Robust Likelihood Ratio Tests for Incomplete Economic Models

Robust Likelihood Ratio Tests for Incomplete Economic Models

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Keywords: Incomplete models, Robust inference, Likelihood ratio tests, Limits of experiments

Introduction

Incomplete structures arise in a wide class of economic models when the researcher's theory does not fully describe how a particular outcome occurs given the primitives of the model. In this study, we consider a class of models in which, given structural parameter $\theta\in \Theta$ and latent variable $u\in U$, the model predicts the set $G(u|\theta)$ of values for discrete outcome $s$.\footnote{Introducing covariates does not fundamentally change the structure. We therefore treat this case in Section (ref) as an extension.} The researcher observes $s$, but his/her theory is silent about the mechanism that determines how $s$ is selected from the predicted set. This class encompasses various models studied in the empirical literature. Examples include models of market entry bresnahan1990entry,bresnahan1991empirical,berry1992estimation,Ciliberto:2009aa where the theory does not specify how a pure strategy Nash equilibrium is selected, models of self-selection Heckman:1990aa,Mourifie:2018aa where an individual's choice of the sector of activity interacts with unobserved skills, and models of English auction Haile:2003to,Aradillas-Lopez:2008ab where the researcher wants to allow solutions that satisfy weak rationality restrictions.\footnote{Other examples include models of voting Kawai:2013aa, choice of product variety EIZENBERG:2014aa, and network formation models Miyauchi:2016aa.} In many of these settings, empirical questions can be investigated by testing hypotheses on the structural parameter. Given the incompleteness of the theory, it is desirable to conduct tests without adding assumptions on how selections operate. Despite the need for such robustness, the theoretical study of robust testing procedures and their properties has been limited.\footnote{Studies of tests of the moment inequality model are related, but their model differs from the one we consider. See Section (ref).} This study develops a framework for hypotheses testing in incomplete models, shows how to construct robust and optimal tests, and provides asymptotic tools to evaluate their performance.

Each of the hypotheses we consider can be written as

for some function $\varphi:\Theta\to \mathbb R^k$ and mutually exclusive sets $K_0,K_1\subset\mathbb R^k$. Such hypotheses naturally arise in applications of incomplete models. For example, in an entry game, a key parameter is the strategic interaction effect, which measures the effect of an opponent firm's entry on a firm's profit. An important empirical question is whether the presence of such interaction effects can be supported by the observed data dePaula2012inference. One way to address the question is to formally test the null hypothesis that the strategic interaction does not exist, namely $H_0:\varphi(\theta)=0$, against an alternative hypothesis that negative externalities exist, $H_1:\varphi(\theta)< 0$, by choosing a suitable functional $\varphi$.

Such a hypothesis testing problem, however, faces several challenges. First, without further assumptions, the model permits multiple distributions of the observables even if each hypothesis fully specifies the value of $\theta$. To see this, consider a simplified problem in which $H_0:\theta=\theta_0~v.s. ~H_1:\theta=\theta_1$. This problem may appear as testing a simple null hypothesis against a simple alternative. However, under each hypothesis, multiple distributions (of $s$) may be compatible with the theory because any distribution $P$ of the outcome is consistent with $\theta$ as long as one can augment the model by finding a suitable selection mechanism that induces $P$. Therefore, even under the simplest setting, both null and alternative hypotheses can be composite (in terms of permitted distributions).\footnote{Furthermore, the hypotheses in (ref) allow the presence of additional nuisance parameters such as sub-components of $\theta$. We address this issue separately as an extension of the base framework in Section (ref).} The problem becomes even more challenging when data are obtained from a sequence of experiments. If one stays agnostic about the selection, the unknown selection mechanism is allowed to be arbitrary across experiments. For example, across experiments, the true selection mechanism may vary with and be correlated through a specific variable; however, the researcher does not even know the identity of this variable. From the researcher's viewpoint, the resulting outcome sequence is then heterogeneous and dependent in an unknown way, which in turn makes it hard to characterize the large-sample distribution of test statistics and apply standard asymptotic tools to analyze the power of the tests.

We develop tests that overcome these challenges. For this, we exploit the fact that the sampling uncertainty and lack of understanding of the selection can be represented by a {\em belief function\/}, a capacity (or non-additive probability), which belongs to a broader class of two-monotone capacities. Capacities in this class are known to have properties useful for conducting robust statistical inference Huber:1973aa.\footnote{See also Huber:1974aa for corrigendum and Huber:1981aa for the broader area of robust statistics.} We start by demonstrating that for testing between simple hypotheses, robust tests can be constructed for any finite sample. The proposed test, which takes the form of a likelihood ratio (LR) test, controls the size in finite samples regardless of the unknown selection mechanism and maximizes a measure of power, which we call {\em lower power\/}. One may wonder how such an LR test can be constructed because incomplete models generally admit infinitely many likelihoods. A key observation is that HS's theory ensures that there exists the least favorable pair (LFP) of likelihoods: one compatible with the null that is the least favorable for size control and the other compatible with the alternative that is the least favorable for power maximization.\footnote{They also show that such a pair is unique up to its Radon-Nikodym derivative.} Distinguishing two such extreme distributions turns out to be the best way to test one parameter value against another while staying agnostic about the selection.

We then develop LR tests for repeated experiments. Our first main contribution is to show that despite the potential heterogeneity and dependence of the data, the LFP consists of product measures as long as the latent variables are independent across experiments. Heuristically, this means that under the least favorable distribution for size control (or power maximization), the observables can be viewed as independent across experiments, while the true data-generating process (DGP) may not satisfy such regularity. This leads to a number of desirable results. In particular, it allows us to construct robust LR tests that are optimal in the minimax sense, provide a simple critical value based on a large-sample Gaussian approximation, and develop an asymptotic framework for evaluating the power of the tests.

Our second contribution is on the practical side. While HS's theory ensures the existence of the LFP, in practice, one needs to find a way to compute it. We show that in the class of models we consider, the LFP can be computed by solving a finite-dimensional convex program in which the constraints of the program are the {\em sharp identifying restrictions\/} studied in the identification literature beresteanu2011sharp,galichon2011set,Chesher:2014aa. These restrictions simplify the constraints by making them linear in the control variable, and they therefore play a crucial role in computing the LFP and implementing the robust optimal tests. While the restrictions are useful for characterizing sharp identified sets, little is known about whether they lead to statistically optimal inference. Our result shows they are indeed crucial for likelihood-based inference that has a certain optimality property. Our theoretical result on the LFP also has a practical implication. In particular, under mild conditions, the distributions forming the LFP are independently and identically distributed (i.i.d.) laws, and hence the researcher only needs to find the LFP in a “single” experiment rather than finding it from the entire sequence of experiments. This result also contributes to a significant reduction in the computational cost of our tests.

Our third main contribution is to provide a framework for analyzing the asymptotic power of the tests by embedding the product LFPs into a model that admits local approximations. Specifically, we show that under regularity conditions, a sequence of experiments characterized by the ratio of the LFPs, obtained from a null parameter value and a local alternative, converge to a limit in the sense of Le-Cam:1972aa,LeCam:1986aa. We use this property to characterize the upper bound of the asymptotic lower power of the tests for one-sided hypotheses. Our approach uses the limits of experiments argument and can potentially be used in other statistical decision problems in incomplete models. The main advantage of this approach is that once the LFPs are embedded into a probabilistic model whose limit is tractable, most of the power analysis can be performed using standard tools.

Our framework, however, also incorporates some non-standard features. First, the underlying model in which we embed the LFPs may not satisfy the well-known differentiability in quadratic mean condition, which is sufficient for the local asymptotic normality (LAN) of the experiments over the entire local parameter space. Instead, the model is typically directionally differentiable (in the $L^2$ sense) and satisfies the LAN property separately on a collection of convex cones that partition the local parameter space. Second, perhaps more importantly, incomplete models may yield alternatives that are not robustly testable. Such an alternative admits a selection mechanism that makes the lower power of any level-$\alpha$ test weakly below the nominal level. We clarify the notion of robust testability and relate it to the observational equivalence concepts studied in the identification literature Chesher:2014aa. To conduct a meaningful power analysis, we then introduce an extended notion of alternatives, which we call {\em shifted local alternatives\/}. The asymptotic power envelope is shown to be non-trivial against such alternatives.

We further extend our analysis to a model that permits the presence of nuisance components of the parameter vector. Setting up a statistical decision problem, we construct a robust LR test that minimizes a certain risk function. We call this test a Bayes--Dempster--Shafer (BDS) test as it minimizes a risk that treats parameter uncertainty in a Bayesian way and incorporates ambiguity due to incompleteness through a belief function. Finally, we establish a minimax theorem for this setting, which suggests that a level-$\alpha$ test that maximizes a weighted average of lower power can be approximated using a sequence of BDS tests.

Relation to the Literature

Our study is most closely related to Epstein:2016qv who developed a theoretical framework for modeling repeated experiments with incompleteness.\footnote{epstein2015exchangeable provided axiomatic foundations for robust subjective inference and decision making in such a setting.} We adopt their framework and use the (product) belief function to characterize the set of joint distributions of outcomes across experiments. This allows us to study the robustness of tests even in settings where selections are heterogeneous and dependent in an unknown way. This study then takes a step further and develop ways to examine the optimality of tests in such settings.

Our study is also related to earlier work on incomplete models.\footnote{The analysis of an incomplete system of equations dates back to the early work of Wald1950. Here, we focus on reviewing more recent developments in models with multiple equilibria.} In particular, our framework for the single experiment builds on that of jovanovic1989observable, who pointed out that models with multiple equilibria lead to incomplete structures and face potential difficulty in identifying structural parameters. tamer2003incomplete studied identifying restrictions in an incomplete simultaneous discrete response model with multiple equilibria. Since his seminal work, it has become common to use partially identifying inequality restrictions to bound parameters of interest. galichon2011set, beresteanu2011sharp, and Chesher:2014aa characterized sharp identifying restrictions for a wide range of incomplete models using the theory of random sets. We also use, as a central tool, the capacities associated with random sets. As discussed above, the sharp identifying restrictions play an important role in the construction of tests that achieve robustness and statistical optimality.

Commonly used identifying restrictions take the form of moment inequalities. As such, inference methods developed for moment inequality models (chernozhukov2007estimation, andrews2010inference, bugni2010bootstrap, andrews2012inference) have been commonly used. Some of them galichon2006inference,galichon2009test,galichon2013dilation use test statistics based on capacities to construct confidence regions. These methods combine the implications of incomplete models on moments with an additional assumption on the sampling process (e.g., i.i.d. sampling). By contrast, our approach uses the model's implications on certain likelihoods and does not restrict the sampling process. Chen:2011aa considered a sieve MLE-based inference, which can be applied to incomplete models. Their approach profiles out a non-parametric nuisance parameter (selection) from the likelihood function using a sieve. Our approach, which picks out the LFP, can also be interpreted as a way to average out the nuisance parameter, in which the weights are the least favorable priors, and averaging is carried out without explicitly introducing a functional space.

The results on the optimality of the tests in related settings are somewhat limited. Within a moment inequality framework, CANAY2010408 found that a test based on the empirical LR statistic is optimal with respect to the large deviations criterion. In a more specialized setting in which moment restrictions are convex in the parameter, Kaido:2014aa showed that a test based on a semiparametrically efficient estimator of the identified set achieves the asymptotic power envelope against some local alternatives. In models characterized by conditional moment inequalities, Armstrong:2014aa,Armstrong:2018aa compared the relative power of the testing procedures based on Cramer--von Mises and weighted Kolmogorov--Smirnov statistics. These studies deal with testing problems in models characterized by moment inequalities, which differ from ours in terms of (i) the hypotheses they test and (ii) how they extend a single experiment to repeated experiments. For the former, these studies consider testing whether $\theta$ is in the identified set, while our focus is on testing hypotheses of the form in (ref), which does not involve identified sets. For the latter, they assume that an i.i.d. sample is available, and hence the robustness issue against heterogeneity and the dependence of selection does not arise.\footnote{See Epstein:2016qv for this distinction as well as MR3753715 (Section 5.3).}

Finally, our framework for inference is related to others that use limit theorems based on the thoery of random sets. As mentioned earlier, we use a Gaussian approximation to compute the critical value for the LR statistic, which is similar in spirit to the central limit theorem (CLT) in Epstein:2016qv, whereas a different tool is used to obtain this result because of the non-trivial difference between the LR statistic we use here and their Kolmogorov--Smirnov-type statistic. In a different class of models, in which observations are set-valued, beresteanu2008asymptotic applied a central limit theorem for random sets to make their inference.

Throughout, for any metric space $A$, we let $\Sigma_A$ denote its Borel $\sigma$-algebra. We then denote the set of Borel probability measures on $A$ by $\Delta(A)$ and equip it with the topology of weak convergence. Let $N(\mu,V)$ denote the law of a normal random vector with mean $\mu\in\mathbb R^k$ and variance-covariance matrix $V\in\mathbb R^{k\times k}$. For any integrable random vector $X$, we let $E_P[X]$ denote its expectation with respect to probability measure $P$.

The remainder of the paper is organized as follows. Section (ref) introduces the model and provides illustrative examples. In Section (ref), after reviewing the theory of Huber:1973aa (Section (ref)), we present our main result on minimax tests in repeated experiments (Section (ref)). We also discuss the robust testability of the hypotheses and computational aspects (Sections (ref)--(ref)). Section (ref) provides results on the local asymptotic power of the tests. Section (ref) then provides simulation evidence. Section (ref) contains extensions of the baseline framework and Section (ref) concludes. Appendices (ref) and (ref) collect the proofs of the theoretical results.

Setup

Let $S$ be a finite set of observable outcomes and let $u\in U$ denote a variable unobservable to the researcher, where $U$ is assumed to be a Polish space. Let $\Theta$ denote the parameter space. We let $m=\{m_\theta,\theta\in\Theta\}$ denote a family of Borel probability measures on $U$. For each $\theta\in\Theta$, let $G(\cdot|\theta):U\twoheadrightarrow S$ be a weakly measurable correspondence. This map shows how latent variable $u$ is mapped to a set of permissible outcomes. Observable outcome $s$ is then a measurable selection of random set $G(u|\theta)$. As such, the model does not impose any restrictions on how $s$ is selected. One may also introduce observable covariates to this model. As the core analysis remains unaffected, we defer the analysis of this case to Section (ref).

The incomplete structure above is summarized by tuple $(S,U,m,\Theta;G)$. Such structures arise in various economic models. To fix the ideas, we present several examples based on simplifications of well-known models. The first example is a binary response game, which is commonly used to analyze environments such as firms' entry into markets and households' joint labor supply decisions bresnahan1990entry,bresnahan1991empirical,berry1992estimation,Ciliberto:2009aa.

The second example is the (binary) Roy model studied in Mourifie:2018aa.

The third example is an incomplete model of an English auction Haile:2003to.

Set of Permitted Distributions and Robustness

To develop tests for incomplete models, we start by defining the family of probability distributions compatible with the model structure. For each $\theta\in\Theta$, define

where $P_u$ is a conditional law of $s$ (supported on $G(u|\theta)$), which represents the unknown {\em selection mechanism\/}. This set collects probability distributions $P$, for which one can find a suitable selection mechanism and make it consistent with a given parameter value $\theta$ and the model structure. Economic theory rarely provides any guidance on selection. The researcher therefore views any distribution in $\mathcal P_\theta$ as consistent with $\theta$.

Within this model, consider testing parameter value $\theta_0$ against another value $\theta_1$ on the basis of observed outcome $s\in S.$ This is equivalent to testing the null hypothesis, $P\in\mathcal{P}_{\theta_0}$, against the alternative hypothesis, $P\in\mathcal{P}_{\theta_1}$. Note that $\mathcal P_{\theta_0}$ and $\mathcal P_{\theta_1}$ may contain multiple (typically infinitely many) elements because the selection is left unspecified. Therefore, even for testing a single value of $\theta$ against another value, the hypotheses are composite in terms of the permitted distributions.\footnote{The composite nature of the hypotheses arises because the unknown selection is a nuisance parameter. It is possible to allow some components of structural parameter $\theta$ to be additional nuisance parameters. We analyze this extension in Section (ref).} Given this challenge, we pursue a robust approach to inference. That is, we construct tests that (i) control the size uniformly across distributions permitted under the null and (ii) maximize certain measures of power under the alternative.

Robust Tests for Incomplete Models

We provide the main theoretical results below. For this, we start with preliminaries including an introduction of the key technical tools and the important extension of the Neyman--Pearson lemma presented by Huber:1973aa. We then discuss the computational aspects of our LR tests, novel results on minimax tests, and LFPs in repeated experiments as well as a local asymptotic power analysis, which builds on our main theorem (Theorem (ref)).

Preliminaries

Belief Functions

For any $\mathcal P\subseteq\Delta(S)$, define the upper and lower probabilities of $\mathcal P$ pointwise by $\nu^{*}(A)\equiv \sup_{P \in \mathcal{P} } P(A)$ and $\nu(A)\equiv\inf_{P \in \mathcal{P}}P(A),A\subset S$, respectively. These functions are conjugate to each other in the sense that $\nu^{*}(A)=1-\nu(A^{c})$ for any $A\subset S$. Under mild restrictions on $\mathcal P$, they define set functions called {\em capacities\/}.\footnote{Appendix A provides the details. Some authors distinguish a capacity from its conjugate (co-capacity). For simplicity, we call both of these “capacities” throughout.}

For each $\theta\in\Theta$ and $A\subset S$, define $\nu_\theta$ and $\nu_\theta^*$ as the lower and upper probabilities of $\mathcal P_\theta$ defined in (ref):

The key factor for our analysis is that the lower probability $\nu_\theta$ of $\mathcal P_\theta$ is a {\em belief function\/} (or infinitely monotone capacity).\footnote{The infinite monotonicity of $\nu_\theta$ follows from philippe1999decision (Theorem 3). The foundations of belief functions are given by dempster1967upper and shafer1982belief. See Gul:2014aa and epstein2015exchangeable for the axiomatic foundations of the use of belief functions in incomplete models.} From Choquet's theorem Choquet1954,philippe1999decision,Molchanov:2006aa, it is related to the probability distribution of random set $G(u|\theta)$ as follows:

This representation allows us to obtain $\nu_\theta$ without explicitly solving the minimization (or maximization) in (ref) by computing the right-hand side of (ref) directly. Another key property of the belief function is that $P\in\mathcal P_\theta$ is equivalent to the following statement:

galichon2011set used the restrictions above to characterize the smallest possible (or “sharp”) identification region of the parameters.\footnote{galichon2011set used the conjugate of $\nu_\theta$, which yields equivalent identifying restrictions.} Following the literature, we call these the {\em sharp identifying restrictions\/} beresteanu2011sharp,Chesher:2014aa.\footnote{While the restrictions play a role in constructing robust tests, the sharp identified set does not play a role as the latter is an object of interest when the sampling processes reveals the unique data generating process in the limit, which is not guaranteed in our setting. See Epstein:2016qv for a discussion.}

Theory of Huber:1973aa

Our starting point is an analog of the Neyman--Pearson framework, which builds upon HS. For $\theta_0, \theta_1 \in \Theta$ such that $\mathcal P_{\theta_0}$ and $\mathcal P_{\theta_1}$ are disjoint, consider testing a simple null hypothesis, $H_0: \theta=\theta_0$, against a simple alternative hypothesis, $H_1:\theta=\theta_1$. In {\em complete\/} models, in which $G$ is singleton-valued, a well-defined reduced form induces a unique likelihood function tamer2003incomplete. In such settings, an optimal test is an LR test, as is well known from the Neyman--Pearson lemma. In incomplete models, however, the model generally admits a (non-singleton) set $\mathcal P_\theta$ of likelihoods, which prevents us from directly applying the Neyman--Pearson lemma.

In this setting, it is useful to consider {\em minimax tests\/} Lehmann:2006aa. Let $\phi: S \mapsto [0,1]$ denote a (possibly randomized) test. For each $P$ on $(S,\Sigma_S)$, the rejection probability of $\phi$ is

Let $\pi_{\theta_1}(\phi)\equiv\inf_{P_1 \in \mathcal{P}_{\theta_1}}E_P [\phi(s)]$ be the {\em lower power\/} of $\phi$ under $\theta_1$. This is the power value certain to be obtained regardless of the unknown selection mechanism. We then call test $\phi$ a {\em level-$\alpha$ minimax test\/} if it satisfies the following conditions:

and

The condition in (ref) imposes a uniform size control requirement. In (ref), tests are ranked in terms of their lower power. This reflects the researcher's preference for tests that exhibit robust power performance across selections.

A belief function (and its conjugate) is a special case of two-monotone (and two-alternating) capacities whose properties have proven powerful for conducting robust inference Huber:1981aa.\footnote{Capacity $\nu$ is said to be {\em monotone of order $k$\/} or, for short, {\em k-monotone\/} if for any $A_i\subset S,i=1\cdots,k$,

Conjugate $\nu^*(A)=1-\nu(A^c)$ is then called a {\em $k$-alternating\/} capacity.} For a class of models whose lower probabilities are two-monotone, HS showed that the rejection region of a minimax test takes the form $\{s:\Lambda(s)>t\}$ for a measurable function, $\Lambda:S\to\mathbb R$, which they called the Radon--Nikodym derivative of $\nu^*_{\theta_1}$ with respect to $\nu^*_{\theta_0}$. Further, they showed that there exists an LFP of distributions $(Q_0,Q_1)\in\mathcal{P}_{\theta_0}\times\mathcal{P}_{\theta_1}$ such that for all $t\in\mathbb{R}_+$,

and

where $\Lambda$ can be taken to be a version of the Radon--Nikodym derivative:

where $\upsilon$ is a measure that dominates $Q_j,j=0,1$. Below, we take $\upsilon$ to be the counting measure.

Heuristically, this means that $Q_0$ is the probability distribution consistent with the null parameter value, under which the size of the test is maximal. Similarly, $Q_1$ is the distribution consistent with the alternative parameter value, which is the least favorable for power maximization. The following extension of the classic Neyman--Pearson lemma (tailored to our setting) then follows from HS.

Lemma (ref) characterizes a level-$\alpha$ minimax test as an LR test in which the ratio is formed by the LFP. Recall that $Q_1$ is the least favorable for maximizing the test's power, while $Q_0$ is the least favorable for controlling the size. Heuristically, a large value of their ratio can then be taken as evidence against the null hypothesis. Lemma (ref) states that it is indeed optimal in the minimax sense to reject $H_0$ when this ratio is sufficiently high.\footnote{In addition, the binary experiment $(S,\Sigma_S,P\in\{Q_0,Q_1\})$ in which one tests $Q_0$ against $Q_1$ is the hardest (or least informative) in terms of Bayes risk among all binary experiments such that $(S,\Sigma_S,P\in\{P_0,P_1\})$ with $P_j\in\mathcal P_{\theta_j},j=0,1$ Bednarski1982.}

Lemma (ref) is an existence and characterization result useful for obtaining the more general results below. To implement LR tests in practice, one needs to compute the LFPs (typically in a single experiment). We discuss the computational aspects in Section (ref).

Testability of Hypotheses

Before proceeding further, we comment on the testability of the hypotheses. The theory of HS requires that $\mathcal P_{\theta_0}$ and $\mathcal P_{\theta_1}$ are disjoint. Otherwise, any test is vacuous from the minimax viewpoint because probability distribution $P\in \mathcal P_{\theta_0}\cap \mathcal P_{\theta_1}$ is consistent with both hypotheses. If this is the case, we say $\theta_1$ is not {\em robustly testable\/} relative to $\theta_0$ because the lower power of any level-$\alpha$ test cannot exceed $\alpha$. This issue does not arise in complete models as long as the likelihood function satisfies $f(s;\theta_0)\ne f(s;\theta_1),a.s.$ for any $\theta_0\ne\theta_1$. One should therefore expect non-trivial lower power only if an alternative hypothesis induces set $\mathcal P_{\theta_1}$ that does not intersect with $\mathcal P_{\theta_0}$. For this, there needs to be an event $\bar A\subset S$ such that $\nu^*_{\theta_0}(\bar A)<\nu_{\theta_1}(\bar A)$ (or $\nu^*_{\theta_1}(\bar A)<\nu_{\theta_0}(\bar A)$).\footnote{In Example (ref), $\bar A=\{(1,1)\}$ (or $\bar A=\{(1,0),(0,1)\}$) constitutes such an event for testing $H_0:\theta=0$ against $H_1:\theta=\theta_1$ with $\theta_1<0$ when $u$ is continuously distributed over $\mathbb R^2$.}

The lack of robust testability is also related to the notion of {\em observational equivalence\/} Chesher:2014aa. Let $s$ follow distribution $P$ and suppose $P$ is known. Consider parameter values $\theta,\theta'\in\Theta$ such that $\theta\ne\theta'$, $P\in \mathcal P_{\theta}$, and $P\in \mathcal P_{\theta'}$. In other words, the true distribution can be justified by structure $\theta$ augmented with some selection or by another structure $\theta'$ (again augmented with some selection). When this holds, $\theta$ and $\theta'$ are said to be observationally equivalent with respect to $P$. In incomplete models, $P$ is not in general identifiable, as the sampling process does not necessarily reveal it even asymptotically Maccheroni:2005aa,Epstein:2016qv. Following Chesher:2014aa, we say that $\theta$ and $\theta'$ are {\em potentially observationally equivalent\/} if two structures are observationally equivalent for some $P$. Clearly, any pair of potentially observationally equivalent parameter values are not robustly testable, as $\mathcal P_{\theta}$ and $\mathcal P_{\theta'}$ share a distribution in common. This feature of the model raises a challenge for analyzing the local power of the tests because some local alternatives may not be robustly testable. Evaluating the power of the tests under such alternatives does not lead to a meaningful comparison. We therefore introduce a suitably modified notion of local alternatives if such an issue arises (see Section (ref)).

Computing LFPs

A key step toward implementing our tests is the computation of the LFPs, in which the sharp identifying restrictions play a role. Let $H:[0,1]\to\mathbb R$ be a twice-continuously differentiable convex function. Our proposal is to find the LFP through the following characterization:

where the constraints on $(P_0,P_1)$ are the sharp identifying restrictions.\footnote{An alternative approach would be to use the sharp identifying restrictions of beresteanu2011sharp, which also yield a finite number of linear restrictions. While we do not pursue that here, the insights presented in this paper may be useful for constructing optimal tests in models with endogeneity. Such models are studied by Chesher:2014aa, who obtained sharp identifying restrictions using generalized instrumental variables.} The number of restrictions can be reduced further by restricting the class of events to the {\em core determining class\/} galichon2011set,Luo:2017aa. This is a convex program with a convex objective function and linear constraints.\footnote{The convexity of the objective function follows from the convexity of the perspective $g(x,t)= tH(x/t)$ on its domain Boyd:2004jk.}

Our proposal builds on Theorem 6.1 in HS, which characterizes the LFP as a solution to a more general and abstract optimization problem in which $(P_0,P_1)$ is constrained to $\mathcal P_{\theta_0}\times\mathcal P_{\theta_1}$. However, because of the presence of an unknown selection in the definition of $\mathcal P_\theta$ (see (ref)), directly imposing such constraints does not lead to a tractable program. Restating the constraints using the sharp identifying restrictions, we may reduce the problem to a convex one with linear constraints, which can then be solved using efficient algorithms Boyd:2004jk. Computing $\nu_{\theta_0}$ and $\nu_{\theta_1}$ in the constraints is often straightforward (see Example (ref) and the supplementary material of Epstein:2016qv).

Emphasizing the role of the sharp identifying restrictions is worthwhile. Instead of using them to characterize the set of identifiable parameter values, we use them to obtain the LFP. To the best of our knowledge, this way of using the sharp identifying restrictions is new. Further, imposing only a subset of them in (ref) does not generally yield an LFP. In this sense, these restrictions are crucial for robust and optimal inference.

We illustrate the computation of an LFP and minimax test using Example (ref).

\setcounter{example}{0}

Minimax Tests in Repeated Experiments

The sequence of outcomes $s^n=(s_1,s_2,\cdots,s_n)$ is commonly generated from repeated experiments. In this section, we present a set of theoretical results that characterize a minimax test in such a setting and provide an asymptotic Gaussian approximation to its (upper) rejection probability.

For any set $B$, let $B^n$ denote the $n$-fold Cartesian product of $B$. For each $n\in\mathbb N$, let $S^n$ and $U^n$ be the sets of outcome sequences $s^n=(s_1,s_2,\cdots,s_n)$ and latent variable sequences $u^n=(u_1,\cdots,u_n)$, respectively. Below, we use the abbreviation $m^n$ to denote the family $\{m^n_{\theta}\}_{\theta \in \Theta}$ of $u^n$'s joint laws permitted by the model. We then make the following assumption on each member of this family.

This assumption requires that $u_i$'s are distributed independently across experiments. A leading case is that $(u_1,\dots,u_n)$ are i.i.d. This can also accommodate heteroskedasticity and other types of heterogeneity across cross-sectional units or clusters of them (e.g., group-specific effects).

Without further assumptions, $s^n$ takes values in the Cartesian product of the sets of permissible outcome values:

where $G(\cdot|\theta)$ is given as in (ref).\footnote{It is possible to allow the functional form of $G(u_i|\theta)$ to vary across $i$ as well. For notational simplicity, we do not explicitly consider this extension here. We introduce the heterogeneity of $G$ due to covariates in Section (ref).} This set collects outcome sequences that are compatible with the model and $\theta$. We represent the repeated experiments by the tuple $(S^n,U^n,\Theta, G^n;m^n).$ Although $u^n$ is assumed to be independent, the outcome sequence $s^n$ can be dependent because the model does not restrict the selection mechanism. Similarly, even if one makes the stronger assumption that the $u_i$'s are i.i.d., the distribution of outcome $s_i$ may be heterogeneous because of the potential heterogeneity of selection across experiments. This feature arises because the joint selection mechanism (across all experiments) is left unspecified and the joint distribution of the outcome sequence depends on this incidental parameter.

For each $\theta\in\Theta$ and $n\in \mathbb N$, the set of distributions compatible with the model is

This set collects all the distributions of $s^n$ consistent with $\theta$. $P_u$ is unrestricted in the sense that the selection may be heterogeneous and dependent across experiments. Hence, $\mathcal P^n_\theta$ contains a broad range of distributions that can exhibit arbitrary dependence and heterogeneity. In particular, $\mathcal P^n_\theta$ allows measures under which the distributions of sample moments are not well approximated by classical limit theorems---even in large samples (see Epstein:2016qv).

Finding the LFP in such a rich set of distributions may be challenging. However, under Assumption (ref) and with the correspondence in (ref), the model has a tractable “product” structure, which significantly simplifies the characterization of the LFPs.

Let $\nu_\theta^{n,*}$ and $\nu_\theta^n$ denote the upper and lower probabilities of $\mathcal P_\theta^n$. For each $i\in\{1,\dots,n\}$ and $\theta\in\Theta$, let

where $m_{\theta,i}$ is the $i$-th marginal distribution of $m_\theta^n.$ The following theorem shows that the minimax test in repeated experiments is an LR test and that the LFPs are product measures.

The LFP consists of the product measures.\footnote{This result does not follow from Corollary 4.2 of HS, who assumed that a sample is independently distributed (p. 258).} Heuristically, this means that either for controlling size or maximizing power, the least favorable distribution in $\mathcal P^n_{\theta_0}$ (or $\mathcal P^n_{\theta_1}$) is a law that multiplies up the least favorable distributions in the individual experiments. When the $u_i$'s are i.i.d., this characterization has a particularly useful implication for the implementation. To construct a minimax test, it suffices to find the LFP $(Q_0,Q_1)\in\mathcal P_{\theta_0}\times\mathcal P_{\theta_1}$ in a {\em single\/} experiment $(S,U,\Theta, G,m)$. One may then obtain the LR statistic by taking the product of their ratios across experiments. We state this result as a corollary.

The LFP consisting of the product measures in Theorem (ref) provides an important link through which we may connect the incomplete model to standard frameworks. Below, we demonstrate this by studying the large-sample approximations and asymptotic local power of the tests. While both issues can be analyzed without assuming identically distributed latent variables, we maintain this assumption to simplify the notation and our analysis.\footnote{If $u_i$ is not identically distributed, one should invoke a central limit theorem for independent and not identically distributed (i.n.i.d.) sequences under $Q_0^n$ White:2001aa to obtian a Gaussian approximation. Similarly, for local approximations of experiments with an i.n.i.d. sequence, Rieder_1994 (Section 2.3) provides a general framework, which can be applied to the sequence $\{\mathsf Q^n_{\theta_{n,\xi,h}}\}$ defined below.}

Gaussian Approximation

One of the consequences of the product structure is that the upper rejection probability of $\phi_n$ admits a Gaussian approximation in large samples. For ease of exposition, we assume that the $u_i$'s are i.i.d., which in turn implies that $Q_0^n$ is an i.i.d. law from Corollary (ref). Hence, properly normalized sample averages follow classical limit theorems under this law. We use this insight to obtain an asymptotically valid critical value. Since $Q_0^n$ is the least favorable, the asymptotic size is controlled under any distribution under the null hypothesis.

Let $z_\alpha$ be the $1-\alpha$ quantile of the standard normal distribution and let $\Lambda\in dQ_1/dQ_0$. For each $n$, let

where $\mu_{Q_0}\equiv E_{Q_0}[\ln \Lambda(s)],$ and $\sigma^2_{Q_0}\equiv\text{Var}_{Q_0}(\ln\Lambda(s))$. Observe that $\mu_{Q_0}$ and $\sigma_{Q_0}$ depend only on the LFP but not on the unknown DGP. Once the LFP is found, computing $\mu_{Q_0}$ and $\sigma_{Q_0}$ is straightforward because $Q_0$ is a discrete distribution and $\Lambda$ is known. The critical value in (ref) is constructed in such a way that the following convergence holds:

where $Z$ is a standard normal random variable. This critical value is computed without any resampling or simulation and therefore can be done so easily. Despite its simplicity, it has the advantage of being asymptotically valid even if the true DGP is highly heterogeneous and dependent.

Let $\phi^*_n$ be a test that rejects the null hypothesis if and only if $\Lambda_n>C^*_n$. The following proposition then follows.

Asymptotic Local Power

Building on Theorem (ref), we analyze the asymptotic local power of the tests. In what follows, suppose that $\Theta$ is a subset of Euclidean space $\mathbb R^d$ and let $\varphi:\Theta\to\mathbb R$ be a continuously differentiable function with gradient $\dot\varphi_{\theta}:\mathbb R^d\to\mathbb R$. Consider the following hypotheses:

Various hypotheses of empirical interest can be formulated in this way. Our goal here is to characterize the upper envelope of the lower power of the tests for (ref) in an asymptotic framework. Theorem (ref) serves as a building block for this purpose, as it allows us to embed our problem into a more standard one. In this section, we assume that $u_i,i=1,\dots,n$ are i.i.d. throughout.

We consider localized experiments. Let $\theta_0\in\Theta$ be a parameter such that $\varphi(\theta_0)=0$ and let $\{\theta_{n,\xi,h}\},(\xi,h)\in\mathbb R^d\times \mathbb R^d$ be a sequence of alternative parameter values, which we specify below. We call $\xi$ a {\em fixed shift\/} and $h$ a {\em local parameter\/}. The sequence of parameters induces a sequence of belief functions $\nu^n_{\theta_{n,\xi,h}}$. Suppose that for each $n$ and $(\xi,h)$, the conditions of Theorem 3.1 hold for $\nu^n_{\theta_0}$ and $\nu^n_{\theta_{n,\xi,h}}$. Then, there exists an LFP $(Q^n_0,Q^n_1)\in \mathcal P^n_{\theta_0}\times \mathcal P^n_{\theta_{n,\xi,h}}$. If there exists model $\theta\mapsto \mathsf Q_\theta$ indexed by $\theta$ defined on a neighborhood of $\theta_0$ such that $Q_0^n=\mathsf Q^n_{\theta_0}$ and $Q_1^n=\mathsf Q^n_{\theta_{n,\xi,h}}$ for all $n$, one may consider the following sequence of experiments:

In this hypothetical environment, observations are generated using a sequence of probability laws that are the least favorable for testing $\theta_0$ against $\theta_{n,\xi,h}$. The fact that the experiments are characterized by probabilities instead of capacities allows us to employ standard asymptotic tools. In particular, we employ the {\em limits of experiments\/} argument in the style of Le-Cam:1972aa,LeCam:1986aa. Heuristically, if one wants to obtain an asymptotic upper envelope of $\pi_{\theta_{n,\xi,h}}$, one may consider the least favorable sequence of DGPs $\{\mathsf Q^n_{\theta_{n,\xi,h}}\}$ for power maximization. It turns out that the lower power of any test can be matched by a power function of a limit experiment, which is often more straightforward to analyze. The limit experiment can then be used to derive an asymptotic power envelope.

While the argument above suggests that we may use the standard limits of experiments framework, a few non-standard features arise. First, the underlying model may not satisfy the differentiability in quadratic mean condition, which is sufficient for the LAN of the experiments. Instead, the model is typically directionally differentiable (in the $L^2$ sense) and satisfies the LAN property separately on a finite number of convex cones that partition the local parameter space, which requires us to consider sub-experiments of (ref). Second, some alternatives are not robustly testable. Hence, to conduct a meaningful power analysis, one needs to construct local alternatives with care.\footnote{We modify the definition of the local alternative so that the sequences of LFPs $\{Q^n_{\theta_0}\}$ and $\{Q^n_{\theta_{n,\xi,h}}\}$ are contiguous.}

Asymptotic Power against Robustly Testable Local Alternatives

Recall that in Example (ref), setting the strategic interaction effects to $\theta_1<0$ made $\mathcal P_{\theta_0}$ and $\mathcal P_{\theta_{1}}$ disjoint no matter how small the deviation from the null $\theta_0=0$ was. Below, we start with a relatively simple setting in which $H_0:\theta=\theta_0$ and $H_1:\theta=\theta_0+h/\sqrt n$ induce sets of distributions that are disjoint for any $n$. In this setting, it suffices to index local alternatives using $h$ only, and hence we use $\theta_{n,h}$ instead of $\theta_{n,\xi,h}$.\footnote{We elaborate on the role of $\xi$ in the next section.}

We define the notion of differentiability below. For this, let $L^2_{P}$ denote the set of square integrable functions on $S$ with respect to measure $P$. For each $f,g\in L^2_P$, we then let $\langle f,g\rangle_{L^2_P}$ and $\|f\|_{L^2_P}$ denote the $L^2$ inner product and $L^2$ norm, respectively. Consider a parametric family of distributions $\{P_\theta,\theta\in V\}$, where $V$ is an open subset of $\Theta.$

This is the $L^2$ differentiability of the square root density commonly used in the literature. If one could embed the LFPs into an $L^2$ differentiable model $\theta\mapsto\mathsf Q_\theta$ with $\mathcal T$ being a suitable limit of the local parameter space $\{h\in \sqrt n(\Theta-\theta_0):\varphi(\theta_0+h/\sqrt n)>0\}$, asymptotic local power can be analyzed in a standard way. However, as we show below, the model is often only directionally differentiable. That is, the form of the derivative, $\dot\ell_\theta$, varies across the subsets of $\mathcal T$ (see the discussions below). The following high-level assumption states this formally.\footnote{For the results that follow, it suffices that a model satisfies Assumption (ref) for an LFP. The LFPs are unique up to the Radon--Nikodym derivative (HS, 1973), and thus they all lead to the same quadratic expansion of the log-LR process. While one could alternatively take (ref) as a high-level condition, we do not do so because Assumption (ref) is often easier to check. A similar comment applies to Assumption (ref).} For this, let $\stackrel{P^n}{\leadsto}$ denote weak convergence under the sequence $\{P^n\}$ of distributions. Let $\mathbb C(0,\epsilon)$ denote an open cube centered on the origin with edges of length $2\epsilon.$ A set $\Gamma\subset\mathbb R^d$ is said to be locally equal to set $\Upsilon\subset\mathbb R^d$ if $\Gamma\cap \mathbb C(0,\epsilon)=\Upsilon\cap \mathbb C(0,\epsilon)$ for some $\epsilon>0$ Andrews:1999aa.

The assumption above imposes a regularity condition on the LFP for testing $H_0:\theta=\theta_0$ against $H_1:\theta=\theta_{0}+\tau h$ for $\tau>0$. While $\theta_0$ is fixed, both the least favorable distributions (under $H_0$ and $H_1$) may depend on deviation $\tau h$. Hence, we index the LFP using $\tau h$. The restrictions in (ref) require that the least favorable distribution $Q_{0,j,\tau h}$ under $H_0$ remains the same for all sufficiently small $\tau$, and this is given by point $\mathsf Q_{j,\theta_0}$ in the $L^2$ differentiable model $\theta\mapsto \mathsf Q_{j,\theta}$. As we demonstrate below through examples, Assumption (ref) (and similarly Assumption (ref)) can be checked by analyzing the LFP. In all our examples, the cardinality of $\mathbb J$ is finite. Appendix (ref) also provides the primitive conditions that ensure the key condition ($L^2$ differentiability) (see Assumption (ref), Proposition (ref), and Corollary (ref)).

Below, we let $\dot \ell_{j,\theta_0}$ denote the $L^2$ derivative defined for $h\in\mathcal T_{j}(\theta_0)$. Under Assumption (ref), one may expand the log-LR for every $h\in \mathcal T_{j}(\theta_0)$ as follows:

where $\Delta_{j,n}\stackrel{\mathsf Q^n_{j,\theta_0}}{\leadsto} \Delta_j\sim N(0,C_{j})$ and $C_j=E_{\mathsf Q_{j,\theta_0}}[\dot\ell_{j,\theta_0}\dot\ell_{j,\theta_0}'].$ In what follows, we call $\Delta_{j,n}$ the central sequence (or normalized score) and $C_{j}$ the information matrix. Consider the following sub-experiments:

When Assumption (ref) holds, for each $j$, the limit of the experiments (as $n\to\infty$) is

which is also equivalent to $(\mathbb R^d,\Sigma_{\mathbb R^d},N( h,C_{j}^{-1}): h\in \mathcal T_{j}(\theta_0))$ if $C_j$ is non-singular Van-der-Vaart:2000aa. In other words, the experiment is equivalent to the one in which the researcher observes a single normal random vector whose mean and variance are $h\in \mathcal T_{j}(\theta_0)$ and $C_j^{-1}$, respectively. The asymptotic local (lower) power of a test is then bounded from above by the corresponding power in the limit experiment. The power envelope can be derived by considering the highest possible power for testing $H_0:\dot\varphi_{\theta_0} h\le 0$ against $H_1:\dot\varphi_{\theta_0} h>0$ at level-$\alpha.$

As is well known, these limit experiments are Gaussian shift experiments defined on suitable subsets of $\mathbb R^d$. If Assumption (ref) holds with $\mathbb J=\{ \textup{\uppercase\expandafter{\romannumeral1}} \}$ and $\mathcal T_{ \textup{\uppercase\expandafter{\romannumeral1}} ,\theta_0}=\mathbb R^d$, we obtain the LAN LeCam:1986aa. Assumption (ref) slightly extends the LAN, and this allows us to consider experiments defined separately on the subsets (cones) of the local parameter space. This extension is motivated by the fact that central sequence $\Delta_{j,n}$ and information matrix $C_{j}$ may differ across the local parameter cones. This is because as $h$ varies (and hence $\nu_{\theta_{n,h}}$ varies), the LFP defined through the convex program in (ref) may change in a non-differentiable (but directionally differentiable) way.\footnote{See Shapiro:1988aa, Dempe_Directional_1993, and the references therein for the directional differentiability of solutions to parametric convex programs. Our primitive conditions (see Appendix (ref)) for Assumptions (ref) and (ref) are based on Shapiro:1988aa.} This may lead to distinct central sequence and information matrix pairs across the cones. Owing to this non-standard feature, our results below concern asymptotically optimal statistical decisions when the underlying model may only be directionally differentiable (in the $L^2$ sense). This complements the recent developments on statistical inference and decisions in non-standard models in which the parameters of interest are only directionally differentiable, whereas the underlying model is regular Hirano_Impossibility_2012,Song2014JMA,Fang:2014aa,Fang2018,HongLi2018JoE.

To characterize the power envelope, we define the tangent cone of the score functions and efficient influence function of $\varphi$. For each $j\in\mathbb J$, let

We call the set above the {\em tangent cone\/} of the model. The influence curve $\varrho_j\in L^2_{\mathsf Q_{j,\theta_0}}(S)$ of $\varphi$ is such that, for any $g\in\mathcal G_{j,\theta_0},$

as $\tau\downarrow 0.$ The {\em efficient influence function\/} (or canonical gradient) $\tilde\varrho_j$ of $\varphi$ is then defined as the projection of $\varrho_j$ on the closure of $\mathcal G_{j,\theta_0}$ (which is often called the {\em tangent set\/}).

The following theorem characterizes the asymptotic upper bound of the lower power and provides a test that achieves the bound (for a given cone).

The asymptotic power envelope in (ref) coincides with that for the one-sided test $\dot\varphi_{\theta_0} h\le 0$ against $\dot\varphi_{\theta_0} h>0$ in the Gaussian shift experiment. The theorem also implies that the test based on the (rescaled) efficient influence function achieves the power envelope toward alternatives with $h\in\mathcal T_{j}(\theta_0)$.\footnote{The power envelope can be expressed using $\dot\varphi h$ and $C_j$ as in Theorem 15.4 in Van-der-Vaart:2000aa if $C_j$ is non-singular and $\mathcal G_{j,\theta_0}$ is a linear subspace. The description above is slightly more general to handle cases in which $C_j$ may be singular and $\mathcal G_{j,\theta_0}$ is a convex cone Rieder2014.} Therefore, the key factor is to find the efficient influence function.

We next revisit Example (ref). \setcounter{example}{0}

Asymptotic Power against Shifted Local Alternatives

As discussed earlier, not all alternatives are robustly testable. Consider the following example. \setcounter{example}{1}

Given the challenge above, we conduct a local power analysis as follows. First, we {\em shift\/} $\theta_0$ by vector $\xi$, which does not depend on $n$. Hence, at $\theta_0+\xi$, certain local deviations can be robustly detectable. We then analyze the limit of experiments constructed from a sequence of LFPs induced by such alternatives. Specifically, the {\em shifted local alternative\/} is

where $\theta_0\in\Theta$ is such that $\varphi(\theta_0)= 0$, and $\xi$ and $h$ take values in the following sets:

In other words, $\Xi_{\theta_0}$ collects the deviations for which $\theta_0+\xi$ are not robustly testable. Set $\mathcal T_n(\theta_0,\xi)$ then collects local deviations that make $\theta_{n,\xi,h}$ robustly testable. Among the points in $\Xi_{\theta_0}$, we focus on those for which $\mathcal T_n(\theta_0,\xi)$ is non-empty.

Under this construction, we may apply Theorem (ref) for any $\tau>0$ to obtain the LFP $(Q_{0,j,\tau h},Q_{1,j,\tau h})\in \mathcal P_{\theta_0}\times\mathcal P_{\theta_0+\xi+\tau h}$. Suppose that the following assumption holds for $\theta_0\in\Theta$ and $\xi\in\Xi_{\theta_0}.$

In what follows, we let $\dot \ell_{j,0},j\in\mathbb J$ denote the $L^2$ derivatives and focus on testing the linear hypotheses, namely $\varphi(\theta)=p'\theta-c$. Define the influence curve $\varrho_j$ of $\varphi(\theta)=p'\theta-c$ as a square integrable function $\varrho_j$ that satisfies, for every $h\in\mathcal T_{j}(\theta_0,\xi)$, $p'h=\langle \varrho_j,g\rangle_{L^2_{\mathsf Q_{j,0}}}$, where $g=h'\dot \ell_{j,0}$. Let $\mathcal G_{j,\theta_0}=\{g\in L^2_{\mathsf Q_{j,0}}:g=h'\dot \ell_{j,0},h\in\mathcal T_j(\theta_0,\xi)\}$. Let $\tilde\varrho_j$ be the efficient influence function, which is the projection of $\varrho_j$ to the closure of $\mathcal G_{j,0}.$ We then obtain the following result.

Below, we again use Example (ref) to illustrate the local power analysis. \setcounter{example}{1}

Monte Carlo Experiments

We conduct Monte Carlo experiments to examine the performance of our tests.

Tests of Strategic Interaction Effects

The first set of experiments evaluates the size and power of the tests on the strategic interaction effects. The design of the experiment is based on Example (ref), in which we generate $u_{i}\stackrel{i.i.d.}{\sim}N(0,I_2)$ for $i=1,\cdots,n.$ Whenever multiple equilibria exist, we select an outcome using one of the three selection mechanisms below. The first one is an i.i.d. selection mechanism, which selects $(1,0)$ out of $G(u|\theta)=\{(1,0),(0,1)\}$ if an i.i.d. Bernoulli random variable $v_i$ takes 1. Otherwise, $(0,1)$ is selected. The second mechanism selects $(1,0)$ when another Bernoulli random variable $\tilde v_i$ takes 1, where $\{\tilde v_i\}$ is a i.n.i.d. sequence. Let $N^*_k$ be an increasing sequence of integers.\footnote{In our simulations, we set $N^*_k=2^{2^k}$.} For each $i$, let $h(i)=N^*_k$, where $N^*_{k-1}<i\le N^*_k$. We define

where $\Psi_n^G(u^{\infty})=\frac{\sum_{i=1}^nI[G(u_i|\theta)=\{(1,0)\}]}{\sum_{i=1}^nI[G(u_i|\theta)=\{(1,0)\} or \{(0,1)\}]}$. One may view observations for which $N^*_{k-1}<i\le N^*_k$ as members of a cluster. Under this selection mechanism, the outcomes are dependent on each cluster, which in turn makes the outcome sequence heterogeneous (and non-ergodic). The third mechanism generates data from the LFP, which draws an outcome sequence from $Q_0^n$ when $\theta=\theta_0$ and from $Q^n_1$ when $\theta=\theta_1$.

We evaluate the size and power of the test in Section (ref) on $H_0:p'\theta=0$ against $H_1:p'\theta>0$ with $p=(-1,-1)'.$ The test based on the statistic in (ref) achieves the power envelope (see Appendix (ref)). We therefore evaluate the size and power of this test. To make a comparison, we consider another test, namely the Wald--Wolfowitz runs test, which non-parametrically tests the i.i.d.-ness of the outcome sequence Wald:1940aa,Cho:2011aa. Since the model is complete under $H_0$ and $u_i$ is i.i.d., the resulting outcome sequence is i.i.d. under the null hypothesis. The test therefore should have power only when the selection introduces heterogeneity and/or dependence.

Figures (ref) and (ref) show the power of the optimal test and runs test, respectively. The power of our test changes little across the selection mechanisms. This can be explained as follows. Our test statistic in (ref) treats the two outcomes, $(0,1)$ and $(1,0)$, symmetrically. While the selection mechanism affects the relative frequencies of the two outcomes, what matters for the statistic is the frequency of $\{(0,1),(1,0)\}$ (relative to $(1,1)$), and hence its power curve is insensitive to the selection mechanism.\footnote{This insensitivity is not a generic feature of the optimal test. See the next example.} The Wald--Wolfowitz test has non-zero power when the selection is i.n.i.d.; it becomes noticeable only when $h$ is very large and its lower power is significantly below that of the optimal test. As expected, it does not have any power when the selection mechanism is i.i.d. or the LFP.

Tests on the Distribution of Potential Outcomes

The second set of experiments is based on Example (ref), which we use to evaluate the performance of the tests when the model is incomplete under $H_0$. The model may be complete under certain alternatives. As described in Section (ref), we consider testing $H_0:p'\theta\le c$ against $H_1:p'\theta>c$, where $p=(0,0,1)'$ and $c=1/6$. As before, we assume that $\theta^{(0,0)}=1/6$ is known and localize the experiment at $\theta_0=(\theta_0^{(0,0)},\theta_0^{(0,1)},\theta_0^{(1,0)})'=(1/6,1/2,1/6)'.$

The set of selection mechanisms is the same as the one in Section (ref). One difference is that the model predicts multiple outcomes when $u=(0,0)$ or $u=(1,1)$. In both cases, an i.i.d. selection mechanism selects one of the outcomes ($s=(0,0)$ when $G(u)=\{(0,0),(0,1)\}$ and $s=(1,0)$ when $G(u)=\{(1,0),(1,1)\}$) when an i.i.d. Bernoulli random variable $v_i$ is 1. Similarly, an i.n.i.d. selection mechanism selects one of the predicted outcomes when $\tilde v_i$ in (ref) is 1. The LFP selection mechanism is defined in the same way as before.

We consider the following shifted local alternatives:

Under Alternative A, we consider a sequence of the parameters that tends to $\theta_0+\xi_A$, where $h\in\mathcal T_{ \textup{\uppercase\expandafter{\romannumeral1}} }(\theta_0,\xi_A)$. Under Alternatives B-I and B-II, we consider the parameters that tend to $\theta_0+\xi_B$, where $h\in\mathcal T_{ \textup{\uppercase\expandafter{\romannumeral1}} }(\theta_0,\xi_B)$ and $h\in\mathcal T_{ \textup{\uppercase\expandafter{\romannumeral2}} }(\theta_0,\xi_B)$, respectively.

Figures (ref) and (ref) report the results with $n=1000$ and $S=2000$, respectively. The right panel of Figure (ref) shows the power curves of the optimal test against Alternative A. Because of the model incompleteness, its performance varies significantly across the selection mechanisms. As predicted by the theory, the power of the test under the LFP selection mechanism essentially coincides with the power envelope. Both the i.i.d. and the i.n.i.d. selection mechanisms are considerably more favorable; that is, the actual power of the test under these mechanisms is much higher than under the LFP selection. In particular, the power of the test under the i.i.d. selection mechanism is already 1 even when $\bar h=0$ because the test can detect deviations from the null under this selection mechanism even for $\theta=\theta_0+\xi$ that are not robustly testable. The left panel of Figure (ref) shows the power curves against alternatives of the form $\theta_0+w\xi_A$ with $w\in[0,1]$. The figure shows that the test may have non-trivial power even for such alternatives when the selection mechanisms are i.i.d. or i.n.i.d.\footnote{Under the LFP selection, the data are drawn from $Q_{\theta_0}\in\mathcal P_{\theta_0+w\xi_A}$, and hence the power of the test does not exceed the nominal level.}

Figure (ref) shows the power curves against Alternative B-I. When $\bar h=0$, the model is indeed complete.\footnote{The model remains incomplete regarding selection from $G(u)=\{(0,0),(0,1)\}$ when $u=(0,0)$, however, these outcomes are not used by the test and hence do not affect the power.} However, as $\bar h$ increases, the region of incompleteness enlarges, leading to differences in the local power across the selection mechanisms. Under Alternative B-II, the model stays complete under the local alternatives. Therefore, the power of the test is essentially the same across the selection mechanisms.

Extensions

Tests in the Presence of Nuisance Components

We now consider testing the hypotheses on subcomponents of $\theta$. Let $\theta=(\beta',\delta')'\in \Theta_\beta\times\Theta_\delta$, where $\beta$ is a $k\times 1$-sub-vector of interest and $\delta$ is a $(d-k)\times 1$ vector of nuisance parameters. Consider the following hypotheses:

This problem can be recast as a special case of (ref) with $\varphi(\theta)=\beta$, $K_0=\{\beta_0\}$, and $K_1=\{\beta\in\mathbb R^k:\beta\ne\beta_0\}.$\footnote{Sub-vector inference has been actively studied in the context of partially identified models, particularly moment inequality models romano2008inference,Shicanaybugni2015inference,Kaido:2016aa. Here, we focus on hypothesis tests on the sub-vectors of the structural parameters.}

In this general setting, both hypotheses are composite in terms of the structural parameters. Therefore, Lemma (ref) is not directly applicable. However, the result is still useful for constructing tests that have desirable properties. To this end, we partition the parameter space into two sets, namely $\Theta_0$ and $\Theta_1$, where $\Theta_0=\{\beta_0\}\times \Theta_\delta$ and $\Theta_1=\{\beta:\beta\ne\beta_0\}\times\Theta_\delta$. We focus on this setting, whereas the theory below applies more generally to the hypotheses of the form in (ref) by taking $\Theta_0=\{\theta:\varphi(\theta)\in K_0\}$ and $\Theta_1=\{\theta:\varphi(\theta)\in K_1\}$.

Throughout, the researcher's action is binary, that is $a=1$ (reject) or $a=0$ (accept). For each $\theta\in\Theta$ and action $a\in\{0,1\}$, define a loss function $\mathsf L:\Theta\times \{0,1\}\to\mathbb R_+$ by

where $\zeta>0$. The loss from the Type-I error is normalized to 1. The trade-off between the Type-I and Type-II errors is determined by parameter $\zeta$.

For each test $\phi$ and $\theta\in\Theta$, define the {\em upper risk\/} by

where the integrals in (ref) are Choquet integrals (see Appendix (ref)). The upper risk determines the trade-off between the size ($R_0(\theta,\phi)\equiv\sup_{P\in\mathcal P_\theta}\int \phi dP=\int\phi d\nu^*_\theta$ for $\theta\in\Theta_0$) and the lower power ($\inf_{P\in\mathcal P_\theta}\int \phi dP= \int\phi d\nu_\theta$ for $\theta\in\Theta_1$). What remains is to incorporate parameter uncertainty. For this, let $\mu$ be a (prior) probability distribution over $\Theta$. We write $\mu$ as $\mu=\tau\mu_0+(1-\tau)\mu_1,$ where $\tau\in (0,1)$ and $\mu_0,\mu_1$ are suitable probability measures supported on $\Theta_0$ and $\Theta_1$, respectively. Define

where $\kappa_0^*=\int_{\Theta_0}\nu_\theta^*d\mu_0$ and $\kappa_1=\int_{\Theta_1}\nu_\theta d\mu_1$.\footnote{The second equality in (ref) is established in the proof of Theorem (ref).} This risk function uses the prior probability to reflect parameter uncertainty, while it uses the belief function (and its conjugate) to incorporate the decision maker's willingness to be robust against incompleteness. In what follows, we call $r$ the {\em Bayes--Dempster--Shafer (BDS) risk\/}. We then call $\phi$ a {\em BDS test\/} if it minimizes the BDS risk.\footnote{The axiomatic foundations for this type of preference (when $S$ is the payoff-relevant state space) is given in Gul:2014aa and epstein2015exchangeable (in the context of repeated experiments).} One of the components of the BDS risk is $\pi_{\kappa_1}(\phi)=\int \phi d\kappa_1$. We call this object the {\em weighted average lower power\/} (WALP). The interpretation of $\pi_{\kappa_1}(\phi)$ is similar to that of the standard weighted average power Andrews:1994aa,Andrews:1995aa. Choosing a suitable $\mu_1$, one may direct the power of a test toward certain alternatives. However, $\pi_{\kappa_1}$ takes the average of the guaranteed power value instead of the actual unknown power.

The following theorem characterizes the BDS test. For this, let $core(\kappa)\equiv \{P\in\Delta(S):P(A)\ge \kappa(A),\forall A\subset S\}$. In what follows, we assume that $core(\kappa_0)\cap core(\kappa_1)=\emptyset$.\footnote{To ensure this condition, it is sufficient to have at least one $\bar A\subset S$ such that $P_0(\bar A)<P_1(\bar A)$ (or $P_0(\bar A)>P_1(\bar A)$) for all $(P_0,P_1)\in \mathcal P_{\theta_0}\times\mathcal P_{\theta_1}$ and $(\theta_0,\theta_1)\in \Theta_0\times\Theta_1$. In Example (ref), one may take $\bar A=\{(1,1)\}$.}

One may view this as an analog of Lemma (ref). A key difference is that the LFP belongs to the product of the cores of the capacities $\kappa_0$ and $\kappa_1$. Hence, $\kappa_0$ and $\kappa_1$ are both belief functions, which in turn allows us to compute the LFP in a tractable way.

The analysis above is useful for constructing optimal tests for minimizing risk. However, the BDS tests are not designed to control size uniformly over $\Theta_0$. Therefore, we consider a test that controls size and maximizes the WALP. For this, we fix $\mu_1$ throughout and follow the developments on the tests in the presence of nuisance parameters Chamberlain:2000aa,Elliott:2015aa,Moreira:2013aa.

The following minimax theorem characterizes the minimax test as a BDS test for the least favorable prior (if it exists). For this, let $\mathcal M(\mu_1)\equiv\{\mu:\mu=\tau\mu_0+(1-\tau)\mu_1,\mu_0\in\Delta(\Theta_0),\tau\in[0,1]\}$, where $\mu_1$ is fixed. In what follows, we drop $\mu_1$ from the argument of $\mathcal M$, but its dependence should be understood. We then let $\mathbf\Phi$ be the set of randomized tests.

Covariates

This section extends the base framework to incorporate observable covariates. Each individual experiment is described by $(S,X,U,G,\Theta;\upsilon,m)$, where $S,U,G,\Theta$ are defined as before. We let $X$ denote the finite set of covariate values and $\{\upsilon_\theta,\theta\in\Theta\}$ be a family of distributions on $X$. Throughout, we assume that each $\upsilon_\theta\in\Delta(X)$ has full support on $X$. Measure $m_\theta(\cdot|x)$ then determines the conditional law of $u$ given $x$. The prediction of the model is then summarized by a weakly measurable correspondence $(u,x)\mapsto F(u,x|\theta)\equiv \{(s,x):s\in G(u|\theta,x) \}\subset S\times X$ for each $\theta\in\Theta$. As before, this correspondence induces a belief function on $S\times X$

If $A$ is a rectangle $A=A_s\times A_x$ for some $A_s\subset S$ and $A_x\subset X$, one may write it as

which can be viewed as the mean of the conditional belief function $\nu_\theta(\cdot|x)$. The subsequent analysis, starting with the Neyman--Pearson lemma, is then essentially the same as before.

A simplification occurs when $\upsilon$ does not depend on $\theta$. Consider $\theta_0,\theta_1\in\Theta$ with $\theta_0\ne\theta_1$. Because of the additivity of $\upsilon$, it suffices to consider sets of the form $B\times \{x\}$, where $B\subset S$ and $x\in X.$ Then, the program that determines the LFP is

Observe that the constraints simplify to

Taking $B=S$, one obtains $\upsilon(x)\le \upsilon_j(x)$ for all $x\in X$ with $j=0,1.$ Since $\upsilon$ is a measure, this implies $\upsilon_0=\upsilon_1=\upsilon,$ and the resulting LR statistic does not depend on $\upsilon$. The LR statistic $dQ_1/dQ_0=q_1(s|x)/q_0(s|x)$ can then be calculated by solving, for each $x$,

Concluding Remarks

This study explored robust likelihood-based inference methods for incomplete economic models. A key result is the existence of an LFP consisting of product measures, through which we may connect incomplete models to standard frameworks and thus obtain asymptotic approximations and analyze optimality properties while remaining agnostic about the selection. However, some problems need further work. They include methods of inference on the subcomponents of $\theta$, especially when the dimension of the parameter is moderately high and a framework that can handle solution concepts involving some form of mixing (e.g., mixed Nash, Bayes correlated equilibria) which are undertaken in ongoing work. Another important avenue for future research is an extension of the current results to statistical inference or decision problems outside hypothesis testing.

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Figures

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Capacities

Let $\Omega$ be a compact metric space and let $\Sigma_\Omega$ denote its Borel $\sigma$-algebra. Let $\mathcal K(\Omega)$ be the set of compact subsets of $\Omega$ endowed with the Hausdorff metric. Let $\mathcal C(\Omega)$ be the set of continuous functions on $\Omega$. Let $\Delta(\Omega)$ be the set of Borel probability measures on $\Omega$ endowed with the weak topology.

A set function $\nu^{*}$ is said to be a {\em capacity\/} if $\nu^{*}$ satisfies the following conditions:

One may define integral operations with respect to capacities as follows. Let $f:\Omega\to\mathbb R$ be a measurable function. The {\em Choquet integral\/} of $f$ with respect to $\nu$ is defined by

where the integrals on the right hand side are Riemann integrals.

The following result due to Choquet follows from Theorems 1-3 in philippe1999decision.

In each experiment characterized by the tuple $(S,U,\Theta,G;m)$, one may apply the lemma above with $\mathcal P=\mathcal P_{\theta}$, $\nu=\nu_\theta$, $K=G(u|\theta)$, and $M$ is the law of $K$ induced by correspondence $G$ and $m_\theta$.

Proofs

Proof of Lemma (ref)

Proof of Theorem (ref), Corollary (ref) and Auxiliary Lemmas

We use Theorem 8.1.1 in Lehmann:2006aa to show Theorem (ref). For ease of reference, we copy their theorem below (with a slight change of notation to avoid conflicts). For this, let $E,E'$ be measurable spaces and let $\mathcal P=\{P_\eta\in \Delta(S):\eta\in E\cup E'\}$ be families of probability distributions on $S$ with densities $p_\eta=dP_\eta/d\upsilon$ parameterized by $\eta\in E\cup E'$. Throughout, we assume that the map $(s,\eta)\mapsto p_\eta(s)$ is jointly measurable.

Proof of Proposition (ref) and Theorems (ref)-(ref)

Proof of Theorems in Section (ref)

In what follows, we repeatedly use the fact that, for any nonnegative measurable function $g$ on $S$, belief function $\nu$ and its conjugate $\nu^*$, one has

where $M_\nu$ is the probability measure on $\mathcal K(S)$ associated with $\nu$ (see Lemma (ref)).

Auxiliary Lemmas

Below, we identify each decision function (randomized test) $\phi$ with a Markov kernel $\phi:S\times \mathcal B_{\{0,1\}}\to [0,1]$ and let $\mathbf\Phi$ be the set of all decision functions. We then equip $\mathbf\Phi$ with the weak topology Hausler:2015aa. The following lemma is an analog of Lemma 46.1 in Strasser:1985aa for the BDS risk. We state it as a lemma because the BDS risk $R$ is defined through Choquet integrals with respect to capacities (instead of measures) and hence Lemma 46.1 in Strasser:1985aa is not directly applicable.\footnote{They also show their results to generalized decision functions, which we do not pursue here.}

Below, a set $M$ is said to be {\em subconvex\/}, if for any $\alpha\in (0,1)$ and $h_1,h_2\in M,$ there exists $h_3\in M$ such that $h_3\le \alpha h_1+(1-\alpha)h_2$.

Examples

Example 1: Binary response game

In this section, we provide details on Example (ref) including the computation of the belief function, least favorable pair, and minimax test. Recall that $S=\{(0,0),(1,1),(1,0),(0,1)\}$. There exist 14 subsets to be considered (without considering the empty set and $S$). One can then compute the the lower and upper bounds of the probability of each event by mimicking the calculation in (ref). The results are summarized in Table (ref).

LFP

The following proposition characterizes the LFP and minimax tests.

Efficient influence function and optimal tests

Example 2: Roy model

In this section, we provide details on Example (ref). Recall that $S=\{(0,0),(0,1),(1,0),(1,1)\}$, and the sharp identifying restrictions are given as (ref)-(ref).

LFP

We start with the following characterization of the LFP for the hypotheses considered in the text. For this, let $\bar c>0$ be a known constant.

Before giving the proof of the claim above, a few remarks are in order. To simplify the analysis, we assume that $\theta^{(0,0)}_0=\theta^{(0,0)}_1$ are known. Recall that the sharp identifying restrictions in (ref)-(ref) imply

The assumption $\theta^{(1,0)}_0<1-\bar c-\theta^{(0,1)}_0$ therefore makes the model incomplete at $\theta_0$. Finally, the assumption $\theta^{(1,0)}_1>1-\bar c-\theta^{(0,1)}_0$ is equivalent to $\nu_{\theta_1}(\{(1,0)\})>\nu_{\theta_0}(\{(1,0)\})$, which ensures that $\mathcal P_{\theta_0}\cap \mathcal P_{\theta_1}\ne\emptyset.$

Efficient influence function, power envelope, and optimal tests

As in Section (ref), we consider testing $H_0:p'\theta=\theta^{(1,0)}\le c$ against $H_1:p'\theta>c$ for some $c\in (0,1)$ with $p=(0,0,1)'.$ Consider the following three configurations analyzed in the text (see also Figure (ref)) with $\bar c=\frac{1}{6}$. In each configuration, the null and alternative parameter values are

where $\xi$ and $h$ are taken from one of the following specifications:

Computing the $L^2$-derivative

In practice, the $L^2$ derivative, a key object for constructing the optimal tests, can be computed by an analytical method or a quadratic-programming method.\footnote{Another possibility is to use numerical differentiation, which is studied by HongLi2018JoE in a related context.} In what follows, let $\theta_1=\theta_0+\xi+\tau h$ with $\tau>0$ and assume that $\mathcal P_{\theta_0}\cap\mathcal P_{\theta_1}=\emptyset.$ A robustly testable alternative is a special case, and the argument below can be applied with $\xi=0.$

Analytical method

Among the two, the analytical method is more straightforward and is recommended whenever possible. Let $h$ be given and assume that Assumption (ref) or (ref) holds. For calculating the $L^2$ derivative, one can take the following steps.

Step 1: For $\tau>0$, derive the LFP $(Q_{0,\tau h},Q_{1,\tau h})$ in closed form by solving (ref). Find a collection of models $\vartheta\mapsto \mathsf Q_{j,\vartheta},j\in\mathbb J$ such that (ref) (or (ref)) holds.

Step 2: For each $j$, calculate $\dot\ell_{j,0}$ using the analytical form of $\mathsf Q_{j,\vartheta}.$

This is the approach taken in the analysis of Examples (ref) and (ref) (see Appendix (ref) and (ref)). Also, in Step 1, one should check if (ref) (or (ref)) is satisfied for $Q_{0,\tau h}$. In Example (ref), it is satisfied because $\mathcal P_{\theta_0}=\{Q_0\}$ is a singleton and by an inspection of the underlying model. In Example 2, the complementary slackness condition determines $Q_{0,\tau h}$ uniquely for all $\tau$ sufficiently small, and again the condition can be checked analytically.

Quadratic-Programming method

Under certain conditions, the $L^2$-derivative of the model can also be calculated as a solution to an auxiliary quadratic programming (QP) problem. Below, we discuss this alternative approach and provide primitive conditions for this method to work.

This approach is based on the stability analysis of a solution to a parametric convex program. Toward this end, we relate the problem in (ref) to a parametric optimization problem studied in Shapiro:1988aa. Let $L$ be the cardinality of $S$ and let $ K$ be the cardinality of $2^S$, the power set of $S$. Let $p_0,p_1\in[0,1]^L$ denote vectors of probability mass functions on $S$. Let $x=(p_0',p_1')'\in [0,1]^{2L}$ denote a vector of control variables, and let $y=\theta_1$ denote a parameter vector in the optimization problem (defined below).

Define

and note that it does not depend on $y$. We impose constraints $\tilde g_\ell(x,y)\le 0$ for $\ell=1,\cdots,2(K+L)$ and $\tilde g_\ell(x,y)= 0$ for $\ell=2(K+L)+1,2(K+L)+2$ with

These restrictions impose the sharp identifying restrictions together with the constraints that restrict $p_j, j=0,1$ in the probability simplex. Before proceeding, one can reduce the constraints by the following procedures.\footnote{This step is not strictly required, but it is recommended as the reduced system is more likely to satisfy Assumption (ref).}

First, reduce the first $2K$ inequality constraints to the ones that are generated by a {\em core determining class\/} $\mathcal C\subset 2^S$ galichon2006inference,galichon2011set.\footnote{galichon2011set provide a tractable characterization of the core determining class for incomplete models with $G$ possessing a certain monotonicity property. Luo:2017aa,Luo:2017ab provide algorithms to construct the core determining class for general incomplete models.} Second, define

which collects events that remain complete both under the null and local alternatives. For any $A\in \mathcal C$, combine the two inequality constraints, $\nu_{\theta_j}(A)-p_j(A)\le 0$ and $\nu_{\theta_j}(A^c)-p_j(A^c)\le 0$, and impose them as an equality constraint $\nu_{\theta_j}(A)=p_j(A).$ Finally, let $\mathcal B\equiv \mathcal C\setminus \mathcal A$, which collects the events associated with the remaining inequality restrictions. We note that some of the inequality constraints associated with events in this collection may still bind when $\tau=0.$ Let $M$ be the number constraints after reducing the restrictions and let the first $M_1\le M$ of the restrictions collect equality constraints.

The resulting convex program can then be written as

for some $g_\ell:\mathbb R^{2L}\times \Theta\to\mathbb R,\ell=1,\dots, M$. This is the setting analyzed in Shapiro:1988aa.

Let $x_\tau=(q_{0,\tau}',q_{1,\tau}')'$ be a solution to the problem above when $y$ is set to $y_\tau=\theta_0+\xi+\tau h$. Let $x_0=(q_{0,0}',q_{1,0}')'$ denote the solution when $\tau=0$. We then construct two sets $\mathbf A$ and $\mathbf B$, which collect gradient vectors of the equality and binding inequality constraints at $x_0$ as follows.

Recall that $H$ is a twice continuously differentiable convex function. The following condition is sufficient for the directional differentiability of the model.

Assumption (ref) (i) is a regularity condition on the objective function, which ensures that the solution of the problem satisfies a weak second-order condition. One can choose $H$ that satisfies these conditions. Assumption (ref) (ii) requires that the solution, hence the LFP (not only its ratio), is unique when $\tau=0$. This condition is satisfied in general when $\mathcal P_{\theta_0}\cap \mathcal P_{\theta_0+\xi}$ is a singleton. A special case is that the local alternatives are robustly testable and the model is complete at $\tau=0$, in which case $\mathcal P_{\theta_0}$ and $\mathcal P_{\theta_0+\tau h}$ coincide at $\tau=0$ and is a singleton. This occurs, for example, in Example (ref). Assumption (ref) (iii) imposes the Mangasarian-Fromovitz constraint qualification (MFCQ). This condition ensures that the set of Lagrange multipliers is bounded for all $\tau\in (0,\bar\tau]$ Gauvin:1977aa,Shapiro:1988aa, which is the key for the stability of the solution to local perturbations. Note that the constraints are linear and the gradient vectors in $\mathbf A\subset \{0,1\}^{2L}$ and $\mathbf B\subset \{0,1\}^{2L}$ do not need to be estimated. Hence, checking this condition is relatively straightforward.

Write the solution to (ref)-(ref) as $x(y)$ and denote its directional derivative with direction $h$ by

Let $\mathcal X_\tau\subset\mathbb R^M$ be the set of Lagrange multipliers under $\tau$ in a neighborhood of 0, which is, under MFCQ, bounded and the convex hull of a finite set $E_\tau$ of extreme points Shapiro:1988aa. For each $\chi\in\mathbb R^M$, let $J_+(\chi)=\{\ell:\chi^{(\ell)}>0,\ell=M_1+1,\dots,M\}$, $J_0=\{\ell:\chi^{(\ell)}=0,\ell=M_1+1,\dots,M\}$, and $J(\chi)=\{1,\dots,M_1\}\cup J_+(\chi)$. Each element $\chi$ in $E_0$ is such that the gradient vectors $\{\nabla_x g_\ell(x_0,y_0),\ell\in J(\chi)\}$ are linearly independent. For each $\ell=1,\dots, M$, define

and note that $\alpha_\ell(u,h)=u'a+h'\nabla_{y}\nu_{y_0}(A)$ for some $a\in \{0,1\}^{2L}$ if $g_\ell$ originated from one of the constraints in (ref) and $\alpha_\ell(u,h)=u'a$ otherwise because the other constraints (in (ref) and (ref)-(ref)) do not involve $\theta_1$. We then have the following directional differentiability result.

Note that each $\alpha_\ell$ is linear in $u$ and the number of constraints is finite because $E_0$ is finite. Hence, $Dx(\theta_0+\xi)[h]$ is a solution to a finite-dimensional (convex) QP.

To satisfy Assumption (ref) (or Assumption (ref)), we also need to ensure (ref) (or (ref)). For this, among the inequality conditions in (ref), let $J_{H_0}\subset \{M_1+1,\dots,M\}$ collect the indices associated with the constraints that restrict $q_{0,\tau}$, i.e. those of the form: $\nu_{\theta_0}(A_\ell)-\sum_{s\in A_\ell}p_{0}(s)\le 0$. We then let $J_{+,H_0}(\chi)\equiv\{\chi^{(\ell)}>0,\ell\in J_{H_0}\}$. The following condition is sufficient for (ref) (or (ref)).

In other words, there is a configuration of the Lagrange multipliers, under which the binding constraints under $H_0$ uniquely determines $Q_0$, which remains constant across $\tau\in (0,\bar\tau]$. As $\tau$ varies, the remaining components of the solution, $q_{1,\tau},\chi_\tau$, are properly adjusted to satisfy the first order conditions (see (ref)).

For each $j\in\mathbb J$ and $h\in\mathcal T_{j}(\xi,\theta_0)$, $Dx(\theta_0+\xi)[h]$ is a $2L$-dimensional vector. Let us write it as $(u_0^{*,(1)},\dots,u_0^{*,(L)},u_1^{*,(1)},\dots,u_1^{*,(L)})'$. The following corollary gives a formula to calculate the $L^2$-derivative for $h\in\mathcal T_j(\xi,\theta_0)$.

The proof of the results above are collected at the end of this appendix.

Proof of Proposition (ref) and Corollary (ref)