Demand estimation without outside good shares

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spacing{1.2} \begin{abstract} The BLP model is the workhorse framework in empirical IO and enables estimation of demand models for differentiated products using aggregate product shares. In practice, however, the share of the outside good is often unobserved. This paper studies identification and inference in the BLP model when the share of the outside good is unobserved. We show that the model is partially identified, and we derive sharp identified sets for structural parameters and equilibrium objects. We also develop inference procedures based on moment inequalities that deliver valid confidence sets for these structural parameters and equilibrium objects. \end{abstract}

KEYWORDS: TBA

JEL classification codes: C01, C12.

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Introduction

Discrete-choice demand models for differentiated products are a central tool in empirical industrial organization and are widely used in other applied fields in economics; see berry/haile:2014,berry/haile:2021 and references therein. Among these, the random-coefficients logit model introduced by berry:1994 and developed by berry/levinsohn/pakes:1995, hereinafter referred to as BLP, has become the workhorse framework for estimating demand systems, conducting counterfactual analyses, and evaluating market power, mergers, and welfare.

The BLP model posits that industry-level demand arises from a discrete-choice model of consumer demand with unobserved heterogeneity. In each market, consumers choose among $J$ differentiated products and one {\it outside good}, representing the option of not choosing any of the $J$ products. The outside good is a key ingredient of the BLP model; see berry/haile:2021. A central feature of BLP is that it allows identification and estimation of the structural demand system using only aggregate product-level market shares.

In many empirical applications, however, the share of the outside good is not observed. This situation arises naturally when only aggregate shares of the $J$ inside goods are observed, or when aggregate purchase data are available but the size or boundaries of the relevant market are not well defined. Identification and estimation results are therefore sensitive to the specification of the share of the outside good. To address this issue, the applied literature often relies on ad hoc assumptions regarding the outside good or, equivalently, market size. A recent survey by Zhang:2024 finds that, among 29 top-five publications using the BLP model, 24 rely on ad hoc assumptions about market size or the outside good, while only 5 assess the robustness of their results to these assumptions.

This paper adopts an agnostic approach to the share of the outside good, or equivalently, to market size. We consider a setting in which the researcher can credibly restrict the share of the outside good in each market to a set $S_0$, which may be random and market-specific. Our framework accommodates varying degrees of missing information about the outside option. At one extreme is the fully agnostic case in which $S_0 = (0,1)$. At the other extreme, the existing literature treats the outside share as observed (possibly up to sampling error), effectively assuming that $S_0$ is a singleton. Intermediate cases allow the researcher to study sensitivity to a proposed value of the outside share by specifying $S_0$ as a neighborhood around that value.

We study identification and inference in the BLP model when the share of the outside good is restricted to the set $S_0$. Our first contribution is to characterize the sharp identified set for the structural parameters of the BLP model in this context. Our analysis shows that once the share of the outside good is no longer pinned down to a single value, the BLP model is typically partially identified, even under otherwise standard assumptions. Using tools from random set theory (molchanov:2005,molchanov/molinari:2018), we derive an explicit characterization of the identified set that fully exploits the structure of the BLP demand system. In essence, our identified set is characterized by an infinite collection of conditional moment inequalities.

Our second contribution is to characterize the identified sets of economically relevant equilibrium objects in the BLP model, such as demand shocks, demand elasticities, markups, diversion ratios, and market shares. We show how partial identification of the structural parameters propagates to these equilibrium objects, and we derive sharp identified sets for them.

To illustrate these results, we present a numerical example that shows how the identified sets of parameters and equilibrium objects evolve as the amount of missing information about the share of the outside good increases. The results demonstrate that although small departures from a singleton outside share eliminate point identification of the structural parameters, the resulting identified sets remain informative enough to support economically meaningful conclusions.

Our third contribution is to develop inference procedures for the BLP model in this partially identified setting. Building on the methods of andrews/shi:2013 and chernozhukov/chetverikov/kato:2015, we transform the conditional moment inequalities into a (potentially large) collection of unconditional moment inequalities and construct confidence sets that are asymptotically valid uniformly over a broad class of data-generating processes. From here, we show how to conduct inference on both the structural parameters and equilibrium objects of the BLP model via projection.

The remainder of the paper is organized as follows. Section (ref) reviews the related literature and situates our contribution. Section (ref) provides a brief review of the BLP model in the standard case in which the outside share is observed. Section (ref) considers the identification of the BLP model when the outside shares are unobserved. We derive sharp identified sets for the structural parameters and equilibrium objects derived from the model. Section (ref) develops inference methods by proposing confidence intervals for the structural parameters and equilibrium objects derived from the model. Section (ref) illustrates the methodology in an empirical application. Section (ref) concludes. Proofs and auxiliary results are collected in the Appendix.

Literature review

When the outside share is unobserved, the existing literature addresses this problem by imposing additional structure to substitute for the missing outside-share information. In the absence of direct data on the outside option, such restrictions are not testable. This paper instead studies identification and inference in the BLP model without imposing parametric or auxiliary structure to recover the outside share.

One strand of the literature incorporates a parametric specification for the outside component within the BLP framework and estimates it jointly with demand parameters. Berry:2006tt introduce additional structure linking the market size — defined as the sum of inside and unobserved outside quantities -- to observed covariates. More recently, Zhang:2024 model the market size as a function of observed market-level variables that is known up to a finite-dimensional parameter vector estimated jointly with demand. Under suitable conditions, this restores point identification using the same data requirements as in standard BLP estimation. Relatedly, Huang:2013vh study a plain logit model without random coefficients and use market fixed effects to eliminate bias from unobserved outside shares. They recover the size of the outside option under the additional assumption that market size is constant across markets. In contrast, our approach does not seek to restore point identification through additional parametric restrictions, but instead allows the model to be set identified.

A second strand of the literature relies on supply-side conditions and auxiliary information, such as cost or accounting data, to estimate market size (e.g., Chu:2011ul, Byrne:2022vz, and Kim:2024ti). These approaches exploit equilibrium pricing conditions to infer market size and, implicitly, the outside share. Identification in these settings therefore depends on additional structural assumptions from the supply side. By contrast, we abstract from supply-side information when identifying the structural parameters and study what can be learned from the demand side alone.

A third approach, increasingly common in applied work, predicts total demand (or proxies for it) in a first step. In grocery and airline applications, for example, researchers often predict total store trips or arriving passengers using stand-alone regressions such as gravity equations or Poisson arrival processes (e.g., Sweeting:2020tw, Li:2022tk, Hortacsu:2022wt, and Backus:2021aa). However, total store trips or arriving passengers do not capture consumers who choose not to visit the store or who use alternative modes of transportation. As a result, the estimated quantity still needs to be scaled by an ad hoc factor to account for these unobserved outside options.

Our paper is also related to the literature on inference in differentiated product demand models. There is a long literature addressing this problem under the assumption that all market shares are observed. berry/linton/pakes:2004 and armstrong:2016 study asymptotics as the number of products grows, whereas freyberger:2015 and hong/li/li:2021 consider asymptotics in the number of markets, as we do. The main difference between these papers and ours is that we treat outside shares as missing data and address the associated identification and inference problem.

The standard BLP model

This paper considers the BLP model when the share of the outside good is unobserved. Before delving into this problem, we briefly review the standard BLP model, in which all shares are observed. We follow the notation and the standard assumptions imposed in berry:1994,berry/levinsohn/pakes:1995,berry/haile:2021.

We observe an i.i.d.\ sample of markets $m=1,\dots ,M$. Market observations typically correspond to different cities and/or periods. That is, $m$ is determined by the pair $(t,c)$, where $t=1,\dots ,T_{c}$ denotes periods observed for city $c$ and $c=1,\dots ,C$ denotes cities observed. Since markets are assumed to be i.i.d., we can drop the index $m$ from the identification analysis.

In each market, there is one “outside” product (labeled as zero) and $J$ “inside” products. We assume that $J$ is a fixed, known number. Throughout this section, we observe the shares of all products, conditional on market characteristics. These are determined by the decisions of a representative consumer in a standard BLP discrete-choice model. We assume that there is a unit mass of consumers $\mathcal{I}$, each choosing a single product.

For each market, the observed random variables are $( s,x,p,z) $, where $s=( s_{j}) _{j=1}^{J}\in (0,1)^{J}$ denotes the shares of the inside products, $x=( x_{j}') _{j=1}^{J}$ with $x_{j}'\in \mathbb{R}^{d_{X}}$ denotes the non-price inside product characteristics, $p=( p_{j}) _{j=1}^{J}$ denotes the prices of the inside products, and $z$ denotes a vector of instruments. In the canonical case of the car market, $j=1,\dots, J$ is a list of the available cars, $x$ would represent car features like fuel efficiency, size, and a constant vector, and $z$ would represent cost shifters and characteristics of other products, which generate changes in the observables but are assumed to be mean independent of the unobserved demand shock. The outside share can be deduced from the inside shares: $s_{0}=1-\sum_{j=1}^{J}s_{j}$. By definition, $( s_{0},s) \in \Delta _{J+1}$, where $\Delta _{J+1}$ denotes the $J+1$-dimensional simplex.

The indirect utility of the (inside) good $j=1,\dots ,J$ for the representative individual $i$ is

align*[align* omitted — 183 chars of source]

where $\tilde{\beta}_{ik}=(\beta _{k}+\zeta _{ik})$ is the random coefficient on $x_{jk}$, $\beta =( \beta _{k}) _{k=1}^{d_{X}}$ is the vector of mean level of taste parameter for $x_{k}$, $\zeta _{ik}$ is a random component of the taste parameter for $x_{k}$, $\zeta _{i} = (\zeta _{ik}) _{k=1}^{d_{X}}$ is the associated vector, $\tilde{\alpha}_{i}=(\alpha +\nu _{i})$ is the random coefficient on $p_{j}$, $\alpha $ is the mean level of distaste parameter for $p_{j}$ (i.e., the price elasticity), $\nu _{i}$ is the random component of the distaste parameter for $p_{j}$, $\xi _{j}$ is the product-specific demand shock, and $\epsilon _{ij}$ is the individual-specific taste shock. In the case of the cars, $\beta _{k}$ represents the mean taste for the observed characteristic $k$, $\zeta _{ik}$ represents customer $i$'s preference shock for this characteristic, $\nu _{i}$ represents customer $i$'s personal sensitivity to price, and $\xi _{j}$ could represent product $j$'s brand prestige. The indirect utility of the outside good for the representative individual $i$ is $u_{i0}=\epsilon _{i0}$.

In this model, the following objects are random and unobserved to the econometrician: the individual-specific taste shocks $((\epsilon _{ij})_{j=0}^{J})_{i \in \mathcal{I}}$, the random components $(\zeta _{i},\nu _{i})_{i \in \mathcal{I}}$, and the demand shocks $\xi = ((\xi _{j})_{j=1}^{J})$. The standard BLP model imposes the following conditions on these shocks. First, $((\epsilon _{ij})_{j=0}^{J})_{i \in \mathcal{I}}$ is independent of the data and other shocks, and i.i.d.\ Type I extreme value.\footnote{The Type I extreme value assumption is commonly adopted in empirical work, but it is not essential for our identification analysis. It can be relaxed, provided the connected substitutes condition in Berry:2013aa holds, which ensures the demand system is invertible under full information.} Second, $(\zeta _{i},\nu _{i})_{i \in \mathcal{I}}$ are also assumed independent of the data and other shocks, and i.i.d.\ with a density function $f( \zeta ,\nu ;\lambda ) $ that is known up to a parameter $\lambda \in \Lambda \subset \mathbb{R}^{d_{\lambda }}$.\footnote{In practice, $f( \zeta ,\nu ;\lambda ) $ is often assumed to be normally distributed with mean zero and variance–covariance matrix $\lambda \in \Lambda $, and so $\Lambda$ represents a space of variance-covariance matrices. In this case, having $\lambda$ equal to zero matrix imposes $\zeta=\mathbf{0}_{d_X}$ and $\nu=0$, which gives rise to the plain logit model.} Finally, the demand shocks are $\xi = ((\xi _{j})_{j=1}^{J})$, and assumed to satisfy

equation[equation omitted — 94 chars of source]

The structural parameters of the model are:

equation*[equation* omitted — 129 chars of source]

where $\Theta$ denotes the parameter space for $\theta$.

For any product $j=1,\dots,J$, this model implies the following conditional market share:

equation[equation omitted — 288 chars of source]

See Lemma (ref) for a derivation. Given that each market contains a continuum of customers, we have that, conditional on $(x,p,\xi)$,

equation[equation omitted — 82 chars of source]

Under mild assumptions in berry:1994 and berry/levinsohn/pakes:1995, this model is point identified. We now briefly outline the argument. For any $\delta \in \mathbb{R}^{J}$, let the function $ \sigma (\delta ,x,p;\lambda ):\mathbb{R}^{J}\times \mathbb{R}^{d_{X}}\times \mathbb{R}\times \mathbb{R}^{d_{\lambda }}\to (0,1)^{J}$ be defined as follows:

equation[equation omitted — 271 chars of source]

berry:1994 and berry/levinsohn/pakes:1995 show that, for any fixed $(s,x,p,\lambda)$, the mapping $\delta \to \delta + \ln{s} - \ln{\sigma (\delta ,x,p;\lambda )}$ is a contraction in the sup-norm. Consequently, for any vector of market shares generated by this model, there exists a unique value of $\delta$ that solves $\sigma (\delta ,x,p;\lambda ) = s$. By combining this with (ref) and (ref), we obtain that

equation[equation omitted — 129 chars of source]

where $\sigma^{-1}_j$ denotes the $j$th coordinate of $\sigma^{-1}( s,x,p;\lambda )$. This and (ref) yield the following equation:

equation[equation omitted — 128 chars of source]

Under standard conditions regarding the variability of the instruments, (ref) point identifies the parameter of interest $\theta =(\alpha,\beta,\lambda )$. In particular, berry/haile:2021 argue that one can use (ref) to produce unconditional moment conditions that point identify $\theta$. Estimation and inference then proceed by standard GMM methods.

Throughout the paper, we will illustrate our results and derivations in the special case of the model without random coefficients, i.e., the plain logit model.

runningexample[Identification in the plain logit model] Let $\bar{\lambda}\in\Lambda$ be the parameter value that makes the distribution of random coefficients $(\zeta,\nu)$ degenerate at zero, i.e., \begin{equation*} f(\zeta,\nu;\bar{\lambda}) = I[\zeta=\mathbf{0}_{d_X}, \nu=0]. \end{equation*} We can specialize the BLP model to the plain logit case by assuming that $\Lambda=\{\bar{\lambda}\}$, leading to a parameter space $\Theta = \mathbb{R}\times \mathbb{R}^{d_{X}}\times \{\bar\lambda\}$. The only parameters left to identify are $\alpha$ and $\beta$. In this special setting, the conditional choice probabilities are given by \begin{equation} P(y=j|x,p,\xi;\theta) = \frac{\exp(\beta x_j - \alpha p_j + \xi_j)}{1+\sum_{b=1}^J \exp(\beta x_b - \alpha p_b + \xi_b)} for j=1,\dots,J, \end{equation} and (ref) becomes \begin{equation*} \sigma_j (\delta ,x,p;\theta) = \frac{\exp ( \delta _{j}) }{1+\sum_{b=1}^{J}\exp ( \delta _{b}) } for j=1,\dots,J. \end{equation*} The function $\sigma(\delta ,x,p;\theta)$ is invertible in the first argument. Using $s_0 = 1-\sum_{j=1}^J s_j$, we obtain \begin{equation} \sigma^{-1}_j(s,x,p;\theta) = \ln ( s_{j}/s_0) for j=1,\dots,J. \end{equation} Substituting into (ref), we obtain the following conditional moment condition: \begin{equation} E[\ln ( s_{j}/s_0) - \beta x_{j} + \alpha p_{j}|z] = 0 for j=1,\dots,J. \end{equation} Provided that the instrument $z$ has sufficient variability, (ref) identifies $(\alpha,\beta)$.

Besides the structural parameters $\theta$, researchers who use the BLP model are often interested in equilibrium objects in this framework, such as demand elasticities and markups. For any pair of products $j,k=1,\dots,J$, the elasticity of demand for product $j$ with respect to the price of product $k$ is given by

equation*[equation* omitted — 93 chars of source]

By combining this with (ref) and (ref), we obtain an expression for the own-price elasticity of product $j$, given by

equation[equation omitted — 523 chars of source]

and an expression for the cross-price elasticity of product $j$ with respect to the price of product $k$, given by

equation[equation omitted — 507 chars of source]

The markup of product $j$ is the difference between $p_{j}$ and the marginal cost of product $j$. Under the assumption that the firms choose prices to maximize profits and that these are unconstrained in equilibrium, we obtain an expression relating the markup and the own elasticity:

equation[equation omitted — 153 chars of source]

Finally, the diversion ratio between products $j$ and $k$ quantifies the extent to which purchases shift from product $k$ to product $j$ in response to an infinitesimal increase in the price of product $k$. Formally, it is given by

equation*[equation* omitted — 150 chars of source]

By combining this with (ref) and (ref), we obtain the following expression:

equation[equation omitted — 931 chars of source]

The expressions in (ref), (ref), (ref), (ref) reveal that the elasticities, markups, and diversion ratios are functions of the structural parameters $\theta$, the observables $(s,p,x)$, and the demand shocks $\xi$.

runningexample[Elasticities and markups in the plain logit model] In the plain logit case (i.e., $\Lambda =\{\bar{ \lambda}\}$), (ref), (ref), and (ref) become particularly simple. For any $j,k=1,\dots,J$ with $j\neq k$, we obtain \begin{align*} e_{jj} = -\alpha p_{j}( 1-s_{j}), e_{jk} = \alpha p_{k}s_{k}, M_{j} = 1/(\alpha (1-s_{j})), and D_{jk} = {s_j}/(1-s_{k}). \end{align*} Notably, these expressions depend only on $p$, $s$, and $\alpha$, and the dependence on the other structural parameters or $\xi$ drops out.

Identification without outside good shares

This section derives the sharp identified set of the BLP model when the outside-good share $s_{0}$ is unobserved. Apart from this, the structure of the model is identical to that described in Section (ref), to which we refer the reader for details. In particular, we observe an i.i.d.\ sample of markets indexed by $m=1,\dots,M$. Since markets are i.i.d., we suppress the market subscript and conduct the identification analysis using a representative market.

Within each market, the model structure and maintained assumptions coincide with those of the standard BLP model in Section (ref). The observed random variables are $(\tilde{s},x,p,z)$, where

equation*[equation* omitted — 124 chars of source]

and $(s,x,p,z)$ are defined as before. Unlike in the standard case, the outside-good share $s_0$ cannot be recovered from the observed data.

Throughout our analysis, we consider the unobserved outside shares $s_0$ to belong to a known, possibly data-dependent, set $S_0$. By definition, $S_0$ denotes the logical domain of $s_0$ based on the researcher's information. The fully agnostic case corresponds to $S_0=(0,1)$, where $s_0=0$ and $s_0=1$ are explicitly ruled out as part of the specification of the BLP model. Of course, if the researcher has additional information about the outside-good share, this can be incorporated by restricting $S_0$ to a narrower, more informative interval.

Under our assumptions, we have that, conditional on $(x,p,\xi)$,

equation[equation omitted — 98 chars of source]

By combining (ref) with the conditional choice probabilities in (ref), we obtain

equation[equation omitted — 546 chars of source]

For any $\delta \in \mathbb{R}^{J}$, the analog of the function $\sigma (\delta ,x,p,\theta )$ in (ref) in the current context is $ \tilde\sigma (\delta ,x,p;\theta ):\mathbb{R}^{J}\times \mathbb{R}^{d_{X}}\times \mathbb{R}^{J}\times \Theta \to (0,1)^{J}$, given by

equation[equation omitted — 478 chars of source]

Unlike $\sigma(\delta ,x,p,\theta )$ in (ref), $\tilde\sigma(\delta ,x,p,\theta)$ is not invertible in the first argument. The next example illustrates this point in the special case of the plain logit model.

runningexample[Non-invertibility $\tilde\sigma(\delta ,x,p,\theta)$ in the plain logit model] In the plain logit case (i.e., $\Lambda =\{\bar{ \lambda}\}$), (ref) becomes \begin{equation} \tilde{\sigma}(\delta ,x,p,\theta ) = \left(\frac{ \exp \left( \delta _{j}\right) }{\sum_{b=1}^{J}\exp ( \delta _{b}) }\right) _{j=1}^{J}. \end{equation} For any $C \in \mathbb{R}$, note that $\tilde{\sigma}(\delta ,x,p,\theta)= \tilde{\sigma}(\delta +{\bf 1}_{J \times 1} C,x,p,\theta)$. Note that $\tilde{\sigma}(\delta ,x,p,\theta)=g( \delta )$, where $g:\mathbb{R}^{J}\to \Delta _{J}$ is many to one. Thus, $\tilde{\sigma}(\delta ,x,p,\theta )$ is not invertible in the first argument.

Therefore, the argument used in Section (ref) to identify the parameters of the model does not apply. In fact, it is not hard to show that the model is not point identified, as we now illustrate.

runningexample[Lack of identification in the plain logit model] Consider the plain logit case (i.e., $\Lambda =\{\bar{ \lambda}\}$), (ref) with $x_{j}=(1,\tilde{x}_{j}')'$ and $\beta =(\beta _{1},\beta _{2})$. Then, (ref) becomes \begin{equation*} \tilde{s} = \left(\frac{ \exp ( \beta _{1}+\beta _{2}\tilde{x} _{j}-\alpha p_{j}+\xi _{j}) }{\sum_{b=1}^{J}\exp ( \beta _{1}+\beta _{2}\tilde{x}_{b}-\alpha p_{b}+\xi _{b}) }\right)_{j=1}^{J}. \end{equation*} First, note that $\beta _{1}$ is not identified. To see this, note that for any $( \alpha ,( \beta _{1},\beta _{2}) ,\bar{\lambda}) \in \Theta $, $( \alpha ,( \beta _{1},\beta _{2}) ,\bar{\lambda}) $ and $( \alpha ,( \beta _{1}+C,\beta _{2}) ,\bar{ \lambda}) $ are observationally equivalent. Next, we argue that $( \beta _{2},\alpha ) $ is identified under mild additional conditions. To see this, note that \begin{equation*} \ln \left( \tilde{s}_{j}/\tilde{s}_{k}\right) = \beta _{2}\left( \tilde{x} _{j}-\tilde{x}_{k}\right) -\alpha (p_{j}-p_{k})+(\xi _{j}-\xi _{k}) for j,k=1,\dots,J. \end{equation*} Then, \begin{equation*} E\left[ \ln \left( \tilde{s}_{j}/\tilde{s}_{k}\right) |z\right] = \beta _{2}E \left[ \tilde{x}_{j}-\tilde{x}_{k}|z\right] -\alpha E\left[ p_{j}-p_{k}|z \right] for j,k=1,\dots,J. \end{equation*} Provided that $E[ \tilde{x}_{j}-\tilde{x}_{k}|z] $ and $E [ p_{j}-p_{k}|z] $ vary sufficiently for some $j,k$, $( \beta _{2},\alpha ) $ is identified.

Example (ref) shows that, when outside-good shares are unobserved, the fixed-coefficient version of the BLP model is not identified. Moreover, that example enables us to characterize the identified set of structural parameters in this special case. Unfortunately, the fixed-coefficient version of the BLP model imposes strong restrictions on demand, thereby limiting its ability to capture empirically relevant substitution patterns. For this reason, the more relevant identification problem is to carry out this analysis for the general BLP model with random coefficients.

Identification of the structural parameters

The following result provides a sharp characterization of the identified set of the BLP model. It derives the sharp identified set for the structural parameters using tools from random set theory.

theorem[Identified set of the structural parameters] Assume that the distribution of $\{(\tilde{s},x,p)|z\}$ is non-atomic a.e.\ $z\in \mathbb{S}_{z}$, and that, for any $\theta =( \alpha ,\beta ,\lambda ) \in \Theta$, the following random set is integrable: \begin{equation} \mathcal{U}( \tilde{s},x,p;\theta ) =\left\{ \xi \in \mathbb{R}^{J}:\tilde{s} = \left( \frac{ \int_{(\zeta ,\nu )}\frac{ \exp ( \beta x_{j}-\alpha p_{j} + \xi_{j} + \zeta x_{j} -p_{j}\nu ) }{ 1+\sum_{b=1}^{J}\exp ( \beta x_{b}-\alpha p_{b} + \xi_{b} +\zeta x_{b} -p_{b}\nu ) }f( \zeta ,\nu ;\lambda ) d( \zeta ,\nu ) }{\int_{(\zeta ,\nu )}\frac{\sum_{l=1}^{J}\exp ( \beta x_{l}-\alpha p_{l} + \xi_{l} +\zeta x_{l}- \nu p_{l}) }{1+\sum_{e=1}^{J}\exp ( \beta x_{e}-\alpha p_{e}+\xi _{e}+\zeta x_{e} -p_{e}\nu ) }f( \zeta ,\nu ;\lambda ) d( \zeta ,\nu ) } \right) _{j=1}^{J} \right\}. \end{equation} Then, the identified set of $\theta =( \alpha ,\beta ,\lambda ) $ is \begin{equation} \Theta _{I}(P) =\left\{ \begin{array}{c} ( \alpha ,\beta ,\lambda ) \in \Theta: \\ E\Big[\sup_{s_{0}\in S_0}v^{\prime }\big( \sigma ^{-1}( (s_{0},\tilde{s}(1-s_{0})),x,p;\theta ) -( \beta x_{j}-\alpha p_{j})_{j=1}^{J}\Big) \Big|z \Big]\geq 0 \\ for all v\in \mathbb{R}^{J}:\Vert v\Vert =1 and a.e. z\in \mathbb{S}_{z} \end{array} \right\} , \end{equation} where $\sigma ^{-1}$ is the inverse of the function in (ref) with respect to the first argument, and $S_0$ represents the logical domain for the outside-good share (in the absence of this, set $S_0 = (0,1)$).

Theorem (ref) requires that the observable random variables be defined on a non-atomic probability space conditional on the instrument; see billingsley:1995. This condition is satisfied whenever at least one component of $(\tilde{s},x,p)$ is continuously distributed conditional on $z$. This is a mild requirement, since in applications we typically assume that either $\tilde{s}$ or $p$ is (conditionally) continuously distributed. This assumption implies that the conditional expectation of the relevant random sets is almost surely convex, which allows us to characterize these random sets via their support function in (ref). See, for example, molchanov/molinari:2018. Theorem (ref) also requires the random set in (ref) to be integrable. By definition, a random set is integrable if at least one of its selections is integrable; see molchanov/molinari:2018. This is another mild requirement that ensures the conditional expectations used in our characterization are well-defined.

Besides characterizing the identified set $\Theta_{I}(P)$, Theorem (ref) also defines the set $\mathcal{U}(\tilde{s},x,p;\theta)$. By construction, $\mathcal{U}(\tilde{s},x,p;\theta)$ collects all values of the demand shocks $\xi$ that are compatible with a given $\theta$ and the observable variables $(x,p)$. Since $\Theta_{I}(P)$ contains all parameter values consistent with the data, the collection of demand shocks $\xi$ compatible with the observables is given by

equation[equation omitted — 107 chars of source]

The set in (ref) can be considered the analog of a singleton set based on the demand shock specified in equation (ref) when outside shares are unobserved.

We now illustrate this in the context of the plain logit model.

runningexample[Computing objects in Theorem (ref) in the plain logit model] Consider the plain logit case (i.e., $\Lambda =\{\bar{ \lambda}\}$), with $x_{j}=(1,\tilde{x}_{j}')'$ and $\beta =(\beta _{1},\beta _{2})$. In this case, (ref) becomes \begin{align*} \mathcal{U}(\tilde{s},x,p;\theta) &= \left\{\xi \in \mathbb{R}^{J}: \tilde{s} = \left(\frac{\exp (\beta x_{j}-\alpha p_{j}+\xi _{j} )}{\sum_{b=1}^{J}\exp (\beta x_{b}-\alpha p_{b}+\xi _{b} )}\right)_{j=1}^{J} \right\} \\ &\overset{(1)}{=} \left\{ \xi \in \mathbb{R}^{J}: \left\{\begin{array}{c} \xi _{j}-\xi _{k}=\ln (\tilde{s}_{j}/ \tilde{s}_{k})-\beta _{2}(\tilde{x}_{j}-\tilde{x}_{k})+\alpha (p_{j}-p_{k})\\ for all j,k=1,\dots,J \end{array}\right\} \right\} , \end{align*} where (1) holds by $x_{j}=(1,\tilde{x}_{j}')'$ and $\beta =(\beta _{1},\beta _{2})$. Given one of the demand shocks (say, $\xi _{1}$), the rest of the demand shocks in $\xi$ are determined by the observables and structural parameters. We now turn to the identified set in $\Theta _{I}(P)$. Of course, $\lambda \in \Lambda =\{\bar{\lambda}\}$ is point identified by assumption, so we now proceed with the characterization of the identified set for $\alpha$ and $\beta$. For concreteness, we focus on the fully agnostic case, i.e., $S_{0}=( 0,1) $. By (ref), the expectation inside of (ref) becomes \begin{equation*} E\Big[ \sup_{s_{0}\in S_{0}}\left(\ln \left(\tfrac{1-s_{0}}{s_{0}}\right)-\beta _{1} \right)\Big(\sum\nolimits_{j=1}^{J}v_{j}\Big) +v^{\prime }\Big(\ln (\tilde{s}_{j})-\beta _{2}\tilde{x}_{j}+\alpha p_{j}\Big)_{j=1}^{J} \Big|z\Big] \geq 0 for a.e. z\in \mathbb{S}_{z}. \end{equation*} Whenever $\sum_{j=1}^{J}v_{j}\neq 0$, one can set $s_{0}$ arbitrarily close to 0 or 1 so that the above equation is trivially satisfied for all values of $\theta$. Then, the only values of $v\in \mathbb{R}^{J}$ that restrict $\theta$ are those with $\sum_{j=1}^{J}v_{j}=0$. If we focus on this case, we get \begin{equation} E\left[ v^{\prime }(\ln (\tilde{s}_{j})-\beta _{2}\tilde{x}_{j}+\alpha p_{j})_{j=1}^{J}|z\right] \geq 0 for all v\in \mathbb{R}^{J}:\Vert v\Vert =1,\sum_{j=1}^{J}v_{j}=0, and a.e. z\in \mathbb{S}_{z}. \end{equation} Note that (ref) exhausts the model information, and this is fully summarized in (ref). Since (ref) does not depend on $\beta _{1}$, we deduce that the model does not restrict $\beta _{1}$. We now proceed to the identification analysis of $\alpha$ and $\beta_2$. If $v\in \mathbb{R}^{J}$ satisfies $\Vert v\Vert =1$ and $ \sum_{j=1}^{J}v_{j}=0$, then $\tilde{v}=-v\in \mathbb{R}^{J}$ also satisfies $\Vert \tilde{v}\Vert =1$ and $\sum_{j=1}^{J}\tilde{v}_{j}=0$. This shows that (ref) implies \begin{equation} E\left[ v^{\prime }(\ln (\tilde{s}_{j})-\beta _{2}\tilde{x}_{j}+\alpha p_{j})_{j=1}^{J}|z\right] \leq 0 for all v\in \mathbb{R}^{J}:\Vert v\Vert =1,\sum_{j=1}^{J}v_{j}=0, and a.e. z\in \mathbb{S}_{z}. \end{equation} Combining (ref) and (ref), we obtain that \begin{equation} E\left[ v^{\prime }(\ln (\tilde{s}_{j})-\beta _{2}\tilde{x}_{j}+\alpha p_{j})_{j=1}^{J}|z\right] =0\text{ for all }v\in \mathbb{R}^{J}:\Vert v\Vert =1,\sum_{j=1}^{J}v_{j}=0,\text{ and a.e. }z\in \mathbb{S}_{z}. \end{equation} By using (ref) with $v_u=0$ for all $u=1,\dots,J$ except for $j\neq k$, we get \begin{equation} E\left[ \ln (\tilde{s}_{j}/\tilde{s}_{k})-\beta _{2}\left( \tilde{x}_{j}-\tilde{x}_{k}\right) +\alpha \left( p_{j}-p_{k}\right) |z\right] =0\text{ for a.e. }z\in \mathbb{S}_{z}. \end{equation} We note that any other restriction produced by (ref) could be represented as a linear combination of (ref) for multiple values of $j,k=1,\dots,J$ with $ j\neq k$. Therefore, (ref) for $j,k=1,\dots ,J$ with $j\neq k$ exhausts all the information in the model regarding $\alpha$ and $\beta_2$. From here, we conclude that the identified set in (ref) can be equivalently expressed as follows: \begin{equation} \Theta _{I}(P)=\left\{ \begin{array}{c} (\alpha ,\beta _{1},\beta _{2},\lambda )\in \Theta =\mathbb{R}\times \mathbb{R}\times \mathbb{R}^{d_{X}-1}\times \{\bar{\lambda}\}: \\ E[ \ln (\tilde{s}_{j}/\tilde{s}_{k})-\beta _{2}( \tilde{x}_{j}-\tilde{x}_{k}) +\alpha ( p_{j}-p_{k}) |z] =0 \\ \text{for all }j,k=1,\dots ,J\text{ with }j\neq k\text{ and a.e. }z\in \mathbb{S}_{z} \end{array} \right\} . \end{equation} Naturally, the conclusions coincide with our earlier ad-hoc derivations in Example (ref). The parameter $\lambda$ is (trivially) point identified, the parameter $\beta _{1}$ is completely unidentified, and $\alpha$ and $\beta_{2} $ are restricted by a collection of conditional moment equalities obtained in Example (ref). Provided that $E[ \tilde{x}_{j}-\tilde{x}_{k}|z] $ and $E[ p_{j}-p_{k}|z] $ vary sufficiently for some $j,k$, $\alpha$ and $\beta_{2} $ are point identified.

In practice, researchers are often interested in a function $g(\theta )$ rather than the full parameter vector $\theta $. This occurs naturally when the economic question of interest involves a component or a subvector of $ \theta =(\beta ,\alpha ,\lambda )$ rather than $\theta $ itself. For instance, if the research question of interest is to determine whether the market corresponds to a pure logit model or a random coefficient model, then the sole parameter of interest is $\lambda $, and, more specifically, whether $\lambda =\bar{\lambda}$ or not, where $f(\zeta ,\nu ;\bar{\lambda})=I[\zeta =\mathbf{0}_{d_{X}},\nu =0]$. In these cases, the parameter of interest can be characterized as $g(\theta )$, and its identified set can be obtained as a by-product of Theorem (ref) through projection. In particular, for any known function $g$, the identified set for $g(\theta )$ is given by

equation[equation omitted — 80 chars of source]

For instance, the identified set of $\lambda $ is $\{\lambda :(\beta ,\alpha ,\lambda )\in \Theta _{I}(P)\}$, which is (ref) with $g(\beta ,\alpha ,\lambda )=\lambda $.

Can we get the identified set by repeated applications of BLP?

Given the result in Theorem (ref), a natural inquiry is whether the identified set $\Theta_I(P)$ can be obtained by simply applying the standard BLP identification arguments to each possible outside share $s_0 \in S_0$ for every market in the data. The next result answers this question affirmatively.

theorem[Alternative characterization of the identified set] Let $w = (\tilde s, x, p , z)$ denote the observed data for each market, and let $\mathbb{S}_{w}$ denote its support. Under the conditions of Theorem (ref), \begin{align*} \Theta_I(P) & = \left\{ \begin{array}{c} ( \alpha ,\beta ,\lambda ) \in \Theta:\exists s_{0}:\mathbb{S}_{w}\to S_0 \\ E\Big[ \sigma ^{-1}( (s_{0} ,\tilde{s}(1-s_{0}(w) )),x,p;\lambda ) -\left( \beta x_{j}-\alpha p_{j}\right) _{j=1}^{J}\Big| z\Big] ={\bf 0}_{J \times 1} a.e. z\in \mathbb{S}_{z} \end{array} \right\} \end{align*}

Theorem (ref) is intuitive. Since the source of partial identification is that $s_0$ can only be restricted to lie in $S_0$, one can generate $\Theta_I(P)$ by considering all possible values of this outside share for each data realization. Unfortunately, this result is of limited practical use, as $w = (\tilde s, x, p, z)$ can take as many values as there are markets, and the number of markets in a dataset can be large. In fact, if $w$ contains a continuously distributed component (e.g., product prices), then there will typically be as many realizations of $w$ as markets in the data. By contrast, the equivalent representation of $\Theta_I(P)$ in (ref) (Theorem (ref)) is more amenable to inference; see Section (ref).

Given this difficulty, one may ask whether $\Theta_I(P)$ can be generated by applying the standard BLP identification arguments separately for each possible outside share $s_0 \in S_0$ and each instrument value. That is, can a result analogous to Theorem (ref) be obtained by restricting attention from the full class of functions $s_0 : \mathbb{S}_{w} \to S_0$ to the simpler class $s_0 : \mathbb{S}_{z} \to S_0$? The following result shows that this replacement yields a subset of $\Theta_I(P)$ that may be too small and can, in fact, even be empty, thus excluding the true parameter value.

theoremUnder the conditions of Theorem (ref), define the following set: \begin{align*} \mathcal{H}(P) & = \left\{ \begin{array}{c} ( \alpha ,\beta ,\lambda ) \in \Theta:\exists s_{0}:\mathbb{S}_{z}\to S_0 \\ E\Big[ \sigma ^{-1}( (s_{0}(z) ,\tilde{s}(1-s_{0}(z) )),x,p;\lambda ) -\left( \beta x_{j}-\alpha p_{j}\right)_{j=1}^{J}\Big| z\Big] ={\bf 0}_{J \times 1} a.e. z\in \mathbb{S}_{z} \end{array} \right\}. \end{align*} Then, $\mathcal{H}(P)\subseteq \Theta _{I}(P)$ and, in general, $\Theta _{I}(P) \neq \mathcal{H}(P)$. In fact, it is possible to have data generating processes in which $\Theta _{I}(P) \neq \emptyset$ and $\mathcal{H}(P) = \emptyset$.

Theorem (ref) underscores that recovering $\Theta_I(P)$ necessitates searching over a rich class of functions. Any simplification of this space can produce a strict subset of $\Theta_I(P)$, potentially even empty. In what follows, we adopt the characterization of $\Theta_I(P)$ in Theorem (ref) (equivalently, Theorem (ref)). Despite its complexity, the representation in (ref) is tractable and amenable to standard inference methods, as shown in Section (ref).

Identification of other equilibrium objects

In addition to structural parameters, one may be interested in equilibrium objects in the BLP model, such as demand elasticities or markups. As shown in Section (ref), these objects are functions of the structural parameters $\theta $, the observables $(s,p,x)$, and the demand shocks $\xi $.

For concreteness, we begin our analysis with the own-price elasticities. Recall from Section (ref) that the own-price elasticity of product $j=1,\dots,J$ is given by

equation*[equation* omitted — 498 chars of source]

The right-hand side expression presents three identification challenges when $s_0$ is unobserved. First, the structural parameter $\theta = (\alpha,\beta,\lambda)$ is partially identified, and can only be restricted to its identified set $\Theta_I(P)$ in (ref). Second, the demand shock $\xi$ is also partially identified, and can only be restricted to $\mathcal{U}(\tilde{s},x,p;\theta) $ in (ref) for each combination of observables and structural parameters $\theta$. These two problems are addressed in Theorem (ref). Third and finally, $s_j = \tilde s_j (1-s_0)$ is unknown. To deal with this issue, we combine (ref) and (ref) to express the unobserved share $s_j$ as follows:

align[align omitted — 287 chars of source]

By plugging this in (ref), we obtain the following formula for the own-elasticity:

equation[equation omitted — 762 chars of source]

By combining this with the identified sets in Theorem (ref), we obtain an identified set for the own-price elasticity. This identified set is denoted by $\mathcal{E}_{jj}(\tilde{s},x,p)$ and specified in the next theorem. The result also includes identified sets for the cross-price elasticities, markups, diversion ratios, and shares.

theorem[Identified set for other equilibrium objects] Assume the conditions in Theorem (ref). Then, the identified set for the own-price elasticity of product $j=1,\dots,J$ in a market with observables $(\tilde{s},x,p)$ is given by: \begin{equation*} \mathcal{E}_{jj}(\tilde{s},x,p) = \left\{ \begin{array}{c} e_{jj} = -p_{j}\tfrac{\int_{(\zeta ,\nu )}(\alpha +\nu ) \left(\begin{array}{c} \tfrac{\exp (\beta x_{j} -\alpha p_{j}+\xi _{j}+\zeta x_{j} -\nu p_{j})}{1+\sum_{b=1}^{J}\exp (\beta x_{b} -\alpha p_{b}+\xi _{b}+\zeta x_{b} -\nu p_{b})}\times \\ \left( 1-\tfrac{\exp (\beta x_{j} -\alpha p_{j}+\xi _{j}+\zeta x_{j} -\nu p_{j})}{1+\sum_{e=1}^{J}\exp (\beta x_{e} -\alpha p_{e}+\xi _{e}+\zeta x_{e} -\nu p_{e})} \right) \end{array}\right) f(\zeta ,\nu ;\lambda )d(\zeta ,\nu )}{\int_{(\zeta ,\nu )}\tfrac{ \exp \left( \beta x_{j}-\alpha p_{j}+\xi_{j}+ \zeta x_{j} -\nu p_{j} \right) }{1+\sum_{l=1}^{J} \exp \left( \beta x_{l}-\alpha p_{l}+\xi _{l}+\zeta x_{l} -\nu p_{l} \right) }f( \zeta ,\nu ;\lambda ) d(\zeta ,\nu )}:\\ \xi \in \mathcal{U}(\tilde{s},x,p;\theta ) and \theta \in \Theta _{I}(P) \end{array} \right\} . \end{equation*} In turn, the identified set for the cross-price elasticity of product $j=1,\dots,J$ with respect to the price of product $k\neq j$ in a market with observables $(\tilde{s},x,p)$ is given by: \begin{equation*} \mathcal{E}_{jk}(\tilde{s},x,p) = \left\{ \begin{array}{c} e_{jk} = p_{k}\tfrac{\int_{(\zeta ,\nu )}(\alpha +\nu ) \left(\begin{array}{c} \tfrac{\exp (\beta x_{j} -\alpha p_{j}+\xi _{j}+\zeta x_{j} -\nu p_{j})}{1+\sum_{b=1}^{J}\exp (\beta x_{b} -\alpha p_{b}+\xi _{b}+\zeta x_{b} -\nu p_{b})}\times \\ \tfrac{\exp (\beta x_{k} -\alpha p_{k}+\xi _{k}+\zeta x_{k} -\nu p_{k})}{1+\sum_{e=1}^{J}\exp (\beta x_{e} -\alpha p_{e}+\xi _{e}+\zeta x_{e} -\nu p_{e})} \end{array}\right) f(\zeta ,\nu ;\lambda )d(\zeta ,\nu )}{\int_{(\zeta ,\nu )}\tfrac{ \exp \left( \beta x_{j}-\alpha p_{j}+\xi_{j}+ \zeta x_{j} -\nu p_{j} \right) }{1+\sum_{l=1}^{J} \exp \left( \beta x_{l}-\alpha p_{l}+\xi _{l}+\zeta x_{l} -\nu p_{l} \right) }f( \zeta ,\nu ;\lambda ) d(\zeta ,\nu )}:\\ \xi \in \mathcal{U}(\tilde{s},x,p;\theta ) and \theta \in \Theta _{I}(P) \end{array} \right\}. \end{equation*} Also, the identified set for the markup of product $j=1,\dots,J$ in a market with observables $(\tilde{s},x,p)$ is given by: \begin{equation*} \mathcal{M}_{j}(\tilde{s},x,p) = \left\{ \begin{array}{c} M_{j} = \tfrac{ \int_{(\zeta ,\nu )}\tfrac{ \exp \left( \beta x_{j}-\alpha p_{j}+\xi_{j}+ \zeta x_{j} -\nu p_{j} \right) }{1+\sum_{l=1}^{J} \exp \left( \beta x_{l}-\alpha p_{l}+\xi _{l}+\zeta x_{l} -\nu p_{l} \right) }f( \zeta ,\nu ;\lambda ) d(\zeta ,\nu ) }{ \alpha \int_{(\zeta ,\nu )}(\alpha +\nu ) \left(\begin{array}{c} \tfrac{\exp (\beta x_{j} -\alpha p_{j}+\xi _{j}+\zeta x_{j} -\nu p_{j})}{1+\sum_{b=1}^{J}\exp (\beta x_{b} -\alpha p_{b}+\xi _{b}+\zeta x_{b} -\nu p_{b})}\times \\ \left( 1-\tfrac{\exp (\beta x_{j} -\alpha p_{j}+\xi _{j}+\zeta x_{j} -\nu p_{j})}{1+\sum_{e=1}^{J}\exp (\beta x_{e} -\alpha p_{e}+\xi _{e}+\zeta x_{e} -\nu p_{e})} \right) \end{array}\right) f(\zeta ,\nu ;\lambda )d(\zeta ,\nu ) }:\\ \xi \in \mathcal{U}(\tilde{s},x,p;\theta ) and \theta \in \Theta _{I}(P) \end{array} \right\}. \end{equation*} The identified set for the diversion ratio of product $j = 1,\dots,J$ with respect to product $k \ne j$ in a market with observables $(\tilde{s},x,p)$ is given by: \begin{equation*} \mathcal{D}_{jk}(\tilde{s},x,p) = \left\{ \begin{array}{c} D_{jk} = \tfrac{\int_{(\zeta ,\nu )}(\alpha +\nu ) \left(\begin{array}{c} \tfrac{\exp (x_{j}\beta -\alpha p_{j}+\xi _{j}+x_{j}\zeta -\nu p_{j})}{1+\sum_{b=1}^{J}\exp (x_{b}\beta -\alpha p_{b}+\xi _{b}+x_{b}\zeta -\nu p_{b})}\times \\ \tfrac{\exp (x_{k}\beta -\alpha p_{k}+\xi _{k}+x_{k}\zeta -\nu p_{k})}{1+\sum_{b=1}^{J}\exp (x_{b}\beta -\alpha p_{b}+\xi _{b}+x_{b}\zeta -\nu p_{b})} \end{array}\right) f(\zeta ,\nu ;\lambda )d(\zeta ,\nu )}{\int_{(\zeta ,\nu )}(\alpha + \nu) \left(\begin{array}{c} \tfrac{\exp (x_{k}\beta -\alpha p_{k}+\xi _{k}+x_{k}\zeta -\nu p_{k})}{1+\sum_{b=1}^{J}\exp (x_{b}\beta -\alpha p_{b}+\xi _{b}+x_{b}\zeta -\nu p_{b})}\times \\ \left( 1-\tfrac{\exp (x_{k}\beta -\alpha p_{k}+\xi _{k}+x_{k}\zeta -\nu p_{k})}{1+\sum_{b=1}^{J}\exp (x_{b}\beta -\alpha p_{b}+\xi _{b}+x_{b}\zeta -\nu p_{b})} \right) \end{array}\right) f(\zeta,\nu;\lambda)d(\zeta,\nu)}:\\ \xi \in \mathcal{U}(\tilde{s},x,p,z;\theta ) and \theta \in \Theta _{I}(P) \end{array} \right\}. \end{equation*} Finally, the identified set for the share of product $j=1,\dots,J$ is given by: \begin{equation*} \mathcal{S}_{j}(\tilde{s},x,p) = \left\{ \begin{array}{c} s_j = \int_{(\zeta ,\nu )}\tfrac{\exp \left( \beta x_{j}-\alpha p_{j}+\xi _{j}+ \zeta x_{j} -\nu p_{j} \right) }{1+\sum_{b=1}^{J} \exp \left( \beta x_{b}-\alpha p_{b}+\xi _{b}+\zeta x_{b} -\nu p_{b} \right) }f( \zeta ,\nu ;\lambda ) d(\zeta ,\nu ):\\ \xi \in \mathcal{U}(\tilde{s},x,p;\theta ) and \theta \in \Theta _{I}(P) \end{array} \right\} , \end{equation*}

We now illustrate this in the context of the plain logit model.

runningexample[Identified sets for equilibrium objects in the plain logit model] Consider the plain logit case (i.e., $\Lambda =\{\bar{ \lambda}\}$), with $x_{j}=(1,\tilde{x}_{j}')'$ and $\beta =(\beta _{1},\beta _{2})$. For each $j,k=1,\dots,J$ with $j\neq k$, \begin{align*} \mathcal{E}_{jj}(\tilde{s},x,p) &= \Big\{e_{jj}=-\alpha p_j \big(1-\tfrac{\exp(\beta_1+\beta_2\tilde x_j-\alpha p_j+\xi_j)} {1+\sum_{b=1}^J \exp(\beta_1+\beta_2\tilde x_b-\alpha p_b+\xi_b)} \big): \xi\in\mathcal U(\tilde s,x,p;\theta) and \theta\in\Theta_I(P)\Big\},\\ \mathcal{E}_{jk}(\tilde{s},x,p) &= \Big\{e_{jk}=\alpha p_k \tfrac{\exp(\beta_1+\beta_2\tilde x_k-\alpha p_k+\xi_k)} {1+\sum_{b=1}^J \exp(\beta_1+\beta_2\tilde x_b-\alpha p_b+\xi_b)}: \xi\in\mathcal U(\tilde s,x,p;\theta) and \theta\in\Theta_I(P)\Big\},\\ \mathcal{M}_{j}(\tilde{s},x,p) &= \Big\{m_{j}=\frac{1}{\alpha}\big(1- \tfrac{\exp(\beta_1+\beta_2\tilde x_j-\alpha p_j+\xi_j)}{1+\sum_{b=1}^J \exp(\beta_1+\beta_2\tilde x_b-\alpha p_b+\xi_b)} \big): \xi\in\mathcal U(\tilde s,x,p;\theta) and \theta\in\Theta_I(P)\Big\},\\ \mathcal{D}_{jk}(\tilde{s},x,p) &= \Big\{d_{jk}=\tfrac{\tfrac{\exp(\beta_1+\beta_2\tilde x_j-\alpha p_k+\xi_j)} {1+\sum_{b=1}^J \exp(\beta_1+\beta_2\tilde x_b-\alpha p_b+\xi_b)} }{\big(1-\tfrac{\exp(\beta_1+\beta_2\tilde x_k-\alpha p_k+\xi_k)} {1+\sum_{b=1}^J \exp(\beta_1+\beta_2\tilde x_b-\alpha p_b+\xi_b)}\big)}: \xi\in\mathcal U(\tilde s,x,p;\theta) and \theta\in\Theta_I(P)\Big\},\\ \mathcal{S}_{j}(\tilde{s},x,p) &= \Big\{s_{j}=\tfrac{\exp(\beta_1+\beta_2\tilde x_j-\alpha p_j+\xi_j)}{1+\sum_{b=1}^J \exp(\beta_1+\beta_2\tilde x_b-\alpha p_b+\xi_b)}: \xi\in\mathcal U(\tilde s,x,p;\theta) and \theta\in\Theta_I(P)\Big\}. \end{align*} By our earlier derivations, $\beta_1$ is completely unidentified and may take any value in $\mathbb{R}$. Holding $(\alpha,\beta_2,\xi)$ fixed with $\xi\in\mathcal U(\tilde s,x,p;\theta)$, we obtain \begin{align*} \lim_{\beta_1\to-\infty}\frac{\exp(\beta_1+\beta_2\tilde x_j-\alpha p_j+\xi_j)}{1+\sum_{b=1}^J \exp(\beta_1+\beta_2\tilde x_b-\alpha p_b+\xi_b)}&=0,\\ \lim_{\beta_1\to\infty}\frac{\exp(\beta_1+\beta_2\tilde x_j-\alpha p_j+\xi_j)}{1+\sum_{b=1}^J \exp(\beta_1+\beta_2\tilde x_b-\alpha p_b+\xi_b)}&=\frac{\exp(\beta_2\tilde x_j-\alpha p_j+\xi_j)} {\sum_{b=1}^J \exp(\beta_2\tilde x_b-\alpha p_b+\xi_b)} \overset{(1)}{=} \tilde s_j, \end{align*} where (1) holds by $\xi\in\mathcal U(\tilde s,x,p;\theta)$. By continuity, as $\beta_1$ ranges over $\mathbb{R}$, this expression ranges over $(0,\tilde s_j)$. By continuity of the mappings involved, we obtain the following identified sets: \begin{align*} \mathcal{E}_{jj}(\tilde{s},x,p) &= \Big\{e_{jj} \in (-\alpha p_j ,-\alpha p_j (1-\tilde s_j)) : (\alpha,\beta_1,\beta_2,\bar\lambda)\in\Theta_I(P)\Big\}\\ \mathcal{E}_{jk}(\tilde{s},x,p) &= \Big\{e_{jk} \in (\alpha p_k \tilde s_k,\alpha p_k ) : (\alpha,\beta_1,\beta_2,\bar\lambda)\in\Theta_I(P)\Big\}\\ \mathcal{M}_{j}(\tilde{s},x,p) &= \Big\{m_{j} \in (\tfrac{1}{\alpha },\tfrac{1}{\alpha (1-\tilde s_j)}) : (\alpha,\beta_1,\beta_2,\bar\lambda)\in\Theta_I(P)\Big\},\\ \mathcal{D}_{jk}(\tilde{s},x,p) &= \Big\{d_{jk} \in (0, \tfrac{\tilde s_j}{1- \tilde s_k}) : (\alpha,\beta_1,\beta_2,\bar\lambda)\in\Theta_I(P)\Big\}\\ \mathcal{S}_{j}(\tilde{s},x,p) &= (0,\tilde s_j). \end{align*} Provided that $E[\tilde{x}_{j}-\tilde{x}_{k}\mid z]$ and $E[p_{j}-p_{k}\mid z]$ vary sufficiently, $\alpha$ was shown to be point identified. If so, the first three identified sets simplify further, with $\alpha$ fixed at its unique identified value.

Numerical illustration

This section presents the results of a numerical illustration in which we compute identified sets for model parameters, unobserved demand shocks, and elasticities in a differentiated-products demand model with partial identification of the outside-good share.

We consider the following simple data-generating process. Markets have three products: $J=2$ inside goods and the outside good. Assume that there is a single discrete instrument

equation*[equation* omitted — 54 chars of source]

The instrument shifts demand for products and affects the prices of inside goods. Prices of the inside goods are generated according to

align*[align* omitted — 130 chars of source]

For each $j=1,2$, the function $g_j$ is strictly increasing, and so prices are expected to increase with the value of the instrument. For concreteness, we set $g_1(z)=z$ and $g_2(z)=\ln z$.

Demand shocks are generated as follows:

equation*[equation* omitted — 136 chars of source]

This induces endogeneity due to

equation[equation omitted — 110 chars of source]

The instrument is valid since

align*[align* omitted — 57 chars of source]

We set the observable covariate to a constant, $x_j = 1$ for all products.

Demand is generated from the BLP model as described in Section (ref). Let

equation*[equation* omitted — 62 chars of source]

where $\beta$ is the mean utility level, $\alpha$ is the price coefficient, and $\lambda$ governs heterogeneity in price sensitivity. The parameter space is

equation*[equation* omitted — 60 chars of source]

Market shares for inside goods $j=1,2$ and the outside option are given by

align*[align* omitted — 329 chars of source]

where $\nu \sim N(0,\lambda^{2})$. We consider both fixed-coefficient designs ($\lambda=0$) and random-coefficient designs ($\lambda>0$). In all simulations, we use the following true parameter values:

equation*[equation* omitted — 56 chars of source]

We study identification under two information structures for the outside good share:

enumerate• Small amount of missing information on $s_0$: $S_0 = (s_0-\varepsilon,\, s_0+\varepsilon)\cap(0,1)$, where $\varepsilon>0$ is small. In the numerical exercise below, we use $\varepsilon=0.05$. This allows us to explore the sensitivity of identification to a relatively small amount of missing information in $s_0$. • No information on $s_0$: $S_0=(0,1)$. This represents a complete lack of information about the outside share. The previous running examples consider this situation in the plain logit model (i.e., $\lambda =0$). The next results explore the possibility of having random coefficients.

We begin with the first case, when there is a small amount of missing information on $s_0$, i.e., $S_0 = (s_0-\varepsilon, s_0+\varepsilon)\cap(0,1)$, where $\varepsilon=0.05$. Figure (ref) depicts the identified set $\Theta _{I}(P)$. This is a three-dimensional object, which we visualize in the $(\alpha,\beta)$ plane for different values of $\lambda$. The figure reports results for $\lambda \in [0.8,1.35]$. We find that the identified set is empty for values of $\lambda \not\in [ 0.8,1.35]$.

If we project this identified set $\Theta _{I}(P)$ on each parameter coordinate, we obtain the parameter-specific identified sets:

align*[align* omitted — 135 chars of source]

Naturally, the true parameter values $(\alpha,\beta,\lambda) = (1,1,1)$ belong to their identified sets.

sidewaysfigure\caption{Identified set $\Theta _{I}(P)$ in a DGP with $(\alpha,\beta,\lambda) = (1,1,1)$ and a small amount of missing information on $s_0$.}

Next, we consider equilibrium objects in a market with $(\tilde s, p)$ set at their median values in our simulation design. These values serve as benchmarks, although other values could also be considered. In this design, the median inside shares are $\tilde s = (0.3211, 0.6789)$ and the median prices are $p = (2.9925, 0.972)$. Figure (ref) shows the identified set for the demand shocks $\mathcal U(\tilde s, p)$ in Theorem (ref). Using this set, we compute the identified sets for equilibrium objects such as elasticities. Finally, we obtain the identified sets for each elasticity in a market with the median $(\tilde s, p)$.

align*[align* omitted — 247 chars of source]

The true elasticity values are $e_{1,1}=-0.307$, $e_{2,2}=-0.599$, $e_{1,2}=0.0336$, and $e_{2,1}=0.0512$, all of which lie within their corresponding identified sets.

sidewaysfigure\caption{Identified set for demand shocks in a market with $(\tilde s, p)$ set at their median values, in a DGP with $(\alpha,\beta,\lambda) = (1,1,1)$ and small amount of missing information on $s_0$.}

Next, we describe the results with no information on $s_0$, i.e., $S_0 = (0,1)$. Given the results presented thus far, it is reasonable to expect much larger identified sets. For the sake of computational ease, we restricted the parameter space to be in

equation*[equation* omitted — 128 chars of source]

Our numerical results produce a much larger identified set than in the previous exercise. Interestingly, we can restrict the identified set for $\lambda$ to the interval $[0.4,4]$. While this set is admittedly very large, it rules out the possibility that the data are generated by a plain logit model (i.e., $\bar\lambda = 0$ is ruled out). On the other hand, our identified set is not able to rule out any value of $\alpha \in [0.5,2]$ or any value of $\beta \in [-3,6]$.

Inference without outside good shares

This section describes how to conduct inference for the BLP model when the outside-good share $s_0$ is unobserved. We conduct inference on both the structural parameters of the BLP model and the equilibrium objects implied by the model. Our approach builds directly on ideas from andrews/shi:2013 and chernozhukov/chetverikov/kato:2015.

To explain our inference strategy, we note that the identified set $\Theta_I(P)$ in Theorem (ref) is defined as an infinite collection of conditional moment inequalities. To see this, let's denote $W=( \tilde{s},x,p,z) $, $\mathcal{V}=\{v\in \mathbb{R}^{J}:\Vert v\Vert =1\} $, and

equation*[equation* omitted — 165 chars of source]

With this notation, it follows that $\Theta_I(P)$ can be equivalently expressed as a collection of parameters satisfying conditional moment inequalities:

equation*[equation* omitted — 154 chars of source]

Our next step is to convert the conditional moment inequalities in $\Theta_I(P)$ into a collection of unconditional moment inequalities, following the approach of andrews/shi:2013. To this end, let $\mathcal G$ denote a collection of nonnegative functions of the instrument $z$, referred to as {\it instrument functions}. We then define the set:

equation[equation omitted — 181 chars of source]

Since each $g\in\mathcal G$ is nonnegative, it follows immediately that $\Theta_I(P)\subseteq \Theta_I(P,\mathcal G)$. Thus, $\Theta_I(P,\mathcal G)$ constitutes an {\it outer identified set} for the parameter of interest. However, andrews/shi:2013 shows that for suitable choices of $\mathcal G$, this outer set coincides with the identified set, that is, $\Theta_I(P,\mathcal G)=\Theta_I(P)$; see andrews/shi:2013. Examples of such choices include collections of indicator functions over hypercubes or hyperrectangles in $S_z$. See Example (ref) for a description of the definition of the hypercubes. Motivated by these results, we choose to conduct inference on $\Theta_I(P,\mathcal G)$. By construction, this set provides an outer identified set for the parameter of interest, and coincides with $\Theta_I(P)$ under appropriate choices of $\mathcal G$.

example[Hypercubes in andrews/shi:2013] The class of instrument functions $\mathcal G_{\mathrm{cube}}$ based on hypercubes is constructed as follows. First, we transform the instrument vector $z=(z_1,\dots,z_{d_z})$ into the unit $d_z$-dimensional hypercube using a componentwise standard normal CDF: \begin{equation*} \tilde z = (\Phi(z_1),\dots,\Phi(z_{d_z})) \in [0,1]^{d_z}. \end{equation*} Second, we partition $[0,1]^{d_z}$ into regular hypercubes. For a given $r\in\mathbb N$, define \begin{equation*} C_{a,r} = \prod_{u=1}^{d_z}\Big(\frac{a_u-1}{2r},\frac{a_u}{2r}\Big], \qquad a=(a_1,\dots,a_{d_z}), \end{equation*} where $a_u\in\{1,2,\dots,2r\}$ for each $u=1,\dots,d_z$. Each $C_{a,r}$ is a hypercube in $[0,1]^{d_z}$ with side length $(2r)^{-1}$. For any tuning parameter $r_0\in\mathbb N$, the collection of hypercubes is defined by \begin{equation*} \mathcal C_{\mathrm{cube}} = \{ C_{a,r} : a_u\in\{1,\dots,2r\},\ u=1,\dots,d_z,\ r=r_0,r_0+1,\dots \}. \end{equation*} Finally, we define the corresponding class of instrument functions as \begin{equation*} \mathcal G_{\mathrm{cube}} = \{ g(z)=I\{\tilde z\in C\} : C\in\mathcal C_{\mathrm{cube}} \}. \end{equation*} Under the regularity conditions described in andrews/shi:2013, the collection of hypercubes $\mathcal C_{\mathrm{cube}}$ generates the Borel $\sigma$-algebra on $[0,1]^{d_z}$, and andrews/shi:2013 show that this choice of $\mathcal G_{\mathrm{cube}}$ is sufficiently rich so that $\Theta_I(P,\mathcal G_{\mathrm{cube}})=\Theta_I(P)$.

The identified set $\Theta _{I}(P,\mathcal{G})$ in (ref) is defined by a collection of moment inequalities indexed by $(v,g)\in \mathcal{V}\times \mathcal{G}$. Since $\mathcal{V}$ is uncountable and $\mathcal{G}$ typically contains countably infinitely many elements, $\Theta _{I}(P,\mathcal{G})$ is effectively characterized by an infinite number of moment inequalities.

To conduct inference, we rely on the methods for many moment inequalities developed in chernozhukov/chetverikov/kato:2015. Specifically, we consider a countable subset $\{(v_{j},g_{j})\}_{j=1}^{p_{n}}\subseteq \mathcal{V}\times \mathcal{G}$, where $p_{n}$ is allowed to grow with the sample size $n$, and conduct inference on the approximating identified set

equation[equation omitted — 181 chars of source]

For any $n\in \mathbb{N}$, $\Theta _{I}(P)\subseteq \Theta _{I}(P,\mathcal{G})\subseteq \tilde{\Theta }_{I}(P,p_{n})$. Hence, $\tilde{\Theta}_{I}(P,p_{n})$ constitutes an outer identified set for the parameter of interest.

The methodology proposed by chernozhukov/chetverikov/kato:2015 is directly applicable to $\tilde{\Theta}_{I}(P,p_{n})$, as it delivers asymptotically valid inference while allowing $p_{n}$ to grow with the sample size $n$ at relatively fast rates. For each $\theta \in \Theta$, the test statistic is given by

equation*[equation* omitted — 124 chars of source]

where, for each $j=1,\dots,p_{n}$,

align*[align* omitted — 238 chars of source]

Our confidence sets are given by

equation*[equation* omitted — 103 chars of source]

where $c_{n}(\theta,\pi)$ are critical values corresponding to a prespecified significance level $\pi\in(0,1)$. chernozhukov/chetverikov/kato:2015 propose several options for $c_{n}(\theta,\pi)$, including self-normalized critical values, bootstrap, and two-step hybrid methods. In terms of implementation, the simplest option is the self-normalized critical value, given by

equation[equation omitted — 135 chars of source]

where $p_{n}$ is the number of moment inequalities in (ref). See chernozhukov/chetverikov/kato:2015 for the other critical values.

Under suitable conditions, any of the inference methods proposed by chernozhukov/chetverikov/kato:2015 generate an asymptotically valid confidence set for the parameter of interest. More formally, for any significance level $\pi \in (0,1)$,

equation[equation omitted — 175 chars of source]

where $\mathcal{P}$ denotes a suitable set of data distributions.

To conduct inference on parameters derived from the model, such as subvectors of $\theta$ or elasticities, we use projections of the confidence sets.\footnote{One could employ more sophisticated inference methods for these parameters, such as those proposed by belloni/bugni/chernozhukov:2019. We nevertheless focus on projection-based inference to keep the exposition simple.} We record this result in Corollary (ref).

corollaryAssume that $C_{n}(1-\pi)$ is an asymptotically valid confidence set for the parameter of interest in the sense of (ref). Then, asymptotically valid confidence sets for the elasticities and markups can be obtained by projecting the parameter values in $C_{n}(1-\pi)$. In particular, the asymptotically valid confidence set for the elasticity of product $j=1,\dots,J$ in a market with observables $(\tilde{s},x,p)$ is given by: \begin{equation*} C_{\mathcal{E}_{jj}}(\tilde{s},x,p; 1-\pi) = \left\{ \begin{array}{c} e_{jj} = -p_{j}\tfrac{\int_{(\zeta ,\nu )}(\alpha +\nu ) \left(\begin{array}{c} \tfrac{\exp (\beta x_{j} -\alpha p_{j}+\xi _{j}+\zeta x_{j} -\nu p_{j})}{1+\sum_{b=1}^{J}\exp (\beta x_{b} -\alpha p_{b}+\xi _{b}+\zeta x_{b} -\nu p_{b})}\times \\ \left( 1-\tfrac{\exp (\beta x_{j} -\alpha p_{j}+\xi _{j}+\zeta x_{j} -\nu p_{j})}{1+\sum_{e=1}^{J}\exp (\beta x_{e} -\alpha p_{e}+\xi _{e}+\zeta x_{e} -\nu p_{e})} \right) \end{array}\right) f(\zeta ,\nu ;\lambda )d(\zeta ,\nu )}{\int_{(\zeta ,\nu )}\tfrac{ \exp \left( \beta x_{j}-\alpha p_{j}+\xi_{j}+ \zeta x_{j} -\nu p_{j} \right) }{1+\sum_{l=1}^{J} \exp \left( \beta x_{l}-\alpha p_{l}+\xi _{l}+\zeta x_{l} -\nu p_{j} \right) }f( \zeta ,\nu ;\lambda ) d(\zeta ,\nu )}:\\ \xi \in \mathcal{U}(\tilde{s},x,p;\theta ) and \theta \in C_{n}( 1-\pi ) \end{array} \right\}. \end{equation*} In turn, the asymptotically valid confidence set for the cross-price elasticity of product $j=1,\dots,J$ with respect to the price of product $k\neq j$ in a market with observables $(\tilde{s},x,p)$ is given by: \begin{equation*} C_{\mathcal{E}_{jk}}(\tilde{s},x,p; 1-\pi)= \left\{ \begin{array}{c} e_{jk} = p_{k}\tfrac{\int_{(\zeta ,\nu )}(\alpha +\nu ) \left(\begin{array}{c} \tfrac{\exp (\beta x_{j} -\alpha p_{j}+\xi _{j}+\zeta x_{j} -\nu p_{j})}{1+\sum_{b=1}^{J}\exp (\beta x_{b} -\alpha p_{b}+\xi _{b}+\zeta x_{b} -\nu p_{b})}\times \\ \tfrac{\exp (\beta x_{k} -\alpha p_{k}+\xi _{k}+\zeta x_{k} -\nu p_{k})}{1+\sum_{e=1}^{J}\exp (\beta x_{e} -\alpha p_{e}+\xi _{e}+\zeta x_{e} -\nu p_{e})} \end{array}\right) f(\zeta ,\nu ;\lambda )d(\zeta ,\nu )}{\int_{(\zeta ,\nu )}\tfrac{ \exp \left( \beta x_{j}-\alpha p_{j}+\xi_{j}+ \zeta x_{j} - \nu p_{j} \right) }{1+\sum_{l=1}^{J} \exp \left( \beta x_{l}-\alpha p_{l}+\xi _{l}+\zeta x_{l} -\nu p_{l} \right) }f( \zeta ,\nu ;\lambda ) d(\zeta ,\nu )}:\\ \xi \in \mathcal{U}(\tilde{s},x,p;\theta ) and \theta \in C_{n}( 1-\pi ) \end{array} \right\}. \end{equation*} Finally, the asymptotically valid confidence set for the markup of product $j=1,\dots,J$ in a market with observables $(\tilde{s},x,p)$ is given by: \begin{equation*} C_{M_{j}}(\tilde{s},x,p; 1-\pi) = \left\{ \begin{array}{c} M_{j} = \tfrac{ \int_{(\zeta ,\nu )}\tfrac{ \exp \left( \beta x_{j}-\alpha p_{j}+\xi_{j}+ \zeta x_{j} -\nu p_{j} \right) }{1+\sum_{l=1}^{J} \exp \left( \beta x_{l}-\alpha p_{l}+\xi _{l}+\zeta x_{l} -\nu p_{l} \right) }f( \zeta ,\nu ;\lambda ) d(\zeta ,\nu ) }{ \alpha \int_{(\zeta ,\nu )}(\alpha +\nu ) \left(\begin{array}{c} \tfrac{\exp (\beta x_{j} -\alpha p_{j}+\xi _{j}+\zeta x_{j} -\nu p_{j})}{1+\sum_{b=1}^{J}\exp (\beta x_{b} -\alpha p_{b}+\xi _{b}+\zeta x_{b} -\nu p_{b})}\times \\ \left( 1-\tfrac{\exp (\beta x_{j} -\alpha p_{j}+\xi _{j}+\zeta x_{j} -\nu p_{j})}{1+\sum_{e=1}^{J}\exp (\beta x_{e} -\alpha p_{e}+\xi _{e}+\zeta x_{e} -\nu p_{e})} \right) \end{array}\right) f(\zeta ,\nu ;\lambda )d(\zeta ,\nu ) }:\\ \xi \in \mathcal{U}(\tilde{s},x,p;\theta ) and \theta \in C_{n}( 1-\pi ) \end{array} \right\}. \end{equation*}

Empirical illustration

This section provides an empirical illustration using data from nevo:2000. This is artificial data generated from a model of demand and supply in the ready-to-eat cereal market. It comprises $J = 24$ products across 47 cities over 2 quarters, yielding a total of $M = 94$ markets. The demand model is derived from the following indirect utility specification:

align*[align* omitted — 179 chars of source]

where $D_{i}$ is a $d_{D} \times 1$ vector of demographic variables drawn from the Current Population Survey, including the log of income, the log of income squared, age, and a child indicator equal to one for individuals under sixteen. For each market, 20 individuals are drawn. $\Pi$ is a $(d_{X} + 1) \times d_{D}$ matrix of coefficients that captures how tastes vary with demographics, and $\Sigma$ is a $(d_{X}+1) \times (d_{X}+1)$ matrix that determines the variance-covariance structure of the random coefficients. The observable covariates $x_{j}$ include a constant, sugar content, and a dummy variable for mushy. Nevo’s specification also includes a full set of product fixed effects. The dataset provides $20$ excluded instruments.

We first estimate all parameters using standard BLP with observed $s_{0}$, imposing the restriction that $\Sigma$ is diagonal, i.e. $\Sigma = \operatorname{diag}(\lambda)$. In an unrestricted model that allows for correlation across unobserved tastes, the estimates of the correlation components of $\Sigma$ are economically small and statistically insignificant. We therefore abstract from such correlation.

We then introduce small amount of missing information on $s_{0}$, letting $S_{0} = (s_{0} - 0.05, s_{0} + 0.05) \cap (0,1)$, and compute the $90\%$ confidence set using the inference procedure described in Section (ref).

To reduce the dimensionality of our problem, we fix product fixed effects at their BLP estimates. This effectively treats the parameter space for the fixed effects as very narrow. We also fix the demographic interaction matrix $\Pi$ and the standard deviations of the random coefficients on Sugar and Mushy at their BLP estimates.\footnote{In Nevo's case, the estimates of $\lambda_{sugar}$ and $\lambda_{mushy}$ are small in magnitude and statistically insignificant. Specifically, $\hat{\lambda}_{sugar} = 0.004$ (S.E.$=0.012$) and $\hat{\lambda}_{mushy} = 0.081$ (S.E.$=0.205$).}

We therefore focus on a four-dimensional parameter vector $\theta = (\beta_{1}, \alpha, (\lambda_{1}, \lambda_{2}))$, where $\beta_{1}$ is the constant term, $\alpha$ is the price coefficient, $\lambda_{1}$ governs heterogeneity in mean utility relative to the outside option, and $\lambda_{2}$ governs heterogeneity in price sensitivity. The parameter space is restricted to

align*[align* omitted — 202 chars of source]

which is centered around the BLP estimates, with each half-width set to three times the standard error of the corresponding BLP estimate.

The class of instrument functions $\mathcal{G}_{cube}$ is constructed based on the excluded instruments provided in Nevo's data. Specifically, we randomly choose $10$ out of the $20$ available instruments and set $r_{0} = 1$. Building $\mathcal{G}_{cube}$ as in Example (ref) requires forming hypercubes of dimension $d_{z} = 10$. Given the small sample size ($M=94$), the average number of observations per cube is small ($3$ or less). Following the practical recommendation in andrews/shi:2013, we instead use all one- and two-dimensional combinations of the $10$ instruments. This yields a final instrument vector of dimension 39.

We implement all inference methods described in Section (ref), including self-normalized critical values, bootstrap, and two-step hybrid methods. All methods produce qualitatively similar confidence sets, with the self-normalized method producing wider confidence sets, as expected. For brevity, we report only the multiplier bootstrap results in Figures (ref) and (ref). The confidence set is four-dimensional. We project it onto the $(\alpha, \beta_{1}, \lambda_{j})$ subspace for $j=\{1,2\}$ and visualize it in the $(\alpha, \beta_{1})$ plane at different values of $\lambda_{j}$.

Projecting the confidence set onto each parameter coordinate yields parameter-specific confidence sets. We find that the confidence set is empty for values of $\alpha \not\in [23,39]$. In contrast, the projected confidence sets for $(\beta_{1},\lambda_{1},\lambda_{2})$ cover their respective parameter spaces.

The lower and upper bounds on $\alpha$ indicate that the data are informative about the price coefficient even when we have some missing information about the outside share. For a given value of $\lambda$, the set of $(\alpha, \beta_{1})$ implies that a higher distaste for price combined with a higher mean utility cannot be distinguished from a lower distaste for price combined with a lower mean utility. However, we cannot rule out any values of $(\lambda_{1}, \lambda_{2})$ within their parameter space, suggesting that these heterogeneity parameters are not well identified in this application. Note that all parameters governing the preference heterogeneity are imprecisely estimated even under standard BLP estimation, which suggests that the current empirical setup and data may have limited ability to deliver an informative confidence set.

sidewaysfigure\caption{Confidence set for $(\beta_{1},\alpha,\lambda_{1})$ and small amount of information on $s_0$ using Nevo's data.}

sidewaysfigure\caption{Confidence set for $(\beta_{1},\alpha,\lambda_{2})$ and small amount of information on $s_0$ using Nevo's data.}

Conclusions

This paper studies identification and inference in the BLP demand model when the share of the outside good is not point identified. We consider a framework in which the researcher specifies a set that contains the share of the outside good. We show that relaxing the standard assumption of a known outside share typically leads to partial identification, and we derive sharp identified sets for both structural parameters and economically relevant equilibrium objects. Our framework can be used to conduct sensitivity analysis by considering a set around the true value. It can also be used to conduct inference without prior information by specifying this set to be the unit interval.

We also develop inference procedures that construct uniformly valid confidence sets for these objects in this partially identified setting. Our results show that even when point identification fails, the model can still yield informative conclusions about demand and market outcomes. Overall, the framework provides a practical approach for conducting demand estimation and counterfactual analysis when market size is uncertain or only partially known.