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Inference for Linear Conditional Moment Inequalities
Moment inequalities are a useful tool in a wide range of fields in empirical economics. As described in recent reviews by HoRosen and Molinari2019, moment inequalities can be used to exploit the most direct implications of utility or profit maximization for inference in both single-agent settings and games. They can also be used to weaken parametric, behavioral, measurement, and selection assumptions in a range of problems. Inference using moment inequalities raises practical challenges, however, particularly when there are nuisance parameters (e.g. coefficients on control variables) that are not of direct interest.
A first challenge is obtaining tests that are computationally tractable. Many available moment inequality methods rely on test inversion over a grid for the full parameter vector (including the nuisance parameters), but the computational costs of such approaches grow exponentially in the dimension of the parameter vector. This has necessitated the development of alternative approaches that either profile out (i.e. optimize over) the nuisance parameters in the computation of the test statistic BugniCanayShi2017 or use computational shortcuts to form projection confidence sets without computing the test for all values of the nuisance parameter Kaidoetal2016. Nevertheless, computation can still be challenging when the dimension of the nuisance parameters is moderate or large.
A second challenge is obtaining tests with good power. When there are nuisance parameters, tests for the parameter of interest can be obtained via projection, but this can lead to conservative tests with poor power BugniCanayShi2017, Kaidoetal2016. Moreover, the power of many existing procedures can be negatively affected by the inclusion of non-binding moments, yet it may not be clear ex ante which of the moments implied by economic theory will be binding. This has prompted a variety of approaches to eliminate or reduce the sensitivity of moment inequality tests to slack moments including work by \citetalias{AndrewsSoares2010}, \citetalias{AndrewsBarwick2012}, Romanoetal2014, Chernozhukovetal2015, BugniCanayShi2017, and Bellonietal2018, among many others.
In this paper, we show that a variety of applications of moment inequalities have a particular structure that can be exploited to address these challenges. Specifically, we study settings with moment inequalities of the form $E[ Y_i(\beta_0) - X_i(\beta_0) \delta | Z_i ] \leq 0$, where $\beta_0$ is the parameter of interest, $\delta$ is a nuisance parameter, and $X_i(\beta_0)$ is a function of $Z_i$. That is, we study conditional moment inequalities that (a) are linear in the nuisance parameters $\delta$, and (b) have conditional variance (given the instruments $Z_i$) that does not depend on the nuisance parameters. In Section (ref), we highlight several recent applications of moment inequalities that have this structure, including interval-valued regression and revealed preference models in industrial organization.
Under this linear conditional structure, the profiled studentized max statistic can be represented as a linear program, and can thus be computed efficiently even when the dimension of the nuisance parameters is large. Linear conditional structure is also helpful for deriving tractable critical values, since it implies that the asymptotic variance of the moments (conditional on the instruments) does not depend on the value of the nuisance parameters. These features allow us to construct profiling-based confidence sets that rely on test inversion only for the target parameter and not for the nuisance parameters, and thus are computationally tractable even when the dimension of the nuisance parameters is large. We exploit this linear conditional structure to develop two tests that have different desirable properties, as well as a third hybrid approach that combines the two and is our preferred approach.
Our first approach is based on the least-favorable (LF) asymptotic distribution of our test statistic. We show that the distribution of the test statistic is increasing (in the sense of first-order stochastic dominance) in the mean of the moments, and thus the least-favorable distribution under the null corresponds with the case where the mean of all of the moments is zero.\footnote{This presumes that the set of data-generating processes considered allows for the possibility that all moments bind simultaneously. If not, then the distribution used for our critical value is an upper bound on the least-favorable distribution under the null.} It is then straightforward to calculate a critical value under the least-favorable distribution via simulation. The LF test has exact asymptotic size when all of the moments are simultaneously binding in population, and thus avoids conservativeness from projection in this case. A downside of the LF test, however, is that its power can be negatively affected by the inclusion of slack moments.
To address sensitivity to slack moments, we introduce a second test based on a novel conditioning argument. We condition on the Lagrange multipliers in the optimization to compute the test statistic, which intuitively correspond with the set of binding moments in sample after profiling out the nuisance parameters. We show that the set of values of the moments for which a particular Lagrange multiplier is optimal is a polyhedron, and we then derive critical values using results from Leeetal2016 on polyhedral conditioning events. We prove that the resulting conditional test is insensitive to slack moments in the strong sense that, as a subset of the moments becomes arbitrarily slack, the conditional test converges to the test that drops these moments ex-ante. A downside of the conditional test, however, is that it may have poor power in settings where multiple moments are approximately equally violated. Finally, given the different relative strengths of the LF and conditional approaches, we introduce a hybrid approach that combines the LF and conditional approaches, while avoiding the conservativeness of Bonferroni approaches.
The critical values for all of our tests are based on a normal approximation to the distribution of the moments conditional on the instruments. If this normal approximation holds exactly with known variance, our proposed tests control size in finite samples. In Section (ref) we provide regularity conditions under which size control in this finite sample normal model translates to uniform asymptotic size control over a large class of data-generating distributions. A desirable feature of our proposed tests is that they they achieve uniform asymptotic size control without having to specify a sequence of tuning parameters that converges at a certain rate. Nevertheless, our tests do require the researcher to make some choices. To use the hybrid test, the researcher must specify the size of the “first-stage” least favorable test $\kappa$, although this choice only affects the power of the test and not its asymptotic validity.\footnote{We recommend using $\kappa=\alpha/10$, and implement this choice in our simulations, following the recommendation for the two-step procedure in Romanoetal2014.} Additionally, although conditional moment inequalities can imply an infinite number of unconditional moments, our tests only exploit the implications of $k$ unconditional moments that must be specified by the researcher. We provide heuristic guidance on the choice of the $k$ moments in Section (ref).
To explore the numerical performance of our methods, we apply our techniques in simulations calibrated to Wollmann's study of the US auto bailout. We consider designs with up to ten nuisance parameters, and find that our proposed tests remain computationally tractable and have good size control in all specifications. The power of the hybrid test is similar to or better than that of the LF and conditional tests in all specifications, and we thus recommend the hybrid approach among our proposed procedures. We also find that the hybrid test has power dominating that of the projection-based tests of \citetalias{AndrewsSoares2010} and Kaidoetal2016 in all specifications for which we are able to compute these tests, and computation time for the hybrid can be over 10 times faster than for either of the projection-based approaches. The hybrid approach is also competitive with the sCC and sRCC tests proposed in concurrent work by cox_simple_2020, although neither approach dominates the other across all specifications in terms of power or computational speed.
\paragraph{Related Literature.} cox_simple_2020 consider the class of linear conditional moment inequalities introduced in this paper and propose tests based on a profiled quasi-likelihood ratio (QLR) statistic, whereas our tests are based on the profiled studentized max statistic. cox_simple_2020 and the present paper independently developed conditional testing approaches, but due to the difference in test statistics, the conditioning events and resulting tests are different. As discussed in Section (ref), we find in our Monte Carlo simulations that our preferred test (the hybrid) has non-nested power with those proposed by cox_simple_2020, which accords with the intuition that tests based on the max and QLR statistics direct power towards different parts of the parameter space.
Subvector inference for moment inequalities with linear parameters is also considered in cho_simple_2021, Garfarov2016 and Flynn2019. The setting in these papers differs from ours in that they consider unconditional moment inequalities, whereas we consider conditional moments; our paper also differs in that we allow the target parameters to potentially enter the moments non-linearly. One advantage of our approach relative to these previous papers is that we do not require a linear independence constraint qualification (LICQ) assumption, which restricts what moments can bind in population; see Section (ref) for further discussion.\footnote{cho_simple_2021 show that LICQ can be guaranteed to hold by adding a stochastic perturbation to the moments, at the expense of obtaining inference on an outer set of the sharp identified set.} Another related paper is kaido_asymptotically_2014, who consider efficient estimation and inference for the support function in settings with convex moment inequalities, which nests the problem of subvector estimation/inference in moment inequality models where all parameters enter linearly. Their approach, however, relies on a Slater constraint qualification that, for example, rules out moment equalities cast as inequalities. Our approach is thus complementary, since we do not require such a constraint qualification but also do not provide any formal efficiency results.
Our approach uses a profiled maximum statistic, and thus is also related to other profiling-based methods for moment inequalities. The profiling-based approach in BugniCanayShi2017 differs from ours in that it accommodates unconditional moment inequalities and does not require that the parameters enter the moments linearly. However, the linear structure that we consider enables highly-tractable computation since the profiled test statistic is computed with a linear program, and also enables us to develop tests that are uniformly asymptotically valid without relying on drifting sequences of tuning parameters. Bellonietal2018 build on the approach of BugniCanayShi2017 to develop methods for subvector inference with high-dimensional unconditional moments. Fangetal2021 propose a test based on the solution to a linear program that is applicable for a large class of problems that nests a high-dimensional version of the conditional linear inequalities considered in this paper, although at the cost of either introducing a sample-size dependent tuning parameter or obtaining a conservative test. Alternative approaches to subvector inference in moment inequality models include projection-based methods kaido_calibrated_2017; sub-sampling approaches RomanoShaikh2008; and quasi-posterior Monte Carlo methods CCT2018.\footnote{The approach of CCT2018 delivers inference on the identified set, rather than on points within the identified set.} We emphasize that the aforementioned methods do not impose the specific linear conditional structure considered in this paper, and thus are applicable in a much wider class of problems. We provide comparisons to the profiling-based approach of cox_simple_2020 as well as two projection-based methods in our Monte Carlo simulations.
One important limitation of our approach is that --- while we assume that conditional moment inequalities are satisfied --- we consider tests that exploit only a fixed number ($k$) of the implied unconditional inequalities. This contrasts with papers that consider asymptotics in which the number of moments grows with the sample size, such as \citetalias{AndrewsShi2013} for full-vector inference, and Chernozhukovetal2015 and Bellonietal2018 for subvector inference.\footnote{Flynn2019 considers a continuum of unconditional moment inequalities.} An interesting open question is whether the tests proposed in this paper can be extended to the setting with a diverging number of moments. See Section (ref) below for additional discussion.
We assume that we observe independent and identically distributed data $D_{i}$, $i=1,...,n$ drawn from an unknown distribution $P \in \mathcal{P}$, for a class $\mathcal{P}$ of distributions. The true values of the parameters $(\beta,\delta)$ are assumed to satisfy the conditional moment inequalities
where $Z_i$ is a subvector of $D_i$, $Y_i(\beta)=y(D_i,\beta)\in \mathbb{R}^k$ and $X_i(\beta)=x(Z_i,\beta)\in \mathbb{R}^{k\times p}$ for known functions $y(\cdot,\cdot)$ and $x(\cdot,\cdot),$ and $P_{D|Z}$ denotes the conditional distribution of $D_i$ given $Z_i$. We are interested in $\beta,$ while $\delta\in\mathbb{R}^p$ is a nuisance parameter. Specifically, we want to test that a given value $\beta_0$ belongs to the identified set for $\beta$, $\widetilde H_0:\beta_0\in B_I(P),$ where
is the set of values $\beta$ such that there exists $\delta$ which makes ((ref)) hold. For the remainder of the paper we omit the phrase “almost surely” for brevity. We call restrictions of the form ((ref)) linear conditional moment inequalities. They have two key properties: first, the nuisance parameter $\delta$ enters linearly and, second, the Jacobian of the moments with respect to $\delta$, $-X_i(\beta)$, is non-random conditional on $Z_i$. This structure implies that the variance of the moments conditional on $Z_i$ does not depend on $\delta$.
It is helpful to compare ((ref)) to the linear regression model
for $Y_i^*\in\mathbb{R}$ and $X^*_i\in\mathbb{R}^p$. Specifically, ((ref)) implies
where $Y_{i}(\beta)\in\mathbb{R}^k$ and $X_i(\beta)\in\mathbb{R}^{k\times p}$. Linear conditional moment inequalities thus generalize the traditional regression model to (a) relax the conditional moment restriction on the errors $\varepsilon_i$ to an inequality, (b) allow the possibility that there are instruments $Z_i$ beyond the regressors $X_i$, (c) allow a vector-valued outcome, and (d) allow $\beta$ to enter the moments non-linearly.
Linear conditional moment inequalities appear in a variety of economic applications.
\paragraph{Example 1} Linear conditional moment inequalities arise naturally from the linear regression model ((ref)), and its instrumental variables generalization, when we observe only bounds on the outcome $Y_i^*$. Consider the model
where $V_i$ is a function of $Z_i$ while $W_i$ may be endogenous. For instance, $\beta$ may be a causal effect of interest whereas $V_i$ represents a set of control variables. This is a linear instrumental variables model where the error is mean-independent of the instrument.
As in e.g. manskitamer2002, suppose that rather than observing $Y_i^*$ we instead observe bounds $Y_i^L$ and $Y_i^U$ where $Y_i^L\le Y_i^*\le Y_i^U$ with probability one. The model ((ref)) implies that $E_{P_{D|Z}}[Y_i^L-W_i\beta-V_i'\delta|Z_i]\le0$ and $E_{P_{D|Z}}[W_i\beta+V_i'\delta-Y_i^U|Z_i]\le0$, so we obtain conditional moment inequalities. To cast these inequalities into our framework, suppose we are interested in inference on $\beta,$ and for any vector of non-negative functions of the instruments $f(Z_i)$ let $Y_i(\beta)=(Y_i^L-W_i\beta,W_i\beta-Y_i^U)'\otimes f(Z_i),$ and $X_i=(V_i', -V_i')'\otimes f(Z_i)$, for “$\otimes$” the Kroneker product. This yields the moments $ E_{P_{D|Z}}[Y_i(\beta)-X_i\delta | Z_i] \leq 0,$ as desired.\footnote{Our approach to this application relies on the conditional moment restriction $E_{P_{D|Z}}\left[\varepsilon_{i}|Z_{i}\right]=0$. As discussed by PonomarevaTamer2011, this means that the identified set may be empty if the linear model is incorrect. For $Z_i=(W_i,V_i')'$, BeresteanuMolinari2008 assume only that $E_{P}[\varepsilon_{i}Z_i]=0$ and conduct inference on the (necessarily nonempty) set of best linear predictors. Bontempsetal2012 study identification and inference, including specification tests, for a class of linear models with unconditional moment restrictions.} $\triangle$
\paragraph{Example 2} Katz2007 studies the impact of travel time on supermarket choice. Katz assumes that utility is additively separable in the basket of goods bought ($B_{i}$), the travel time to the supermarket chosen ($T_{i,s}$), and the cost of the basket ($\pi(B_{i},s)$). Normalizing coefficient on cost to one, agent $i$'s realized utility is $$ U_i(B_{i},s)= U_i(B_{i}) + C_s'\delta - (\beta + \nu_i) T_{i,s} - \pi(B_{i},s), $$ where $C_s$ are observed characteristics of the supermarket, $T_{i,s}$ is the travel time for $i$ going to $s$, and $\beta+\nu_i$ is its impact on utility, where $\nu_i$ has mean zero given supermarket characteristics and travel times.
Katz assumes travel times and store characteristics are known to the shopper. For $\tilde{s}$ a supermarket with $T_{i,\tilde{s}} > T_{i,s}$ that also marketed $B_{i}$, he divides the difference $U_i(B_{i},s)-U_i(B_{i},\tilde{s})$ by $T_{i,s}-T_{i,\tilde{s}}$ and notes that a combination of expected utility maximization and revealed preference implies that $ E_{P_{D|Z}}[Y_i(\beta)-X_i\delta | Z_i] \le 0, $ for $$ Y_i(\beta) \equiv -\beta-\frac{[\pi(B_{i},s)- \pi(B_{i},\tilde{s})]}{T_{i,s}-T_{i,\tilde{s}}}, \; X_i \equiv -\frac{C_s'-C_{\tilde{s}}'}{T_{i,s}-T_{i,\tilde{s}}}, \hbox{ and } Z_i \equiv (T_{i,s},T_{i,\tilde{s}}, C_s',C_{\tilde{s}}')'. $$ Together with an analogous inequality which uses a store closer to the agent, Katz obtains both upper and lower bounds for $\beta$. $\triangle$
\paragraph{Example 3} Wollmann considers the bailout of GM and Chrysler's commercial truck divisions during the 2008 financial crisis and asks what would have happened had they instead been allowed to either fail or merge with another firm. This example is the basis for our simulations below.
Merger analysis focuses on price differences pre- and post-merger. Wollmann notes that some commercial truck production is modular (it is possible to connect different cab types to different trailers), so some products would likely have been repositioned after the change in the environment. To analyze product repositioning he requires estimates for the fixed costs of marketing a product. His estimated demand and cost systems enable him to estimate counterfactual profits from adding or deleting products. Assuming firms maximize expected profits, differences in expected profits from adding or subtracting products imply bounds on fixed costs.
To illustrate, let $J_{f,t}$ be the set of models that firm $f$ marketed in year $t$ and let $J_{f,t}\setminus j $ be that set excluding product $j$, while $\Delta \pi(J_{f,t}, J_{f,t}\setminus j)$ is the difference in expected profits between marketing $J_{f,t}$ and $J_{f,t}\setminus j$. The fixed cost to firm $f$ of marketing product $j$ at time $t$ is given by $(\delta_{c,f} + \delta_g g_j)$ if the product was not marketed previously ($j\not\in J_{f,t-1}$), and $\beta (\delta_{c,f} + \delta_g g_j)$ if it was previously marketed. Here $\delta_{c,f}$ is a firm-specific intercept, $g_j$ is the weight of product $j$, $\delta_g$ is the cost of adding additional weight (assumed common across firms), and $\beta$ captures the cost savings of marketing a pre-existing product. We can write the fixed cost as $X_{j,f,t}^*(\beta)\delta$, where $X_{j,f,t}^*(\beta)$ contains a firm indicator and the product's weight, possibly multiplied by $\beta$ depending on whether $j \in J_{j,f,t-1}$. For $Z_{f,t}$ a set of variables known to the firm when marketing decisions were made, including the variables used to form $X_{j,f,t}^*(\beta)$,
by the firm's equilibrium conditions, where $$ Y_{j,f,t} \equiv \Delta \pi(J_{f,t}, J_{f,t} \setminus j) \cdot 1\{ j \in J_{f,t}, j \in J_{f, t-1}\}, \; \hspace{0.5cm} X_{j,f,t}(\beta) \equiv X_{f,j,t}^*(\beta) \cdot 1\{ j \in J_{f,t}, j \in J_{f, t-1}\} $$ and $1\{A\}$ is an indicator for the event $A$. Additional inequalities can be added for marketing a product that was not marketed in the prior period, for withdrawing products, and for combining the withdrawal of one product with adding another. $\triangle$
cox_simple_2020 note that moment inequalities in Eizenberg2014 and Gandhietal2019 also have linear conditional structure. Further recent examples appear in HoPakes2014, Moralesetal2017, rambachan_more_2022, and rambachan_identifying_2021.
In addition to arising frequently in applications, the structure of linear conditional moment inequalities can be exploited to develop simple and computationally tractable tests of ((ref)). We begin by describing an asymptotic framework frequently used to test moment inequalities, and some challenges it generates. We then describe how linear conditional structure can be used to circumvent some of these issues. We focus on the intuition here, deferring formal results to the following sections.
\paragraph{Unconditional asymptotics} Conditional moment inequalities are often tested indirectly. In particular, ((ref)) implies that $ E_P[ Y_i(\beta) - X_i(\beta) \delta ] \leq 0. $ To test $\widetilde{H}_0:\beta_0\in B_I(P),$ we may therefore test that there exists a value of $\delta$ such that $E_P[ Y_i(\beta_0) - X_i(\beta_0) \delta] \leq 0$. Letting $Y_{n,0}=\frac{1}{\sqrt{n}}\sum_i Y_i(\beta_0)$ and $X_{n,0} = \frac{1}{\sqrt{n}}\sum_i X_i(\beta_0)$, the central limit theorem implies that for each $\delta$, $Y_{n,0}-X_{n,0}\delta- \mu_{U,n,0}(\delta) \to_d N(0,\Sigma_{U,0}(\delta)),$ for $\mu_{U,n,0}(\delta) = \sqrt{n}E_P[Y_{i}(\beta_0)-X_i(\beta_0)\delta]$ and $\Sigma_{U,0}(\delta)=Var_P(Y_i(\beta_0)-X_i(\beta_0)\delta)$. This suggests the approximation
where $\approx^d$ denotes approximate equality in distribution. The normal approximation ((ref)) may be used to test $H_0^{joint}(\delta_0):\mu_{U,n,0}(\delta_0)\le 0$, which jointly restricts $(\beta,\delta)$. This allows a projection test of $\widetilde{H}_0:\beta_0\in B_I(P)$, which rejects if and only if we reject $H_0^{joint}(\delta_0)$ for all $\delta_0$. Simple projection tests can be quite conservative, however, which has motivated approaches based on the joint limiting distribution across different values of $\delta$ Kaidoetal2019.
Even if we are happy to use the projection method, projection tests based on ((ref)) are complicated by the dependence of the variance matrix $\Sigma_{U,0}(\delta_0)$ on the value of $\delta_0$, since critical values for tests of $H_0^{joint}(\delta_0)$ will typically depend on $\delta_0$ as well. When the nuisance parameter $\delta$ has even moderate dimension, calculating the critical value for many values of $\delta_0$ can become computationally burdensome, necessitating careful attention to algorithms to mitigate the computational cost Kaidoetal2019.
\paragraph{Conditional asymptotics} Linear conditional structure allows an alternative asymptotic approximation, which avoids complications discussed above by conditioning on the sequence of realized instrument values $\{Z_i\} = \{Z_i\}_{i=1}^{\infty}$. For $\mu_i(\beta,{P_{D|Z}}) = E_{P_{D|Z}}[Y_i(\beta) | Z_i]$ and $\mu_{n,0} = \frac{1}{\sqrt{n}}\sum_i \mu_i(\beta_0,{P_{D|Z}})$, the Lindeberg-Feller central limit theorem implies that under mild conditions $Y_{n,0} - \mu_{n,0} | \{Z_i\}\rightarrow_d N(0,\Sigma_0)$, where $\Sigma_0 = E_{P}[Var_{P_{D|Z}}(Y_i(\beta_0)|Z_i)]$. Since $X_{n,0}$ is non-random conditional on $\{Z_i\}$, this suggests the approximation
Importantly, and in contrast to ((ref)), the variance $\Sigma_0$ in ((ref)) does not depend on the value of $\delta$. This substantially simplifies the problem of constructing tests. Further, since $X_{n,0}$ is non-stochastic conditional on $\{Z_i\}$, ((ref)) holds jointly across values of $\delta.$
To construct tests based on this conditional approximation, observe that if $\widetilde{H}_0: \beta_0 \in B_I(P)$ holds, then there exists (almost surely) a value of $\delta$ such that $\mu_{n,0}- X_{n,0}\delta \leq 0$. The null $\widetilde{H}_0:\beta_0\in B_I(P)$ thus implies the null $H_0: \mu_{n,0} \in \mathcal{M}_{n,0}$, where \[\mathcal{M}_{n,0} = \{ \mu\in\mathbb{R}^k : \text{ there exists } \delta \text{ such that } \mu - X_{n,0} \delta \leq 0\}\] is non-stochastic conditional on $\{Z_i\}$.\footnote{In fact, $\widetilde{H}_0$ implies that $\mu_{n,0} \in \mathcal{M}_{n,0} \cap \mathcal{M}_{n,0,\mathcal{P}_{D|Z}}$, where $\mathcal{P}_{D|Z}$ is the family of conditional distributions implied by $\mathcal{P}$, while $\mathcal{M}_{n,0,\mathcal{P}_{D|Z}} = \left\{ \frac{1}{\sqrt{n}} \sum_i E_{P_{D|Z}}[Y_i(\beta_0) |Z_i] \,|\, P_{D|Z} \in \mathcal{P}_{D|Z} \right\} $. For tractability, we focus on the implied null that $ \mu_{n,0} \in \mathcal{M}_{n,0} $ rather than $\mu_{n,0} \in \mathcal{M}_{n,0} \cap \mathcal{M}_{n,0,\mathcal{P}_{D|Z}}$. This yields valid but potentially conservative tests if $ X_{n,0}\delta \not\in \mathcal{M}_{n,0,\mathcal{P}_{D|Z}}$ for all $\delta$, i.e. if $\mathcal{P}_{D|Z}$ does not allow all moments to simultaneously bind; see Section (ref) for additional discussion.} Equation ((ref)) with $\delta=0$ further implies that $Y_{n,0} | \{Z_i\} \approx^d N(\mu_{n,0}, \Sigma_0)$, so testing $H_0$ reduces, asymptotically, to testing a restriction on the mean of a multivariate normal vector.
\paragraph{Indirect Tests} While indirect tests of $\widetilde{H}_0:\beta_0\in B_I(P)$ are natural, they can entail a loss of consistency. The original null hypothesis $\widetilde{H}_0:\beta_0\in B_I(P)$ implies that there exists a $\delta$ such that $\frac{1}{\sqrt{n}}\sum_i (E_{P_{D|Z}}[Y_i(\beta_0) | Z_i] - X_i(\beta_0) \delta) \otimes f(Z_i) \leq 0$ for all non-negative functions $f(Z_i)$, whereas $H_0: \mu_{n,0} \in \mathcal{M}_{n,0}$ only tests that this is satisfied for $f(Z_i)=1$.\footnote{Note that if one starts with $(Y_i,X_i)$ satisfying ((ref)), then $E_{P_{D|Z}}[\tilde{Y}_i - \tilde{X}_i \delta | Z_i]\leq0$ for $ (\tilde{Y}_i, \tilde{X}_i) = (Y_i,X_i) \otimes f(Z_i)$ and any non-negative finite instrument function $f(Z_i)$. Thus, a key restriction imposed in our framework is that the researcher chooses a finite set of instruments with which to interact the initial moments.} Indeed, conditional moment inequalities based on continuously distributed instruments $Z_i$ generate an infinite number of unconditional inequalities, as discussed in e.g. \citetalias{AndrewsShi2013}, Armstrong2014, Chernozhukovetal2015, and Chetverikov2013. As a result, the tests we develop do not in general yield consistent tests when the instruments are continuously distributed. This contrasts with the aforementioned papers, which develop consistent tests by checking an (asymptotically) infinite number of moment restrictions.
Inference based on a finite, researcher-selected set of inequalities nonetheless appears widespread in applications, and is the approach adopted in all the empirical applications discussed above save Gandhietal2019. This raises the question of how to select the finite set of moments (i.e, which restrictions to include in $Y_i$), which we discuss informally in Section (ref) below. Whether one can go further, either characterizing an optimal selection of moments or combining our results with those in the previous literature on conditional moment inequalities to ensure consistent inference in settings with continuously distributed $Z_i$, is an interesting question for future work.
We now introduce our tests. Motivated by the asymptotic approximation ((ref)), we begin with tests of $H_0:\mu_{n,0}\in\mathcal{M}_{n,0}$ in the exact normal model
The next section presents sufficient conditions for feasible versions of our tests, based on non-normal data and estimates of $\Sigma_0$, to uniformly control asymptotic size.
Given $Y_{n,0} \sim N(\mu_{n,0}, \Sigma_0)$ for known $\Sigma_0$, we construct tests for the hypothesis $H_0:\mu_{n,0} \in\mathcal{M}_{n,0}$, that is, that there exists some $\delta$ such that $\mu_{n,0}-X_{n,0}\delta \le 0$. We eliminate the nuisance parameter $\delta$ by using the profiled max statistic, $$ \hat{\eta}_{n,0} = \min_\delta \hspace{0.1cm} \max_{j} \,\, \left\{ e_j'\left( Y_{n,0}-X_{n,0}\delta\right)/\sigma_{0,j} \right\} \label{eq: max statistic} $$ for $e_j$ the $j$th standard basis vector and $\sigma_{0,j}=\sqrt{e_j'\Sigma_{0}e_j}$.\footnote{We define $\frac{c}{0}=\infty$ for all $c>0$.} Our test statistic thus profiles the maximum-criterion statistic ($S_3$ in the notation of \citetalias{AndrewsSoares2010}). By a profiled test statistic, we mean one that optimizes over the nuisance parameter $\delta$ to find the value that makes the test statistic as small as possible. Specifically, note that $\max_{j} \left\{ e_j'\left( Y_{n,0}-X_{n,0}\delta\right)/\sigma_{0,j} \right\}$ calculates the maximum studentized violation of the sample moments at a given $\delta$, so $\hat{\eta}_{n,0}$ corresponds to the maximum violation at the value of $\delta$ that makes this violation the smallest. One could profile test statistics other than the max statistic --- e.g. cox_simple_2020 study profiled QLR statistics and BugniCanayShi2017 study profiled modified method of moments (MMM) statistics (among others) --- but it will be helpful for our analysis that the profiled max statistic admits an equivalent representation as the solution to the linear program,
for $\sigma_{0}=(\sigma_{0,1},...,\sigma_{0,k})'$. This allows for tractable computation of $\hat{\eta}_{n,0}$ even when the dimension of $\delta$ is large, and the linear structure plays a key role in the construction of our tests.
To derive critical values, we will make use of the dual representation of the linear program ((ref)). Standard results in linear programming (e.g., Chapter 7.4 of Schrijver1986) imply that when $\hat{\eta}_{n,0} > -\infty$ it is the solution of the dual linear program,\footnote{Observe that $\hat{\eta}_{n,0}$ is equal to $-\infty$ if and only if $\min_\delta \max_j e_j' X_{n,0} \delta =-\infty$, in which case $H_0$ is satisfied regardless of the value of $\mu_{n,0}$, so the testing problem is trivial. Finiteness of $\hat{\eta}_{n,0}$ implies that $X_{n,0}$ does not have full row rank, for instance because $k>p$. }
Moreover, the maximum is obtained at one of the finite set of vertices of the feasible set. Intuitively, the set of feasible values $F(X_{n,0}, \sigma_0) = \{ \gamma \geq 0 | \gamma' X_{n,0} = 0, \gamma'\sigma_0 = 1 \}$ is a polyhedron, i.e. a convex set with flat sides, and a vertex corresponds with a “corner” of this set. More formally, as described in e.g. Schrijver1986, $\gamma\in F(X_{n,0}, \sigma_0)$ is a vertex if it can be realized as a unique solution to ((ref)) for some value of $Y_{n,0}$:
As a simple example, if $\Sigma =I$ and $X_{n,0}=0$, then $V(X_{n,0}, \sigma_0)$ is the set of standard basis vectors in $\mathbb{R}^k$. In Lemma (ref) in the appendix, we give an alternative characterization of the set of vertices, which shows that $\gamma \in F(X_{n,0}, \sigma_0)$ is a vertex if and only if $\gamma$ is the solution to the system of equations defined by a full-rank subset of the constraints in ((ref)). Since there are a finite number of constraints in ((ref)), this immediately implies that $V(X_{n,0},\sigma_0)$ is finite. It is neither necessary nor recommended to enumerate all of the elements of $V(X_{n,0},\sigma_0)$ to compute our test statistic and critical values (see Section (ref) for details on computation), but this representation will be useful for explaining our approach.
The dual representation for $\hat\eta_{n,0}$ implies that in the finite sample normal model the test statistic $\hat{\eta}_{n,0}$ is the maximum of a multivariate normal vector, $\hat{\eta}_{n,0}=\max_{\gamma\in V(X_{n,0}, \sigma_0)}\gamma'Y_{n,0} = \max\{\gamma_{(1)}'Y_{n,0},...,\gamma_{(J)}'Y_{n,0} \}$, for $\gamma_{(1)},...,\gamma_{(J)}$ the elements of $V(X_{n,0} , \sigma_{0})$. Our critical values will then be based on properties of the maximum of a correlated Gaussian vector.
Our first test is based on the “least-favorable” value of $\mu_{n,0}$ under the null hypothesis $H_0$. Recall that $\hat{\eta}_{n,0} = \max_{\gamma \in V(X_{n,0}, \sigma_0)} \gamma' Y_{n,0} $. Hence $$\hat{\eta}_{n,0} = \max_{\gamma \in V(X_{n,0}, \sigma_0)} \left\{\gamma' \mu_{n,0} + \gamma' (Y_{n,0} -\mu_{n,0})\right\} \leq \max_{\gamma \in V(X_{n,0}, \sigma_0)} \gamma' \mu_{n,0} + \max_{\gamma \in V(X_{n,0}, \sigma_0)} \gamma' (Y_{n,0} -\mu_{n,0}).$$ Under $H_0$, however, there exists $\delta$ such that $\mu_{n,0} - X_{n,0} \delta \leq 0$. Since every $\gamma \in V(X_{n,0}, \sigma_{0})$ is feasible in ((ref)) by construction, we also have that $\gamma'X_{n,0} = 0$ and $\gamma \geq 0$ for all $\gamma \in V(X_{n,0}, \sigma_0)$. It follows that under the null, $ \gamma'\mu_{n,0} = \gamma'(\mu_{n,0} - X_{n,0} \delta) \leq 0$ for all $\gamma \in V(X_{n,0}, \sigma_{0})$. Combined with the previous display, this implies that under $H_0,$
Since $Y_{n,0} -\mu_{n,0} \sim N(0,\Sigma_0)$, we define the least-favorable critical value $c_{\alpha,LF}=c_{\alpha,LF}(X_{n,0}, \sigma_{0})$ as the $1-\alpha$ quantile of $ \max_{\gamma \in V(X_{n,0}, \sigma_0)} \gamma' \xi$ for $\xi \sim N(0,\Sigma_0)$ and consider the test that rejects when $\hat{\eta}_{n,0}$ exceeds this critical value, $\phi_{LF} = 1\left\{ \hat{\eta}_{n,0} > c_{\alpha,LF} \right\}$. It follows immediately from the inequality ((ref)) that under the finite sample normal model $E[\phi_{LF}] \leq \alpha$ whenever $H_0: \mu_{n,0} \in \mathcal{M}_{n,0}$ holds. Moreover, the inequality ((ref)) reduces to an equality if $\gamma' \mu_{n,0} = 0$ for all $\gamma \in V(X_{n,0}, \sigma_0)$, as for example occurs if $\mu_{n,0} =0$ or more generally if $\mu_{n,0}=X_{n,0}\delta$ for some $\delta$, in which case $E[\phi_{LF}] = \alpha$. Thus, the LF test has exact size in the finite sample normal model if it is possible for all moments to bind simultaneously. We note, however, that this may not be possible for some data-generating processes (e.g., if certain pairs of moments correspond to upper and lower bounds that cannot simultaneously bind), in which case the least favorable test may have size strictly less than $\alpha$.\footnote{ In such cases, where $0\not\in\mathcal{M}_{n,0}\cap\mathcal{M}_{n,0,\mathcal{P}_{D|Z}}$ for $\mathcal{M}_{n,0,\mathcal{P}_{D|Z}}$ as defined in footnote (ref), tests based on the critical value $ c_{\alpha,LF}+\psi$ for $\psi=\max_{\mu_{n,0}\in\mathcal{M}_{n,0}\cap\mathcal{M}_{n,0,\mathcal{P}_{D|Z}}}\max_{\gamma\in V(X_{n,0},\sigma_0)}\gamma'\mu_{n,0}$ will also control size. These tests have (weakly) improved power since $\psi\le 0$ by definition. The adjustment factor $\psi$ depends on the class of conditional data generating processes $\mathcal{P}_{D|Z}$ considered, however, so we focus on results using $c_{\alpha,LF}$ for simplicity. }
\paragraph{Sensitivity to slack moments} An undesirable feature of the LF test is that it may be sensitive to the inclusion of slack moments. That is, the power of the test may be negatively affected if one includes in $Y_{n,0}$ moments that are very far from binding (i.e. elements $j$ with $\mu_{n,0,j}\ll 0$). The reason is that the critical value $c_{\alpha,LF}$ is based on the distribution of the test statistic when $\mu_{n,0}=0$, and thus generally increases when adding additional moments, even though the test statistic $\hat{\eta}_{n,0}$ will generally not be affected by the inclusion of very slack moments. Motivated by this fact, \citetalias{AndrewsSoares2010}, \citetalias{AndrewsBarwick2012}, Romanoetal2014, and related papers propose techniques that use information from the data to either select moments or shift the mean of the distribution from which the critical values are calculated. This yields tests with higher power in cases where many of the moments are slack. Unfortunately, applying these existing methods in our setting breaks the linear structure, and hence the computational advantages from using linear programming, which motivates us to introduce an alternative approach.
We next introduce a test that is less sensitive to the inclusion of slack moments than the LF test while also exploiting the linear conditional structure in our context. This test is based on the distribution of $\hat{\eta}_{n,0}$ conditional on the identity of the optimal vertex in the dual problem, $\hat\gamma=\arg\max_{\gamma \in V(X_{n,0},\sigma_0)}\gamma'Y_{n,0}$.\footnote{$\hat\gamma$ depends on $n$ and $\beta_0$, but we leave this dependence implicit for simplicity of notation.} For simplicity of exposition, we begin by assuming that $\hat\gamma$ is unique, in the sense that $\arg\max_{\gamma \in V(X_{n,0},\sigma_0)}\gamma'Y_{n,0}$ is a singleton; we will discuss the case of a non-unique dual below.\footnote{Our asymptotic results in the next section impose a sufficient condition for uniqueness to hold with probability one asymptotically.} If $\hat\gamma'\Sigma_0\hat\gamma=0$ then we define the conditional test to reject if and only if $\hat\eta_{n,0}>0.$ For the remainder of this section, we thus assume that $\hat\gamma'\Sigma_0\hat\gamma>0.$ For any $\gamma \in V(X_{n,0},\sigma_0),$ note that $\hat\gamma=\gamma$ only if $\gamma'Y_{n,0}\ge \tilde\gamma'Y_{n,0}$ for all $\tilde\gamma\in V(X_{n,0},\sigma_0)$. Hence, $\hat\gamma=\gamma$ is optimal only if $Y_{n,0}$ lies in the polyhedron $\{y \,|\, (\gamma - \tilde\gamma)' y \geq 0, \, \forall \tilde\gamma \in V(X_{n,0}, \sigma_0)\}$. This representation allows us to characterize the distribution of $\hat{\eta}_{n,0}$ conditional on $\hat\gamma=\gamma$ using Lemma 5.1 in Leeetal2016, which characterizes the behavior of Gaussian random variables conditional on polyhedral events.
Recall that under $H_0$, $\gamma'\mu_{n,0} \leq 0$ for all $\gamma \in V(X_{n,0}, \sigma_0)$. Additionally, Lemma A.1 in Leeetal2016 shows that the $TN(\mu,\sigma^2;[a,b])$ distribution is increasing in $\mu$ in the sense of first order stochastic dominance. It follows that the distribution on the right-hand side of ((ref)) is weakly dominated by the $TN(0,\hat\gamma' \Sigma_0 \hat\gamma, [\mathcal{V}^{lo}_{n,0}, \mathcal{V}^{up}_{n,0}])$ distribution under the null. We therefore base our test on this distribution. Letting $\bar{c}_{\alpha,C}$ be the $1-\alpha$ quantile of the $TN(0,\hat\gamma' \Sigma_0 \hat\gamma, [\mathcal{V}^{lo}_{n,0}, \mathcal{V}^{up}_{n,0}])$ distribution, we define the conditional critical value as $c_{\alpha,C} = c_{\alpha,C}(Y_{n,0},X_{n,0},\Sigma_0) = \max\{ \bar{c}_{\alpha,C}, 0\}$ and reject if $\hat{\eta}_{n,0}$ exceeds it, $\phi_C= 1\left\{ \hat{\eta}_{n,0} > c_{\alpha,C} \right\}$.\footnote{The censoring of the critical value at 0 is unnecessary for size control in the finite-sample normal model, but simplifies asymptotic arguments. It is also substantively reasonable as it prevents the test from rejecting when all of the moment inequalities are satisfied in sample ($\hat{\eta}_{n,0}\leq0$).} It follows immediately that $\phi_C$ controls size conditionally in the finite sample normal model, with $E[\phi_{C} | \hat\gamma =\gamma, S_{n,0,\gamma} ] \leq \alpha$ whenever $\mu_{n,0}\in \mathcal{M}_{n,0}$.\footnote{As for the least favorable test, if $X_{n,0}\delta\not\in\mathcal{M}_{n,0}\cap\mathcal{M}_{n,0,\mathcal{P}_{D|Z}}$ for all $\delta,$ we can potentially use smaller critical values, replacing $\bar{c}_{\alpha,C}$ with the $1-\alpha$ quantile of a $TN(\psi_{\hat\gamma},\hat\gamma' \Sigma_0 \hat\gamma, [\mathcal{V}^{lo}_{n,0}, \mathcal{V}^{up}_{n,0}])$ distribution for $\psi_{\hat\gamma}=\max_{\mu_{n,0}\in\mathcal{M}_{n,0}\cap\mathcal{M}_{n,0,\mathcal{P}_{D|Z}}}\hat\gamma'\mu_{n,0}$. As before, $\psi_{\hat\gamma}$ will depend on the specification of $\mathcal{P}_{D|Z}$, and we focus on tests based on $\bar{c}_{\alpha,C}$ for simplicity.} Unconditional size control follows by the law of iterated expectations.
\paragraph{Example (uncorrelated moments)} Consider the case where $Y_{n,0}\sim N(\mu_{n,0},I)$, and $X_{n,0}=0$, so that there is no nuisance parameter $\delta$. Then $V(X_{n,0},\sigma_0)$ is simply the set of standard basis vectors, so $\hat{\eta}_{n,0} = \max_j e_j'Y_{n,0}$ is the maximum component of $Y_{n,0}$. In this case $\mathcal{V}^{lo}_{n,0}$ corresponds to the second-largest component of $Y_{n,0}$, i.e. $\max_{j\neq \hat j} e_j'Y_{n,0}$, for $\hat{j}$ the location of the maximum, and $\mathcal{V}^{up}_{n,0}=\infty$. The conditional test thus rejects if $\hat{\eta}_{n,0}$ exceeds the $1-\alpha$ quantile of the standard normal distribution truncated to $[\mathcal{V}^{lo}_{n,0}, \infty]$.
\paragraph{Non-unique dual solutions.} So far we have assumed the existence of a unique dual solution, $\hat\gamma = \gamma$. If $\Sigma_0$ is not full-rank, however, then there may be multiple solutions to the dual problem with positive probability.\footnote{Since the dual objective is $\hat\eta_{n,0} = \max\{\gamma_{(1)}'Y_{n,0},..., \gamma_{(J)}'Y_{n,0} \}$ and $\gamma_{(j)}\neq \gamma_{(j')}$ for $j\neq j',$ the dual has a unique solution with probability 1 so long as $\Sigma_0$ is full rank.} In Appendix (ref), we consider a version of the conditional test that, when the dual solution is non-unique, calculates $(\mathcal{V}^{lo}_{n,0}, \mathcal{V}^{up}_{n,0})$ via ((ref)) by selecting an element of the dual solution set, $\gamma = h(\hat\gamma)$. We show that in the finite sample normal model, with probability 1 the critical values do not depend on how the optimal vertex is chosen, so the test obtained does not depend on the choice of $h(\cdot)$. Further, we show in Appendix (ref) that this test controls size in the finite-sample normal model. Our sufficient conditions for uniform asymptotic size control in Section (ref) below imply that the dual solution will be unique with probability tending to 1, however, so we focus primarily on the case where the dual solution is unique.
\paragraph{Insensitivity to Slack Moments} In contrast with the LF test, the conditional test has the desirable property that it is insensitive to the inclusion of slack moments. Specifically, our next result shows that the conditional test is insensitive to slack moments in the strong sense that as a moment becomes arbitrarily slack the conditional test converges to the conditional test that drops that moment ex-ante. Intuitively, this happens because (under mild conditions) sufficiently slack moments make no contribution to $\hat{\eta}_{n,0},$ $\mathcal{V}^{lo}_{n,0},$ or $\mathcal{V}^{up}_{n,0},$ and so have no impact on the conditional test. To state this result formally, define $Y_{n,0}^{j,d}=Y_{n,0}-e_{j}\cdot d$ as a version of $Y_{n,0}$ which decreases the $j$th moment by $d$. Let $Y_{n,0}^{-j}$ collect the rows of $Y_{n,0}$ other than the $j$th, and define $X_{n,0}^{-j}$ and $\Sigma^{-j}_0$ accordingly. Define $\hat{\eta}_{n,0}^{j,d}$ and $\hat{\eta}_{n,0}^{-j}$ as versions of $\hat{\eta}_{n,0}$ based on $\left(Y_{n,0}^{j,d},X_{n,0},\Sigma_0\right)$ and $\left(Y_{n,0}^{-j},X_{n,0}^{-j},\Sigma^{-j}_0\right)$, respectively, and let $\phi_{C}^{j,d}$ and $\phi_{C}^{-j}$ denote the corresponding tests.
The conditions of Lemma (ref) hold for Lebesgue almost every $Y_{n,0}$, and hold with probability 1 under ((ref)) provided that $\gamma' \Sigma_{0} \gamma > 0$ and $(\gamma-\tilde\gamma)' \Sigma_0 (\gamma-\tilde\gamma) > 0$ for all distinct $\gamma,\tilde\gamma \in V(X_{n,0}, \sigma_0)$, so that the variables $\gamma' Y_{n,0}$ have positive variance and are not perfectly correlated with one another. The only other tests we are aware of that both control size in the finite-sample normal model and are unaffected by the inclusion of arbitrarily slack moments in the sense of Lemma (ref) are those of cox_simple_2020.
\paragraph{Power with Multiple Violated Moments.} Although the conditional test exhibits a desirable insensitivity to the inclusion of slack moments, it may exhibit poor power in cases where two (or more) moments are approximately equally violated. This is most easily seen in the example of uncorrelated moments from above, where $\mathcal{V}^{lo}_{n,0}$ corresponds with the value of the second-largest sample moment, and the critical value is the $1-\alpha$ quantile of the standard normal distribution truncated to $[\mathcal{V}^{lo}_{n,0}, \infty]$. If two moments are approximately equally violated, then the largest and second largest sample moments ($\hat{\eta}_{n,0}$ and $\mathcal{V}^{lo}_{n,0}$, respectively) may be close together, so the conditional test need not reject even if both of these are large. This phenomenon is highlighted in parts of the parameter space in our simulations in Section (ref).
To mitigate the possible power losses of the conditional test when multiple moments are approximately equally violated, we next introduce a hybrid test that combines the least favorable and conditional approaches. For some $0<\kappa<\alpha$, we define the size-$\alpha$ hybrid test to reject whenever the size-$\kappa$ least favorable test does. If the least favorable test does not reject, we then consider a size-$\frac{\alpha-\kappa}{1-\kappa}$ test that conditions on both $\hat\gamma=\gamma$ and the event that the least-favorable test did not reject. Specifically, the same argument used to prove Lemma (ref) yields that $$ \hat{\eta}_{n,0} \,|\, \left\{ \hat\gamma=\gamma, S_{n,0,{\gamma}} = s, \phi_{LF,\kappa} =0 \right\} \sim TN( \gamma'\mu_{n,0}, \gamma' \Sigma_0 \gamma, [\mathcal{V}^{lo}_{n,0}, \mathcal{V}^{up,H}_{n,0}] ), $$ where $ \mathcal{V}^{up,H}_{n,0} = \min\{\mathcal{V}^{up}_{n,0}, c_{\alpha,LF} \}$. We then construct the second-stage critical value $\bar{c}_{\frac{\alpha-\kappa}{1-\kappa}, H}=\bar{c}_{\frac{\alpha-\kappa}{1-\kappa}, H}(Y_{n,0},X_{n,0},\Sigma_0)$ analogously to the conditional critical value $c_{\frac{\alpha-\kappa}{1-\kappa}, C}$ except using the modified truncation point $ \mathcal{V}^{up,H}_{n,0}$. Letting $c_{\frac{\alpha-\kappa}{1-\kappa}, H} = \min\{c_{\kappa,LF}, \bar{c}_{\frac{\alpha-\kappa}{1-\kappa}, H} \}$, the hybrid test is then $\phi_H = 1\{ \hat{\eta}_{n,0} > c_{\frac{\alpha-\kappa}{1-\kappa}, H} \}$. Observe that the critical value for the hybrid test approaches that of the LF test as $\kappa \rightarrow \alpha$, while it approaches that of the conditional test as $\kappa \rightarrow 0$.
As argued above, the first-stage LF test for the hybrid rejects with probability not exceeding $\kappa$ under the null in the finite-sample normal model. Likewise, by arguments analogous to those for the conditional test, the second stage test rejects with probability no more than $\frac{\alpha - \kappa}{1-\kappa}$ conditional on the first stage not rejecting. It follows that when $\mu_{n,0} \in \mathcal{M}_{n,0}$, the hybrid test rejects with probability $$E[\phi_{LF,\kappa}] + \left(1-E[\phi_{LF,\kappa}]\right)E\left[\hat{\eta}_{n,0} > \bar{c}_{\frac{\alpha-\kappa}{1-\kappa}, H} | \phi_{LF,\kappa} = 0\right] \leq \kappa + (1-\kappa )\frac{\alpha-\kappa}{1-\kappa} = \alpha,$$ and so controls size in the finite sample normal model.
The hybrid test proposed above always rejects whenever a simple Bonferroni combination of a size-$\kappa$ LF test and size-($\alpha - \kappa$) conditional test would reject, and can reject in cases where the simple Bonferroni does not. The proposed method improves upon the simple Bonferroni approach in two ways, first modifying the second-stage test to condition on the event that the LF test does not reject (which truncates the distribution above and so reduces the critical value), and then using a size $\frac{\alpha-\kappa}{1-\kappa} > \alpha-\kappa$ critical value. This helps to reduce the conservativeness usually associated with Bonferroni approaches.
\paragraph{Sensitivity to Slack Moments} The hybrid test will be sensitive to the inclusion of slack moments via its dependence on the LF critical values. However, this sensitivity will be small when $\kappa$ is close to zero, since in this case the critical values will tend to be close to those of the conditional test, which as shown above do not depend on the inclusion of slack moments. Similar to Romanoetal2014, we consider $\kappa=\alpha/10$ in our simulations below.
We conduct our analysis conditional on a sequence of values for the instruments, $\left\{ Z_{i}\right\} =\left\{ Z_{i}\right\} _{i=1}^{\infty},$ where the data are independent but potentially not identically distributed conditional on $\left\{ Z_{i}\right\}$, $ D_{i}\protect\mathpalette{\protect\independenT}{\perp} D_{i'}|\left\{ Z_{j}\right\} \mbox{ for all }i\neq i'$. Recall that $\mathcal{P}_{D|Z}$ is the class of conditional distributions for $D_i$ given $Z_i$, and let $B_I(P_{D|Z})$ denote the conditional identified set for $\beta$ given $\left\{ Z_{i}\right\},$ \[ B_I(P_{D|Z})=\left\{ \beta :\text{ there exists } \delta~s.t.~E_{P_{D|Z}}\left[Y_{i}(\beta)-X_i(\beta)\delta|Z_{i}\right]\le0\mbox{ for all }i\right\} . \] Note that for $B_I(P)$ as defined in (ref), $B_I(P)\subseteq B_I(P_{D|Z})$ for almost every $\left\{ Z_{i}\right\}.$ We provide conditions under which our tests uniformly control asymptotic rejection probabilities over $P_{D|Z}\in\mathcal{P}_{D|Z}$ and $\beta_0\in B_I(P_{D|Z})$. For brevity, we will leave the conditioning on $\left\{ Z_{i}\right\}$ implicit when this is without loss of clarity.
Our first assumption is that, conditional on $Z_i$, $Y_i(\beta_0)$ can be written as a known linear transformation of a vector $U_i(\beta_0)$, whose average conditional variance given $Z_{i}$ converges uniformly to a bounded and full-rank limit.
Note that if the variance of $Y_i(\beta_0)$ is full-rank (as in Examples 2 and 3 above) then the moments can trivially be written as $Y_{i}(\beta_0)=T U_{i}(\beta_0)+\zeta_{i}(\beta_0)$ for $T=I$, $U_{i}(\beta_0)=Y_i(\beta_0)$, and $\zeta_i(\beta_0)=0$. The structure in Assumption (ref) also commonly arises in moment inequality settings where the variance of $Y_i(\beta_0)$ is not full-rank. For example, consider the case of interval-valued regression (Example 1 above) where the upper- and lower-bounds of the interval are perfectly collinear, $Y_i^U = Y_i^L + c$ for fixed constant $c$. Then $Y_{i}(\beta_0)=T U_{i}(\beta_0)+\zeta_{i}(\beta_0)$ with $T = [I,~ -I]'$, $U_i(\beta_0) = Y_i^L - W_i\beta_0$, and $\zeta_i(\beta_0) = [0,~ -c]'$. Settings with moment equalities represented as inequalities can similarly be expressed in this form --- if all the moments are of this form, for example, then we can take $T=[I,~-I]'$ and $\zeta_i(\beta_0)=0$.
Assumption (ref) implies that the average conditional variance of $Y_i(\beta_0)$ given $Z_i$ converges, $\frac{1}{n}\sum Var_{P_{D|Z}}\left(Y_{i}(\beta_0)|Z_{i}\right)\to \Sigma(P_{D|Z},\beta_0)=T\Omega\left(P_{D|Z},\beta_0\right) T'$. Although $\Omega\left(P_{D|Z},\beta_0\right)$ has full rank, $\Sigma(P_{D|Z},\beta_0)$ may have reduced rank since e.g. the dimension of $\Sigma\left(P_{D|Z},\beta_0\right)$ may exceed that of $\Omega\left(P_{D|Z},\beta_0\right)$. We next assume that we have a uniformly consistent estimator for $\Omega(P_{D|Z},\beta_0)$, and thus for $\Sigma(P_{D|Z},\beta_0)$.
We discuss sufficient conditions for uniform consistency of $\widehat{\Omega}_{n,0}$ in Appendix (ref). Note that $\widehat{\Omega}_{n,0}$ depends on the null parameter value $\beta_0$ considered, where we again suppress this dependence for brevity of notation.
We further assume that the scaled sample average of $U_i(\beta_0)$ is uniformly asymptotically normal once recentered around its mean. To state this assumption we use the fact that uniform convergence in distribution is equivalent to uniform convergence in bounded Lipschitz metric (see e.g. Theorem 1.12.4 of van der Vaart and Wellner, 1996).
Under Assumption (ref), the following lower-level condition is sufficient for Assumption (ref).
Our final assumption, which is needed for the conditional and hybrid approaches, restricts $T$ and $X_{n,0}$. Before stating this assumption, we note that the structure imposed by Assumption (ref) allows us to consider a subset of the vertices $V(X_{n,0}, \sigma_0)$ discussed in the previous section. Intuitively, the optimal vertex $\hat\gamma$ corresponds to a vector of Lagrange multipliers for the primal problem ((ref)), and thus $\hat\gamma$ must satisfy the complementary slackness conditions. Assumption (ref) then implies that certain vertices can never be optimal when the test rejects -- for example, if the matrix $T$ encodes moment equalities as inequalities, then the positive and negative copies of a given moment cannot bind simultaneously unless $\hat\eta_{n,0}=0$, in which case our tests do not reject. The following lemma shows that we can ignore such “never-optimal” vertices when establishing size control.
With the definition of $V_\dagger(X_{n,0},\hat{\sigma}_{n,0})$ in hand, we can now state our final assumption.
Together with the structure for the variance matrix $\Sigma$ imposed in Assumption (ref), Assumption (ref) ensures that (i) $\gamma'Y_{n,0}$ has nonvanishing asymptotic variance for all dual vertices $\gamma \in V_\dagger(X_{n,0},\hat{\sigma}_{n,0})$, and (ii) for distinct dual vertices $\gamma$ and $\tilde\gamma$ in $V_\dagger(X_{n,0},\hat{\sigma}_{n,0})$, $\gamma'Y_{n,0}$ and $\tilde\gamma'Y_{n,0}$ are not perfectly positively correlated asymptotically. The former implies that $\hat\eta_{n,0}$ is continuously distributed in large samples, while the latter ensures that the dual problem $\max_{\gamma\in V_\dagger(X_{n,0},\hat{\sigma}_{n,0})}\gamma'Y_{n,0}$ has a unique solution with probability tending to one.
In Appendix (ref), we provide lower-level sufficient conditions for Assumption (ref) in settings where either $\Sigma(P_{D|Z},\beta_0)$ is full-rank or degeneracy in $\Sigma(P_{D|Z},\beta_0)$ arises from matching moments of opposite signs (e.g. moment equalities cast as inequalities). In these settings, we show that Assumption (ref) holds automatically when $X_{n,0}$ is constant up to scale (as occurs, e.g., in the difference-in-differences setting of rambachan_more_2022). When $X_{n,0}$ is non-constant, a sufficient condition is that $X_{n,0}$ lies in a set $\mathcal{X}$ such that the distance between distinct vertices of $V(X,\upsilon)$ is bounded away from zero over $X \in \mathcal{X}$, where again $\upsilon = \sqrt{diag(TT')}$. Intuitively, this assumption requires that distinct vertices in $V(X_{n,0}, \upsilon)$ not “converge to each other.”
We also note that we do not require any additional assumptions about how $V(X,\sigma)$ depends on $\sigma$, since the proof of Lemma (ref) shows that $\sigma$ affects $V(X,\sigma)$ only through a continuous re-scaling of the vertices of $V(X,\upsilon)$. This enables us to establish size control when $\sigma_{n,0}$ is replaced with a consistent estimate $\hat\sigma_{n,0}$ without further assumptions.
It is worth highlighting that Assumption 4 involves the variance of $Y_{n,0}$ but not its mean $\mu_{n,0}$. This contrasts with linear independence constraint qualification (LICQ) assumptions that have been considered in other work cho_simple_2021, Garfarov2016, which restrict the set of moments that can bind in population and thus the value of $\mu_{n,0}$ (see kaido_constraint_2021 for discussion). In the simplest case without nuisance parameters ($X_{n,0} = 0$), for example, Assumption 4 holds if all of the elements of $Y_{n,0}$ have positive variance and are not perfectly correlated, whereas a standard LICQ condition would impose that $\mu_{n,0}$ has a unique maximum element.\footnote{rambachan_more_2022 show that in a special setting where $\beta_0$ enters the moments linearly, a population version of LICQ implies that our conditional test has optimal local asymptotic power.} We explore the connections between LICQ and Assumption (ref) more formally in Appendix (ref), where we show that LICQ implies that there is a unique solution to a “population version” of the dual for $\hat\eta_{n,0}$, whereas Assumption (ref) only implies uniqueness of the sample version of the problem (but not necessarily the population version). The tests proposed in cox_simple_2020, as well as our LF test, do not require Assumption (ref) for uniform asymptotic validity, and thus may be attractive in settings where the researcher is not comfortable with this assumption.
Under these assumptions, feasible versions of our tests, based on the observed $(Y_{n,0},X_{n,0})$, and the estimated variance $\widehat{\Sigma}_{n,0}$, are uniformly asymptotically valid.
We next provide practical guidance on implementing the tests described above. We also provide Matlab code to facilitate implementation.\footnote{The code is available at \href{https://github.com/jonathandroth/LinearMomentInequalities/}{https://github.com/jonathandroth/LinearMomentInequalities/}.}
Researchers can use our methods whenever their model implies conditional moment inequalities of the form ((ref)). As discussed in Section (ref), if the model ((ref)) holds for a given $(Y,X)$ pair, then it also holds if $Y$ and $X$ are interacted with any non-negative function of the instruments -- i.e., if we replace $Y$ and $X$ with $\tilde Y = Y \otimes f(Z)$ and $\tilde X = X \otimes f(Z)$. An important choice in implementing our methods is thus the choice of the $k$ moments (i.e., the choice of $Y$). A formal analysis of how to optimally choose the $k$ moments is beyond the scope of this paper, but we offer some heuristic guidance.
Intuitively, including more informative moments can tighten the identified set based on the included moments, but including too many moments relative to the sample size can harm the quality of the normal approximation. Including uninformative moments (that are not infinitely slack) can also reduce the finite-sample power of our tests. The multivariate Berry-Esseen theorem bentkus_dependence_2003 suggests that the normal approximation to the distribution of the sample average should perform well when the number of moments included is sufficiently small relative to the sample size.\footnote{Specifically, as discussed in chernozhukov_central_2017, we need the dimension of the moments ($k$) to be smaller than $o(n^{\frac{2}{7}})$ for the approximation to hold uniformly over all convex sets. If the moments are of the form $Y=TU$, as in Assumption (ref), then the relevant dimension is $dim(U)$ rather than $dim(Y)$.} As a heuristic, cox_simple_2020 suggest that one should ensure there are at least 15 observations per cell in cases where the instruments $f(Z)$ are binary indicators for whether $Z$ falls in a particular cell. In our Monte Carlo simulations below, where the instrument functions are continuous, we find that our proposed tests have good size control with 500 observations and up to 110 moments, although we caution that the quality of the normal approximation may depend on the specific data-generating process.
Regarding the choice of which $k$ moments to use, researchers should include the moments that they think will be most informative about the parameter of interest. Note that interacting an original set of moments with an instrument function $f(Z)$ will only add identifying information to the extent that $f(Z)$ is correlated with $Y$ and $X$, since if $(Y,X)$ and $f(Z)$ are uncorrelated $E_P[f(Z) (Y- X \delta) ] = E_P[f(Z)] E_P[Y-X\delta] \propto E_P[Y-X\delta]$, so adding the interaction does not shrink the set of values where the moment inequalities are satisfied on average. Heuristically, researchers should therefore include instrument functions that are likely to be strongly related to $(Y,X)$.\footnote{ As noted in Section (ref) above, our approach does not deliver consistent tests in settings with continuously distributed $Z_i$. Hence, to derive general optimality results one would have to go beyond our finite-dimensional analysis. Armstrong2014,Armstrong2018 and Chetverikov2013 establish convergence rates for inference on the full parameter in partially identified settings, including rate-optimality results for procedures using particular kernel-based instruments and bandwidths. Their analysis could provide a natural starting point for the study of asymptotic optimality in our setting. We thank Tim Armstrong for bringing these connections to our attention. } Consistent with this intuition, HoPakes2014 use instrument functions based on the distance of an individual to a hospital, since their $Y$ and $X$ relate to individuals' choices of hospitals, and distance to the hospital is known to be an important determinant of hospital choice; see Section VI.B of HoPakes2014 for an intuitive discussion of how economic knowledge can inform the choice of moments. We also emphasize that applied researchers frequently conduct inference based on a finite set of unconditional moments implied by conditional moment inequalities, so the use of our methods does not introduce a new choice relative to this common practice in empirical work.
Researchers often wish to compute confidence sets for the target parameter $\beta$. This can be achieved by discretizing the parameter space for $\beta$ as $\{\beta_{(1)}, \ldots, \beta_{(L)}\}$ and testing the null hypothesis $H_0: \beta = \beta_{(l)}$ for each $l$ using the tests described above. A confidence set can then be formed by collecting the grid points for which the test fails to reject. If the researcher is interested in a subvector of $\beta$ -- e.g. the first component of $\beta$ is of interest, whereas the remaining components are nuisance parameters that enter the moments non-linearly -- then the researcher can first form a confidence set for the full parameter vector $\beta$, and then obtain a confidence set for the parameter of interest by projection. We emphasize that test inversion is only required for $\beta$, and not for the nuisance parameters $\delta$, which can lead to substantial computational simplifications when the dimension of $\delta$ is large. For the remainder of the section, we focus on the implementation of our tests for a particular null value $\beta_0$.
Our tests require an estimate of the average conditional variance, $\Omega_0 = E_P[ Var( U_i(\beta_0) | Z_i ) ]$. We briefly describe how a matching procedure proposed by AbadieImbensZheng2014 can be used to estimate $\Omega_0$ when the data are i.i.d. across $i$; see Chetverikov2013 and horowitz_adaptive_2001 for alternative estimators. Let $\widehat\Sigma_Z$ be the sample variance of $Z_i$.\footnote{The matching procedure described below assumes that $\widehat\Sigma_Z$ is non-singular. In certain applications, such as in our Monte Carlo, elements of $Z_i$ may be linearly dependent by construction, leading $\widehat\Sigma_Z$ to be singular. In this case conditioning on a maximal linearly independent subset of $Z_i$ is equivalent to conditioning on the full vector, so one can drop dependent elements from $Z_i$ until $\widehat\Sigma_Z$ is non-singular.} For each $i$, find the nearest neighbor using the Mahalanobis distance for $Z_i$: $$ \ell_{Z}\left(i\right)=\mbox{argmin}_{j\in\left\{ 1,...,n\right\} ,j\neq i}\left(Z_{i}-Z_{j}\right)' \widehat\Sigma_Z^{-1} \left(Z_{i}-Z_{j}\right). $$ For ease of exposition we assume that $Z_{i}$ has at least one continuously distributed dimension, so that $\ell_{Z}\left(i\right)$ is unique for all $i$.\footnote{If instead $Z_{i}$ is entirely discrete, one can estimate $\widehat{\Omega}_{n,0}$ using the average of the sample conditional variances across $Z_i$ cells.} The estimate of $\Omega_0$ is then:
Appendix (ref) provides regularity conditions under which $\widehat{\Omega}_{n,0} $ is uniformly consistent for $\Omega_0$.
To test the null hypothesis for a particular null value $\beta_0$, one needs to compute the test statistic $\hat\eta_{n,0}$ and the critical value for the relevant test ($c_{\alpha,LF}, c_{\alpha,C},$ or $c_{\alpha,H}$). We discuss computation of each component in turn.
The test statistic $\hat\eta_{n,0}$ can be computed by solving the linear program ((ref)). This can be achieved using standard software, such as Matlab's linprog command. We recommend using the dual-simplex method in Matlab, which conveniently returns both the optimal value $\hat\eta_{n,0}$ as well as the optimal vector of Lagrange multipliers $\hat\gamma$, which is used for computing the conditional and hybrid critical values.
Recall that the LF critical value $c_{\alpha,LF}$ is the $1-\alpha$ quantile of $\max_{\gamma \in V(X_{n,0},\hat{\sigma}_{n,0})} \gamma' \xi$ for $\xi \sim N(0,\widehat\Sigma_{n,0})$. By duality results for linear programming, we have that $$\hat\eta(\xi) = \max_{\gamma \in V(X_{n,0},\hat{\sigma}_{n,0})} \gamma' \xi = \left(\min_{\eta,\delta}~\eta \mbox{ subject to }\xi-X_{n,0}\delta\le\eta\cdot \hat\sigma_{n,0} \right),$$ where $\hat\sigma_{n,0} = \sqrt{Diag(\widehat{\Sigma}_{n,0})}$. To compute $c_{\alpha,LF}$, one can simulate $\xi_{(1)},...,\xi_{(S)} \sim N(0,\widehat\Sigma_{n,0})$, compute $\hat\eta(\xi_{(s)})$ using the linear program in the previous display and then take the $1-\alpha$ quantile of $\hat\eta(\xi_{(1)}),...,\hat\eta(\xi_{(S)})$.\footnote{To increase computational speed and stability across different values of $\beta$, one can fix $Z_1,...,Z_S \sim N(0,I)$, and then set $\xi_s = \widehat\Sigma_{n,0}^{\frac{1}{2}} Z_s$.} We use $S=1000$ in our simulations below.
To compute the conditional and hybrid critical values, one needs to compute $\mathcal{V}^{lo}_{n,0}$ and $\mathcal{V}^{up}_{n,0}$. Equation ((ref)) gives an analytical formula for these quantities that involves a minimum and maximum over the set of dual vertices $V(X_{n,0}, \hat\sigma_{n,0})$. Enumerating all of the vertices is, however, computationally prohibitive when there are many moments or nuisance parameters. Fortunately, we show in Appendix (ref) that there are two computational shortcuts available that allow for computation of $\mathcal{V}^{lo}_{n,0}$ and $\mathcal{V}^{up}_{n,0}$ without vertex enumeration. First, when the problem for $\hat\eta_{n,0}$ has a non-degenerate solution, $\mathcal{V}^{lo}_{n,0}$ and $\mathcal{V}^{up}_{n,0}$ can each be written as the maximum/minimum of a set of at most $k$ easy-to-compute elements.\footnote{The solution to the primal problem is said to be non-degenerate if $W_{n,0,B}$ is invertible, where $W_{n,0} = (\hat\sigma_{n,0} ,~ X_{n,0})$ and $B$ indexes the set of binding moments in the primal. To use this approach, we also require that $e_1'W_{n,0,B}\geq 0$.} Second, if the problem for $\hat\eta_{n,0}$ is degenerate, $\mathcal{V}^{lo}_{n,0}$ and $\mathcal{V}^{up}_{n,0}$ can be solved using a computationally-tractable bisection approach. We thus recommend to first check whether the solution to the primal problem ((ref)) is non-degenerate, and if so, use the formula given in Lemma (ref); if not, then use the bisection approach described in Appendix (ref). We implement this approach in our publicly-available Matlab code, and find that it yields computationally tractable tests with as many as 110 moments and 11 parameters in our simulations below.
In some settings, we may have inequalities of the form $$ E_{P_{D|Z}}[Y_i - X_{\beta,i} \beta - X_{\delta,i} \delta | Z_i ] \leq 0,$$ where $\beta$ is the parameter of interest, $\delta$ is again a nuisance parameter, $X_{\beta,i}$ and $X_{\delta,i}$ are non-random conditional on $Z_i$, and the value of $(Y_i,X_{\beta,i},X_{\delta,i})$ does not depend on $\beta$ or $\delta$. This structure arises, for example, in interval-valued regression if we are interested in the coefficient on an exogenous variable. This structure also arises in rambachan_more_2022, who consider bounds on treatment effects in difference-in-differences settings under linear constraints on the possible violations of parallel trends. Moment inequalities of this sort can be cast into the form ((ref)) by setting $Y_i(\beta) = Y_i - X_{\beta,i} \beta$ and $X_{i}(\beta) = X_{\delta,i}$. The methods described above can thus be applied directly.
The additional linear structure allows for multiple computational shortcuts, however. First, the conditional covariance matrix $E_{P}[Var_{P_{D|Z}}(Y_i(\beta) | Z_i)]$ does not depend on $\beta$, and thus the estimated variance $\widehat\Sigma_{n}$ need only be calculated once, rather than for every candidate value of $\beta$.\footnote{We write $\widehat \Sigma_n$ instead of $\widehat \Sigma_{n,0}$, since the value does not depend on the null hypothesis. We apply an analogous convention for other variables, e.g. writing $X_n$ instead of $X_{n,0}$ and $\hat\sigma_n$ instead of $\hat\sigma_{n,0}$.} Second, the LF critical value $c_{\alpha,LF}( X_{n}, \widehat\Sigma_{n} )$ likewise does not depend on the value of $\beta$. As a result, a confidence set for the LF test can be computed by solving a linear program for each of the upper and lower bounds, without any test inversion at all. For instance, the lower bound of the confidence set for the LF test can be calculated by solving $$ \min_{\beta,\delta} \hspace{.1cm}\beta \text{ subject to } Y_{n}- X_{n,\beta} \beta - X_{n,\delta} \delta \leq c_{\alpha,LF} \cdot \hat\sigma_n ,$$ where $Y_{n} = \frac{1}{\sqrt{n}} \sum_i Y_i$, and $X_{n,\beta}$ and $X_{n,\delta}$ are defined analogously. Computation of confidence sets for the conditional and hybrid tests still requires test inversion over a grid for $\beta$, but will be faster because $\widehat\Sigma_n$ and the first-stage LF critical value for the hybrid need only be computed once.
Our simulations are calibrated to Wollmann's study of the bailouts of GM and Chryslers' truck divisions. As discussed in Example 3 above, Wollmann obtains bounds on the fixed cost of marketing a product using moment inequalities derived from revealed preference arguments. The fixed cost to firm $f$ of marketing product $j$ at time $t$ is $\beta(\delta_{c,f}+\delta_g g_j)$ if the product was marketed at time $t-1$, and $\delta_{c,f}+\delta_g g_j$ otherwise. Consistent with ((ref)), the parameter $\delta=(\delta_g,\{\delta_{c,f}\})$ enters the moments linearly for a fixed value of $\beta$.
The moments we consider take the form of the example given in equation ((ref)) for the case where a product was marketed in both periods. To illustrate how performance varies with the number of parameters, we consider specifications where the intercept $\delta_{c,f}$ is constant across firms, specifications where it is allowed to vary across three groups of firms, and specifications where each of the nine firms in the data has its own intercept. In each case, we average the moment inequalities involving $\delta_{c,f}$ across firms assumed to have the same coefficient. We also vary the instruments used. See Appendix (ref) for details on the exact construction of the moments. Overall, the number of moments varies between 6 and 110 across our specifications.
We consider inference on three parameters of interest: the cost of marketing the truck of mean weight when it was not marketed in the prior year;\footnote{When we assume $\delta_{c,f}$ is common across firms this is $\delta_c + \delta_g \mu_g$, where $\mu_g$ is the population average weight of trucks. When we allow the estimated $\delta_c$ parameters to vary across groups, we estimate $l'\delta$, for $l = (\frac{1}{G} , \ldots, \frac{1}{G}, \mu_g)'$, where $G$ denotes the number of groups and $\delta=(\delta_{c,1},...,\delta_{c,G},\delta_g)'$. Note that since the simulation DGP holds the true value of $\delta_c$ constant across groups, the true value of the parameter is the same in all specifications.} the incremental cost of changing the weight of a product, $\delta_g$; and the non-linear parameter $\beta$, where $1-\beta$ represents the proportional cost savings from marketing a product that was previously marketed relative to a new product. For the first two target parameters, which can be written in the form $l'\delta$, we hold $\beta$ fixed at its true value and treat the component of $\delta$ orthogonal to $l'\delta$ as the nuisance parameter. This allows us to examine performance in the linear case discussed in Section (ref). In Wollman's setting the parameter $\beta$ might be calibrated based on industry knowledge about the relative cost of marketing a new versus pre-existing product. As discussed in Section (ref), if we instead treated $\beta$ as unknown we could form joint confidence sets for $\beta$ along with the linear combination of interest and obtain confidence sets for the linear parameter alone by projection. For inference on $\beta$ we treat the entire vector $\delta$ as a nuisance parameter. Overall, the number of unknown parameters varies between 2 and 11 across our specifications.
We calibrate the data-generating process in our simulations using moments reported in Wollmann -- see Appendix (ref) for details. In each simulation draw, we generate data from a cross-section of 500 independent markets.\footnote{The data in Wollmann are a time-series but his variance estimates assume no serial correlation, so we adopt a simulation design consistent with this.} This is substantially larger than the 27 observations used by Wollmann, but allows us to consider specifications with a widely varying number of moments. All results are based on 500 simulations.
We consider the performance of the LF, Conditional, and Hybrid tests and compare these to several benchmarks. First, we compare to a studentized-max-statistic-based projection test which we label the least favorable projection, or LFP, test. Second, we compute the sCC and sRCC tests proposed in cox_simple_2020. The sRCC test, which is a refinement of the sCC test, can be computationally difficult when there are many parameters. For the specifications with 10+ parameters and 100+ moments, we therefore report an upper bound for the power of the sRCC test using the fact that the refinement to the sCC test can only matter when the test statistic falls in a certain range.\footnote{Specifically, the sRCC test always rejects when the sCC test does, and can only differ from the sCC test when one moment is active ($k=1)$ and the test statistic falls between the $1-\alpha$ and $1-\alpha/2$ quantiles of the chi-squared distribution. When there are 10 or more parameters, we thus report the power of the test that rejects whenever either the sCC test rejects or the refinement could potentially lead the sRCC test to reject.} Third, we compute the projection tests of \citetalias[][AS]{AndrewsSoares2010} and Kaidoetal2019 using the EAM algorithm implemented in Matlab by kaido_calibrated_2017. The AS and KMS tests can be computationally taxing when there are many parameters, and at present, the Matlab implementation of KMS by kaido_calibrated_2017 is only written for settings where the parameters enter in an additively separable way. We therefore compute the AS and KMS tests only for the specifications when the parameters enter linearly and there are fewer than 10 parameters. See Appendix (ref) for additional details on the implementation of these comparisons.
Table (ref) reports the maximum null rejection probability (size) over a conservative estimate of the identified set. Since we do not have an analytical characterization of the identified set, we approximate it by the set satisfying the sample (unconditional) moment inequalities based on a simulation run with five million observations. To ensure that our estimate of the identified set is conservative, we follow chernozhukov2007estimation and add a correction factor to the moments of $\log(n)/\sqrt{n} \approx .003$. Our estimate of the identified set is thus conservative due to both (a) the chernozhukov2007estimation correction factor and (b) the use of unconditional rather than conditional moment inequalities. All of the procedures nevertheless approximately control size on this set, with rejection probabilities never exceeding 0.08 for any of the procedures.
We next turn to comparisons of power. Figure (ref) shows the rejection rates for each of our three main tests in the simulation design where the target parameter is the cost of the mean-weight truck. The vertical dashed lines denote conservative estimates of the bounds of the identified set, and the remaining curves show the probability that each of the tests rejects given a null value of the parameter of interest (holding fixed the DGP). Since the rejection probability is near-zero for all procedures in the interior of the identified set, we omit the portion of the $x$-axis well inside the identified set bounds so as to focus on the most relevant parts of the parameter space; the omitted part is grayed out in Figure (ref) and subsequent figures.
Overall, the figure indicates that the hybrid approach performs best among our three procedures, with rejection probabilities comparable to or above those of the LF and conditional approaches at all points in the parameter space. To understand the superior performance of the hybrid approach, it is worth highlighting that the rejection curves for the LF and conditional approaches cross: in some specifications, the conditional approach has power substantially above that of the LF test at all parameter values (e.g. panel (e) of Figure (ref)). In other specifications, however, the conditional approach exhibits poor power relative to the LF test in some areas of the parameter space -- e.g., in the area above the identified set in panel (d) of Figure (ref). We have confirmed that in this simulation design for some parameter values there are two vertices which are optimal with approximately equal probability in this part of the parameter space, which as discussed in Section (ref) can lead to poor power for the conditional test. Indeed, this feature can even lead the power curves for the conditional approach to be non-monotonic, since moving farther away from the identified set can push the mean values of a pair of vertices closer together. The hybrid approach has similar power to the conditional approach in most of the parameter space, while mitigating the issues in regions of the parameter space where multiple vertices are close to binding, thus leading to better performance overall. Appendix Figures (ref)-(ref) show results when the parameter of interest is $\delta_g$ or $\beta$: the qualitative patterns are similar, with the hybrid exhibiting power comparable to or above the other two methods throughout the parameter space.
Table (ref) provides a comparison of our three procedures relative to the other benchmarks. We report the median excess length for confidence sets formed based on each approach, where excess length is defined as the length of the confidence set minus the length of the identified set. For reference, we also report the length of the identified set. We find that the median excess length of the hybrid confidence set is below that for the AS and KMS sets in all specifications. The median excess length for the hybrid is also better or equal to that for the sCC and sRCC sets in most specifications, although the sRCC set outperforms the hybrid for three of the specifications with target parameter $\beta$.\footnote{Appendix Figures (ref)-(ref) show a comparison of the power curves of the hybrid and the sCC and sRCC tests. The figures show that for several specifications the rejection curves for the hybrid and sRCC tests cross.} The ranking of the hybrid and sRCC approaches in these results differs from that in the simulations in cox_simple_2020, who find better performance for sRCC. One potential factor is that the hybrid test is based on the max statistic whereas the sRCC test uses a QLR statistic, so the hybrid may be more powerful in settings where one moment is violated to a large extent, whereas the sRCC test may be more powerful when several moments are locally violated. Finally, it is worth highlighting that all of the procedures considered have better power than the LFP test in nearly all specifications. Appendix Figures (ref)-(ref) display comparisons of the full power curves of the hybrid relative to the LFP, sCC, sRCC, AS, and KMS tests.
In our simulations the excess length of KMS intervals sometimes exceeds that of AS intervals. This is potentially surprising, since by construction the KMS test should reject whenever the AS test rejects, and thus should yield confidence intervals with uniformly shorter excess length. In practice, however, the bounds of the projected confidence intervals are approximated using a finite number of objective evaluations of the Evaluation-Approximation-Maximization algorithm studied by KMS, and thus are subject to optimization error. As a consequence of these optimization errors we find the median excess length of AS to be slightly smaller than that of KMS in two of our specifications (although by less than 2%). We have verified in an example where these issues arise that providing the EAM algorithm for AS with the optimal solution for KMS as a starting point leads to an AS interval that is a superset of the KMS interval. For simplicity, however, we report results from applying the EAM algorithm for AS directly.\footnote{We also found that reducing the objective tolerance to half the default value reduced (but did not fully eliminate) this issue, but were unable to reduce the tolerance further owing to computational constraints.}
Lastly, Table (ref) reports runtimes in minutes to calculate confidence sets for each parameter, averaging over 20 runs on a 2022 MacStudio (with M1 Ultra processor, 64GM RAM) without parallelizing the test inversion. Perhaps the most remarkable feature of the table is that our proposed tests are computationally tractable even in settings with as many as 11 parameters and 110 moments. Our preferred test, the hybrid, has runtimes under 5 minutes for all specifications in panels (a) and (b), where all of the parameters enter the moments linearly, and under 2 hours in all specifications in panel (c), where the target parameter enters the moments non-linearly. We emphasize that these runtimes could be further improved by parallelizing the test inversion.
We highlight a few noteworthy comparisons of runtimes across both procedures and specifications. First, the runtime of the hybrid test can be either faster or slower than the runtime of the sCC and sRCC tests proposed by cox_simple_2020 depending on the specification.\footnote{The refinement for the sRCC test is needed relatively rarely, and thus the reported runtimes for the sRCC and sCC test are identical to two decimal places.} The hybrid test is faster in the majority of simulations where all parameters enter the moments linearly; this is because the LF test used in the first-stage of the hybrid is particularly fast for these specifications, as the LF confidence set can be calculated without any test inversion (see Section (ref)). The cox_simple_2020 tests are faster in most of the specifications in panel (c), where the target parameter enters the moments non-linearly and thus the LF critical value must be re-calculated for each candidate value of $\beta$, with the exception of the specification with the most moments and parameters in which the hybrid is faster. Second, the runtimes for the hybrid tests are faster than for the AS and KMS projection tests in nearly all specifications, with larger differences in settings with more moments/parameters.\footnote{Runtimes between the hybrid and sCC/sRCC tests are directly comparable, since both tests use test inversion over the same grid. Comparing runtimes between the hybrid and AS/KMS projection confidence sets is somewhat more difficult, since the former depends on the grid resolution while the latter depend on the stopping criteria for the EAM algorithm. Given that the EAM algorithm relies on several stopping criteria kaido_calibrated_2017, it is not entirely obvious how to align these parameters so that the computational accuracy of the tests is comparable. Note, however, that if the lower bound for the AS confidence set computed by the EAM algorithm is larger than that for the KMS confidence set, then the computational error in the former must be at least as large as the difference between the two computed endpoints. In the specification corresponding with the first row in Table (ref), this difference is larger than the grid resolution used for the hybrid test in 13 percent of the cases, which provides suggestive evidence that the computational errors of the two approaches are often of a similar order of magnitude.} In the specification in the fourth row of panel (b), for example, the hybrid test is over 14 times faster than both AS and KMS.\footnote{We ran a single iteration of AS for the specification with 10 parameters and 38 moments, which took 5.5 hours to complete (and the EAM algorithm for the upper bound reached the maximum of 1000 iterations without converging).} It is intuitive that the computation time is faster for the hybrid since it exploits the linear conditional structure present in our setting, whereas the EAM algorithm used to calculate the AS/KMS CIs is designed for a larger class of potentially non-linear problems and thus does not make use of this additional structure. Third, both the conditional and hybrid tests are somewhat slower when the target parameter is $\delta_g$ (panel b) relative to the cost of the mean-weight truck (panel a). The reason is that the primal solution for $\hat\eta_{n,0}$ is often degenerate, and thus we must use the slower bisection method to calculate the $\mathcal{V}^{lo}_{n,0}$ and $\mathcal{V}^{up}_{n,0}$, as described in Appendix (ref).
This paper considers the problem of inference based on linear conditional moment inequalities, which arise in a wide variety of economic applications. Using linear conditional structure, we develop inference procedures which remain both computationally tractable and powerful in the presence of nuisance parameters. We find good performance for our procedures under a variety of simulation designs based on Wollmann, with especially good performance for our recommended hybrid procedure.