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Penalized GMM Framework for Inference on Functionals of Nonparametric Instrumental Variable Estimators

Edvard Bakhitov

Abstract

This paper develops a penalized GMM (PGMM) framework for automatic debiased inference on functionals of nonparametric instrumental variable estimators. We derive convergence rates for the PGMM estimator and provide conditions for root-n consistency and asymptotic normality of debiased functional estimates, covering both linear and nonlinear functionals. Monte Carlo experiments on average derivative show that the PGMM-based debiased estimator performs on par with the analytical debiased estimator that uses the known closed-form Riesz representer, achieving 90-96% coverage while the plug-in estimator falls below 5%. We apply our procedure to estimate mean own-price elasticities in a semiparametric demand model for differentiated products. Simulations confirm near-nominal coverage while the plug-in severely undercovers. Applied to IRI scanner data on carbonated beverages, debiased semiparametric estimates are approximately 20% more elastic compared to the logit benchmark, and debiasing corrections are heterogeneous across products, ranging from negligible to several times the standard error.

Penalized GMM Framework for Inference on Functionals of Nonparametric Instrumental Variable Estimators

Abstract

This paper develops a penalized GMM (PGMM) framework for automatic debiased inference on functionals of nonparametric instrumental variable estimators. We derive convergence rates for the PGMM estimator and provide conditions for root-n consistency and asymptotic normality of debiased functional estimates, covering both linear and nonlinear functionals. Monte Carlo experiments on average derivative show that the PGMM-based debiased estimator performs on par with the analytical debiased estimator that uses the known closed-form Riesz representer, achieving 90-96% coverage while the plug-in estimator falls below 5%. We apply our procedure to estimate mean own-price elasticities in a semiparametric demand model for differentiated products. Simulations confirm near-nominal coverage while the plug-in severely undercovers. Applied to IRI scanner data on carbonated beverages, debiased semiparametric estimates are approximately 20% more elastic compared to the logit benchmark, and debiasing corrections are heterogeneous across products, ranging from negligible to several times the standard error.

Paper Structure

This paper contains 68 sections, 31 theorems, 299 equations, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Flexible estimation under endogeneity
  3. Nonparametric IV framework
  4. Practical concerns and MLIV estimators
  5. Learning functionals of MLIV estimators
  6. Functionals of interest and economic examples
  7. Orthogonal moment condition
  8. Estimation of the Riesz representer
  9. Choice of the weight matrix
  10. Informal preview of estimation and inference results
  11. Properties of the PGMM estimator
  12. Inference for linear functionals
  13. Nonlinear functionals
  14. Finite Sample Performance
  15. Application to nonparametric demand estimation
  16. ...and 53 more sections

Key Result

Theorem 1

Suppose Assumptions ass:weight_matrix--ass:m_bound are satisfied. Let $\varepsilon_{n} = o(\lambda_{n})$ and $\bar{s}\lambda_n = o(1)$, then $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (58)

  • Example 1
  • Example 2
  • Example 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Lemma 6.1
  • Theorem 4
  • Corollary 2
  • ...and 48 more