arXiv subjects

Christopher D. Walker

Publications and source records attributed to Christopher D. Walker.

8 recordsLinked to original sources

arXiv2026

Nonparametric Bayesian Inference for Partially Identified Discrete Response Models

This paper proposes a nonparametric Bayesian inference framework for partially identified discrete response models. The key observation is that these models map a reduced-form conditional choice probability to an identified set. Consequently, nonparametric Bayesian inference for the conditional probability mass function leads to Bayesian inference for the identified set. The inference framework nests conditional moment inequalities and linear systems with unknown coefficients as special cases. Importantly, our proposal does not require converting conditional moments into unconditional moments or discretizing covariates. We show that the posterior is consistent for the true identified set when the model is correctly specified, show that the posterior can consistently detect model misspecification, and show posterior consistency for a pseudo-identified set that is valid under misspecification. We also verify the assumptions for a class of priors based on Gaussian processes that we use to implement our proposal. These priors offer similar flexibility to frequentist partial identification methods, and are computationally attractive because posterior sampling can be performed in closed-form. We also show that many of the ideas in this paper extend to continuous responses and aggregated discrete responses (e.g., market shares).

econ.EM↗

arXiv2026

Semiparametric Bayesian Inference for a Conditional Moment Equality Model

I propose a semiparametric Bayesian inference framework for conditional moment equalities. The core idea is that these models deterministically map a conditional distribution of data to a structural parameter via the restriction that a conditional expectation equals zero. Consequently, a posterior for the conditional distribution leads to a posterior for the structural parameter by minimizing the distance of the conditional moments to zero. The method has similar flexibility to frequentist semiparametric estimators and does not require converting the conditional moments into unconditional moments. I also establish frequentist asymptotic optimality of my proposal via a semiparametric Bernsteinvon Mises theorem (BvM), which establishes that the posterior for the structural parameter is asymptotically normal and matches the Chamberlain (1987) semiparametric efficiency bound. The BvM conditions are verified for Gaussian process priors and complement the numerical aspects of the paper in which these priors are used to estimate welfare effects.

econ.EM↗

arXiv2025

Parametrization, Prior Independence, and the Semiparametric Bernstein-von Mises Theorem for the Partially Linear Model

I prove a semiparametric Bernstein-von Mises theorem for a partially linear regression model with independent priors for the low-dimensional parameter of interest and the infinite-dimensional nuisance parameters. My result avoids a challenging prior invariance condition that arises from a loss of information associated with not knowing the nuisance parameter. The key idea is to employ a feasible reparametrization of the partially linear regression model that reflects the semiparametric structure of the model. This allows a researcher to assume independent priors for the model parameters while automatically accounting for the loss of information associated with not knowing the nuisance parameters. The theorem is verified for uniform wavelet series priors and Matérn Gaussian process priors.

math.ST↗

arXiv2024

A Bayesian Perspective on the Maximum Score Problem

This paper presents a Bayesian inference framework for a linear index threshold-crossing binary choice model that satisfies a median independence restriction. The key idea is that the model is observationally equivalent to a probit model with nonparametric heteroskedasticity. Consequently, Gibbs sampling techniques from Albert and Chib (1993) and Chib and Greenberg (2013) lead to a computationally attractive Bayesian inference procedure in which a Gaussian process forms a conditionally conjugate prior for the natural logarithm of the skedastic function.

econ.EM↗

arXiv2020

Inference for Moment Inequalities: A Constrained Moment Selection Procedure

Inference in models where the parameter is defined by moment inequalities is of interest in many areas of economics. This paper develops a new method for improving the performance of generalized moment selection (GMS) testing procedures in finite-samples. The method modifies GMS tests by tilting the empirical distribution in its moment selection step by an amount that maximizes the empirical likelihood subject to the restrictions of the null hypothesis. We characterize sets of population distributions on which a modified GMS test is (i) asymptotically equivalent to its non-modified version to first-order, and (ii) superior to its non-modified version according to local power when the sample size is large enough. An important feature of the proposed modification is that it remains computationally feasible even when the number of moment inequalities is large. We report simulation results that show the modified tests control size well, and have markedly improved local power over their non-modified counterparts.

econ.EM↗

arXiv2010

Hall Algebras as Hopf Objects

One problematic feature of Hall algebras is the fact that the standard multiplication and comultiplication maps do not satisfy the bialgebra compatibility condition in the underlying symmetric monoidal category Vect. In the past this problem has been resolved by working with a weaker structure called a `twisted' bialgebra. In this paper we solve the problem differently by first switching to a different underlying category Vect^K of vector spaces graded by a group K called the Grothendieck group. We equip this category with a nontrivial braiding which depends on the K-grading. With this braiding, we find that the Hall algebra does satisfy the bialgebra condition exactly for the standard multiplication and comultiplication, and can also be equipped with an antipode, making it a Hopf algebra object in Vect^K.

math.QA↗

arXiv2010

Higher-Dimensional Algebra VII: Groupoidification

Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of "degroupoidification": a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present three applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang-Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field with q elements. The third application is to Hall algebras. We explain how the standard construction of the Hall algebra from the category of representations of a simply-laced quiver can be seen as an example of degroupoidification. This in turn provides a new way to categorify - or more precisely, groupoidify - the positive part of the quantum group associated to the quiver.

math.QA↗

arXiv2008

Groupoidification Made Easy

Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of 'degroupoidification': a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present two applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang-Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field with q elements.

math.QA↗