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Nonparametric Identification of Demand without Exogenous Product Characteristics

Kirill Borusyak, Jiafeng Chen, Peter Hull, Lihua Lei

TL;DR

The paper tackles the challenge of identifying price counterfactuals in differentiated-product demand when product characteristics may be endogenous. It introduces recentered instruments that combine exogenous price variation with endogenous characteristics and a faithfulness condition to identify price counterfactuals under a weaker index restriction on $δ$. It shows that, under exogenous prices, completeness suffices for identification, while with endogenous prices faithfulness serves as the central identifying condition, closely linked to completeness. The findings offer practical guidance for robust parametric estimation using recentered instruments and highlight market-level data as a viable source for identification, potentially broadening the toolkit for structural demand analysis.

Abstract

We study identification of differentiated product demand from market-level data when product characteristics can be endogenous. Past work suggests nonparametric identification may be impossible: that is, in addition to standard price instruments, exogenous characteristic-based instruments are essentially necessary to identify sufficiently flexible demand models with standard index restrictions. We show, however, that price counterfactuals are nonparametrically identified using recentered instruments -- which combine exogenous price instruments with possibly endogenous product characteristics -- under a weaker index restriction and a new condition we term faithfulness. We argue that faithfulness, like the usual completeness condition for nonparametric instrumental variable identification, is best viewed as a technical requirement on the strength of identifying variation rather than a substantive economic or statistical restriction. We show the two conditions are closely related, though generally distinct. We conclude with several practical implications for the parametric estimation of demand counterfactuals.

Nonparametric Identification of Demand without Exogenous Product Characteristics

TL;DR

The paper tackles the challenge of identifying price counterfactuals in differentiated-product demand when product characteristics may be endogenous. It introduces recentered instruments that combine exogenous price variation with endogenous characteristics and a faithfulness condition to identify price counterfactuals under a weaker index restriction on . It shows that, under exogenous prices, completeness suffices for identification, while with endogenous prices faithfulness serves as the central identifying condition, closely linked to completeness. The findings offer practical guidance for robust parametric estimation using recentered instruments and highlight market-level data as a viable source for identification, potentially broadening the toolkit for structural demand analysis.

Abstract

We study identification of differentiated product demand from market-level data when product characteristics can be endogenous. Past work suggests nonparametric identification may be impossible: that is, in addition to standard price instruments, exogenous characteristic-based instruments are essentially necessary to identify sufficiently flexible demand models with standard index restrictions. We show, however, that price counterfactuals are nonparametrically identified using recentered instruments -- which combine exogenous price instruments with possibly endogenous product characteristics -- under a weaker index restriction and a new condition we term faithfulness. We argue that faithfulness, like the usual completeness condition for nonparametric instrumental variable identification, is best viewed as a technical requirement on the strength of identifying variation rather than a substantive economic or statistical restriction. We show the two conditions are closely related, though generally distinct. We conclude with several practical implications for the parametric estimation of demand counterfactuals.
Paper Structure (41 sections, 26 theorems, 239 equations, 1 figure)

This paper contains 41 sections, 26 theorems, 239 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Theory
  3. Setup
  4. Identification strategy
  5. A surprisingly insightful case: exogenous prices
  6. Faithfulness and identification with endogenous prices
  7. Relationship between completeness and faithfulness
  8. When does completeness imply faithfulness?
  9. Restrictions on price-setting
  10. Restrictions on the $\delta$ index
  11. When does faithfulness imply completeness?
  12. Practical Implications
  13. Conclusion
  14. Detailed Statements and Proofs
  15. Setup
  16. ...and 26 more sections

Key Result

lemma 1

Let $h$ be $\mathbb{R}^J$-valued, invertible, and square-integrable. Then $h\in \Theta_I$ if and only if $\mathbb{E}[h(S, P) \left\{{R( X, Z) - \mathbb{E}[R( X,Z) \mid X]}\right\} ] = 0$ for all square-integrable $R$.

Figures (1)

  • Figure B.1: DAG for $(X,P,Z,\delta, H(\delta, P))$.

Theorems & Definitions (54)

  • lemma 1
  • lemma 2
  • lemma A.1
  • proof
  • lemma A.2
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 44 more