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This paper establishes consistency and non-standard rates of convergence for set estimators based on contour sets of criterion functions for a semiparametric binary response model under a conditional median restriction. The model may be partially identified due to potentially limited-support regressors. A set estimator analogous to the maximum score estimator is essentially cube-root consistent for the identified set when a continuous but possibly bounded regressor is present. Arbitrarily fast convergence occurs when all regressors are discrete. We also establish the validity of a subsampling procedure for constructing confidence sets for the identified set. As a technical contribution, we provide more convenient sufficient conditions on the underlying empirical processes for cube root convergence and a sufficient condition for arbitrarily fast convergence, both of which can be applied to other models. Finally, we carry out a series of Monte Carlo experiments which verify our theoretical findings and shed light on the finite sample performance of the proposed procedures.

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... Horowitz (1992) developed a smoothed maximum score estimator that converges faster than the n −1/3 rate and is asymptotically normal under some additional smoothness assumptions. Additional papers that study large sample estimation and inference in the maximum score context include Manski and Thompson (1986), Delgado, Rodríguez-Poo, and Wolf (2001), Abrevaya and Huang (2005), Léger and MacGibbon (2006), Komarova (2013), Blevins (2015), Chen andLee (2017, 2018), and Cattaneo, Jansson, and Nagasawa (2018). ...

... To do this we employ a conditional moment inequality characterization of the observable implications of the binary response model in the finite sample. Moment inequality characterizations of the model's implications have been previously used by Komarova (2013), Blevins (2015), and Chen and Lee (2017), but none of these papers proposed a method for conducting finite sample inference. As was the case in the analysis provided in these papers, we do not require that β is point identified. ...

... Among the aforementioned papers from the literature on maximum score, the most closely related is that of Chen and Lee (2017), who also cast the implications of Manski's (1985) model as conditional moment inequalties for the sake of delivering a new insight, albeit one that is entirely different from ours. Chen and Lee (2017) expand on the conditional moment inequalities used by Komarova (2013) and Blevins (2015) to develop a novel conditional moment inequality characterization of the identified set which involves conditioning on two linear indices instead of on the entire exogenous covariate vector. They apply intersection bound inference from Chernozhukov, Lee, and Rosen (2013) to this conditional moment inequality characterization to achieve asymptotically valid inference. ...

We provide a finite sample inference method for the structural parameters of a semiparametric binary response model under a conditional median restriction originally studied by Manski (1975, 1985). Our inference method is valid for any sample size and irrespective of whether the structural parameters are point identified or partially identified, for example due to the lack of a continuously distributed covariate with large support. Our inference approach exploits distributional properties of observable outcomes conditional on the observed sequence of exogenous variables. Moment inequalities conditional on this size n sequence of exogenous covariates are constructed, and the test statistic is a monotone function of violations of sample moment inequalities. The critical value used for inference is provided by the appropriate quantile of a known function of n independent Rademacher random variables. We investigate power properties of the underlying test and provide simulation studies to support the theoretical findings.

... 100-108) suggested a computational approach that can be used to compute the identified set for β when these conditions do not hold. Komarova (2013) developed Horowitz's approach into a more analytic argument, while Blevins (2015) considered estimation of this identified set. ...

I show that sharp identified sets in a large class of econometric models can be characterized by solving linear systems of equations. These linear systems determine whether, for a given value of a parameter of interest, there exists an admissible joint distribution of unobservables that can generate the distribution of the observed variables. The joint distribution of unobservables is not required to satisfy any parametric restrictions, but can (if desired) be assumed to satisfy a variety of location, shape, and/or conditional independence restrictions. To prove sharpness of the characterization, I generalize a classic result in copula theory concerning the extendibility of subcopulas to show that related objects—termed subdistributions—can be extended to proper distribution functions. I describe this characterization argument as partial identification by extending subdistributions, or PIES. One particularly attractive feature of PIES is that it focuses directly on the sharp identified set for a parameter of interest, such as an average treatment effect, without needing to construct the identified set for the entire model. I apply PIES to univariate and bivariate binary response models. A notable product of the analysis is a method for characterizing the sharp identified set for the average treatment effect in Manski's (, ,) semiparametric binary response model.

... There is a large body of the literature that studies maximum score estimation in various other aspects since the seminal work by Manski (1975Manski ( , 1985). In the context of binary response models with the conditional median restriction , advances of the maximum score approach have been made in terms of point identification (Manski, 1988), partial identification (Manski and Tamer, 2002; Komarova, 2013; Blevins, 2015; Chen and Lee, 2015), asymptotic distribution (Kim and Pollard, 1990), panel data (Manski, 1987; Charlier, Melenberg, and van Soest, 1995; Abrevaya, 2000), time series (Moon, 2004; Guerre and Moon, 2006; de Jong and Woutersen, 2011), nonparametrically generated regressors (Chen, Lee, and Sung, 2014), and so on. The numerical approach taken in this paper can be adapted to these contexts. ...

We consider a variable selection problem for the prediction of binary outcomes. We study the best subset selection procedure by which the explanatory variables are chosen by maximising Manski (1975, 1985)'s maximum score type objective function subject to a constraint on the maximal number of selected variables. We show that this procedure can be equivalently reformulated as solving a mixed integer optimization (MIO) problem, which enables computation of the exact or an approximate solution with a definite approximation error bound. In terms of theoretical results, we obtain non-asymptotic upper and lower risk bounds that are minimax rate-optimal when the dimension of potential covariates is possibly much larger than the sample size ($n$) but the maximal number of selected variables is fixed and does not increase with $n$. We illustrate usefulness of the best subset maximum score binary prediction rule in Horowitz (1993)'s application of the work-trip transportation mode choice.

... Recently, Komarova (2013) and Blevins (2015) use this type of characterization to partially identify β. Both papers consider estimation and inference of the identified set Θ using a maximum score objective function; however, they do not develop inference methods for the parameter value β based on the conditional moment inequalities in (1.2). ...

This paper studies inference of preference parameters in semiparametric
discrete choice models when these parameters are not point-identified and the
identified set is characterized by a class of conditional moment inequalities.
Exploring the semiparametric modeling restrictions, we show that the identified
set can be equivalently formulated by moment inequalities conditional on only
two continuous indexing variables. Such formulation holds regardless of the
covariate dimension, thereby breaking the curse of dimensionality for
nonparametric inference based on the underlying conditional moment
inequalities. We also extend this dimension reducing characterization result to
a variety of semiparametric models under which the sign of conditional
expectation of a certain transformation of the outcome is the same as that of
the indexing variable.

In many micro‐data studies, the dependent variable often involves ordered categories and at least one regressor is measured by the interval rather than the precise value. This paper considers partial identification of such an ordered response model when point identification fails. We show the identified set of non‐intercept coefficients is the intersection of those for composite binary response models. We also propose a generalized modified maximum score set (GMMS) estimator. A practical implication of our finding is researchers can shrink the identified set and obtain more precise inference by designing as many as categories of response in a questionnaire during data collection. Another advantage is our theoretical finding can be used to infer the identified region in the multinomial choice model. A Monte Carlo study is conducted to illustrate the main finding in a finite sample. Finally, we apply GMMS estimator to a job satisfaction study using US data with the interval income.

This chapter reviews the microeconometrics literature on partial identification, focusing on the developments of the last thirty years. The topics presented illustrate that the available data combined with credible maintained assumptions may yield much information about a parameter of interest, even if they do not reveal it exactly. Special attention is devoted to discussing the challenges associated with, and some of the solutions put forward to, (1) obtain a tractable characterization of the values for the parameters of interest which are observationally equivalent, given the available data and maintained assumptions; (2) estimate this set of values; (3) conduct test of hypotheses and make confidence statements. The chapter reviews advances in partial identification analysis both as applied to learning (functionals of) probability distributions that are well-defined in the absence of models, as well as to learning parameters that are well-defined only in the context of particular models. A simple organizing principle is highlighted: the source of the identification problem can often be traced to a collection of random variables that are consistent with the available data and maintained assumptions. This collection may be part of the observed data or be a model implication. In either case, it can be formalized as a random set. Random set theory is then used as a mathematical framework to unify a number of special results and produce a general methodology to carry out partial identification analysis.

We propose an estimation method for the conditional mode when the conditioning variable is high-dimensional. In the proposed method, we first estimate the conditional density by solving quantile regressions multiple times. We then estimate the conditional mode by finding the maximum of the estimated conditional density. The proposed method has two advantages in that it is computationally stable because it has no initial parameter dependencies, and it is statistically efficient with a fast convergence rate. Synthetic and real-world data experiments demonstrate the better performance of the proposed method compared to other existing ones.

This paper considers the problem of inference for partially identified econometric models. The class of models studied are defined by a population objective function Q(θ,P) for θ∈Θ. The second argument indicates the dependence of the objective function on P, the distribution of the observed data. Unlike the classical extremum estimation framework, it is not assumed that Q(θ,P) has a unique minimizer in the parameter space Θ. The goal may be either to draw inferences about some unknown point in the set of minimizers of the population objective function or to draw inferences about the set of minimizers itself. In this paper, the object of interest is some unknown point θ∈Θ0(P), where , and so we seek random sets that contain each θ∈Θ0(P) with at least some prespecified probability asymptotically. We also consider situations where the object of interest is the image of some point θ∈Θ0(P) under a known function. Computationally intensive, yet feasible procedures for constructing random sets satisfying the desired coverage property under weak assumptions are provided. We also provide conditions under which the confidence regions are uniformly consistent in level.

This paper evaluates a pilot program run by a company called OPOWER, previously known as Positive Energy, to mail home energy reports to residential utility consumers. The reports compare a household’s energy use to that of its neighbors and provide energy conservation tips. Using data from randomized natural field experiment at 80,000 treatment and control households in Minnesota, I estimate that the monthly program reduces energy consumption by 1.9 to 2.0 percent relative to baseline. In a treatment arm receiving reports each quarter, the effects decay in the months between letters and again increase upon receipt of the next letter. This suggests either that the energy conservation information is not useful across seasons or, perhaps more interestingly, that consumers’ motivation or attention is malleable and non-durable. I show that “profiling,” or using a statistical decision rule to target the program at households whose observable characteristics suggest larger treatment effects, could substantially improve cost effectiveness in future programs. The effects of this program provide additional evidence that non-price “nudges” can substantially affect consumer behavior.

This paper introduces a novel bootstrap procedure to perform inference in a wide class of partially identified econometric models. We consider econometric models defined by finitely many weak moment inequalities, -super-2 which encompass many applications of economic interest. The objective of our inferential procedure is to cover the identified set with a prespecified probability. -super-3 We compare our bootstrap procedure, a competing asymptotic approximation, and subsampling procedures in terms of the rate at which they achieve the desired coverage level, also known as the error in the coverage probability. Under certain conditions, we show that our bootstrap procedure and the asymptotic approximation have the same order of error in the coverage probability, which is smaller than that obtained by using subsampling. This implies that inference based on our bootstrap and asymptotic approximation should eventually be more precise than inference based on subsampling. A Monte Carlo study confirms this finding in a small sample simulation. Copyright 2010 The Econometric Society.

This paper introduces a new stochastic process, a collection of $U$-statistics indexed by a family of symmetric kernels. Conditions are found for the uniform almost-sure convergence of a sequence of such processes. Rates of convergence are obtained. An application to cross-validation in density estimation is given. The proofs adapt methods from the theory of empirical processes.

We establish a new functional central limit theorem for empirical processes indexed by classes of functions. In a neighborhood of a fixed parameter point, an $n^{-1/3}$ rescaling of the parameter is compensated for by an $n^{2/3}$ rescaling of the empirical measure, resulting in a limiting Gaussian process. By means of a modified continuous mapping theorem for the location of the maximizing value, we deduce limit theorems for several statistics defined by maximization or constrained minimization of a process derived from the empirical measure. These statistics include the short, Rousseeuw's least median of squares estimator, Manski's maximum score estimator, and the maximum likelihood estimator for a monotone density. The limit theory depends on a simple new sufficient condition for a Gaussian process to achieve its maximum almost surely at a unique point.

This paper provides conditions under which the inequality constraints generated by either single agent optimizing behavior, or by the Nash equilibria of multiple agent problems, can be used as a basis for estimation and inference. We also add to the econometric literature on inference in models defined by inequality constraints by providing a new specification test and methods of inference for the boundaries of the model's identified set. Two applications illustrate how the use of inequality constraints can simplify the problem of obtaining estimators from complex behavioral models of substantial applied interest.

This paper extends Imbens and Manski's (2004) analysis of confidence intervals for interval identified parameters. For their final result, Imbens and Manski implicitly assume superefficient estimation of a nuisance parameter. This appears to have gone unnoticed before, and it limits the result's applicability. I re-analyze the problem both with assumptions that merely weaken the superefficiency condition and with assumptions that remove it altogether. Imbens and Manski's confidence region is found to be valid under weaker assumptions than theirs, yet superefficiency is required. I also provide a different confidence interval that is valid under superefficiency but can be adapted to the general case, in which case it embeds a specification test for nonemptiness of the identified set. A methodological contribution is to notice that the difficulty of inference comes from a boundary problem regarding a nuisance parameter, clarifying the connection to other work on partial identification.

A general approach to constructing confidence intervals by subsampling was presented in Politis and Romano (1994). The crux of the method is recomputing a statistic over subsamples of the data, and these recomputed values are used to build up an estimated sampling distribution. The method works under extremely weak conditions, it applies to independent, identically distributed (i.i.d.) observations as well as to dependent data situations, such as time series (possibly nonstationary), random fields, and marked point processes. In this article, we present some theorems showing: a new construction for confidence intervals that removes a previous condition, a general theorem showing the validity of subsampling for data-dependent choices of the block size, and a general theorem for the construction of hypothesis tests (not necessarily derived from a confidence interval construction). The arguments apply to both the i.i.d. setting and the dependent data case.

Many estimation problems in econometrics involve an unknown function or an unknown function and an unknown finite-dimensional parameter. Models and estimation problems that involve an unknown function are called nonparametric. Models and estimation problems that involve an unknown function and an unknown finite-dimensional parameter are called semiparametric.

In this paper, we propose an instrumental variable approach to constructing confidence sets (CS's) for the true parameter in models defined by conditional moment inequalities/equalities. We show that by properly choosing instrument functions, one can transform conditional moment inequalities/equalities into unconditional ones without losing identification power. Based on the unconditional moment inequalities/equalities, we construct CS's by inverting Cramér-von Mises-type or Kolmogorov-Smirnov-type tests. Critical values are obtained using generalized moment selection (GMS) procedures. We show that the proposed CS's have correct uniform asymptotic coverage probabilities. New methods are required to establish these results because an infinite-dimensional nuisance parameter affects the asymptotic distributions. We show that the tests considered are consistent against all fixed alternatives and typically have power against n-1/2-local alternatives to some, but not all, sequences of distributions in the null hypothesis. Monte Carlo simulations for five different models show that the methods perform well in finite samples.

This paper is concerned with tests and confidence intervals for parameters that are not necessarily point identified and are defined by moment inequalities. In the literature, different test statistics, critical-value methods, and implementation methods (i.e., the asymptotic distribution versus the bootstrap) have been proposed. In this paper, we compare these methods. We provide a recommended test statistic, moment selection critical value, and implementation method. We provide data-dependent procedures for choosing the key moment selection tuning parameter κ and a size-correction factor η.

We propose a classical Laplace estimator alternative for a large class of –consistent estimators, including isotonic regression, monotone hazard, and maximum score estimators. The proposed alternative provides a unified method of smoothing; easier computation is a byproduct in the maximum score case. Depending on input parameter choice and smoothness, the convergence rate of our estimator varies between and (almost) and its limit distribution varies from Chernoff to normal. We provide a bias reduction method and an inference procedure which automatically adapts to the correct convergence rate and limit distribution.

This paper studies the semiparametric binary response model with interval data investigated by Manski and Tamer (2002). In this partially identified model, we propose a new estimator based on MT’s modified maximum score (MMS) method by introducing density weights to the objective function, which allows us to develop asymptotic properties of the proposed set estimator for inference. We show that the density-weighted MMS estimator converges at a nearly cube-root-n rate. We propose an asymptotically valid inference procedure for the identified region based on subsampling. Monte Carlo experiments provide supports to our inference procedure.

In this paper, we consider estimation of the identified set when the number of moment inequalities is large relative to sample size, possibly infinite. Many applications in the recent literature on partially identified problems have this feature, including dynamic games, set-identified IV models, and parameters defined by a continuum of moment inequalities, in particular conditional moment inequalities. We provide a generic consistency result for criterion-based estimators using an increasing number of unconditional moment inequalities. We then develop more specific results for set estimation subject to conditional moment inequalities: we first derive the fastest possible rate for estimating the sharp identification region under smoothness conditions on the conditional moment functions. We also give rate conditions for inference under local alternatives.

In semiparametric binary response models, support conditions on the regressors are required to guarantee point identification of the parameter of interest. For example,one regressor is usually assumed to have continuous support conditional on the other regressors. In some instances, such conditions have precluded the use of these models; in others, practitioners have failed to consider whether the conditions are satisfied in their data. This paper explores the inferential question in these semiparametric models when the continuous support condition is not satisfied and all regressors have discrete support. I suggest a recursive procedure that finds sharp bounds on the components of the parameter of interest and outline several applications, focusing mainly on the models under the conditional median restriction, as in Manski (1985). After deriving closed-form bounds on the components of the parameter, I show how these formulas can help analyze cases where one regressor's support becomes increasingly dense. Furthermore, I investigate asymptotic properties of estimators of the identification set. I describe a relation between the maximum score estimation and support vector machines and also propose several approaches to address the problem of empty identification sets when a model is misspecified. Finally, I present a Monte Carlo experiment and an empirical illustration to compare several estimation techniques.

We provide a tractable characterization of the sharp identification region of the parameters θ in a broad class of incomplete econometric models. Models in this class have set valued predictions that yield a convex set of conditional or unconditional moments for the observable model variables. In short, we call these models with convex moment predictions. Examples include static, simultaneous move finite games of complete and incomplete information in the presence of multiple equilibria; best linear predictors with interval outcome and covariate data; and random utility models of multinomial choice in the presence of interval regressors data. Given a candidate value for θ, we establish that the convex set of moments yielded by the model predictions can be represented as the Aumann expectation of a properly defined random set. The sharp identification region of θ, denoted Θ1, can then be obtained as the set of minimizers of the distance from a properly specified vector of moments of random variables to this Aumann expectation. Algorithms in convex programming can be exploited to efficiently verify whether a candidate θ is in Θ1. We use examples analyzed in the literature to illustrate the gains in identification and computational tractability afforded by our method. This paper is a revised version of CWP27/09.

This paper studies the problem of estimating the set of finite-dimensional parameter values defined by a finite number of moment inequality or equality conditions and gives conditions under which the estimator defined by the set of parameter values that satisfy the estimated versions of these conditions is consistent in Hausdorff metric. This paper also suggests extremum estimators that with probability approaching 1 agree with the set consisting of parameter values that satisfy the sample versions of the moment conditions. In particular, it is shown that the set of minimizers of the sample generalized method of moments (GMM) objective function is consistent for the set of minimizers of the population GMM objective function in Hausdorff metric.

This article studies identification of the threshold-crossing model of binary response. Most research on binary response has considered specific estimators and tests. The study of identification exposes the foundations of binary response analysis by making explicit the assumptions needed to justify different methods. It also clarifies the connections between reduced-form and structural analyses of binary response data. Assume that the binary outcome z is determined by an observable random vector x and by an unobservable scalar u through a model z = 1[xβ + u ≤ 0]. Also assume that Fu|x, the probability distribution of u conditional on x, is continuous and strictly increasing. Given these maintained assumptions, we investigate the identifiability of β given the following restrictions on the distributions (Fu|x, x ∈ X): mean independence, quantile independence, index sufficiency, statistical independence, and statistical independence with the distribution known. We find that mean independence has no identifying power. On the other hand, quantile independence implies that β is identified up to scale, provided that the distribution of x has sufficiently rich support. Index sufficiency can identify the slope components of β up to scale and sign, again provided that the distribution of x has a rich support. Statistical independence subsumes both quantile independence and index sufficiency and so implies all of the positive findings previously reported. If u is statistically independent of x with the distribution known, identification requires only that the distribution of x have full rank.

This paper provides computationally intensive, yet feasible methods for inference in a very general class of partially identified econometric models. Let P denote the distribution of the observed data. The class of models we consider is defined by a population objective function Q(θ, P) for θ∈Θ. The point of departure from the classical extremum estimation framework is that it is not assumed that Q(θ, P) has a unique minimizer in the parameter space Θ. The goal may be either to draw inferences about some unknown point in the set of minimizers of the population objective function or to draw inferences about the set of minimizers itself. In this paper, the object of interest is Θ0(P)=argminθ∈ΘQ(θ, P), and so we seek random sets that contain this set with at least some prespecified probability asymptotically. We also consider situations where the object of interest is the image of Θ0(P) under a known function. Random sets that satisfy the desired coverage property are constructed under weak assumptions. Conditions are provided under which the confidence regions are asymptotically valid not only pointwise in P, but also uniformly in P. We illustrate the use of our methods with an empirical study of the impact of top-coding outcomes on inferences about the parameters of a linear regression. Finally, a modest simulation study sheds some light on the finite-sample behavior of our procedure.

This paper develops a framework for performing estimation and inference in econometric models with partial identification, focusing particularly on models characterized by moment inequalities and equalities. Applications of this framework include the analysis of game-theoretic models, revealed preference restrictions, regressions with missing and corrupted data, auction models, structural quantile regressions, and asset pricing models.Specifically, we provide estimators and confidence regions for the set of minimizers ΘI of an econometric criterion function Q(θ). In applications, the criterion function embodies testable restrictions on economic models. A parameter value θthat describes an economic model satisfies these restrictions if Q(θ) attains its minimum at this value. Interest therefore focuses on the set of minimizers, called the identified set. We use the inversion of the sample analog, Qn(θ), of the population criterion, Q(θ), to construct estimators and confidence regions for the identified set, and develop consistency, rates of convergence, and inference results for these estimators and regions. To derive these results, we develop methods for analyzing the asymptotic properties of sample criterion functions under set identification.

Recently a growing body of research has studied inference in settings where parameters of interest are partially identified. In many cases the parameter is real-valued and the identification region is an interval whose lower and upper bounds may be estimated from sample data. For this case confidence intervals (CIs) have been proposed that cover the entire identification region with fixed probability. Here, we introduce a conceptually different type of confidence interval. Rather than cover the entire identification region with fixed probability, we propose CIs that asymptotically cover the true value of the parameter with this probability. However, the exact coverage probabilities of the simplest version of our new CIs do not converge to their nominal values uniformly across different values for the width of the identification region. To avoid the problems associated with this, we modify the proposed CI to ensure that its exact coverage probabilities do converge uniformly to their nominal values. We motivate this modified CI through exact results for the Gaussian case.

Identification in econometric models maps prior assumptions and the data to information about a parameter of interest. The partial identification approach to inference recognizes that this process should not result in a binary answer that consists of whether the parameter is point identified. Rather, given the data, the partial identification approach characterizes the informational content of various assumptions by providing a menu of estimates, each based on different sets of assumptions, some of which are plausible and some of which are not. Of course, more assumptions beget more information, so stronger conclusions can be made at the expense of more assumptions. The partial identification approach advocates a more fluid view of identification and hence provides the empirical researcher with methods to help study the spectrum of information that we can harness about a parameter of interest using a menu of assumptions. This approach links conclusions drawn from various empirical models to sets of assumptions made in a transparent way. It allows researchers to examine the informational content of their assumptions and their impacts on the inferences made. Naturally, with finite sample sizes, this approach leads to statistical complications, as one needs to deal with characterizing sampling uncertainty in models that do not point identify a parameter. Therefore, new methods for inference are developed. These methods construct confidence sets for partially identified parameters, and confidence regions for sets of parameters, or identifiable sets.

This paper extends Imbens and Manski's (2004) analysis of confidence intervals for interval identified parameters. The extension is motivated by the discovery that for their final result, Imbens and Manski implicitly assumed locally superefficient estimation of a nuisance parameter. Copyright 2009 The Econometric Society.

This book is an introduction to the field of asymptotic statistics. The treatment is both practical and mathematically rigorous. In addition to most of the standard topics of an asymptotics course, including likelihood inference, M-estimation, the theory of asymptotic efficiency, U-statistics, and rank procedures, the book also presents recent research topics such as semiparametric models, the bootstrap, and empirical processes and their applications. The topics are organized from the central idea of approximation by limit experiments, which gives the book one of its unifying themes. This entails mainly the local approximation of the classical i.i.d. set up with smooth parameters by location experiments involving a single, normally distributed observation. Thus, even the standard subjects of asymptotic statistics are presented in a novel way. Suitable as a graduate or Master's level statistics text, this book will also give researchers an overview of research in asymptotic statistics.

Asymptotic distribution theory is the primary method used to examine the properties of econometric estimators and tests. We present conditions for obtaining cosistency and asymptotic normality of a very general class of estimators (extremum estimators). Consistent asymptotic variance estimators are given to enable approximation of the asymptotic distribution. Asymptotic efficiency is another desirable property then considered. Throughout the chapter, the general results are also specialized to common econometric estimators (e.g. MLE and GMM), and in specific examples we work through the conditions for the various results in detail. The results are also extended to two-step estimators (with finite-dimensional parameter estimation in the first step), estimators derived from nonsmooth objective functions, and semiparametric two-step estimators (with nonparametric estimation of an infinite-dimensional parameter in the first step). Finally, the trinity of test statistics is considered within the quite general setting of GMM estimation, and numerous examples are given.

This paper considers estimation of a fixed-effects version of the generalized regression model of Han (1987, Journal of Econometrics 35, 303–316). The model allows for censoring, places no parametric assumptions on the error disturbances, and allows the fixed effects to be correlated with the covariates. We introduce a class of rank estimators that consistently estimate the coefficients in the generalized fixed-effects regression model. The maximum score estimator for the binary choice fixed-effects model is part of this class. Like the maximum score estimator, the class of rank estimators converge at less than the rate. Smoothed versions of these estimators, however, converge at rates approaching the rate. In a version of the model that allows for truncated data, a sufficient condition for consistency of the estimators is that the error disturbances have an increasing hazard function.

Under a quantile restriction, randomly censored regression models can be written in terms of conditional moment inequalities. We study the identified features of these moment inequalities with respect to the regression parameters where we allow for covariate dependent censoring, endogenous censoring and endogenous regressors. These inequalities restrict the parameters to a set. We show regular point identification can be achieved under a set of interpretable sufficient conditions. We then provide a simple way to convert conditional moment inequalities into unconditional ones while preserving the informational content. Our method obviates the need for nonparametric estimation, which would require the selection of smoothing parameters and trimming procedures. Without the point identification conditions, our objective function can be used to do inference on the partially identified parameter. Maintaining the point identification conditions, we propose a quantile minimum distance estimator which converges at the parametric rate to the parameter vector of interest, and has an asymptotically normal distribution. A small scale simulation study and an application using drug relapse data demonstrate satisfactory finite sample performance.

This paper provides estimators of discrete choice models, including binary, ordered, and multinomial response (choice) models. The estimators closely resemble ordinary and two-stage least squares. The distribution of the model's latent variable error is unknown and may be related to the regressors, e.g., the model could have errors that are heteroscedastic or correlated with regressors. The estimator does not require numerical searches, even for multinomial choice. For ordered and binary choice models the estimator is root N consistent and asymptotically normal. A consistent estimator of the conditional error distribution is also provided.

For the class of single-index models, I construct a semiparametric estimator of coefficients up to a multiplicative constant that exhibits -consistency and asymptotic normality. This class of models includes censored and truncated Tobit models, binary choice models, and duration models with unobserved individual heterogeneity and random censoring. I also investigate a weighting scheme that achieves the semiparametric efficiency bound.

This paper addresses the issue of optimal inference for parameters that are partially identified in models with moment inequalities. There currently exists a variety of inferential methods for use in this setting. However, the question of choosing optimally among contending procedures is unresolved. In this paper, I first consider a canonical large deviations criterion for optimality and show that inference based on the empirical likelihood ratio statistic is optimal. Second, I introduce a new empirical likelihood bootstrap that provides a valid resampling method for moment inequality models and overcomes the implementation challenges that arise as a result of non-pivotal limit distributions. Lastly, I analyze the finite sample properties of the proposed framework using Monte Carlo simulations. The simulation results are encouraging.

The paper considers estimation of a model yi = D · F(x′iβ0, ui), where the composite transformation D · F is only specified that is non-degenerate monotonic and is strictly monotonic in each of its variables. The paper thus generalizes standard data analysis which assumes that the functional form of D · F is known and additive. The estimator which it proposes is the maximum rank correlation estimator which is non-parametric in the functional form of D · F and non-parametric in the distribution of the error terms, ui. The estimator is shown to be strongly consistent for the parameters β0 up to a scale coefficient.

The topic of this paper is inference in models in which parameters are defined by moment inequalities and/or equalities. The parameters may or may not be identified. This paper introduces a new class of confidence sets and tests based on generalized moment selection (GMS). GMS procedures are shown to have correct asymptotic size in a uniform sense and are shown not to be asymptotically conservative. The power of GMS tests is compared to that of subsampling, m out of n bootstrap, and “plug-in asymptotic” (PA) tests. The latter three procedures are the only general procedures in the literature that have been shown to have correct asymptotic size (in a uniform sense) for the moment inequality/equality model. GMS tests are shown to have asymptotic power that dominates that of subsampling, m out of n bootstrap, and PA tests. Subsampling and m out of n bootstrap tests are shown to have asymptotic power that dominates that of PA tests.

We provide a tractable characterization of the sharp identification region of the parameters θ in a broad class of incomplete econometric models. Models in this class have set-valued predictions that yield a convex set of conditional or unconditional moments for the model variables. In short, we call these models with convex predictions. Examples include static, simultaneous move finite games of complete information in the presence of multiple mixed strategy Nash equilibria; random utility models of multinomial choice in the presence of interval regressors data; and best linear predictors with interval outcome and covariate data. Given a candidate value for θ, we establish that the convex set of moments yielded by the model predictions can be represented as the Aumann expectation of a properly defined random set. The sharp identification region of θ, denoted ΘI, can then be obtained as the set of minimizers of the distance from a properly specified vector of moments of random variables to this Aumann expectation. We show that algorithms in convex programming can be exploited to efficiently verify whether a candidate θ is in ΘI. We use examples analyzed in the literature to illustrate the gains in identification and computational tractability afforded by our method.This paper is a revised version of cemmap working paper CWP15/08

This paper offers a glimpse into the theory of empirical processes. Two asymptotic problems are sketched as motivation for the study of maximal inequalities for stochastic processes made up of properly standardized sums of random variables--empirical processes. The exposition develops the technique of Gaussian symmetrization, which is the least technical of the techniques to have evolved during the last decade of empirical process research. The resulting maximal inequalities are useful because they depend on quantities that can be bounded using simple methods. These methods, which extend the concept of a Vapnik-Cervonenkis class of sets, are demonstrated by use of the two motivating asymptotic problems. The paper is not intended as a complete survey of the state of empirical process theory; it certainly does not present the whole range of available techniques. It is written as an attempt to convey the look and feel of a very powerful, very useful, and tractable tool of contemporary mathematical statistics.

When \(\mathfrak{F}\) is a universal Donsker class, then for independent, indetically distributed (i.i.d) observation
\(\mathbf{X}_1,\ldots,\mathbf{X}_n\)
with an unknown law P, for any
\(\mathfrak{f}_i\)in \(\mathfrak{F},\)
\(i=1,\ldots,m,\quad n^{-1/2}\left\{
\mathfrak{f}_1\left(\mathbf{X}_1\right)+\ldots+\mathfrak{f}_i\left(\mathbf{X}_n\right)\right\}_{1\leq
i\leq m}\)
is asymptotically normal with mean Vector
\(n^{1/2}\left\{\int\mathfrak{f}_i\left(\mathbf{X}_n\right)d\mathbf{P}\left(x\right)\right\}_{1_\leq i\leq m}\)
and covariance matrix \(\int\mathfrak{f}_i\mathfrak{f}_j d\mathbf{P}-\int\mathfrak{f}_id\mathbf{P}\int\mathfrak{f}_jd\mathbf{P},\)
uniformly for \({\mathfrak{f}_i}\in \mathfrak{F}.\)
Then, for certain Statistics formed frome the
\(\mathfrak{f}_i\left(\mathbf{X}_k\right),\)
even where \(\mathfrak{f}_i\) may be chosen depending on the \(\mathbf{X}_k\) there will be asymptotic distribution as \(n \rightarrow \infty.\)
For example, for
\(\mathbf{X}^2\) statistics, where \(f_i\) are indicators of disjoint intervals, depending suitably on \(\mathbf{X}_1,\ldots,\mathbf{X}_n\), whose union is the real line, \(\mathbf{X}^2\) quadratic forms have limiting distributions [Roy (1956) and Watson (1958)] which may, however, not be \(\mathbf{X}^2\) distributions and may depend on P [Chernoff and Lehmann (1954)]. Universal Donsker classes of sets are, up to mild measurability conditions, just classes satisfying the Vapnik–Červonenkis comdinatorial conditions defined later in this section Donsker the Vapnik-Červonenkis combinatorial conditions defined later in this section [Durst and Dudley (1981) and Dudley (1984) Chapter 11]. The use of such classes allows a variety of extensions of the Roy–Watson results to general (multidimensional) sample spaces [Pollard (1979) and Moore and Subblebine (1981)]. Vapnik and Červonenkis (1974) indicated application of their families of sets to classification (pattern recognition) problems. More recently, the classes have been applied to tree-structured classifiacation [Breiman, Friedman, Olshen and Stone (1984), Chapter 12].

We consider games with incomplete information a la Harsanyi, where the payoff of a player depends on an unknown state of nature as well as on the profile of chosen actions. As opposed to the standard model, players' preferences over state--contingent utility vectors are represented by arbitrary functionals. The definitions of Nash and Bayes equilibria naturally extend to this generalized setting. We characterize equilibrium existence in terms of the preferences of the participating players. It turns out that, given continuity and monotonicity of the preferences, equilibrium exists in every game if and only if all players are averse to uncertainty (i.e., all the functionals are quasi--concave). We further show that if the functionals are either homogeneous or translation invariant then equilibrium existence is equivalent to concavity of the functionals.

This paper provides a survey on studies that analyze the macroeconomic effects of intellectual property rights (IPR). The first part of this paper introduces different patent policy instruments and reviews their effects on R&D and economic growth. This part also discusses the distortionary effects and distributional consequences of IPR protection as well as empirical evidence on the effects of patent rights. Then, the second part considers the international aspects of IPR protection. In summary, this paper draws the following conclusions from the literature. Firstly, different patent policy instruments have different effects on R&D and growth. Secondly, there is empirical evidence supporting a positive relationship between IPR protection and innovation, but the evidence is stronger for developed countries than for developing countries. Thirdly, the optimal level of IPR protection should tradeoff the social benefits of enhanced innovation against the social costs of multiple distortions and income inequality. Finally, in an open economy, achieving the globally optimal level of protection requires an international coordination (rather than the harmonization) of IPR protection.

We empirically test existing theories on the provision of public goods, in particular air quality, using data on sulfur dioxide (SO2) concentrations from the Global Environment Monitoring Projects for 107 cities in 42 countries from 1971 to 1996. The results are as follows: First, we provide additional support for the claim that the degree of democracy has an independent positive effect on air quality. Second, we find that among democracies, presidential systems are more conducive to air quality than parliamentary ones. Third, in testing competing claims about the effect of interest groups on public goods provision in democracies we establish that labor union strength contributes to lower environmental quality, whereas the strength of green parties has the opposite effect.

We investigate identification in semi-parametric binary regression models, y = 1(xβ+υ+ε > 0) when υ is either discrete or measured within intervals. The error term ε is assumed to be uncorrelated with a set
of instruments z, ε is independent of υ conditionally on x and z, and the support of −(xβ + ε) is finite. We provide a sharp characterization of the set of observationally equivalent parameters β. When there are
as many instruments z as variables x, the bounds of the identified intervals of the different scalar components βk of parameter β can be expressed as simple moments of the data. Also, in the case of interval data, we show that additional
information on the distribution of υ within intervals shrinks the identified set. Specifically, the closer the conditional
distribution of υ given z is to uniformity, the smaller is the identified set. Point identified is achieved if and only if υ is uniform within intervals.

In 1971, President Nixon declared war on cancer. Thirty years later, many declared this war a failure: the age-adjusted mortality rate from cancer in 2000 was essentially the same as in the early 1970s. Meanwhile the age-adjusted mortality rate from cardiovascular disease fell dramatically. Since the causes that underlie cancer and cardiovascular disease are likely dependent, the decline in mortality rates from cardiovascular disease may partially explain the lack of progress in cancer mortality. Because competing risks models (used to model mortality from multiple causes) are fundamentally unidentified, it is difficult to estimate cancer trends. We derive bounds for aspects of the underlying distributions without assuming that the underlying risks are independent. We then estimate changes in cancer and cardiovascular mortality since 1970. The bounds for the change in duration until death for either cause are fairly tight and suggest much larger improvements in cancer than previously estimated. Copyright The Econometric Society 2006.

This paper examines inference on regressions when interval data are available on one variable, the other variables being measured precisely. Let a population be characterized by a distribution "P"("y", "x", "v", "v"-sub-0, "v"-sub-1), where "y" is an element of "R"-super-1, "x" is an element of "R-super-k", and the real variables ("v", "v"-sub-0, "v"-sub-1) satisfy "v"-sub-0≤"v"≤"v"-sub-1. Let a random sample be drawn from "P" and the realizations of ("y", "x", "v"-sub-0, "v"-sub-1) be observed, but not those of "v". The problem of interest may be to infer "E"("y"|"x", "v") or "E"("v"|"x"). This analysis maintains Interval (I), Monotonicity (M), and Mean Independence (MI) assumptions: (I) "P"("v"-sub-0≤"v"≤"v"-sub-1)=1; (M) "E"("y"|"x", "v") is monotone in "v"; (MI) "E"("y"|"x", "v", "v"-sub-0, "v"-sub-1)="E"("y"|"x", "v"). No restrictions are imposed on the distribution of the unobserved values of "v" within the observed intervals ["v"-sub-0, "v"-sub-1]. It is found that the IMMI Assumptions alone imply simple nonparametric bounds on "E"("y"|"x", "v") and "E"("v"|"x"). These assumptions invoked when "y" is binary and combined with a semiparametric binary regression model yield an identification region for the parameters that may be estimated consistently by a "modified maximum score (MMS)" method. The IMMI assumptions combined with a parametric model for "E"("y"|"x", "v") or "E"("v"|"x") yield an identification region that may be estimated consistently by a "modified minimum-distance (MMD)" method. Monte Carlo methods are used to characterize the finite-sample performance of these estimators. Empirical case studies are performed using interval wealth data in the Health and Retirement Study and interval income data in the Current Population Survey. Copyright The Econometric Society 2002.