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25
EL Inference for Partially Identified Models: Large Deviations Optimality and Bootstrap Validity
, 2008
"... 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. ..."
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Cited by 60 (5 self)
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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. This finding is a direct analog to that in Kitamura (2001) for moment equality models. 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.
On the Testability of Identification in Some Nonparametric Models with Endogeneity
, 2013
"... This paper examines three distinct hypothesis testing problems that arise in the context of identification of some nonparametric models with endogeneity. The first hypothesis testing problem we study concerns testing necessary conditions for identification in some nonparametric models with endogenei ..."
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Cited by 11 (1 self)
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This paper examines three distinct hypothesis testing problems that arise in the context of identification of some nonparametric models with endogeneity. The first hypothesis testing problem we study concerns testing necessary conditions for identification in some nonparametric models with endogeneity involving mean independence restrictions. These conditions are typically referred to as completeness conditions. The second and third hypothesis testing problems we examine concern testing for identification directly in some nonparametric models with endogeneity involving quantile independence restrictions. For each of these hypothesis testing problems, we provide conditions under which any test will have power no greater than size against any alternative. In this sense, we conclude that no nontrivial tests for these hypothesis testing problems exist.
Instrumental Variables Methods for Recovering Continuous Linear Functionals
- Journal of Econometrics
, 2011
"... This paper develops methods for estimating continuous linear functionals in a nonparametric instrumental variables (IV) setting. Examples of such functionals include consumer surplus and applications to tests for shape restrictions like monotonicity, concavity and additive separability. The estimati ..."
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Cited by 7 (2 self)
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This paper develops methods for estimating continuous linear functionals in a nonparametric instrumental variables (IV) setting. Examples of such functionals include consumer surplus and applications to tests for shape restrictions like monotonicity, concavity and additive separability. The estimation procedure is robust to a setting where the underlying model is not identified but the linear functional of interest is. In order to attain such robustness, it is necessary to use a nuisance parameter that is not identified. A procedure is proposed that circumvents this challenge and delivers a √ n asymptotically normal estimator for the linear functional of interest. A Monte Carlo study examines the finite sample performance of the procedure.
Estimating Demand for Differentiated Products with Error in Market Shares: A Moment Inequalities Approach
, 2013
"... In this paper we introduce a new approach to estimating differentiated product de-mand system that allows for error in market shares as measures of choice probabilities. In particular, our approach allows for products with zero sales in the data, which is a frequent phenomenon that arises in product ..."
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Cited by 7 (2 self)
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In this paper we introduce a new approach to estimating differentiated product de-mand system that allows for error in market shares as measures of choice probabilities. In particular, our approach allows for products with zero sales in the data, which is a frequent phenomenon that arises in product differentiated markets but lies outside the scope of existing demand estimation techniques. Although we find that error in market shares generally undermine the standard point identification of discrete choice models of demand, we exploit shape restrictions on demand implied by discrete choice to gener-ate a system of moment inequalities that partially identify demand parameters. These moment inequalities are fully robust to the variability in market shares yet are also adaptive to the information revealed by market shares in a way that allows for informa-tive inferences. In addition, we construct a profiling approach for parameter inference with moment inequalities, making it feasible to study models with a large number of parameters (as typically required in demand applications) by focusing attention on a profile of the parameters, such as the price coefficient. We use our approach to study consumer demand from scanner data using the Dominick’s Finer Foods database, and find that even for the baseline logit model, demand elasticities nearly double when the full error in market shares is taken into account.
Penalized Sieve Estimation and Inference of Semi-nonparametric Dynamic Models: A Selective Review
, 2011
"... In this selective review, we …rst provide some empirical examples that motivate the usefulness of semi-nonparametric techniques in modelling economic and …nancial time series. We describe popular classes of semi-nonparametric dynamic models and some temporal dependence properties. We then present pe ..."
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Cited by 6 (2 self)
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In this selective review, we …rst provide some empirical examples that motivate the usefulness of semi-nonparametric techniques in modelling economic and …nancial time series. We describe popular classes of semi-nonparametric dynamic models and some temporal dependence properties. We then present penalized sieve extremum (PSE) estimation as a general method for semi-nonparametric models with cross-sectional, panel, time series, or spatial data. The method is especially powerful in estimating di ¢ cult ill-posed inverse problems such as semi-nonparametric mixtures or conditional moment restrictions. We review recent advances on inference and large sample properties of the PSE estimators, which include (1) consistency and convergence rates of the PSE estimator of the nonparametric part; (2) limiting distributions of plug-in PSE estimators of functionals that are either smooth (i.e., root-n estimable) or non-smooth (i.e., slower than root-n estimable); (3) simple criterion-based inference for plug-in PSE estimation of smooth or non-smooth functionals; and (4) root-n asymptotic normality of semiparametric two-step estimators and their consistent variance estimators. Examples from dynamic asset pricing, nonlinear spatial VAR, semiparametric GARCH,
Supplement to “Quasi-Bayesian analysis of nonparametric instrumental variables models
, 2013
"... ar ..."
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Sieve Wald and QLR Inferences on Semi/nonparametric Conditional Moment Models1
, 2013
"... This paper considers inference on functionals of semi/nonparametric conditional moment re-strictions with possibly nonsmooth generalized residuals, which include all of the (nonlinear) non-parametric instrumental variables (IV) as special cases. For these models it is often difficult to verify wheth ..."
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Cited by 6 (1 self)
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This paper considers inference on functionals of semi/nonparametric conditional moment re-strictions with possibly nonsmooth generalized residuals, which include all of the (nonlinear) non-parametric instrumental variables (IV) as special cases. For these models it is often difficult to verify whether a functional is regular (i.e., root-n estimable) or irregular (i.e., slower than root-n estimable). We provide computationally simple, unified inference procedures that are asymptot-ically valid regardless of whether a functional is regular or not. We establish the following new useful results: (1) the asymptotic normality of a plug-in penalized sieve minimum distance (PSMD) estimator of a (possibly irregular) functional; (2) the consistency of simple sieve variance estimators of the plug-in PSMD estimator, and hence the asymptotic chi-square distribution of the sieve Wald statistic; (3) the asymptotic chi-square distribution of an optimally weighted sieve quasi likelihood ratio (QLR) test under the null hypothesis; (4) the asymptotic tight distribution of a non-optimally weighted sieve QLR statistic under the null; (5) the consistency of generalized residual bootstrap sieve Wald and QLR tests; (6) local power properties of sieve Wald and QLR tests and of their bootstrap versions; (7) Wilks phenomenon of the sieve QLR test of hypothesis with increasing di-
2013): “Specification Tests for Partially Identified Models Defined by Moment Inequalities,” CeMMAP working paper CWP01/13
"... Abstract This paper studies the problem of specification testing in partially identified models defined by a finite number of moment equalities and inequalities (i.e., (in)equalities). Under the null hypothesis, there is at least one parameter value that simultaneously satisfies all of the moment ( ..."
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Cited by 4 (4 self)
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Abstract This paper studies the problem of specification testing in partially identified models defined by a finite number of moment equalities and inequalities (i.e., (in)equalities). Under the null hypothesis, there is at least one parameter value that simultaneously satisfies all of the moment (in)equalities whereas under the alternative hypothesis there is no such parameter value. While this problem has not been directly addressed in the literature (except in particular cases), several papers have suggested implementing this inferential problem by checking whether confidence intervals for the parameters of interest are empty or not. We propose two hypothesis tests that use the infimum of the sample criterion function over the parameter space as the test statistic together with two different critical values. We obtain two main results. First, we show that the two tests we propose are asymptotically size correct in a uniform sense. Second, we show our tests are more powerful than the test that checks whether the confidence set for the parameters of interest is empty or not.
IDENTIFICATION AND SHAPE RESTRICTIONS IN NONPARAMETRIC INSTRUMENTAL VARIABLES ESTIMATION
, 2012
"... This paper is concerned with inference about an unidentified linear functional, Lg (), where the function g satisfies the relation Y = g ( X) + U; EU ( | W) = 0. In this relation, Y is the dependent variable, X is a possibly endogenous explanatory variable, W is an instrument for X, and U is an un ..."
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Cited by 4 (1 self)
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This paper is concerned with inference about an unidentified linear functional, Lg (), where the function g satisfies the relation Y = g ( X) + U; EU ( | W) = 0. In this relation, Y is the dependent variable, X is a possibly endogenous explanatory variable, W is an instrument for X, and U is an unobserved random variable. The data are an independent random sample of ( Y, XW,). In much applied research, X and W are discrete, and W has fewer points of support than X. Consequently, neither g nor Lg ( ) is nonparametrically identified. Indeed, Lg ( ) can have any value in ( −∞, ∞). In applied research, this problem is typically overcome and point identification is achieved by assuming that g is a linear function of X. However, the assumption of linearity is arbitrary. It is untestable if W is binary, as is the case in many applications. This paper explores the use of shape restrictions, such as monotonicity or convexity, for achieving interval identification of Lg (). Economic theory often provides such shape restrictions. This paper shows that they restrict Lg ( ) to an interval whose upper and lower bounds can be obtained by solving linear programming problems. Inference about the identified interval and the functional Lg ( ) can be carried out by using the bootstrap. An empirical application illustrates the usefulness of shape restrictions for carrying out nonparametric inference about Lg ().
