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Information Based Inference in Models with Set-Valued Predictions and Misspecification

Hiroaki Kaido, Francesca Molinari

TL;DR

An information-based inference method for partially identified parameters in incomplete models that is valid both when the model is correctly specified and when it is misspecified and relies on Rao's score statistic, which is shown to be asymptotically pivotal.

Abstract

This paper proposes an information-based inference method for partially identified parameters in incomplete models that is valid both when the model is correctly specified and when it is misspecified. Key features of the method are: (i) it is based on minimizing a suitably defined Kullback-Leibler information criterion that accounts for incompleteness of the model and delivers a non-empty pseudo-true set; (ii) it is computationally tractable; (iii) its implementation is the same for both correctly and incorrectly specified models; (iv) it exploits all information provided by variation in discrete and continuous covariates; (v) it relies on Rao's score statistic, which is shown to be asymptotically pivotal.

Information Based Inference in Models with Set-Valued Predictions and Misspecification

TL;DR

An information-based inference method for partially identified parameters in incomplete models that is valid both when the model is correctly specified and when it is misspecified and relies on Rao's score statistic, which is shown to be asymptotically pivotal.

Abstract

This paper proposes an information-based inference method for partially identified parameters in incomplete models that is valid both when the model is correctly specified and when it is misspecified. Key features of the method are: (i) it is based on minimizing a suitably defined Kullback-Leibler information criterion that accounts for incompleteness of the model and delivers a non-empty pseudo-true set; (ii) it is computationally tractable; (iii) its implementation is the same for both correctly and incorrectly specified models; (iv) it exploits all information provided by variation in discrete and continuous covariates; (v) it relies on Rao's score statistic, which is shown to be asymptotically pivotal.
Paper Structure (21 sections, 15 theorems, 126 equations, 4 figures, 4 tables, 1 algorithm)

This paper contains 21 sections, 15 theorems, 126 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and Motivating Example
  3. Information-Based Inference Robust to Misspecification
  4. The Set of Model-Implied Density Functions
  5. Correct Specification, Misspecification, and Pseudo-True Set
  6. The Score Function
  7. Geometry of the Pseudo-True Set
  8. Asymptotic Distribution of the Average Score and Rao's Test Statistic
  9. Role of Sharp Identifying Restrictions
  10. Computation of the Score Function
  11. Empirical Illustration
  12. Monte Carlo Experiments
  13. Conclusions
  14. Proofs of Main Theorems
  15. Lemmas Used in Proofs of Main Theorems
  16. ...and 6 more sections

Key Result

Theorem 3.1

Under Assumption ass:finite_support, (i) $L(\theta|x)$ is differentiable with respect to $\theta$ on $\mathrm{int}(\Theta)$, $P_0-a.s.$ (ii) There exists a function $s:\Theta\times \mathcal{Y}\times\mathcal{X}\times\Delta\to \mathbb{R}^{d_\theta}$, with $\Delta$ the unit-simplex in $\mathbb{R}^{|\ma

Figures (4)

  • Figure 1: Stylized depiction of $G(\cdot|x;\theta)$ in Example \ref{['example:CT']} with $\delta_1< 0,\delta_2< 0$.
  • Figure 2: Confidence Intervals for $\Phi(x_{LCC,m}^\top\beta^*_{LCC}+\delta^*_{LCC} d)$ for $d=1$ (orange) and $d=0$ (blue). Panel (a): Rao score test-based inference with $X_{LCC,m}^{pres}$ set equal to the $\tau$ quantile of its distribution and $X_m^{size}$ set equal to its median. Panel (b): Chen_2018 projection-based inference with $\mathbf{1}(X_{j,m}^{pres}> Med(X_j^{pres}))$.
  • Figure 3: Projections of $\Theta^*(p_{0})$ (left, correctly specified; right, misspecified with $\gamma=-0.4$)
  • Figure 4: Stylized depictions of $G(\cdot|x;\theta)$ in Example \ref{['example:BCMT']} (Panel (a), with $\pi(X_{j},U;\theta)=\pi(X_{j};\theta)+U_j$, $\mathcal{J}=\{1,2,3\}$, $\kappa=2$, and $\bar{\pi}(j,k;x) \equiv \pi(x_{k};\theta)-\pi(x_{j};\theta)$) and Example \ref{['example:panel']} (Panel (b), with $\beta\ge 0$).

Theorems & Definitions (40)

  • Example 1: Static entry game
  • Definition 3.1: Correctly Specified Model & Misspecified Model
  • Remark 3.1
  • Definition 3.2: KLIC for set of density functions
  • Definition 3.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 3.1
  • Theorem 3.4
  • ...and 30 more