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Shrinkage Bayesian Causal Forest with Instrumental Variable
\maketitle
\begin{abstract}
Discovering interpretable subgroups whose complier effects deviate from the average is a central goal of instrumental variable analysis under imperfect compliance, yet existing tree-based methods degrade when most covariates are irrelevant to the effect.
We propose Shrinkage Bayesian Causal Forest with Instrumental Variable (SBCF-IV) for discovering and estimating subgroups with heterogeneous Complier Average Causal Effects (CACE) in sparse high-dimensional settings.
SBCF-IV places a sparsity-inducing Dirichlet prior on the splitting probabilities of the Bayesian Additive Regression Trees that estimate the conditional intention-to-treat and the complier share, concentrating posterior mass on the few covariates that moderate the complier effect and thereby regularizing effect estimation.
The posterior split frequencies additionally enter a downstream CART as variable-level costs that steer the partition toward relevant moderators, providing an interpretable division of the covariate space.
Monte Carlo experiments show that, as the share of irrelevant covariates grows, SBCF-IV recovers the true partition more reliably than its non-sparse predecessor BCF-IV at the tree and unit level, and retains nominal coverage where BCF-IV's intervals deteriorate. We apply the method to the Oregon Health Insurance Experiment and the 401(k) eligibility data.
\end{abstract}
\begin{keywords}
Heterogeneous treatment effects, instrumental variables, Bayesian shrinkage, subgroup discovery
\end{keywords}
\begin{jelclass}
C11, C14, C21, C26
\end{jelclass}
\section{Introduction}\label{ch:intro}
The heterogeneous treatment effect (HTE) literature has expanded along two complementary dimensions: estimating the conditional average treatment effect (CATE) and discovering interpretable subgroups whose effects deviate from the population average \citep{lipkovich_tutorial_2017, kunzel_metalearners_2019, dwivedi_stable_2020}. For CATE estimation, nonparametric machine learning methods have become standard, including the causal forest \citep{athey_generalized_2019}, Bayesian Additive Regression Trees (BART) \citep{hill_bayesian_2011}, \gls{bcf} \citep{hahn_bayesian_2020, caron_shrinkage_2022}, and doubly robust meta-learners \citep{kennedy_towards_2023, semenova_debiased_2021}. In parallel, a growing strand of work has focused on the data-driven discovery of interpretable subgroups, with decision-tree-based methods \citep{athey_recursive_2016, bargagli_stoffi_causal_2020, bargagli-stoffi_heterogeneous_2022, lee_discovering_2021} being particularly prominent due to their interpretability. Ensemble-based extensions such as the causal rule ensemble \citep{bargagli-stoffi_causal_2024} and causal distillation trees \citep{huang_distilling_2025} address the shortcomings of unstable single-tree methods by aggregating decision rules across many trees, yielding more stable and expressive subgroup representations when estimating CATE.
A limitation of recent work on HTE is that the identification for CATE rests on the assumption of regular assignment mechanisms, which is rarely defensible in observational studies with unobserved heterogeneity in treatment uptake. When a valid instrument is available, a relevant target estimand becomes the \gls{cace}, or Local Average Treatment Effect (LATE), nonparametrically identified for the subpopulation of compliers under the standard \gls{iv} assumptions \citep{imbens_identification_1994, angrist_identification_1996}. The interpretation of complier effects has been the subject of ongoing debate: because compliance status is a counterfactual quantity that is never directly observed, critics have argued that the CACE pertains to an unidentified subgroup and is therefore of limited policy relevance \citep{deaton_instruments_2010, swanson_think_2014}. In contrast, the LATE is often the most one can learn nonparametrically in the presence of unmeasured confounding without imposing restrictive effect homogeneity assumptions, and it remains informative about the underlying causal structure. Crucially, when covariates explain a substantial share of the variation in compliance, complier effects effectively coincide with conditional effects in identifiable subgroups, and the concerns about an unknown target population largely dissolve \citep{kennedy_sharp_2020}. This observation provides direct motivation for studying the \gls{ccace}. The \gls{ccace} is the \gls{cace} as a function of observed characteristics. Characterizing how complier effects vary along interpretable covariate profiles transforms the CACE from a property of an unobserved subgroup into a set of policy-relevant statements about identifiable populations.
Several methods have been developed to recover heterogeneous \gls{iv} effects, each targeting a distinct inferential object. Forest-based estimators such as the instrumental variable forest of \citet{wang_instrumental_2022} and the instrumental forest within the \gls{grf} framework \citep{athey_generalized_2019} target a unit-level conditional \gls{iv} function, delivering pointwise-consistent estimates with valid asymptotic inference but no explicit partition of the covariate space. The drivers of heterogeneity are recovered post hoc through variable importance scores or best linear projections. Partition-based procedures like the \gls{iv} tree of \citet{wang_instrumental_2022} and the matching procedure of \citet{johnson_detecting_2022} return an interpretable partition over which subgroup-level effects can be read off directly, paired with closed-testing inference.
We propose \gls{sbcf-iv}, a generalization of \gls{bcf-iv} tailored to settings in which the share of covariates that drive effect heterogeneity is small relative to $P$. The \gls{bcf-iv} algorithm of \citet{bargagli-stoffi_heterogeneous_2022} combines a \gls{bart}-based \citep{chipman_bart_2010} sum-of-trees estimator for the \gls{citt} with a shallow \gls{cart} post-processing step \citep{breiman_classification_1984}, yielding interpretable subgroup-level estimates of \gls{ccace} within a two-step procedure based on stratification and \gls{iv} estimation. By doing so, it reconciles the predictive accuracy of ensemble methods \citep{athey_generalized_2019, hartford_deep_2017} with the interpretability of single-tree approaches \citep{athey_recursive_2016, bargagli_stoffi_causal_2020, johnson_detecting_2022}. However, \gls{bcf-iv} inherits \gls{bart}'s uniform split-variable prior and therefore performs no targeted feature selection: when the covariate vector contains many irrelevant variables, as is typical in modern administrative or biomedical data, both the ensemble and the downstream \gls{cart} can spread splits across spurious moderators, weakening the discovered subgroups.
Our contribution is twofold. First, we diagnose the failure mode of \gls{bcf-iv} in high dimensions. \gls{bcf-iv} increasingly splits on spurious moderator variables as the share of irrelevant covariates grows, which degrades subgroup recovery and effect estimation.
We address this by estimating the \gls{citt} with the \gls{sbcf} of \citet{caron_shrinkage_2022}, whose sparsity-inducing Dirichlet prior, in the spirit of SoftBART \citep{linero_bayesian_2018, linero_bayesian_2018-1}, concentrates posterior mass on the few covariates that moderate the effect. The resulting estimation gains in our setting are attributable to this prior.
Second, we feed the posterior split frequencies into the subgroup-discovery \gls{cart} as variable-level costs, a component that steers the partition toward ensemble-relevant covariates and is most beneficial when the ensemble is not already sparse. Throughout, our scope follows that of \citet{bargagli-stoffi_heterogeneous_2022}: a binary randomized instrument, a binary treatment, and standard \gls{iv} identification assumptions.
Section~\ref{ch:sim_study} accordingly benchmarks \gls{sbcf-iv} against \gls{bcf-iv} as the direct ancestor and natural comparator under this setting, with a supplementary comparison to the
instrumental forest of \citet{athey_generalized_2019} and a cost-weighting ablation in Appendix~\ref{append:further_precision_results}.
The paper proceeds as follows. Section~\ref{ch:PO_irreg} sets up the potential outcomes framework and identifying assumptions for the \gls{ccace} under an irregular assignment mechanism. Section~\ref{ch:BCF_IV} introduces \gls{sbcf-iv}. Section~\ref{ch:sim_study} reports Monte Carlo evidence on its performance relative to \gls{bcf-iv} in high-dimensional settings based on tree-level and unit-level performance criteria. Section~\ref{ch:emp_appl} applies \gls{sbcf-iv} to two empirical studies: the Oregon Health Insurance Experiment (OHIE) \citep{finkelstein_oregon_2012, johnson_detecting_2022} and the 401(k) retirement plans dataset \citep{poterba_401k_1992, poterba_401k_1995, chernozhukov_doubledebiased_2018}. Section~\ref{ch:conclusion} concludes.
\section{Potential outcomes and irregular assignment}
\label{ch:PO_irreg}
We follow \citet{bargagli_stoffi_causal_2020} and \citet{bargagli-stoffi_heterogeneous_2022} and adopt Rubin's causal model, working within the irregular assignment framework of \citet{imbens_causal_2015}. For $N$ units indexed $i = 1, \dots, N$, let $Y_i \in \mathbb{R}$ denote the observed outcome, $Z_i \in \{0,1\}$ a binary instrument (assignment), $W_i \in \{0,1\}$ the actual treatment received, and $X_i \in \mathbb{R}^P$ the $i$-th row of an $N \times P$ matrix $X$ of pre-treatment covariates. Each unit is endowed with potential outcomes $Y_i(Z_i=z, W_i=w)$ and potential treatments $W_i(z)$ for $z, w \in \{0,1\}$, related to observed quantities by the consistency relations $Y_i = Y_i(Z_i, W_i)$ and $W_i = W_i(Z_i)$. The instrument $Z_i$ is unconfounded but the receipt $W_i$ may be confounded. This is the canonical \gls{iv} setting, and our inferential target is the \gls{ccace} in Definition \ref{defn:cCACE}, based on the latent subpopulation of compliers.
\begin{defn}[Compliance subgroups]
\label{defn:compliance}
Each unit belongs to one of four latent compliance subgroups, defined by the joint values of its potential treatments:
\[
G_i =
\begin{cases}
C, & W_i(0) = 0,\ W_i(1) = 1 \quad \text{(compliers)} \\
D, & W_i(0) = 1,\ W_i(1) = 0 \quad \text{(defiers)} \\
AT, & W_i(0) = 1,\ W_i(1) = 1 \quad \text{(always-takers)} \\
NT, & W_i(0) = 0,\ W_i(1) = 0 \quad \text{(never-takers)},
\end{cases}
\]
with conditional subgroup proportions $\pi_G(x) = \Pr(G_i = G \mid X_i = x)$ for $G \in \{C, D, AT, NT\}$.
\end{defn}
\begin{defn}[Conditional CACE]
\label{defn:cCACE}
The conditional Complier Average Causal Effect is the estimand
\[
\tau^{\text{CACE}}(x) \;\coloneqq\; \mathbb{E}\bigl[Y_i(1, W_i(1)) - Y_i(0, W_i(0)) \,\bigm|\, G_i = C,\, X_i = x\bigr],
\]
which, under the exclusion restriction in Assumption \ref{assump:identification_irregular}(d), reduces to $\mathbb{E}[Y_i(1) - Y_i(0) \mid G_i = C, X_i = x]$.
\end{defn}
Definition~\ref{defn:cCACE} fixes the target as a property of the latent complier subpopulation and it is not, by itself, an object computable from the observed distribution of $(Y_i, W_i, Z_i, X_i)$. Identification proceeds through the conditional \gls{itt} effect and the conditional complier share, both of which admit clean expressions in terms of observed conditional means.
\begin{defn}[Conditional ITT and complier share]
\label{defn:cITT}
The conditional \gls{itt} effect and the conditional complier share are
\begin{align*}
\text{ITT}_Y(x) &\;\coloneqq\; \mathbb{E}[Y_i \mid Z_i = 1, X_i = x] - \mathbb{E}[Y_i \mid Z_i = 0, X_i = x], \\
\pi_C(x) &\;\coloneqq\; \Pr(G_i = C \mid X_i = x).
\end{align*}
By a mixture argument over the compliance subgroups of Definition~\ref{defn:compliance},
\begin{align*}
\text{ITT}_Y(x) \;=\; \pi_C(x)\,\text{ITT}_{Y,C}(x) + \pi_D(x)\,\text{ITT}_{Y,D}(x) + \pi_{AT}(x)\,\text{ITT}_{Y,AT}(x) + \pi_{NT}(x)\,\text{ITT}_{Y,NT}(x),
\end{align*}
where $\text{ITT}_{Y,G}(x)$ denotes the conditional ITT among units of compliance types $G$ in Definition \ref{defn:compliance}.
\end{defn}
Definition~\ref{defn:cITT} makes explicit that $\text{ITT}_Y(x)$ is a covariate-weighted mixture and isolating $\tau^{\text{CACE}}(x) = \text{ITT}_{Y,C}(x)$ requires assumptions that neutralize the contributions of defiers, always-takers, and never-takers. The classical \citet{angrist_identification_1996} conditions for \gls{iv} estimation deliver point identification.
\begin{assump}[IV identification under irregular assignment]
\label{assump:identification_irregular} \leavevmode
\begin{enumerate}[label=(\alph*)]
\item \textit{\gls{SUTVA} / consistency:} $Y_i = Y_i(Z_i, W_i)$ and $W_i = W_i(Z_i)$, with no interference between units and no hidden treatment variants.
\item \textit{Relevance:} $\pi_C(x) > 0$ almost surely.
\item \textit{Unconfounded instrument:} $Z_i \perp\!\!\!\perp \bigl(\{Y_i(z, w)\}_{z,w \in \{0,1\}^2},\, W_i(0),\, W_i(1)\bigr) \,\bigm|\, X_i$.
\item \textit{Exclusion restriction:} $Y_i(z, w) = Y_i(w)$ for all $z, w \in \{0,1\}$.
\item \textit{Monotonicity:} $W_i(1) \geq W_i(0)$.
\end{enumerate}
\end{assump}
Assumption~\ref{assump:identification_irregular}(a) ensures consistency and rules out interference and hidden treatment variants. Relevance (b), together with monotonicity (e), guarantees $\pi_C(x) > 0$ almost surely so that the identification ratio below is well-defined. Monotonicity also rules out defiers ($\pi_D(x) = 0$), which must be defended on substantive grounds or enforced through a one-sided non-compliance design. The exclusion restriction (d) confines the effect of $Z_i$ on $Y_i$ to the channel through $W_i$. Under Assumption~\ref{assump:identification_irregular}, the complier estimand of Definition~\ref{defn:cCACE} is identified from the observed distribution.
\begin{prop}[Identification of $\tau^{\text{CACE}}(x)$]
\label{prop:identification}
Under Assumption~\ref{assump:identification_irregular},
\[
\tau^{\text{CACE}}(x) \;=\; \frac{\text{ITT}_Y(x)}{\pi_C(x)} \;=\; \frac{\mathbb{E}[Y_i \mid Z_i = 1, X_i = x] - \mathbb{E}[Y_i \mid Z_i = 0, X_i = x]}{\mathbb{E}[W_i \mid Z_i = 1, X_i = x] - \mathbb{E}[W_i \mid Z_i = 0, X_i = x]}.
\]
The proof, adapting \citet{angrist_identification_1996} to the conditional target as in \citet{bargagli-stoffi_heterogeneous_2022}, is given in Appendix~\ref{append:proof_cCACE}.
\end{prop}
Given a partition $\{\mathbb{X}_j\}_j$ of the covariate space, Proposition~\ref{prop:identification} motivates a sample-moment estimator targeting the subgroup-averaged complier effect
\begin{align}\label{eq:subgroup-cace}
\tau^{\text{CACE}}_{\mathbb{X}_j} \;\coloneqq\; \mathbb{E}\bigl[Y_i(W=1) - Y_i(W=0) \,\bigm|\, G_i = C,\, X_i \in \mathbb{X}_j\bigr],
\end{align}
obtained by replacing population conditional means in Proposition~\ref{prop:identification} with subgroup sample analogues.
\begin{defn}[Subgroup-wise 2SLS estimator]
\label{defn:cCACE_estimator}
For $X_i =x \in \mathbb{X}_j$, with $N_{z,j} = \sum_{l:\, X_l \in \mathbb{X}_j} \mathbf{1}\{Z_l = z\}$ the count of $\mathbb{X}_j$-units assigned to $Z_l = z \in \{0,1\}$,
\begin{align*}
\widehat\tau^{\text{CACE}}(x) \coloneqq \widehat\tau^{\,\text{2SLS}}_{\mathbb{X}_j} \;=\; \frac{\dfrac{1}{N_{1,j}} \sum_{l:\, X_l \in \mathbb{X}_j} Y_l Z_l \;-\; \dfrac{1}{N_{0,j}} \sum_{l:\, X_l \in \mathbb{X}_j} Y_l (1 - Z_l)}{\dfrac{1}{N_{1,j}} \sum_{l:\, X_l \in \mathbb{X}_j} W_l Z_l \;-\; \dfrac{1}{N_{0,j}} \sum_{l:\, X_l \in \mathbb{X}_j} W_l (1 - Z_l)}
\end{align*}
targets $\tau^{\text{CACE}}_{\mathbb{X}_j}$.
\end{defn}
Equivalently, $\widehat\tau^{\,\text{2SLS}}_{\mathbb{X}_j}$ is the Two-Stage Least Squares estimator on the subgroup-restricted simultaneous system
\begin{equation}\label{eq:simultaneous_equations}
Y_{i, \mathbb{X}_j} \;=\; \kappa_{\mathbb{X}_j} + \tau^{\text{CACE}}_{\mathbb{X}_j}\,W_{i, \mathbb{X}_j} + \varepsilon_{i, \mathbb{X}_j}, \qquad W_{i, \mathbb{X}_j} \;=\; \pi_{0, \mathbb{X}_j} + \pi_{C, \mathbb{X}_j}\,Z_{i, \mathbb{X}_j} + \nu_{i, \mathbb{X}_j},
\end{equation}
with intercepts $\kappa_{\mathbb{X}_j}$ and $\pi_{0, \mathbb{X}_j}$, error terms $\varepsilon_{i, \mathbb{X}_j}$ and $\nu_{i, \mathbb{X}_j}$, while $\mathbb{E}[\varepsilon_{i, \mathbb{X}_j}] = \mathbb{E}[\nu_{i, \mathbb{X}_j}] = 0$ and $\mathbb{E}[Z_{i, \mathbb{X}_j}\,\nu_{i, \mathbb{X}_j}] = 0$. Under Assumption~\ref{assump:identification_irregular} and a sufficient number of i.i.d.\ observations within each $\mathbb{X}_j$, $\widehat\tau^{\,\text{2SLS}}_{\mathbb{X}_j}$ is consistent and asymptotically normal for $\tau^{\text{CACE}}_{\mathbb{X}_j}$, with the reduced form and formal asymptotic results collected in Appendix~\ref{append:theorem_2SLS}.
Definition~\ref{defn:cCACE_estimator} presumes that the partition $\{\mathbb{X}_j\}_j$ is known. The contribution of \gls{bcf-iv} \citep{bargagli-stoffi_heterogeneous_2022} is to discover this partition from the data through (i) an honest split of the sample into disjoint discovery and inference subsets $\mathcal{I}_{\text{disc}}$ and $\mathcal{I}_{\text{inf}}$; (ii) interpretable discovery of heterogeneity on $\mathcal{I}_{\text{disc}}$; and (iii) inference for $\tau^{\text{CACE}}_{\mathbb{X}_j}$ on $\mathcal{I}_{\text{inf}}$. Our \gls{sbcf-iv} algorithm in Algorithm~\ref{alg:sbcf-iv} inherits this three-step structure of \citet{bargagli-stoffi_heterogeneous_2022} with adaptations for a high-dimensional covariate setup described in the next section.
\section{Shrinkage Bayesian Causal Forest with Instrumental Variable}
\label{ch:BCF_IV}
We propose \gls{sbcf-iv}, an extension of \gls{bcf-iv} \citep{bargagli-stoffi_heterogeneous_2022} to settings with many irrelevant covariates. The overall structure of honest sample splitting, data-driven discovery of heterogeneous subgroups on $\mathcal{I}_{\text{disc}}$, and \gls{2sls} inference on $\mathcal{I}_{\text{inf}}$ is preserved. Our adaptations concern the discovery step where we replace BCF with the sparsity-inducing Shrinkage BCF (SBCF) of \citet{caron_shrinkage_2022} and feed its posterior variable-selection frequencies into the subgroup-finding tree as variable-level costs. Algorithm~\ref{alg:sbcf-iv} summarizes the full procedure.
Working on $\mathcal{I}_{\text{disc}}$, we separately model the numerator and denominator of the identification ratio in Proposition~\ref{prop:identification}. Following \citet{hahn_bayesian_2020}, we adopt the semi-parametric specification
\footnote{Background on CART, BART, BCF, and SBCF is collected in Appendix~\ref{append:bart_bcf_cart}.}
\begin{equation}\label{eq:cond_exp_Y}
\mathbb{E}[Y_i \mid Z_i = z, X_i = x] \;=\; \mu\bigl(e(x), x\bigr) \;+\; \text{ITT}_Y(x)\,z,
\end{equation}
where $e(x) = \Pr(Z_i = 1 \mid X_i = x)$ is the instrument's propensity score, included as a covariate in the control function $\mu(e(x), x)$ to mitigate regularization-induced confounding and targeted selection. Independent BART priors are placed on $\mu(e(x), x)$ and $\text{ITT}_Y(x)$, with depth-penalty parameters $(\eta, \beta) = (0.25, 3)$ on $\text{ITT}_Y(x)$ favoring shallow trees and hence simpler heterogeneity patterns. The compliance component is modeled analogously via
\begin{equation}\label{eq:cond_exp_W}
\mathbb{E}[W_i \mid Z_i = z, X_i = x] \;=\; \delta(z, x),
\end{equation}
with a BART probit prior on $\delta(z, x)$ \citep{hill_bayesian_2011}. Combining estimates yields a pointwise complier-share estimate $\widehat\pi_C^{\,\text{SBCF}}(x) = \widehat\delta(1, x) - \widehat\delta(0, x)$ and a pointwise complier-effect estimate $\widehat\tau^{\,\text{SBCF}}(x) = \widehat{\text{ITT}}_Y^{\,\text{SBCF}}(x) / \widehat\pi_C^{\,\text{SBCF}}(x)$. We emphasize that $\widehat\tau^{\,\text{SBCF}}(x)$ is a posterior estimate on $\mathcal{I}_{\text{disc}}$ used solely as a heterogeneity signal for the tree-fitting step, while we conduct final inference for the subgroup target $\tau^{\text{CACE}}_{\mathbb{X}_j}$ with $\widehat\tau^{\,\text{2SLS}}_{\mathbb{X}_j}$ from Definition \ref{defn:cCACE_estimator} on $\mathcal{I}_{\text{inf}}$.
\gls{sbcf-iv} departs from \gls{bcf-iv} in the prior on the split-variable selection probabilities $s = (s_1, \ldots, s_P)$. BCF uses a uniform $s_j = 1/P$. Instead, SBCF imposes a sparsity-inducing Dirichlet prior,
\begin{equation}\label{eq:sbcf_prior}
s \sim \text{Dirichlet}\!\left(\tfrac{\alpha}{P}, \ldots, \tfrac{\alpha}{P}\right), \qquad \frac{\alpha}{\alpha + \rho} \sim \text{Beta}(a, b),
\end{equation}
with defaults $(a, b, \rho) = (0.5, 1, P)$ \citep{caron_shrinkage_2022}. Small $\alpha$ concentrates mass on few covariates, and the hyperprior on $\alpha/(\alpha + \rho)$ lets the data determine the degree of sparsity, with the preference for sparsity strengthening as $P$ grows. We place separate Dirichlet priors on the split probabilities $s_\mu$ and $s_{\text{ITT}_Y}$ of the two components of \eqref{eq:cond_exp_Y}, using $\rho_\mu = P + 1$ (to accommodate the propensity score as an extra covariate in $\mu$) and $\rho_{\text{ITT}_Y} = P/2$ (to further concentrate mass on few active splits in the treatment effect component). The same sparsity-inducing prior in \eqref{eq:sbcf_prior} is applied in the SoftBART probit model used for $\delta(z, x)$ in \eqref{eq:cond_exp_W}, with $\rho_\delta = P+1$. Appendix~\ref{append:sbcf_priors} provides more information on the full sparsity-inducing prior specifications. We fit a shallow CART \citep{breiman_classification_1984} to the pointwise estimates $\widehat\tau^{\,\text{SBCF}}(X_i)$ to recover an interpretable partition $\{\mathbb{X}_j\}_j$. The posterior split frequencies $\widehat{s}_{\text{ITT}_Y}$ indicate which covariates drive heterogeneity and we pass these split frequencies to \texttt{rpart} \citep{rpart} through the \texttt{cost} argument, setting variable-level costs to
\begin{equation}\label{eq:cost}
c_{\text{psp}} \;=\; \frac{\max\{\widehat{s}_{\text{ITT}_Y}\}}{\widehat{s}_{\text{ITT}_Y}}.
\end{equation}
Because \texttt{rpart} divides split-improvement by the candidate variable's cost, \eqref{eq:cost} upweights covariates with a higher posterior inclusion probability at each split, in a manner analogous to variable-importance weighting in random forests \citep{breiman_random_2001}.
The cost vector $c_{\text{psp}}$ is defined via $\widehat s_{\text{ITT}_Y}$, the posterior
split frequencies of the cITT component, mirroring the ITT-anchored discovery step of \gls{bcf-iv} \citep{bargagli-stoffi_heterogeneous_2022}. Posterior split frequencies can be exactly zero for covariates the ensemble never selects given finitely many draws. Therefore, we floor $\widehat s_{\text{ITT}_Y}$ at a small $\varepsilon_c > 0$ before inversion in \eqref{eq:cost}, capping the cost of a never-selected covariate at $\max\{\widehat s_{\text{ITT}_Y}\}/\varepsilon_c$. Equation \eqref{eq:cost} is thus always well-defined and assigns such covariates a large but finite penalty.
The choice of $\widehat s_{\text{ITT}_Y}$ for the cost vector is justified by a coherence property of the two-step procedure: within a leaf, the discovery average and the \gls{2sls} estimand differ only by the within-leaf covariance between the complier share and the complier effect, a leaf-level instance of the compliance-weighting of instrumental variable estimands \citep{angrist_twostage_1995, abadie_semiparametric_2003, frolich_nonparametric_2007}.
When within-leaf variation of $\pi_C(x)$ is small, heterogeneity in $\tau^{\text{CACE}}(x)$ is inherited almost entirely from $\text{ITT}_Y(x)$ and $\widehat s_{\text{ITT}_Y}$ is the natural cost weight (see Corollary~\ref{cor:coherence} in Appendix~\ref{append:proof_cCACE}). Aggregating $\widehat s_{\text{ITT}_Y}$ with the compliance component's split frequencies, for regimes with substantial denominator-driven heterogeneity, is an extension left to future work.
Inference on the discovered partition follows \gls{bcf-iv} without modification. For each node $\mathbb{X}_j$ of the tree learned on $\mathcal{I}_{\text{disc}}$, we compute $\widehat\tau^{\,\text{2SLS}}_{\mathbb{X}_j}$ on $\mathcal{I}_{\text{inf}}$ via the simultaneous system in \eqref{eq:simultaneous_equations}. Consistency and asymptotic normality under subgroup-level moment conditions are collected in Appendix~\ref{append:theorem_2SLS}. To guard against spurious heterogeneity, nodes flagged by a first-stage $F$-test for weak instruments are discarded, and $p$-values across leaves are adjusted for the familywise error rate using Holm-corrections for adjusted $p$-values \citep{holm_simple_1979, bargagli-stoffi_heterogeneous_2022}.
\begin{algorithm}[t]
\caption{Shrinkage Bayesian Causal Forest with Instrumental Variable (SBCF-IV)}\label{alg:sbcf-iv}
\begin{algorithmic}[1]
\Require $N$ units $\{(X_i, Z_i, W_i, Y_i)\}_{i=1}^N$.
\Ensure Tree-structured partition of the covariate space with node-level CACE estimates.
\Statex
\Statex \textbf{Step 1: Honest splitting.}
\State Randomly partition the sample into $\mathcal{I}_{\text{disc}}$ and $\mathcal{I}_{\text{inf}}$ (defaults to half-size splits).
\Statex
\Statex \textbf{Step 2: Discovery} (on $\mathcal{I}_{\text{disc}}$).
\State Estimate $\widehat{\text{ITT}}_Y^{\,\text{SBCF}}(x)$ via SBCF under the sparsity prior \eqref{eq:sbcf_prior}; save posterior split frequencies $\widehat{s}_{\text{ITT}_Y}$.
\State Estimate $\widehat\pi_C^{\,\text{SBCF}}(x)$ via a SoftBART probit with similar sparsity prior.
\State Form the pointwise heterogeneity signal $\widehat\tau^{\,\text{SBCF}}(x) = \widehat{\text{ITT}}_Y^{\,\text{SBCF}}(x) / \widehat\pi_C^{\,\text{SBCF}}(x)$.
\State Fit a shallow CART to $\{(\widehat\tau^{\,\text{SBCF}}(x), X_i)\}$ with variable-level costs $c_{\text{psp}}$ as in \eqref{eq:cost}; the resulting partition is $\{\mathbb{X}_j\}_j$.
\Statex
\Statex \textbf{Step 3: Inference} (on $\mathcal{I}_{\text{inf}}$).
\State For every node $\mathbb{X}_j$, compute $\widehat\tau^{\,\text{2SLS}}_{\mathbb{X}_j}$ via \eqref{eq:simultaneous_equations}, targeting $\tau^{\text{CACE}}_{\mathbb{X}_j}$.
\State Run first-stage weak-instrument tests and adjust leaf-level $p$-values for the familywise error rate.
\Statex
\State \Return The pruned tree with node-level CACE estimates and adjusted inference.
\end{algorithmic}
\end{algorithm}
\section{Simulation study}
\label{ch:sim_study}
We combine the design of \citet{bargagli-stoffi_heterogeneous_2022} with the high-dimensional setup of \citet{caron_shrinkage_2022} to assess \gls{sbcf-iv} and \gls{bcf-iv} in settings with many irrelevant covariates. Appendix~\ref{append:sim_design} presents a detailed description of the complete simulation design. For each of $N = 1{,}000$ units, we generate a binary instrument $Z_i \sim \text{Bernoulli}(0.5)$, a covariate vector $X_i \in \mathbb{R}^P$ with $P \in \{10, 50, 100\}$ (half binary, half continuous), and potential outcomes and treatments according to
\begin{align}\label{eq:dgp}
\begin{split}
W_i(0) = 0, \quad W_i(1) &\sim \text{Bernoulli}(\pi_{\text{comp}} = 0.75), \\
Y_i(0) = \mu(X_i) + \epsilon_i, \quad Y_i(1) &= Y_i(0) + W_i(1)\,\tau^{\text{CACE}}(X_i),
\end{split}
\end{align}
with observed values $W_i = Z_i W_i(1)$ and $Y_i$ defined analogously under \gls{SUTVA}. The Gaussian error term reads $\epsilon_i \sim \mathcal{N}(0,1)$. Three features of \eqref{eq:dgp} are central to the estimand of interest. First, $W_i(0) = 0$ imposes one-sided non-compliance, ruling out defiers and always-takers by design and making monotonicity (Assumption~\ref{assump:identification_irregular}(e)) hold exactly. Second, we choose the compliance rate $\pi_{\text{comp}} = 0.75$ to control the strength of the instrument. Third, heterogeneity in the conditional CACE is confined to the first two binary covariates,
\begin{equation}\label{eq:cace_truth}
\tau^{\text{CACE}}(X_i) =
\begin{cases}
\phantom{-}k, & X_i \in l_1 = \{X_{i,1} = 0,\, X_{i,2} = 0\}, \\
-k, & X_i \in l_2 = \{X_{i,1} = 1,\, X_{i,2} = 1\}, \\
\phantom{-}0, & \text{otherwise},
\end{cases}
\end{equation}
with effect size $k \in \{0, 0.2, 0.4, \dots, 2\}$.\footnote{We use $l_1, l_2$ to denote the true heterogeneity subgroups in the \gls{DGP}, distinct from the discovered subgroups $\{\mathbb{X}_j\}_j$ produced by the CART step of \gls{sbcf-iv}. A successful run recovers $l_1$ and $l_2$ as leaves of the tree, i.e., $\mathbb{X}_j = l_j$ for $j \in \{1, 2\}$ up to labeling.} The remaining covariates, including all continuous ones, are irrelevant for $\tau^{\text{CACE}}(x)$, so the true sparsity level in the treatment effect component grows with $P$. This isolates the setting \gls{sbcf-iv} is designed for: relevance concentrated on two covariates while $P - 2$ noise variables compete for splits. The control function $\mu(X_i)$ is adopted from \citet{caron_shrinkage_2022} and depends only on three continuous covariates through a nonlinear combination of sine, quadratic, and absolute-value terms. Its explicit form is given in Appendix~\ref{append:sim_design}.
We evaluate \gls{sbcf-iv} along three complementary dimensions: tree-level subgroup detection, unit-level classification, and unit-level estimation precision with uncertainty quantification. Tree-level recovery is captured by the \gls{DR} and \gls{FDR}, which record whether the true heterogeneity regions $l_1, l_2$ are recovered. Unit-level classification is assessed on $\mathcal{I}_{\text{inf}}$ via Recall, Precision, False Positive Rate (FPR), and $F$-score, translating structural detection into how reliably individual observations are sorted into statistically significant leaves. Estimation precision is then quantified by the per-unit bias, MSE, and $95\%$ coverage of the resulting conditional CACE estimates. Formal definitions and further discussions are deferred to Appendix~\ref{append:sim_metrics}.
Figure~\ref{fig:dr_fdr} shows that \gls{sbcf-iv} dominates \gls{bcf-iv} on both tree-level criteria across all three covariate dimensions. For detection, \gls{sbcf-iv}'s \gls{DR} rises steeply once $k \geq 0.8$ and saturates near one for $k \geq 1.4$, essentially uniformly in $P$: adding irrelevant covariates does not meaningfully degrade the algorithm's ability to recover $l_1$ and $l_2$ as leaves. \gls{bcf-iv}, by contrast, only begins to detect the true subgroups for $k \geq 1$ in the low-dimensional case ($P = 10$), and its \gls{DR} flattens near 0.3 at $k = 2$. For $P \in \{50, 100\}$, \gls{bcf-iv}'s \gls{DR} stays close to zero throughout the grid, indicating that the uniform split-variable prior fails to concentrate on the two binary covariates that drive heterogeneity once they are buried among many noise variables. The ordering on \gls{FDR} is equally clear. \gls{sbcf-iv} maintains an \gls{FDR} near zero across the entire $(k, P)$ grid, so its detection gains do not come at the cost of spurious discoveries. \gls{bcf-iv}, by contrast, exhibits an \gls{FDR} that rises with $k$ and is largest at $P=10$, reaching roughly $0.75$ at $k = 2$ with $P = 10$ and remaining above $0.25$ for larger $P$. Most of \gls{bcf-iv}'s rare "discoveries" in high-dimensional settings are false positives. The two panels indicate that the sparsity-inducing prior improves both detection and false-discovery control, recovering true heterogeneity where \gls{bcf-iv} misses it while controlling spurious leaves where \gls{bcf-iv} generates them.
\begin{figure}[h]
\centering
\caption{Tree-level subgroup detection across effect size $k$ and covariate dimension $P$.}
\label{fig:dr_fdr}
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\begin{notes}
The top row reports the Detection Rate (DR), defined in \eqref{eq:DR} as the average share of true heterogeneity subgroups $l_1, l_2$ recovered as leaves of the discovered tree. The bottom row reports the False Detection Rate (FDR), defined in \eqref{eq:FDR} as the share of replications in which at least one spurious leaf is flagged as significant at level $\alpha = 0.05$. Results are averaged over $M=500$ Monte Carlo replications with $N = 1{,}000$, and compare \gls{sbcf-iv} (orange) with \gls{bcf-iv} (gray). Full results in Table \ref{tab:rule} of Appendix \ref{append:full_precision_results}.
\end{notes}
\end{figure}
Figure~\ref{fig:precision.Fscore} confirms that \gls{sbcf-iv}'s tree-level advantages in Figure~\ref{fig:dr_fdr} translate directly into sharper unit-level sorting. Here, we report Precision and the $F$-score. The full set of classification metrics is reported in Figure \ref{fig:full_classification} of Appendix~\ref{append:supmat_simstudy}. Precision follows the same pattern across both algorithms: for $k \geq 1$, \gls{sbcf-iv}'s Precision rises steeply and saturates near one by $k = 1.4$ across all three values of $P$, meaning that nearly every unit assigned to a significant leaf is in a truly heterogeneous subgroup. \gls{bcf-iv} reaches roughly $\text{Precision} = 0.4$ at $k = 2$ under $P = 10$ and stays close to zero for $P \in \{50, 100\}$. In the high-dimensional regime, nearly all of \gls{bcf-iv}'s unit-level positive classifications are misclassifications, which is a consequence of \gls{FDR} results in Figure \ref{fig:dr_fdr}. The $F$-score mirrors this ranking and is the more informative single summary since it penalizes both missed heterogeneity and spurious flags. \gls{sbcf-iv}'s $F$-score converges to one for $k \geq 1.4$ uniformly in $P$, indicating that the algorithm both identifies the correct units and avoids misclassification. \gls{bcf-iv}'s $F$-score remains below $0.4$ throughout the $(k, P)$ grid and flatlines near zero once $P \geq 50$.
\begin{figure}[h]
\centering
\caption{Unit-level classification performance across effect size $k$ and covariate dimension $P$.}
\label{fig:precision.Fscore}
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\begin{notes}
The top row reports Precision and the bottom row the $F$-score, both defined in \eqref{eq:class_metrics} and computed over all units in $\mathcal{I}_{\text{inf}}$ with significance evaluated at $\alpha = 0.05$ using Holm-adjusted $p$-values. Results are averaged over $M$ Monte Carlo replications with $N = 1{,}000$, and compare \gls{sbcf-iv} (orange) with \gls{bcf-iv} (gray). We report full results in Table \ref{tab:full_classification} of Appendix \ref{append:full_precision_results}.
\end{notes}
\end{figure}
Table~\ref{tab:short_precision_uncertainty} reports estimation precision and uncertainty quantification for $\widehat\tau^{\text{CACE}}(x)$ and $k \in \{0, 1, 2\}$, while qualitatively similar results for the full $k$-range are deferred to Appendix~\ref{append:full_precision_results}. We summarize three emerging patterns.
First, point estimation accuracy diverges with effect size. For $k \geq 1$, \gls{sbcf-iv} consistently delivers lower MSE and absolute bias than \gls{bcf-iv}, with the gap widening as $k$ grows: \gls{sbcf-iv}'s MSE is about a quarter of BCF-IV's at $(k, P) = (2, 10)$ and about a tenth at $(k, P) = (2, 100)$.
\gls{bcf-iv}'s MSE grows with $P$, while \gls{sbcf-iv}'s remains flat, confirming that the dimension-invariance seen in Figures~\ref{fig:dr_fdr} and~\ref{fig:precision.Fscore} extends to point estimation accuracy.
Second, \gls{bcf-iv}'s absolute bias grows from $0.29$ at $(k, P) = (0, 10)$ to $1.5$ at $(k, P) = (2, 100)$ because its uniform split-variable prior dilutes the estimated effect across noise covariates. \gls{sbcf-iv}'s absolute bias stays below $0.5$ throughout the grid.
Third, uncertainty quantification degrades with increasing $k$ and $P$ for \gls{bcf-iv}. \gls{sbcf-iv} maintains coverage near the nominal $0.95$ level across the entire $(k, P)$ grid, so its confidence intervals remain well-calibrated even when the true effect is large or the covariate space is high-dimensional. \gls{bcf-iv}'s coverage, by contrast, drops with $k$. Taken together, the three simulation dimensions indicate that the sparsity prior improves discovery, classification, point estimation, and inference simultaneously, with \gls{sbcf-iv} delivering near-nominal coverage at comparable interval width.
\begin{table}[H]
\centering
\caption{Estimation precision and uncertainty quantification for $\widehat{\tau}^{\text{CACE}}(x)$.}
\label{tab:short_precision_uncertainty}
\centering
\resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
\begin{tabular}[t]{ll>{\raggedleft\arraybackslash}p{0.25cm}rrrrr>{\raggedleft\arraybackslash}p{0.25cm}rrrrr}
\toprule
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{5}{c}{\textbf{BCF-IV}} & \multicolumn{1}{c}{} & \multicolumn{5}{c}{\textbf{SBCF-IV}} \\
\cmidrule(l{3pt}r{3pt}){4-8} \cmidrule(l{3pt}r{3pt}){10-14}
$P$ & $k$ & & \multicolumn{1}{c}{MSE} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{Abs. Bias} & \multicolumn{1}{c}{Coverage} & \multicolumn{1}{c}{Width} & & \multicolumn{1}{c}{MSE} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{Abs. Bias} & \multicolumn{1}{c}{Coverage} & \multicolumn{1}{c}{Width}\\
\midrule
& 0 & & \makecell{0.188 (0.163)} & \makecell{-0.004 (0.225)} & \makecell{0.286 (0.140)} & \makecell{0.948 (0.176)} & \makecell{1.434 (0.219)} & & \makecell{0.191 (0.161)} & \makecell{-0.004 (0.222)} & \makecell{0.314 (0.131)} & \makecell{0.957 (0.131)} & \makecell{1.600 (0.176)}\\
& 1 & & \makecell{0.473 (0.333)} & \makecell{0.000 (0.286)} & \makecell{0.569 (0.240)} & \makecell{0.695 (0.337)} & \makecell{1.600 (0.210)} & & \makecell{0.298 (0.327)} & \makecell{-0.002 (0.308)} & \makecell{0.426 (0.256)} & \makecell{0.899 (0.235)} & \makecell{1.794 (0.160)}\\
\multirow{-3}{*}{\raggedright\arraybackslash 10} & 2 & & \makecell{0.931 (0.729)} & \makecell{0.014 (0.446)} & \makecell{0.811 (0.354)} & \makecell{0.478 (0.335)} & \makecell{1.697 (0.174)} & & \makecell{0.214 (0.211)} & \makecell{0.012 (0.337)} & \makecell{0.368 (0.199)} & \makecell{0.953 (0.149)} & \makecell{1.840 (0.141)}\\
\cmidrule{1-14}
& 0 & & \makecell{0.205 (0.244)} & \makecell{-0.003 (0.223)} & \makecell{0.273 (0.145)} & \makecell{0.953 (0.178)} & \makecell{1.399 (0.244)} & & \makecell{0.214 (0.151)} & \makecell{-0.003 (0.222)} & \makecell{0.358 (0.140)} & \makecell{0.960 (0.111)} & \makecell{1.769 (0.143)}\\
& 1 & & \makecell{0.785 (0.472)} & \makecell{0.012 (0.241)} & \makecell{0.744 (0.288)} & \makecell{0.454 (0.404)} & \makecell{1.495 (0.232)} & & \makecell{0.213 (0.232)} & \makecell{-0.003 (0.309)} & \makecell{0.362 (0.203)} & \makecell{0.949 (0.151)} & \makecell{1.830 (0.140)}\\
\multirow{-3}{*}{\raggedright\arraybackslash 50} & 2 & & \makecell{2.088 (1.446)} & \makecell{-0.008 (0.299)} & \makecell{1.291 (0.508)} & \makecell{0.202 (0.256)} & \makecell{1.593 (0.255)} & & \makecell{0.204 (0.203)} & \makecell{0.002 (0.320)} & \makecell{0.359 (0.192)} & \makecell{0.957 (0.145)} & \makecell{1.836 (0.138)}\\
\cmidrule{1-14}
& 0 & & \makecell{0.196 (0.234)} & \makecell{-0.020 (0.226)} & \makecell{0.253 (0.143)} & \makecell{0.957 (0.180)} & \makecell{1.325 (0.215)} & & \makecell{0.217 (0.162)} & \makecell{-0.021 (0.226)} & \makecell{0.360 (0.143)} & \makecell{0.951 (0.111)} & \makecell{1.785 (0.136)}\\
& 1 & & \makecell{0.954 (0.418)} & \makecell{0.019 (0.220)} & \makecell{0.872 (0.251)} & \makecell{0.273 (0.351)} & \makecell{1.417 (0.243)} & & \makecell{0.230 (0.215)} & \makecell{0.005 (0.319)} & \makecell{0.387 (0.199)} & \makecell{0.950 (0.149)} & \makecell{1.826 (0.139)}\\
\multirow{-3}{*}{\raggedright\arraybackslash 100} & 2 & & \makecell{2.551 (1.522)} & \makecell{0.013 (0.257)} & \makecell{1.454 (0.526)} & \makecell{0.157 (0.222)} & \makecell{1.535 (0.235)} & & \makecell{0.230 (0.238)} & \makecell{0.022 (0.340)} & \makecell{0.376 (0.212)} & \makecell{0.942 (0.164)} & \makecell{1.835 (0.146)}\\
\bottomrule
\end{tabular}}
\begin{notes}
MSE, Bias, Absolute Bias, as well as Coverage and Width of 95\% confidence intervals. All values are averaged over $M = 500$ Monte Carlo replications with $N = 1{,}000$ and standard deviations across Monte Carlo replications are reported in parentheses. Results pool subgroups $l_1$ and $l_2$ and compare BCF-IV (left) with SBCF-IV (right). We focus on $k \in \{ 0, 1, 2\}$ and $P \in \{10, 50, 100\}$ for visibility. Table \ref{tab:full_precision_results} of Appendix~\ref{append:full_precision_results} reports full results.
\end{notes}
\end{table}
To check whether the improvements above are specific to the comparison of \gls{sbcf-iv} with \gls{bcf-iv} or to the cost weighting of \eqref{eq:cost}, Appendix~\ref{append:further_precision_results} reports an analysis that considers the instrumental forest of the \gls{grf} framework as an additional benchmark method \citep{athey_generalized_2019}. We further cross each method that estimates the heterogeneity signal (\gls{bcf}, \gls{sbcf}, \gls{grf}) in line four of Algorithm \ref{alg:sbcf-iv} with the split-frequency cost of \eqref{eq:cost}. Table \ref{tab:competitors} provides an overview of all six methods as variants of Algorithm \ref{alg:sbcf-iv}. The simulation shows that the sparsity-inducing prior, not the cost weighting, is the major source of the improvement. The two \gls{sbcf-iv} variants (with and without cost weighting) are nearly indistinguishable across the $(k,P)$ grid, consistent with Corollary~\ref{cor:coherence}. In contrast, \gls{grf-iv} degrades with $P$ similary to \gls{bcf-iv}.
The cost weighting helps the non-sparse methods that are based on either \gls{grf} or \gls{bcf}, for which it partially restores detection and coverage while improving precision of the final estimates but without matching the performance of \gls{sbcf-iv}. We present full tables and figures in Appendix~\ref{append:further_precision_results}.
\section{Empirical application}
\label{ch:emp_appl}
We apply \gls{sbcf-iv} to two empirical datasets, the Oregon Health Insurance Experiment (OHIE) \citep{finkelstein_oregon_2012, johnson_detecting_2022} and the 401(k) dataset \citep{poterba_401k_1992, chernozhukov_doubledebiased_2018, belloni_program_2017}. The OHIE was a randomized controlled trial conducted in 2008 to assess the effects of expanding Medicaid coverage on health outcomes, financial security, and healthcare utilization. The US state Oregon used a lottery system to allocate a limited number of Medicaid spots to uninsured, low-income adults \citep{finkelstein_oregon_2012, johnson_detecting_2022}. This created a natural experiment, allowing researchers to compare those who received Medicaid to those who did not. Following \citet{johnson_detecting_2022}, we apply \gls{sbcf-iv} to the OHIE data. The outcome is the number of days in the past month on which poor physical or mental health did not impair the respondent's usual activities (a survey count between 0 and 30, with higher values indicating better health). \gls{sbcf-iv} identifies a single complier subgroup with a positive, statistically significant Medicaid effect: English-preferring individuals aged between 38 and 59 (effect: $2.263$; adjusted $p$-value: $0.0945$), in contrast to the two subgroups reported by \citet{johnson_detecting_2022}. Estimated subgroup compliance rates range from $19\%$ to $32\%$, in line with \citet{johnson_detecting_2022} and exhibiting only modest variation relative to the heterogeneity in conditional \gls{itt} effects. Both analyses thus locate the dominant source of complier-effect heterogeneity in the variation of the \gls{ccace} numerator rather than in compliance probabilities. Appendix \ref{sec:OHIE} re-investigates the empirical analysis presented in \citet{johnson_detecting_2022} in more detail.
On the 401(k) data, \gls{sbcf-iv} partitions eligible households into seven subgroups, of which two possess adjusted $p$-values at the 10\% level: the bulk subgroup with income below \$68{,}810 (89\% of the inference sample, $\widehat{\tau}^{\text{CACE}}(x) = \$17{,}818$) and a small upper-middle-income subgroup with household income between $\$92{,}690$ and $\$110{,}400$ with $\widehat{\tau}^{\text{CACE}}(x) = \$53{,}422$.
The remaining five leaves show adjusted $p$-values without statistical significance. Four carry inference-sample shares at or below 1\% and the fifth below 7\%, which we read as small-sample artifacts.
The discovered splits run almost entirely along income, aligning with the lifecycle and earnings-gradient emphasis of the classical 401(k) saving literature \citep{poterba_401k_1992, engen_effects_2000}. As in the OHIE application, complier shares vary far less than the conditional CACE across leaves. The high-mass leaves cluster near
a complier share of 0.70, so the heterogeneity is driven by variation in the conditional \gls{itt} numerator rather than in the complier denominator. This mirrors the simulation evidence in Section~\ref{ch:sim_study} that \gls{sbcf-iv} recovers \gls{itt}-driven partitions reliably. Both statistically significant leaf estimates exceed the overall cross-fitted
\gls{cace} of $\$9{,}000$--$\$13{,}000$ reported by \citet{chernozhukov_doubledebiased_2018}, the bulk leaf modestly (by $\$5{,}000$--$\$9{,}000$) and the upper-middle-income leaf substantially. We present this as a descriptive comparison only: the gap
is consistent with several explanations (genuine
income-based heterogeneity in the \gls{cace}, post-selection effects from
reporting leaves after screening, and the
upward bias from unobserved saver heterogeneity flagged by
\citet{engen_illusory_1996}) which the present design cannot separate. We provide an extended discussion of this second empirical application in Appendix~\ref{sec:401k}.
\section{Conclusion}
\label{ch:conclusion}
This paper introduces \gls{sbcf-iv}, an extension of \gls{bcf-iv} \citep{bargagli-stoffi_heterogeneous_2022} that combines Bayesian shrinkage with subgroup-level estimation of the conditional CACE under imperfect compliance in high-dimensional covariate settings. \gls{sbcf-iv} preserves the interpretive structure of \gls{bcf-iv} with an explicit partition over compliers and subgroup-level \gls{cace} while introducing regularization needed to make that structure stable in high dimensions \citep{caron_shrinkage_2022}. The simulation study in Section \ref{ch:sim_study} demonstrates for varying effect sizes $k$ that, as the covariate dimension $P$ grows, \gls{sbcf-iv} preserves subgroup discovery rates, precision, and nominal coverage. In comparison, \gls{bcf-iv} exhibits a degradation along most criteria, leading to a loss of inferential reliability. Applied to two empirical applications, \gls{sbcf-iv} recovers interpretable partitions of compliers whose \gls{ccace} estimates differ across subgroups, illustrating its practical value beyond the simulated regime.
Several limitations warrant emphasis. First, we establish no selection- or partition-consistency guarantee for the discovery step. The sparsity prior's variable-selection benefit is that of \citet{caron_shrinkage_2022}, which we demonstrate empirically in our \gls{ccace} setting. The discovery--inference coherence we invoke is a population-target alignment rather than a finite-sample recovery result. Second, the current formulation is restricted to binary instruments and binary treatments, and the honest sample split between $\mathcal{I}_{\text{disc}}$ and $\mathcal{I}_{\text{inf}}$ reduces the effective sample size that may hinder subgroup discovery in smaller empirical studies. Extensions to continuous instruments and multi-valued treatments combined with cross-fitting are natural directions for future work.
\begin{ack}
We thank Christoph Hanck and four anonymous referees for valuable comments that improved the paper. The paper has been presented at the Statistical Week 2025 and the ICSDS 2025. We thank all participants for insightful discussions that enhanced the focus of the paper. The authors acknowledge partial financial support from TRR 391 Spatio-temporal Statistics for the Transition of Energy and Transport (520388526) by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) and from the Rhine-Ruhr Center for Scientific Data Literacy (DKZ.2R) by the German Federal Ministry of Education and Research (BMBF). We acknowledge the use of Anthropic's Claude Opus 4.8 for proofreading and editorial assistance in preparing this manuscript. All content was checked and verified by the authors, who remain responsible for any errors. Replication code for simulations and both empirical applications is available at \url{https://github.com/jens-klenke/SBART-IV}.
\end{ack}
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