Latent Propensity Scores Expose Entry Bias in Demand Estimation

Demand estimation for differentiated products usually observes only products that firms choose to offer. That choice is not random: firms enter markets where they expect demand, costs, or competitive conditions to make entry profitable, and some of that information is hidden from the econometrician. Aguirregabiria, Iaria, and Sokullu study the resulting selection problem in oligopoly settings where multiple products may enter or exit together. Their central point is that the usual scalar propensity score can lose the latent market information that makes entry decisions correlated across firms.

Core Contribution

The paper’s new object is a selection-bias representation based on latent propensity scores rather than only the observed, ordinary propensity score. The ordinary score is $\bar P_j(x_t)=\Pr(a_{jt}=1\mid x_t)$, the probability that product $j$ is offered conditional on observables. The selected demand equation contains a bias term $\lambda_j(x_t)=E(\xi_{jt}\mid x_t,a_{jt}=1)$, and the authors show why simply controlling for $\bar P_j(x_t)$ is generally insufficient in oligopoly entry.

The mechanism is economic, not cosmetic. A firm’s entry rule depends on expected profits, which depend on its own demand shock, rivals’ shocks, variable costs, fixed costs, and a common latent market state. Averaging over that latent state can collapse too much information into one scalar. The paper gives a two-product example in which the ordinary propensity score fails because a rival’s demand shock changes the entrant’s profitability threshold; the conditional selection term is then not a function of the ordinary score alone.

Technical Approach

The authors derive a mixture representation in which entry probabilities are conditioned on both observables and a latent market state, $P_j(x_t,\kappa_t)=\Pr(a_{jt}=1\mid x_t,\kappa_t)$. In the finite-mixture implementation, the latent state takes $L$ types. The selection correction then uses type-specific propensity scores and type-specific conditional means, rather than forcing selection into one scalar control.

Estimation is deliberately separated from a full joint solution of demand, pricing, and entry. The first step estimates the entry model with a semiparametric finite mixture. In the airline application, the entry model includes market size, distance, own hub size, competitors’ hub sizes, airline-quarter indicators, and airport indicators; the mixture logit is estimated by maximum likelihood using EM. The second step estimates demand with correction terms constructed from the fitted latent propensity scores, while price endogeneity is handled with BLP-style excluded instruments based on characteristics of other potential products.

This is the practical contribution: richer information at entry is allowed without repeatedly solving the full oligopoly model during demand estimation. The trade-off is that the correction now depends on the quality of the first-step mixture approximation and on exclusion restrictions that separate demand shifters, price instruments, and entry-cost determinants.

Results and Analysis

The empirical setting is U.S. airline routes in 2012–2013. A demand market is a directional airport pair in a quarter; an entry market is the corresponding non-directional airport pair in that quarter. The sample retains 2,652 non-directional airport pairs and 20,859 entry markets, then excludes 1,315 markets with no modeled potential entrant, leaving 19,544 markets for entry estimation. Market concentration is high: Table 2 reports that 35.19% of retained markets have no modeled airline, 44.87% have one airline, and 13.86% have two. Among markets served by nonstop flights, the authors state that more than 90% are monopolies or duopolies.

The first-step evidence supports latent heterogeneity in entry. In Table 4, moving from a probit without latent types to a two-type mixture logit lowers the first-step BIC from 31,749.6 to 30,069.1 and raises the log-likelihood from -13,493.6 to -11,047.8. Adding a third type raises the likelihood again to -10,002.2 but worsens the first-step BIC to 31,189.1; four types perform worse on that criterion at 32,933.4 and become imprecise. The type probabilities are economically sized, not negligible: 0.55 and 0.45 for $L=2$, and 0.38, 0.34, and 0.28 for $L=3$.

Figure 1 shows the downstream implication for estimated own-price elasticities: the selection-corrected specifications shift the airline-level distributions relative to 2SLS without entry correction, with the finite-mixture correction reported separately from a continuous-$\kappa$ correction.

The appendix figures show the same comparison for cross-price elasticities and marginal costs. Their role is diagnostic: the paper’s main empirical claim is not only that demand slopes change, but that the correction changes the implied competitive conduct objects built from those slopes. The authors interpret the corrected estimates as more elastic demand and less market power than estimates that ignore endogenous entry. That interpretation is credible within the model, especially given the entry-model fit, but it remains an application-based result rather than a design with known ground truth.

Caveats in Practice

The method weakens some supply-side commitments by avoiding full joint estimation, but it does not eliminate structure. It assumes the latent market-state mixture is a good enough approximation to the information driving entry, and the application must choose the number of latent types. The airline implementation also relies on instruments and exclusion restrictions: rival hub-share variables shift expected variable profits but are excluded from airline fixed costs, while price endogeneity is handled separately. Those are plausible in the paper’s setting, but they are the points a practitioner would need to defend before using the estimator elsewhere.