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\vskip20pt \footnotetext[1]{This research was partly carried out while the first author was visiting the Bendheim Center for Finance, Princeton University and financial support from NSF grant SES 0350770 to Princeton University, from NSF grant SES 0532398, from the Program for Economic Research at Columbia University and from Chaire EDF-Calyon “Finance et D\'eveloppement Durable” is gratefully acknowledged. We are grateful to Victor Chernozhukov, Pierre-Andr\'e Chiappori, Guido Imbens and Bernard Salani\'e for encouragement, support and many helpful discussions. We also thank three anonymous referees, whose detailed and insightful comments helped significantly improve the paper, and we thank conference participants at Econometrics in Rio and seminar participants at Berkeley, Chicago, Columbia, \'Ecole polytechnique, Harvard-MIT, MIT Sloane OR, Northwestern, NYU, Princeton, SAMSI, Stanford, the Weierstrass Institut and Yale for helpful comments (with the usual disclaimer). Correspondence address: D\'epartement d'\'economie, \'Ecole polytechnique, 91128 Palaiseau, France and D\'epartement de sciences \'economiques, Universit\'e de Montr\'eal, C.P. 6128, succursale Centre-ville, Montr\'eal QC H3C 3J7, Canada. E-mail: [email removed] and [email removed].}
\vskip6pt {\scriptsize JEL Classification: C10, C12, C13, C14, C52, C61 \\Keywords: partial identification, mass transportation, specification test.}
In several rapidly expanding areas of economic research, the identification problem is steadily becoming more acute. In policy and program evaluation (Manski:90) and more general contexts with censored or missing data (Molinari:2003, MM:2005) and measurement error (CHT:2005), ad hoc imputation rules lead to fragile inference. In demand estimation based on revealed preference (BBC:2005) the data is generically insufficient for identification. In the analysis of social interactions (BD:2005, Manski:2004), complex strategies to reduce the large dimensionality of the correlation structure are needed. In the estimation of models with complex strategic interactions and multiple equilibria (Tamer:2003, ABJ:2003, PPHI:2004), assumptions on equilibrium selection mechanisms may not be available or acceptable.
More generally, in all areas of investigation with structural data insufficiencies or incompletely specified economic mechanisms, the hypothesized structure fails to identify a unique possible generating mechanism for the data that is actually observed. Hence, when the structure depends on unknown parameters, and even if a unique value of the parameter can still be construed as the true value in some well defined way, it does not correspond in a one-to-one mapping with a probability measure for the observed variables. We then call the structural restrictions non-identifying. In other words, even if we abstract from sampling uncertainty and assume the distribution of the observable variables is perfectly known, no unique parameter but a whole set of parameter values (hereafter called identified set in the terminology of Manski:2005) will be compatible with it.
Once a theoretical description of an economic system is given, a natural question to consider is whether the structure can be rejected on the basis of data on its observable components. MA:44 construct a collection of production functions that are compatible with structural restrictions and are not rejected by the data. We extend this approach within the general formulation of KR:50, who define a structure as the combination of a binary relation between observed socioeconomic variables (market entry, insurance coverage, winning bids in auctions, etc...) and unobserved ones (productivity shocks, risk level, or risk attitude, valuations or information depending on the auction paradigm, etc...) and a generating mechanism for the unobserved variables. This setup is employed by Roehrig:88 and Matzkin:94, who analyze conditions for nonparametric identification of structures where the endogenous observable variables are functions of unobservable variables and exogenous observable ones.
Here, following Jovanovic:89, we allow the relation between observable and unobservable variables to be many-to-many, thereby including structures with multiple equilibria (when a value of the latent variables is associated with a set of values of the observable variables) and censored endogenous observable variables (where a value of the observable variable is associated with set of values of the latent variables). We do not strive for identification conditions, but rather for the ability to reject such structures that are incompatible with data, as in the original work of MA:44.
We show that such a goal can be attained in all generality (ie. for any structure, involving discrete as well as continuous observable variables), through an appeal to the duality of mass transportation (see Villani:2003 for a comprehensive account of the theory). Given any set of (possibly non-identifying) restrictions on the relation between latent and observable variables, and given the distribution $\nu$ of latent variables, the structure thus defined is compatible with the true distribution $P$ of the observable variables if and only if there exists a joint distribution with marginals $P$ and $\nu$ and such that the restrictions are almost surely respected. Otherwise, the data could not have been generated in a such a way. We show that the latter condition can be formulated as a mass transportation problem (the problem of transporting a given distribution of mass from an initial location to a different distribution of mass in a final location while minimizing a certain cost of transportation, as originally formulated by Monge:1781). We show that this optimization problem has a dual formulation, an empirical version of which is a generalized Kolmogorov-Smirnov test statistic. We base a test of the restrictions in the structure on this statistic, whose asymptotic distribution we derive, and approximate using the bootstrapped empirical process.
Once we have a test of the structure, we can form confidence regions for unknown parameters using the methodology of AR:49, which consists in collecting all parameter values for which the structure is not rejected by the test at the desired significance level. The construction of such confidence regions has been the focus of much research lately (see for instance the thorough literature review in CHT:2007). Unlike much of the econometric research on this issue, we do not restrict the analysis to models defined by moment inequalities. On the other hand, we consider structures in the sense of KR:50, and hence parametric distributions for the latent variables. This, however, is a common assumption in empirical work with game theoretic models, as exemplified by ABJ:2003, CT:2006, and more generally ABBP:2007.
The paper is organized as follows. The next section is divided in four subsections. The first describes the setup; the second defines the hypothesis of compatibility of the structure with the data; the third explains how to construct a confidence region for the identified set, and the fourth reviews the related literature. The second section is divided in three subsections. The first subsection describes and justifies the generalized Kolmogorov-Smirnov test of compatibility of the structure with the data; the second shows consistency of the test, and the third investigates size properties of the test in a Monte Carlo experiment. The last section concludes.
Consider the model of an economy which is composed of an observed variable $Y$ and a latent, unobserved variable $U$. Formally, $(Y,U)$ is a pair of random vectors defined on a common probability space. The pair $(Y,U)$ has probability law $\pi$ which is unknown. $Y$ represents the variables that are observable, and $U$ the variables that are unobservable. $Y$ may have discrete and continuous components. $Y$ may include variables of interest in their own right, and randomly censored or otherwise transformed versions of variables of interest. We call the law of the observable variables $P$. It is unknown, but the data available is a sample of independent and identically distributed vectors $(Y_1,\ldots,Y_n)$ with law $P$. $U$ includes random shocks and other unobserved heterogeneity components. The law $\pi$ of $(Y,U)$ can be decomposed into the unconditional distribution $P$ of $Y$ and the conditional distribution of $U$ given $Y$, namely $\pi_{U\vert Y}$. Throughout the paper it is supposed that $\pi_{U\vert Y}$ is unknown but fixed across observations.
The distribution of $U$ is parameterized by a vector $\theta_1\in\Theta_1$, where $\Theta_1$ is an open subset of $\mathbb{R}^{d_1}$, and the law of $U$ is denoted $\nu_{\theta_1}$. Finally, an economic model is given to us in the form of a set of restrictions on the vector $(Y,U)$, which can be summarized without loss of generality by the relation $U\in\Gamma_{\theta_2}(Y)$ where $\Gamma_{\theta_2}$ is a many-to-many mapping, which is completely given except for the vector of structural parameters $\theta_2\in\Theta_2$, where $\Theta_2$ is an open subset of $\mathbb{R}^{d_2}$. $\theta_1$ and $\theta_2$ may contain common components. We call $\theta$ the combination of the two, so that $\theta\in\Theta$, with $\Theta$ an open subset of $\mathbb{R}^{d_\theta}$, and $d_{\theta}\leq d_1+d_2$. From now on, we shall therefore denote the distribution of $U$ by $\nu_\theta$ and the many-to-many mapping by $\Gamma_\theta$. In all that follows, we assume that $\Gamma_\theta$ is measurable (a very weak requirement which is defined in the appendix), and has non-empty and closed values.
We are interested in testing the compatibility of the observed variables $Y$ with the model described by $(\Gamma,\nu)$. A related question is set-inference in a parametric model $(\Gamma_\theta,\nu_\theta)$: a confidence region for $\theta$ can be obtained by inverting the specification test, namely retaining the values of $\theta$ which are not rejected. Note that if $\theta_2=(\beta,\eta)$, where $\beta$ are the parameters of interest and $\eta\in H$ are nuisance parameters, we can redefine the economic model restrictions as $U\in \Gamma_\beta(Y)$ where $\Gamma_\beta$ is defined by $\Gamma_\beta(y)=\bigcup_{\eta\in H}\Gamma_{(\beta,\eta)}(y)$ for all $y\in\mathbb{R}^{d_y}$. Hence we can assume again without loss of generality that $\theta_2$ is indeed the parameter of interest. As the main focus of the present paper is to derive a specification test, whenever there is no ambiguity we shall implicitly fix the parameter $\theta$ and drop it from our notations.
A special case of the latter example is given in Jovanovic:89 and will serve as our first illustrative example:
A second example illustrates the case with continuous observable variables:
Identification of the parameter $\theta$ would require the correspondence between the law of the observations $P$ and the parameter vector $\theta$ to be a function. Compared to the setup described in Roehrig:88, there is the added complexity of the possibility that the observable variables have discrete components, and that the structure allows multiple equilibria. Conditions ensuring identification are likely to prove complicated and restrictive, and will often rule out multiple equilibria, which is the norm rather than the exception in example (ref). We therefore eschew identification, and allow the relation between $P$ and $\theta$ to be many-to-many. Our objective is to conduct inference on the set $\Theta_I$ of parameter values that are compatible with the true law of the observable variables $P$.
Let us formally define compatibility of a given value $\theta_0$ of the parameter vector with a law $P$ for the observable variables $Y$. When $\theta_0$ is fixed, all the elements in the model are completely known. We therefore have a structure in the terminology of KR:50 extended by Jovanovic:89. The structure is given by the law $\nu_{\theta_0}$ for $U$, and the many-to-many mapping $\Gamma_{\theta_0}$ linking $Y$ and $U$. We denote this structure by the triple $(P,\Gamma_{\theta_0},\nu_{\theta_0})$. Consider now the restrictions that $(P,\Gamma_{\theta_0},\nu_{\theta_0})$ imposes on the unknown $\pi$, the law of the vector of variables $(Y,U)$.
A probability law $\pi$ that satisfies the restrictions above may or may not exist. If and only if it does, we say that the structure $(P,\Gamma_{\theta_0},\nu_{\theta_0})$ is internally consistent, or simply that the value $\theta_0$ of the parameter is compatible with the law $P$ of the observable variables. If no value $\theta_0$ is found such that the structure is internally consistent, then the model restrictions are rejected.
We can now define the identified set as the set of values of the parameters that achieve this internal consistency. They are observationally equivalent, since even though they may correspond to different $\pi$'s, they correspond to the same $P$.
We illustrate the previous definitions with our pilot example:
The previous example illustrates the fact that definition (ref) is not very easy to apply to derive the identified set in specific problems. We therefore propose a characterization of internal consistency which will prove more practical, and which, as we shall see in the next section, will motivate the construction of the statistic to test internal consistency.
This proposition shows that checking internal consistency of a structure is equivalent to checking that the $P$-measure of a set is always dominated by the $\nu$-measure of the image of this set by $\Gamma$ (recall that the image of a set $A$ by a many-to-many mapping is defined by $\Gamma(A)=\bigcup_{a\in A}\Gamma(a)$). Note that it is relatively easy to show necessity, i.e. that the existence of $\pi$ satisfying the constraints (the definition of internal consistency) implies that $\sup_{A\in{\cal B}} [P(A)-\nu(\Gamma(A))]=0$. Indeed, the definition of internal consistency implies that ${Y\in A}\Rightarrow U\in\Gamma(A)$, so that $1_{\{Y\in A\}}\leq1_{\{U\in\Gamma(A)\}}$, $\pi$-almost surely. Taking expectation, we have $\mathbb{E}_\pi(1_{\{Y\in A\}})\leq \mathbb{E}_\pi(1_{\{U\in\Gamma(A)\}})$, which yields the result, since $\pi$ has marginals $P$ and $\nu$. The converse (proved in the appendix) is far more involved, as it relies on mass transportation duality, where mass $P$ is transported into mass $\nu$ with 0-1 cost of transportation associated with violations of the restrictions $U\in\Gamma(Y)$.
Given a sample $(Y_1,\ldots,Y_n)$ of independently and identically distributed realizations of $Y$, our objective is to construct a sequence of random sets $\Theta_{n}^{\alpha}$ such that for all $\theta\in\Theta_I$, $\lim_{n\rightarrow\infty}\mbox{Pr}\left(\theta\in\Theta_n^{\alpha}\right)= 1-\alpha$. In other words, we are concerned with constructing a region $\Theta_n^{\alpha}$ that covers each value of the identified set, as opposed to a region $\tilde\Theta$ that covers the identified set uniformly, i.e. such that Pr$(\Theta_I\subseteq\tilde{\Theta})=1-\alpha$. We do so by including in $\Theta_n^{\alpha}$ all the values of $\theta$ such that we fail to reject a test of internal consistency of $(P,\Gamma_\theta,\nu_\theta)$ with asymptotic level $1-\alpha$. We shall demonstrate the construction of a test statistic $T_n(\theta)$ and a sequence $c_n^\alpha(\theta)$ such that, conditionally on the structure $(P,\Gamma_\theta,\nu_\theta)$ being internally consistent, the probability that $T_n(\theta)\leq c_n^\alpha(\theta)$ is $1-\alpha$ asymptotically, i.e.
Hence we define our confidence region in the following way.
The full procedure is summarized in table 1. It is clear from equation (ref) and the above definition that our confidence region covers each element of the identified set with probability $1-\alpha$ asymptotically. Hence, after a section devoted to discussing in detail our contribution within the literature on the topic, the remainder of this paper will be concerned with the construction of the statistic $T_n$ and sequence $c_n^\alpha$ with the required property ((ref)).
This paper appears to be the first to cast partial identification as a mass transportation problem. Somewhat related is the specific use of Fr\'echet-Hoeffding bounds on cell probabilities in HSC:97 and CM:2002.
The literature on specification testing in econometrics is quite extensive (see the many references in Andrews:88 for Cram\'er-von Mises tests and Andrews:97 for the Kolmogorov-Smirnov type). Jovanovic:89 proposes to consider testing specifications with multiple equilibria and possible lack of identification with a generalization of the Kolmogorov-Smirnov specification test, which is exceedingly conservative unless the structure is nearly identified. The stochastic dominance tests of McFadden:89 (see also LMW:2005 and references within) are also related to tests of partially identified structures based on the Kolmogorov-Smirnov statistic. The feasible version of our testing procedure and the use of the bootstrapped empirical process is related to Andrews:97.
The incompleteness of the structure to be tested raises boundary problems, which appear also in the estimation of models defined by moment inequalities (see IM:2004 and the link drawn by Rosen:2006 with the literature on constrained statistical testing, surveyed in SS:2004) and stochastic dominance testing (see LMW:2005). Here the asymptotic analysis is carried out via a localization of the empirical processes to treat the boundary problem, which is another major innovation of this paper. Also related is the analysis in LS:2003 of the likelihood ratio test when the likelihood is maximized on a set as opposed to a single point.
The related problem of constructing confidence regions for partially identified structural parameters is the focus of considerable recent research, following the recognition (advocated in Manski:2005) that ad-hoc identification conditions can considerably weaken inference drawn on their basis. HM:98 propose confidence intervals that asymptotically cover interval identified sets with fixed probability. Beyond the interval case, CHT:2007 propose a criterion function based method, where the criterion is maximized on a set, as opposed to a single point. The method allows the construction of confidence regions for the identified set and for each parameter value in the identified set. CHT:2007 also specialize their method to the case of models defined by moment inequalities, with a quadratic criterion function.
The case of moment inequalities is also considered as a special case by GH:2006c, RS:2006a and RS:2006b (see also Rosen:2006 and Bugni:2007). The present paper complements CHT:2007 in that it justifies, via a mass transportation argument, the use of a generalized Kolmogorov-Smirnov criterion function in the extended KR:50 setup presented here. Note that our proposed use of the bootstrap only concerns the empirical process, as in Andrews:2000, so that issues of validity related to bootstrapping the test statistic itself do not arise.
The AR:49 approach taken here to construct confidence regions for parameter values within the identified set is also adopted in CHT:2007, ABJ:2003, RS:2006a among many others. ABJ:2003 work in a similar framework to the present paper (they consider example (ref)), but restrict their analysis to discrete dependent variables, and use a projection method, so that their inference is likely to be more conservative.
Since confidence regions are asymptotically validated, as emphasized by IM:2004, uniformity of the confidence region for parameter values is a desirable property for small sample accuracy. AG:2006 analyze uniformity of sub-sampling procedures. RS:2006a and RS:2006b give high level conditions for uniformity of sub-sampling procedures in the criterion-based approach, with specific conditions under which these results hold in case of regression with interval outcomes. Here, we propose to invert a test, which is shown to be asymptotically uniform in level in GH:2008.
In related research, BM:2007 propose a direct analogy to central limit theorem based confidence regions in best linear prediction problems. The confidence region they propose for the identified set, in a problem of best linear prediction with interval outcomes, is the union of a collection of random sets that contain the identified set with pre-specified probability. The latter is obtained from central limit theorems for random sets (see Molchanov:2005 for a comprehensive account of the theory). They propose one-sided and two-sided versions of their test. The BM:2007 two-sided procedure does not suffer from discontinuity at the limit where the identified set is a singleton. However, by construction, BM:2007 only provide confidence regions for the whole set, which are typically larger than identified regions for each point in the identified set.
As explained in the previous section, the construction of the confidence region relies on a test of internal consistency of the structure $ (P,\Gamma_\theta,\nu_\theta)$ for a fixed $\theta$. We now explain the construction of our test statistic and decision rule, for the hypothesis of internal consistency of a structure $(P,\Gamma,\nu)$ defined by a a probability law $\nu$ for $U$ and a set of constraints $U\in\Gamma(Y)$. The hypothesis that $(P,\Gamma,\nu)$ is internally consistent is equivalent to the existence of a law $\pi$ for $(Y,U)$ with marginals $P$ and $\nu$ and such that the constraints $U\in\Gamma(Y)$ hold $\pi$-almost surely. By proposition (ref), this null hypothesis is also equivalent to \[\mathbb{H}_0:\;\sup_{A\in{\cal B}}[P(A)-\nu(\Gamma(A))]=0.\]
We propose the following statistic to test the null described above:
where $P_n$ is the empirical distribution of the sample (so that for any measurable set $A$, $P_n(A)=(1/n)\sum_{i=1}^n1_{\{Y_i\in A\}}$) and where $\mathcal{C}$ is defined in table 3.
This statistic is a generalized Kolmogorov-Smirnov specification test statistic in the sense that when $\Gamma$ has disjoint images (i.e. $\Gamma^{-1}$ is a function), $T_n$ is a multivariate Kolmogorov-Smirnov statistic for the test of the hypothesis that the structure is correctly specified, i.e. that the probability law $A\mapsto\nu(\Gamma(A))$ is indeed equal to the true law $P$ generating the observable variables $Y$. In the general case where $\Gamma$ is a many-to-many mapping, $A\mapsto\nu(\Gamma(A))$ is no longer a probability measure, since two sets $A$ and $B$ may be disjoint, and yet their images $\Gamma(A)$ and $\Gamma(B)$ are not, so that $\nu(\Gamma(A\cup B))$ may be strictly smaller than $\nu(\Gamma(A))+\nu(\Gamma(B))$. This introduces significant complications in the asymptotic analysis of the statistic $T_n$ as explained in the following discussion.
We can write
where $\mathbb{G}_n(A):=\sqrt{n}[P_n(A)-P(A)]$ is the empirical process. In the case of the classical Kolmogorov-Smirnov statistic (i.e. if $\Gamma^{-1}$ were a function), the term $P(A)-\nu(\Gamma(A))$ would vanish under the null hypothesis. Here, however, under the null we only have $P(A)\leq\nu(\Gamma(A))$, so that the term $\sqrt{n}[P(A)-\nu(\Gamma(A)]$ will also contribute. Indeed, for any set $A\in\mathcal{C}$ such that $P(A)=\nu(\Gamma(A))$ (i.e. $A\in\mathcal{C}_b$ as defined in table 3), the only remaining term in the right-hand-side of equation ((ref)) is the empirical process. On the other hand, for any set $A\in\mathcal{C}$ such that $P(A)<\nu(\Gamma(A))$, $\sqrt{n}[P(A)-\nu(\Gamma(A))]$ will take increasingly large negative values and eventually dominate the expression inside the supremum in the right-hand-side of equation ((ref)) and such a set $A$ will not contribute to the supremum. We show in the proof of theorem (ref) that under a very mild assumption on the structure, the limit will only involve a supremum over sets in $\mathcal{C}_b$. Since $\mathcal{C}_b$ depends on $P$, it is unknown, and needs to be approximated by a data dependent class $\mathcal{C}_{h_n}$ defined in table 3 (namely $\mathcal{C}_h$ with $h=h_n$).
Theorem (ref) is not applicable directly for two reasons:
We show now that $T_n$ can be replaced by $\hat{T}_n$ defined in table 2, and that $c_n^\alpha$ can be replaced by $c_\ast^\alpha$, which is the $1-\alpha$ quantile of $T^\ast:=\sup_{A\in\mathcal{C}_{n,h_n}}\mathbb{G}^\ast(A)$, where $\mathbb{G}^\ast:=\sqrt{n}[P^\ast-P_n]$ is the bootstrapped empirical process. We thereby justify the fully implementable procedure described in table 1. This feasible version of the test mirrors the feasible version of the conditional Kolmogorov-Smirnov test proposed by Andrews:97, albeit in generalized form (multivariate and incompletely specified).
To that end, we need a large support assumption and a log concavity assumption for the distribution of observable variables and a continuity assumption on the mapping $\Gamma$ to ensure that $\hat{T}_n$ has the same limit as $T_n$.
To complete the analysis of the test of internal consistency we give conditions under which the test is consistent. The class of alternatives we consider is the following: \[\mathbb{H}_a:\;\sup_{A\in{\cal C}}[P(A)-\nu(\Gamma(A))]\neq0,\] where $\mathcal{C}$ is defined in table 3. We choose this class of alternatives since it simplifies to the set of alternatives in a multivariate Kolmogorov-Smirnov goodness-of-fit test when $P$ is absolutely continuous with respect to Lebesgue measure and when $\Gamma^{-1}$ is a function.
We have
We investigate the small sample properties of out test, and compare it to the properties of the Kolmogorov-Smirnov specification test in the identified case in a small Monte Carlo experiment based on a special case of illustrative example (ref).
We consider the following setup illustrated in figure 1: the structure is given by the correspondence $\Gamma(Y)=[\underline{s}(Y), \overline{s}(Y)]$ with $\underline{s}(Y)=\max(0,Y+s)$ and $\overline{s}(Y)=\min(1,Y+s)$, $s=0.15$, and the latent variable $U$ has law $\nu$, which is the uniform distribution over $[0,1]$. $Y$ has cumulative distribution function defined on $[0,1]$ by
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We perform 1000 repetitions of the following testing procedure, and we report the proportions of rejections out of these 1000 repetitions. We first generate\footnote{We use MATLAB version 7.1 with random seed 777.} a sample $(U_1,\ldots,U_n)$ of iid uniform $[0,1]$, with $n=100,500,1000$ and compute the sample of observable variables $(Y_1,\ldots,Y_n)$ as $(F^{-1}(U_1),\ldots,F^{-1}(U_n))$. $P_n$ is the empirical law of $(Y_1,\ldots,Y_n)$, and $\mathcal{C}_{n,h_n}$ is the collection of sets of the form $[0,Y_i]$, $i=1,\ldots,n$ with $P_n[0,Y_i]=(1/n)\sum_{j=1}^n1_{\{Y_j\leq Y_i\}} \geq \nu(\Gamma([0,Y_i]))-h_n=\min[1,Y_i+s]-h_n$ or $[Y_i,1]$, $i,\ldots,n$ with $P_n[Y_i,1] \geq \nu(\Gamma([Y_i,1]))-h_n=\min[1,1-Y_i+s]-h_n$.
For each sample, we draw 1000 bootstrap samples $(Y_1^b,\ldots,Y_n^b)$, and call $P^b$ the law of the bootstrap sample. For each bootstrap sample, we calculate the maximum of the quantities $P^b[0,Y_i]-P_n[0,Y_i]$ for all $i$ such that $[0,Y_i]\in\mathcal{C}_{n,h_n}$ and $P^b[Y_i,1]-P_n[Y_i,1]$ for all $i$ such that $[Y_i,1]\in\mathcal{C}_{n,h_n}$, and call this maximum $\max\mathbb{G}^b$. Order the $\max\mathbb{G}^b$ obtained for all bootstrap draws, and call $c_\ast^\alpha$ the $(1-\alpha)1000$ largest, for $\alpha=0.01,0.05,0.1$. Reject if $c_\ast^\alpha$ is smaller than the maximum of the quantities $P_n[0,Y_i]$ and $P_n[Y_i,1]$ for $i=1\ldots,n$.
The results are given in table 4 for the partially identified case ($s=0.15$) and in table 5, we give the benchmark of the exactly identified case ($s=0$ and $h_n=1$), so that the test is a traditional Kolmogorov-Smirnov specification test. The results are given for $h_n$ on the boundary of the admissible rate, i.e. $h_n=\sqrt{\ln\ln n/n}$. This rate was chosen as a power maximizing rate (the rate that will ensure smaller quantiles, hence larger rejection rates). This is the only justification for a choice of rate that we can provide at this stage, as optimal rate choice is beyond the scope of this paper. In applications, it is recommended to provide results for different choices of rates, as one would typically do in density, nonparametric regression or spectral estimation. The rejection rates are low for small sample sizes and improve sharply when sample size increases. To give a sense of the sensitivity of rejection rates to the choice of the tuning parameter $h_n$, table 6 reports rejection rates in the case of $\alpha=0.01,0.05,0.1$ and $n=100,500,1000$ and choices of tuning parameter $h_n$ that are significantly above, and significantly below the initial choice of $h_n=\sqrt{\ln\ln n/n}$. For $n=1000$, $\sqrt{\ln\ln n/n}=0.044$, so we report results for $h_n=0.010,0.070$. For $n=500$, $\sqrt{\ln\ln n/n}=0.060$, so we report results for $h_n=0.020,0.100$. For $n=100$, $\sqrt{\ln\ln n/n}=0.120$, so we report results for $h_n=0.050,0.150$. Notice that we decrease the investigated range of tuning parameter with sample size, which corresponds to the fact that the tuning parameter converges to zero. For $n=100$, the rejection rates are sensitive to the choice of rate within the theoretical range (assumption (ref)) of tuning parameters. For $n=500$, there is still sensitivity to the choice of $h_n$, somewhat less so for $n=1000$. However, as in the case of bandwidth in kernel estimation or in local spectral estimation of time series, it is highly recommended to report empirical results with a good range of values of the tuning parameter $h_n$. Figure (ref) graphs rejections rates against tuning parameter to give a better sense of this sensitivity for sample size 500 and level 0.05. It is important also to note that higher values of the tuning parameter lead to less filtering, i.e. more sets are used in the computation of the supremum of the bootstrap empirical process, leading to larger quantiles, hence smaller rejection rates. Hence it also shows how crucial the filtering procedure is, since without it, the power of the test would be very poor.
We propose a test of the specification of a structure in the sense of KR:50, extended by Jovanovic:89, where observable variables and latent variables are related by a many-to-many mapping, thereby allowing censored observable variables and multiple equilibria. We apply mass transportation duality to derive a simple necessary and sufficient condition for compatibility of such structures and data in complete generality, and to justify the use of a generalized Kolmogorov-Smirnov test statistic. We propose a generically applicable and easily implementable procedure to test compatibility of structure and data, and to construct confidence regions for partially identified parameters specifying the structure. This work therefore complements other proposals, which tend to focus on models defined by moments inequalities. The small sample performance of the test is investigated in a Monte Carlo experiment, and is found to be comparable to the performance of the traditional Kolmogorov-Smirnov specification test statistic.