Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
76,569 characters · 11 sections · 64 citation commands
Inference for Linear Systems with Unknown Coefficients
KEYWORDS: Linear programming, linear (in)equalities, partial identification, uniform inference, treatment effects, nonparametric instrumental variables
JEL classification codes: C31, C35, C36
\thispagestyle{empty} \setcounter{page}{1}
Given an independent and identically distributed (i.i.d.) sample $\{Z_i\}_{i=1}^n$ with $Z_i$ distributed according to $P \in \mathbf P$, this paper studies the hypothesis testing problem
where $\mathbf P$ is a “large” set of distributions satisfying conditions described below and
Here, “$x_1 \geq 0$” signifies that all coordinates of $x_1 \in \mathbb R^{d_1}$ are non-negative, $A_0(P)$ is a $p \times d_0$ matrix with $d_0 \geq 0$, $A_1(P)$ is a $p \times d_1$ matrix, and $\beta(P)$ is a $p \times 1$ vector.
As discussed further in Section (ref), the testing problem described above arises naturally in many settings of empirical interest, including (i) inference for linear functionals of structural functions in nonparametric instrumental variables (NPIV) models with shape restrictions, as in freyberger2015identification; (ii) inference for average marginal effects in nonlinear models with random coefficients, as in fox2011simple; (iii) inference for treatment effect parameters that are partially identified through the marginal treatment response (MTR) framework of mogstad2018using; (iv) inference under “synthetic parallel trends” with convex weights as considerd in liu2025synthetic; and (v) inference for counterfactual choice probabilities in the distribution-free binary choice model in gurussell2023joe. We also note that the null hypothesis (ref) subsumes as a special case the setting where $A_0(P)$ and $A_1(P)$ are known, for which there exist many other examples fang2023inference.
We demonstrate in Section (ref) that testing (ref) in certain special cases is impossible (in the sense that the power of any test is bounded by its size), by observing that the null set $\mathbf{P}_0$ is dense in $\mathbf{P}$ with respect to the total variation metric. Accordingly, the first contribution of the paper is to obtain a characterization of the closure of $\mathbf P_0$ which guides the construction of our test. Exploiting Farkas' lemma, we show that (a subset of) the closure of $\mathbf P_0$ with respect to the total variation metric can be described in terms of a set of linear inequality restrictions involving the projection of $(A_1(P), \beta(P))$ onto the orthogonal complement of the column span of $A_0(P)$. Specifically, the characterization amounts to verifying if, for every unit vector, the minimum of a collection of linear inequalities is non-positive.
Building on this observation, in Section (ref) we develop an inference procedure based on sample splitting. Using the first subsample, we construct a unit vector for which we expect that our characterization of the null is most strongly violated. We then use the second subsample to formally test whether or not our inequalities are violated. By virtue of this sample-splitting construction, the resulting test is straightforward to implement and involves no simulation: in particular, the construction of the “violating” unit vector requires solving two linear programs, and the test statistic is provided in closed form with rejection based on a comparison with the appropriate quantile of a standard normal distribution. Moreover, the uniform asymptotic validity of our test is established under weak and interpretable regularity conditions while allowing the dimensions of $p$, $d_0$, $d_1$ to grow with the sample size $n$.
We propose two variants of our test. The first, which we term the “direct” method, tests if the union of all the inequalities in our characterization hold. The second, which we term the “screening” method, tests whether or not a single inequality in our characterization is non-positive, under the restriction that the other inequalities are positive with high probability. In our simulation study, the screening method typically yields shorter confidence intervals through test inversion, at the cost of introducing one tuning parameter when selecting the unit vector in the first subsample.
Inference procedures for testing (ref) and closely related null hypotheses have gained increasing attention in the literature. bai2022testing, andrews2023inference, cox2023simple, and fang2023inference all propose inference procedures which could be applied to (ref) whenever $A_0(P)$ and $A_1(P)$ do not depend on $P$ (i.e. they are known quantities). The method proposed in our paper immediately applies to this setting as a special case. There is also a literature for the closely related problem of inference for the value of a linear program: see in particular freyberger2015identification, cho2024simple, gafarov2025simple, voronin2025linear, goff2025inference. As explained in cox2025testing, these methods can be used to address the same empirical problems as those that we consider in Section (ref), but there is a subtle technical difference between the two settings, and in general these methods are neither tuning parameter nor simulation-free.
Two recent papers that propose inference procedures which could be used to test (ref) are cox2025testing and goff2025inference. The theoretical results in cox2025testing and goff2025inference require rank conditions which, as argued in liu2025synthetic, could be difficult to verify or may even be violated in certain settings of empirical interest. In contrast, we are able to establish the asymptotic validity of our procedures under weak and interpretable regularity conditions on the ranks of $A_0(P)$ and $A_1(P)$. Moreover, neither paper establishes the validity of their tests in a high-dimensional regime where $p$, $d_1$ and $d_0$ are allowed to grow with sample size, as we do in this paper.\footnote{We note that goff2025inference explain that they derive their results in a semi high-dimensional regime, where the number of unknown coefficients in the linear system must remain bounded.} We illustrate the finite-sample performance of our procedures using simulation designs based on the ones considered previously in these papers, as well as some high-dimensional counterparts that these papers do not consider.
The remainder of the paper is organized as follows. In Section (ref), we describe several examples of empirical problems of interest that can be accommodated in our framework. Our main results are contained in Sections (ref) and (ref): Section (ref) presents our characterization of the closure of the null hypothesis, whereas Section (ref) describes the test and establishes its uniform asymptotic validity. In Section (ref), we study the finite-sample behavior of our proposed tests in a simulation study. Proofs of all results are collected in the Appendix.
In this section, we present a collection of motivating examples, two of which we revisit in the simulation study in Section (ref). goff2025inference and cox2025testing discuss several additional examples that provide further motivation.
In this section, we provide a characterization of $\mathbf P_0$ that informs the construction of the test we present in Section (ref). Before discussing the characterization formally, we motivate the need for such a characterization by demonstrating the impossibility of testing the null hypothesis that $P\in \mathbf P_0$ in a seemingly simple example. In particular, consider the case in which, for some scalars $a(P)$ and $b(P)$, the set $\mathbf{P}_0$ is given by \[\mathbf{P}_0 = \{P \in \mathbf{P}: a(P)x = b(P) \text{ for some } x \in \mathbb{R}\}~.\] This is a special case of (ref) where $d_0 = 1$ and $A_1(P)$ does not exist. In Figure (ref)(i) we plot the set \[\mathbf{C}_0 = \{(a, b) \in \mathbb{R}^2: ax = b \text{ for some } x \in \mathbb{R}\}~,\] which represents the set of points $(a(P),b(P)) \in \mathbb{R}^2$ for which there exists a distribution $P \in \mathbf{P}_0$. From Figure (ref)(i) we notice immediately that the closure of $\mathbf{C}_0$ (as a subset of $\mathbb{R}^2$) is the entire space $\mathbb{R}^2$. This observation suggests that the closure of $\mathbf{P}_0$ (with respect to the total variation metric) coincides with the entire set of distributions $\mathbf{P}$, and indeed we discuss conditions under which this is the case below. In contrast, Figure (ref)(ii) depicts the analogous set $\mathbf{C}_0$ for testing the null hypothesis given by \[\mathbf{P}_0 = \{P \in \mathbf{P}: a(P)x = b(P) \text{ for some } x \in \mathbb{R}, x \ge 0\}~,\] which is a special case of (ref) where $d_1$ = 1 and $A_0(P)$ does not exist. Here, we see that the closure of $\mathbf{C}_0$ (as a subset of $\mathbb{R}^2$) is a strict subset of $\mathbb{R}^2$, which suggests that the closure of $\mathbf{P}_0$ in this example is a strict subset of $\mathbf{P}$.
Because it is impossible to test a null hypothesis for which $\mathbf{P}_0$ is dense in $\mathbf{P}$ with respect to the total variation metric, and more generally that it is impossible for any test to have non-trivial power against alternatives which lie on the boundary of $\mathbf P_0$ romano2004non, our test is based on a characterization of the closure of $\mathbf P_0$ relative to the total variation metric, which we denote by $\mathrm{cl}(\mathbf P_0)$. Towards that end, define
so that $\mathbf P_0 = \{P \in \mathbf P: (A_0(P),A_1(P),\beta(P)) \in \mathbf{C}_0\}$. Accordingly, we begin by deriving a characterization of the closure of $\mathbf {C}_0$ with respect to the Euclidean topology, and then relate this characterization back to $\mathrm{cl}(\mathbf P_0)$. First, consider the following alternative representation of $\mathbf{C}_0$ based on pre-multiplying the equation in (ref) by the annihilator of $A_0$, which we denote by $M_0$.
Let $a_1 , \dots, a_{d_1}$ denote the columns of $A_1$. The columns of $M_0A_1$ are then given by $M_0a_1, \ldots, M_0a_{d_1}$. Given this alternative representation of $\mathbf C_0$, we can apply Farkas' lemma to conclude that $(A_0,A_1,b) \in \mathbf C_0$ if and only if for all $y \in \mathbb R^p$, either there exists some $1 \le j \le d_1$ such that $a_j'M_0y < 0$ or $b' M_0 y \geq 0$. Consider the set of triples $(A_0, A_1, b)$ obtained by weakening the strict inequalities $a_j'M_0y < 0$ to weak inequalities:
Theorem (ref) formalizes the sense in which replacing these strict inequalities with weak inequalities relates to the closure of the set $\mathbf{C}_0$.
Theorem (ref) shows that the closure of $\mathbf{C}_0$ can be characterized by combining the set $\bar{\mathbf{C}}_0$, which describes the set of triples obtained by weakening the inequalities in the conclusion of Farkas' lemma, along with the set of triples for which $A_0$ is rank deficient. Note the theorem implies that, if $p < d_0$, then the closure of $\mathbf{C}_0$ becomes $\mathbb R^{p \times d_0}\times \mathbb R^{p \times d_1} \times \mathbb R^p$. As a result, we implicitly assume that $p \ge d_0$ for the rest of the paper. This characterization of the closure forms the basis for the test which we present in Section (ref).
Finally, we relate $\mathrm{cl}(\mathbf{C}_0)$ to the closure $\mathrm{cl}(\mathbf{P}_0)$ in the total variation distance. Let $\widetilde{\mathbf P}_0 := \{P \in \mathbf P: (A_0(P), A_1(P), \beta(P)) \in \mathrm{cl}(\mathbf C_0)\}$ denote the pre-image of $\mathrm{cl}(\mathbf C_0)$ in $\mathbf{P}$. We claim that in general $\mathbf P_0 \subseteq \widetilde{\mathbf P}_0 \subseteq \mathrm{cl}(\mathbf P_0)$. Indeed, it follows by construction that $\mathbf{P}_0 \subseteq \widetilde{\mathbf P}_0$. As a result, any test that controls size on $\widetilde{\mathbf P}_0$ will necessarily control size on $\mathbf{P}_0$. Meanwhile, we generally expect $\widetilde{\mathbf P}_0 \subseteq \mathrm{cl}(\mathbf P_0)$, in which case we will not have power against any distribution in $\widetilde{\mathbf P}_0$. By definition, this will be the case whenever $\mathbf P$ is “rich” enough in the sense that for every $P \in \mathbf P$ such that $(A_0(P), A_1(P), \beta(P)) \in \mathrm{cl}(\mathbf C_0)$, there exists a sequence $P_n \in \mathbf P_0$ such that $P_n$ converges to $P$ in the total variation metric.
Low-level sufficient conditions for this “richness" property could be obtained, for example, if we view $P$ as the distribution of a random vector in $\mathbb R^{p(d_0 + d_1 + 1)}$ and define $\mathrm{vec}(A_0(P), A_1(P), \beta(P))$ to be the corresponding vector of means, where $\mathrm{vec}$ is the vec-operator magnus2019matrix. In this case, $\mathbf P$ is rich enough if it contains a sufficiently large collection of normal location families. To illustrate, let $\mu = \operatorname*{vec}(A_0(P), A_1(P), \beta(P))$ where $(A_0(P), A_1(P), \beta(P)) \in \mathrm{cl}(\mathbf{C}_0)$. Then by definition there exists a sequence of vectors $\mu_n$ corresponding to triplets in $\mathbf{C}_0$ such that $\mu_n \rightarrow \mu$. Let $H = \mathrm{span}\{\mu_n - \mu: n \ge 1\}$ and let $\Sigma$ be a positive semi-definite matrix whose range is exactly $H$, so that $\mu_n - \mu \in \mathrm{range}(\Sigma)$ for all $n \ge 1$. Then, the sequence of distributions $N(\mu_n, \Sigma)$ converges to $N(\mu, \Sigma)$ in the total variation metric by Lemma (ref) in the appendix.
Let $\{Z_i\}_{i=1}^n$ be i.i.d.\ with $Z_i$ distributed according to $P \in \mathbf P$. Recall from Section (ref) that the closure of the null space can be characterized as the set of distributions $P$ for which either $A_0(P)$ is rank-deficient, or
Recognizing that whether condition (ref) holds is not affected by norm constraints on $y$, and defining $b_j(P) := a_j(P)$ for $1 \le j \le d_1$, $b_{d_1+1}(P) := -\beta(P)$, and $J = \{1, 2, \ldots, d_1 + 1\}$, we may rewrite (ref) as
where $\|y\|_1 = \sum_{i=1}^p |y_i|$ for any $(y_1,\ldots, y_p)\in \mathbb R^p$. In what follows, we consider the following equivalent formulation of (ref): For any $J^* \subseteq J$ and $(J^*)^c := J \setminus J^*$, condition (ref) is equivalent to the statement
Our test uses a sample-splitting procedure in which one sample split is used to select a $y\in \mathbb R^p$ and a second split is used to test whether the inequalities in (ref) hold at the selected $y$. We consider two proposals for $(J^*)^c$. Our first proposal takes $(J^*)^c$ to be those $j\in J$ for which $b_j(P)'M_0(P)$ is known deterministically -- i.e.\ for which $b_j(P)^\prime M_0(P)$ does not depend on $P$. In other words, we test the condition $\min_{j \in J^*} b_j(P)'M_0(P)y \le 0$ using a unit vector $y$ that is known to satisfy $\min_{j \in (J^*)^c}b_j(P)'M_0(P)y > 0$. We call this method the “direct” method in what follows. Our second proposal sets $J^*=\{j^*\}$ for some non-random $j^*$. In this case, we test the condition $b_{j^*}(P)'M_0(P)y \le 0$ using a unit vector $y$ such that $\min_{j \in (J^*)^c}b_j(P)'M_0(P)y > 0$ holds with high probability. We call this method the “screening” method. In all of our examples, we set $j^* = d_1 + 1$, which is natural in settings where we perform test inversion to construct a confidence set for a scalar parameter whose null value only enters the vector $b_{d_1 + 1}(P)$; see, e.g., the examples in Section (ref). We show via simulation in Section (ref) that the screening method often generates shorter confidence intervals than the direct method, at the cost of introducing an additional tuning parameter which determines the amount of “screening” that is performed in the first sample split.
We next present a high-level description of the test and defer the details of the construction of its specific components to Sections (ref) and (ref). To construct the test, we first randomly split the data into two samples $\{Z_i\}_{i \in I_{1, n}}$, $\{Z_i\}_{i \in I_{2, n}}$ of sizes $n_1$ and $n_2$, where $I_{1, n} \cup I_{2, n} = \{1, \dots, n\}$ and $I_{1, n} \cap I_{2, n} = \emptyset$. In what follows, we always assume that $n_2 \to \infty$ as $n \to \infty$ and allow $n_1$ to be fixed for the direct method but require $n_1 \to \infty$ for the screening method. Throughout, we use the superscript $(k)$ to denote when a given quantity is a function of only the $k$th split. Using the first sample split $\{Z_i\}_{i\in I_{1,n}}$, we construct a vector $\hat y_n^{(1)}$ which represents a direction in which the weak inequality in (ref) appears to be “most violated” --- we discuss how to construct such a vector in Section (ref).
Next, given suitable estimators $\hat{b}_{j, n}^{(2)}$ and $\hat{M}_{0, n}^{(2)}$ for $b_j(P)$ and $M_0(P)$ computed in the second sample split $\{Z_i\}_{i\in I_{2,n}}$, we define the test statistic
where $\hat{\sigma}_{j, n}^{(2)}(\hat y_n^{(1)})$ is an estimator for the asymptotic standard deviation of $\sqrt{n_2}(\hat b_{j, n}^{(2)})'\hat{M}_{0, n}^{(2)} \hat y_n^{(1)}$. Finally, we set
where, for $\alpha \in (0, 1)$, $z_{1 - \alpha}$ denotes the $1 - \alpha$ quantile of a standard normal distribution.
In this section, we present results establishing the asymptotic validity of our test and a detailed construction of the standard deviation in (ref). When stating our assumptions, we will suppress the superscript $(k)$, with the understanding that all assumptions stated on the entire sample will also hold when applied on the sample splits $\{Z_i\}_{I_{1, n}}$ and $\{Z_i\}_{i\in I_{2, n}}$, under suitable scaling.
Our first assumption imposes conditions on the estimators $\hat A_{0,n}$ for $A_0(P)$ and $\hat b_{j,n}$ for $b_j(P)$ with $1\leq j \leq d_1+1$. In its statement, $\|\cdot\|_2$ denotes the Euclidean norm, $\|\cdot\|_{2, 2}$ denotes the operator norm of a matrix when the domain and range are endowed with $\|\cdot\|_2$, and $a \vee b = \max(a, b)$ for any $a, b \in \mathbb R$.
Assumption (ref)(a) requires our estimators for $A_0(P)$ and $b_j(P)$ for $1 \leq j \leq d_1 + 1$ to be asymptotically linear with influence functions whose moments are disciplined by Assumption (ref)(b). Assumption (ref)(a) is automatically satisfied with $a_n = 0$ whenever the entries of $A_0(P)$ and $b_j(P)$ are expectations and the entries of $\hat A_{0,n}$ and $\hat b_{j,n}$ are the corresponding sample means.
Our second assumption imposes boundedness and non-degeneracy conditions on $A_0(P)$ and $b_j(P)$. In its statement, $\bar s(A_0(P))$ and $\underline{s}(A_0(P))$ denote the maximum and minimum singular value of $A_0(P)$.
Assumption (ref)(a) requires the maximum singular value of $A_0(P)$ is bounded above uniformly in $\mathbf{P}$, with the bound possibly depending on $p$. Assumption (ref)(b) ensures that $A_0(P)'A_0(P)$ is bounded away from degeneracy. This rules out the set of rank deficient matrices $\mathbf{C}^{\rm RD}$ in our characterization of the closure of the null hypothesis, as defined in Theorem (ref). Assumption (ref)(c) requires that the Euclidean norm of $b_j(P)$ is bounded uniformly in $P \in \mathbf P$, with the bound possibly depending on $p$.
We note that Assumption (ref)(b) can be dropped if we let $n_1 \to \infty$ and apply the test $\phi_n$ in (ref) only when $\underline s(\hat{A}^{(1)}_{0,n}) > \tau$ for some pre-specified small value $\tau > 0$ (and do not reject the null hypothesis when $s(\hat{A}^{(1)}_{0,n}) \leq \tau$). We emphasize, however, that even if we maintain Assumption (ref)(b), our assumptions impose no requirements on the rank of $A_1(P)$. In contrast, the assumptions underlying cox2025testing and goff2025inference implicitly restrict the ranks of both $A_0(P)$ and $A_1(P)$.
Assumptions (ref) and (ref) are instrumental in obtaining an asymptotic expansion for our test statistic. In particular, letting $A_0^\dagger(P)$ denote the Moore-Penrose pseudoinverse of $A_0(P)$, we will show that for every $1\leq j \leq d_1+1$ the vector $\sqrt n(\hat b_{j, n}' \hat M_{0, n} - b_j(P)' M_0(P))'$ is asymptotically linear with influence function
Our third assumption imposes moment restrictions on the influence function $\xi_j(Z,P)$.
Assumption (ref) ensures that we are able to couple an influence function for our test statistic to a Gaussian random variable uniformly in $P\in \mathbf P$. The requirement of Assumption (ref) is satisfied, for example, if the third moments of the entries of $\xi_j(Z,P)$ are uniformly bounded across $1\leq j\leq d_1+1$ and $P\in \mathbf P$.
Our fourth assumption ensures that the inequalities not examined by our test statistic (i.e., those in $(J^*)^c$) are indeed positive when evaluated at a $\hat y^{(1)}_n$ not equal to zero, as required by the characterization of the null hypothesis in (ref). We describe methods to construct such a $\hat y_n^{(1)}$ in Section (ref) below.
Assumptions (ref) automatically holds, for instance, for the direct method which either sets $J^*$ to equal $J$ (so $(J^*)^c$ is empty) or $(J^*)^c$ to only contains coordinates $j$ for which $b_j^\prime M_0(P)$ is known. In this case, $\hat y_n^{(1)}$ can be chosen to belong to $\mathcal{Y}(P;J^*)$ with probability one for any $n_1$, and our asymptotics only require that $n_2 \to \infty$. In contrast, the screening method intuitively conducts a pre-test in the first fold to ensure that $\hat y_n^{(1)}\in \mathcal Y(P;J^*)$ with high probability. In this case, our asymptotics therefore require $n_1\to \infty$ in order for Assumption (ref) to be satisfied. We also note that Assumption (ref) allows $\hat y_n^{(1)}$ to equal zero when it does not belong to $\mathcal Y(P;J^*)$. This flexibility is important because our test never rejects when $\hat y_n^{(1)}$ is zero (since then $T_n =0$; see (ref)). Therefore, setting $\hat y_n^{(1)} =0$ allows us to decide not to reject after examining the first sample split --- e.g., if we fail to reject in the screening method pre-test.
We now turn to the construction of the standard error $\hat \sigma_{j,n}(y)$ employed in the construction of our test statistic. To this end, we let $\operatorname*{vec}$ denote the vec-operator and $\otimes$ denote the Kronecker product magnus2019matrix. We further define the asymptotic covariance matrix \[ V_j(P) := \operatorname{Var}_P \bigg [
\bigg ] \] for our estimator $\hat A_{0,n}$ and $\hat b_{j,n}$ and let $\hat V_{j, n}$ denote an estimator for $V_j(P)$. For each fixed $y \in \mathbb R^p$ and $1\leq j \leq d_1+1$, we apply the Delta method to the function $(A_0(P), b_j(P)) \mapsto b_j(P)' M_0(P) y$ to obtain the asymptotic variance of the estimator $\hat b_{j,n}^\prime \hat M_{0,n}y$. Accordingly, defining the gradient \[ D_j(P; y) :=
, \] it is possible to show that the asymptotic variance of $\hat b_{j,n}^\prime \hat M_{0,n}y$ equals $\sigma_j^2(P; y) := D_j(P; y)' V_j(P) D_j(P; y)$. By analogy, for $\hat A_{0,n}^\dagger$ the Moore-Penrose pseudoinverse of $\hat A_{0,n}$, we estimate $D_j(P; y)$ by setting \[ \hat D_{j, n}(y) =
. \] As an estimator for the asymptotic variance we then set $\hat{\sigma}^2_{j, n}(y) = (\hat D_{j, n}(y)' \hat{V}_{j, n} \hat D_{j, n}(y)) \vee \underline{\sigma}^2$ for $1 \leq j \leq d_1 + 1$, where, as previously discussed in Remark (ref), $\underline{\sigma} > 0$ is a small positive constant which we include to address potential (near) degeneracies in $\sigma^2_j(P;y)$.
Our fifth assumption imposes that our estimator $\hat V_{j,n}$ is suitably uniformly consistent.
Assumption (ref) enables us to show that the standard errors $\hat \sigma_{j,n}(y)$ are suitably uniformly consistent in both $P\in \mathbf P$ and $1\leq j \leq d_1+1$. It requires that $\hat V_{j,n}$ converge to $V_j(P)$ at a rate faster than $K_{2,p}^2$, where $K_{2,p}$ depends on $p$ and is specified in Assumption (ref)(c). We state Assumption (ref) as a high level condition because the structure of $V_j(P)$ is dictated by the influence functions of our estimators (as introduced in Assumption (ref)). In applications for which our estimators are sample means, and hence $\hat V_{j,n}$ are sample covariance matrices, sufficient conditions for Assumption (ref) are readily available from the literature; see, e.g., Chapter 6 in wainwright2019high.
Our final assumption imposes conditions on the rates of convergence of our moment and error bounds.
In Remark (ref) below, we discuss high-level sufficient conditions which guarantee Assumption (ref) holds, as well as how these conditions differ when $A_0(P)$ needs to be estimated versus when it is known (or does not exist). At this point we emphasize, however, that Assumptions (ref) are automatically satisfied in asymptotic regimes in which $p$ and $d_1$ are fixed. Moreover, we note that Assumption (ref) only restricts the dimension $d_1$ through Assumptions (ref) and (ref)(c), which impose that the linearization error, the second moment of the influence function $\varphi_j(Z,P)$, and the norm of $\|b_j(P)\|_2$ be bounded uniformly in $1\leq j \leq d_1$. If these bounds do not grow with $d_1$, then Assumption (ref) in fact leaves the dimension $d_1$ unrestricted. As we discuss in the next section, however, certain approaches for selecting $\hat y_n^{(1)}$ may impose restrictions on the dimension $d_1$ relative to the sample size in the first split $\{Z_i\}_{i\in I_{n,1}}$.
Our next, main, result establishes that our proposed test is uniformly consistent in level over $\mathbf P_0$.
In this section we describe a procedure for selecting $\hat y_n^{(1)}$ that satisfies Assumption (ref). Given suitable estimators $\hat{b}_{j, n}^{(1)}$ and $\hat{M}_{0, n}^{(1)}$, we select $\hat y_n^{(1)}$ as the solution to the following optimization problem:
where $\hat{\omega}_{j, n} > 0$ are positive scaling parameters that we specify below. If the optimization problem in (ref) is found to be infeasible, then we set $\hat y_n^{(1)} = 0$. In Appendix (ref), we explain how this optimization problem can be re-formulated as a linear program. Note that the specific choice of $\hat\omega_{j,n}$ for $j \in J^*$ will not affect the validity of the procedure, although these should be carefully selected to ensure the test has good power; see the discussion following Lemma (ref) for details. In contrast, as we demonstrate in Lemma (ref), the choice of $\hat \omega_{j,n}$ for $j \in (J^*)^c$ is relevant for the screening method. In the case of the direct method, that is, when $(J^*)^c$ contains only those $j$ for which $b_j(P)'M_0(P)$ is known, the specific choice of $\hat{\omega}_{j, n}$ for $j \in (J^*)^c$ is immaterial in practice: we can simply set the constraints in (ref) to ensure that $b_j(P)'M_0(P)y$ is strictly positive for every $j \in (J^*)^c$.
Part (a) of Lemma (ref) formally states that for the direct method, the sole constraint on the weights $\hat {\omega}_{j,n}$ is that they be positive. Parts (b) and (c) specify the requirements on the weights $\hat \omega_{j,n}$ when the coordinates $j\in (J^*)^c$ may be such that $b_j(P)^\prime M_0(P)$ depends on $P$, as in the screening method. In particular part (b) shows that Assumption (ref) is satisfied under rate restrictions on the growth of $d_1$ and that $\hat \omega_{j,n}$ diverge to infinity faster than $(pd_1)^{1/3}$. As we show in part (c), however, these restrictions can be considerably weakened provided we strengthen our moment conditions by assuming that (ref) holds. Under such a requirement, part (c) weakens the rate restrictions on $d_1$ and the rate at which $\hat {\omega}_{j,n}$ must diverge to be logarithmic.
Although the conditions of Lemma (ref) are satisfied for a wide range of choices of $\hat \omega_{j,n}$, careful choices of these weights should be used to ensure that our test has good power properties. First, the weights $\hat{\omega}_{j, n}$ for $j \in J^*$ should be selected in a way that “appropriately weights” the degree of violation of each component. For this we set $\hat \omega_{j,n} = \hat \sigma_{j,n}^{(1)}(\hat y_{0,n})$ for $\hat \sigma_{j,n}^{(1)}(y)$ our (truncated) estimate of the asymptotic standard deviation of $\sqrt{n_1}(\hat b_{j,n}^{(1)})^\prime \hat M_{0,n} y$ evaluated at $\hat y_{0,n}$. To maintain the linear structure of the optimization problem, we employ a preliminary direction $\hat y_{0,n}$ computed by solving the linear program
The weights $\hat \omega_{j,n}$ for $j \in (J^*)^c$ should further be selected so that the assumptions of Lemma (ref) are satisfied and so that, under the alternative, (ref) is feasible with probability tending to one: for this we specify $\hat \omega_{j,n}$ to satisfy $\hat \omega_{j,n} = o_P(\sqrt{n_1})$. To this end, we've found that in simulations it works well to set $\hat \omega_{j,n} = c_n \cdot \hat \sigma_{j,n}^{(1)}(\hat y_{0,n})$ for some sequence $c_n$ diverging to infinity. We have found that the following choices work well in practice: in a low dimensional regime where $p$ and $d_1$ could be considered fixed, we set $c_n = \sqrt{\log(\log(n_1))}$; in a high dimensional regime where $p$ or $d_1$ diverges with sample size, we set $c_n = \sqrt{\log(\log(\log(n_1)))\times \log(p + d_1)}$.
In this section, we illustrate the finite-sample performance of our procedure via simulation. We consider three distinct settings, each inspired by earlier related papers. In all designs, we examine the performance of the direct and the screening method with $n_1 = n_2 = n/2$. In all cases we set $\hat \omega_{j,n}$ for $j \in J^*$ to equal $\hat \sigma^{(1)}_{j,n}(\hat y_{0n})$ for $\hat y_{0,n}$ as defined in (ref) and $\hat \omega_{j,n}$ for $j \in (J^*)^c$ to equal $c_n \cdot \hat \sigma^{(1)}_{j,n}(\hat y_{0n})$ with $c_n = \sqrt{\log(\log(\log(n_1)))\times \log(p + d_1)}$.
We first consider the simple one-sided model described in cox2025testing. Let $C := (C_1, \ldots C_H)'$ and $X := (X_1, \ldots, X_H)'$ be random vectors and let $H \ge 0$ index the number of inequalities to be considered. In our notation, the design can be described as $A_0(P) = \nu(P) := (E_P[C_1], \dots, E_P[C_H])'$, $A_1(P) = \mathbf{I}_H$, $\beta(P) = -\mu(P) - \mathbf v \theta$, where $\mu(P) := (E_P[X_1], \ldots, E_P[X_H])'$ and $\mathbf v := (1, 1, 0, \dots, 0)' \in \mathbb R^H$. It can be shown that the identified set of $\theta$ is $(-\infty, 0]$. Under $P$, $(C,X)$ are distributed according to
with $\mu(P) = (-1, 1, 1, \ldots, 1)'$, $\nu(P) = (1, -1, -1, \ldots, -1)'$. Accordingly, given a random sample from $(X,C)$, $\hat{A}_{0,n}$ and $\hat{b}_{j,n}$ are computed by taking sample averages.
Figures (ref) and (ref) present the rejection probabilities from $1,000$ draws for both of our proposed tests for the null hypothesis that $P \in \mathbf P_0$ at a $5\%$ significance level, over a grid of values of $\theta$ and across different choices of dimension $H$ and sample size $n$ (the direct method is represented by the solid line and the screening method by the dashed line). We see that both the direct and screening methods control size in all cases within the identified set for $\theta$. Outside of the identified set, the rejection probabilities for the screening method are slightly larger than for the direct method, although the differences become negligible for large $H$ and/or large $n$.
We revisit a simple instance of Example (ref) proposed by goff2025inference, which is loosely based on the application in mogstadsantostorgovitsky2017nwp to the bed net data used by dupas2014e. The outcome $Y$ is a binary indicator of using a new type of anti-malarial bed net, the treatment $D$ is an indicator for purchasing the bed net, and the instrument $Z$ is a randomly-assigned price for the bed net. The context implies that $Y(0) = 0$. Suppose that the instrument is binary and that there are no covariates. The researcher assumes that $E[Y(1) \vert U = u] = \theta_{0} + \theta_{1}u + \theta_{2}u^{2}$ is a weakly decreasing, quadratic function of $u$, which must be contained within $[0,1]$ because $Y(1)$ is binary. These shape restrictions are imposed through four linear constraints:
which imply $E[Y(1) \vert U = u] \leq 1$ for all $u$, because $E[Y(1) \vert U = 0] = \theta_{0} \leq 1$ and the derivative of $E[Y(1) \vert U = u]$ is $\theta_{1} + 2\theta_{2} \leq 0$.
It will be convenient to use an alternative parameterization. Set $\tilde{\theta}_{1} = -\theta_{1}$, $\delta = \theta_{0} + \theta_{1} + \theta_{2}$, so that $\theta_{0}, \tilde{\theta}_{1}, \delta \geq 0$. Then the constraints (ref) can be simplified to
where $s_1, s_2 \geq 0$ are slack variables. The researcher matches the moments $E_{P}[YD \vert Z = z]$ for $z = 0,1$, creating the two restrictions
The null hypothesis of interest is $H_{0}: E_{P}[Y(1)] = \tau_{0}$, where
We set $x_{1} = (\theta_{0}, \tilde{\theta}_{1}, \delta, s_{1}, s_{2})'$, so that the problem defined by (ref)--(ref) fits in the form (ref) with
and with both $x_{0}$ and $A_{0}(P)$ null.
The simulation design is specified as $P\{Z = 1\} = 0.5 = P\{Z = 0\}$, $p(0) = 1/3$, $p(1) = 2/3$, $Y(0) = 0$, $Y(1) = \mathbf{1}\{V \le \theta_{0} + \theta_{1} U + \theta_{2} U^2\}$ where $V|Z,U \sim U[0,1]$ and $(\theta_{0},\theta_{1},\theta_{2}) = (1, -1, 0.5)$. Given this design, the identified set for the ATE is $[0.58,0.67]$. Using a random sample from $(Y,D,Z)$, $\hat{b}_{j,n}$ is once again computed using sample analogs (recall that $A_0(P)$ does not exist given this parametrization).
Figure (ref) presents the rejection probabilities from $1,000$ draws for tests of the null hypothesis $H_0: E[Y(1)] = \tau_{0}$ at a $5\%$ significance level, for a grid of values of $\tau_0$. Both tests control size within the identified set for all sample sizes, however, the rejection probability remains below $5\%$ well outside of the identified set in small samples. At all sample sizes, we find that the direct method is more conservative outside of the identified set than the screening method in this example.
Finally, we consider the model presented in freyberger2015identification, as described in Example (ref). The simulation design is specified as follows: $\mathcal{X} = \{2,3,4,5,6,7\}$ and $\mathcal{W} = \{0,1\}$, with $\pi_{h,k} = \Pr(X = x_h, W = w_k)$ given by \[ \Pi =
. \] The vector $g = (g(2), \ldots, g(7))'$ is given by $(23, 17, 13, 11, 9, 8)'$. The instrument is distributed as $Z_i \sim N(0,1)$, independently of $(X_i, W_i)$. Finally, the outcome equation is \[Y_i = g(X_i) + U_i~,\] where the error is given by \[U_i = X_iZ_i^2 - E_P[X_i|W_i]~.\] The shape constraint we maintain is that the structural function $g$ is decreasing, i.e., $Sg \leq 0$ for \[S =
, \] and our functional of interest is $L(g) = g(2)$. Given this design the identified set for $L(g)$ is $[20.21, 24.61]$. Using a random sample from $(Y, X, W, Z)$, $\hat{A}_{0,n}$ and $\hat{b}_{j,n}$ are computed using sample analogs, using the known value of $S$ and $c$.
Figure (ref) presents the rejection probabilities from $1,000$ draws for tests of the null hypothesis $H_0: L(g) = L_0$ at a $5\%$ significance level, for a grid of values of $L_0$. This design seems particularly challenging, with the rejection probabilities outside of the identified set being highly asymmetric. One possible source for this asymmetry is that the shape constraints are violated to the left of the identified set, but are not violated to the right. Beyond this observation, our findings are similar to the previous two examples; both tests control size within the identified set for all sample sizes, and we find that the direct method is more conservative outside of the identified set than the screening method.