Estimation of random coefficients logit demand models with interactive fixed effects

Abstract: We extend the Berry, Levinsohn and Pakes (BLP, 1995) random coefficients discrete-choice demand model, which underlies much recent empirical work in IO. We add interactive fixed effects in the form of a factor structure on the unobserved product characteristics. The interactive fixed effects can be arbitrarily correlated with the observed product characteristics (including price), which accommodates endogeneity and, at the same time, captures strong persistence in market shares across products and markets. We propose a two-step least squares-minimum distance (LS-MD) procedure to calculate the estimator. Our estimator is easy to compute, and Monte Carlo simulations show that it performs well. We consider an empirical illustration to US automobile demand.

• The paper introduces a two-step LS-MD estimator that incorporates interactive fixed effects into the random coefficients logit model to mitigate endogeneity issues.
• It demonstrates through simulations that including latent factors reduces bias, accurately recovers structural parameters, and improves estimator efficiency.
• The empirical analysis on the U.S. automobile market shows that modeling persistent unobserved heterogeneity leads to more responsive and robust elasticity estimates.

Estimating Random Coefficients Logit Demand Models with Interactive Fixed Effects

Overview and Motivation

This paper introduces a significant extension to the canonical Berry, Levinsohn, and Pakes (BLP, 1995) framework for discrete-choice demand estimation by incorporating interactive fixed effects into the model's unobserved product characteristics. The factor structure on unobserved heterogeneity allows for richer forms of correlation than product- or market-specific additive fixed effects, accommodating persistent unobserved shocks that are both product and market specific. This generalization is particularly relevant for empirical Industrial Organization, where endogeneity and strong unobservable correlation structures often hinder identification and proper inference in demand systems.

The estimation strategy leverages recent advances in panel data econometrics, offering a tractable and computationally efficient two-step procedure termed least squares-minimum distance (LS-MD). The method aims to address issues of endogeneity stemming from unobserved heterogeneity that is potentially correlated with observed product characteristics, especially price. The practical relevance is underscored by both simulation evidence and an empirical application to the U.S. automobile market.

Model Specification

The authors generalize the standard random coefficients logit model by specifying the unobserved product characteristic $\xi_{jt}$ as a factor model:

$\xi_{jt} = \lambda_j' f_t + e_{jt}$

where $\lambda_j$ are product-specific factor loadings, $f_t$ are market- or time-specific latent factors, and $e_{jt}$ is an idiosyncratic error. The core innovation is allowing both $\lambda_j$ and $f_t$ to be arbitrarily correlated with observed product characteristics $X_{jt}$, including price, thus capturing strong and potentially endogenous unobserved persistence in market shares.

The market share equations remain in the BLP form, with consumer $i$’s utility being conditional on observed and unobserved characteristics and random taste shocks. The model reduces to BLP when $R=0$ (no factors), but for $\xi_{jt} = \lambda_j' f_t + e_{jt}$0 the factor structure can absorb correlations typically ascribed to endogeneity and serial correlation in unobservables.

Identification Analysis

Identifying the structural parameters in the presence of interactive fixed effects and random coefficients is challenging due to the nonlinear structure and the incidental parameter problem when both the number of products $\xi_{jt} = \lambda_j' f_t + e_{jt}$1 and markets $\xi_{jt} = \lambda_j' f_t + e_{jt}$2 grow. The authors demonstrate identification of the parameters of interest $\xi_{jt} = \lambda_j' f_t + e_{jt}$3—which govern the distribution of random coefficients and the coefficients on observed characteristics—as well as the product $\xi_{jt} = \lambda_j' f_t + e_{jt}$4 (since $\xi_{jt} = \lambda_j' f_t + e_{jt}$5 and $\xi_{jt} = \lambda_j' f_t + e_{jt}$6 are not separately identified absent further restrictions).

Crucially, the identification argument highlights that while interactive fixed effects may "soak up" a large part of the endogeneity between, for instance, price and demand shocks, identification of the nonlinear random coefficient distribution still necessitates suitable instruments. The authors formalize identification conditions, ensuring that the explanatory power of instruments and exogenous regressors exceeds that of any potential set of (misspecified) factor loadings.

Estimation Methodology: LS-MD Procedure

The LS-MD estimator is a two-step procedure:

1. Step 1 (Least Squares): For a fixed value of the random coefficient parameter $\xi_{jt} = \lambda_j' f_t + e_{jt}$7, project mean utility onto observed regressors, interactive fixed effects, and instrument variables using least squares.
2. Step 2 (Minimum Distance): Minimize the norm of the coefficients on the instruments from Step 1 over candidate $\xi_{jt} = \lambda_j' f_t + e_{jt}$8 values, effectively implementing the instrument exclusion restriction in a minimum distance framework.

The solution for $\xi_{jt} = \lambda_j' f_t + e_{jt}$9 and $\lambda_j$0 at each iteration is provided by principal components, paralleling approaches in the linear factor models literature. The method remains tractable computationally, as the principal components have closed forms and the key nonlinearities are encountered only in the lower-dimensional step focused on $\lambda_j$1.

Extensions are provided for cases with endogenous regressors (e.g., price), by shifting the estimation of coefficients on endogenous variables into the second step and using extra instruments. In all cases, the procedure generalizes the GMM approach employed in BLP to accommodate interactive fixed effects.

Asymptotic Properties and Bias Correction

Establishing consistency and deriving the asymptotic distribution requires both $\lambda_j$2 and $\lambda_j$3 to grow large. The authors demonstrate that the LS-MD estimator is consistent and asymptotically normal under standard regularity conditions, with a limit distribution characterized by terms analogous to the GMM variance and additional bias terms arising from the estimation of high-dimensional incidental parameters (the factors and loadings). Explicit formulas for these bias and variance components are provided, allowing for analytical bias correction.

Notably, the bias vanishes if regressors are strictly exogenous and errors are homoscedastic, indicating that the incidental parameter problem can be effectively managed with suitable modeling and estimation choices.

Simulation Evidence

Monte Carlo simulations consider data-generating processes mirroring the empirical challenges of demand estimation (notably, one where a latent factor induces correlation between price and demand shocks). The simulations demonstrate:

• Substantial bias in estimating model parameters when the interactive fixed effects (factors) are omitted ($\lambda_j$4), even when the number of products and markets is moderate.
• Accurate and unbiased recovery of structural parameters when the number of factors matches or exceeds the true number in the data-generating process.
• The standard errors of the estimators decline at the rate $\lambda_j$5, validating the asymptotic theory.

Overestimating the number of factors yields negligible deterioration in efficiency, while under-specification induces bias and inflated standard errors.

Instrument relevance is investigated both analytically and numerically. The authors show that, even when exogeneity conditions are only approximately satisfied (as is typically the case in practice), suitably chosen nonlinear functions of the regressors (e.g., squared price) as instruments perform robustly in identification.

Empirical Application

Applying the LS-MD estimator to U.S. automobile market data, the authors contrast models with and without interactive fixed effects and price endogeneity. The findings are striking:

• Inclusion of interactive fixed effects (factors) renders the estimates robust to the method used to handle price endogeneity. The point estimates and standard errors are similar whether price is treated as exogenous or instrumented, suggesting that the factors capture persistent heterogeneity correlated with both price and demand.
• Models without factors exhibit attenuated coefficients on price and product characteristics and generally imprecise estimates, with some counterintuitive signs.
• Elasticity estimates derived under the factor model are quantitatively larger (in magnitude) than those from the standard BLP estimator—implying greater demand responsiveness when persistent, unobserved product-market heterogeneity is correctly accounted for.

This empirical exercise underscores the importance of modeling complex unobserved structure in demand estimation, especially in settings susceptible to high-dimensional endogeneity.

Implications and Future Research

The integration of interactive fixed effects with random coefficients logit models offers several practical and theoretical advantages:

• Instrumental variable requirements are relaxed: Endogeneity between observed characteristics and unobserved heterogeneity, a persistent issue in demand estimation, can be captured within the factor structure; when endogeneity remains, the exclusion restriction need only hold relative to the residual error.
• Broader modeling potential: The approach extends beyond discrete choice demand, with direct applicability to dynamic models, auction models, and other nonlinear settings with high-dimensional unobserved heterogeneity.
• Methodological advancements: The identification and estimation arguments formalize how high-dimensional nonlinear factor structures can be incorporated—previously an open challenge in econometric theory.

Future work will likely focus on:

• Model selection for the number of factors and formal inference procedures given possible cross-sectional heteroskedasticity and dependence,
• Extensions to richer dynamic models and alternative nonlinear panel specifications,
• Broadening the class of valid instruments or leveraging flexible machine learning approaches for instrument construction.

Conclusion

The paper offers an integrated framework for discrete choice demand systems incorporating interactive fixed effects using a tractable LS-MD estimator. Accounting for high-dimensional unobserved heterogeneity via a factor structure significantly reduces endogeneity concerns, sharpens identification, and improves empirical inference—addressing long-standing issues in applied demand analysis. The combination of theoretical rigor and empirical feasibility positions this approach as a robust tool for modern empirical IO and related fields.