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Nonparametric Identification of Demand without Exogenous Product Characteristics
\title{Nonparametric Identification of Demand \\ without Exogenous Product Characteristics \vspace{0.6cm}}
\author{Kirill Borusyak \and Jiafeng Chen \and Peter Hull \and Lihua Lei
\vspace{0.1cm}
\footnote{\textsuperscript{*}February 2026 (first draft: December 2025). Borusyak: University of California, Berkeley,
[email removed]; Chen: Stanford University,
[email removed]; Hull: Brown University, peter\[email removed]; Lei: Stanford
University, [email removed]. We thank Nikhil Agarwal, Isaiah Andrews, Steve Berry, Jimbo Brand, Giovanni Compiani, Jos\'e Ignacio Cuesta, Matt Gentzkow, Jeff Gortmaker, Phil Haile, Nail Kashaev, Toru Kitagawa, Ashesh Rambachan, Adam Rosen, Jon Roth, Jesse Shapiro, Paulo Somaini, and Daniel Yi Xu for helpful comments. Google Gemini and OpenAI's ChatGPT contributed valuable insights. Refine.ink was used to check for consistency and
clarity.}}
\maketitle
\begin{abstract}
\singlespacing
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 \emph{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.
\vspace{1em}
\noindent \textsc{JEL codes.} C14, C36, L13
\noindent \textsc{Keywords.} Demand, nonparametric identification, recentered IV, faithfulness
\noindent\thispagestyle{empty}
\end{abstract}
\noindent\newpage{}
\noindent
\global\long\def\expec#1{\mathbb{E}\left[#1\right]}
\global\long\def\var#1{\mathrm{Var}\left[#1\right]}
\global\long\def\cov#1{\mathrm{Cov}\left[#1\right]}
\global\long\global\long\global\long
\pagestyle{plain}
\section{Introduction}
Conventional wisdom holds that exogenous supply-side instruments trace out demand curves and identify important counterfactuals: how quantities would change under hypothetical price changes \citep{wright1928tariff}. Indeed, when considering demand for a single product, \cite{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 \emph{particular} market \citep{berry2021foundations}. These analyses usually incorporate other kinds of instruments: functions of the observed characteristics of competing products \citep[e.g.,][] {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 \citep{berry1995auto,petrin2022identification} and they can moreover reduce robustness to model misspecification \citep{andrews2025structural}.
Can other instruments be found to avoid these issues? Are exogenous shocks to prices enough to identify market-specific counterfactuals? In influential work, \citet{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:
\begin{align}\label{eq:simpleBH}
\sigma^{-1}(S,P)=X+\xi \equiv \delta,
\end{align}
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.}
\citet{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, \citet[p.~6]{berry2021foundations}: ``having valid instruments for all $J$ prices will not generally suffice for identification of [\ldots] the \emph{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{\citet{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. \citet{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 \citep{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 \emph{can} suffice when combined with possibly \emph{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 \cite{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---\emph{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: \emph{(i)} $Z$ is a strong instrument for $P$, given $X$, and \emph{(ii)} $X$ is a strong \emph{proxy} for $\delta$. Under \emph{(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 \emph{(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 \cite{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 \emph{(i)} $X$ and $Z$ combine in a nonparametric index $\lambda$ such that $P$ is independent of $(X, Z)$ given $(\lambda,\delta)$ or \emph{(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 \emph{(i)} $\delta$ and $X$ have finite support with the same number of values or \emph{(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 \citep{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 \citet{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 \citet{borusyak2025estimating} and \citet{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 \cite{berry1994} and \cite{berry1995auto,berry1999voluntary}; most closely related is the nonparametric identification analysis of \cite{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 \citet{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 \cite{ackerbergcrawford} which study the estimation of parametric demand models with endogenous characteristics; \cite{borusyak2025estimating} propose recentered instruments for this case.
From this literature, our focus on nonparametric identification of price counterfactuals aligns most closely with \citet{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 \citet{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 \cite{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 \cite{brownmatzkin}, \cite{npv03},
\cite{newey2003instrumental}, \cite{altonjimatzkin}, \cite{cik07}, \citet{chiappori2015nonparametric},
\cite{imbensnewey},
\citet{torgovitsky2015identification},
\citet{d2015identification}, and \citet{blundell2017individual}.
The model we study imposes an index restriction in the spirit of \cite{berryhaile}, albeit a substantially weaker one. Thus, it is still restrictive relative to the fully-general potential outcomes model that \cite{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 \citep{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
\citep{spirtes2000causation}, which considers whether a joint distribution of variables is
``faithful'' to an underlying causal graph; \Cref{sub:faithfulness_in_causal_graphs} details this connection. \Cref{sub:triangular_model} shows that some
nonparametric identification results in \cite{imbensnewey}, \cite{torgovitsky2015identification}, and \cite{d2015identification} can be interpreted as verifying an analog of
faithfulness.}
The rest of this paper is organized as follows. \Cref{sec:theory} develops the main identification results. \Cref{sec:sufficient_conditions} relates faithfulness to completeness via a series of sufficient and necessary conditions. \Cref{sec:practical} discusses key practical insights for parametric estimation. \Cref{sec:conclusion} concludes. For expositional ease we omit various technical details in the main text (e.g., regularity conditions on measurability, existence of moments, and support); \Cref{sec:proofs} gives precise theoretical statements and proofs.
\section{Theory}\label{sec:theory}
\subsection{Setup}
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:
\begin{align}
S=\sigma(\delta,\tilde{X},P),\quad \delta=\delta(\bar{X},\xi)\label{eq:demand}
\end{align}
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.
\cref{eq:demand} follows \citet
{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 \cite{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 \cite{berryhaile} by assuming demand is invertible in this index:
\smallskip
\begin{as}
\label{as:invert}
For every $(\tilde{x},p)$, the map $d\mapsto \sigma(d, \tilde{x},p)$ is invertible.
\end{as}
\smallskip
\noindent Invertibility follows here from a ``connected substitutes'' condition \citep{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). \citet{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 $\delta$.
\cite{berryhaile} consider identification of $\sigma$ with the additional assumptions of
\emph{(i)} exclusion of $\tilde X$ from the utility index, $\delta(\bar X, \xi) = \delta(X ,\xi)$, \emph{(ii)} a linear index, $\delta (X, \xi) = X +\xi$ with $\xi\in\mathbb{R}^J$, and \emph{(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, \emph{(i)}--\emph{(iii)} imply:
\begin{align}\label{eq:linear_exogenous}
\E[\delta \mid \bar{X},Z] = X.
\end{align}
\cref{eq:linear_exogenous} identifies $\sigma$ under a completeness condition \citep{newey2003instrumental,ai2003efficient,andrews2011examples,d2011completeness,miao2018identifying}:
\smallskip
\begin{as}
\label{as:completeness}
The distribution of $(S, P, \tilde X) \mid (X,Z, \tilde X)$ is complete: For all $h$ with
$\E[\norm{h(S,P, \tilde X)}] < \infty$, if $
\E[h(S,P, \tilde X) \mid X,Z,\tilde X] = 0$ then $ h(S,P,\tilde X) = 0$. Equivalently, under \cref{as:invert}, the distribution of $(\delta, P, \tilde X) \mid (X, Z, \tilde X)$ is complete.
\end{as}
\smallskip
\noindent Identification follows from the fact that, under \cref{as:invert,eq:linear_exogenous},
\[
X - \E[\sigma^{-1}(S, P, \tilde X) \mid \bar{X},Z] = 0,
\numberthis \label{eq:bh_integral_eqn}
\] while \cref{as:completeness} ensures the solution to this integral equation in $\sigma^{-1}$ is
unique.
\cite{berryhaile}'s identification result leads to a standard intuition on the need for exogenous variation in the special characteristic $X$. Assumptions \emph{(i)}--\emph{(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 \cref{eq:linear_exogenous} that only imposes conditional exogeneity of the price instruments:
\smallskip
\begin{as}
\label{as:exogenous}
$Z \indep \delta \mid \bar X.$
\end{as}
\smallskip
\noindent This restriction strengthens the mean independence in \eqref{eq:linear_exogenous} to full independence of $Z$ given $\bar{X}$ but crucially drops the exogeneity of $\bar{X}$.\footnote{\Cref{as:exogenous} 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.
}
\Cref{as:exogenous} 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 \citep{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 \cref{as:exogenous}. See \citet{ackerbergcrawford} and \citet{borusyak2025estimating} for further discussion;
Appendix A of \citet{berryhaile24} shows how \cref{as:exogenous} can arise for different types of price instruments in various economic models.}
Our relaxation of \Cref{eq:linear_exogenous} 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 \eqref
{eq:linear_exogenous} need not hold, and model-implied price counterfactuals relying on
\eqref{eq:linear_exogenous} may be incorrect \citep{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 $S$ \citep{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 \citet{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.}
\subsection{Identification strategy}
Under \cref{as:exogenous},
\begin{align}
\E[\delta \mid \bar{X},Z] = \E[\delta \mid \bar
{X}] \equiv k_0(\bar{X}) \numberthis \label{eq:moment_condition_with_z}
\end{align}
for some unknown $k_0$.
This motivates our identification strategy. The true inverse demand function, which returns $\delta$, is mean-independent of $Z$ given $\bar{X}$:
\begin{equation}\label{eq:cond_moment} \E[\sigma^{-1}(S,P,\tilde{X}) \mid \bar{X},Z] = \E[\sigma^{-1}(S,P,\tilde{X}) \mid \bar{X}].\end{equation}
We can thus rule out some candidate inverse demand functions $\check\sigma^{-1}(S,P,\tilde X)$: those that violate the
conditional moment restriction \eqref{eq:cond_moment}.\footnote{\cref{as:exogenous} 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 \cite{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 $\sigma^{-1}$:
\begin{align*}
\Theta_I \equiv
\br{h(s,p), \text{ $\R^J$-valued and invertible in $s$}\colon \E
[h(S,P) \mid X,
Z] = k(X) \text{ for some $k$}}.
\end{align*}
This set is nonempty because \eqref{eq:cond_moment} 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 $R$ \citep{borusyak2023nonrandom}. The set of functions in $\Theta_I$ is exactly the set of functions that are orthogonal to any recentered instrument:
\begin{lemma}
Let $h$ be $\mathbb{R}^J$-valued, invertible, and square-integrable. Then $ h\in \Theta_I$ if and only if $\E[h(S, P) \br{R( X, Z) - \E[R( X,Z) \mid X]} ] = 0$ for all square-integrable $R$.
\end{lemma}
\noindent 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 \cref{as:exogenous}, 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 \cref{lemma:transform} are also intermediate steps in \citet[][p. 950]{matzkin2008identification}, \citet[][p. 1190]{torgovitsky2015identification}, and \citet[][p. 1143-1144]{berryhaile24}. In particular, \citet{berryhaile24} normalize their index to avoid considering analogous indeterminacy as with $T$.}
\smallskip
\begin{restatable}{lemma}
{lemmatransform}
\label{lemma:transform}
Suppose all $\check{\sigma}^{-1}
\in \Theta_I$ have the form $\check{\sigma}^{-1}(S, P) = T(\sigma^
{-1}(S,P))$ for some $T : \R^J \to \R^J$. Then price counterfactuals are identified.
\end{restatable}
\smallskip
\noindent 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:
\begin{align*}
\check{\sigma}(\check{\sigma}^{-1}(S,P),p^\prime)=\sigma(T^{-1}(T(\sigma^{-1}(S,P))),p^\prime)=\sigma(\delta,p^\prime).
\end{align*}
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.
\subsection{A surprisingly insightful case: exogenous prices}
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 \citet{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 \citet{berryhaile}---despite our substantial relaxation of their model.
Indeed, all characteristics can be fully endogenous in this case:
\begin{prop}
\label{prop:exogenous_price}
Suppose $P = Z$ and \cref{as:completeness,as:exogenous,as:invert}
hold.
Then price counterfactuals are identified.
\end{prop}
\noindent 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 \cref{prop:exogenous_price_x_first} and \cref{prop:p-index_x} for a formal argument.} we then have:
\begin{align*}
0 &= \E[H(\delta,P)\mid P,X]-\E[H(\delta,p_0)\mid P=p_0,X]\\
&= \E[H(\delta,P)\mid P,X] - \E[H(\delta,p_0)\mid P,X]\\
\implies 0 &= H(\delta, p) - H(\delta, p_0)
\end{align*}
where the first line uses the fact that $P$ does not enter $k(X)$,
the second line uses \cref{as:exogenous}, and the third line uses \cref{as:completeness}. 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 \Cref{lemma:transform}.
\cref{prop:exogenous_price} 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 (\cref{lemma:transform}), 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 \emph{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 \citet{berryhaile}. Hence, to trace out \emph{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 \emph{proxy} for $\delta$.\footnote{A related argument by \citet{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 \eqref{eq:demand} 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 \citet{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 \citet{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 \citet{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 \cref{as:exogenous}), no such arguments are needed for $X$.
Of course, one can always make $X$ ``technically exogenous'' by rewriting \eqref{eq:cond_moment} as:
\begin{equation*}
0 = \sigma^{-1}(S, P) - k_0(X) + \tilde\xi, \qquad \E[\tilde\xi \mid X, P] = 0.
\end{equation*}
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 \citet{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 \eqref{eq:demand}. 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 \eqref{eq:demand}.
We next generalize these results and intuitions to settings with endogenous prices.
\subsection{Faithfulness and identification 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 \cref{as:completeness}:
\begin{as}[\textbf{Faithfulness}]
\label{as:faithfulness} The distribution of $(\delta, P) \mid(X,Z)$ is \emph
{faithful}: for all $\R^J$-valued $H(\delta,P)$, if $
\E[H(\delta,P) \mid X,Z] = k(X)$ does not depend on $Z$ then $H$ does not depend on $P$, i.e., $H(\delta,P) = H(\delta).
$
\end{as}
Under faithfulness, price counterfactuals are identified even with endogenous prices:\begin{restatable}{prop}{lemmamain}
\label{lemma:main}
Under \Cref{as:invert,as:exogenous,as:faithfulness}, price counterfactuals are identified.
\end{restatable}
\noindent The proof of \cref{lemma:main} follows the same steps as for \cref{prop:exogenous_price}: 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 \Cref{lemma:transform}.
The two key intuitions from the exogenous price case are retained in \cref{lemma:main}. 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{\Cref{sec:no_x} 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 \cite{borusyak2025estimating}), and where $J=1$ with $\sigma$ monotone in $\delta$ (as in \cite{imbensnewey}, \cite{torgovitsky2015identification}, and \cite{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 \citep[see, e.g.,][]{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., \citet{d2011completeness} and \citet{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.
\section{Relationship between completeness and faithfulness}
\label{sec:sufficient_conditions}
We first provide a useful calibration: when $Z$ is a perfect instrument (i.e., the exogenous price case), faithfulness and completeness are exactly equivalent.
\begin{prop}
\label{prop:equiv}
Suppose $P = Z$ and \cref{as:exogenous} holds. Then \cref{as:completeness} is equivalent to \cref{as:faithfulness}. Hence \cref{prop:exogenous_price} is a special case of \cref{lemma:main}.
\end{prop}
\noindent
In this sense, faithfulness is not an exotic new assumption: it is the natural technical condition for leveraging the conditional moment restriction \eqref{eq:cond_moment}.
Outside of the $P=Z$ case, faithfulness and completeness are distinct conditions. Two counterexamples illustrate this: \Cref{sub:fwithoutC} exhibits a class of demand models where faithfulness holds but $(\delta, P) \mid (X, Z)$ needs not be complete, while \Cref{sub:failure_of_faithfulness} 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.
\subsection{When does completeness imply faithfulness?}
\label{subsec:C_to_F}
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.
\subsubsection{Restrictions on price-setting}\label{sec:model-p}
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, \Cref{eq:price_eqn_main} 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 $Z$:
\smallskip
\begin{as}
\label{as:p-index} $P \indep (X, Z) \mid (\lambda(X, Z), \delta)$, for some $\lambda(x,
z)$ that is invertible in $z$. Equivalently, in \eqref{eq:price_eqn_main}, $ f(X, Z,
\delta,\tilde \omega) = f(\lambda
(X,Z), \delta, \tilde \omega)$.
\end{as}
\smallskip
\noindent This assumption is satisfied, in particular, when $X$
enters price in \eqref{eq:price_eqn_main} only through $\delta$---paralleling how it enters $\sigma(\cdot)$. Under this index restriction for price, \cref{as:p-index} is satisfied with
$\lambda(X, Z) = Z$. By further setting $Z=P$, this special case also nests exogenous prices and generates \cref{prop:exogenous_price} as a corollary. In general we have:
\smallskip
\begin{restatable}{prop}{proppindex}
\label{prop:p-index}
\cref{as:p-index,as:exogenous,as:completeness} imply \cref{as:faithfulness}.
\end{restatable}
\smallskip
To see how this result follows, note that for any realization $\lambda_0$ of $\lambda$:
\begin{align*}
k(X) &= \E[H(\delta, P) \mid X, Z]
\\
& = \E[\E[H(\delta, P) \mid \delta, \lambda] \mid
X, Z] \tag{Iterated expectations, \cref{as:p-index}}
\\
&= \E[\E[H(\delta, P) \mid \delta, \lambda = \lambda_0] \mid X, Z = \lambda^{-1}
(X, \lambda_0)] \tag{$Z$ does not enter $k(X)$}\\
&= \E[\E[H(\delta, P) \mid \delta, \lambda = \lambda_0] \mid X] \tag{\cref{as:exogenous}} \\
&= \E[\E[H(\delta, P) \mid \delta, \lambda = \lambda_0] \mid X, Z]. \tag{\cref{as:exogenous}}
\end{align*} 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 (\cref{as:completeness}), $H
(\delta, P) = H(\delta)$.
The second sufficient condition instead imposes a separability condition on the derivatives of price with respect to $Z$:
\smallskip
\begin{as}
\label{as:model-p}
In \eqref{eq:price_eqn_main}, $f$ is continuously differentiable in $Z$ with Jacobian
$D_z f$, which satisfies the following separability condition: for measurable functions
$A, B$,
\[
\underbrace{D_z f(X, Z, \delta, \tilde\omega)}_{J \times d_z} = \underbrace{A(f(X, Z, \delta,
\tilde \omega),
\delta)}_{J \times J} \cdot \underbrace{B
(X, Z)}_{J \times d_z} \numberthis \label{eq:separable_maintext}
\]
where $A(f(X, \delta,
Z, \tilde \omega),
\delta)$ and $B(X,Z)$ are full row rank with $\dim(Z)\equiv d_z \ge J$.
\end{as}
\smallskip
\noindent \cref{as:model-p} 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 \cref{lemma:regular}). The
restriction in \eqref{eq:fn_form} 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 \eqref{eq:fn_form} and suitable regularity conditions on $f_0, f_1, f_2$, faithfulness is \emph{equivalent} to completeness. To see this, suppose faithfulness holds but completeness fails. By \cref{prop:fimpliesc_maintext}, 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.
}
\smallskip
\begin{prop}
\label{prop:model-p}
\cref{as:model-p,as:completeness,as:exogenous} imply \cref{as:faithfulness}.
\end{prop}
\smallskip
The logic for this result is as follows: under \Cref{eq:separable_maintext}, 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 \cref{prop:model-p_x}, which redefines faithfulness and
ensures that differentiation is exchangeable with expectation.} we have
\begin{align*}
0_{J\times d_z} &= \diff{}{z} \E[H(\delta, P) \mid X, Z=z]=\diff{}{z}\E[H(\delta,f(X,Z,\delta,\omega))\mid X,Z=z] \\
&= \E\bk{H_p(\delta, P) A(P, \delta) \mid X,
Z=z}
B(X, z) \tag{\cref{as:model-p}}\\
\implies 0_{J\times J} &= \E\bk{H_p(\delta, P) A(P, \delta) \mid X,
Z} \tag{$B$ is full-rank} \\
\implies 0_{J\times J} &= H_p(\delta, P) A(P, \delta) \tag{\cref{as:completeness}} \\
\implies \hspace{0.4cm} 0_J &= H_p(\delta, P). \tag{$A$ is nonsingular}
\end{align*}
By the fundamental theorem of calculus, $H_p(\delta, P) = 0$ implies $H(\delta, P)=H(\delta)$.
\Cref{as:p-index,as:model-p} are non-nested.
\Cref{as:p-index} allows richer interactions between
how $Z$ and $\tilde \omega$: e.g., $P=f(Z,\tilde \omega)$ always satisfies
\cref{as:p-index} but not necessarily \cref{as:model-p}. \Cref{as:model-p} allows richer interactions between $X$ and $\tilde \omega$: e.g., $P=Z+f(X,\tilde \omega)$ always
satisfies \cref{as:model-p} but not necessarily \cref{as:p-index}.
\cref{as:p-index,as:model-p} 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
\begin{equation}
P=g(C,\delta)\numberthis \label{eq:bertrand-nash}
\end{equation}
for some function $g$ (see \cref{sub:BertrandNash}). 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
\cref{as:p-index} follows. Similarly, \cref{as:model-p} 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.
\subsubsection{Restrictions on the $\delta$ index}
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 \emph{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.
\begin{restatable}{prop}{lemmafinite}
\label{lemma:finite}
Fix $M \in \N$. Assume the support of $\delta$ and $X$ are both finite sets of size $M$.
Then \cref{as:completeness,as:exogenous} imply \cref{as:faithfulness}.
\end{restatable}
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 \eqref{eq:location-scale-maintext} 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 $q(\cdot)$:
\begin{as}
\label{as:smoothness}
Fix some integers $s > J+1$ and $M = J+2s+1$.
Let \[\mathbb{K}^s = \br{u : \R^J \to \R^J \colon u(x) = Ax + r(x), A \in \R^{J\times J},r \in
\mathbb{W}^{s,\infty}(\R^J)}\] where $\mathbb{W}^{s,p}(\R^J)$ is a Bessel potential space of smoothness parameter $s$ and integrability parameter $p$, defined in \eqref{eq:bessel}.
Let $\mathcal K = \mathbb{K}^M$. Assume
\begin{enumerate}
\item $a,b$ in \eqref{eq:location-scale-maintext} are invertible.
\item (Density smoothness for $\epsilon$) The Fourier transform of $q$ and its derivatives have bounded tails obeying
\cref{as:location-scale-assns}(2).
\item (Limited heteroskedasticity) $\Sigma(x)$ is
uniformly close to some fixed $\Sigma_0$, in the sense that certain distances in
\cref{as:location-scale-assns}(3) are bounded above by a sufficiently small $\psi$.
\item (Smooth $b$ with Lipschitz inverse) It is known that $b\in \mathcal K$ and that \\ $\sup_
{x\in \R^J}\| (D b(x))^{-1}\|_{\mathrm{op}} <\infty
$
where $D b(x)$ is the Jacobian of $b$.
\end{enumerate}
\end{as}
The key requirement in \cref{as:smoothness} 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 \cref{as:smoothness} 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{\Cref{as:smoothness}(2) is satisfied by
distributions with Gamma-like tails, though it rules out Gaussian distributions. See
\cref{lem:B2} and \cref{rmk:radial_gamma}. 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 \cref{lemma:main} continues to hold.
\begin{prop}
\label{prop:model_delta}
Under \cref{as:completeness,as:exogenous,as:invert,as:smoothness}, for any integrable $H
(\delta,
P)$, if $\E[H(\delta, P) \mid X, Z] \in \mathcal K$, then $H
(\delta,
P) = H(\delta)$ for some $H(\delta)$. As a result, price counterfactuals are identified.
\end{prop}
Unlike \cref{lemma:finite}, which requires an exact equivalence in the number of support points of $\delta$ and $X$, \cref{as:smoothness} 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 \Cref{subsec:C_to_F} 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.
\subsection{When does faithfulness imply completeness?}
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 \cref{sub:fwithoutC}---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:
\begin{restatable}{prop}{propfimpliesc}
\label{prop:fimpliesc_maintext}
Suppose \cref{as:exogenous} holds and $\delta \mid X$ is complete. Then \cref{as:faithfulness} implies
\cref{as:completeness}.
\end{restatable}
\noindent 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 $H(\delta) = 0$.
\section{Practical Implications}\label{sec:practical}
Our nonparametric identification results yield several insights for applied researchers estimating \emph{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 \citet{berryhaile} and \citet{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 \eqref{eq:demand}. For mixed logit models, this would mean $X$ enters demand without a random coefficient; \citet{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 \cite{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 \citep{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 \citep{borusyak2025estimating} or deviations of realized input prices from futures markets values \citep{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 \citet{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 \citet{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 \citep{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 \citep[][Section 4]{chen2025reinterpreting} which our identification strategy does not require.
\section{Conclusion}\label{sec:conclusion}
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 \citep{compiani2022market,chen2015sieve}. As in \cite{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.
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