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Finite sample inference in partially identified and incomplete models
\address{Johns Hopkins and Penn State}
In this paper, we study a class of incomplete econometric models that combines (i) a restriction on the support of the random variables involved in the model specification, and (ii) a restriction on the distribution of those variables in the model, that the analyst cannot observe. The support restriction is implied by economic theory, and usually involves the implications of behavioral assumptions, equilibrium concepts and structural features of the economic environment. A game of perfect information with a pure strategy equilibrium concept, as in Jovanovic:89 and Tamer:2003 is a prime example. Other examples include models of choice with limited attention, as in BCMT:2021, discrete choice with endogeneity, as in CRS:2011, auction models, as in HT:2003, network formation, as in dPRST:2018, and structural vector autoregressions, as in GK:2021 and GKR:2021. Molinari:2020 and CR:2020 provide comprehensive surveys of the literature on incomplete structural models. The procedure we propose in this paper applies to parametric incomplete structural models: Both the support restriction and the distribution of unobserved heterogeneity are known up to a finite dimensional parameter vector. It is ill suited to semiparametric extensions, where either the support restriction or the distribution of unobserved heterogeneity depends on an infinite dimensional parameter.
Incomplete structural models are called {\em incomplete} because the model structure predicts a set of possible values for the outcome variables. Incompleteness arises because of multiple equilibria in games, unobserved heterogeneity in choice sets in limited attention models, interval predictions in auctions, and unknown sample selection mechanisms. Model incompleteness generally leads to partial identification, where more than one value of the model parameter could have given rise to the true data generating process for the observed variables. However, model incompleteness and partial identification are distinct concepts.
The current state of the art in deriving confidence regions for the parameters of incomplete structural models involves the BMM:2011-GH:2011 characterization of the sharp identified region as a collection of conditional moment inequality restrictions, and the application of one of the existing inference methods with conditional moment inequality models, surveyed in CS:2018, Molinari:2020 and shi2025inference. This method, however, results in a very large, possibly infinite, number of conditional moment inequalities. Even in cases, where the endogenous variable is discrete, such as discrete games, the cardinality of the number of moment inequalities increases exponentially in the number of strategy profiles.
The challenge is both computational and statistical, as the number of inequalities may be much larger than the sample size, requiring new methods, such as CCK:2019. Basing inference on a non sharp reduced collection of inequalities leads to low power and loss of robustness to misspecification. See li2024discordant for a discussion. Methods to reduce the number of conditional moment inequalities without losing sharpness exist. They are based on core determining classes, as proposed in GH:2011 and further developed in CRS:2011, CR:2017e, LW:2017, MM:2018 and Ponomarev:2022.\footnote{GRS:2022 characterize identified set in incomplete models using minimal relevant partitions, as an alternative to core determining classes.} However, these methods are complex, model specific, and only partially alleviate the problem. In addition, when the conditional moment inequalities are transformed into unconditional ones, as in AS:2013, sharpness is preserved only when the number of moment inequalities increases with sample size, which induces an extra layer of computational burden.\footnote{The dimensionality of the conditioning set in such models generally precludes the alternative approach to conditional moment inequalities, which involves estimating them, as in CLR:2009.} Moreover, inference methods in moment inequalities rely on asymptotic arguments and some user chosen tuning parameter to preselect inequalities that are close to binding in the sample and thereby avoid overly conservative inference.
We propose an alternative method to construct confidence regions for the parameters of incomplete structural models that circumvents the many moments and conditioning issues, allows for continuous outcome variables, and avoids tuning parameters and asymptotic arguments. As is customary with moment inequality models, we construct our confidence region by inverting a test. However, the test statistic is based on a different characterization of the sharp identified region, and we show that it controls size in finite samples. Our testing procedure relies on two key ingredients. First, the test statistic is based on an optimal transport characterization of the sharp identified region, inspired by formulations in GH:2006 and EGH:2010. As a result, the test statistic is the solution of a discrete optimal transport problem, which is a special kind of linear programming problem, the computation of which has a long history. Second, the test generalizes Monte Carlo tests of Dwass:57 and Barnard:63\footnote{See also Dufour:2006 and DK:2001.} to incomplete models to control size in finite samples. The test statistic and critical values are based on simulation draws from the conditional distribution of latent variables.
Our test controls size, hence coverage probability of the confidence region for any finite sample size. To the best of our knowledge, this is the first inference method for structural parameters of incomplete models that is valid and exact in finite samples. KZ:2019 (KZ:2019,kaido2025universal) derive small sample valid and asymptotically exact inference procedures in incomplete models with finite outcome space $\mathcal Y$. The test is based on the ratio of worst case likelihood pairs. However, the test can be very conservative in finite samples (as shown in Section 5 of kaido2025universal) and derivation of the worst case likelihoods requires solving a convex program with a number of linear constraints that grows exponentially with the cardinality of $\mathcal Y$ for each value of the conditioning variable $x$. Non asymptotic results in the broader partial identification literature includes CS:2022, who provide an inference method for a class of partially identified models that requires no tuning parameter, and achieves exact finite sample size in normal models. CLR:2009 and CCK:2019 derive non asymptotic bounds on the rejection probabilities of their confidence regions. These bounds are useful to derive asymptotic rates of convergence, not for finite sample inference. rosen2025finite provide finite sample inference for the maximum score. CCT:2018 is also related to our procedure, as their asymptotically exact inference for identified sets (for full or subvector of parameters) is based on Monte Carlo simulations from quasi-posteriors.
Finite sample validity has several advantages, beyond the obvious benefit of avoiding reliance on often questionable asymptotic approximations. First, the support constraint and the dimension of the vector of unobservables may change with sample size, as would arise in applications to games on networks and network formation games. Second, our finite sample validity result requires no restriction on the dependence between observations in the sample. This property is particularly desirable with incomplete models. As discussed in EKS:2016, it is hard to reconcile the customary independence or mixing assumptions across units of observation with total ignorance of the mechanism that selected each realization from the model prediction set. The degree of dependence between observations does not affect size control of our procedure, but it does affect the power of the test, hence informativeness of the confidence region. However, a simple ergodicity condition is sufficient to ensure that parameter sequences that violate the optimal transport characterization of the sharp identified region ultimately lie outside the confidence region.
Our method requires a search in the space of parameters. At each value of the parameter in the search, we must compute a test statistic and a critical value. This computational burden is shared by inference methods in partially identified models, where the objective is coverage of the true value of the parameter. In order to accelerate the search, we also propose a conservative superset of our confidence region. The conservative superset is based on a parameter free critical value, and hence covers the sharp identified region. Once this conservative confidence region is computed, all values of the parameter that lie outside of it can be excluded a priori from the exact confidence region in our main proposal. Given that the full vector of structural parameters is rarely of interest in itself, the search in a high dimensional parameter space can and should be averted because of the computational cost and the conservativeness of confidence regions on subvectors obtained from projecting confidence regions for the full vector. See kaido2019confidence and Section 3.2.2 of Molinari:2020 for a discussion of this issue and a survey of recent approaches. An approach that is applied to inference in moment inequality models in romano2008inference, BCS:2017, and belloni2018subvector is profiling. We propose a profiled version of our test, to perform finite sample inference on a subset of the structural parameters, or more generally on a low dimensional transformation of the full structural parameter vector.
We provide an extensive simulation study of our procedure to illustrate exact coverage, to analyze its power properties, to investigate the effect of the choice of discrepancy in the definition of the test statistic, and to compare our procedure in terms of size, power and computing time, with the current state of the art procedures proposed for the case of many moment inequalities in CCK:2019 and andrews2017inference. The study of power is particularly important, since our theoretical results on power are limited to consistency of the test, and we make no claim to optimality of our testing procedure. Recent progress has been made on inference in incomplete models with optimality properties in KZ:2019 (KZ:2019,kaido2025universal). KZ:2019 derive minimax optimal tests for simple hypotheses in incomplete models with finite outcome space and derive their asymptotic local power. Minimax optimality in the context of simple hypothesis testing for parameters of incomplete models means that the test maximizes worst case power under the alternative (i.e., among all possible DGPs predicted under the alternative parameter value), while controlling worst case size under the null (i.e., such that the largest rejection probability among DGPs predicted under the null parameter value is smaller or equal to nominal size). KZ:2019 also derive asymptotic local power properties of their minimax optimal tests, with extensions to composite hypotheses. CK:2022 apply the theory of KZ:2019 to test model incompleteness, thereby leveraging the completeness of the model under the null hypothesis. kaido2025universal apply the framework of KZ:2019 to develop a finite sample valid and asymptotically exact subvector inference procedures for incomplete models with finite outcome space $\mathcal Y$. In related work, KM:2024 propose mispecification robust inference in incomplete models based on a relative entropy projection of the empirical distribution on the set of predicted data generating processes.
Finally, we illustrate the implementation of our procedure in the structural model of airline entry and price competition in CMT:2021. We apply our procedure to the empirical model of CMT:2021 without modification, and we use exactly the same data set. We apply our procedure to test to a large number of structural parameter values drawn randomly from the confidence region reported in CMT:2021. We strongly reject all of them, pointing to the possibility of mis-specification of the structural model that the inference procedure implemented in CMT:2021 does not have sufficient power to detect.
Random vectors are defined on the same complete probability space $(\Omega,\mathcal F,\mathbb P)$. All vectors are written as row vectors throughout. We don't use transposition notation when the format is clear from the context. Throughout the paper, $(Y,X,U)$ will denote a random vector on $\mathcal Y \times \mathcal X\times \mathcal U$, and $\theta \in\Theta$ a fixed parameter vector, where $\mathcal Y\subseteq \mathbb R^{d_Y}$, $\mathcal X\subseteq \mathbb R^{d_X}$, $\mathcal U\subseteq \mathbb R^{d_U}$, and $\Theta\subseteq \mathbb R^{d_\theta}$. We will denote $\mathcal Q$ and $\mathcal P$ the collections of Borel probability measures on $\mathcal U\times \mathcal X$ and $\mathcal Y\times \mathcal X$ respectively. $\mathcal M(Q,P)$ is the set of probability measures on $(\mathcal U\times \mathcal X)\times(\mathcal Y\times \mathcal X)$ with marginals $Q$ on $\mathcal U \times \mathcal X$ and $P$ on $\mathcal Y \times \mathcal X$. The symbol $\oplus$ is used for the sum of two sets, i.e., $A\oplus B:=\{(a,b): a\in A, b\in B\}$, and by slight abuse of notation, $A\oplus\{b\}$ is denoted $A\oplus b$. The convex hull of a set $A$ is denoted co$A$. We denote $\mathcal M_n^+$ the set of $n\times n$ non negative matrices, and $\Pi_n$ the subset of $\mathcal M_n^+$ containing matrices $\pi$ such that $n\pi$ is doubly stochastic, i.e., such that $\Sigma_i\pi_{ij} = \Sigma_j\pi_{ij} = 1/n$, for all $i,j \leq n.$ Finally, $\mathcal S_n$ is the set of permutations $\sigma$ on $\{1,\ldots,n\}$, and $\delta_x$ denotes the Dirac mass concentrated at $x$. Let $\lfloor a \rfloor$ denote the component-wise integer part of a vector $a$.
Section (ref) defines the model and characterizes the sharp identified region. We present the finite sample inference procedure and its properties in Section (ref). Section (ref) proposes refinements of the procedure. It discusses the case of parametric latent variables, discrete outcomes and subvector inference. Section (ref) is a simulation analysis of the informativeness and computational intensiveness of the proposed procedure, and section (ref) illustrates the procedure on the model and data from CMT:2021. Proofs are collected in the appendix, together with an extended simulation exercise based on CMT:2021.
We restrict attention to the class of parametric incomplete structural models introduced in Jovanovic:89. The vector of variables of interest $(Y,X,U)\in\mathcal Y \times \mathcal X \times \mathcal U$ satisfies support constraint $(Y,X,U) \in \Gamma(\theta)\subseteq \mathcal Y \times \mathcal X \times \mathcal U$, and $U$ has fixed and known distribution $Q_U$.\footnote{This is without loss of generality as we explain in section (ref).} The object of inference is the finite dimensional parameter $\theta\in\Theta$. Both vectors or variables $Y$ and $X$ are observed, in the sense that available data consists in a sample $((Y_1,X_1),\ldots,(Y_n,X_n)))$. Variables in vector $U$ are unobserved. Variables in vector $X$ are exogenous\footnote{We show in section (ref) that the distribution of $U$ may depend on $X$ as long as it is known up to a finite dimensional parameter vector.} (in the sense that $U\perp X$), and there are no restrictions on the process generating $(X_1,\ldots,X_n)$. All endogenous variables are subsumed in vector $Y$.
The model is incomplete in that multiple values of endogenous variables may be consistent with a single value of exogenous and unobserved variables. This can be seen in the fact that the set $\{ y\in \mathcal Y: (y,x,u)\in\Gamma(\theta) \}$ may not be a singleton for all $(u,x)\in\mathcal U\times\mathcal X$. This corresponds to the fact that the model fails to produce a unique prediction.
Incomplete models as described above encompass examples as diverse as static simultaneous move games with complete information and pure strategy equilibrium concepts, choice models with limited attention or partially observed consideration sets, and auctions with independent private values. Section 3 in Molinari:2020 gives a detailed account of such incomplete structural models with extensive references. In what follows, we concentrate on two recent examples and illustrate precisely how they fit into this theoretical framework.
The final example in this section is a simple parametric regression with interval censored covariates, which will serve as a running example to illustrate several aspects of our inference procedure.
The sample $((y_1,x_1),\ldots,(y_n,x_n)))$ of observed data is assumed to be a sample of $n$ realizations of the random vector $(Y,X)$ with true distribution $P_0$. The model stipulates that the latter is an element of a subset $\mathcal P_\theta$ of the set of distributions on $\mathcal Y\times \mathcal X$. The set $\mathcal P_\theta$ is defined as follows.
The compatibility between the structural model of Definition (ref) and the true data generation process is defined as the fact that $P_0$ is an element of $\mathcal P_\theta$. The true data generating process $P_0$ may be compatible with the structural model, i.e., $P_0\in\mathcal P_\theta$, for more than one value of the parameter $\theta\in\Theta$. Hence the parameter vector $\theta$ can be partially identified. The sharp identified region $\Theta_I$ is defined as the set of values of the parameter $\theta$ such that our model is compatible with the true data generating process.
Definition (ref) is equivalent to the definition of the identified set for incomplete models in BMM:2011 and GH:2011 and the subsequent literature.
We discuss a characterization of the sharp identified region that motivates our test statistic. The existing characterization of the sharp identified region, derived in BMM:2011 and GH:2011, takes the form of a collection of conditional moment inequalities of typically very large cardinality. Our inference strategy is based on a different characterization of the sharp identified region as the solution of an optimal transport problem, and as such, is related to the characterization in GH:2006 and EGH:2010. The fundamental idea of this characterization is that the existence of a joint distribution $\tilde\pi$ for $(Y,X,U)$ satisfying the model constraints is equivalent to finding a minimum value of $\tilde\pi((Y,X,U)\notin\Gamma(\theta))$ equal to $0$ among all joint distributions $\tilde \pi$ that meet the specified marginal constraints.
The way we treat dependence on exogenous variables $X$ is crucially different from those of the previous proposals. It relies on a reformulation of the support constraint in the model. Define the correspondences $\Gamma_u$ and $\Gamma_y$ between $\mathcal Y \times \mathcal X$ and $\mathcal U \times \mathcal X$ by:
Correspondence $\Gamma_y$ defines the set of model predictions for the endogenous variables, whereas correspondence $\Gamma_u$ defines the set of latent variables that can rationalize the data. We define the correspondences between $\mathcal Y\times\mathcal X$ and $\mathcal U\times\mathcal X$ instead of simply $\mathcal Y$ and $\mathcal U$ in order to avoid conditioning on $X$.
With the notation of ((ref)), and writing $V=(U,X)$ and $W=(Y,X)$, the distributional constraint (Constraint (2) in Definition (ref)) can be written $V\sim Q:=Q_U\times P_{X,0}$. Moreover, for this model to be consistent with the true data generating process, we need $W\sim P:=P_0$. Let $\mathcal M(Q,P)$ be defined as the set of joint distributions with marginals $Q$ and $P$ (see notations and preliminaries). Then, the above two restrictions imply that the joint distribution $\pi$ of $(V,W)$ must satisfy $\pi\in\mathcal M(Q,P)$. Finally, the support constraint in the definition of the structural model (Definition (ref)) is $\pi(V\in\Gamma_u(W;\theta))=1$, or, equivalently, $\int \delta(v,w;\theta)d\pi(v,w)=0$, for any discrepancy $\delta$ that satisfies the following assumption.\footnote{The choice of discrepancy is discussed with the definition of the test statistic in section (ref).}
Therefore, under assumption (ref), if the model and parameter $\theta$ are compatible with the true data generating process, the following must hold:
Here $\mathcal D(Q,P;\theta)$ can be viewed as an optimal transport problem (see Villani:2003) with cost function $(v,w)\mapsto \delta(v,w;\theta)$. The following theorem shows that condition ((ref)) is not only necessary, but also sufficient.
An immediate benefit of characterizing the sharp identified region in Theorem (ref) with the optimal transport formulation ((ref)) is that a sample analogue, where $P_0$ is replaced with the sample empirical distribution, readily provides a test statistic. There are two essential differences between our identification characterization result and EGH:2010 and GH:2011. First, we define the model correspondence $\Gamma_y:\mathcal U\times\mathcal X\rightrightarrows\mathcal Y\times\mathcal X$ instead of $\mathcal U\times\mathcal X\rightrightarrows\mathcal Y$. Second, we replace the binary discrepancy advocated in EGH:2010 and GH:2011 with a more general discrepancy. Both modifications, taken together, allow us to balance detection of violations of the model for given values of the covariates, with nonparametric matching on covariates to avoid the conditioning approach that increases the computational burden and reduces informativeness in competing methods.
The objective of this section is to provide a confidence region for the parameters of interest $\theta$. The confidence region $CR_n$ is obtained by test inversion, as in AR:49. For each value of $\theta$, we test the null hypothesis
The hypothesis is rejected and $\theta$ deemed outside the confidence region if and only if the test statistic $T_n(\theta)$, a function of the sample $((Y_1,X_1),\ldots,(Y_n,X_n))$, is larger than a corresponding critical value. We consider two kinds of critical values: A parameter free critical value $c_{n,1-\alpha}^0$ and parameter dependent critical values $c_{n,1-\alpha}(\theta)$. Hence, we consider two confidence regions
We achieve finite sample valid inference with parameter-free critical values and exact inference with parameter-dependent critical values by extending the traditional Monte Carlo tests of Dwass:57 and Barnard:63 to incomplete models. Our proposed inference on the true value of the structural parameter combines the parameter free critical value with the parameter dependent critical value. The former allows very fast initial search in the parameter space, since the critical value is only computed once for all values of $\theta$. Then, given that $c_{n,1-\alpha}(\theta)\leq c_{n,1-\alpha}^0$ the parameter dependent critical value $c_{n,1-\alpha}(\theta)$ need only be computed for $\theta\in CR_n^0$, which greatly reduces the computational burden needed to achieve exact finite sample inference. The rest of this section is devoted to constructing the test statistic and the critical values.
Our test statistic is based on a sample analogue of the optimal transport problem $\mathcal D(Q_U\times P_{X,0},P_{0};\theta)$, which characterizes the sharp identified region in Theorem (ref). The sample analogue is $\mathcal D(Q_U\times \hat P_{X,n},\hat P_n;\theta)$, where the estimated distributions $\hat P_{X,n}$ and $\hat P_n$ are empirical distributions based on the data sample $((Y_1,X_1),\ldots,(Y_n,X_n))$. For computational tractability, we replace the resulting semi-discrete optimal transport problem with an approximation, based on a discretization of the latent variable distribution $Q_U$.\footnote{In section (ref) we propose a variant of the procedure in case the outcome space $\mathcal Y$ is finite, where the discretization involves no approximation.} In this approximation, the latent variable distribution is replaced with the empirical distribution based on a low discrepancy sequence $\tilde u^{(n)}:=(\tilde u_1,\ldots,\tilde u_n)$ (see section (ref) for details). Our chosen test statistic $T_n(\theta)$, therefore, is the discrete optimal transport solution
Computation of the test statistic is discussed in Section (ref) below. For now, note that ((ref)) solves a discrete optimal transport problem, which is a special kind of linear programming problem.
The test statistic also requires a specific choice of discrepancy $\delta$ such that $\delta(v,w;\theta)\ge 0$ with equality if and only if $v\in\Gamma_u(w;\theta)$. Such a discrepancy can be constructed in multiple ways. Unless specified otherwise, we recommend to construct the discrepancy as follows:
In the expression above, the matrix $\hat\Sigma$ is an approximation of the covariance matrix of $V=(U,X)$. Let $\Sigma_U$ denote the (known) covariance matrix of the random vector $U$ with distribution $Q_U$ and $\hat\Sigma_X$ the empirical covariance matrix of the sample $(X_1,\ldots,X_n)$. Since the distribution of $U$ conditional on $X$ is given as a primitive of the model, the best choice for $\hat\Sigma$ is
A useful alternative definitions of the discrepancy $\delta$ is the analogue of ((ref)) in the outcome space, namely
In the expression above, the matrix $\tilde\Sigma$ is an approximation of the covariance matrix of $W=(Y,X)$ by simulation. An advantage of this alternative discrepancy, is that $\Gamma_y$ may be easier to characterize than $\Gamma_u$, as will be the case in the empirical illustration of Section (ref). The major drawback is that $\tilde\Sigma$ is more complicated to obtain than $\hat\Sigma$. It is also likely to be a poor approximation of the covariance of $(Y,X)$ because of the incompleteness of the model. Indeed, simulating values of $Y$ from the model require assuming an equilibrium selection mechanism (a way to select within $\Gamma_y$), which may be incorrect. In the latter case, however, our inference procedure remains valid.
Our critical values rely on simulated samples of unobservables.
A Monte Carlo latent sample $\tilde U^{\prime(n)}$ is designed to be generated from the same distribution as the true latent variables $U^{(n)}$.
To ensure valid coverage of the true parameter with confidence region $CR_n$, we choose as critical value $c_{n,1-\alpha}(\theta)$, the $1-\alpha$ quantile of a distribution that first order stochastically dominates $T_n(\theta)$ for each $n$. We then show exact coverage by exhibiting a data generating process in $\mathcal P_\theta$ such that $T_n(\theta)$ has $1-\alpha$ quantile $c_{n,1-\alpha}(\theta)$. Let $\tilde U^{\prime(n)}$ be a Monte Carlo latent sample. Let $\tilde y^{(n)}=(\tilde y_1,\ldots,\tilde y_n)$ be the notation of a generic vector in $\mathcal Y^n$. Define the cost matrix $C(\tilde y^{(n)};\theta)$ with elements
Finally, define the set
The critical value we propose is the $1-\alpha$ quantile $c_{n,1-\alpha}(\theta)$ of the distribution of
The statistic $\tilde T_n(\theta)$ of equation ((ref)) differs from the test statistic $T_n(\theta)$ in two critical ways. First, the sample realization $Y_j$ in the element $C_{ij}$ of the cost matrix is replaced with a value $\tilde y_j$. This value $\tilde y_j$ is constrained by the Monte Carlo draw $\tilde U_j^\prime$ from $Q_U$ and the constraint $(\tilde{y}_j,X_j)\in\Gamma_y(\tilde U^\prime_j,X_j;\theta)$ to enforce the null hypothesis $H_0(\theta)$. Second, the supremum in expression ((ref)) ensures that $\tilde y_j$ is chosen to achieve the worst-case scenario, i.e., the largest possible value of $\tilde T_n(\theta)$ under the null $H_0(\theta)$.
For computational convenience, we also provide a more conservative confidence region $CR_n^0$ based on a critical value $c_{n,1-\alpha}^0$ which is independent of the parameter value $\theta$. Let $\delta^0$ be a discrepancy on $\mathcal U\times\mathcal X$ that satisfies the following.
The recommended choice of discrepancy is the following:
where $\hat\Sigma$ is defined in ((ref)). Denote $C^0$ the cost matrix with elements
where $(\tilde u_i)_{i\leq n}$ is the same low discrepancy sequence, and $(U_j^\prime)_{j\leq n}$ is a Monte Carlo latent sample of definition (ref). The critical value $c_{n,1-\alpha}^0$ is chosen to be the $1-\alpha$ quantile of the distribution of
By construction, for any $\tilde y$ such that $(\tilde y,X_j)\in\Gamma_y(U_j^\prime,X_j;\theta)$, we have $(U_j^\prime,X_j)\in\Gamma_u(\tilde y,X_i;\theta)$. Hence, if discrepancy $\delta$ in section (ref) is consistent with $\delta^0$ in the sense that $\delta(u,v;\theta)=\inf\{ \delta^0(v,v^\prime): v^\prime\in\Gamma_u(v;\theta)\}$, then we have $\delta((\tilde u_i,X_i),(\tilde y,X_j);\theta)\leq C_{ij}^0$. It follows that for all $\theta\in\Theta$, $\tilde T_n(\theta)\leq\tilde T_n^0$, and, therefore:
From Statement ((ref)), we deduce three advantages of the outer confidence region $CR_n^0$. First, the critical value is independent of the parameter value. Hence, it needs to be computed only once, and only the test statistic $T_n(\theta)$ needs to be computed for each value of the parameter $\theta$. Second, the outer confidence region $CR_n^0$ covers the whole identified set as opposed to each value in the identified set\footnote{See Section 4.3.1 of Molinari:2020 for a discussion of the distinction between coverage of the identified set and coverage of each of its elements.}. Third, given that $CR_n\subseteq CR_n^0$, the computation of Confidence region $CR_n$ can be performed with a search limited to $CR_n^0$ as opposed to the whole parameter space $\Theta$.
Note that replacing $C_{ij}^0$ with
would yield a valid and weakly smaller parameter free critical value. However, it would come at the cost of computational convenience.
The next theorem shows finite sample validity and exactness of our procedure. Validity is achieved because both $\tilde T_n(\theta)$ and $\tilde T_n^0$ first order stochastically dominate the test statistic $T_n(\theta)$. Hence, both confidence regions $CR_n^0$ and $CR_n$ have valid coverage in finite samples. Exact finite sample inference is achieved because we can construct a data generating process under which both $T_n(\theta)$ and $\tilde T_n(\theta)$ have the same distribution. Hence, our proposed confidence region $CR_n$ has the correct coverage probability in finite samples.
Since our testing procedure relies on Monte Carlo sampling from distribution $Q_U$ to mimic the latent sample $U^{(n)}$, we must make an assumption on the sampling process for the latter.
Assumption (ref) ensures that the conditional distribution of the Monte Carlo samples $\tilde U^{\prime(n)}$ given $X^{(n)}$ is identical to the conditional distribution of the true latent sample $U^{(n)}$ given $X^{(n)}$ under the null hypothesis $H_0(\theta)$. Importantly, this assumption does not require independence of the observations $(Y_i,X_i)$ across different indices $i$, thus permitting the selected values of $Y_i$ within $\Gamma(X_i, U_i;\theta)$ to be correlated across observations in an arbitrary manner. This property is particularly desirable with incomplete models. As discussed in EKS:2016, it is hard to reconcile the customary independence or mixing assumptions across units of observation with total ignorance of the mechanism that selected each realization from the model prediction set.
The formal proof of Theorem (ref) is given in the appendix, where we also give a version of the theorem conditional on $X^{(n)}$ (see appendix (ref)). Proof heuristics are as follows. By construction, under the null hypothesis, the Monte Carlo latent sample $\tilde U^{\prime(n)}$ has the same distribution as the true latent sample $U^{(n)}:=(U_1,\ldots,U_n)$. Now, if for each $j\leq n$, the process $P_j^{(n)}$ generating $(Y_j,X_j)$ is in $\mathcal P_\theta$, then each realization $(Y_j,X_j)$, $j\leq n$, falls in $\Gamma_y(U_j,X_j;\theta)$ almost surely (according to the support restriction in the model). Hence the test statistic $T_n(\theta)$ is smaller than $\sup\{ \mathcal D_n(C(\tilde y^{(n)};\theta)): \tilde y^{(n)}\in\Gamma_y^{(n)}(U^{\prime(n)},X^{(n)};\theta)\}$. Since the latter is identically distributed to $\sup\{ \mathcal D_n(C(\tilde y^{(n)};\theta)): \tilde y^{(n)}\in\Gamma_y^{(n)}(\tilde U^{\prime(n)},X^{(n)};\theta)\}$, size control follows. To see that the inequality in ((ref)) is an equality, we find $(Y^{(n)},X^{(n)})$ that achieves the maximum of $T_n(\theta)$ under the constraint $(Y_i,X_i)\in\Gamma_y(\tilde U^\prime_i,X_i;\theta)$.
In this section, we theoretically assess informativeness of the confidence region, as sample size increases. We characterize sequences of data generating processes and parameters that violate the model, and show that such parameter sequences are outside the confidence region, eventually. We prove this consistency result for the conservative outer region $CR_n^0$. Since the latter includes our proposed confidence region $CR_n$, the result also holds for $CR_n$.
For each $n\geq1$, let $P_0^{(n)}$ be a probability distribution on $\left(\mathcal Y\times\mathcal X\right)^n$ with identical marginals $P_{0n}:=P_{Y\vert X,0n}\times P_{X,0n}$. Let $(Y_{i,n}, X_{i,n})_{i\le n}$ be a triangular array where, for any $n\ge 1$, the size $n$ sample $(Y_{i,n}, X_{i,n})_{i\le n}$ follows distribution $P_0^{(n)}$. We consider parameters $\theta$ that violate the condition that characterizes the sharp identified region in Theorem (ref). Formally, the alternative is defined as follows, where $\mathcal D$ is defined as in ((ref)).
Since, by Theorem (ref), $\theta\in\Theta_I(P_{0n})$ if and only if $\mathcal D(Q_U\times P_{X,0n},P_{0n};\theta)=0$, Assumption (ref) means that $\theta$ is eventually outside the identified set under the sequence of alternative data generating processes.
In order to detect violations defined in Assumption (ref), or equivalently, to make sure such a parameter ultimately falls outside the confidence region, the data sequence must be sufficiently informative to identify the marginal distributions $P_{0n}$. Independence across observations or strong mixing assumptions are sufficient, but not necessary, as any dependence structure that allows estimation of $P_{0n}$ from the sequence of empirical distributions $\hat P_n:=\Sigma_{i\leq n}\delta_{(Y_i,X_i)}/n$ is suitable.
Detection of violations of the type ((ref)) also requires continuity of the cost function in the optimal transport problem.
The condition is stated in its most general form. However, sufficient conditions on the model structure can be derived. For instance, by Lemma 16.30 page 538 of AB:99, Assumption (ref) holds if the discrepancy is chosen according to ((ref)) and if $\Gamma_u$ is a continuous correspondence (i.e., both upper- and lower-hemicontinuous) with non empty and compact values.
If discrepancy $\delta$ in section (ref) is consistent with $\delta^0$ in the sense that $\delta(u,v;\theta)=\inf\{ \delta^0(v,v^\prime): v^\prime\in\Gamma_u(v;\theta)\}$, then we have ((ref)). Hence, the exact critical value $c_{n,1-\alpha}(\theta)$ is uniformly smaller than the conservative critical value $c_{n,1-\alpha}^0$. Therefore, Theorem (ref) also implies that any parameter value $\theta$ satisfying in Assumption (ref) eventually falls outside the confidence region $CR_n$.
Most structural models of interest involve latent variables, whose distribution depends on a vector of unknown parameters. The distribution of the latent variables may also depend on the exogenous variable $X$. The structural model then stipulates that $(Y,X,U)\in\mathcal Y \times \mathcal X \times \mathcal U$ satisfies support constraint $(Y,X,U) \in \Gamma(\theta)\subseteq \mathcal Y \times \mathcal X \times \mathcal U$, and $U$ has distribution $Q_{U\vert X;\theta}$ conditionally on $X$. For ease of notation, we use the same $\theta$ notation for the parameter of the latent variable distribution and the parameter of the correspondence $\Gamma$, even though they will generally be disjoint.
The objective of this section is twofold. First we show that the procedure we proposed is without loss of generality, because the model with parametric latent variable distribution can always be transformed into a model with fixed latent variable distribution. The transformation is particularly straightforward when the vector of latent variables is multivariate normal. Second, when the transformation is more involved, we propose a variant of the proposed inference method that doesn't require the transformation to a model with fixed latent variable distribution.
The basic ingredient in the reformulation is a transformation that recovers the vector of unobservable variables $U$ from a vector of latent variables $U^\ast$ with fixed distribution $Q_U^\ast$ appropriately chosen. We fix the distribution $Q^\ast_U$ on $\mathcal U^\ast$. We then find a function $h$ such that the random vector $U:=h(U^\ast,X;\theta)$ has distribution $Q_{U\vert X;\theta_2}$. When $U$ is scalar, the conditional quantile transform is an example of such a function $h$. In Example (ref), $Q_{U\vert X;\theta_2}$ is a multivariate normal with mean zero and variance covariance matrix $\Sigma$. In that case, we can simply let $Q^\ast_U$ be the standard multivariate normal and $h$ be defined by $U=\Sigma^{\frac{1}{2}}U^\ast$. More generally, such a transformation always exists, as long as $Q_U^\ast$ is chosen to be absolutely continuous, by Theorem 2.1 of carlier2016vector. It can also be computed as the solution of an optimal transport problem. Given the $h$ function above, the structural incomplete model can be reformulated as the combination of the support constraint $(Y,X,U^\ast)\in\Gamma^\ast(\theta)$, where
and the marginal constraint $U^\ast\sim Q_U^\ast$ and $U^\ast\perp X$.
For cases, where the transformation $h$ is difficult to compute, we propose a variant of our inference procedure without transformation. Inference proceeds as in Section (ref) with the following two modifications to account for the fact that the latent vectors $U_i$ in sample $U^{(n)}$ are drawn independently from $Q_{U\vert X_i;\theta}$ conditionally on $X^{(n)}$.
We now propose a refinement of our procedure, which bypasses the need for a low discrepancy approximation of the distribution $Q_U$ of latent variables.\footnote{We are grateful to Francesca Molinari for suggesting this refinement. The usual disclaimer applies.} This refinement applies to models, where the space $\mathcal Y$ of endogenous variables is finite. This refinement is easier to implement with parametric unobservable distribution, so we will use the framework and notation of Section (ref), where the distribution of latent variables $U$ conditional on $X$ is denoted $Q_{U\vert X;\theta}$. We start with the traditional illustration of incomplete models with finite outcomes.
With finite outcomes, the sample analogue $\mathcal D(Q_{U\vert X;\theta}\times \hat P_{X,n},\hat P_n;\theta)$ of the characterization in Theorem (ref) can be computed without relying on a low discrepancy sequence approximation. Call $\mathcal Y(u,x;\theta)$ the set of $y$'s predicted by $u$. Precisely, $\mathcal Y(u,x;\theta)$ is defined by the fact that $y^\prime\in\mathcal Y(u,x;\theta)$ if and only if $(y^\prime,x;\theta)\in\Gamma(\theta)$. Since $\mathcal Y$ is finite, for any given $x$ and $\theta$, $\mathcal Y(u,x;\theta)$, as a subset of $\mathcal Y$, can only take a finite number of distinct values. Call them $\mathcal Y^1(x;\theta),\ldots,\mathcal Y^K(x;\theta)$.\footnote{We suppress the dependence of $K$ on $x$ for ease of notation.} Now call $(\mathcal U^1(x;\theta),\ldots,\mathcal U^K(x;\theta))$ the partition of $\mathcal U$ defined for each $k\leq K$ by $u\in\mathcal U^k(x;\theta)$ if and only if $\mathcal Y(u,x;\theta)=\mathcal Y^k(x;\theta)$. Distribution $Q_{U\vert X;\theta}$ on $\mathcal U$ induces mass $q^k(x;\theta):=Q_{U\vert X;\theta}(\mathcal U^k(x;\theta)\vert X=x;\theta)$ on each element $\mathcal U^k(x;\theta)$ of this partition. Unlike GRS:2022, we allow this partition to depend on $X$, so that $X$ can have many support points, or be continuous and multivariate.
We can therefore directly use the sample analogue $\mathcal D(Q_{U\vert X;\theta}\times \hat P_{X,n},\hat P_n;\theta)$ of the characterization in Theorem (ref) as a test statistic. By construction, all values of $u$ within a element $\mathcal U^k(x;\theta)$ predict the same set $\mathcal Y^k(x;\theta)$. Therefore, we can replace the distribution $Q_{U\vert X;\theta}\times \hat P_{X,n}$ with the probability mass function with $nK$ support points. Define $l_i := \lfloor i / K \rfloor$, and $k_i := i \mbox{ mod } K$. The probability mass function that characterizes $Q_{U\vert X;\theta}\times \hat P_{X,n}$ has $nK$ support points $\mathcal U^{k_i}(X_{l_i};\theta)$ and probability $q^{k_i}(X_{l_i};\theta)$, where $1\leq l_i\leq n$ and $1\leq k_i \leq K$ for each $i\leq n$, and $l_i\neq l_{i'}$, $k_i\ne k_{i'}$ all $i\ne i'$ (so as to range over all $nK$ possible pairs $(k,l)$). A natural way to choose the test statistic would be to choose an arbitrary point $u_{k_i}(X_{l_i};\theta)$ in each $\mathcal U^{k_i}(X_{l_i};\theta)$ and use discrepancy ((ref)). However, such a definition would make the test statistic, and hence the resulting inference, depend on the arbitrary choice of representative $u_{k_i}(X_{l_i};\theta)$ in each $\mathcal U^k(X_{l_i};\theta)$. Instead, we use the fact that all $u\in\mathcal U^k(X_{l_i};\theta)$ predict the same outcome $\mathcal Y^k(X_{l_i};\theta)$, and use discrepancy ((ref)) to define the test statistic. Here, discrepancy ((ref)) can be written
where $\tilde\Sigma$ is defined as in ((ref)).
The test statistic is:
Similarly, the critical values are obtained as the quantiles of the following statistic.
where, for each $j\leq n$, $\tilde k_j$ is a random draw from the distribution with probability mass function $(k,q^k(X_{l_j};\theta))_{k=1}^{K}$, and the draws are independent conditionally on $X^{(n)}$. As before, the quantiles of statistic $\tilde T_n$ are approximated with $S$ Monte Carlo samples $(\tilde k_1^s,\ldots,\tilde k_n^s)$.
We now propose a profiled version of our test, to perform finite sample inference on a subset of the structural parameters, or more generally on a low dimensional transformation of the full structural parameter vector. This reduces the computational burden associated with the search in the parameter space. It also removes the concern that a confidence region for a subvector obtained via projection can be very conservative.
We therefore consider the problem of testing $H_0(S):\Theta_I\cap S\ne\varnothing$, for some region $S\subseteq\Theta$ of the parameter space. In the most common case, where a component, say $\theta_1$, of the vector of structural parameters, with true value $\theta_{01}$, is of interest, $S:=\{\theta\in\Theta: \theta_1=\theta_{10}\}$, and a confidence region for $\theta_1$ is obtained by inverting the test, i.e., including all values $\theta_{10}$ such the test fails to reject at the chosen significance level.
By definition, the profiled test statistic is equal to
Critical values $c_{n,1-\alpha}(S)$ are obtained from the worst case distribution under the null hypothesis:
where the outer supremum is taken over collections $(\tilde y_1,\ldots,\tilde y_n)$ that enforce the null hypothesis as follows: $\tilde U^{\prime(n)}$ is a Monte Carlo sample and the collection $(\tilde y_1,\ldots,\tilde y_n)$ satisfies $(\tilde y_j,X_j)\in\Gamma_y(\tilde U_j^\prime,X_j;\theta')$ for each $j\leq n$, and for some $\theta^\prime\in S$.
The test based on the profiled statistic for subvector inference is also valid and exact in finite samples, as we show in the following theorem.
Theorem (ref) is proved by showing that, under the null hypothesis $\Theta_I\cap S\ne\varnothing$, $\tilde T_n(S)$ stochastically dominates $T_n(S)$ and there is a $\theta\in S$ and a worst-case DGP in $\mathcal P_\theta$ such that $T_n(S)$ and $\tilde T_n(S)$ have the same distribution.
As for full vector inference, we need only search within the parameter free confidence region $CR_n^0$. This will be implied and omitted from the notation in the rest of this section.
The test statistic $T_n(S)$ in ((ref)) can be computed recursively with alternating minimizations over $\Pi_n$ and $S$. Start with an arbitrary $\theta^1$ and at each iteration $l>1$, find
until convergence, which is guaranteed by the fact that the objective is non negative and decreases weakly at each step.
To compute the critical values, the worst case statistic $\tilde T_n(S)$ of ((ref)) can be replaced with the weakly larger statistic
where the profiling minimization of $\theta$ over $S$ is removed and $\theta^\prime$ is used as an approximate minimizer. The resulting statistic, up to the outer maximization over $S$, is identical to the statistic $\tilde T_n(\theta^\prime)$ used for critical values in the full vector inference problem, and can hence be computed in the same way.
The objective of this simulation exercise is to compare size, power and computing time of our inference procedure with existing methods, and to evaluate the robustness to changes in the choice of discrepancy $d$ in the test statistic.
We analyze the performance of our inference procedure with a simulation design based on Example (ref). We test $H_0:\theta=1$. In the size analysis, the true value of $\theta$ is set to $1$. In the power analysis, it ranges between $-1$ and $2.5$. In each simulation replication, we draw an i.i.d. sample $((\underline W_1,\overline W_1),\ldots,(\underline W_n,\overline W_n))$ of replications of $(\min\{W_1,W_2\}-c/2,\max\{W_1,W_2\}+c/2)$, where $c>0$, \[ \left(
\right) \sim N\left(\left(
\right), \left(
\right) \right), \] and $\rho$ takes values $0$, $0.5$ and $1$. In one simulation design (hereafter {\em random} or R), each $W^\ast_j$ is selected randomly (i.e., drawn independently from a uniform distribution) from $[\underline W_j,\overline W_j]$, all $j\leq n$. In a second simulation design (hereafter {\em worst case} or WC), each $W^\ast_j$ is selected from $[\underline W_j,\overline W_j]$, all $j\leq n$, so as to maximize the value of the resulting test statistic. In the latter design, exact size control is expected. In this design, the vector $X_j$ of exogenous variables contains $\underline{W}$ and $\overline{W}$. A sample of latent variables $(U_1,\ldots,U_n)$ is drawn independently from $N(0,1)$ and $Y_j$ is set equal to $\theta W^\ast_j+U_j$ for each $j\leq n$.
The inference strategy proposed in Section (ref) requires a choice of discrepancy $d$ for the test statistic. This is akin to the choice of statistic (Cramer-von Mises vs. Kolmogorov-Smirnov, for instance) in other proposals. We compare the performance of our procedure with three competing discrepancies:
Given the choice of discrepancy, the numerical implementation of the procedure is discussed in appendix (ref).
We compare results using our inference strategy and results using existing strategies explicitly designed for cases with many moment inequalities, with an emphasis on incomplete models, i.e., CCK:2019 (hereafter CCK) and andrews2017inference (hereafter AS) .\footnote{We are grateful to the authors for sharing their simulation code.} Both strategies involve three steps:
Sample sizes range over $n\in\{10,50,100,500\}$. Nominal levels range over $\alpha\in\{0.90,0.95,0.99\}$. Coverage probabilities are based on quantiles of $\tilde T_n(\theta)$ as critical values. For each sample size, nominal level and choice of critical value, we conduct $5,000$ replications of the test. For each test, the critical values are based on $S=1000$ Monte Carlo samples. The time columns show total computation time for all $5,000$ replications. We run all our simulations on a single server with dual CPU AMD EPYC 7702 @ 2.0 GHz.
Table (ref) shows coverage probabilities for our testing procedure under three different correlation levels $\rho$ for $(\underline W,\overline W)$, two values of $c$ and three different choices of discrepancy for the test statistic. Coverage probabilities are equal to the nominal levels in all cases and all sample sizes. As expected from the theory, the correlation $\rho$, the degree of incompleteness $c$ and the choice of distance have no effect on size, except for two coverage probabilities of $0.92$ and $0.933$ for a nominal level of $0.90$ and one coverage probabilities of $0.967$ for a nominal level of $0.95$ in case the discrepancy is in the outcome space. So far, we have no explanation for this slight but unexpected under rejection.
Table (ref) compares the performance of our procedure, with the recommended weighted discrepancy in the latent variable space OTW, with that of AS and CCK, with the recommended test statistics and the recommended values of tuning parameters. Procedures are compared on coverage probabilities and computation times. We use the random (R) simulation design, and $c=0$, which is the most challenging case for size control. As expected, our method based on optimal transport is computationally attractive for small sample sizes (below $500$), but less so for high sample sizes (above $500$). Coverage probabilities are equal to nominal level for our performance. AS undercover, while CCK are conservative.
In our power analysis, we fix nominal size to $\alpha:=0.05$, and the null hypothesis to $H_0:\theta=1$. We vary the true value of $\theta$ from $-1$ to $2.5$, and for each true value of $\theta$, we obtain rejection frequencies from $5,000$ replications of the dgp simulation and testing procedure, and $1,000$ Monte Carlo samples in each instance of the testing procedure. With these rejection frequencies, we trace power curves. Unless specified otherwise, the sample size is $n=500$, $c=1$, the correlation $\rho$ between $\underline W$ and $\overline W$ is $0$, the DGP involves the random selection of $W^\ast$ from $[\underline W,\overline W]$. The benchmark procedure is OTW, i.e., our testing procedure with the recommended weighted discrepancy in the latent variable space in the definition of the test statistic.
Figure (ref) compares the power curves for our procedure, AS and CCK. Our procedure clearly dominates both AS and CCK. More surprisingly, the parameter free procedure, that we expected to be quite conservative, since it covers the whole identified set, is quite competitive relative to AS and clearly dominates CCK.
Figure (ref) shows power curves for all three choices of discrepancy in the definition of the test statistic, i.e., unweighted OTE, the recommended weighted OTW and weighted discrepancy on the outcome space OTY. We don't see a significant difference in power performance between the three choices of discrepancy. If anything, OTY is appears less powerful than the alternatives OTE and OTW.
Finally, figure (ref) achieves two objectives. First, we trace power curves for $n=100$ and $n=500$ to see the effect of sample size on power. The effect of sample size is significant, but the procedure still has good power for sample sizes as low as $n=100$. Second, we try to disentangle the effects of partial identification and sampling uncertainty. To that end, we approximate the identified set based on the characterization from BMM:2011 and GH:2011. Applied to the present simulation design, this characterization is specified in ((ref)). The grey area in figure (ref) covers the set of values of $\theta$ such that $1$ belongs to the identified set when $\theta$ is the true value. We also disentangle the specific effect of sampling of covariates in the following way. We approximate the identified set with random draws of size $n=100$ and $500$ of the conditioning variables. We trace two curves, the dashed green curve for $n=100$ and the dashed purple curve for $n=500$. At each value of $\theta$, the dashed curve gives the probability that $1$ satisfies ((ref)) based on a random draw of the conditioning covariates (drawn according to their known DGP).
We illustrate our testing procedure with an application to the data and results in CMT:2021, described in example (ref). We replicated their model exactly and used our methodology on the same data.\footnote{We replicated table 5 page 3023 in CMT:2021 to check that our modeling and coding replicates their paper perfectly.} We first recall the details of the structural model in CMT:2021. We then explain our empirical exercise and discuss results.
The sample size $n$ is the number of regional markets in which $J$ firms, indexed by $j$, potentially operate. In each regional market, indexed by $i$, the firms simultaneously decide whether or not to enter, and the firms who enter compete in prices.
Exogenous variables are the following. The size of market $i$ is $M_i$. The identity of potential entrants\footnote{A firm is a potential entrant in market $i$ if it operates in one of the origin or destination airport.} in market $i$ is collected in $\mathcal G_i$. Each firm $j$ is associated with three types of exogenous covariates in each market $i$: First, the vector of demand relevant firm characteristics is $A_{ij}$ (in CMT:2021, the notation is $X$). Second, the vector of production cost relevant firm characteristics is $W_{ij}$. Finally, the vector of market entry cost relevant firm characteristics is $Z_{ij}$. We call $X_{i}:=(M_i,\mathcal G_i,A_{i},W_{i},Z_{i})$ the vector of all exogenous variables in market $i$, with $A_i:=(A_{ij})_{j}$, $W_i:=(W_{ij})_{j}$ and $Z_i:=(Z_{ij})_{j}$.
We follow the model structure and parametric specifications in CMT:2021. Firm $j$ has fixed cost of entry $\exp(\gamma Z_j+\nu_j+\nu_c)$, where $\gamma$ is an unknown vector of parameters of interest, $\nu_c$ is a common entry cost shock, and $\nu_j$ is an idiosyncratic entry cost shock. Both common and idiosyncratic shocks are normally distributed latent variable and independent of each other. Firm $j$ has marginal unit cost of production $\exp(\delta W_j+\eta_j)$, where $\delta$ is an unknown vector of parameters of interest and $\eta_j$ is a normally distributed latent variable. The market share is generated by a nested logit demand model. Specifically, consumer $l$'s indirect utility from choosing the outside option is $\epsilon_{l0}$, whereas their utility from choosing firm $j$'s product is $u_{lj} = \beta A_j - \rho P_j + \xi_{j} + (1 - \lambda)\epsilon_{lj},$ where $P_j$ is the product price, $\rho$ and $\lambda$ are unknown parameters of interest , $\xi_j$ is a normally distributed latent variable, and $\epsilon_{l0}$ and $(\epsilon_{lj})_j$ are independent type I extreme value preference shocks. The vector of latent variables $U:=(\xi, \eta,\nu,\nu_c)$ is assumed to follow the normal distribution $N(0,\Sigma)$ where $\xi = (\xi_j)_j$, $\eta = (\eta_j)_j$, $\nu = (\nu_j)_j$ and
We now describe the support restriction on the joint distribution of $(Y,X,U)$ derived from the structural entry and competition model in CMT:2021. The endogenous variable $D_j$ is equal to $1$ if the firm enters the market, and $0$ otherwise. Firms who enter then choose price $P_j$ and realize their market share $S_j$. Market shares $S:=(S_j)_{j}$ are determined by the entry profile $D:=(D_j)_{j}$ and the price chosen by firms who entered. Given the nested logit specification, market shares are given by
where
The profit maximizing price for firm $j$ satisfies the following first order condition:
Equations (ref) and (ref) are joint equations for prices and market shares of firms that enter the market. The solution for equations, together with the normalization that $P_j = 1$ and $S_j = 0$ if $D_j = 0$, determines $P$ and $S$ as functions of $(D,A,\xi)$. Finally, firm $j$ enters the market if and only if it makes non negative profit, i.e.,
The entry profile $D$ is determined as a pure strategy Nash equilibrium of the full information simultaneous entry game, with payoffs given by ((ref)) for a firm who enters, and normalized to zero otherwise.
We use data from CMT:2021 and repeat the description here for the reader's convenience. The data are drawn from the second quarters of 2012’s Airline Origin and Destination Survey, the T-100 Domestic Segment Data Set’s Aviation Support Tables, available from the Department of Transportation’s National Transportation Library, and the US Census for demographic data. The basic unit of observation is an airline in a market (a market carrier). A market as a unidirectional trip between two airports, irrespective of intermediate transfer points. The data set includes the markets between the top 100 USMSAs ranked by their population. There are 8,163 unidirectional markets. There are six carriers in the data set: AA, DL, UA, US, WN, and a low-cost carrier denoted LCC. The LCCs include Alaska, JetBlue, Frontier, Allegiant, Spirit, Sun Country, and Virgin. There are 22,445 market-carrier observations for which prices and market shares are observed. An airline is considered a potential entrant if it is serving at least one market out of both of the endpoint airports. Demand relevant firm characteristics $A_{ij}$ include origin presence, which is defined as the number of markets served by an airline out of the origin airport, and the distance between the origin and destination airports. Entry cost relevant firm characteristics $Z_{ij}$ include nonstop origin (the number of nonstop routes that an airline serves out of the origin airport) and nonstop destination (the number of nonstop routes that an airline serves out of the destination airport). The marginal cost relevant firm characteristics $W_{ij}$ are distance between origin and destination, and a dummy for LCCs and Southwest.
In column 3 of table 4 page 3019, CMT:2021 report intervals for each of the $20$ parameters $\theta:=(\beta,\gamma,\delta,\lambda,\rho, \Sigma)$ in the model. Each such interval is the projection of a $95\%$-level confidence region for the parameter vector. We evaluate their results by testing $10,000$ values of the parameter vector drawn at random in the hyper-rectangle defined by the cartesian product these intervals. For each value of the parameter vector $\theta$, we implement the test as described in section 2, with the choice of discrepancy in the outcome space to define the cost matrix. We do this because of computational convenience in this particular class of applications. Specifically, we implement the following algorithm. The vector of latent variables $U_i=(\xi_i,\eta_i,\nu_i,\nu_{ci})$ has dimension $19$ and a normal distribution with mean $0$ and covariance $\Sigma$.
We call $Y_i$ the vector of all observed endogenous variables $(D_{ij})_j$, $P_{ij})_j$ and $(S_{ij})_j$ pertaining to a market $i$, where $(D_{ij})_j$ is the entry profile of firms in market $i$, $(P_{ij})_j$ is the price profile and $(S_{ij})_j$ is the market share profile, with the normalizations $P_{ij}=1$ and $S_{ij}=0$, if $j$ does not enter market $i$. The vector $X_i$ of exogenous variables for a given market includes $M_i$, $A_i$, $W_i$ and $Z_i$. The parameter vector $\theta$ includes $\beta$, $\gamma$, $\delta$, $\lambda$, $\rho$ and $\Sigma$.
We compute a low discrepancy sequence\footnote{We use the generalized golden sequence from the GoldenSequences.jl Julia package.} of size equal to the sample size on $[0,1]^{19}$, and transform each element by the component-wise quantiles of the standard normal composed with the linear transformation $a\mapsto \Sigma^{1/2}a$ (where $\Sigma^{1/2}$ is obtained via Choleski decomposition) to obtain the low discrepancy sequence $(\tilde u_i)_i$ that approximates the distribution $Q_{U\vert\theta}$ of $U$. An alternative would be to draw a random sample from the distribution $Q_{U\vert\theta}$. The test statistic $T_n(\theta)$ is computed with cost matrix is defined by
and $\hat\Sigma$ is the simulated approximation (from section (ref)) of the covariance matrix of $(Y_j,X_j)$. Cost $C_{ij}$ is set to $+\infty$ if markets $i$ and $j$ have different sets of potential entrants.\footnote{CMT:2021 define potential entrants in a market as the firms that already operate in either origin or destination airport.} The numerical procedure to compute the test statistic and the corresponding critical values is discussed in appendix (ref).
We draw $10,000$ values of $\theta$ uniformly in the hyper-rectangle defined by the cartesian product of the confidence intervals in table 4 of CMT:2021. We compute the $p$-value for the test of each of these $10,000$ values of $\theta$. The total computation time was $146$ hours on a single server with dual CPU AMD EPYC 7702 @ 2.0 GHz. We find a $p$-value of $0.000$ in $9,699$ cases, a $p$-value of $0.001$ in $296$ cases, and a $p$-value of $0.002$ in $5$ cases. This strong rejection can be attributed to mis-specification of the structural model and low power of the inference method used in CMT:2021.
We have proposed a procedure to compute confidence regions in incomplete models with exact coverage in finite samples. Compared to existing approaches, our procedure has many advantages, some straightforward and others more subtle. First, finite sample validity avoids reliance on asymptotic approximations, which are often suspect. It also removes the need for user-chosen tuning parameters, that inference results are often very sensitive to. Second, our procedure removes the need for transforming conditional into unconditional moment inequalities, and for reducing the very large number of moment inequalities with complex and model-specific core determining classes. Third, finite sample validity allows us to conduct inference in models, where the specification depends on the sample size. This includes possible future applications to games on networks and network formation games, when a single network is observed. In such cases, the support constraint in the model specification depends on the sample size, and so does the dimension of the latent variable, which involves an individual's neighbors in the network. Finally, although we haven't developed it here, our method extends to specifications, where the structural support constraint is individual-specific, thereby allowing us to conduct inference with the structural vector autoregressions proposed in GK:2021 and GKR:2021. This paper has contributed to a growing literature that shows how optimal transport theory provides a rich set of tools in econometrics in general, and incomplete models in particular. We expect these tools to underlay an extension to semiparametric incomplete models with independence constraints.