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Non-Standard Rates of Convergence of Criterion-Function-Based Set Estimators for Binary Response Models

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Abstract

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. ...
Preprint
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. ...
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... 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. ...
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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). ...
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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].
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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.
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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.
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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.
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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.
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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.
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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.