sense that the map from latent probabilities to observed probabilities is not unique. We propose a linear programming formulation to obtain bounds on different average treatment effect parameters. Different restrictions that the literature has commonly used can be easily incorporated into the linear programming as a form or restricting the support of latent probabilities. In this way, their joint or separate identifying power can be gauged against their impact on the width of the identified set. Future work will consider inference, Monte Carlo simulations, and an empirical application.
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