Nonparametric Identification of Demand without Exogenous Product Characteristics (2512.23211v1)

Abstract: We study the identification of differentiated product demand with exogenous supply-side instruments, allowing product characteristics to be endogenous. Past analyses have argued that exogenous characteristic-based instruments are essentially necessary given a sufficiently flexible demand model with a suitable index restriction. We show, however, that price counterfactuals are nonparametrically identified by recentered instruments -- which combine exogenous shocks to prices with 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 identification with instruments, can be viewed as a technical requirement on the richness of identifying variation rather than a substantive economic restriction, and we show that it holds under a variety of non-nested conditions on either price-setting or the index.

• The paper's main contribution is showing that nonparametric identification of price counterfactuals can be achieved without exogenous product characteristics by leveraging a faithfulness condition.
• It develops a recentered instrument approach that isolates exogenous price variation, thereby removing the need for characteristic-based instruments and relaxing classic completeness requirements.
• The theoretical findings extend identification strategies in demand analysis and offer robust empirical implications even when product characteristics are endogenously determined.

Nonparametric Identification of Demand without Exogenous Product Characteristics

Introduction and Motivation

The structure of demand systems for differentiated products underlies empirical industrial organization, guiding counterfactual analysis of pricing and welfare. Identification of demand without imposing parametric restrictions is central to credible inference. Seminal works have argued that, in the standard market-level data setup, nonparametric identification of flexible demand models requires both exogenous price instruments and exogenous, excluded product characteristics—chiefly due to the index restriction in random utility models and completeness requirements. This paradigm has rendered characteristic-based instruments (CBIs) essential, but these are problematic if characteristics are endogenously selected by firms, which is plausible in many settings.

This work re-examines the necessity of exogenous CBIs and demonstrates that nonparametric identification of price counterfactuals is possible without exogenous product characteristics, under weaker conditions, provided that the supply-side instruments are sufficiently rich and a technical property called faithfulness is satisfied. The analysis substantially broadens the scope of viable identification strategies for market demand, removing a perceived roadblock for nonparametric analysis in settings with endogenous characteristics.

Background: The Limitation of Classic Completeness-Based Arguments

In Berry and Haile's framework, the key challenge is to invert a general index-space structural equation relating shares, observed and unobserved characteristics, and prices. With endogenous characteristics, the direct application of completeness theorems (e.g., Newey-Powell-Vella, 1999) is blocked. Prior literature argued that in the absence of exogenous CBIs, the supply-side instruments alone cannot trace out the full counterfactual response surface, as price instruments only generate price variation intertwined with equilibrium outcomes, entangling variation in $\delta$ and $P$ in the index. Empirically, this meant the pursuit of excluded exogenous characteristics even though they might not be credible or available.

Main Contributions and Faithfulness Condition

The paper's central technical innovation is to show that, for the identification of price counterfactuals (i.e., for evaluating the effect of exogenous price changes on quantity demanded), it suffices to identify the underlying index function up to an unknown transformation. Provided that this transformation does not depend on price, counterfactual analysis remains valid. The crucial property enabling this is termed faithfulness.

Faithfulness, in this context, requires that the distribution of unobservable demand heterogeneity and prices, conditional on observed characteristics and instruments, is sufficiently rich so that any mean-independence between candidate functions of $(\delta, P)$ and the instruments given observed characteristics implies that function must not depend on price. In essence, faithfulness asserts that every causal effect of price is detectable via the available exogenous price variation, beyond what is captured by the observed characteristics—even if those characteristics are endogenous.

This refines and sharply reduces the stringency of the completeness condition long regarded as essential for nonparametric identification in this context. Notably, faithfulness is interpreted as a technical, not economic, requirement and holds under a wide variety of non-nested scenarios related to either price-setting conduct or the nature of the index.

Identification Strategy and Recentered Instruments

A core technical tool is the recentered instrument (RI) approach: instruments formed by removing conditional means (relative to observed characteristics) from functions of price instruments and characteristics. This recentering strips out the endogenous components while exploiting all exogenous variation in the instruments, sidestepping the invalidity engendered when observed characteristics are endogenous. The paper proves that, for any candidate model, orthogonality of candidate index functions to all recentered instruments is both necessary and sufficient for satisfying the conditional moment restrictions guaranteeing correct price counterfactuals.

Under faithfulness, the set of index functions orthogonal to all recentered instruments can only differ from the true index function by a transformation constant in prices, thereby enabling full nonparametric identification of price counterfactuals.

Sufficient Conditions for Faithfulness

The sufficient conditions under which faithfulness holds are broad and non-nested:

1. Discrete Support: When the dimension of unobserved heterogeneity matches that of observed characteristics and both are discretely supported (with the same cardinality), completeness implies faithfulness.
2. Exogenous Prices: In the knife-edge case where prices themselves are as-good-as-randomly assigned, completeness of the $\delta|X$ distribution suffices. This verifies intuition that excluded characteristics are unnecessary for price counterfactuals when price randomization is present.
3. Price-Setting Index Restrictions: If $P$ is generated from $X$ and $Z$ only via a function of an index that is invertible in $Z$, faithfulness follows.
4. Separable Derivative Structure: For richer price-setting mechanisms, faithfulness is implied when the Jacobian of the price function with respect to the instruments is of a separable form.
5. Location-Scale Models for Heterogeneity: If the demand unobserved heterogeneity, conditional on observed characteristics, can be written in a flexible location-scale family (with mild smoothness and regularity on the scale), then completeness guarantees faithfulness.

Importantly, these conditions cover a wide array of empirical models, including Bertrand-Nash pricing under common cost structures, and permit substantial heterogeneity and endogeneity in characteristics.

Implications, Contradictory Claims, and Connections

The strong claim advanced is that identification of price counterfactuals does not require exogenous excluded characteristics even in highly flexible demand models, in contradiction to the dominant paradigm in empirical IO rooted in the "2$J$-instruments" argument of Berry and Haile. The result holds even with endogenous product characteristics, requiring only that the price instruments generate enough variation for faithfulness.

The analysis further connects the new faithfulness condition to classic results on nonseparable triangular models (e.g., Imbens & Newey 2009, Torgovitsky 2015), and to faithfulness concepts in causal discovery—highlighting the broad applicability beyond demand estimation. Substantively, the new condition is demonstrated to be nearly as weak as completeness in finitely supported cases, with counterexamples arising only in artificial infinite-dimensional scenarios.

Numerical results are not the focus, as the contribution is theoretical; the main empirical implication is that well-designed supply-side (price) instruments, when combined with recentered instrument strategies, are sufficient for credible nonparametric identification of demand responses to price, without the need to find exogenous observed characteristics.

Practical, Theoretical, and Future Considerations

Practically, this work justifies empirical estimation approaches that use supply-side recentered instruments for demand systems, enhancing the robustness of estimation to both endogeneity and model misspecification concerns. It removes previous constraints faced by applied researchers who could not credibly argue for exogeneity of observed characteristics.

Theoretically, the recognition that transformations irrelevant for price counterfactuals do not undermine identification is likely to extend to other counterfactual analyses wherever similar index restrictions and exogeneity structures arise. Future work could extend these identification arguments to broader classes of nonparametric models (including those with micro-level data linking choices and characteristics) and develop concrete nonparametric estimators built on these identification strategies, with a careful analysis of the associated regularity and convergence properties.

Conclusion

This study rigorously establishes that nonparametric identification of price counterfactuals in differentiated product demand models does not require exogenous product characteristics, provided sufficiently rich exogenous price variation and the technical faithfulness condition are satisfied. The necessity of characteristic-based instruments is thus sharply diminished for counterfactual analysis in large classes of models, relaxing a defining constraint in empirical industrial organization. These results open the path to more flexible and robust estimation strategies in modern demand analysis, and the faithfulness condition introduced may have wide implications for identification across applied econometrics and causal inference (2512.23211).