[Paper Review] Simple Adaptive Size-Exact Testing for Full-Vector and Subvector Inference in Moment Inequality Models

Gregory A. Cox, Xiaoxia Shi|arXiv (Cornell University)|Jul 14, 2019

Statistical Methods and Inference4 citations

TL;DR

This paper proposes a simulation-free, tuning-parameter-free test for moment inequalities that achieves exact or uniformly asymptotically exact size by using a chi-squared critical value with degrees of freedom equal to the rank of active inequalities. The method enables fast, reliable inference for full-vector and subvector parameters in conditional moment inequality models, significantly reducing computational burden while maintaining validity and power.

ABSTRACT

We propose a simple test for moment inequalities that has exact size in normal models with known variance and has uniformly asymptotically exact size more generally. The test compares the quasi-likelihood ratio statistic to a chi-squared critical value, where the degree of freedom is the rank of the inequalities that are active in finite samples. The test requires no simulation and thus is computationally fast and especially suitable for constructing confidence sets for parameters by test inversion. It uses no tuning parameter for moment selection and yet still adapts to the slackness of the moment inequalities. Furthermore, we show how the test can be easily adapted for inference on subvectors for the common empirical setting of conditional moment inequalities with nuisance parameters entering linearly.

Motivation & Objective

• To address the high computational cost of existing moment inequality tests that rely on simulated critical values.
• To eliminate the need for tuning parameters in moment selection, which can compromise test validity.
• To develop a computationally efficient method for constructing confidence sets via test inversion in high-dimensional parameter spaces.
• To extend the method to subvector inference in models with linearly entering nuisance parameters, improving power and reducing dimensionality.
• To ensure exact or uniformly asymptotically exact size under general regularity conditions, especially in normal models with known variance.

Proposed method

• The test uses the quasi-likelihood ratio statistic and compares it to a chi-squared critical value with degrees of freedom equal to the rank of the active inequalities—those holding with equality at the restricted moment estimator.
• Active inequalities are identified as those binding at the restricted estimator, and their rank determines the data-driven degrees of freedom.
• The method avoids simulation entirely, relying only on the observed sample to compute the test statistic and critical value.
• For subvector inference, the nuisance parameters are eliminated via projection, reducing the parameter space to the vector of interest and enabling efficient grid-based confidence set construction.
• The approach leverages linear programming to compute identified sets and projected confidence sets, ensuring computational feasibility.
• The test is applied via test inversion to construct confidence sets, with a derivative-free local optimization algorithm used to check feasibility of moment inequalities under estimated parameters.

Experimental results

Research questions

• RQ1 Can a moment inequality test be constructed that is both computationally efficient and free of simulation and tuning parameters?
• RQ2 Does the proposed test maintain exact or uniformly asymptotically exact size in finite samples and asymptotically?
• RQ3 How does the method perform in subvector inference settings with nuisance parameters entering linearly?
• RQ4 Can the test be adapted to reduce the dimensionality of the parameter grid in confidence set construction?
• RQ5 How does the power of the subvector test compare to the full-vector test and projection-based alternatives?

Key findings

• The proposed test achieves exact size in normal models with known variance and uniformly asymptotically exact size in general models, ensuring valid inference without simulation.
• The method is computationally efficient, avoiding simulation and tuning parameters, which reduces computation time by hundreds of times compared to existing simulation-based tests.
• For subvector inference, the test is less conservative and more powerful than the projection of the full-vector test, as shown in Monte Carlo experiments.
• The identified set for the parameter of interest was computed as [-1.203, -0.757] using linear programming on conditional expectations derived from simulated error draws.
• The algorithm for projected confidence sets uses a derivative-free minimization procedure with multiple random restarts to ensure robustness in finding feasible parameter values.
• The test statistic and critical value are derived directly from the data, with degrees of freedom determined by the rank of active inequalities at the restricted estimator, ensuring data-driven adaptivity to inequality slackness.