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Reinterpreting demand estimation
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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 splawa1990application,rubin1974estimating generally focuses on recovering 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 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, 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.” 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.” 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 berry1995automobile,berry2014identification and settings with demographic-specific market shares 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 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{vytlacil2002independence shows that the selection model $D_i = \one (\alpha + \beta z > \xi_i)$, for $\beta>0$, is equivalent to the monotonicity restriction $\P (D_i(1) \ge D_i(0)) = 1$ 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 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 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 molinari2020microeconometrics are popular in the literature on entry games ciliberto2009market and revealed preference pakes2015moment. It is also possible to partially relax counterfactual homogeneity by demanding that only certain counterfactuals---e.g., counterfactual in prices---are identified andrews2023structural,borusyak2025estimating,newpaper. } Additional functional form assumptions in 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 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 kline2019heckits. We formalize how 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 which ATEs to extrapolate from.
This paper contributes to a literature that bridges causal inference and structural modeling andrews2023structural,kline2019heckits,borusyak2025estimating,kong2024nonparametric,humphries2025conviction,torgovitsky2019nonparametric,mogstad2024instrumental,angrist2000interpretation,conlon2021empirical. This paper is also related to transformation models and other simultaneous equation models chiappori2015nonparametric,vuong2017counterfactual,benkard2006nonparametric,matzkin2008identification. Counterfactual homogeneity is related to a literature on omitted parameter heterogeneity chesher1984testing,hahn2014neglected,qian2025testing. Like 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 berry2014identification,berry2024nonparametric with additional parametric assumptions. compiani2018nonparametric studies nonparametric estimation for berry2014identification. }
This paper proceeds as follows. (ref) derives equivalent assumptions to berry2014identification. (ref) discusses counterfactual homogeneity, derives its necessity, and derives an equivalence between structural model predictions and extrapolations from average treatment effects. (ref) derives equivalent assumptions to berry2024nonparametric and examines the extent to which models with micro-data allow for counterfactual heterogeneity.
We start with a standard model of differentiated products berry1995automobile,berry2014identification in potential outcomes notation. Markets are i.i.d. draws from a population $F^*$, following 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, 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 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, imbensangrist) to ensure that instrument-level comparisons recover proper comparisons over endogenous treatments.
On the other hand, the structural demand literature is concerned with 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 (ref), 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 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 doksum1974empirical,heckman1997making.\footnote{This is even termed the “fundamental problem of causal inference” 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 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 $F$.
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 (ref), 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 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 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 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.
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(ref) is stated as Assumption 5.1 in berry2021foundations. It is an implication of Assumption 1 in berry2014identification, which is a similar index restriction on an underlying random utility model. (ref) is a conclusion of Lemma 1 in berry2014identification, justified via a “connected substitutes” condition in 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, (ref) allow for identification of the function $\mathfrak{s}$ by exploiting an “index-inversion-instruments” recipe 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 newey2003instrumental; see Theorem 1 in 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 (ref) 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 (ref) 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.
Our first assumption imposes that $Y(\cdot)$ satisfy counterfactual homogeneity.
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} (ref) 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 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, (ref) restricts the heterogeneity across markets by restricting the intrinsic dimension of the support of potential outcomes $\br{Y(a)}_ {a \in \mathcal A}$. The relationship between $Y (a)$ and $Y(a_0)$ is kept homogeneous across all markets. We refer to it as 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. (ref) is also a generalization of rank invariance in standard treatment effect settings.\footnote {There, rank invariance 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, (ref) extends to non-binary treatment and multidimensional outcomes.
Counterfactual homogeneity rules out heterogeneity 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 $i$, (ref) generates flexible substitution patterns for any given market. What (ref) does restrict is how consumer populations can be different across markets.
The second assumption imposes some functional form restriction on the map $C_0$. {
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(ref) 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 $Y (a)$: \[ \diff{Y(a)}{x_1} = \pr{\diff{h(Y(a), p, x_2)}{y}}^{-1} \quad \diff{Y(a)}{(p, x_2)} = - \diff{Y(a)} {x_1} \diff{h(Y(a), p, x_2)}{(p, x_2)}. \numberthis \label{eq:derivatives} \]
(ref) can be combined as the following homogeneity assumption on some transformation of potential outcomes.
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(ref) 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 $H$.} (ref) makes clear how (ref) restrict treatment effect heterogeneity. Viewed as assumptions on some transformation of potential outcomes, (ref) are exactly constant treatment effects assumptions. (ref) 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 berry2014identification assumptions, in the same spirit as vytlacil2002independence,vytlacil2006ordered's results for instrumental variable models. The equivalence is easy to derive, once we link $(\mathfrak{s}, \xi)$ in (ref) to $(h,\phi, C_0)$ in (ref) and $H$ in (ref): \[ \mathfrak{s} = h^{-1}, \quad \xi = \phi^{-1}(Y(a_0)), \quad h(y,p,x_2) = H(y, p, x_2). \]
Reformulating assumptions this way retells the progress in demand models with market share data. In the standard telling ackerberg2007econometric, different generations of structural demand models (e.g., vertical models, simple logit, nested logit, BLP, 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 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 $H(a)$.
(ref) clarifies that structural demand models do restrict unobserved heterogeneity. The need to restrict unobserved heterogeneity is not specific to these particular demand models either. Any model that identifies unit-level counterfactuals necessarily has to impose counterfactual homogeneity: (ref) is necessary for identification in the sense of (ref).
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, berry2014identification impose little more than what is necessary. The functional form assumption, (ref), 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 (ref) 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 berry2014identification are close to minimal for point-identification.
Nevertheless, counterfactual homogeneity is likely misspecified: The zero-variance implication (ref) is implausible in many applications. Economic models allowing for markets that differ in terms of their consumer populations, like (ref), 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 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 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 in prices, then structural models can be relaxed to allow for misspecification in characteristics $x_1, x_2$ andrews2023structural. We show in (ref) that such a relaxation exactly corresponds to allowing for counterfactual heterogeneity in characteristics. Ongoing work \citep*{newpaper} additionally shows that price counterfactuals in nonparametric versions of these relaxations are identified by recentered instruments 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 which ATEs to extrapolate from.
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:
We formalize this in (ref) and call such predictions $\tilde Y$ 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 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 $\mathcal H = \br{H(y,a) \in [0,1] : H(\cdot, a) \text{ is strictly increasing}}$ chernozhukov2005iv.
Through this lens, 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 berry2014identification extrapolating from ATEs through $\mathcal H$ as well. Compared to extrapolating using rules (ref), 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 (ref). By (ref), 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 (ref).
(ref) 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 (ref). 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.
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 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, tebaldi2023nonparametric assume that prices are exogenously assigned\footnote {In tebaldi2023nonparametric, prices (insurance premiums) are deterministic functions of consumer age and income. 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” 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 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.
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 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 (ref), 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 $w$.
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 berry2024nonparametric, (ref) 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 (ref) directly. } These assumptions nest parametric versions like microblp (see (ref)).
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(ref) are equivalently represented in counterfactual outcomes. The first of these equivalent assumptions is analogous to (ref).
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(ref) posits that a deterministic, invertible function maps $Y (a)$ to $Y(a_0)$. Analogously, (ref) posits that such a function maps the profile of market shares $Y (a)[\cdot]$ to $Y (a_0) [\cdot]$. The mapping in (ref) acts identically along the profile $w \mapsto Y(a) [w]$ and does not depend on $w$.
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(ref) 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, (ref) effectively assumes away the idiosyncratic shocks $\epsilon_i[w]$. } Thus, in addition to restricting heterogeneity in the relationship $a\mapsto Y (a)$, (ref) restricts the heterogeneity of the relationship $w \mapsto Y(a_0)[w]$, at some fixed $x_0$, across markets.
(ref) are further equivalent to the following assumption by choosing $h(\cdot, a) = \phi(C_0(\cdot, a))$. {
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Analogous to (ref), (ref) 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 (ref), 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.
We conclude this section---and the paper---by explaining the identification argument in berry2024nonparametric, from the perspective of (ref). This exposition highlights the strength of the homogeneity assumptions in delivering identification results. In short, the homogeneity structure embedded in (ref) 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 $h(y_0, \cdot)$.
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, (ref) 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 $J=1$, (ref)(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 (ref)(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 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 $h(y_0, a)$. (ref) implies that, for any fixed $w$,
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 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 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). (ref) additionally highlights that {homogeneity}---in the sense of individual parallel trends---is also important, relative to standard assumptions in panel settings. (ref), 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 (ref) 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.