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Reinterpreting demand estimation
\maketitle
\begin{abstract}
This paper clarifies how and why structural demand models
\citep{berry2014identification,berry2024nonparametric} predict unit-level counterfactual
outcomes. We do so by casting structural assumptions equivalently as restrictions on
the joint distribution of potential outcomes. Our reformulation highlights a
\emph{counterfactual homogeneity} assumption underlying structural demand models: The
relationship between counterfactual outcomes is assumed to be identical across markets.
This assumption is strong, but cannot be relaxed without sacrificing identification of
market-level counterfactuals. Absent this assumption, we can interpret model-based
predictions as extrapolations from certain causally identified average treatment
effects. This reinterpretation provides a conceptual bridge between structural modeling
and causal inference.
\end{abstract}
\pagestyle{plain}
\newpage
\section{Introduction}
Predicting counterfactual outcomes for individual units is central to many areas of
economics. In industrial organization, for instance, prices set by firms depend on market
shares at counterfactual prices; thus, predictions of these counterfactuals yield
markups and marginal costs. Structural econometric methods are often motivated by their
ability to predict counterfactual outcomes for individual units. Once researchers fit a
model to observed data, the model implies counterfactual outcomes for all units. By
contrast, the literature on causal inference
\citep{splawa1990application,rubin1974estimating} generally focuses on recovering \emph
{average} counterfactuals. For unit-level counterfactuals, causal inference methods are
typically informal---e.g., extrapolating from average treatment effects (ATEs) among
observably similar units---if they are produced at all.
These ``two cultures'' for predicting counterfactuals, to quote
\citet{breiman2001statistical}, face parallel critiques. The causal inference literature
shows that certain average counterfactuals are identified through credible treatment
variation. However, these averages often are not themselves of economic interest.
Extrapolating them to individuals would only be valid under constant treatment effects,
which severely restricts unobserved heterogeneity. In contrast, structural methods impose
modeling assumptions up front to directly target unit-level counterfactuals. But these
predictions seem to hinge on the model: It can be unclear how to interpret them without
the model.\footnote{As examples of these respective critiques in the literature,
\citet{berry2021foundations} write of the treatment effects literature, ``In empirical
settings with endogeneity and multiple unobservables, economists often settle for
estimation of particular weighted average responses (e.g., a local average treatment
effect); but this is a compromise poorly suited to the economic questions that motivate
demand estimation, as these typically require the levels and slopes of demand at specific
points.'' \citet{nevo2010taking} argue that heterogeneity is sufficiently strong for
average effects over past mergers to not be informative: ``As our discussion of merger
analysis illustrates, industrial organization economists seem far more concerned than
labor economists that environmental changes are heterogeneous, so that useful estimates of
average treatment effects in similar situations are not likely to be available.''
\citet{angrist2010credibility} write of structural industrial organization, ``In this
framework, it’s hard to see precisely which features of the data drive the ultimate
results.''}
To reconcile and bridge the two cultures, we ask: First, do structural models avoid
restricting unobserved heterogeneity, or do they too extrapolate from averages? Second,
how should we interpret structural model predictions when the model is only an
approximation? This paper studies these questions in the context of canonical structural
demand models, in both settings with market-level shares \citep
{berry1995automobile,berry2014identification} and settings with demographic-specific
market shares \citep{berry2024nonparametric}.
In either case, we cast the structural model equivalently as restrictions on the joint
distribution of potential outcomes. This equivalent reformulation---in the spirit of
\citet{vytlacil2002independence,vytlacil2006ordered}---does not simply declare that the
potential outcomes are generated from the corresponding structural model. Rather, the
model is reinterpreted as restricting the joint distribution of potential
outcomes.\footnote{\citet {vytlacil2002independence} shows that the selection model $D_i
=
\one (\alpha + \beta z > \xi_i)$, for $\beta>0$, is equivalent to the monotonicity
restriction $\P (D_i(1) \ge D_i(0)) = 1$ \citep{imbensangrist}. The former is stated as
a generative structural model of an endogenous treatment $D$, whereas the latter is
stated as a restriction on the joint distribution of $ (D_i(1), D_i(0))$. }
We find that a key restriction that structural demand models impose is what we term \emph
{counterfactual homogeneity}: If the structural model holds, then two counterfactual
outcomes are deterministically related to each other through a function that is identical
across markets. Concretely, let $Y_i (a)$ denote the counterfactual market shares of a
given market $i$ under a bundle $a$ of product characteristics and prices. Counterfactual
homogeneity restricts that $\var(Y_i(a) \mid Y_i(a')) = 0$ for any bundle $a' \neq a$,
where the variance is taken over draws of markets over its population. This is a strong
restriction on the joint distribution of $(Y_i(a), Y_i(a'))$. In this sense, these
structural
demand models \emph{do} restrict unobserved heterogeneity.
This restriction is not a flaw of these particular structural models---if we want models
that point-identify unit-level counterfactuals. Restricting unobserved heterogeneity is
necessary for identifying unit-level counterfactuals. Thus, counterfactual homogeneity
cannot be relaxed, unless we give up point-identification as well.\footnote{Of course,
point-identification is convenient but not necessary to make effective use of data.
Partial identification strategies \citep {molinari2020microeconometrics} are popular in
the literature on entry games \citep {ciliberto2009market} and revealed preference
\citep{pakes2015moment}. It is also possible to partially relax counterfactual homogeneity
by demanding that only certain counterfactuals---e.g., counterfactual in prices---are
identified \citep{andrews2023structural,borusyak2025estimating,newpaper}. } Additional
functional form assumptions in \citet{berry2014identification,berry2024nonparametric},
which are sufficient but not necessary, ensure that the homogeneous relationship linking
counterfactuals is uniquely recovered by instrument variation. Modulo these additional
functional forms, nonparametric structural demand models are indeed minimally restrictive
for point-identified unit counterfactuals.
Nevertheless, just as we are uneasy with homogeneous treatment effects when extrapolating
from ATEs, counterfactual homogeneity should also give us pause. Counterfactual
homogeneity meaningfully restricts how markets may be different from each other.\footnote
{This differs from within-market consumer heterogeneity allowed by BLP \citep
{berry1995automobile}. In our notation, consumer heterogeneity corresponds to whether $a
\mapsto Y_i(a)$ is a flexible function. In contrast, counterfactual homogeneity are
restrictions on how different the demand curves $Y_i(\cdot)$ and $Y_j(\cdot)$ for two
markets can be.} It rules out, for instance, settings in which each market aggregates a
population of consumers with heterogeneous preferences, but different markets have
unobservably different populations of consumers. It also imposes that markets with the
same observed conditions necessarily have identical counterfactuals everywhere---ruling
out demand surfaces that intersect nontrivially.
These implications of counterfactual homogeneity are demanding. This reflects
that the structural models are simplifications and are unlikely to hold literally. Thus,
unit-level counterfactuals under counterfactual homogeneity are better interpreted as
extrapolated predictions rather than as point-identified treatment effects \citep
{kline2019heckits}. We formalize how \citet{berry2014identification} extrapolate from
average effects. We also show that this kind of extrapolation is essentially what large
classes of structural models do. This exercise clarifies the value of structural models.
Many counterfactual predictions are effectively extrapolating from certain ATEs---acting
as if every unit has the same treatment effect. Structural models are additionally helpful
in motivating \emph{which} ATEs to extrapolate from.
This paper contributes to a literature that bridges causal inference and structural
modeling
\citep{andrews2023structural,kline2019heckits,borusyak2025estimating,kong2024nonparametric,humphries2025conviction,torgovitsky2019nonparametric,mogstad2024instrumental,angrist2000interpretation,conlon2021empirical}.
This paper is also related to transformation
models
and other simultaneous equation
models
\citep
{chiappori2015nonparametric,vuong2017counterfactual,benkard2006nonparametric,matzkin2008identification}.
Counterfactual homogeneity is related to a literature on omitted parameter
heterogeneity
\citep{chesher1984testing,hahn2014neglected,qian2025testing}.
Like
\citet
{berry2014identification,berry2024nonparametric,vytlacil2002independence,kline2019heckits},
this paper's primary focus is conceptual---the identification and expressivity of
workhorse models.\footnote{The models estimated in practice are typically versions of
\citet{berry2014identification,berry2024nonparametric} with additional parametric
assumptions. \citet{compiani2018nonparametric} studies nonparametric estimation for
\citet{berry2014identification}. }
This paper proceeds as follows. \Cref{sec:market_level_data} derives equivalent
assumptions to \citet{berry2014identification}. \Cref{sec:discussion} discusses
counterfactual homogeneity, derives its necessity, and derives an equivalence between
structural model predictions and extrapolations from average treatment effects. \Cref
{sec:demographics_specific_market_shares} derives equivalent assumptions to
\citet{berry2024nonparametric} and examines the extent to which models with micro-data
allow for counterfactual heterogeneity.
\section{Market-level data}
\label{sec:market_level_data}
We start with a standard model of differentiated products \citep
{berry1995automobile,berry2014identification} in potential outcomes notation. Markets are
i.i.d. draws from a population $F^*$, following \citet
{freyberger2015asymptotic}. Each market contains the same $J \in \N$ inside options. The
observed data in each market take the form $(Y, A, Z)$. Here $Y \in
\mathcal Y \subset [0,1]^J$ is the vector of observed market shares,
$A \in
\mathcal A\subset \R^{J \times d_a}$ is the bundle of prices and characteristics
associated with each of the $J$
goods, and $Z \in \mathcal Z \subset \R^{d_z}$ is a vector of external instruments that
includes the exogenous entries of $A$. For concreteness, we may write $A =
(A_1,\ldots,A_J)$ for each product, for $A_j = (P_j, X_j)$ the prices and
characteristics of a product. We view $A$ as a treatment acting on $Y$.
To embed the setup in the potential outcomes framework, let the random variable $Y
(a)$ denote the potential outcome for a given market, were the bundle set to some
counterfactual value $a$. The observed market shares $Y$ are generated from underlying
potential outcomes, $Y = Y(A)$. We condition on other observed market covariates and
omit them from notation.
Structural demand models posit that counterfactuals $Y (a)$ are generated
through \[ Y(a) = \mathfrak{s}(a, \xi) \numberthis.
\label{eq:structure}
\]
For instance, \citet{berry1995automobile} posit that market shares aggregate
heterogeneous consumers with Gumbel idiosyncratic preferences and
heterogeneous valuations for attributes ($\beta \sim G$): \[
Y_j(a) = \mathfrak{s}_j(a, \xi) = \int
\frac{e^{a_j'\beta + \xi_j}}{1 + \sum_{k=1}^J e^{a_k'\beta + \xi_k}} d G(\beta)
\text{ for some distribution $G$.}
\]
Here, the map $\mathfrak{s}$ is indexed by the random coefficient distribution $G$.
The causal inference and structural demand literatures differ in their typical workflow.
The former usually focuses on average effects like $\E[Y(a_1) - Y(a_0)], \frac{d}{da} \E
[Y(a)]$, or conditional-on-covariates versions thereof
\citep{angrist2000interpretation}. If $a_1$ represents a price increase in good $j$
relative to $a_0$, these parameters measure the average response of market shares to
this price increase across some (sub)population of markets. These averages are in turn
identified through various comparisons that exploit variation in $A$ induced by the
instruments. Care is taken on restricting how $A$ responds to instruments
(e.g., monotonicity, \citet{imbensangrist}) to ensure that instrument-level comparisons
recover proper comparisons over endogenous treatments.
On the other hand, the structural demand literature is concerned with \emph
{unit-level} counterfactuals and views average effects as insufficient for scientific and
policy objectives. These unit-level counterfactuals are $Y(a_1) - Y(a_0)$, representing a
particular market's response to changes in $a$. Typically, models impose restrictions
on \eqref{eq:structure}, such that the structural error $\xi =
\mathfrak{s}^{-1}(A, Y)$ can be recovered from observed variables with knowledge of $\mathfrak{s}$, which
is itself identified through instrument variation.\footnote {In the case of
\citet{berry1995automobile}, the random coefficient distribution $G$ is identified under
additional parametric assumptions, implying that $\mathfrak{s}$ is.} The model then identifies
counterfactual outcomes through $Y(a) =
\mathfrak{s} (a, \mathfrak{s}^ {-1} (A, Y))$. Identification of $\mathfrak{s}$ requires assumptions on instrument
strength, but need no monotonicity-type restrictions on the selection of $A$.
A standard intuition in causal inference is that unit-level counterfactuals---or even the
distribution of individual treatment effects---are not identified even with a randomized
experiment, absent assumptions like rank invariance
\citep{doksum1974empirical,heckman1997making}.\footnote{This is even termed the
``fundamental
problem of causal inference'' \citep {holland1986statistics}.} Consequently, predictions
of unit-level counterfactuals are rare and often informal in causal inference. For
instance, a unit's treatment effect may be approximated by the conditional average
treatment effect (CATE) among observably similar units, under implicit assumptions
ruling out unobserved heterogeneity.
This lack of focus on individual counterfactuals---as well as concerns about unobserved
heterogeneity---in part explains limited takeup of standard causal inference tools and
language in subfields that rely on structural demand models. On the other hand, the
complexity of structural models makes it difficult to see how its predictions depend on
modeling assumptions. It is thus useful to understand what drives identification of
unit-level counterfactuals. We do so by interpreting structural models as explicit
restrictions on the joint distribution of $Y(\cdot)$.
We now set up notation to discuss identification formally and to introduce the assumptions
in \citet{berry2014identification}. We let $F \in \mathcal P$ denote the distribution of
the observed variables, and we let $F^* \in \mathcal P^*$ denote the distribution of $
(\br{Y (\cdot): a
\in
\mathcal A}, A, Z)$. Each $F^*$ generates a particular $F$ through $Y = Y(A)$, and thus
$\mathcal P^*$ generates $\mathcal P$. Let $\mathcal S \subset \mathcal Y \times
\mathcal A$ denote the support of $(Y(A), A)$.\footnote{For simplicity, we assume throughout that
all
members of $\mathcal P$ have common support: $\P_{F}((Y, A, Z) \in E) = 0 \iff \P_{F'}(
(Y, A, Z)
\in E) = 0$ for all $F, F'
\in \mathcal P$ and all events $E \subset \mathcal Y \times
\mathcal A \times \mathcal Z$.}
We define identification for unit-level counterfactuals: A unit-level counterfactual
$Y(a)$ is identified if we can compute it from any other $(Y(a'), a')$, with a function
$m(\cdot; F)$ that is known given the observed distribution $F$.
\begin{defn}
\label{defn:id} We say that a counterfactual $Y(a)$ is identified\footnote{This notion is
slightly stronger than what may be natural. We require the function $m$ to link any two
potential outcomes. An alternative definition could just require that $m$ link the
observed outcome $ (Y, A)$ to counterfactual outcomes. When $A$ is randomly assigned,
these two notions are identical.} at $F$ if for all $F^* \in
\mathcal P^*$ that generates $F$, there is some function $m(a, \cdot, \cdot; F) :
\mathcal S \to \mathcal Y$
such that \[
\P_{F^*}\br{Y(a) = m(a, Y(a'), a'; F)} = 1
\]
for all $(Y(a'), a') \in \mathcal S$. We say that all counterfactuals are identified
under $\mathcal P^*$ if, for all $a \in \mathcal A$, $Y(a)$ is identified at all $F
\in
\mathcal P$.
\end{defn}
\noindent If counterfactuals are identified, then the function $m (\cdot; F)$ can be
obtained from $F$. Any counterfactual for any market can then be computed by substituting
the observed $(Y,A)$ into this function, $Y (a) = m (a, Y,A; F)$. Under
\eqref{eq:structure}, if we identify the function $\mathfrak{s}$ and can compute $\xi$ from any $
(Y(a'), a')$ with the knowledge of $F$, then we can identify counterfactuals $Y(a)$ by
applying $Y(a) = \mathfrak{s} (a,
\xi(Y(a'), a'))$.
The seminal paper by \citet{berry2014identification} shows identification in this sense
for a flexible class of structural demand models. Their result nests parametric demand
models like logit, nested logit, or BLP \citep{berry1995automobile}. To introduce their
result, we partition characteristics and prices of option $j$ into $a_j =(x_{1j}, p_j, x_
{2j})$. We write $a = (x_1, p, x_2)$. Here, $x_ {1j}
\in \R$ is a special scalar characteristic,\footnote{To nest BLP in this framework, $x_1$
can
be chosen to be any characteristic that does not have a random coefficient
\citep{berry2014identification}.} $p_j$ is price, and $x_ {2j}$ collects other
characteristics.
In their identification argument, prices $p$ and characteristics $x_2$ do not play
distinct roles. Let $\mathcal X$ denote the space in which $p, x_2$ take values.
{\begin{as}[Linear index]
\label{as:bh_linear_index}
For some random variable $\xi \in \Xi \subset \R^J$ and some map $\mathfrak{s} = \mathfrak{s}_{F^*}$, the
potential outcomes $F^*$ satisfy \[\P_{F^*}\br{Y (a) = \mathfrak{s}(x_1 + \xi, p, x_2)} = 1
\quad \text{for all $a =
(x_1, p, x_2) \in \mathcal A$}.\]
\end{as}
\begin{as}[Invertible demand]
\label{as:bh_invertible}
The function $\mathfrak{s}(\cdot, p, x_2)$ is invertible in its first argument: There exists
some measurable function $\mathfrak{s}^ {-1}: \mathcal Y \times \mathcal X \to \R^J$ where \[\P_{F^*}\br{x_1 + \xi = \mathfrak{s}^{-1}(Y(a), p,
x_2)} = 1 \quad \text{ for
all $a = (x_1, p,x_2) \in \mathcal A$}. \]
\end{as}
}
\Cref{as:bh_linear_index} is stated as Assumption 5.1 in \citet{berry2021foundations}. It
is an implication of Assumption 1 in \citet{berry2014identification}, which is a similar
index restriction on an underlying random utility model. \Cref{as:bh_invertible} is a
conclusion of Lemma 1 in \citet{berry2014identification}, justified via a ``connected
substitutes'' condition in \citet{berry2013connected}. Since the identification of demand
only relies on this implication, we impose it as a high-level assumption instead.
Combined with assumptions on instruments, \cref{as:bh_invertible,as:bh_linear_index} allow
for identification of the function $\mathfrak{s}$ by exploiting an
``index-inversion-instruments'' recipe \citep{berry2021foundations}, which returns the
following moment condition:
\[\E[\xi \mid Z] = \E[\mathfrak{s}^{-1} (Y, P, X_2)\mid Z] - X_1 = 0.\] The function
$\mathfrak{s}^{-1}$ is then identified through nonparametric instrumental variables
\citep{newey2003instrumental}; see Theorem 1 in \citet{berry2014identification}. Upon
identification of $\mathfrak{s}$, the structural shock $\xi =
\mathfrak{s}^ {-1}(Y,P, X_2) - X_1$ can be computed and unit-level counterfactuals are recovered.
The map $m$ in \cref{defn:id} can be chosen as \[Y (a) = \mathfrak{s} (x_1 +
\underbrace{\mathfrak{s}^{-1}(Y (a'), p', x_2') - x_1'}_{\text{model-implied $\xi$}}, p, x_2)
\quad a= (x_1, p, x_2),
a'= (x_1',p', x_2'),
\] which depends on the data only through the identified structural function $\mathfrak{s}$.
This identification argument is mathematically simple. It shows that parametric
restrictions in BLP, for instance, are not crucial for identification. Nevertheless, it
can be somewhat mysterious how the index and invertibility assumptions allow for
identification of $\mathfrak{s}$, and what distributions over $Y(a)$ they rule out. Our central
exercise is to restate
\cref{as:bh_invertible,as:bh_linear_index} equivalently only in terms of counterfactuals
$Y(\cdot)$, without presuming a generative model of $Y (\cdot)$. This restatement
precisely clarifies the restrictions on counterfactuals made by the generative model.
\subsection{Equivalent assumptions in potential outcomes}
Our first assumption imposes that $Y(\cdot)$ satisfy \emph{counterfactual
homogeneity}.
{\begin{as}[Counterfactual homogeneity]
\label{as:latent_homogeneity}
For each $F^* \in \mathcal P^*$, there exists some mapping $C_{\cdot \to \cdot} = C_
{\cdot \to \cdot , F^*}$ such that \[\P_{F^*}\br{Y
(a') = C_{a \to a'}(Y(a))} =1
\text{
for all $a, a' \in \mathcal A$.}
\numberthis \label{eq:conversion}\]
Equivalently, for some baseline treatment $a_0
\in
\mathcal A$, there exists $C_0(y,a) = C_{a \to a_0}(y)$, invertible in its first
argument, such
that for all $a \in \mathcal A$,
\[
\P_{F^*} \br{Y(a_0) = C_0(Y(a), a)} = 1. \]
\end{as}
}
\Cref{as:latent_homogeneity} states that there is a {deterministic} mapping $C_{a \to a'}$
that converts one counterfactual $Y(a)$ into another $Y(a')$. This mapping is common to
\emph{all markets} in the population $F^*$. Equivalently,
counterfactuals $Y(a')$ have zero conditional
variance given any other counterfactual outcome $Y(a)$, over draws of markets in $F^*$:
\[\var_ {F^*} \pr{Y (a') \mid Y
(a) } = 0_{J\times J} \quad \text{ for all $a, a' \in \mathcal A$.}
\numberthis \label{eq:zero_variance}
\] Also equivalently, we can first convert all counterfactuals $Y(a)$ into some baseline
outcome $Y(a_0)$, and then generating counterfactuals $Y(a')$ from $Y(a_0)$. In these
senses, \cref{as:latent_homogeneity} restricts the heterogeneity across markets by
restricting the intrinsic dimension of the support of potential outcomes $\br{Y(a)}_
{a \in \mathcal A}$. The
\emph{relationship} between $Y (a)$ and $Y(a_0)$ is kept homogeneous across all markets.
We refer to it as \emph {counterfactual homogeneity} for this reason.
An implication of counterfactual homogeneity is that all markets that have identical
conditions in the data $(Y, A) = (y,a)$ must then also have identical counterfactual
outcomes $Y(a') = C_{a \to a'}(y, a)$, for all counterfactual characteristics and
prices $a' \in
\mathcal A$: Geometrically, if two markets have crossing demand curves $a \mapsto Y
(a)$, then the two demand curves must be identical. \Cref {as:latent_homogeneity} is
also a generalization of rank invariance in standard treatment effect settings.\footnote
{There, rank invariance
\citep{doksum1974empirical} imposes that $Y(0) = C(Y(1))$ for some monotone $C$, and if
both outcomes are continuously distributed, $C$ can be taken to be $F_{Y(0)}^{-1} \circ
F_{Y(1)}$ and invertible, for $F_ {Y(j)}$ the CDF of $Y (j)$.} Relative to rank
invariance, \cref{as:latent_homogeneity} extends to non-binary treatment and
multidimensional outcomes.
Counterfactual homogeneity rules out heterogeneity \emph{across markets}. It is not an a
priori restriction on how a particular market, say a realization $y_i (a) = Y_i(a)$ drawn
from $F^*$, may respond to counterfactual bundles $a
\mapsto y_i(a)$. Thus, to the extent that we think of $y_i(a)$ as aggregations of
consumers within market $i$, \cref{as:latent_homogeneity} generates flexible
substitution patterns for any given market. What \cref{as:latent_homogeneity} does
restrict is how consumer populations can be different across markets.
\begin{exsq}[An economic model that violates counterfactual homogeneity]
\label{ex:model}
Suppose each market aggregates BLP-style preferences: \[
Y_i(a; \xi_i, \zeta_i) = \int \frac{e^{a_j'\beta+\xi_{ij}}}{1+\sum_{k=1}^J e^
{a_k'\beta +
\xi_{ik}}} dG
(\beta; \zeta_i).
\] However, instead of assuming that the consumer taste distributions $G
(\cdot; \zeta_i)$ are identical across markets, perhaps certain markets $(\zeta_i = 1)$
are more price sensitive than others $(\zeta_i = 0)$. The type of the market $\zeta_i$
is either unobserved or insufficiently proxied by observables. Then $\zeta_i$ cannot be
recovered from the observed data and thus unit-level counterfactuals are not
identified, even with randomized $A$. An example with $J=1$ is shown in
\cref{fig:demandcurves}.
\end{exsq}
\begin{figure}[tb]
\centering
\includegraphics[width=0.7\textwidth]{example.pdf}
\begin{proof}[Notes]
All market shares follow random coefficient logit $Y(p) = \int \Lambda
\pr{-\alpha p + \xi} \,d G (\alpha)$, for $\Lambda(t) = 1/(1+e^{-t})$ and randomly
assigned prices. Markets are randomly blue or orange, corresponding to $\zeta_i$ in
\cref{ex:model}. The {\color{blue}blue} markets
have {$\color{blue} G_ {\text{blue}}
\sim
\mathrm{Lognormal}(0, 0.5^2)$}. The {\color{orange}orange} markets have {$ \color
{orange}G_ {
\text{orange}} \sim \mathrm{Lognormal}(-0.5, 2^2)$}. For each market realization $
(P, Y)$, we plot its own counterfactual shares at nearby price values (for {\color
{blue}blue} markets, this is the {\color{blue}blue} curve). We also compute the
$\xi$ value for a hypothetical market of opposite color such that its hypothetical
demand curve passes through $(P, Y)$ (for {\color{blue}blue} markets, this is the
{\color{orange}orange} curve). Because demand curves cross in this setting, this
setup
violates counterfactual homogeneity. When the colors of the markets are not
observed, the population distribution of $ (P, Y)$ cannot perfectly distinguish
whether a particular market is blue or orange. Since different colors imply
different counterfactuals---including price elasticities, the counterfactuals are
not identified. \end{proof}
\caption{A parametric demand model with $J=1$ where counterfactual homogeneity fails
to hold}
\label{fig:demandcurves}
\end{figure}
The second assumption imposes some functional form restriction on the map $C_0$.
{\begin{as}[Latent partial linearity]
\label{as:latent_partial_linearity}
For all $F^* \in \mathcal P^*$, there exists a function $h = h_{F^*}:
\mathcal Y \times \mathcal X \to \R^J$ where, for all $a = (x_1, p, x_2) \in
\mathcal A$, invertible in its first argument, such that \[
\phi^{-1}\pr{C_0(y, a)} = h(y, p, x_2) - x_1,
\numberthis \label{eq:linearity}
\]
for $\phi^{-1}(y) = h(y, p_0, x_{20}) - x_{10}.$
\end{as}}
\Cref{as:latent_partial_linearity} states that, up to some invertible transformation
$\phi$, $C_0$ is partially linear in $x_1$. This functional form restriction is
important for identification using instrumental variables. It is also substantive,
imposing, e.g., that $x_1$ is excluded from elasticities: the Jacobian of $Y(a)$ with
respect to $a$ depends on $x_1$ only through $Y (a)$:
\[
\diff{Y(a)}{x_1} = \pr{\diff{h(Y(a), p, x_2)}{y}}^{-1} \quad \diff{Y(a)}{(p, x_2)} = - \diff{Y(a)}
{x_1} \diff{h(Y(a), p, x_2)}{(p, x_2)}. \numberthis \label{eq:derivatives}
\]
\Cref{as:latent_homogeneity,as:latent_partial_linearity} can be combined as
the following homogeneity assumption on some transformation of potential outcomes.
{\begin{as}[Homogeneous effects in a transformed outcome]
\label{as:homogeneity}
There exists some function $H(y, p, x_2) = H_{F^*}(y,p,x_2)$, invertible in $y$, such
that the transformed potential outcome $H(a)$, for $H(a) \equiv H(Y(a), p, x_2)$,
satisfies: \begin{enumerate}
\item (No treatment effect in $(p, x_2)$) For all $(x_1, p_1, x_{2,1}),
(x_1,p_2,x_{2,2})\in \mathcal A$, \[\P_{F^*}\br{H
(x_1,p_1,x_{2,1}) = H (x_1, p_2, x_{2,2})} = 1\]
\item (Homogeneous linear effects in $x_1$) For all $(x_{1,1}, p, x_2),
(x_ {1,2}, p, x_2)
\in \mathcal A$, \[
\P_{F^*}\br{H(x_{1,1}, p, x_2) - H(x_{1,2}, p, x_2) = x_{1,1} - x_{1,2}}=1.
\]
\end{enumerate}
\end{as}}
\Cref{as:homogeneity} states that for some unknown transformation of the potential outcome
$H(a) = H(Y(a), p, x_2)$, if we treat $H(a)$ as a new potential outcome, then it admits
no treatment effects in $(p, x_2)$ and linear treatment effects in $x_1$.\footnote
{The slope of the $x_1$-treatment effect on $H(a)$ can be normalized through $H$.}
\Cref{as:homogeneity} makes clear how
\cref{as:latent_homogeneity,as:latent_partial_linearity} restrict treatment effect
heterogeneity. Viewed as assumptions on some transformation of potential outcomes,
\cref{as:latent_homogeneity,as:latent_partial_linearity} are exactly constant treatment
effects assumptions. \Cref{as:homogeneity} is weaker than standard constant treatment
effects by not specifying which transformed outcome satisfies homogeneity---only that
some transformation does.
Our main result is that these assumptions are equivalent to the
\citet{berry2014identification} assumptions, in the same spirit as
\citet{vytlacil2002independence,vytlacil2006ordered}'s results for instrumental variable
models. The equivalence is easy to derive, once we link $(\mathfrak{s}, \xi)$ in \cref
{as:bh_invertible,as:bh_linear_index} to $(h,\phi, C_0)$ in
\cref{as:latent_homogeneity,as:latent_partial_linearity} and $H$ in \cref{as:homogeneity}:
\[
\mathfrak{s} = h^{-1}, \quad \xi = \phi^{-1}(Y(a_0)), \quad h(y,p,x_2) = H(y, p, x_2).
\]
\begin{restatable}{theorem}{thmmainequiv}
\label{thm:mainequiv}
The following are equivalent:
\begin{enumerate}
\item \cref{as:bh_invertible,as:bh_linear_index},
\item \cref{as:latent_homogeneity,as:latent_partial_linearity},
\item \cref{as:homogeneity}.
\end{enumerate}
\end{restatable}
Reformulating assumptions this way retells the progress in demand models with market share
data. In the standard telling \citep{ackerberg2007econometric}, different generations of
structural demand models (e.g., vertical models, simple logit, nested logit, BLP, \citet
{berry2014identification}) all maintain random utility models of consumer behavior and
treat market shares as aggregations of consumer choices. They differ in the flexibility
of the utility model and of implied substitution patterns. In this retelling, all such
demand models instead maintain counterfactual homogeneity and latent partial linearity of
market shares. They specify different parametrized classes of $h$, which governs
model-implied substitution patterns. These two perspectives---making the random utility
model increasingly flexible versus enlarging the function class for $h$---meet at the
nonparametric model in \citet {berry2014identification}.
This reformulation also clarifies why nonparametric structural demand models are able to
identify unit-level counterfactuals. It likewise explains why these models avoid
selection assumptions on how $A$ responds to instruments. Unit-level counterfactuals are
identified because of counterfactual homogeneity. Counterfactual homogeneity likewise
means that heterogeneity in the first stage does not matter for how $A$ affects $Y$,
since different types of compliers trace out exactly the same response in $H(a)$.
\section{Discussion}
\label{sec:discussion}
\subsection{The curse of unobserved heterogeneity}
\Cref{thm:mainequiv} clarifies that structural demand models \emph{do} restrict unobserved
heterogeneity. The need to restrict unobserved heterogeneity is not specific to these
particular demand models either. Any model that \emph{identifies} unit-level
counterfactuals necessarily has to impose counterfactual homogeneity: \cref
{as:latent_homogeneity} is necessary for identification in the sense of \cref
{defn:id}.
\begin{restatable}[Necessity of counterfactual homogeneity]{prop}{lemmalatent}
\label{lemma:lemmalatent}
Suppose all counterfactuals are identified under $\mathcal P^*$ in the sense of
\cref{defn:id}, then \cref{as:latent_homogeneity} is satisfied.
\end{restatable}
No nonparametric model can relax counterfactual homogeneity without giving up
identification. Thus, the difference between the two cultures---structural demand
modeling and causal inference---is when each incurs this curse of unobserved
heterogeneity. Structural demand models incurs it up front, whereas causal inference
approaches implicitly incurs it when extrapolating from average treatment effects. In
either case, the fundamental problem of causal inference remains.
Given the goal of identifying unit-level counterfactuals, \citet
{berry2014identification} impose little more than what is necessary. The functional form
assumption, \cref {as:latent_partial_linearity}, is strictly speaking not
necessary.\footnote{As a simple example, suppose we instead assumed a different,
multiplicative functional form: \[ Y_j(a_0) = \phi_j\pr { g_j(Y(a), x) \exp(-w_j) }.
\numberthis \label{eq:multiplicative}
\] When $g_j(y, x)$ can take on zero or negative values, this multiplicative formulation
is different from \cref{as:latent_partial_linearity} because $\log(g_j(Y(a), x) \exp
(-w_j))$ is undefined. However, we may continue to exploit $\E[g_j(Y, X) \mid W, Z] =
c_0 \exp(W_j)$ to identify $g_j(\cdot, \cdot)$.} But it is not relaxable without
imposing additional assumptions, since many distinct mappings among the potential
outcomes are observationally equivalent and satisfy counterfactual homogeneity.\footnote
{This is clear with two treatments $(a_0, a_1)$, the set of observationally equivalent
$C_0$ corresponds to the set of transport maps between the distributions $F_{Y
(a_0)}$ and $F_{Y(a_1)}$. One would need some other assumption to rule out all but one
transport map for identification. } In this sense, the assumptions in \citet
{berry2014identification} are close to minimal for point-identification.
Nevertheless, counterfactual homogeneity is likely misspecified: The zero-variance
implication \eqref{eq:zero_variance} is implausible in many applications. Economic
models allowing for markets that differ in terms of their consumer populations,
like \cref{ex:model}, would violate this assumption. We may have little compelling
reason to rule out these models---other than that ruling them out makes unit-level
counterfactuals identified. In parametric models, these restrictions are also testable
if overidentifying moments are nonlinear in parameters \citep
{chesher1984testing,hahn2014neglected,qian2025testing}. Omitted heterogeneity may
explain rejection of overidentification restrictions.
If researchers do not find counterfactual homogeneity credible, what are their options?
One option is to avoid imposing counterfactual homogeneity altogether---conceding that
point-identification of unit-level counterfactuals is too ambitious. In some structural
contexts, researchers are willing to settle for partial identification rather than
imposing stronger assumptions \citep
{molinari2020microeconometrics,ciliberto2009market,tebaldi2023nonparametric,kalouptsidi2020partial,pakes2015moment}.\footnote{However,
the identified set for $Y(a)$ for a unit with $(Y,A, Z)$ cannot be smaller than the
conditional support $Y(a) \mid Y,A,Z$ under $F^*$. If counterfactual homogeneity does not
hold, then this conditional support can in principle be large. Thus, partial
identification alone is unlikely to be informative of individual counterfactual
outcomes.} Another alternative is to report a posterior predictive $\pi(Y(a) \mid
(Y,A,Z))$ for $\pi$ a prior on $\mathcal P^*$, where $\mathcal P^*$ allows for
counterfactual heterogeneity. Yet another option is to focus on a smaller set of
unit-level counterfactuals. If one only demands point-identification of
counterfactuals \emph{in prices}, then structural models can be relaxed to allow for
misspecification in characteristics $x_1, x_2$ \citep{andrews2023structural}. We show
in
\cref{sub:counterfactuals_in_prices} that such a relaxation exactly corresponds to
allowing for counterfactual \emph{heterogeneity} in characteristics. Ongoing
work \citep*{newpaper} additionally shows that price counterfactuals in nonparametric
versions of these relaxations are identified by recentered instruments \citep
{borusyak2025estimating}.
A second option treats the model as misspecified and interprets unit-level predictions as
extrapolations \citep*{andrews2025purpose}. The next subsection formalizes an
equivalence---in a context broader than demand---between extrapolation from ATEs and
making unit-level predictions under a structural model that identifies unit-level
counterfactuals. This result then allows us to separate quasi-experimental identification
of average effects from extrapolation in structural models. We can thus interpret
structural models as extrapolating from ATEs identified through instrument variation,
thus retaining an interpretation when the model does not hold. Structural modeling serves
as an informative prior over \emph{which} ATEs to extrapolate from.
\subsection{Reinterpretation of predicted unit-level counterfactuals}
\label{sub:extrap_equiv}
Consider a generic context where one observes outcomes, treatments, and instruments $
(Y,A,Z)$, where $Y$ need not be market shares. A common recipe for extrapolating from
ATEs is:
\begin{enumerate}[wide]
\item Researchers specify a class $\mathcal H$ of extrapolation rules $H(Y,A)$,
invertible in $Y$. Each function implicitly defines a potential outcome $H(a) = H(Y
(a), a)$.
\item Researchers posit that some outcome $H(A) = H(Y,A)$ is independent of
the instrument $Z$, in the sense that certain transforms $m(H(A))$ is mean
independent of $Z$.\footnote{Mean independence takes $m(\cdot)$ to be the identity.
Full independence takes $m(\cdot)$ to be all bounded measurable functions. This is
formalized in \cref{defn:extrapolate_from_ATE}} With some caveats, we may interpret
this orthogonality as a lack of average treatment effect on the transformed outcome
$H (a)$.\footnote{When the treatment itself is randomly assigned ($Z=A \indep Y
(a)$), then $\E[H (A) \mid A] = \E[H(a)] = 0$ means that $a$ has no average treatment
effects
on $H(a)$. When only the instrument is randomly assigned, then this condition can be
interpreted as a lack of treatment effects that are detectable through instrument
variation.}
\item When $(Y, A, Z) \sim F_0$, suppose the data $F_0$ identifies a unique member $H_
{F_0}
\in \mathcal H$ through the orthogonality restriction in (2). Researchers then
extrapolate from the knowledge that $H_{F_0}(a)$ has no ATEs---by making a leap of
faith that $H_{F_0}(a)$ also has no individaul treatment effects. This results in
predictions of the form $\tilde Y (a; Y,A) = H_{F_0}^{-1}(H_{F_0}(Y, A), a)$.
\end{enumerate}
We formalize this in \cref{defn:extrapolate_from_ATE} and call such predictions $\tilde Y$
\emph{extrapolated from averages} with respect to extrapolation rules $\mathcal H$, since
they fundamentally extrapolate a lack of average effects to a lack of individual
effects.
This recipe rationalizes many informal extrapolation rules. For instance, a researcher
who extrapolates by estimating the average treatment effect in some transformation $f
(Y)$ (e.g. $\log Y$) implicitly takes $\mathcal H$ to be demeaned outcomes: \[
\mathcal H = \br{H(y,a) = f(y) - \mu(a) : \mu(\cdot)}. \numberthis
\label{eq:ATE_class}
\]
Independence with instruments pins down the average
structural function $\mu (a) =
\E[Y(a)]$.\footnote{The uniqueness holds, for instance, under completeness
\citep{newey2003instrumental}.} Predictions under this model act as if individual
treatment effects are equal to differences in $\mu(\cdot)$: \[
\tilde Y(a) = f^{-1}(f(Y) + \underbrace{\mu(a) - \mu(A) }_{\text{ATE in $f(Y)$}}),
\qquad \mu (a) =
\E[Y(a)].
\] Predictions from quantile treatment effects similarly extrapolate by choosing
$\mathcal H = \br{H(y,a) \in [0,1] : H(\cdot, a) \text{ is strictly
increasing}}$ \citep{chernozhukov2005iv}.
Through this lens, \citet{berry2014identification} choose partially linear
extrapolation rules $
\mathcal H = \br{H(y,x_1, p, x_2) = h(y, p, x_2) - x_1 : h(\cdot)} . $ We may
thus interpret \citet{berry2014identification} extrapolating from ATEs through
$\mathcal H$ as well. Compared to extrapolating using rules
\eqref{eq:ATE_class}, these rules essentially trade flexibility with respect to the
average structural function $\mu (a)=\mu (x_1, p, x_1)$ for flexibility with respect to
$h(y, p, x_2)$.
This dual interpretation for structural models holds more broadly: Extrapolation from
averages implicitly specify structural models that identify unit-level counterfactuals,
and structural models that identify unit counterfactuals implicitly specify extrapolation
rules.
Indeed, we could instead extrapolate by positing a structural model $\mathcal P^*$ that
rationalizes the data---in which $Y = \mathfrak{s} (A,
\xi)$ and unit-level counterfactuals are identified in the sense of \cref
{defn:id}. By \cref{lemma:lemmalatent}, the model $\mathcal P^*$ must satisfy
counterfactual homogeneity. We can thus view a member $F^* \in \mathcal P^*$ as indexed
by a joint distribution $ (Y(a_0), A, Z) \sim Q
\in \mathcal Q$ and a mapping $C_0(y,a) \in \mathcal C$, since any $Y(a)$ is obtained by
$C_0^{-1}(Y(a_0), a)$. We can likewise view a structural model as specifying a class of
$(Q, C_0) \subset \mathcal Q \times \mathcal C$ pairs.
The following result shows that imposing such a model generates predictions equivalent to
extrapolation using some extrapolation rules $\mathcal H$. That is, any prediction that
extrapolates from averages can be equivalently cast under a (possibly misspecified)
structural model. Conversely, any structural model $\mathcal P^*$ can be thought of as
choosing extrapolation rules---with the technical caveat that $\mathcal P^*$ allows for
combining $C_0$ with arbitrary distributions $(Y(a_0), A, Z)$ satisfying instrument
exogeneity, which we formalize in \cref{defn:complete}.
\begin{restatable}{prop}{propequivextrapolate}
\label{prop:equivalence_extrapolation} Fix a class of distributions $\mathcal P$ over
observables $(Y,A,Z)$. Extrapolation from averages and structural models are equivalent
in the following sense: For any $F \in \mathcal P$, let $(Y, A, Z) \sim F$ and let
$\tilde Y_F(a; Y,A)$ be a prediction of the counterfactual $Y(a)$ for some observed unit
$
(Y,A)$.
\begin{enumerate}[wide]
\item If $\tilde Y_F(a; Y,A)$ is extrapolated from averages with respect to $\mathcal H$ in the sense of
\cref{defn:extrapolate_from_ATE}, then there exists some $\mathcal P^*$ that
identifies unit-level counterfactuals, generates $\mathcal P$, and rationalizes
$\tilde Y$ as identified unit-level counterfactuals.
\item Conversely, if the predictions $\tilde Y_F(a; Y,A)$ arise from some structural
model $\mathcal P^*$ that identifies unit-level counterfactuals, rationalizes $\mathcal
P$, and is \emph{only restricted by exogeneity and $\mathcal C$} in the sense of \cref
{defn:complete}, then there exists some $\mathcal H$ that rationalizes $\tilde Y$ as
extrapolated averages in the sense of \cref{defn:extrapolate_from_ATE}.
\end{enumerate}
\end{restatable}
\Cref{prop:equivalence_extrapolation} thus allows us to separate quasi-experimental
identification from extrapolation in structural models. Models identifying unit-level
counterfactuals fundamentally extrapolate from averages, and vice versa. The averages
themselves are identified through standard quasi-experimental research designs and do
not require restricting the joint distribution of potential outcomes. Tools and language
from causal inference can also be helpful in assessing the internal validity of these
average effects.
The value of structural models lies in providing economically motivated extrapolation
rules $\mathcal H$, which improve on intuitively reasonable but ad hoc ones like
\eqref{eq:ATE_class}. These rules are exactly correct under the model, but can be viewed
as approximately correct when counterfactual homogeneity approximately holds. Separating
identification from extrapolation in this way thus clarifies what one can credibly learn
from data and what one needs to believe to extrapolate to economically relevant
quantities.
\medskip
So far, we have shown that market-level counterfactuals are only identified under
counterfactual homogeneity when we only observe market-level data. Their prediction
requires extrapolation from average effects over markets in some way. This motivates
considering whether richer data can restore identification of market-level
counterfactuals without strong assumptions.
As an idealized benchmark, since markets aggregate populations of consumers, market-level
causal effects are also average causal effects for consumers within a given market. Thus,
with exogenous treatment variation
\emph{within} a given market at the consumer level, counterfactual outcomes for individual
markets are identified as average treatment effects among consumers. Close to this
idealized benchmark,
\citet{tebaldi2023nonparametric} assume that prices are exogenously assigned\footnote
{In \citet{tebaldi2023nonparametric}, prices (insurance premiums) are deterministic
functions of consumer age and income.
\citet{tebaldi2023nonparametric} assume that consumers with different ages and incomes do
not have systematically different latent preferences, given the market that they reside
in.} for consumers participating in the California healthcare market and partially
identify counterfactual market shares.
The additional value of richer data similarly motivates the literature on ``micro BLP''
\citep{berry2024nonparametric,microblp,conlon2025incorporating}, where we observe market
shares by demographic subgroups within a given market, though these subgroups are
subjected to the same bundle of products. Do identification results these settings avoid the curse of unobserved heterogeneity? We conclude this paper by
deriving an analogous equivalence for identification results
with micro-data
\citep{berry2024nonparametric}. We find that identification with micro-data continues to
impose counterfactual homogeneity. In fact, since these results are primarily motivated
by relaxing dependence on instruments, they use even stronger forms of homogeneity
instead.
\section{Demographics-specific market shares}
\label{sec:demographics_specific_market_shares}
We observe market shares for different demographic subgroups $w \in \mathcal{W}
\subset \R^J$. $a \in \mathcal A$ continues to denote treatment. Each market's potential
outcome is a \emph{process} indexed by $w \in \mathcal{W}$: $Y(a)[\cdot]: \mathcal{W}
\to [0,1]^J$. In this notation, $Y(a) [w]$ denotes market shares among
demographics $w$ in a randomly drawn market, when prices and characteristics are
counterfactually set to some value $a$. Analogous to \cref{defn:id}, we are interested
in identifying the profile of market shares for a given market, at counterfactual
values of treatment: $Y(a)[\cdot]$ for some $a\neq A$. It is useful to think of $w$ as
analogous to a time index in panel settings. Consistent with that analogy, we use
square brackets for $w$ to emphasize that comparisons in $w$ are not causal comparisons
that represent counterfactual assignment of $w$.
\citet{berry2024nonparametric} consider a structural model in which \[
Y(a) [w] = \mathfrak{s}(w, a, \xi)
\] for some function $\mathfrak{s}$ and market demand shock $\xi$, under the following assumptions.\footnote{Relative to
Assumption 1 in \citet{berry2024nonparametric}, \cref{as:index-micro} normalizes the
index directly, following their Section 2.5. Relative to their setting, we suppressed
other market-level interventions (their $X_t$) that may enter $\gamma$. Doing so makes
the normalization in their Section 2.3 unnecessary, which we impose in \cref
{as:index-micro} directly. } These assumptions nest parametric versions like \citet
{microblp} (see
\cref{ex:micro-blp}).
{ \begin{as}[Index]
\label{as:index-micro} $\mathfrak{s}(w, a, \xi) = \sigma(\gamma(w, \xi), a)$, where $\gamma$ has
codomain $\R^J$, and for all $j$, $
\gamma_j(w,\xi) = g_j(w) + \xi_j.
$
For some fixed $w_0$, $g(w_0) = 0$ and $\frac{dg(w_0)}{dw} = I_J$.
\end{as}
\begin{as}[Invertible demand]
\label{as:invertible-micro}
For all $a \in \mathcal A$, $\sigma(\cdot, a)$ is injective on the support of
$\gamma(w, \xi)$.
\end{as}
\begin{as}[Injective index]
\label{as:invertible-index-micro}
For all $\xi$ in its support, $\gamma(\cdot, \xi)$ is injective on $\mathcal W$.
\end{as}}
\cref{as:index-micro,as:invertible-micro,as:invertible-index-micro} are equivalently
represented in counterfactual outcomes. The
first of these equivalent assumptions is analogous to
\cref{as:latent_homogeneity}.
{\begin{as}[Counterfactual homogeneity of market share profiles]
\label{as:lat_homogeneity_paths}
For some baseline treatment $a_0 \in \mathcal A$, there exists some invertible function
$C_0 (\cdot , a): \mathcal Y \to \mathcal Y$ such that for all $w \in \mathcal{W}$ and all $a \in
\mathcal A$, \[ Y (x_0) [w] = C_0 (Y (a) [w], a) \quad \text{$P^*$-almost surely}.
\]
\end{as}}
\cref{as:latent_homogeneity} posits that a deterministic, invertible function maps $Y
(a)$ to $Y(a_0)$. Analogously, \cref{as:lat_homogeneity_paths} posits that such a
function maps the \emph{profile} of market shares $Y (a)[\cdot]$ to $Y (a_0)
[\cdot]$. The mapping in \cref{as:lat_homogeneity_paths} acts identically
along the profile $w \mapsto Y(a) [w]$ and does not depend on $w$.
{
\begin{as}[Latent individual parallel trends]
\label{as:latent-PT}
Fix baseline values $a_0, w_0$. For some invertible mapping $\phi: \mathcal Y \to
\R^J$, the profiles $w\mapsto \phi(Y(a_0)[w])$ are parallel almost surely: There exists
an invertible and differentiable function $g:
\mathcal W \to \R^J$ such that differences in $\phi(Y(a_0)[\cdot])$ are equal to
differences in $g(w)$
\[
\phi(Y(a_0)[w]) - \phi(Y(a_0)[w_0]) = g(w) - g(w_0) \quad \text{ $P^*$-almost
surely for all $w
\in \mathcal{W}$.}
\] Redefining $\phi(\cdot)$ if necessary, we normalize $g(w_0) = 0$ and $\frac{d}{dw}g
(w_0) = I_J$.
\end{as}}
\Cref{as:latent-PT} states that, up to some invertible transformation $\phi(\cdot)$, the
market share profiles at some baseline treatment $w \mapsto Y(a_0)[w]$ are parallel
almost surely. This is an individual version of the parallel trends assumption, though
here the ``time index'' is the demographic values $w$. It imposes that trends are not
only parallel in expectation, but are parallel almost surely.\footnote
{In difference-in-differences applications, where $w$ is a time index, parallel trends
is usually stated as \[
\E[Y(x_0)[w] - Y(x_0)[w_0] \mid A=a] = g(w)
\]
and does not depend on the realized treatment $a$. This does not require that $Y(a_0)[w] -
Y (a_0)[w_0] = g(w)$ almost surely.
Similarly, suppose $w$ is a time-index, if potential outcomes are generated through a
two-way fixed effects model $Y_i(a)[w] = \alpha_i +
\beta[w] + f(a) + \epsilon_i[w] $, then the individual-level trends are only parallel to
$w \mapsto \beta[w] + \epsilon_i[w]$, which depends on the path of idiosyncratic shocks
$\epsilon_i[\cdot]$. Relative to this, \cref{as:latent-PT} effectively assumes away the
idiosyncratic shocks $\epsilon_i[w]$. } Thus, in addition to restricting heterogeneity
in the relationship $a\mapsto Y (a)$, \cref{as:latent-PT} restricts the heterogeneity of
the relationship $w \mapsto Y(a_0)[w]$, at some fixed $x_0$, across markets.
\Cref{as:lat_homogeneity_paths,as:latent-PT} are further equivalent to the following
assumption by choosing $h(\cdot, a) = \phi(C_0(\cdot, a))$.
{\begin{as}[Individual parallel trends in a transformed outcome]
\label{as:micro-data-combined}
For some fixed $w_0$, there is an invertible function $h(\cdot, x)$ such that for some
invertible and differentiable function $g$, \[
h(Y(x)[w], x) - h(Y(x_0)[w_0], x_0) = g(w) - g(w_0) \text{ for all $x, w, x_0$},
\]
$P^*$-almost surely. Redefining $h$ if necessary, we normalize $g(w_0) = 0,
\frac{d}{dw}g(w_0) = I_J$.
\end{as}}
Analogous to \cref{as:homogeneity}, \cref{as:micro-data-combined} states that individual
parallel trends hold for transformed outcome profiles $H[w] = H(a)[w] \equiv h(Y(a)
[w], x)$, which do not depend on the treatment $x$. Thus, under \cref
{as:micro-data-combined}, there is some transformed outcome profile $H(a)[\cdot]$ that
receives no treatment effect from $a$ and has parallel sample profiles.
We collect these equivalences in the following theorem.
\begin{restatable}{theorem}{thmequivmicro}
\label{thm:equivmicro}
The following are equivalent:
\begin{enumerate}
\item \cref{as:index-micro,as:invertible-micro,as:invertible-index-micro}
\item \cref{as:lat_homogeneity_paths,as:latent-PT}
\item \cref{as:micro-data-combined}.
\end{enumerate}
\end{restatable}
We conclude this section---and the paper---by explaining the identification argument in
\citet{berry2024nonparametric}, from the perspective of \cref
{as:micro-data-combined}. This exposition highlights the strength of the homogeneity
assumptions in delivering identification results. In short, the homogeneity structure
embedded in \cref{as:micro-data-combined} is already powerful enough to identify $g
(w)$ and identify $h$ up to level shifts,\footnote{That is, for some fixed baseline
$y_0$, $h(\cdot, x) - h(y_0, x)$ can be identified.} given the distribution of observed
data $(Y[\cdot], A) \sim F$---without any restrictions on treatment assignment. Randomly
assigned instruments then identify the remaining unknown $h(y_0, \cdot)$.
\begin{figure}[tb]
\centering
(a) Distribution of market share \emph{profiles} under some $H(Y)$, rejected by the
data
\includegraphics[width=0.8\textwidth]{sample_paths_incorrect.pdf}
(b) Distribution of market share \emph{profiles} under the true $H(Y)$
\includegraphics[width=0.8\textwidth]{sample_paths_correct.pdf}
\caption{We show sample paths of $Y[\cdot] \mid A=a$ when $J=1$ and $H(Y[\cdot]) \mid
A=a$ for candidate $H(\cdot)$. A candidate is rejected by the data if the sample
paths of $H(Y[\cdot])$ are not almost surely parallel.}
\label{fig:sample_paths}
\end{figure}
To see this, for a given value $a$, consider the conditional distribution $Y
[\cdot] \mid A=a$. Since $A$ is not randomly assigned, this is the distribution of demand
profiles for markets that select into the product bundle $a$. On this
subpopulation, \cref{as:micro-data-combined} states that there is some function $h
(\cdot) = h(\cdot, a)$, such that the sample paths $w
\mapsto h(Y[w])$ are almost surely parallel: \[ h \in \br{h: \P_{F}\br{h (Y[w]) - h(Y
[w_0]) =
g(w) \mid A=a} = 1 }.
\]
Intuitively, this requirement is highly constraining: There should not be many
transformations $h$ that result in parallel profiles. In a setting with $J=1$,
\cref{fig:sample_paths}(a) illustrates for an arbitrary candidate $H(y)$, the sample paths
post-transformation are unlikely to be almost surely parallel, leading us to reject this
candidate. Making the sample paths parallel seems to require getting $H$
exactly right, as in \cref{fig:sample_paths}(b). This rigidity locks in certain features
of $h (\cdot)$. In fact, under mild smoothness and support restrictions, this rigidity
identifies $h(\cdot)$ up to a vertical shift and $g(w)$: Lemma 2, Lemma 3, and
Corollary 1 in
\citet{berry2024nonparametric} show that $g(w)$ and $h (\cdot, a) - h(y_0, a)$, for some
baseline value $y_0$, are identified.
Instruments eliminate this last indeterminacy in $h(y_0, a)$.
\cref{as:micro-data-combined} implies that, for any fixed $w$, \begin{align*} h(y_0, X) &=
\overbrace{g(w) - (h(Y[w], X) - h(y_0, X))}^{\text{Identified through parallel trends}}
-
h (Y (a_0) [w_0], a_0) \\&\equiv Q (Y, w, X) - h
(Y(a_0)[w_0], a_0)
\end{align*} for an identified function $Q(Y, w, X)$. Given some instrument $Z \indep Y
(a_0)$, we then have a moment condition that identifies $h(y_0, \cdot)$ under
completeness \citep{newey2003instrumental}, since $
\E[h(y_0, X) \mid Z] - \E[Q(Y, w, X) \mid Z]$ is constant in $Z$.
This intuition concurs with that in \citet{berry2024nonparametric} on the value of
micro-data and instruments. They argue that micro-data $w$ provide variation akin to
within-unit comparisons in panel data settings (p.1152).
\cref{as:latent-PT} additionally highlights that {homogeneity}---in the sense of
\emph{individual} parallel trends---is also important, relative to standard assumptions
in panel settings.
\Cref{as:latent-PT}, interpreted as a panel assumption, additionally imposes that the unit
fixed effect is the only heterogeneity across units; absent the fixed effect, all units
have the same evolution over $w$.
The equivalence \cref{thm:equivmicro} reveals that in this model, the availability of
micro-data does not relax requirements on counterfactual homogeneity. In fact, additional
homogeneity assumptions---those with respect to $w \mapsto Y(a)[w]$---are imposed to
instead weaken requirements on instruments. Thus, whether identification results
exist---without these cross-market homogeneity assumptions and without within-market
treatment variation---remains a question for future research.
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