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Nonparametric Identification of Demand without Exogenous Product Characteristics
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Conventional wisdom holds that exogenous supply-side instruments trace out demand curves and identify important counterfactuals: how quantities would change under hypothetical price changes wright1928tariff. Indeed, when considering demand for a single product, agrist2000interpretation show such instruments are enough to nonparametrically identify certain average price elasticities across markets. Modern demand analyses, however, involve multiple differentiated goods and target counterfactuals that change prices in a particular market berry2021foundations. These analyses usually incorporate other kinds of instruments: functions of the observed characteristics of competing products berry1995auto,berry1999voluntary,gandhi2019measuring. It is well-known that such characteristic-based instruments can be invalid if firm entry is strategic or characteristics are otherwise endogenous berry1995auto,petrin2022identification and they can moreover reduce robustness to model misspecification andrews2025structural.
Can other instruments be found to avoid these issues? Are exogenous shocks to prices enough to identify market-specific counterfactuals? In influential work, berryhaile argue that the answer is generally no: i.e., that exogenous characteristic-based instruments are essentially necessary to flexibly estimate demand with market-level data. Specifically, they consider a nonparametric demand model of the form:
where $S$ and $P$ are $J$-vectors of the observed quantity shares and prices of $J$ goods in a market, $X$ is a $J$-vector of an observed product characteristic, and $\xi$ is an unobserved $J$-vector of demand shocks. The demand shocks and characteristics combine linearly in an index $\delta$. The unknown $\sigma^{-1}(\cdot)$ is the inverse of a demand function $\sigma(d,p)$, which returns the shares that would arise if $\delta$ were set to $d$ and $P$ were set to $p$.\footnote{$\sigma(\cdot)$ may also depend on other observable characteristics $\tilde{X}$, which we suppress here for exposition.}
berryhaile argue that observing a $J$-vector of exogenous price instruments $Z$---or even having exogenous prices---generally does not identify $\sigma(\cdot)$ or even price counterfactuals, i.e. the effects of changing $P$ while holding $\delta$ fixed.\footnote{This is directly stated in a review, berry2021foundations: “having valid instruments for all $J$ prices will not generally suffice for identification of [\ldots] the ceteris paribus effects of price changes.”} Intuitively, the inverse demand function $\sigma^{-1}(\cdot)$ depends separately on $S$ and $P$ but $Z$ affects market shares only through prices; this suggests needing other instruments that shift $S$ holding $P$ fixed.\footnote{berryhaile also consider a model in which $P$ enters the index and is excluded from $\sigma^{-1}(\cdot)$, concluding that price instruments suffice for identification, though again assuming that characteristics are exogenous. borusyak2025estimating show that recentered instruments identify price counterfactuals in this case, allowing endogenous characteristics.} The characteristic $X$ that enters the index uniquely fits this bill, provided it is also exogenous. All functions of $(X,Z)$ can then serve as instruments and identify $\sigma(\cdot)$ so long as they induce enough variation in $(S,P)$---a standard technical condition known as completeness newey2003instrumental.
We reexamine this setting and reach a more optimistic conclusion: while exogenous price shocks are not generally enough for nonparametric identification of market-specific counterfactuals on their own, they can suffice when combined with possibly endogenous characteristics. This is because combinations of an as-good-as-randomly assigned $Z$ and an endogenous $X$ that are mean-zero given $X$---what borusyak2023nonrandom call recentered instruments---are valid instruments with identifying power. In particular, they yield a simple test: for any candidate $\check\sigma(\cdot)$, if the implied index $\check \delta=\check \sigma^{-1}(S,P)$ correlates with a recentered instrument, then $\check \sigma(\cdot)$ cannot be the true demand function.
We introduce a new technical condition---faithfulness---under which all price counterfactuals are identified by candidate inverse demand functions that survive this recentered instrument test. Under faithfulness, any surviving $\check{\sigma}^{-1}(\cdot)$ equals an invertible transformation of the true $\sigma^{-1}(\cdot)$. The transformation is unknown, making demand not fully identified; we cannot, for example, learn counterfactuals that change the endogenous characteristics. But this is not a problem for price counterfactuals, as the unknown transformation is normalized away in those calculations.
Faithfulness requires the variation in $Z$ and $X$ to be rich enough to make all price effects on any functions of the form $H(\delta,P)$ detectable. More precisely, it says that functions of the form $H(\delta, P)$ cannot be mean-independent of the exogenous $Z$, given the possibly endogenous $X$, unless they are constant in $P$. This is a useful condition because any candidate $\check\sigma^{-1}(\cdot)$ can be written as such a function: $\check\sigma^{-1} (S,P)=\check\sigma^{-1}(\sigma(\delta,P),P)\equiv\check{H}(\delta,P)$. Thus, if faithfulness holds, any $\check\sigma^{-1}(S,P)$ that survives the recentered instrument test (and is thus mean-independent of $Z$ given $X$) must be a transformation of the true $\sigma^{-1}(S,P)=\delta$: i.e., $\check H(\delta, P) = \check H(\delta)=\check H (\sigma^{-1}(S,P))$.
Intuitively, faithfulness is satisfied when two conditions hold: (i) $Z$ is a strong instrument for $P$, given $X$, and (ii) $X$ is a strong proxy for $\delta$. Under (i), we have identification of all conditional-on-$X$ average price effects on any $\check H(\delta,P)$. This is not generally enough to make all price effects on $\check H(\delta,P)$ detectable, however, since the identified effects average over the unobserved $\delta\mid X$ distribution. Complex interactions between $\delta$ and $P$ could potentially average out to zero conditional on $X$, making a candidate $\check\sigma^{-1}(\cdot)$ survive the recentered instrument test despite not being a transformation of the true $\sigma^{-1}(\cdot)$. The proxy condition (ii) prevents this by ensuring the variation in $X$ is sufficiently predictive of $\delta$ such that it is impossible for such interactions to average out across all the strata of $X$. Importantly, this argument does not require $X$ to causally vary the distribution of $\delta$: the strong proxy condition can hold even when all characteristics are endogenous.\footnote{Allowing characteristics to be endogenous further relaxes the index restriction in berryhaile by allowing $\delta$ to be an arbitrary function of $X$ and $\xi$ and for $\xi$ to have an arbitrary dimension.}
This intuition highlights the conceptually distinct roles of $Z$ (as an instrument) and $X$ (as a proxy) in our identification results. Unlike with $Z$, researchers do not need to justify why the characteristics $X$ are as-good-as-randomly assigned or otherwise independent of unobserved demand shocks; indeed, faithfulness is even more plausible when firms strategically choose characteristics using partial information on $\delta$. This distinction is also reflected in how $X$ and $Z$ would be used in forming “technical instruments” for estimation: while $X$ can be useful when combined with $Z$ and recentered, it has no identifying power on its own (i.e., in conventional “BLP instruments”) as recentered functions of $X$ only are identically zero.
While faithfulness is a novel condition, we argue it is a close cousin to completeness. Completeness says that functions of the form $\check\sigma^{-1}(S,P)$ cannot be mean-independent of $(X,Z)$ unless they are constant in $(S,P)$. Like completeness, faithfulness is therefore best viewed as a requirement on the strength of identifying variation rather than a substantive economic or statistical assumption.
To explore the close relationship between faithfulness and completeness, we study when one follows from the other. In the limit where $Z$ is a perfect instrument for price (i.e., $P=Z$), we show faithfulness and completeness are equivalent. Outside this limit, faithfulness follows from completeness under several conditions on either pricing or the utility index. On the pricing side, it suffices that either (i) $X$ and $Z$ combine in a nonparametric index $\lambda$ such that $P$ is independent of $(X, Z)$ given $(\lambda,\delta)$ or (ii) the derivatives of $P$ with respect to $Z$ satisfy a particular separability condition. Both conditions are compatible with firms engaging in Bertrand--Nash pricing with certain forms of marginal costs. On the $\delta$ index side, we show faithfulness follows from completeness when (i) $\delta$ and $X$ have finite support with the same number of values or (ii) $\delta \mid X$ can be transformed to follow a location-scale model satisfying certain smoothness conditions---regardless of how prices are determined. In the other direction, completeness of $ (S,P) \mid (X, Z)$ follows from faithfulness when $\delta \mid X$ is complete.
Importantly, researchers do not need to take a stand on the specific sufficient conditions---our identification arguments are the same so long as faithfulness holds. Our menu of conditions shows faithfulness can hold without strong economic or statistical restrictions. Given completeness, faithfulness is not a big additional leap.
Overall, our analysis clarifies the types of variation that are useful for identifying different demand counterfactuals. The model specifies potential outcomes in two “treatments”: prices and characteristics chen2025reinterpreting. If counterfactuals in both treatments are of interest, then it is unsurprising that exogenous variation in both $X$ and $P$ is needed---leading to the familiar “$2J$” instrument requirement in berryhaile. If only the average causal effects of price were of interest, then $J$ instruments for price would intuitively suffice. Price counterfactuals in individual markets have an intermediate requirement: only average price effects are needed, but a sufficiently large and diverse set of them. This is satisfied by having $J$ strong instruments and $J$ strong proxies.
Our analysis also yields several practical insights for applied researchers estimating parametric demand models. Most importantly, it strengthens the recommendation---made previously in the parametric analyses of borusyak2025estimating and andrews2025structural---that researchers use recentered instruments in order to make their price counterfactual estimates more robust to characteristic endogeneity and model misspecification concerns. Our nonparametric results reassure practitioners that such robustness is not tied to any particular functional form or distributional assumptions. Our results also suggest a new role for potentially endogenous variation in $X$, as proxying for unobserved model heterogeneity, which practitioners can assess alongside the conceptually distinct requirement of price instrument exogeneity.
This paper contributes to two main literatures. First, we add to an influential literature on the identification of differentiated product demand with market-level data started by berry1994 and berry1995auto,berry1999voluntary; most closely related is the nonparametric identification analysis of berryhaile. We depart from most of this literature by allowing product characteristics to be endogenous and by focusing on the identification of price counterfactuals. In this way, our market-level identification results complement Proposition 1 in borusyak2025estimating which shows nonparametric identification of price counterfactuals without exogenous characteristics in a model with an index restriction on price and a standard completeness condition. Our analysis also relates to papers like ackerbergcrawford which study the estimation of parametric demand models with endogenous characteristics; borusyak2025estimating propose recentered instruments for this case.
From this literature, our focus on nonparametric identification of price counterfactuals aligns most closely with berryhaile24, who call such counterfactuals “conditional demand.” Their analysis is also similar in allowing product characteristics $X$ to be endogenous and using price instruments that are exogenous conditional on $X$. The key difference is that berryhaile24 leverage “micro data”: i.e., variation in market shares across consumer characteristics. Our results show that price counterfactuals can be identified with market-level data only. This entails a very different identification strategy: while berryhaile24 essentially condition on $X$ and do not require variation on it, we use $X$ as a proxy for unobserved heterogeneity.
Second, we contribute to a large literature studying nonparametric identification with instrumental variables, including brownmatzkin, npv03, newey2003instrumental, altonjimatzkin, cik07, chiappori2015nonparametric, imbensnewey, torgovitsky2015identification, d2015identification, and blundell2017individual. The model we study imposes an index restriction in the spirit of berryhaile, albeit a substantially weaker one. Thus, it is still restrictive relative to the fully-general potential outcomes model that agrist2000interpretation consider for linear instrumental variables (IV) estimation. The index restriction is important for identifying market-specific counterfactuals, which are generally not given by the local average demand elasticities that linear IV identifies berry2021foundations,chen2025reinterpreting. Beyond demand, our results can be restated to apply to triangular models in which unobserved heterogeneity enters the second stage through an index with a potentially endogenous covariate.\footnote{Specifically, our results apply to the model characterized by $Y = g(W_1, \delta(W_2, \xi))$ and $W_1 = h(Z, W_2, \omega)$ with $Z \indep (\xi, \omega) \mid W_2 $ and $(Y, W_1, \delta, W_2) \in \R^J$. } Our faithfulness condition appears new and may prove useful for identifying other nonparametric models with this structure.\footnote{Our faithfulness condition derives its name from and relates conceptually to a condition in the causal discovery literature spirtes2000causation, which considers whether a joint distribution of variables is “faithful” to an underlying causal graph; (ref) details this connection. (ref) shows that some nonparametric identification results in imbensnewey, torgovitsky2015identification, and d2015identification can be interpreted as verifying an analog of faithfulness.}
The rest of this paper is organized as follows. (ref) develops the main identification results. (ref) relates faithfulness to completeness via a series of sufficient and necessary conditions. (ref) discusses key practical insights for parametric estimation. (ref) concludes. For expositional ease we omit various technical details in the main text (e.g., regularity conditions on measurability, existence of moments, and support); (ref) gives precise theoretical statements and proofs.
We consider a demand system in a market with $J$ products. The product quantity shares (or some other quantity measures) $S\in\mathbb{R}^{J}$ are given by:
for prices $P\in\mathbb{R}^{J}$, observed characteristics $\bar{X}=(X, \tilde X)\in\mathbb{R}^ {J}\times\mathcal{X}$, and unobserved demand shocks $\xi\in\Xi$ which capture latent consumer tastes or unobserved product characteristics. Focusing on identification, we suppress market subscripts.
(ref) follows berryhaile by imposing an index restriction in which the demand shocks and characteristics enter together through the unobserved $\delta\in \mathbb {R}^J$ with the “special characteristic” $X$ otherwise excluded from $\sigma$.\footnote{Note that the meaning of the $\delta$ index differs from that in the literature on parametric demand models following berry1995auto where it usually denotes the mean utility vector over products. Most importantly, here $\delta$ does not include any effects of price on demand.} We also follow berryhaile by assuming demand is invertible in this index:
Invertibility follows here from a “connected substitutes” condition berry2013connected.
We are interested in identifying price counterfactuals: the quantities $\sigma(\delta, \tilde X, p^\prime)$ that would be observed if prices were set to some counterfactual $p^\prime$ while holding $\delta,\tilde{X}$ fixed at their realized values. These capture both local changes in prices (i.e. own- and cross-price elasticities) and global counterfactuals that can inform many positive and normative analyses---such as merger simulations, conduct testing, and studying the effects of product entry and exit.\footnote{Under standard assumptions on price-setting, firm optimization only involves demand at counterfactual prices---such that our counterfactuals are enough for computing marginal costs, markups, and welfare changes from mergers or entry/exit (modeled by prices moving from/to infinity). borusyak2025estimating discuss how $\delta$ can be obtained for new products if characteristics are endogenous. } It is straightforward to extend our analysis to counterfactuals in other product attributes by interpreting $P$ as those non-price variables of interest.
To identify price counterfactuals, we assume that a set of price instruments $Z\in \R^{d_z}$, $d_z \ge J$, is observed in addition to $S$, $\bar{X}$ and $P$. We formalize the sense in which these instruments are exogenous below. The identification challenge stems from the unobservability of $\xi$ and the unknown nonparametric functions $\sigma$ and $\delta$.
berryhaile consider identification of $\sigma$ with the additional assumptions of (i) exclusion of $\tilde X$ from the utility index, $\delta(\bar X, \xi) = \delta(X ,\xi)$, (ii) a linear index, $\delta (X, \xi) = X +\xi$ with $\xi\in\mathbb{R}^J$, and (iii) mean-independence of the unobserved demand shocks from the characteristics and instruments: $\E [\xi \mid \bar{X},Z] = 0$. The last assumption captures the exogeneity of both $\bar{X}$ and $Z$. Together, (i)--(iii) imply:
(ref) identifies $\sigma$ under a completeness condition newey2003instrumental,ai2003efficient,andrews2011examples,d2011completeness,miao2018identifying:
Identification follows from the fact that, under (ref), \[ X - \E[\sigma^{-1}(S, P, \tilde X) \mid \bar{X},Z] = 0, \numberthis \label{eq:bh_integral_eqn} \] while (ref) ensures the solution to this integral equation in $\sigma^{-1}$ is unique.
berryhaile's identification result leads to a standard intuition on the need for exogenous variation in the special characteristic $X$. Assumptions (i)--(iii) yield a structural equation, $X = \sigma^{-1}(S, P, \tilde X) - \xi$, with $2J$ endogenous variables $(S,P)$ on the right-hand side after conditioning on $\tilde X$. This suggests a need for $2J$ exogenous instruments: $J$ “for the prices” $P$ and an additional $J$ “for the shares” $S$. When $X$ is exogenous, it can serve as the latter set of instruments since it is excluded from the right-hand side. The role of exogenous characteristics, then, is not to generate variation in prices but to generate variation in shares conditional on prices so that one can disentangle the two arguments of $\sigma^{-1}$. The completeness condition ensures that transformations of $X$ and $Z$ provide sufficient variation for this.
For price counterfactuals, we work under a partial relaxation of (ref) that only imposes conditional exogeneity of the price instruments:
This restriction strengthens the mean independence in (ref) to full independence of $Z$ given $\bar{X}$ but crucially drops the exogeneity of $\bar{X}$.\footnote{(ref) is implied by the more standard $Z \indep \xi \mid \bar X$ but is weaker when $\xi$ has a higher dimension than $\delta$. We do not view the strengthening from mean- to full independence as meaningfully restrictive, since there is little reason to distinguish $\xi$ from some transformation $g(\xi)$. Full independence amounts to mean-independence of any $g(\xi)$, treating all functions symmetrically. } (ref) can be motivated by viewing $Z$ as a set of supply-side shocks drawn in some true or natural experiment after product characteristics are determined borusyak2025estimating.\footnote{For example, in the U.S. automobile market, $Z$ may contain exchange rate shocks in cars' countries of production that are excluded from demand and drawn randomly after cars are designed but before prices are set. In this case $Z\indep (\xi,\bar X)$, implying (ref). See ackerbergcrawford and borusyak2025estimating for further discussion; Appendix A of berryhaile24 shows how (ref) can arise for different types of price instruments in various economic models.}
Our relaxation of (ref) is motivated by three related misspecification concerns. First, suppose the functional form $\delta (\bar X, \xi) = X + \xi$ is correctly specified but characteristics $\bar X$ are chosen strategically by firms with partial knowledge of demand conditions as captured by $\xi$. Then $\E [\xi\mid\bar{X}]=0$ is unlikely to hold. Second, $\xi$ may include physical product characteristics chosen by firms along with $\bar{X}$ but unobserved by the econometrician. Again, there is little reason they would be uncorrelated with $\bar{X}$. Finally, suppose that characteristics $\bar{X}$ are fully independent of $\xi$ but that the functional form of $\delta$ is not linear or does not exclude $\tilde X$: i.e., $\delta(\bar X, \xi) \neq X+\xi$. Then (ref) need not hold, and model-implied price counterfactuals relying on (ref) may be incorrect andrews2025structural. This in particular accommodates the view of $\xi$ as a “statistical disturbance” rather than an economically meaningful object, in which case $\bar{X}\indep \xi$ by definition of $\xi$ but exclusion and linearity are restrictive.\footnote{Any two random vectors $(\delta, \bar X)$ can be represented as $\delta = \tilde\delta(\bar X, \tilde\xi)$ for some $\tilde\xi \indep \bar X$ and some measurable function $\tilde\delta$---for instance by using Knothe--Rosenblatt (conditional quantile) transforms. In such a representation, $\tilde\xi$ need not have a straightforward economic interpretation. } Relaxing the linear index functional form substantively weakens restrictions on the heterogeneity of causal effects of $\bar X$ on $S$ chen2025reinterpreting; it also allows $\xi$ to be multi-dimensional for each product, in contrast to most demand models in the literature.\footnote{For instance, Appendix B of berryhaile contains a nonseparable model in which $\delta = \delta(X, \xi)$, but restricts the map coordinate-wise: $\delta_j = \delta_j(X_j,\xi_j)$ with the function increasing in $\xi_j$. This requires $\xi$ to be of dimension $J$, in addition to other restrictions that we remove.}
Under (ref),
for some unknown $k_0$. This motivates our identification strategy. The true inverse demand function, which returns $\delta$, is mean-independent of $Z$ given $\bar{X}$:
We can thus rule out some candidate inverse demand functions $\check\sigma^{-1}(S,P,\tilde X)$: those that violate the conditional moment restriction (ref).\footnote{(ref) implies conditional independence of higher moments of $\sigma^{-1}(S,P,\tilde X)$ too, which could in principle be used for identification. We focus on identification via first moments, which is more aligned with berryhaile as well as common parametric estimation strategies.}
To develop this strategy, we first ease some notation. Our identification results fully condition on the “non-special” characteristics $\tilde X$. To simplify notation, we suppress $\tilde X$ and replace $\bar{X}$ with $X$ throughout.
We next define a (potentially non-sharp) identified set for $\sigma^{-1}$:
This set is nonempty because (ref) implies it contains at least the true $\sigma^{-1}$.
The identified set can be more constructively defined via recentered instruments, which take the form $R( X,Z) - \E[R( X,Z) \mid X]$ for some function $R$ borusyak2023nonrandom. The set of functions in $\Theta_I$ is exactly the set of functions that are orthogonal to any recentered instrument:
Intuitively, a candidate $\check\sigma^{-1}$ can only be conditionally mean-independent of $Z$, and thus in $\Theta_I$, if it is uncorrelated with all conditionally mean-zero functions of $Z$.
The identified set is not a singleton that contains $\sigma^{-1}$ only: under (ref), all transformations of the true $\sigma^{-1}$ are independent of $Z$ given $X$ and hence in $\Theta_I$. However, this does not necessarily impede point-identification of price counterfactuals: in fact, it suffices for $\sigma^{-1}$ to be identified only up to some transformation.\footnote{Statements like (ref) are also intermediate steps in matzkin2008identification, torgovitsky2015identification, and berryhaile24. In particular, berryhaile24 normalize their index to avoid considering analogous indeterminacy as with $T$.}
Intuitively, counterfactual quantities given hypothetical prices $p^\prime$ under a candidate model $\check{\sigma}^{-1}=T(\sigma^{-1})$ are computed from the observed $(S,P)$ as:
The transformation $T$ cancels, so it does not affect the counterfactual $\sigma(\delta,p^\prime)$.\footnote{Note that $T$ is implicitly invertible because $\Theta_I$ only includes invertible $\check\sigma^{-1}$ candidates.}
Our identification strategy is therefore successful when $\Theta_I$ contains nothing but transformations of $\sigma^{-1}$. We next consider when this condition holds.
We start with the case of $Z=P$: i.e., where prices are themselves exogenous ($P\indep \delta\mid X$). This is unrealistic in many settings, but it is an illuminating baseline case for developing our strategy (and notably one which berry2021foundations call “surprisingly difficult”). In particular, it clarifies why exogenous variation in $P$ can be enough and what the endogenous variation in $X$ nevertheless contributes. This case can also be viewed as the limit of our general setting with perfect price instruments.
With exogenous prices, price counterfactuals are identified under the same completeness condition as in berryhaile---despite our substantial relaxation of their model. Indeed, all characteristics can be fully endogenous in this case:
To see this result, consider any candidate $\check\sigma^{-1}\in\Theta_I$. We can write $\check\sigma^{-1}(S,P)=\check\sigma^{-1}(\sigma(\delta,P),P)\equiv H(\delta,P)$ for an unknown $H$ such that $\E[H(\delta,P) \mid P, X] = k (X)$. Fixing some price value $p_0$,\footnote{When $P$ is continuously distributed, conditioning on a realization of it requires measure-theoretic care. See (ref) and (ref) for a formal argument.} we then have:
where the first line uses the fact that $P$ does not enter $k(X)$, the second line uses (ref), and the third line uses (ref). Thus $H(\delta, p) = H(\delta, p_0) $ does not depend on $p$: $H(\delta,P)=H(\delta)$. In turn, this implies that the candidate inverse demand function is a transformation of the true inverse demand function: \[\check\sigma^{-1}(S,P)=H(\delta,P)=H(\delta)=H(\sigma^{-1}(S,P)).\] Since this holds for any $\check\sigma^{-1}\in\Theta_I$, price counterfactuals are identified by (ref).
(ref) contains two key intuitions. First, for identifying price counterfactuals, it is enough to detect causal effects of prices only. Indeed, since it suffices to identify $\sigma^{-1}(S,P)$ only up to a transformation ((ref)), we only need to find some function $h(S,P)$ such that $h(\sigma(\delta,P),P)\equiv H(\delta,P)$ is a transformation of $\delta$ only with no dependence on $P$. From a causal inference perspective, this means finding some outcome $H=h(S,P)$ which is fully unaffected by $P$.
Importantly, here we are only interested in whether $P$ has any causal effects on $H$ and not in disentangling the effects that come through $S$ versus $P$. We thus sidestep the need for separate exogenous variation in $S$ given $P$, which is what compels characteristic exogeneity in berryhaile. Hence, to trace out average price effects, we require only $J$-dimensional exogenous price variation---not $2J$ instruments.
The second key intuition is on the role of the additional $J$-dimensional variation in $X$ as a proxy for $\delta$.\footnote{A related argument by ahn2025prognostic, in a context with an exogenous binary treatment $P$, uses $X$ in a similar way and calls it a “prognostic variable.”} Tracing out average effects of $P$ is not generally enough: while (ref) implies that the effect of $P$ on $S$ (and therefore on $H$) is fully governed by the index $\delta$, this index is unobserved. Zero average price effects, which average across the unobserved $\delta$, do not by itself imply zero price effects for a given $\delta$.\footnote{Borrowing an example from benkard2006nonparametric, if $X$ is independent of $\delta$ and thus variation in $X$ is not useful, then one can choose $\delta \sim \Norm(0, I_J)$ and $H(\delta, p)$ to be a $p$-dependent rotation of $\delta.$ This construction implies that $H(\delta, P) \indep (X, P)$, and thus, absent other conditions that rule out such $H$, the set $\Theta_I$ would include inverse demand candidates that are not just transformations of $\delta$---leading to incorrect price counterfactuals.} This gap can be filled by leveraging variation in $X$, so long as it strongly correlates with $\delta$---i.e., “proxies” for it. For each observed value of $X$, exogenous price variation reveals the conditional average price effects on $H$. Completeness of $(\delta, P) \mid (X, Z)$---which here, with $Z=P$ and $\delta\indep P\mid X$, is equivalent to completeness of $\delta \mid X$---ensures that this set of identified effects is rich enough to span the set of price effects for each unobserved value of $\delta$. Hence, under completeness, the finding of no conditional-on-$X$ average price effects ensures no causal dependence of $H$ on $P$.\footnote{Section 3 in chen2025reinterpreting shows that many counterfactual predictions from structural models analogously extrapolate from a lack of average causal effects to a total lack of causal effects.}
This proxy role of $X$ is substantively different from the role that $X$ plays in berryhaile, as an instrument “for the shares.” Here, $X$ need not be exogenous with respect to the economically meaningful unobserved demand shock $\xi$, nor must it causally shift $\delta$. For instance, it is perfectly fine if $X$ correlates with $\delta$ because of selection only. As a result, while practitioners should justify why $Z$ is as-good-as-randomly assigned and why it is not strategically chosen (in order to satisfy (ref)), no such arguments are needed for $X$.
Of course, one can always make $X$ “technically exogenous” by rewriting (ref) as:
But since $k_0$ is unknown, $X$ is not excluded from this model: variation in it informs both $k_0$ directly and $\sigma^{-1}$ indirectly through $S$. This highlights that we cannot merely redefine residuals, make $X$ exogenous without loss, and appeal to the argument of berryhaile; fundamentally, our results rely on a distinct use of $X$.
The proxy role of $X$ is also substantively different from the role of other characteristics $\tilde{X}$, which---being fully conditioned on---might be viewed here as “controls.” While there are no restrictions on the amount or nature of variation in $\tilde{X}$ or its effects on demand, $X$ must be strongly correlated with $\delta$ and excluded from demand in the sense of the index restriction in (ref). Indeed, the proxy role of $X$ is what gives the index restriction bite: with a constant or completely randomly generated $X$, any demand model could be written in the form of (ref).
We next generalize these results and intuitions to settings with endogenous prices.
When $P\neq Z$, we continue to need sufficiently rich variation in $X$ to proxy for the unobserved $\delta$. We additionally need rich enough variation in the exogenous instruments $Z$ to trace out the causal effects of the now-endogenous prices. Perhaps surprisingly, the usual completeness condition is not the appropriate formalization of these two requirements. Instead, we consider a new condition to replace (ref):
Under faithfulness, price counterfactuals are identified even with endogenous prices:
The proof of (ref) follows the same steps as for (ref): write $\check\sigma^ {-1} (S,P)=\check\sigma^{-1}(\sigma(\delta,P),P) \equiv H(\delta,P)$ and note that $\E[H(\delta,P)\mid X,Z]=\E[\check\sigma^{-1}(S,P)\mid X,Z]=k(X)$ for some $k$ if $\check\sigma^{-1}\in\Theta_I$. Under faithfulness, this means that $\check\sigma^{-1}(S,P)=H (\delta,P)=H(\delta)=H(\sigma^{-1}(S,P))$ is a transformation of the true inverse demand function. All price counterfactuals are thus identified by (ref).
The two key intuitions from the exogenous price case are retained in (ref). First, because we are only interested in whether $P$ has any effects on $H(\delta,P)=h(S,P)$, and not in disentangling its direct effects from indirect effects through $S$, we avoid the need for separate exogenous variation in $S$ given $P$. Second, though variation in $X$ need not be exogenous, it is still critical for proxying for the unobserved $\delta$. As in the exogenous price case, complex interactions between $\delta$ and $P$ can potentially average out and threaten identification by obscuring true price effects on $H(\delta,P)$. Here faithfulness, rather than completeness, rules out such scenarios via rich variation in $X$ that strongly predicts $\delta$, together with strong price instruments $Z$.\footnote{(ref) shows how, under additional restrictions, analogs of faithfulness can hold without an $X$ that satisfies the index restriction and proxies for $\delta$. These include the case where $P$ combines linearly with $\delta$ (as in Proposition 1 of borusyak2025estimating), and where $J=1$ with $\sigma$ monotone in $\delta$ (as in imbensnewey, torgovitsky2015identification, and d2015identification).}
How different is faithfulness from the usual completeness condition? They are similar in kind: both are high-level conditions that yield identification by directly asserting that certain operations are injective. Namely, they link the variation in some $H(\delta,P)$ that is detectable via conditional expectations in $(X,Z)$ to the structural dependence of $H$ on $(\delta,P)$. Completeness requires that if $H(\delta, P)$ varies then $\E[H(\delta,P) \mid X, Z]$ also varies. In this sense, conditional expectations “faithfully reflect” changes in $(\delta,P)$. Faithfulness instead says that if prices truly affect a function of demand primitives, these effects must show up in the instrument-induced price variation conditional on $X$. Conversely, functions with no instrument-induced conditional-on-$X$ variation cannot depend on $P$.\footnote{Note that faithfulness is different than conditional-on-$X$ completeness---i.e., completeness of $(\delta,P,X)\mid(X,Z)$. This condition cannot hold here as $Z$ generates no variation in $\delta$.}
In the nonparametric instrumental variables literature, completeness is often treated as a technical condition which encodes a sense of instrument strength while not imposing substantive restrictions newey2003instrumental,ai2003efficient,darolles2011nonparametric.\footnote{This is especially true when nonparametric identification is viewed as a theoretical argument for how particular parametric structure does not “drive” conclusions. In any given parametric model, one could posit a strong-instrument-type condition where all non-constant functions $w \in \mathcal{W}$ of endogenous variables $w(S, P)$ are assumed to correlate with some function of $(X,Z)$, over some parametrized class $\mathcal{W}$. Completeness is the limit of such conditions as we enlarge $\mathcal{W}$ to include all (integrable) functions. } Lower-level conditions are given by, e.g., d2011completeness and andrews2011examples. We argue that faithfulness should be similarly treated as a technical condition which encodes a sense of instrument strength for $Z$ as well as proxy strength for $X$.
To bolster this argument, we next detail connections between the two conditions---showing, in particular, that faithfulness follows from completeness under a wide range of different lower-level conditions. Together these results suggest faithfulness and completeness, while non-nested in general, are close cousins. Thus, to the extent nonparametric identification under completeness reassures practitioners that demand can be flexibly estimated with exogenous characteristics and price instruments, our results under faithfulness should likewise reassure that price counterfactuals can be flexibly estimated with recentered instruments.
We first provide a useful calibration: when $Z$ is a perfect instrument (i.e., the exogenous price case), faithfulness and completeness are exactly equivalent.
In this sense, faithfulness is not an exotic new assumption: it is the natural technical condition for leveraging the conditional moment restriction (ref).
Outside of the $P=Z$ case, faithfulness and completeness are distinct conditions. Two counterexamples illustrate this: (ref) exhibits a class of demand models where faithfulness holds but $(\delta, P) \mid (X, Z)$ needs not be complete, while (ref) presents a distribution $(\delta, P) \mid (X,Z)$ that is complete but not faithful.
We next show that the two conditions are nevertheless closely related, in the sense that each condition implies the other under additional restrictions.
We first present four non-nested conditions under which completeness implies faithfulness; these conditions are sufficient but not necessary. The sufficient conditions upper-bound the extent to which faithfulness is “stronger” than completeness.
We start from two sufficient conditions that restrict how price depends on the observables and unobservables of the model. Both conditions extend the case of exogenous prices. For some function $f$, write \[ P = f(X, Z, \delta, \tilde\omega), \qquad \tilde \omega \indep (X, Z, \delta) \numberthis \label{eq:price_eqn_main}. \] Here $\tilde \omega$ is an unobservable of arbitrary dimension that captures residual variation in $P$ that is independent of $(X, Z, \delta)$. So far, (ref) is without loss of generality.\footnote{Note that $f$ is not a structural function because of the parametrization of $\tilde\omega$, but it is consistent with any structural formulation. Indeed, for any structural shock $\omega$ possibly correlated with $(X,Z,\delta)$ and $P=\tilde f(X,Z,\delta,\omega)$, one can represent $\omega = f_\omega(X,Z,\delta, \tilde\omega)$ and $P=\tilde f(X,Z,\delta,f_\omega(X,Z,\delta,\tilde\omega))$.} In what follows we place different restrictions on $f$. At the end of this subsection, we show that these restrictions can be restated as similar conditions on marginal costs under Bertrand--Nash pricing.
The first sufficient condition is that $X$ and $Z$ enter price only through the utility index $\delta$ and an index $\lambda(X, Z)=\lambda$ that is invertible in $Z$:
This assumption is satisfied, in particular, when $X$ enters price in (ref) only through $\delta$---paralleling how it enters $\sigma(\cdot)$. Under this index restriction for price, (ref) is satisfied with $\lambda(X, Z) = Z$. By further setting $Z=P$, this special case also nests exogenous prices and generates (ref) as a corollary. In general we have:
To see how this result follows, note that for any realization $\lambda_0$ of $\lambda$:
Hence for $H(\delta) = \E[H(\delta, P) \mid \delta, \lambda = \lambda_0]$ we have $\E[H(\delta) \mid X, Z] = k(X)$. Finally, by completeness ((ref)), $H (\delta, P) = H(\delta)$.
The second sufficient condition instead imposes a separability condition on the derivatives of price with respect to $Z$:
(ref) holds, in particular, when \[ P = f_0\bigg( f_1(X,Z) + f_2(X, \delta , \tilde \omega), \delta \bigg) \numberthis \label{eq:fn_form} \] (with regularity conditions on $f_0$ and $f_1$ given in (ref)). The restriction in (ref) is that $Z$ enters $P$ through an index $f_1(X,Z) + f_2 (X, \delta,\tilde \omega)$, which is a form of separability between observed and unobserved cost shifters. Clearly, it is satisfied when $P=Z$.\footnote{Moreover, under (ref) and suitable regularity conditions on $f_0, f_1, f_2$, faithfulness is equivalent to completeness. To see this, suppose faithfulness holds but completeness fails. By (ref), below, $\delta \mid X$ is not complete: there exists $h(\delta)\ne 0$ such that $\E[h(\delta)\mid X]=0$. Consider $\E[h(\delta) f_0^{-1}(P, \delta) \mid X, Z] = \E[h(\delta) \mid X]f_1(X, Z) + \E[h(\delta) f_2(X, \delta, \tilde\omega) \mid X, Z]$. The first term is 0 and the second does not vary with $Z$. Yet, $h(\delta)f_0^{-1}(P,\delta)$ depends on $P$, contradicting faithfulness. }
The logic for this result is as follows: under (ref), given any differentiable $H(\delta, P)$ with $H_p(\delta,P)\equiv\diff{H}{P}$,\footnote{Because we assume $H$ is differentiable here, we have to modify faithfulness to restrict only differentiable functions. These complications are resolved in (ref), which redefines faithfulness and ensures that differentiation is exchangeable with expectation.} we have
By the fundamental theorem of calculus, $H_p(\delta, P) = 0$ implies $H(\delta, P)=H(\delta)$.
(ref) are non-nested. (ref) allows richer interactions between how $Z$ and $\tilde \omega$: e.g., $P=f(Z,\tilde \omega)$ always satisfies (ref) but not necessarily (ref). (ref) allows richer interactions between $X$ and $\tilde \omega$: e.g., $P=Z+f(X,\tilde \omega)$ always satisfies (ref) but not necessarily (ref).
(ref) can be economically grounded with more primitive conditions on marginal costs. Under Bertrand--Nash pricing with constant marginal costs that are represented without loss of generality as $c(X,Z,\delta,\tilde\omega)\equiv C\in\mathbb R^J$, one can always write the equilibrium prices as
for some function $g$ (see (ref)). Thus, if one assumes that $X$ and $Z$ enter marginal costs via the index $\lambda$, i.e., $C=c(\lambda(X,Z),\delta,\tilde\omega),$ then (ref) follows. Similarly, (ref) holds if $C=f_0(f_1(Z, X) + f_2(\delta, X,\tilde \omega),\delta)$ with $g$ and $f_0,f_1$ satisfying certain regularity conditions.
Our second set of sufficient conditions leverage statistical restrictions on the conditional distribution of $\delta \mid X$. With these conditions, it is possible to show that every function of $X$ belonging to some known class $\mathcal{K}$ can be written as $\E[H(\delta) \mid X]$ for some function $H$. If this is true, we can then conclude from completeness that $H(\delta, P) = H(\delta)$ for some $H(\delta)$, so long as $\E[H(\delta, P) \mid X, Z] \in \mathcal{K}$. To be sure, these conditions are not fully general: our main goal is to demonstrate the existence of restrictions that are purely statistical; these restrictions should not necessarily be viewed as recommended modeling choices.
One simple case is if $\delta$ and $X$ are discrete and known to take the same finite number of values. Here, completeness directly implies that any function $k (X)$ can be represented as a projection of some function $H(\delta)$ to $(X,Z)$-space.
The same strategy can be used in the case of continuously-distributed $X$ and $\delta$. Suppose that $\delta\mid X$ can be transformed into a location-scale model of the form: \[ a(\delta) = b(X) + \Sigma(X) \epsilon, \quad \epsilon \sim q(\cdot), \quad \epsilon \indep X, \numberthis \label{eq:location-scale-maintext} \] for some invertible $a(\cdot)$ and continuously distributed $J$-dimensional $\epsilon$ with density $q(\cdot)$; $\epsilon$ can be seen as reparametrizing the component of $\delta$ that is independent of $X$. Note that (ref) is a purely statistical assumption as $\epsilon$ is generally distinct from $\xi$.
We consider the following assumptions on $a(\cdot)$, $b(\cdot)$, $\Sigma(\cdot)$, and $q(\cdot)$:
The key requirement in (ref) is that $b$ is known to fall in a “well-behaved” function class $\mathcal K$, which---loosely speaking---consists of functions that deviate from a linear map by a suitably smooth function with $s$th-order derivatives in $L^p$. Because of this, we can limit $\Theta_I$ to just those functions $h$ with conditional expectations in $\mathcal K$: \[ \Theta_I(\mathcal K) = \br{h : \E[h(S, P) \mid X, Z] \in \mathcal K} \ni a(\sigma^ {-1}(\cdot, \cdot)). \]
The other regularity conditions in (ref) establish that $\mathcal K$ is sufficiently small and the operator $u\mapsto\E[u(\delta) \mid X]$ is sufficiently well-behaved so that $\mathcal K$ can be entirely populated by conditional expectations of functions of $\delta$.\footnote{(ref)(2) is satisfied by distributions with Gamma-like tails, though it rules out Gaussian distributions. See (ref) and (ref). We strongly suspect that these restrictions are not essential and can be further relaxed.} That is, for any $k \in \mathcal K$, there exists some $H (\delta)$ such that $ \E[H(\delta) \mid X] = k(X). $ Completeness would then imply a version of faithfulness with respect to $\mathcal K$. That is, for any candidate $H(\delta, P)$ where $\E[H(\delta, P) \mid X, Z] \in \mathcal K$, it follows that for some $H(\delta)$, we have \[ \E[H(\delta,P) \mid X, Z] = \E[H(\delta) \mid X, Z], \] and therefore $H(\delta, P) = H(\delta)$ by completeness. We summarize this argument in the following result and verify that the conclusion of (ref) continues to hold.
Unlike (ref), which requires an exact equivalence in the number of support points of $\delta$ and $X$, (ref) is not knife-edge in nature---suggesting that it can be further relaxed to allow for more flexible models for $\delta \mid X$. Overall, the combination of results in (ref) confirms that faithfulness can follow from completeness without strong economic or statistical assumptions. We expect that many other sufficient conditions for faithfulness exist as well.
The above arguments verify faithfulness from completeness by, intuitively, using $X$ as a proxy for $\delta$. Since faithfulness only requires ruling out the possibility of certain $\delta$ and $P$ interactions averaging out, it is possible---indeed shown by the example in (ref)---that faithfulness does not fully use the completeness of $\delta \mid X$. In other words, completeness of $\delta \mid X$ is sufficient for faithfulness but may not be necessary. It turns out this is the only reason that faithfulness does not always imply completeness:
The proof is simple: Given $\E[H(\delta, P) \mid X, Z] = 0$, faithfulness implies that $H(\delta, P) = H(\delta)$ since $0$ is constant in $Z$. Exogeneity of $Z$ further implies that $\E [H(\delta) \mid X, Z] = \E[H(\delta) \mid X] = 0$. Finally, completeness of $\delta \mid X$ implies that $H(\delta) = 0$.
Our nonparametric identification results yield several insights for applied researchers estimating parametric demand models. Namely, they suggest key conditions that practitioners can scrutinize in order to use our results to argue (perhaps informally) that their parametric assumptions serve the usual role of filling gaps left by insufficient identifying variation. These conditions, discussed here, concern the counterfactuals of interest, the available variation in the price instruments and characteristics, and the estimation method. We also discuss how the conditions differ from what researchers appealing to berryhaile and berryhaile24 would need to argue.
First, in order to appeal to our results, a researcher should be interested in estimating counterfactuals that change a product attribute for which some plausibly exogenous variation is available. Most commonly, this attribute is the product's price. Price counterfactuals arise, for instance, in merger simulations, when studying product exit or entry (which can be understood as moving prices to or from infinity), and when price elasticities are fundamentally of interest. In each of these cases, one can plausibly observe a set of supply-side shocks $Z$ which exogenously vary prices $P$.
Second, a researcher should argue there exists at least one “special” characteristic $X$ which satisfies the index restriction in (ref). For mixed logit models, this would mean $X$ enters demand without a random coefficient; berryhaile argue this requirement is satisfied in most applications. Note that no assumptions are generally needed on how the other characteristics $\tilde{X}$ enter demand.
The researcher should further argue that the special characteristic is a strong proxy for the index $\delta$. In practice, this means that $X$ is strongly predictive of demand holding the other product attributes fixed. Again, no such requirement is generally imposed on the other observed characteristics $\tilde X$.
Importantly, and in contrast with the berryhaile results, researchers motivated by our identification results need not justify that either $X$ or $\tilde X$ are exogenous. Several distinct forms of endogeneity are allowed: firms can strategically design products (i.e., choose characteristics both observed and unobserved by the researcher) or choose which markets to enter with a given product with some knowledge of local demand conditions. Moreover, the functional form restrictions that the researcher imposes on how characteristics enter demand need not be correct; for instance, the model is still able to predict price counterfactuals correctly if quadratic terms in $X$ are incorrectly excluded, violating their exogeneity andrews2025structural.
Third, a researcher should scrutinize the exogeneity of the price instruments $Z$. These should be fully independent of the $\delta$ index conditional on $(X,\tilde X)$ which in practice will generally involve certain timing assumptions. The cleanest scenario is when $Z$ is a set of unconditionally as-good-as-random shocks that have some variation across products, are realized after the product design is chosen and entry decisions are made but before $P$ is set, and do not affect demand except through $P$. Short-term unanticipated fluctuations in the exchange rate of countries producing different goods borusyak2025estimating or deviations of realized input prices from futures markets values ackerbergcrawford are possible examples of such price instruments. However, there are also scenarios where only the conditional exogeneity of $Z$ is plausible; see Appendix A of berryhaile24 for a detailed discussion.
Fourth, to avoid any bias from the potentially endogenous $(X,\tilde{X})$, the moment conditions used in estimation must be based on recentered instruments. Functions of $Z$ alone are not typically enough to identify parametric demand models beyond pure logit, so other instruments have to be chosen. This paper shows that recentered instruments can be sufficient for this goal nonparametrically, while borusyak2025estimating propose specific constructions of powerful recentered instruments in parametric models. Importantly, other instruments---such as “BLP instruments” that are functions of characteristics only---are not generally valid without characteristic exogeneity.\footnote{In fact, functions of own and rivals' characteristics can be controlled for and potentially improve estimation efficiency by absorbing some residual variation borusyak2025estimating.}
We conclude here by noting that with our identification strategy works with only market-level data, which are widely available. Other types of data---e.g., on how market shares vary by consumer characteristics within the market or on second choices of consumers---are known to be helpful for demand estimation. However, these types of data are not always available and our results show that they are not necessary for identification even nonparametrically. Moreover, even when microdata are available, the identification strategies developed for them impose additional homogeneity assumptions chen2025reinterpreting which our identification strategy does not require.
We have shown that price counterfactuals are identified without exogenous product characteristics in a nonparametric demand model with a weak index restriction, given exogenous instruments that induce sufficient variation in prices and a strong proxy for the index. The richness of this identifying variation is captured by a new faithfulness condition, which essentially requires that the instruments and the proxies make all causal price effects detectable. We show through a variety of non-nested sufficient conditions that faithfulness can follow from a standard completeness condition without strong statistical or economic restrictions. These results reassure practitioners that price counterfactuals can be reliably identified with recentered instruments, without needing to justify that observed characteristics are as-if-randomly assigned or chosen non-strategically. We suspect the new faithfulness condition and identification results may also prove useful in other nonparametric models.
We have not provided a theoretical analysis of nonparametric estimation based on these results, which would naturally require additional regularity conditions compiani2022market,chen2015sieve. As in berryhaile, we leave developing these conditions to future research. We nevertheless hope that our identification analysis will help guide empirical researchers towards more robust and credible estimation strategies, by clarifying the kinds of variation that can reveal counterfactuals in flexible demand models. Most importantly, in contrast to widely-held intuition, our results suggest researchers can generally avoid conventional characteristic-based instruments---and the potential biases associated with them---by looking for plausibly exogenous supply shocks and leveraging them via recentered instruments.