Xiaoxia Shi

Inference Based on Conditional Moment Inequalities

In this paper, we propose an instrumental variable approach to constructing confidence sets (CSs) 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 CSs 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 CSs 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 have power against n^{-1/2}-local alternatives to some, but not all, sequences of distributions in the null hypothesis. Monte Carlo simulations for four different models show that the methods perform well in finite samples.

Nonlinear Cointegrating Regression Under Weak Identification

An asymptotic theory is developed for a weakly identified cointegrating regression model in which the regressor is a nonlinear transformation of an integrated process. Weak identification arises from the presence of a loading coefficient for the nonlinear function that may be close to zero. In that case, standard nonlinear cointegrating limit theory does not provide good approximations to the finite sample distributions of nonlinear least squares estimators, resulting in potentially misleading inference. A new local limit theory is developed that approximates the finite sample distributions of the estimators uniformly well irrespective of the strength of the identification. An important technical component of this theory involves new results showing the uniform weak convergence of sample covariances involving nonlinear functions to mixed normal and stochastic integral limits. Based on these asymptotics, we construct confidence intervals for the loading coefficient and the nonlinear transformation parameter and show that these confidence intervals have correct asymptotic size. As in other cases of nonlinear estimation with integrated processes and unlike stationary process asymptotics, the properties of the nonlinear transformations affect the asymptotics and, in particular, give rise to parameter dependent rates of convergence and differences between the limit results for integrable and asymptotically homogeneous functions.

A nondegenerate Vuong test

In this paper, I propose a one‐step nondegenerate test as an alternative to the classical Vuong (1989) tests. I show that the new test achieves uniform asymptotic size control in both the overlapping and the non‐overlapping cases, while the classical Vuong tests do not. Meanwhile, the power of the new test can be substantially better than the two‐step classical Vuong test and is not dominated by the one‐step classical Vuong test. An extension to moment‐based models is also developed. I apply the new test to the voter turnout data set of Coate and Conlin (2004) and find that it can yield model comparison conclusions different from those of the classical tests. The implementation of the new test is straightforward and can be done using the MATLAB and STATA routines that accompany this paper.

Inference for subvectors and other functions of partially identified parameters in moment inequality models

This paper introduces a bootstrap-based inference method for functions of the parameter vector in a moment (in)equality model. These functions are restricted to be linear for two-sided testing problems, but may be non-linear for one-sided testing problems. In the most common case, this function selects a subvector of the parameter, such as a single component. The new inference method we propose controls asymptotic size uniformly over a large class of data distributions and improves upon the two existing methods that deliver uniform size control for this type of problem: projection-based and subsampling inference. Relative to projection-based procedures, our method presents three advantages: (i) it weakly dominates in terms of nite sample power, (ii) it strictly dominates in terms of asymptotic power, and (iii) it is typically less computationally demanding. Relative to subsampling, our method presents two advantages: (i) it strictly dominates in terms of asymptotic power (for reasonable choices of subsample size), and (ii) it appears to be less sensitive to the choice of its tuning parameter than subsampling is to the choice of subsample size.

Can Words Get in the Way? The Effect of Deliberation in Collective Decision-Making

We quantify the effect of deliberation on the decisions of US appellate courts. We estimate a model in which strategic judges communicate before casting their votes and then compare the probability of mistakes in the court with deliberation with a counterfactual of no communication. The model has multiple equilibria, and preferences and information parameters are only partially identified. We find that there is a range of parameters in the identified set--when judges tend to disagree ex ante or their private information is imprecise--in which deliberation can be beneficial; otherwise, deliberation reduces the effectiveness of the court.

Estimating Semi-Parametric Panel Multinomial Choice Models Using Cyclic Monotonicity

This paper proposes a new semi‐parametric identification and estimation approach to multinomial choice models in a panel data setting with individual fixed effects. Our approach is based on cyclic monotonicity, which is a defining convex‐analytic feature of the random utility framework underlying multinomial choice models. From the cyclic monotonicity property, we derive identifying inequalities without requiring any shape restrictions for the distribution of the random utility shocks. These inequalities point identify model parameters under straightforward assumptions on the covariates. We propose a consistent estimator based on these inequalities.

Simple Two-Stage Inference for a Class of Partially Identified Models

This paper proposes a new two-stage estimation and inference procedure for a class of partially identified models. The procedure can be considered an extension of classical minimum distance estimation procedures to accommodate inequality constraints and partial identification. It involves no tuning parameter, is nonconservative, and is conceptually and computationally simple. The class of models includes models of interest to applied researchers, including the static entry game, a voting game with communication, and a discrete mixture model. Besides, a technical contribution is an implicit correspondence lemma which generalizes the implicit function theorem to multivalued implicit maps.

On the Empirical Content of the Beckerian Marriage Model

Abstract This note studies the empirical content of a simple marriage matching model with transferable utility, based on Becker (J Polit Econ 81:813–846, 1973). Under Becker’s conditions, the equilibrium matching is unique and assortative. However, this note shows that when the researcher only observes a subset of relevant characteristics, the unique assortative matching does not uniquely determine a distribution of observed characteristics. This precludes standard approaches to point estimation of the underlying model parameters. We propose a solution to this problem, based on the idea of “random matching.”

Inference on Estimators defined by Mathematical Programming

We propose an inference procedure for estimators defined by mathematical programming problems, focusing on the important special cases of linear programming (LP) and quadratic programming (QP). In these settings, the coefficients in both the objective function and the constraints of the mathematical programming problem may be estimated from data and hence involve sampling error. Our inference approach exploits the characterization of the solutions to these programming problems by complementarity conditions; by doing so, we can transform the problem of doing inference on the solution of a constrained optimization problem (a non-standard inference problem) into one involving inference based on a set of inequalities with pre-estimated coefficients, which is much better understood. We evaluate the performance of our procedure in several Monte Carlo simulations and an empirical application to the classic portfolio selection problem in finance.

Model Selection Tests for Conditional Moment Restriction Models

We propose a Vuong (1989)-type model-selection test for models defined by conditional moment restrictions. The moment restrictions that define the models can be standard equality restrictions that point-identify the model parameters, or moment equality or inequality restrictions that partially identify the model parameters. The test uses a new average generalized empirical likelihood criterion function designed to incorporate full restriction of the conditional model. We also introduce a new adjustment to the test statistic that makes it asymptotically pivotal whether the candidate models are nested or nonnested. The test uses simple standard normal critical values and is shown to be asymptotically similar, to be consistent against all fixed alternatives, and to have nontrivial power against n−1=2-local alternatives. Monte Carlo simulations demonstrate that the finite sample performance of the test is in accordance with the theoretical prediction. This article is protected by copyright. All rights reserved