Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
Authors: Xichen Guo
Feng Xie
Yan Zeng
Hao Zhang
Zhi Geng
Venue: ICML 2025 poster (2025)
Keywords: Instrumental Variable; Testability; Causal Effect; Unmeasured Confounders; Causal Graphical Models
Primary area: general_machine_learning->causality
TL;DR: We first propose a novel testable condition that is both necessary and sufficient for identifying valid instrumental variable sets within additive non-linear, constant effects models under the milder assumption.

Abstract: We consider the problem of selecting instrumental variables from observational data, a fundamental challenge in causal inference.  
Existing methods mostly focus on additive linear, constant effects models, limiting their applicability in complex real-world scenarios.
In this paper, we tackle a more general and challenging setting: the additive non-linear, constant effects model. We first propose a novel testable condition, termed the Cross Auxiliary-based independent Test (CAT) condition, for selecting the valid IV set. We show that this condition is both necessary and sufficient for identifying valid instrumental variable sets within such a model under milder assumptions. Building on this condition, we develop a practical algorithm for selecting the set of valid instrumental variables. Extensive experiments on both synthetic and two real-world datasets demonstrate the effectiveness and robustness of our proposed approach, highlighting its potential for broader applications in causal analysis.

Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
Xichen Guo! Feng Xie 1
Yan Zeng'
Hao Zhang²
Zhi Geng'
Abstract
restriction); and (C3) the instrument is not related to measured variables that affect both the exposure and the outcome (exogeneity) (see Section 3.1 for formal defini- tion of valid IVs). For example, Figure is an illustration of the IV conditions. Here, Quarter of Birth is a valid IV rel- ative to the causal relationship Education Level + Income. In some scenarios, researchers can directly select suitable variables IVs through expert knowledge or background information. However, in many practical applications, such information is often lacking, making it difficult to select valid IVs and obtain the unbiased causal effect of interest. Therefore, developing statistical methods for selecting valid IV's from observational data is crucial.
We consider the problem of selecting instrumental variables from observational data, a fundamental challenge causal inference. Existing methods mostly focus on additive linear, constant effects models, limiting their applicability in complex real-world scenarios. In this paper, tackle more general and challenging setting: the additive non-linear, constant effects model. We first pro- pose a novel testable condition, termed the Cross Auxiliary-basee independent Test (CAT) condi- tion, for selecting the valid IV set. We show that this condition is both necessary and sufficient for identifying valid instrumental variable sets within such a model under milder assumptions. Building on this condition, we develop a practical algo- rithm for selecting the set of valid instrumental variables. Extensive experiments on both syn- thetic and two real-world datasets demonstrate the effectiveness and robustness of our proposed approach, highlighting its potential for broader applications in causal analysis.
Unmeasured
Variables
Quarter of Birth
C1
Education Level
Income
Figure I. Graphical illustration of valid IV model, where dashed lines indicate the absence of arrows. The variable Quarter of Birth valid IV relative the causal relationship Education Level-Income. Unmeasured Variables, such as ability, are in- cluded unobserved confounders (Angrist & Krueger, 1991).
1. Introduction
Instrumental variables (IVs) are powerful tool for estimat- ing unbiased causal effects in the presence of unobserved confounders. They have been widely applied in various fields, such as economics (Imbens, 2014; Imbens & Ru- bin, 2015), cpidemiology (Hernán & Robins, 2006; Baioc- chi al., 2014), and sociology (Pearl, 2009; Spirtes al., 2000). Informally speaking, a valid instrumental variable needs satisfy the following three conditions: (C1) the instrument is related to the exposure (relevance); (C2) the instrument has no direct effect on the outcome (exclusion
The problem of IV selection has been extensively studied in finding a single valid IV, and several necessary criteria have been established (Pearl, 1995; Manski, 2003; Palmer et al., 2011; Kitagawa, 2015; Wang et al., 2017; Kédagni & Mourifié, 2020), such Pearl's 's instrumental inequality and generalized instrumental inequality. Although these methods have been used in a range of fields, these methods typically require discrete treatment variables and assume that condition C2 holds, i.e., the instrument has no direct effect on the outcome, which is not always feasible in prac- tical settings. For instance, in Mendelian randomization scenarios, the genetic variants' pleiotropic effects the outcome (Burgess et al., 2017), violating C2 and making it challenging to find suitable IVs. Additionally, in cases where the treatment variable is continuous, such as vitamin levels (Skaaby et al., 2013), these methods are less appli- cable. Recently, two novel necessary conditions, IV-GIN
DDepartment of Applied Statistics, Beijing Technology and Business University, Beijing, China "SIAT, Chinese Academy of Sciences, Shenzhen, China. Correspondence to: Feng Xie dicngxie@btbu.cdu..co
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
(Generalized Independent Noise) (Xie et al., 2022) and AIT (Auxiliary-based Independence Test) (Guo al., 2024), have been proposed for continuous variables. However, both conditions face challenges when it comes to testing condition C2 (see the example in Section 4.1).
no guidance on how select the initial set of valid IVs. Significant efforts have been devoted to identifying valid IVs solely from observational data, without relying on prior knowledge of any valid IVs. Existing approaches can gener- ally be categorized into two main strategies.
Another line of research attempts to select valid IV sets (i.e., all IVs in the set are valid) in the additive linear, constant effects (ALICE) model. Interesting developments along this line include Kang al. (2016); Bowden et al. (2016); Hartwig et al. (2017); Guo et al. (2018); Windmeijer et al. (2021); Lin et al. (2024). However, these methods typically rely on the ratio of valid IVs within the potential IV set, adhering to assumptions such as the majority rule or the plurality rule. Silva & Shimizu (2017) relaxed the afore- mentioned conditions and demonstrated that an IV set can be identified when at least two or more valid IVs are present in the system, under the assumption of runk-faithfulnesss. This assumption asserts that every rank constraint on sub- covariance matrix holding in distribution P is implied by any free-paraun ree-parmeeter linear structural model whose path di- agram corresponds to the causal graph G. Although these methods allow for continuous treatment variables and do not require Condition C2 hold, they often assume an additive linear model, which severely limits their applicability.
Single IV. One typical strategy is to focus on discrete vari- able setting. Well-known methods along this line include instrumental inequality (Pearl, 1995), and its various exten- sions (Manski, 2003; Palmer et al., 201 1; Kitagawa, 2015; Wang et al., 2017; Kédagni & Mourifié, 2020). Another line of work addresses continuous- variable settings, including the IV-GIN method, which leverages the Generalized In- dependent Noise (GIN) condition (Xie et al., 2020) within linear non-Gaussian acyelic models (Xie et al., 2022), the IV-PIM method, which is based on the Principle of Inde- pendent Mechanisms (PIM) (Jonas et al., 2017; Janzing & Schölkopf, 2018) under the linear IV framework (Burauel, 2023); and the AIT condition for additive nonparametric models (Guo et al., 2024). Methods in the above two strate- gies typically focus on selecting a single valid IV, while our approach targets the valid IV set. Moreover, these methods struggle to evaluate the exclusion restriction condition (C2). We demonstrate that IV sets impose additional information to identify invalid IVs that single IV methods cannot detect (see Section 4).
In this paper, we focus on violations of the exclusion re- striction and exogeneity of IV. We address the challenge of IV identification in more complex scenarios, referred to as the Additive NonlInear, Constant Effects (ANICE) model, where the relationship between the treatment and IVs can be nonlinear. Specifically, we make the following contributions:
IV Set. Based on the assumed constraints, approaches along this line can be categorized into the following types: (I) Proportion of Invalid IVs. Approaches under this category include the Majority Rule constraint, which assumes that over 50% of the candidate IVs are valid (Han, 2008; Kang et al., 2016; Bowden et al., 2016; Windmeijer et al., 2019; Hartford et al., 2021), and the Plurality Rule constraint, which states that the number of valid IVs exceeds that of any group of invalid IVs sharing the same ratio estimator limit (Hartwig et al., 2017: Guo et al., 2018; Windmeijer et al., 2021; Lin al., 2024); (2) The InSIDE constraint. This constraint assumes that the pleiotropic effects of IVs the outcome are uncorrelated with their effects on the treatment (Bowden et al., 2015; Kolesár et al., 2015; Sander- son et al., 2022); (3) Two Valid IVs constrains. This requires that there are at least two valid IVs and assumes the Rank- faithfulness assumption. Methods based on rank constraints include those in (Silva & Shimizu, 2017: Cheng al., 2023). Compared to existing methods based on IV sets, which typ- ically focus on the additive linear, constant effect models, our work tackles the more challenging scenario of additive non-linear, constant effect models.
We introduce a testable condition, termed the Cross Auxiliary-based independent Test (CAT) condition for selecting the valid IV set within an ANICE model. We demonstrate that the CAT condition is a necessary and sufficient condition to detect all observable viola- tions of Conditions C2 (exclusion restriction) and C3 (exogeneity) under milder assumptions.
We propose a practical algorithm for selecting valid IV sets by leveraging the CAT condition.
We validate the efficacy of our algorithm in assessing IV validity through experiments on both synthetic data and real-world datasets.
2. Related Works
The Durbin-Wus Wu-Hausman test (Durbin, 1954; Wu, 1973;
Hausman, 1978; Nakamura & Nakamura, 1981) is a well- established method for evaluating instrumental variable (IV) models. It allows for testing whether set of potential can- didates can be considered valid IVs, assuming subset of valid IVs is already identified. However, the test provides
In summary, existing methods are either limited to identi- fying single valid IVs or rely on strong assumptions within linear models. Our work extends to the more general setting of additive nonlinear, constant-effect models and introduces testable condition for identifying valid IV sets.
2

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Data Meln-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
T
7A2
U
Z
Z1
Z1
Z1
(a) Valid IV set model
(b) IV set model that violates C2
(c) set model that violates C3
Figure 2. Graphical illustration of TV set Z (Z1, 7z} models, where U is the set of unmeasured confounders. (a) Z is valid IV set. (b) is an invalid set due to the edge Zı - Y (Violate C2). (c) Z is an invalid IV set due to the edge U + Zz (Violate C3).
3. Notations and Background
the generation of the ANICE model is as follows
3.1. Notations and Definitions
X g(Z) + + tex,
This study builds upon the framework of causal graphical models (Pearl, 2009; Spirtes et al., 2000). Specifically, we represent causal relationships using the directed acyclic graph (DAG), denoted as g, where nodes represent variables and directed edges (arrows) indicate causal links between those variables. We denote the treatment (exposure) by X, outcome by Y, the candidate IVs by Z, and the unmeasured confounders between X and Y by U. Let (X) denote the expected value of the random variable X. For statistical independence, we use the notation X IL Y to denote "X is statistically independent of Y", and X J. Y to denote "X not statistically independent of Y We use "w.rt." as a shorthand to denote "with respect to."
(I)
Y = 3X + fZZ) + py (y +ey
where the parameter 3 represents the causal effect of interest, functions g(-), [-), and yp are smooth functions from R R, R + R, and RIU| - R. respectively. All noise terms of variables are mutually independent. Note that the non-zero f() function indicates that subset Z directly affects the outcome Y, thereby violating the exclusion re- striction condition (C2). Furthermore, if there exist Z, E Z that is dependent on U, this indicates the violation of the exogeneity condition (C3). In the remainder of this paper, we denote Z ZZy Zz), where Zy and Z7, represent the valid and invalid IVs, respectively.
Definition (Instrumental Variable (IV) (Pearl, 2009)). A variable is said be instrumental variable w.r.t. X - Y, if a variable Zi satisfies the following three conditions:
C1. (Relevance). IV Z; is associated with the treatment X; C2. (Exclusion Restriction). IV Zr does not directly affect the outcome Y.
In contrast to the additive linear, constant effects model (ALICE) studied in Holland (1988); Bowden et al. (2015); Kang al. (2016); Silva & Shimizu (2017); Guo al. (2018); Windmeijer al. (2021); Lin al. (2024), our work explores more challenging scenario, where 9()), f(),, and (.(.) may be non-linear functions.
C3. (Exogeneity or Randommess). IV Z; independent of the unmeasured confounders U;
Definition 2 (IV Set). A set Z is said be valid IV set w.r.t X + +Y ifechh instrument Z, E Z satisfies Conditions C1 ~ C3. Otherwise, we say that Z is an invalid IV set W.E. X Y.
Our Task. The goal of this paper is to identify valid IV sets that satisfy the conditions C1 ~ C3 for a given causal relationship X - Y. and estimate the causal effect of treatment X on outcome Y simultaneously.
Figure 2 (a) shows an example of valid IV set (Z1, Z2). J. In contrast, Figure (b) presents an example of an invalid IV set (Zi, Z2), where Z, violates Condition C2, while Figure 2 (c) shows an example of invalid IV set [Z1, Z2), where
Remark 1. Using statistical independence test, one can easily verify Condition C1 (relevance). Therefore, our fo- cus is on addressing the remaining Conditions, C2 C3. Notably, causal discovery methods based on conditional independence tests, such as the FCI (Fast Causal Inference) algorithum (Spirtes et al., 1995) and its extensions (Colombo et al., 2012; Akbari et al., 2021), ), often produce fully connected graph over {X, Y, Zi} due to unmeasured con- founders U (Z, 1. X Hence, it becomes challenging to verify Conditions C2 ~ C3 and identify valid IVs.
Z/ violates Condition C3.
3.2. Additive Nonlinear, Constant Effects Model
In this paper, focus the Additive NonlInear, Constant Effects (ANICE) model with variables (X,Y, Y, Z, UJ. With- out loss of generality, we assume that all variables have a zero mean (otherwise can be centered) and that there are no baseline covariates (We will briefly discuss the situation where covariates are included Section 5.1.). Specifically,
II some studies, the unmeasured confounders U are often im- plicitly represented by 8 and which not independent. In this work, we explicitly represent to facilitate subsequent analysis.
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Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
3.3. Instrumental Variable Estimator
Definition 4 (CAT Condition). Let X, Y, and (Z, Zn} S Z denote the treatment, outcome, and candidate IV set, re- spectively. We say that {X, Zi, Z follows the Cross Auxiliary-basse baaiaarry-aaddddd Independence Test (CAT) condition if and only if the following independent relationships hold:
Below, we briefly demonstrate that, given valid IV Z; E Zy w.r.t the causal relationship X - Y, the causal effect 3 of interest in the ANICE model can be consistently es- timated via the instrumental variable formula (Bowden & Turkington, 1990; Pearl, 2009; Wooldridge et al., 2016)²:
Ax-Y112, 11 Zz. and AX-Y 1L Zz-
(6)
B. YE(Y [ilYdo(XX- = x)) Cou(Y, 2,) CovlX X, Zi) = B. (2)
In general, the CAT condition describes the independence between candidate IVs and auxiliary variables, similar to a "cross-test". Specifically, given reference IV Z;, we test the independence between the auxiliary variable Ax-r112 and another candidate IV Zz.. Similarly, using Zj as the reference, we test the independence between A Ax- and Z₁-
dx
However, if an invalid IV Z₁ E Zi is used, the estimated causal effect 3, becomes biased, as shown in the following formula:
B Y do(X = x))] Cou(Y, Z,)
(3)
dx
Cov(X, 2z)
= 3 + (f + py (U)(U)+ + EY. Zz) Cov(X, 2z))
(4) We next give an example to illustrate that there is a connec- tion between the CAT condition and the validity of an IV set. Let's consider the causal diagram in Figure 2. Assume that the data are generated from a linear causal model with Gaussian noise terms. We have the following observations:
Deins
4. CAT Condition and Its Implications
In the subgraph (a), (2,, 32} is a valid IV set w.r.t. X + Y, we have that {X, YIK 2z}} follows the CAT condition, as explained below. The gener- ation mechanisms are defined as: U = ey, Z = EZ,, Z/2 EZg X = 7, + b2 + exU + Ex, Y = BX + cyU &y, where the normal noise terms EU, E2, E2, Ey are assumed to be indepen- dent of each other. According to the above equations, E[Ax-rI1Z; [1] = 0 = 3n = B. Then we can see that Ax = crU + and further we have Ax
In this section, we first formulate the Cross Auxiliary-based Independent Test (CAT) condition, which acts as a neces- sary criterion for determining valid IV sets. We further illustrate the implications of the CAT condition in the AN- ICE model, showing that it is both necessary and sufficient for identifying valid IV sets under mild assumptions.
4.1. CAT Condition: A Brief Formulation
We introduce the key concept of the "auxiliary variable" alongside the definition of the CAT condition for pairwise IVs, which characterizes the independent relationships be- tween the "auxiliary variable" and a separate, distinct IV.
11 Zz. Similarly, for Zz. we have Ax-v1I = CyU + EY 1L Z1. Hence, these imply that {X, YI,,, 2,}} follows the CAT condition.
Definition 3 (Auxiliary Variable). Let X, Y, and Z; E Z denole the treatment, outcome, and candidate IV, respec- tively. The auxiliary variable for the causal relationship X -Y Y relative to Z. is defined as:
In the subgraph (b). {Z1, Z2] is an invalid IV set w.r.t. X + Y. Compared to subgraph (a), the gen- eration mechanism of Y changes to: Y == 3X+ di
+ cy U EY. E[Ax-vijzz Zi] 0
B1
Ax-Y1Iz, == Y B,X,
(5)
B + di
# B. Then, we can see that Ax-YlIzz =
bi
deb2 Zz + (cr _xx)U + + 4xx,, and fur-
where B, satisfies E[Ax-yY|12 Z,] = O and B, o
b1
ther we have Ax-Y1I2, L Zz. This implies that {X, YX,,, 2,}} violates the CAT condition.
It is worth noting that the concept of the "auxiliary variable" or similar ideas have been explored in the context of various tasks (Drton& & Richardson, 2004; Chen et al., 2017: Cai et al., 2019; Guo et al., 2024). However, our formalization differs from these prior works (see Equation (6)). To the best of our knowledge, the wross-indeeendnece property involving such an auxiliary variable has not been previously identified as a criterion for assessing the validity of the IV set in the ANICE model.
In summary, the above facts show that lack of edge Z₁ - Y, i.e., IV does not directly affect the outcome (exclusion re- striction condition C2), has a testable implication. It is noteworthy that the validity of a single IV Zi (or Z2) cannot be verified in subfigure (b). This is because the property of being an IV imposes no constraints on the joint marginal distribution of the observed variables (X, Y, Z1) (see the discussion in Section 3 of Chu et al. (2001) or Proposi- tion 3 of Guo et al. (2024)). This also explains why the identifiability condition for the IV set is necessary.
Here, do(X- = x) denotes an intervention that sets the vari- able X to the value x, removing any influence of other variables on X.
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Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
Building on the above observations, we now present the CAT condition as a necessary criterion for identifying valid IV set.
Remark 2. CAT's core characterization is the indepen- dence between two variables. We leverage the linear sepa- rability of the logarithm of the joint density of independent variables, which, as shown in Lin ( (1997), states that for set of independent, twice-differentiable random variables, the Hessian of their logarithmic density is diagonal, leading to Assumption 1.
Theorem (Necessary Condition for IV Set). Let X, Y, and Z be the treatment, outcome, and candidate IV set in an ANICE model, respectively. Suppose that X, Y, and Z are correlated. If the candidate IVs (Zi, 2 S a valid IV set W.F.l X x Y, then {X, YX,YIHZ, Zi, 45}} always satisfies the CAT condition.
Proposition 1 (Sufficient Condition for IV Set). Let X, Y, and Z be the treatment, outcome, and candidate IV set in an ANICE model, respectively. Suppose that X, Y, Z are correlated, and Assumption 1 holds. If the candidate IV set
Theorem states that if (X, Y (2, 2,}} violates the CAT condition, then the candidate IV set 25} w.r.t. X + Y is invalid. Otherwise, [2, Z may or may not be valid.
{Z, Z5} is invalid, then the (X,YIH22 Zz}} violates the CAT' condition.
4.2. Implications of CAT Condition in ANICE Models
Proposition I states that under Assumption I, all invalid IV sets can be identified. Next, we provide a counterexample to illustrate that violating Assumption 1 is an extreme case.
In the above section, we have shown that the CAT condition is a necessary condition for identifying an IV set. Below, we investigate the sufficient conditions that render the IV set identifiable in ANICE models.
Example 1. (Counterexample) Consider the graph in Figure 3, where [Z, Z2} is an invalid IV set. Let X = 91 (1) + 92 (2) + 9x (U) + Ex and Y = =3X- + (z) + [2 + yr (U) + EY. If the direct causal ef- fects of Z; + Y share the same coefficients as the direct causal effects of Z4 + X for each candidate IV Za (i.e., (11 = a 91 (71) + b, f2(72) = a 922 (22) + b2), then (X, 2,, 2 satisfies the CAT condition. Proof in Appendix B.
Before presenting the theoretical results, we state the key assumption: the algebraic equation condition.
Assumption 1 (Algebraic Equation Condition). As- sume that the probability densities plsx) and plEz.)
E {L,,..,IZZi [Z(} are twice differentiable, and positive on (-00, oc). At least one of the cross second-order ooddr partial aFlog Aorgpxx ZZ)) derivatives, 8 log pogplAx-yii ((x-11--)1) or 0Ax 82:
-Ax-Y112,02 is non-zero. Define the algebraic equation condition char- acterizing the cross second- second-order partial derivative below:
91 (1,)
ga (Zz)
0² log p(Ax -112,,) 0AX-Y112,02;
(K1 + K2 + K3)
Zi
Zz
X
# O,
fzl 12(
(2
(7)
i.e.,
KT dēy der + Ki 04,0Z
PEY + K²² 0A
087,
087,
Figure 3. Invalid IV set Z = (21, Z2} that cach candidate IV solely violates the exclusion restriction condition (C2).
0A: ozj
0Z;
Pez, + K3 DA d7;
0|J| 0|
8 81|
Based on Theorem and Proposition 1, we present the necessary and sufficient conditions for identifying valid IV sets within the ANICE models.
+ K3 0A,07
+ Kg
# 0,
0A,07
(8)
where Ax-Y112. (0) Brins Xx Ki logp(Ey), ), K2 = log p(E and K3 = [J. Here, [J] represents the Jacobian matrix of the transformation f from (Ax-Y112.:22) () to (zy, and the second-order partial derivatives (w.r.. auxiliary variable and candidate IV) of the Jacobian exist.
Theorem 2 (Necessary and Sufficient Condition for IV Set). Let X, Y. and Z be the treatment, outcome, and candidate IV set in an ANICE model, respectively. Suppose that X, Y, and Z are correlated, and Assumption holds. The candidate IV set {Zi, Zz} is a valid IV set relative to X + Y if and only if {X, Y [2 2,}} always satisfies the CAT condition.
Assumption 1 is natural condition that one expects to hold to identify the invalid IV set. Although its formu- lation may seem complex due to the nonlinearities and interaction effects inherent in the model structure (i.e., gZ)) and fZ)), this complexity is expected. When ad- ditional information, such the linearity of the IV model, is available, the algebraic equation condition can be written Bay
Above, we discussed identifiability from joint density under arbitrary distributions. Below, we focus on the second-ord -oder moments, assuming that the variables follow a joint nor- mal distribution, where uncorrelation implies independence. This leads to a less stringent identifiability assumption and corresponding conclusion about the identification.
as: JAx-v 02,
8 (o K"
BeY O2,
862, dEz, Ký BAA
Assumption (Distinct Causal Effect Biases). For an
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Data-Driver Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
invalid IV set {Zz Zi) C Z, the causal effect bias estimates obtained using the two IVs differ, i.e., 3: - 3 + 33 B. Proposition (Sufficient Condition for IV Set Using Sec- ond-Order Moments). Let X, Y. and Z be the treatment, outcome, and candidate IV set in an ANICE model, re- spectively. Suppose that X, Y, and Z are correlated, and Assumption 2 holds. If the candidate IV set (Zi, Zz} S Z is invalid, then the {X, YIK 23}} violates the CAT con- dition.
5.1. CAT Condition with Covariates
The generation of the ANICE model with covariates be- comes:
X = x + tx (W) + 9x (U) +EX.
(9)
Y 3X + by (W) + [(Z) + (0)
where W denotes the covariates. Below, we show that the CAT condition in Definition can be easily extended to address the issue of covariates through regression, as stated below.
Intuitively, Assumption ensures that E[Ax-xX (2,2)) + 0 for any invalid IV set {Z,, 2n} (i.e., they are correlated). This implies that Ax-+Y Z, and Zj are not independent, so the CAT condition does not hold.
Definition 5 (CAT Condition with Covariates). Let X, Y. W, and { (Z, Zz) S Z denote the treatment, outcome, covariates, and candidate IV set, respectively. Further- more, let X. y, Z represent the residual from the regres- sions of X, Y, on W, respectively (e.g., for each IV, We say that {X, YI((,,,, 4z}, W)} follows the CAT condition if and only if the following inde- pendent relationships hold:
Corollary 1. Let X, Y, and Z be the treatment, outcome, and candidate IV set in a linear Gaussian model, respec- tively. Suppose that X, Y. and Z are correlated, and As- sumption 2 holds. The candidate IV set (Zi, Zz} is a valid IV set if and only if (,,,,, } always satisfies the CAT condition.
Z: = (/
Ax->1I(Z..)) 1L Zj, and Ax-MVI(Z, W) TL 2, (10)
Based Definition 5 and Theorem 1, we derive the nec- essary condition for an IV set in the presence of covariates W, as stated in the following corollary.
In Corollary 1, we provide the necessary and sufficient conditions for identifying valid IV sets within the linear Gaussian model, special case of the ANICE model.
Corollary 2 (Necessary Condition for IV Set with Co- variates). Let X, Y. W, and Z be the treatment, outcome, covariates, and candidate IV set in an ANICE model, re- spectively. Suppose that X, Y. W, and Z are corvelated. If the candidate IVs {Z, 7 C Z is valid IV set relative to X + Y given W. then {X, (X,), Zz}. W)} always satisfies the CAT condition.
Although the second-order moment is typically sufficient to identify most invalid IV sets, Proposition 2 may not capture all such sets, as shown in Example 2.
Example 2. (Counterexample) Consider the graph in Fig- ure 4. Let U- = EU. Z1 = U² EZ,, Zz = U + EZ2 X = Z1 + Z2 + U + and Y = X + U2 Ey, where all noise terms are standard Gaussian distribution. We found that Assumption 2 does not hold, and in this case,
Corollary 2 states that if (x, YI,([,,, W)) violates CAT condition, then {Z,, 2,} is invalid IV set.
[[Ax-Y1 2, Z22) B(Ax-Y]|2, Zn) 0 (Uncorre- lated). However {X,Y]I (1, } violates the CAT condi-
5.2. CAT Algorithm with Finite Data
tion because Assumption 1 is satisfied.
In this section, we provide a practical method to identify the valid IV set. Theorems ~ 2, and Corollary 2 have paved the way to discover the foundation for identifying a valid IV set. However, in practice, we face two key issues:
Z
L2
How to efficiently search for the IV set.
Figure 4. Invalid IVs set (Z1, [23 that variables solely vio- late exogeneity condition (C3).
How to test the CAT condition.
First, we address the first issue. To avoid a combinatorial search, we introduce a parametric value K, which represents the number of valid IVs to be selected from the candidate IV set Z. In practical applications, we treat K as prior knowledge. If this knowledge is unavailable, a small value of K can be used, such K = 2. Hence, for an IV set S of length K, the number of subsets to be tested is (K) the number of ways to choose 2 elements from K.
This example highlights that algebraic equation conditions are stricter than second-order moment conditions and em- phasizes the need for Assumption in identifying valid IV sets.
(2) (i.c.,
5. Practical Algorithm
In this section, we first address how to apply the CAT con- dition when covariates are present. We then present our method, which leverages the CAT condition and is suitable for limited sample sizes.
Remark 3. Theoretically, when K exceeds the true number of valid IVs, the candidate IV set should fail to satisfy the CAT condition. Therefore. in the absence of prior knowl- edge, we suggest that users validate IVs incrementally (in
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Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
ascending order) to ensure robustness and prevent the inclu- sion of unnecessary invalid IVs.
Algorithm 1 CAT
Input: Observed dataset D = (X,Y, Y. W, Z); K, the num- ber of valid IVs consider;
Next, we address the second issue. Since we do not assume Gaussian distributions, we here use non-parametric inde- pendence method. Specifically, we apply the distance cor- relation test (Székely et al., 2007; Székely & Rizzo, 2009). Unlike the classical definition of correlation, distance corre- lation is zero only if the random vectors are independent. Let S. be the set to be tested, where [S.] K. Let dCor(.-)) denote the distance correlation between two variables. Our basic idea for selecting valid IV set is as follows: given the candidate set S., we first evaluate the pairwise CAT condition among the variables using the distance correlation.
1: Initialize: valid IV set S + 0;
2: if W * 0 then
3: X. 2 + the residuals from the regressions of X,
Y.Z on W, respectively;
4: else
X. V, 2 X.
6: end if
Step I: Find the valid IV set
7: for each subset S. S 2 with SSel = K do
8: Ts. +0
9: repeat
We then sum these pairwise distance correlations. Finally,
10: Select pairwise IV set {2,, 2,} from Sei
we choose the set S. that yields the smallest total distance correlation the valid IV set. This procedure is motivated by Theorem 1, which indicates that perfect distance cor- relation of O is achieved when S. is valid. The detailed procedure is given in Lines 7 ~ 16 of Algorithm 1.
11: Ax-Vilz, + y - 3:X, where Br + Coo(Y) 21).
CoulX,2,]
12:
Ax- + 3 where B, t Coul (x,
Cov() (y,2,)
13:
Ts. = Ts. + (Cor(Ax-)llz Zj) +
dCor(Ax->>12z) Z,)
14: until all pairwise IV sets in S. are selected.
We now present the complete algorithm, which primarily comprises two key steps: First, for each candidate IV sub- set Se with [S.] K, we identify the valid IV subset S. that minimizes the cross distance correlation between the auxiliary variables and the IV subset (Step I). Second, we estimate the causal effect & using the IV set S (Step II). For causal effect estimation, we apply the point estimator (PE) from Guo et al. (2018) in linear models, and for non- linear models, we use the generalized method of moments (GMM) as described by Hansen (1982). The entire process summarized in Algorithm 1.
15: end for
16: Find a IV set S that yields the smallest total distance
correlation S f arg ming, CP Ts., where P is set containing all subsets of length K from the set Z; Step II: Estimate the causal effect /
17: The causal effect & + Causal Effect Estimator
(X, Y, W,S);
Output: 3, the causal effect of X on Y.
6. Experiments
Below, we demonstrate that, with limited sample sizes, our algorithm identifies the valid IV set and provides an unbi- ased estimate of the causal effect for the relationship.
In this section, we begin by validating the proposed method using synthetic data. We then apply our algorithm to two real-world datasets to highlight its practical benefits. The source code is available in the Supplementary Material.
Theorem 3 (Correctness). Assume that the input data {X, Y, W, Z} followss the ANICE model and at least two valid IVs are present in the system. Furthermore, as- sume Assumption 1 holds. Given infinite samples, the CAT algorithu outputs the valid IV set and true causal effect B correctly.
6.1. Synthetic Data
Here, we evaluate the performance of the proposed meth- ods in estimating causal effects from synthetic data. We consider four typical cases, each involving two valid IVs. Specifically, in Case 1, three invalid IVs violate the exclu- sion restriction (C2); in Case 2, three invalid IVs violate the exogeneity (C3); in Case 3, three invalid IVs violate both the exclusion restriction (C2) and exogeneity (C3); and in Case 4, four candidate IVs are included, where two are in- valid IVs that violate Assumption 2 but satisfy Assumption (see Example 2), demonstrating the benefits of algebraic equation condition to identify the IV set. In all four cases, the data are generated by the ANICE model, and the causal effect of X on Y is set to 3 = 1, as in Guo et al. (2018). Due to space constraints, the detailed causal graphs and data generation mechanism are provided in Appendix D.I.
We finally analyze the complexity of the CAT algorithm. Let n denote the sample size, m IZ, and [W) The time complexity of our algorithm consists of three components:
Covariates residual calculation: Onn m.mmm. m-p²):
2. Step (find the valid IV set): On (m) K));;
3. Step II (estimate the causal effect):
(1) for the PE method, O(n (K + p)));
(2) for the GMM method, O(n K²).
Hence, the overall computational complexity is: O(r2.
(m) K2) + On m p²) + O(n (K p)2)
7

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Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
2.2
1.8-
2.0
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3000
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0.6
NAIVE MR-Egger TSHT
ciiv
IVitetrad
0.8
NAIVE MR-Egger
TSHT
CiVV
sisVIVE
IV-tetrad
(a) Case 1
(b) Case 2
2.8
2.0-
2.6-
1.8
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TSHT
citv
sisVIVE
IV-tetrdd
0.4
NAIVE MR-Egger
TSHT
Citv
sisVIVE
IV-tetrad
CAT
(c) Case 3
(d) Case
Figure 5. Performance of NAIVE, MR-Egger, TSHT, CIIV, ViVE,VE, IV-tetrad, and CAT across four cases the ANICE model.
We compare the proposed method, the CAT algorithm, against the following approaches: 1) NAIVE, the least- squares regression coefficient of Y on ; 2) the MR-Egger algorithm (Bowden al., 2015); 3) the TSHT algorithm (Guo et al., 2018); 4) the CIIV algorithm (Windmeijer al., 2021); 5) the sis VisV algorithm (Kang al., 2016); and 6) the IV-tetrad algorithm (Silva & Shimizu, 2017). Note that approaches 3) to 6) first select valid IV set before estimat- ing the causal effect, whereas the NAIVE and MR-Egger algorithms directly estimate the causal effect. To ensure a fair comparison, for methods 3) to 6) and our algorithm, we use the same IV estimator after selecting the valid IV Each experiment is repeated 100 times using randomly generated data, and the results are averaged. The sample sizes are chosen from 1, 000 (1 k), 3, 000 (3k), 5, 000 (5k).
method requires the InSIDE assumption, which states that the instruments' plciotropic effects on the outcome Y are uncorrelated with their effects on the exposure X. Further- more, we also provide comparison results for linear models and the ANICE model with covariates W Appendix D.I (see Figure
for details). We found that our method performs well here well, yielding results consistent with methods 3) to 6) in the linear model, and outperforming all others in the ANICE model with covariates.
6.2. Real-world Data
In this section, evaluate the performance of the CAT algorithm on two real-world datasets. More details of the real-world data described in Appendix D.2.
Colonial Origins Data (Acemoglu et al., 2001). This dataset examines the impact of social systems on economic development. It contains five key variables across 63 coun- tries (excluding those with missing data): Mortality Mor- Euro1990 Euro, Latitude Lat, Institutions Inss and Eco- nomic Development Eg. We test the validity of the candidate IV set (Mor, Furo) by applying our CAT condition while conditioning on Lat. Notably, because the dataset contains only two candidate IVs, we here use distance correlation- based independence tests from Székely et al. (2007) to as- sess the CAT conditions. Specifically, we first regress Mor- Furo. Ins, and Fg on the covariate Tage to obtain Mor, Euro. Ins. and Ed respectively. We then conduct independence tests for A Euro- which yields a p-value of 0.21, and
Results. Figure 5 summarizes the causal effect estimates of each method in the additive nonlinear, constant effects (ANICE) model. As expected, the proposed CAT algorithm consistently outperforms the other methods across all four cases and sample sizes, exhibiting minimal variance and pro- ducing estimates closest to the true causal effect. In contrast, the NAIVE method performs poorly in all cases due to un- measured confounders U. We observed that all comparison methods perform poorly across cases because they rely on the assumption of linearity, whereas the data generation process is nonlinear. Additionally, we found that the MR- Egger algorithm yields inaccurate results. A possible reason for this is that, in addition to the linearity assumption, this
8

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Data- Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
for AB..o Mor. which yields p-value of 0.25. These results suggest that we cannot reject the hypothesis that {Mor, Euro] is valid IV set with respect to Ins + Ea. consistent with the findings of Acemoglu et al. (2001).
greatly helped to improve the paper.
Impact Statement
Estimating causal effects is a key research challenge from observational data and has broad implications across vari- ous disciplines. Our study addresses the critical problem of selecting instrumental variables purely from observational data, a challenge central to reliable causal inference. By focusing on the more complex additive non-linear, constant effects model, we extend the applicability of existing meth- ods, which often fall short in real-world scenarios due to their reliance on linear assumptions. The new testable con- dition we propose offers a robust, necessary, and sufficient criterion for identifying valid instrumental variables in such models, without requiring strong prior assumptions. Our practical algorithm for instrumental variable selection al- lows researchers to make unbiased causal inferences from purely observational data, advancing the field of causal anal- ysis in domains like economics, public health, and social sciences, where such methods can be particularly impactful.
Children and Mothers' Labor Supply Data (Angrist & Evans, 1996). This dataset comes from an empirical study on the effect of childbearing on mothers' labor supply. Af- ter applying filtering criteria, it includes 254,652 obser- vations. We used 21 variables: week worked (weeksml), more than two children (morekids), 12 candidate instru- mental variables (CandIVs): Two boys (boys2), Two girls (girls2), AGEQK, AGEO2ND, KIDCOUNT, YOBM, non- momil, educm, hsormore, nonmomi, ageqm, agefstd, and 7 covarites W: Mother's 's age at first birth (agem/), Father's age at first birth (agefstm), etc. Angrist & Evans (1996) demonstrated that boys2 and girls2 are valid IVs. Due to the large sample size, distance correlation (dCor) could not be computed directly, so we randomly selected 5% of the data and averaged the results over 10 repeated tests. We tested the candidate IV set with K = 2 using the CAT method. We find that the smallest distance correlation dCor 0.022 w.n.t {boys2, girls2], confirming its validity as an IV set for morekids weeksml, consistent with Angrist & Evans (1996).
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7. Conclusion
In this paper, we explored the identifiability of the IV set in the additive nonlinear, constant effects model. Specifi- cally, we introduced a testable condition, termed the CAT Condition, to detect valid IV sets. Additionally, we outlined the necessary and sufficient conditions for identifying valid IV sets in the ANICE model. Experimental results using both simulation data and real datasets have further validated the practicality of our condition and algorithm. Currently, we assume that the causal effect of interest is constant. A future research direction is to extend the CAT condition to address more general cases, such as the non-linear causal effect model of Newey & Powell (2003); Horowitz (201 1), where causal effects can be effectively estimated using tech- niques such as kernel- based or moments- based IV estima- tors (Singh et al., 2019; Bennett et al., 2019).
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Acknowledgements
This research was supported by the National Natural Sci- ence Foundation of China (62306019, 62472415). Yan Zeng would like to acknowledge the support of the Beijing Municipal Education Commission Science and Technology Program General Project (KM20241001 1016). Feng Xie and Yan Zeng were supported by the Beijing Key Labora- tory of Applied Statistics and Digital Regulation, and the BTBU Digital Business Platform Project by BMEC. We appreciate the comments from anonymous reviewers, which
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Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
A. Notations
Symbol
Description
G
directed acyclic graph
IV
Instrumental Variable (or instrument)
X
Treatment (exposure)
Y
Outcome
candidate (potential)
Z ZN ZT U
candidate (potential) set
A candidate (potential) valid IV set A candidate (potential) invalid IV set
The unmeasured confounders
Covariates
N
The residual of variable X after regressing on covariates W
X
The residuals after regressing on covariates W real-world datasets
A B
A is statistically independent B A is statistically dependent on B The additive nonlinear, constant effects model
A M. B
ANICE
The number of variables in Z
((Z)
The causal effect of Z Y
E
The noise of variable The field of real numbers A mapping from the real numbers to the real numbers
R
R
E(X)
The expected value of random variable X The covariance between random variables X and
Coo(X,Y)
Ax-
The auxiliary variable of causal relationship X relative to
Table 1. The list of main symbols used in this paper
B. More Details on the Examples in Section
B.1. More Details of Example
Proof. Suppose the candidate IVs {41 422 violate only the exclusion restriction condition, as shown in Figure 3, The generation mechanism can be expressed as follows:
U EU, Z1 EZ,, Z2 EZ2
X gi (2,) + (22((2))) + xxx (U) + Ex,
(11)
Y 3X + fi(Zi) + f2(22) + (U) +
where fi = a (1((()) + b, fe(Z2) = a 92(Z2) + b2. Next, we construct a new causal structure, as shown in Figure 2 (a), where (Z1, [2] forms a valid IV set. Specifically, let B' B + a, Zi Z1, Z Zz, X' X, and Y. This transformation ensures that (X, Y, Z1, Zz) has the same distribution (X', Y", Zi, Zz), implying that the candidate variable set Z (71, Z2) is valid IV set without imposing any constraints the joint distribution of the observed variables. Since {Z Σ32 [21, Ze} is valid IV set, and according to the Theorem 1, we know that {X, Y Z1. ] always satisfies the CAT condition.
Y' -''''' (U) +sy -a(px (U)+E))+6, + -
B.2. More Details of Example 2
Proof. For the candidate invalid IV set {Z1, Z2} in Figure 4, the data generation mechanism is described as follows:
U = EU,
Z 12 +
Zz U2² + E22:
(12)
X = 2, + Z, 2 + +ex,
== X + 2 +EY.
12

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Data- Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
where all noise terms follow the Gaussian distribution (0, 1). According to Equation (2), the estimated causal effects for 71 and Zz are:
Var(U2)
31
CoolY,Z1) 21)
1+ Var( (Zi) + 2V 2V)r(U2)
Paine Var(U²)
(13)
32 Coe(Y.Zz)
1 V (2) + 2V (V))(U2)
(,2)
Since all noise terms follow the Gaussian distribution Ni 1), Var(Zi) Var(U²) + Var(ez) = Var(U²) +
Var(sz,) (2r( it follows that Bi -B2, violates Assumption 2. According the definition of the auxiliary
variable, we have the auxiliary variables are
3i = 02 + Pbiae X, Ax-Y112 = -Y -
(14)
A =Y B2X = U² + P X.
And, we know that
E[AA- -Z1)= = E [U2 + - -Baass X) -1] = 0,
(15)
E[Ax-Y -Z2)= = E [U2 ey - 3 B X): Z2] = 0,
which lead to:
E +r) 2,] E(3 X Zi), bias
(16)
E[ + -2] B( X. Zz).
The cross second-order moments between auxiliary variables and candidate IVs are:
E(Ax-Y1IZ: Z2) E [(U2 + er 31 biass X) Zz] E EE[
31
bias ) Z/] B(A 91,2 X -Z2);
bing
(17)
E(Ax-y|1z Z)) = E (U2 t - :2ias X): Z [B8baa
B Z1| E( A Biiss X -z))
Combine the estimate causal effect 31 = 32. we know that Equation (17) is zero, i.e., E Zz) 1EAAxxy1 Z1) O. Below, we compute the cross second-order partial derivatives. Combining Equations (12) and (14), we observe that the transformation from (Ax-Y11z,,72 72) (ev,Ez,) is:
EY = Ax- + Bbius X - X
(18)
EZ. Zz -
For the sake of conciseness, we denote Ax-Yl as Aı when there is no ambiguity. Let |J] denote the Jacobian matrix of this transformation, giving by |J| of (Ax-YlIz,: Z2) Then, we have p(Ax-Y Z22 p(5)) p(ey) p(e2,) |J]. Let K₃ A logp(sy), ) K₂ & ((2,() and K3 A log((J)). Since the densities p(ey) and p(Ez,) are twice differentiable and positive on (-00,00) 00), we have
dsy 8522 dey 0zzz Jdi TZz
Define p(Ax--rizz Zz) as the joint density a2g add
logp(Ax-H Izz: 22) = log(p(er) p(ez)) 17])
= =logp(ěr) + loga($z)) + log({
(19)
= K1 + K2 + K3.
One can find the 2) entry of the Hessian matrix of log p( 11,, 72):
8a log (0 72) = 0Ax 0Z2
@K(Ki + K₂ + Kz)
= KK" + (2 + Var(Zı)
Var(U2) OK²
0 0Ax+r112, 0Z2
(20)
Var(sy) (2+ Var(Zi)
Var(U2) Var(sz,) - Var(U2)*
8² Lloggp(A ,Y1IZ 2n) 0Ax -112, 02g # O, which confirms that Assumption is satisfied, and (X, || Z2}} violates
Then, we find that the CAT condition. In summary, the IV model (12) violates Assumption 2, satisfies Assumption 1, and {X, [X,,][[z, ZzH} violates the CAT condition.
13

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Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
C. Prooff
In this section, we provide detailed proofs for the theorems, propositions, and corollaries. We quote a local geometric information theorem that characterizes the independence of two nonlinear statistics (Lin, 1997), which serves as the foundation for the subsequent results.
Theorem 4. The Hessian H of function I is block diagonal everywhere, 0,0, = for all points so and all i S k, j > k, if and only if separable into sum ($1, 8n) g(51, 82) + (($k+11 ... 8,) for some functions g and h.
The above proposition states that function f' is separable if and only if its mixed second-order partial derivative is zero.
C.I. Proof of Theorem 1
Proof. To prove this theorem, need to show that if candidate IV set {Z,, Z,} is a valid IV set relative to X Y in the ANICE model, then (x, Y Zi } will satisfy the CAT condition. According to the definition of auxiliary variable Y - where 3, satisfies B[Ax-Y112, Z,] = 0 and B; * Since Z; and Zj are valid IVs relative to X Y and following the IV estimator, e.g., the instrumental variable formula (Bowden & Turkington, 1990; Pearl, 2009; Wooldridge et al., 2016) (Equation (2) of Section 3.3), B,, 3, are the unbiased causal effect / of X on Y, i.e.
w.r.t. X + Y, Ax-YIIz
3. 8 = B. Thus, we can express the auxiliary variable as:
Ax-Y11z Y - 8, Y BX = f(Z) + (U) +Ey, 3x = f(z) + pr (U) +
Ax-v = = - -xx
(21)
Below, we prove this theorem using the linear separability of the logarithm of the joint density of independent variables, which states the fact that for a set of independent random variables whose joint density is twice differentiable, the Hessian of the logarithm of their density is diagonal everywhere from Lin (1997) (see Theorem 4 for further details).
Combining Zy E2, with Equation (21), we observe that the transformation from (Ax-rIıZ: 2,) to (EY, is:
EY = Ax-riIz 1(2) pr)
(22)
E2, = Zj.
For the sake of conciseness, we denote Ax-F112z 112, as A, when there is no ambiguity. Let [J| denote the Jacobian matrix of this transformation, giving by |J| Ut. J2, Jh, Jx. Define p(AA--YlIz) Zς) as the joint density of (Ax-y1122, Z,). Then, we have p(Ax 2n) p(er, EZ, |J| = p(er) p(=z,) |J]. Let K₁ logp(sy), K₂ - log togp(Ez,), ). and K3 A log((J)). Since the densities p(er) and p(Ez,) ) are twice differentiable and positive on (-o0, oo), we have
dey
d22, dsy. dez.
log
Z,) = log(p(Ey) p(E2,) (VJ), = -logp(er) + log + log(IJ)
(23)
Ki + K2 + K3.
One can find the 2)-th entry of the Hessian matrix of log p(A 11, Zj):
8 logp(Ax-r11 -r1
OK1 Kg K3)
dev 0(K)
oz,
+ K22 az,
deZ;
+ Kg (2022)
0Ax-11122002
dAx 2Ax-,112,0Z 0Z,
0.A
K" der der + K| 0A; Oz, 0A, 82Y
+ K2 3E73 057,
(24)
0A 00,
SEZ,
K"d Ø|J|
Ø|
+ K2 aA,07,
K' 0A,07
0A
O7,
dey = O. Pey = 0, Dzz, = 0,
aZ, Øe2,, = 0, ||J = 0. JA,0Z,
For valid IV set (Z, Zi}, the following conditions hold:
o2, JA,2 8² lo -x-11
|= and a4.Z = 0. Consequently, the cross second-order partial derivative Ax-YYIz, and Z, are statistically independent. Likewise, for pairwise (Ax-y1) 2, Z,), we can derive that Ax Ax- is independent of Z. In other words, (X,Y (1, ] always satisfies the CAT condition.
0, which implies that
0Ax -1
14

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Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
C.2. Proof of Proposition 1
Proof. Since the candidate IV set {Zz, 2z} violates the IV conditions, the data generation mechanism can be described as follows:
U EU, 74 (U) EZn Zi P2, (U) + EZ,: X = (Z) + px (U) + Y= 3X + f(z) + pr te,,
(25)
where function g(), f(),, and (-(+) twice differentiable. Hence, A satisfies E((Y - -1,) Z,] - and 3g + 0. According to the definition of the auxiliary variables, have
Ax Ax-Miiz = = f(²) + (y + - -Pbias X, Ax Ax-YIIZ, = = 8,X [(Z) (U) + EY - Pliax X,
(26)
where Biaas = 3, 1/aas Bj 33 Below, we prove this proposition using the linear separability of the logarithm of the joint density of independent variables, which states the fact that for set of independent random variables whose joint density is twice differentiable, the Hessian of the logarithm of their density is diagonal everywhere from Lin (1997) (see Theorem for further details).
-
Combining Equations (25) and (26), we observe that the transformation from (Ax- 2z) to (EY, is:
Ey = Ax-YY1zz + Bisas X - - -u)
(27)
E2, = Zj - -32, (U).
For the sake of conciseness, we denote Ax-Y as A, when there no ambiguity. Let |J| denote the Jacobian matrix of this 85k dez, Define p(Ax-Y11z, 2₃) as the joint density of (Ax- Zj). transformation, giving by |J| = Bey BA. 07, 87, HA.
8cz;
Then, we have p(Ax-t11z. -,) = p(Er, EZ, |||| = p(Ex) plzz,) |J]. Let K & logp(sy), K₂ - logp(sz,), and K3 e log(IJ)). Since the densities p(er) ) and p(Ez,) ) are twice differentiable and positive on (-o0, oo), we have
log logr(Ax-y112: 2z) = -lo((((s)) (((b((sv)pp(zz))))) -V)) <logp(Ey) + logp(zz,) + log(IJ)
(28)
= K1 + K2 + K3.
One can find the (1, 2)-th entry of the Hessian matrix of log p(Ax 2,):
8 logg((Axyy]
O21 K₂
K3)
0(Kg dey. 82,
K2 az,
dEz; + K2 0 OZ,
0Ax-r112,0
0Ax dAx-YrIzz 0Z,
DEy DEy
J.EY
+ K4 JEZ3 0E7;
+ K1 K" 0A: 0zj 0A,00
(29)
0A 0z,
8eEz
0|.|
J|J|
K2 0A,07,
K3 0A.
K'
OZ,
DA,07,
8 log logp(Ax Z,) 8Ax -11z,, a2,
According to Assumption 1, the condition that the cross second-order partial derivative
+ 0
follows that Ax-YYIzZ H Zj. Similarly, we can test the independence relationship between auxiliary variable Ax x and candidate IV Zi, i.e., Ax-Y1Iz, HL Z. This implies that {X, YIK Z;, Z violates the CAT condition.
'It worth noting that for an invalid IV set {7,, 7z} that violates the IV conditions, some terms in Equation (29) may not he identically zero. Specifically, for candidate IV set (Zx, Zz} is invalid, there are two scenarios: 1. Violation the exclusion restriction
condition C2 while satisfying the exogeneity condition C3; 2. Violation the exogeneity condition C3; we have day + 0. Uzz Dez, B.A # 0, and Bey # O. However, even in such cases, it is still possible for the entire expression in Equation (29) to be equal to zero. BAA; Therefore, we need to explicitly ensure that Equation (29) is nonzero, which guaranteed by Assumption! 1.
de2, 0.
15

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Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
C.3. Proof of Theorem 2
Proof. Below, we prove the necessary and sufficient conditions for identifying valid IV sets in the ANICE model.
(i): Assume the candidate IV set (Z,, 2 is valid IV set relative to X - Y. By Theorem 1, it directly follows that if the candidate IV set (Z,, Z is a valid IV set relative to X - Y, then {x, Yll 2j3} always satisfies the CAT condition.
(ii): Assume the candidate IV set {Z, 2z} is an invalid IV set relative to X + Y. By Proposition 1, under Assumption 1, if the candidate IV set {Z, 2 ) is invalid, then 2 2 consequently violates the CAT condition.
From (i) and (ii), the theorem is proven.
C.4. Proof of Proposition 2
Proof. Since the candidate IV set {Z,, 2 violates the IV conditions, the data generation mechanism can be described as follows:
U EU; 74 = ( EZ Z = 42, (U) + EZ₁:
(30)
X = g(Z) + (U) + Y= Bx (Z) + py (U) +
where function g(-). f(), and y (.) are twice differentiable. According to the definition of the auxiliary variable, we know that
B(Ax-yi12 -))= (f(Z) + (x +ey X) Zi 0 C[Ax-YY122 -4,)= (1(Z) + (U) + B X) -,} 0
(31)
We can conclude that
[[(f(Z) + (U) _z] E(Shias X- Zz);
(32)
(Z) +
2z] = E(3 7z))
By Equation (32), we have the cross second-order moment for i,j E {l...., | with + j
E E(Ax- Zz) E + (U) + Ex Piias X) Zj E (Baas -Bb )x [Zj E(A
gi.
X
Zz);
bias
E R[Ax-Y|12 Z,) = E [ + - Braas X) 2g
EE (Bhies
BB X -,
E
A
L.jj
X
Zi).
bias
(33)
Since Assumption holds, we have Bhas = 33 - 3 + 33 - 13 = Bras such that A A * O, follows that Equation (33) does not equal zero, i.c.Ax-Y1I2, + Z, or Ax-r112z 1 Z. Thus, (X,YlI[2, Z,, 2j}} violates the CAT condition.
C.5. Proof of Corollary
Proof. Below, we prove the necessary and sufficient conditions for identifying valid IV sets in the linear Gaussian model.
(i): Assume the candidate IV set (7z, Zz) is a valid IV set relative X - Y. By Theorem 1, since the linear Gaussian model is special case of the ANICE model, it follows directly that if the candidate IV set {Z, 2,} is valid IV set relative to X + Y. then {X, ||| (2, 2,}} always satisfies the CAT condition.
(ii): Assume the candidate IV set (Z, 2 is an invalid IV set relative to X - Y. By Proposition 2, under Assumption 2, if the candidate IV set (Z,, 2 is invalid, then {X,Y Z,, 2,1} consequently violates the CAT condition.
From (i) and (ii), the corollary is proven.
C.6. Proof of Corollary 2
Proof. Suppose {Z,, 2,} is valid pairwise IV set relative to X + Y given W, the generation mechanism can be expressed
as follows:
U EU. = (PAw) + tew, Z, tz, (W) + EZ, Z, = tz, (W) + =Z
(34)
X = (Z) + (W) +Ex,
Y = 3X + by (W) + f(ž) + y (U) +
16

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Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
where g()), [()))(()))))))))))) ((-), and (.(+) smooth functions, PAw denotes the of parent variables for each variable in W, and PAw _ W.
Let X' V. = = 2,} represent the residual from the regressions of X, Y. Z on W, respectively. We observe that X, V and Z correspond to the same structure as the ANICE model without covariates. Thus, the proof follows directly from Theorem 1.
C.7. Proof of Theorem 3
Proof. The correctness the CAT-Condition originates from the following observations: assume that the input data {X, Y, w, 3 strictly follow the ANICE model and at least two valid IVs are present the system.
Step I: As the sample size approaches infinity, for valid IV set Se, the total distance correlation Ts, tends to 0 (indicating statistical independence), whereas for an invalid IV set, it does not converge to O. According to Assumption and Theorem 2, the valid IV set S C Z is identified precisely (Lines 7-16 of Algorithm 1).
Step II: Given valid IV set, unbiased causal effects can be accurately estimated using estimator methods, such the point estimator from Guo et al. (2018) or the generalized method of moments (Hansen, 1982), based on the identified IV sets (Line 17 of Algorithm 1).
Based on the above analysis, can identify all valid IV sets and obtain the causal effects correctly.
D. More Details on Experiments in Section 6
D.1. More Details on Simulation Experiments in Section 6.1
In this section, we provide detailed descriptions of the data generation mechanisms and causal graphs for all cases. All experiments were performed with Intel 2.90 GHz and 2.89 GHz CPUs and 128 GB of memory.
Data Generation Mechanisms: We consider four typical cases, each involving two valid IVs, as shown in Figure 6, which are described as follows:
In Case 1, three invalid IVs violate the exclusion restriction (C2);
In Case 2, three invalid IVs violate the exogeneity (C3);
In Case 3, three invalid IVs violate both the exclusion restriction (C2) and exogeneity (C3);
In Case 4, four candidate IVs are included, where two are invalid IVs that violate Assumption 2 but satisfy Assumption (see Example 2).
In all four cases, the data are generated by the ANICE model, and the causal effect of X on Y is set to 8 = 1. In all four cases, the data generated by the ANICE model, and the causal effect of X on Y is set and related works. Across all cases, under both the linear and non-linear models, the noise terms are drawn from uniform distribution with parameters min =1 and max = 1, except in Case 4, where the noise follows Gaussian distribution A( 1), as in Example 2. All constant coefficients in the model generation process are randomly selected from uniform distribution between [-1.5, -0.5] U [0.5, 1.5].
1, in Guo et al. (2018)
Here, we consider three different scenarios based on the aforementioned structure: a linear model, a nonlinear model, and a model with covariates. Note that, since Case is a specific example, its data generation mechanism is only detailed in the nonlinear model scenario. The specific data generation settings follows:
a) The linear model setups are as follows:
-Case 1: = EU, Z1 = EZ Ze = EZg, Za = =2g, Z4 = E2,, ZN = 1Zg + EZg, X = 7 + T2 7
7577 + cU Ex, Y = 3X + dU + Z3Z3 1.5- f474 +s77s + EY.
17

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Data-Driver Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
Z2
1
U
Za
Z
Zs
21
Zr2
(a) Case 1: {Za, Za, Z violates exclusion restriction condition.
(b) Case 2: {Za, Za, Zo J violates exogeneity condition.
75
U
Za
7a
Z1
Zz
Zz
Zz
(c) Case 3: {Zs, Za, Zo} violates either exclusion restriction or exogencity condition.
(d) Case 4: {Za, Za} violates exogeneity condition.
Zr
Z
Z3
X
(c) Case 5: {Za, Z4} solely violates exclusion restriction condition.
Figure 6. The causal diagram used in our simulation studies illustrates the candidate IV sets under various violation conditions. The candidate IV set is denoted Z, where (Z,, 22) represents valid IV and the remaining IVs considered invalid.
Case U EU. 21 EZ,, Za Ezg Z3 azl + EZ, Z4 aqU + EZi Zo 1.2 anU EZs X
Zi + 72Z2 Zz + 7a4 Ts cU Ex, Y 3 dU +
Case U EU. Zı EZ, Ze 52,, Zz = eZs + Z4 asU + E2,, Zs azU + +EZ, X= 71 + 72 TaZa 74Z4 + Zs + cl +8x, Y = BX + dU +2 1373 f4Z4 &v.
b) The nonlinear model setups are as follows:
Case U EU. = =2, 22 52n, Z3 82, 24 2 2,
EZ. X = 2 22 3 23 24
Z5 + +Zz Z, Zs)² +ex Y BX
+1. Xx
+ U
Z53 +EY.
Case U cu. = =2 24= +2, 7, = + Z5 = 13 E2 X 7 log (z) +
Z3 +
22 = +U8
Case U su, 2i = = -,
+ E2,, 24 = -0- + 55,, 21 = + 32 X 21 + 222 1.5
Zg² log(Zz) +2 Z33 (2,)+ +xx 3X + 0 +2 Zgz 4 tyy
Case 4: U = cv. 2, = E%,, Zn = E2,, Z, 0.5 1 E Z4 = 0.5 12 + X= Z 3 + Z₂ + 0.1 (Z 2,)3 +2 Z3 + Z, + aU + Ex, Y 3X + U + EY, where all noise terms follow Gaussian distribution \\ (consistent with Example 2).
18

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Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
Case 5: U Ev, Z1 E2, Z2 Ezn, Z3 = E2. Z4 E2. X sin(Z1) + sin 0.3 sin(Z1) sin Z2) +
sin((Z) 74 sin(U)+ +x. Ex, 3x 1.5 sin sin( 1.5 7 + sim sin(U) ) ey, consistent with Example
<) The nonlinear model setups with covariates W are as follows: we consider two covariates, ( W2), which influence the treatment, the outcome, and all candidate instrumental variables across the first three cases in the ANICE model.
Case I: = Ev, Wi = , W2 = EW,, Z1 = 0, + W2 + Z2 0 0.5 W1 + p2 Z3 = p3 U + +
- + -00
0.:-b:W:+0.5-b,WW; 3 x 22
Y =
Case 2: U = EU. Wi = ,,,, W2 =w,, = =, P:W: te,, Zu 0.5 0.5 +08 P2 +E
Z3 03 1 0.8 P3W2 &ZZ. 74 = =a,W, + 0.8 0,W, + 82. 73- W, 0.8 P3W2 7 + E
+ x= Z + log( Zz) Z33 -Zs 4.2. 7 + +Ex,Y= Y + +
Case 3: W1 Ew1, W2 = 21 0.5
a,W, 0,
0.8 p,W, + 6 Z2 0.5 02 a2W1 + 0.8 02 P2W2 + 822
74 0.5 i + +0.8 +.88p4 + 02 + 82, 75 0.6 asWi +.5 P5W2 U + 82,, 73 0.6 03Wi +
0.5 23 + 1.2 Z32 + X = 0.3 W + 0.5 22 + =2,² + +Z2² Zz² Zi² 2/² + 0.5 0 +
Y = 3X + 0.3 - b +.2.b,, W,W - Za² + Za + 0.4 U +ey.
30
26
2.6
24.
22"
26
O8-
00-
-52
-2.6
0.5
-0.2
MA VRcgger TRIT
CBV
CHNVVE Ninezas
CAT
SAAE Ma-ioger TRiTT
MOME (MMrrrdd
CAT
NAVE Hi-Egper TRIT
ON
CAT
(a) Case
(b) Case 2
(c) Case 3
Figure 7. Performance of NAIVE, MR-Fgger, TSHT, CIVV SisVIVE, VVtetrrd, and CAT across three different cases in the linear model.
144
10:
53
2.8
20-
22-
10.
62-
-42
-08
MA VR-Gger TRIT
CBV
CHAVVE Nineras
CAT
WANE
VP- uP-Toper
TRIT
CIV
on Voorndd OA
NAVE NarEgper
ON
CAT
(a) Case
(b) Case
2
(c) Case 3
Figure 8. Performance of NAIVE, MR-Egger, TSHT, CITV, sisVIVE, IV-tetrad, and CAT across three different cases in the ANICE model with covariates W.
Simulation Results: Below, we present additional experimental results about Cases I ~ 3, including scenarios under the linear model and non-linear model with covariates. The results are shown in Figures and 8. Figure 7 illustrates the causal effect estimates for each method in linear models. The proposed CAT-Condition algorithm performs comparably to methods such as TSHT CIIV SiSVIVE 6, and IV-tetrad 7 The results of the CAT condition outperform those of NAIVE
'For the TSHT algorithm, we used the implementation in the R RobustIV package, available at https: org/wes wwwpbpppckkgeesseee paackaees/kobust..... RRobust IV/-
For the CITV method, we used the implementation in the R package, available at http:///gggttht.c.mm///brrrttttt FFrr the sisVIVE algorithm, we used the implementation in the R package, available at packages/ / sisVVV. SVIVE/.
Forr the IV-tetrad method, we used the implementation in the R package, available https: //ww://www//////////// www.hhhrepagg..uu.. ucl sucgtrbd/code/ iv vv_di scovery/.
19

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Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
and MR-Egger* Figure shows the causal effect estimates for each method in ANICE models with covariates W. As expected, method demonstrates superior performance compared the others. This is mainly due to the fact that the first six methods assume linear model, which leads to suboptimal performance when applied to the non-linear model.
D.2. More Details of Real- World Application in Section 6.2
In this section, will provide more details of the real-world data described in Section 6.2. The distance correlation independence test of variables A and B has the following hypotheses Székely et al. (2007) - Hg: A is independent of B; H1: A is not independent of B. Additionally, the significance level a of the distance correlation independence test is set to 10 divided by the sample size of the dataset.
D.2.1. COLONIAL ORIGINS DATA
Data Description The Colonial Origins of Comparative Development dataset, derived from an empirical study on the impact of colonial history on the economic development of various regions, described in Acemoglu et al. (2001). The dataset includes key variables across 63 countries, after excluding samples with missing data. These variables are: Mortality or. Euro1990 Fuuro- Latitude Lat, Institutions ne. and Economic Development Egg. The hypothesized model proposed by Acemoglu et al. (2001) is illustrated in Figure 9, and the hypothesized data generation mechanism is described as follows:
Ins = 7 Mor + Euro + Yabat + 5,
(35)
Eg = 33+ Ine + 8 B2I c.
where ó and are dependent.
Cultural
difference
Ina
Ed
Euro
Mor
Lidt
Figure 9. Graphical illustration of IV model for estimating the causal effect of institutions (Tns) on economic development (Fa) (Acemoglu et al., 2001).
Results: Acemoglu al. (2001) demonstrated that both Mar and Euvr can serve as valid IVs, conditional on Tats with respect to Lns and Ea. To verify this, test their validity using the CAT condition. Specifically, we first obtain the residuals Mar. Furo, Tns. and F.a by regressing Mar, Furo. Tus, and Fa on the covariate T.at. respectively. Then, we conduct distance correlation independence tests from Szěkely et al. (2007) between auxiliary variables and the candidate set { (Mor Euor J. The cross test for .uro yields a P-value of 0.21, and the test for A E.... Mor yields P-value of 0.25. All the tests pass the test, which means cannot reject the hypothesis that (Mory Euro} is valid IV set. These results suggest that the IV set (Mor, }uro} can be considered valid. Our findings are consistent with those of Acemoglu et al. (2001).
D.2.2. CHILDREN AND MOTHERS' LABOR SUPPLY DATA
Data Description: The dataset used in this analysis is derived from an empirical study on the effect of childbearing on mothers' labor supply, as described Angrist & Evans (1996). We focus on women aged 21 21-35 with two or more children, excluding those whose second child less than year old, using data from the 1980 PUMS. After applying the filtering criteria and excluding samples with missing data, the dataset includes 254,652 observations. Our analysis focuses on the following 21 key variables: the outcome, Weeks Worked (weeksml), 1); the treatment, More than Two Children (morekids); AGEQ2ND, 'oor the MR-Egger algorithm, used the implementation in the R package, available https: / http///aaammccca/ com. je/ article/ 44 / 52/ 754653/.
12 candidate instrumental variables (CandIVs), including (Two boys (boys2), Two girls (girls2),
20

---

Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models
KIDCOUNT, YOBM, nonmomil, educm, hsormore, nonmomii, ageqm, agef and covariates W (Mother" age at first birth (agem.l) ) Father' age at first birth (agef stm). Whether the first child is boy (boy1st), Whether the second child is boy (boy2nd), black of mother indicator (blackm), Hispanic of mother indicator (hispm)) another race of mother indicator (othracem)). In this dataset, we select weeksml as measure to characterize mothers' labor supply. The valid IVs hypothesized model proposed by Angrist & Evans (1996) is illustrated in Figure 10, and the valid IVs hypothesized data generation mechanism is described as follows:
morekids = raboys2 + rigirls2 + 7/2 W + 6, weeksml morekids + (W boy2nd) +.
(36)
where å and e are dependent, W boy2nd represents the set all elements in covariates W after removing the variable boy2nd.
Unobserved counfounder
morekids
weeksml
boys2
girls2
covariates
W
Figure Graphical illustration of valid model estimating the causal effect of childbearing (morekids) on mother's labor supply (weeksml) (Angrist & Evans, 1996).
Results: Angrist & Evans (1996) showed that both boys2 and girls2 can serve as valid IVs, while controlling for the covariates W, with respect to morekids and weeksm.. Due to the large sample size, distance correlation (dCor) could not be computed directly, so we randomly selected 5% of the data and averaged the results over 10 repeated tests. To verify the validity of these IVs, tested the candidate IV set with K = 2 using the CAT method. The procedure proceeds as follows: We applied the CAT algorithm and found that when K = 2, the IV set {boys2, girls2] had the smallest distance correlation, with oCor 0.022. At the same time, we also conduct distance correlation independence tests between the auxiliary variables and the IV set girls. The cross test for Aboyee girls2 yields a P-value of 0.32. Similarly, the
test. for girle2" boys2 yields a P-value of 0.34. All the above tests pass, these suggest that we cannot reject the hypothesis } is valid IV set with respect to morekids weeksm1. These findings are consistent with those of Angrist & Evans (1996), confirming that the IV set (boys2, girls2) is valid.
that {boys22 girls2}
21


Reviews:
[{"review_id": "XxturlmoMz", "reviewer": "Reviewer_SYgt", "summary": "This work proposes a method for identifying valid instrumental variables from observational data under the additive nonlinear model with constant effects. The authors introduce a new testable condition that is necessary and sufficient for selecting a valid IV set (the CAT condition). The proposed algorithm leverages the CAT condition to identify a valid IV set from finite data with a single hyperparameter $K$: the number of expected valid IVs. In step 1, $K$ valid IVs are discovered. In step 2, causal effect estimation for the exposure-outcome pair is performed using the valid IVs. Experimental validation on multiple synthetic settings and two real-world data sets show favorable results.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your helpful comments. Please find our responses below.\n\n> **S1.** time complexity analysis\n\n**A1:** Let $n$ denote the sample size, $m=|\\mathbf{Z}|$, and $p=|\\mathbf{W}|$. The time complexity of our algorithm consists of three components: \n1. Caculate the covariates' residual: $\\mathcal{O}(n \\cdot m\\cdot p^2)$; \n2. Find the valid IV set: $\\mathcal{O}(n^2 \\cdot \\binom{m}{K} \\cdot K^2)$;\n3. Estimate the causal effect: (1) for the TSHT method, $\\mathcal{O}(n\\cdot (K+p)^2)$; (2) for the GMM method, $\\mathcal{O}(n \\cdot K^2)$.\n\nHence, the overall computational complexity is $\\mathcal{O}(n^2 \\cdot \\binom{m}{K} \\cdot K^2 + n\\cdot m\\cdot p^2 + n\\cdot (K+p)^2)$.\n\n> **S2.** run-time scaling... some error analysis\n \n**A2:** We first present run-time and error rate comparisons with baseline methods. Partial results (case 2, 1000 samples) are shown below:\n\n| Methods | MSE of $\\beta$ | Error rate |Run time（sec）|\n| -------- | -------- |--------|--------|\n|NAIVE     |0.0123    | -    |0.0012  |\n|MR-Egger  |0.0741    | -    |0.0228  |\n|TSHT      |0.1638    | 0.97  |0.0035  |\n|CIIV      |0.2211    | 0.78   |0.0300  |\n|sisVIVE   |0.2598    | 0.92   |0.1344  |\n|IV-tetrad |0.2295    | 0.92   |0.0689  |\n|CAT       |0.0051    | 0.01   |1.4587 |\n\nwhere the error rate is the proportion of invalid IV set selections, and “–” indicates no output from the method.\n\nThe results above indicate that, although the runtime is longer—mainly due to distance correlation—our method achieves the lowest MSE and error rate.\n\nNext, we present an experiment with four IVs, where $\\{Z_1,Z_2\\}$ are valid and $\\{Z_3,Z_4\\}$ violate Assumption 1. The results are as follows：\n|Sample sizes |MSE of $\\beta$ |Error rate |Run time（sec）|\n|---|---|---|---|\n|1000 | 0.116039 | 0.06 |0.81 |\n|3000 | 0.000699 | 0    |14.68 |\n|5000 | 0.000419 | 0    |42.83 |\n\nThese results indicate our method also performs well in this case.\n\n> **S3.** For CAT Condition, provide a natural language explanation\n\n**A3:** In general, the CAT condition describes the independence between candidate IVs and auxiliary variables, similar to a \"cross-test\". Specifically, given a reference IV $Z_i$, we test the independence between the auxiliary variable  $\\mathcal{A} _ {X \\to Y || Z_i}$ and another candidate IV $Z_j$. Likewise, using $Z_j$ as the reference, we test the independence between $\\mathcal{A}_{X \\to Y||Z_j}$ and $Z_i$. If both $Z_i$ and $Z_j$ are valid, these conditions hold simultaneously, confirming the CAT condition.\n\n> **S4.** Figure 5 with a dashed/dotted line\n\n**A4:**  Following your suggestion, we updated Figure 5 with a dashed line to clearly show the true causal effect.\n\n\n> **Q1.** Under what settings...be prior knowledge? \n\n**A1:** In Mendelian randomization studies [Burgess et al., (2017)], multiple candidate genes often serve as valid IVs. Thus, a small K (e.g., K=3) can be used. We would like to clarify that introducing  K aims to avoid combinatorial search. If prior knowledge about K is unavailable, one may start with K=2.\n\n> **Q2.** ...have prior knowledge of which covariates W form a valid adjustment set... \n\n**A2:** Here, we assume the covariate set is known by default. If such prior knowledge is absent, we can use other variables as covariates for $Z_i$, typically including the remaining IVs $\\mathbf{Z}\\setminus Z_i$ [Silva & Shimizu (2017)]. In practice, the choice of some covariates can often be straightforward. For example, factors such as age and sex commonly influence the effectiveness of drugs on disease recovery.\n\n> **Q3.** What if K is greater than the true number of IVs? \n\n**A3:** You are right. Theoretically, when K exceeds the true number of valid IVs, the candidate IV set should fail to satisfy the CAT condition. Therefore, in the absence of prior knowledge, we recommend setting K=2. Hence, in practice, we suggest that users validate IVs incrementally (in ascending order) to ensure robustness and prevent the inclusion of unnecessary invalid IVs.\n\n> **Q4.** If multiple IVs are used for effect estimation, ... inappropriately large K? ... \n\n**A4:** Yes, if K exceeds the true number of valid IVs, the estimated causal effect using $\\mathcal{S}$ will be biased. We conducted a robustness experiment for cases where K is larger than the true number of valid IVs. As expected, we observed an increased MSE in causal effect estimation, compared to when K is set correctly. Thus, we recommend validating IVs incrementally (in ascending order) to ensure robustness and avoid introducing unnecessary invalid IVs. \n\n> **Q5.** If K=1, what benefits does your method provide that single-IV methods do not?\n\n**A5:** We would like to clarify that our method applies only to cases where $K\\ge 2$. If K=1, our method will fail, as the CAT condition relies on \"cross test\" to exclude invalid IV sets.\nHowever, if K>1, the most significant advantage of our method is its ability to identify IVs violating the exclusion restriction assumption, whereas single-IV methods cannot.", "claims_and_evidence": "Claims appear well supported.", "methods_and_evaluation_criteria": "Methods and evaluation criteria appear sound.", "theoretical_claims": "Theoretical claims appear sound.", "experimental_designs_or_analyses": "Experimental design covers several relevant settings and real-world data. All existing experiments appear well-designed. In addition, I might suggest additional robustness checks where the assumptions of the proposed method are violated, with some error analysis on performance in these cases.", "relation_to_broader_scientific_literature": "This work is a novel and well-motivated contribution to the literature on IVs, causal discovery, and effect estimation in the presence of latent variables. This might be of interest to the Mendelian randomization community.", "essential_references_not_discussed": "None to suggest.", "other_strengths_and_weaknesses": "**Strengths:**\n\n- This work is well motivated, well organized, and clearly written. Experiments are thorough and convincing.\n\n**Suggestions:**\n\n- Newly introduced algorithms should provide a time complexity analysis.\n- Empirics showing run-time scaling wrt sample size or baseline methods might also be nice.\n- For Definition 4 (CAT Condition), please provide a natural language explanation of this condition for further intuition. The illustrative example is very helpful, but the formal mathematical condition itself could benefit from plain English explanation.", "other_comments_or_suggestions": "- I recommend augmenting Figure 5 with a dashed/dotted line indicating the true causal effect.", "questions_for_authors": "- The authors state, \"In practical applications, we treat $K$ as prior knowledge.\" Under what settings would we expect this to be prior knowledge? I cannot imagine a setting where I would have too little domain expertise to select the valid IVs manually and yet would somehow know how many of them exist.\n- Similarly, if we do not know the graphical structure such that we cannot manually select valid IVs, how do we have prior knowledge of which covariates $\\mathbf{W}$ form a valid adjustment set? If we use pretreatment assumptions and adjust for everything pretreatment, wouldn't this include $\\mathbf{Z}$ as well? Please provide practical examples where these conditions would arise such that the estimation setting is realistic.\n- What if $K$ is greater than the true number of IVs? Since you would inadvertently retain at least one invalid IV, this must incentivize the user to choose a very small $K$, in which case the IV set might be hardly better than the single-IV setting.\n- If multiple IVs are used for effect estimation, what impacts might we see empirically if $\\mathcal{S}$ (from algorithm 1) is \"polluted\" with varying proportions of invalid IVs due to an inappropriately large $K$? This could be a useful robustness experiment to perform.\n- If $K = 1$, what benefits does your method provide that single-IV methods do not?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "eO2LOLdXRG", "reviewer": "Reviewer_RhNr", "summary": "This paper addresses the challenge of selecting instrumental variables (IVs) for causal inference in the Additive Nonlinear, Constant Effects (ANICE) model. \n\nUnlike traditional methods that assume linearity, the proposed approach generalizes IV selection to nonlinear settings, making it applicable to real-world scenarios where standard exclusion restrictions and exogeneity conditions may be violated.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for acknowledging the significance of our theoretical contributions and the novel of the algorithm. We hope the following response addresses your concerns.\n\n> **W1.** The Auxiliary Variable...using Eq.(5).\n\n**A1.** Yes, the auxiliary variable can be viewed as a pseudo-residual. We would like to emphasize that identifying valid IVs is not always straightforward due to unmeasured confounders, often requiring substantial domain knowledge [Pearl, 2009; Imbens & Rubin, 2015]. This highlights the importance of a data-driven approach for testing IV validity. To the best of our knowledge, the independence property involving such an auxiliary variable and a reference IV has not been previously recognized as a criterion for assessing the validity of an IV set.\n\n> **W2.** the title of the paper \n\n**A2.** Our paper’s title was based on prior work addressing IV set selection [Guo et al., 2018; Windmeijer et al., 2021; Silva & Shimizu, 2017]. **To avoid potential misunderstanding, we will update the title to “Data-Driven Selection of Instrumental Variable Sets for Additive Nonlinear, Constant-Effects Models.”** Thanks to you–hope it is clear. \n\n> **W3.** The ANICE model...which may limit the applicability... \n\n**A3.** We would like to mention that linear models are common in the social sciences and ought to be more common in economics and elsewhere [Bollen (1989); Angrist \\& Evans (1996); Acemoglu et al., (2001); Spirtes et al., (2000)]. Furthermore, a series of articles have focused on studying the Additive Linear, Constant Effects (ALICE) model, including works by [Bowden et al. (2015); Kang et al. (2016); Silva & Shimizu (2017); Guo et al. (2018); Windmeijer et al. (2021)]. Unlike the ALICE model, our approach extends to a more general framework—Additive Nonlinear, Constant Effects (ANICE) models. In other words, our work explores a more challenging scenario, where $g(\\cdot)$, $f(\\cdot)$, and $\\varphi_*(\\cdot)$ may be non-linear functions. \n\n\n> **W4.** real-word datasets...suggesting the need for further validation and robustness checks.\n\n**A4.** According to your suggestion, we conducted two additional real-world experiments:\n1. **Fulton Fish Market Data.** This data study on the price elasticity of demand for fish. Our analysis focuses on the 111 samples and 10 key variables: the outcome logquantit ($logq$); the treatment logprice ($logp$), 3 candidate IVs (\\{$wave$, $wind$, $rainy$\\}), and covariates (\\{monday, tuesday, etc\\}). Cunningham, (2021) showed that both $wind$ and $wave$ can serve as valid IVs w.r.t. $logp \\to logq$. Using the CAT method with K = 2, we found that \\{$wind, wave$\\} had the smallest distance correlation $dCor = 0.21$. Furthermore, distance correlation independence tests yielded a p-value of 0.98 for $\\mathcal{A} _ {\\widetilde{wind}}, \\widetilde{wave}$ and a p-value of 0.95 for $\\mathcal{A}_{\\widetilde{wave}}, \\widetilde{wind}$. These results suggest that we cannot reject \\{$wind, wave$\\} as a valid IV set w.r.t. $logp \\to logq$, aligning with Cunningham’s findings.\n \nCunningham, Scott. Causal inference: The mixtape. Yale university press, 2021.\n\n2. **Education Wage Data.** This dataset studies the effect of education on wages. Our analysis consisted of 663 individuals and 13 variables: the outcome logarithm of wage ($lwage$), the  treatment years of education ($educ$), 8 candidate IVs, \\{father’s education ($feduc$), mother’s education ($meduc$), $urban$, $tenure$, $age$, $married$, $black$, $hours$\\}; and covariates, \\{$IQ$, $exper$, $expersq$\\}. Wooldridge et al., (2016) showed that both $feduc$ and $meduc$ can serve as valid IVs w.r.t. $educ$ and $lwage$. Using the CAT method with K = 2, we found that \\{$feduc, meduc$\\} yielded the smallest distance correlation (dCor=0.16). The distance correlation independence tests yielded a p-value of 0.03 for $\\mathcal{A} _ {\\widetilde{feduc}}, \\widetilde{meduc}$ and a p-value of 0.12 for $\\mathcal{A}_{\\widetilde{meduc}}, \\widetilde{feduc}$. These results imply that we cannot reject \\{$feduc, meduc$\\} as a valid IV set,  consistent with Wooldridge et al., (2016). \n\nWooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach 6rd ed. Cengage learning, 2016.\n\n> **Q1.** how can you verify...on real-world datasets? Is there a reliable method for this?\n\n**A1.** Verifying whether a variable is a valid IV in real-world datasets is inherently challenging, as it cannot be directly tested. Typically, domain knowledge is used to determine whether a variable qualifies as an IV, but such information is often absent, making it difficult to select valid IVs and achieve unbiased causal estimates. Hence, a data-driven approach is needed. In general, one can rule out a variable's validity as an IV using necessary conditions proposed by existing studies (e.g., Pearl's instrumental variable inequality or our CAT condition). However, confirming that a variable is truly valid requires additional assumptions—such as Assumption 1 introduced in our paper.", "claims_and_evidence": "The paper claims to offer a new theoretical condition (CAT) for IV selection and a novel algorithm.\n\nThese claims are well-supported through theoretical proofs and comparative experiments.", "methods_and_evaluation_criteria": "The proposed CAT algorithm relies on statistical independence tests and optimization techniques.", "theoretical_claims": "The paper provides formal proofs.", "experimental_designs_or_analyses": "Four synthetic data cases covering various IV violation scenarios.\n\t\nReal-world datasets from economics and labor studies, ensuring applicability beyond simulations.", "relation_to_broader_scientific_literature": "This work builds on and extends prior research in instrumental variable selection and causal inference.", "essential_references_not_discussed": "n/a", "other_strengths_and_weaknesses": "Strengths:\n\nThe CAT condition is a significant conceptual contribution.\n\nThe experiments are well-structured and extensive.\n\nThe proposed method is computationally feasible.\n\nWeaknesses:\n \nThe Auxiliary Variable is not a directly measured variable but is instead calculated using Eq. (5). It might seem like an error term, but in reality, it serves as a constructed quantity that captures residual dependencies in the system.\n\nDefinition 4 requires at least two candidate IVs. However, the title of the paper, “Data-Driven Selection of Instrumental Variables”, may be misleading, as it does not explicitly convey this requirement.\n\nThe ANICE model relies on strong assumptions, which may limit the applicability of the proposed method in real-world scenarios.\n\nAdditionally, the experimental results on the two real-world datasets appear too weak to draw meaningful conclusions about instrumental variables (IVs). The current findings do not provide strong evidence to validate the proposed approach in practical settings. Moreover, the results presented in Acemoglu et al. (2001) and Angrist & Evans (1996) could also be questionable, suggesting the need for further validation and robustness checks.", "other_comments_or_suggestions": "see weaknesses\n\n-------- Post rebuttal: ----------\n\nThe authors have confirmed and adequately addressed my concerns. In light of the revisions they have committed to making, I am happy to raise my score from 2 to 4 and support the acceptance of this paper.", "questions_for_authors": "Selecting instrumental variables (IVs) is indeed interesting, but how can you verify that the IVs chosen are correct on real-world datasets? Is there a reliable method for this?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "R3jRKQmXHD", "reviewer": "Reviewer_ifxU", "summary": "This paper studies the testability of instrumental variables (IV), or in other words, helps researchers find the correct set of IVs using observational data.\n\nFor this purpose, most existing methods assume a simple linear model or discrete treatment variables, and still, the exclusion restriction condition (C2; that the IV does not directly cause the effect) is usually untestable.\n\nIn this work, the authors propose to testify all C1 to C3 conditions in the general setting called the Additive Nonlinear, Constant Effects (ANICE) model. The key assumption in the model is that the causal effect from treatment to the outcome remains linear (as the term \"constant\"). Under this assumption, the authors develop the Cross Auxiliary-based Independence Test (CAT).\n\nRoughly speaking, the \"auxiliary\" variable is the part of the outcome left over after accounting for the treatment’s effect that is removed relative to the instrument. When two instruments are both valid, each instrument's auxiliary outcome should be independent of each other. And vice versa.\n\nBased on this, a practical algorithm for IV selection is provided.\n\n---\n\n## update after rebuttal:\n\nI have decided to raise my score from 3 to 4. My concerns regarding the linearity assumption, two IVs requirements, and the identifiability difference to the existing works have been well addressed by the authors' rebuttal.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your comments and suggestions, and we hope the following response addresses your concerns.\n\n> **W1.** The linear assumption from treatment X to effect Y is still strong...Under this assumptions, actually the core condition (CAT condition) is a direct consequence of existing conditions based on generalized independence noise (GIN) condition. \n\n**A1.** **Regarding the linear assumption**, we would like to mention that linear models are common in the social sciences and ought to be more common in economics and elsewhere [Bollen (1989); Angrist \\& Evans (1996); Acemoglu et al., (2001); Spirtes et al., (2000)]. Furthermore, a series of articles have focused on studying the Additive Linear, Constant Effects (ALICE) model, including works by [Bowden et al. (2015); Kang et al. (2016); Silva & Shimizu (2017); Guo et al. (2018); Windmeijer et al. (2021)]. Unlike the ALICE model, our approach extends to a more general framework—Additive Nonlinear, Constant Effects (ANICE) models. In other words, our work explores a more challenging scenario, where $g(\\cdot)$, $f(\\cdot)$, and $\\varphi_*(\\cdot)$ may be non-linear functions. \n\n**Regarding the GIN condition and CAT condition**, both the proposed CAT and GIN conditions use auxiliary variables to test independence among variables. However, their strategies differ: GIN’s basic approach is that, given a reference variable, it tests the independence between the auxiliary variable and that reference. In contrast, the CAT condition is similar to a “cross-test”: given a reference IV $Z_i$, it tests the independence between the auxiliary variable and another candidate IV $Z_j$. Moreover, the GIN condition is designed to identify the causal structure of latent variables within a linear non-Gaussian model, whereas the CAT condition specifically evaluates the validity of IV sets within the ANICE model.\n\n> **W2.** To testify the validity of one (truly valid) IV, it seems that at least one another truly valid IV is needed. Then when there is only one truly valid IV, can this algorithm still correctly identify it? \n\n**A2.** You are correct: in order to test the validity of one IV, at least one other truly valid IV is needed. Notably, we do not need to know in advance whether that other IV is truly valid. If K=1, our method will fail, as the CAT condition relies on \"cross test\" to exclude invalid IV sets. In such cases, one can use single-IV methods, such as those proposed by Xie et al. (2022), Burauel (2023), and Guo et al. (2024).\n\n\n> **Q1.** ...Does that then yield weaker identifiability (e.g., the exclusion restriction condition being untestable)? Except for this, could the authors please discuss more on how these two works connect to each other, e.g., from the technical side? \n\n**A1:** **Yes, we would like to mention that, if at least two valid IVs exist in the system, our method offers a key advantage in identifying IVs violating the exclusion restriction assumption—a capability Guo’s method lacks.** Generally, although both proposed conditions use auxiliary variables to test independence, Guo et al. (2024) focus on determining whether a single variable is a valid IV, whereas our approach validates an entire IV set. Roughly speaking, given a reference IV $Z_i$，Guo et al. (2024) test the independence between the auxiliary variable and $Z_i$ itself. By contrast, the CAT condition is more akin to a “cross-test”: given a reference IV $Z_i$, it tests the independence between the auxiliary variable and a different candidate IV $Z_j$. It is precisely the information provided by the “cross-test” that gives the CAT condition a broader capacity to test the exclusion restriction assumption.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes. I read the theorems and assumptions. They look correct to me but I cannot guarantee.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "/", "essential_references_not_discussed": "/", "other_strengths_and_weaknesses": "Strengths:\n\n1. The problem studied is crucial and necessary. Prior methods could either handle only one instrument at a time or require the linear model assumptions. This paper provides the solution in a more general setting. Several existing condition can also be shown as the specific cases of the proposed CAT condition.\n\n2. The paper is generally well written. The technical development is rigorous and solid. This is reflected in, for example, the algebraic equation condition (Assumption 1) and the discussion on the corresponding counterexamples. It would be better if authors could introduce more motivations in layman words before directly giving final formulations (e.g., Eq. 5).\n\n\nWeaknesses:\n\n1. The linear assumption from treatment X to effect Y is still strong. Though claimed to be nonlinear, these nonlinear parts are only allowed for hidden confounders and IVs. The core part of X to Y being linear is still needed. Under this assumptions, actually the core condition (CAT condition) is a direct consequence of existing conditions based on generalized independence noise (GIN) condition. The nonlinear part will not affect the regression residual for the linear part.\n\n2. To testify the validity of one (truly valid) IV, it seems that at least one another truly valid IV is needed. Then when there is only one truly valid IV, can this algorithm still correctly identify it? Or please correct me if I am wrong.", "other_comments_or_suggestions": "/", "questions_for_authors": "This work seems very related to https://arxiv.org/abs/2411.12184, as also discussed in the paper. The setting in that work seems more general (allowing X to Y to be also nonlinear). Does that then yield weaker identifiability (e.g., the exclusion restriction condition being untestable)? Except for this, could the authors please discuss more on how these two works connect to each other, e.g., from the technical side?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SVAqtzT2Ij", "reviewer": "Reviewer_Wfgc", "summary": "This manuscript presents a novel testable condition for identifying valid instrumental variable sets within Additive Nonlinear, Constant Effects Models using observational data. The proposed Cross Auxiliary-based Independent Test (CAT) condition is shown to be both necessary and sufficient under mild assumptions. The authors also explore the application of the CAT condition in the presence of covariates, which is common in practice. Building on this foundation, they propose a practical algorithm for selecting valid instrumental variable sets. The effectiveness and robustness of the approach are demonstrated through both synthetic and real-world datasets, highlighting its potential for broader applications in causal inference.\n\n## update after rebuttal\nAfter reviewing the authors' rebuttal, I've decided to raise my score from 3 to 4. The authors addressed my concerns thoughtfully and constructively. They justified the focus on the constant effects model by referencing key studies and highlighting the novelty of their approach in extending it to a more general framework. Additionally, they provided detailed explanations of how the CAT condition applies in scenarios with non-constant causal effects, showing a willingness to improve the paper based on feedback. They also clarified the validation of assumptions, emphasizing the necessity of Assumption 1 and the verifiability of Assumption 2. These responses demonstrated a strong commitment to addressing my feedback, which led me to adjust my score upward.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your inspiring positive feedback and suggestions. Please see below for our responses to your specific comments. \n\n> **W1.** The article focuses on the constant effects model. However, this is not a weakness per se, nor do I believe it constitutes a reason for rejection.\n\n**A1:** **we would like to mention that linear models are common** in the social sciences and ought to be more common in economics and elsewhere [Bollen (1989); Angrist \\& Evans (1996); Acemoglu et al., (2001); Spirtes et al., (2000)]. Furthermore, a series of articles have focused on studying the Additive Linear, Constant Effects (ALICE) model, including works by [Bowden et al. (2015); Kang et al. (2016); Silva & Shimizu (2017); Guo et al. (2018); Windmeijer et al. (2021)]. Unlike the ALICE model, our approach extends to a more general framework—Additive Nonlinear, Constant Effects (ANICE) models. In other words, our work explores a more challenging scenario, where $g(\\cdot)$, $f(\\cdot)$, and $\\varphi_*(\\cdot)$ may be non-linear functions. \n\n> **S1.** I suggest that the authors discuss the research approach for the additive nonlinear, non-constant effects model, as this would make the article more comprehensive. \n\n\n**A1:** This is a great point! We have examined the applicability of the CAT condition in scenarios with **non-constant causal effects and fully additive relationships among variables.** Our results suggest that the CAT condition can still hold under these settings. Specifically, we consider two data-generation mechanisms illustrated in Figure 2(a) and 2(b) of the manuscript: \n\n(a) **Valid IV set \\{$Z_1, Z_2$\\}:** \n$$\nU = \\varepsilon_U, \\quad\nZ_1 = \\varepsilon_{Z_1}, \\quad\nZ_2 = \\varepsilon_{Z_2}, \\quad\nX = {Z_1}^2 + {Z_2}^2 + U + \\varepsilon_X, \\quad\nY = X^2 + U^3 + \\varepsilon_Y,\n$$where the noise terms $\\varepsilon_U$, $\\varepsilon_{Z_1}$, $\\varepsilon_{Z_2}$, $\\varepsilon_X$, $\\varepsilon_Y$ are independent. Following the kernel-based or moments-based IV estimators, we have $\\hat{f} _ 1(X) = \\hat{f} _ 2(X) = f(X) = X^{2}$.  Consequently,\n$$\n\\mathcal{A} _ {X \\to Y \\parallel Z_1} = U^3 + \\varepsilon_Y,\n$$which is independent of $Z_2$. Likewise, for $Z_2$,\n$$\n\\mathcal{A}_{X \\to Y \\parallel Z_2} = U^3 + \\varepsilon_Y,\n$$which is independent of $Z_1$. These imply that \\{$X, Y|| \\{ Z_1, Z_2 \\}$ \\} satisfies the CAT condition.\n\n(b) **Invalid IV set \\{$Z_1, Z_2$\\}:**\n\nCompared to (a), the generation mechanism for $Y$ changes to\n$$\nY = X^2 + Z_2 + U^3 + \\varepsilon_Y.\n$$\nAccording to the IV formula, $\\hat{f} _ 1(X) = f(X) = X^{2}$. Hence,\n$$\n\\mathcal{A}_{X \\to Y \\parallel Z_1} = U^3 + Z_2 + \\varepsilon_Y,\n$$which depends on $Z_2$. Therefore, \\{$X, Y|| \\{ Z_1, Z_2 \\}$ \\} violates the CAT condition.\n\nWe will incorporate the above discussion into the main text.\n\n> **Q1:** Is it possible to validate the algebraic equation condition (Assumption 1)? Additionally, the distinct causal effect biases (Assumption 2) seem to be verifiable.\n\n**A1:** **Firstly**, because the probability density of noise terms cannot be fully determined from observational data, Assumption 1 cannot be directly validated. In practice, violating Assumption 1 imposes an extremely strict condition on an invalid IV set. Notably, one may not need to explicitly verify Assumption 1 or 2. Once \\{$X, Y|| \\{ Z_1, Z_2 \\}$ \\} violates the CAT condition, it indicates that  \\{$Z_i, Z_j$\\} is an invalid IV set. In our work, we include Assumption 1 to show the necessity and sufficiency of the CAT condition. \n\n**Next**, you are correct that Assumption 2 can be validated using observational data. While it holds for most invalid IV sets, it does not guarantee soundness, making it weaker than Assumption 1. As shown in Example 2 (line 278), this also highlights why Assumption 1 is necessary.", "claims_and_evidence": "The claims made in the submission are supported by clear and convincing evidence. The article provides a robust theoretical framework for the CAT condition, and the authors thoroughly demonstrate its necessity and sufficiency for identifying instrumental variable sets. Additionally, the authors effectively apply the CAT condition in the presence of covariates, further strengthening their argument. The theoretical development is well-supported, and the practical algorithm is validated through both synthetic and two real-world datasets.", "methods_and_evaluation_criteria": "The proposed methods make sense for the problem and application at hand. The paper clearly articulates a strong motivation for the problem, and the CAT condition is well-suited for selecting valid instrumental variable sets based solely on observational data, addressing a practically important case. Overall, the methods are appropriate and well-aligned with the research goals.", "theoretical_claims": "Yes, I have quickly reviewed the proofs related to the theoretical claims in the paper and found no issues with the logical structure or the correctness of the proofs.", "experimental_designs_or_analyses": "Yes, the experimental designs are sound and valid. The authors provide various cases (exclusion restriction and exogeneity conditions) within the additive nonlinear, constant effects model, including both with and without covariates, and linear model setting. Additionally, the authors compare the performance of the CAT condition against six other methods across all settings. The results demonstrate that the CAT condition performs well across these different scenarios. Furthermore, the authors evaluate the performance of the CAT algorithm on two real-world datasets, providing additional evidence of its practical applicability.", "relation_to_broader_scientific_literature": "The key contributions of this paper are highly innovative in the context of selecting valid instrumental variable sets. The related works addressed in this paper primarily focus on the constant effects model. Previous studies, such as those by Guo et al. (JRSSB, 2018), Windmeijer et al. (JRSSB, 2021), Silva and Shimizu (JMLR, 2017), and Lin et al. (JRSSB, 2024), have concentrated on selecting valid IV sets and providing identification theorems and estimation methods within linear models, often requiring at least two or more valid instruments. In contrast, this paper tackles the more complex challenge of identifying IV sets in the context of an \\textbf{Additive Nonlinear, Constant Effects model}. The proposed CAT condition is both necessary and sufficient, and it does not rely on assumptions about the ratio of valid instruments, setting it apart from previous work in this field.", "essential_references_not_discussed": "No, the related works are thoroughly summarized.", "other_strengths_and_weaknesses": "**Strengths**:\n-The paper is well-written and clearly articulated. The claims are backed up with theoretical results and proofs. The experimental results are promising. Overall this work seems to be technically solid work. \n\n**Weaknesses**:\n-The article focuses on the constant effects model. However, this is not a weakness per se, nor do I believe it constitutes a reason for rejection.", "other_comments_or_suggestions": "I suggest that the authors discuss the research approach for the additive nonlinear, non-constant effects model, as this would make the article more comprehensive.", "questions_for_authors": "Is it possible to validate the algebraic equation condition (Assumption 1)? Additionally, the distinct causal effect biases (Assumption 2) seem to be verifiable.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper introduces a novel, testable condition called the Cross Auxiliary-based Test (CAT) for selecting valid instrumental variables (IVs) in additive nonlinear, constant effect models, and proposes a practical two-step algorithm leveraging the CAT condition to identify IVs and estimate causal effects from observational data. 

Pros:

+ Introduces a necessary and sufficient condition (CAT) for valid IV selection in nonlinear models, which is a significant theoretical advancement.

+ Provides a practical, data-driven algorithm with empirical validation across diverse settings.

Cons:

+ Relies on the assumption of a constant treatment effect, which may limit applicability in some real-world scenarios.

+ Requires at least two valid IVs for the CAT condition to be testable, reducing effectiveness in sparse IV settings.

+ Real-world experimental results are relatively weak and may not conclusively demonstrate practical effectiveness.
Decision: Accept (poster)