A Multinomial Probit Model for Asymmetric Choice Responses

Technical Summary: A Multinomial Probit Model for Asymmetric Choice Responses

Problem Statement
Standard Multinomial Probit (MNP) models impose symmetric choice responses, implying that positive and negative shifts in covariates of equal magnitude affect choice probabilities symmetrically. This symmetry arises from the assumption of a multivariate normal distribution for latent utilities. However, in many empirical settings—such as pricing strategies or welfare evaluations—this assumption is implausible. Behavioral theories like Prospect Theory suggest agents react differently to gains and losses (loss aversion), leading to asymmetric responses. Existing literature often addresses this by modifying the observable utility component (e.g., splitting attributes into gains and losses) or using flexible predictor functions. Alternatively, some approaches introduce asymmetry via the error distribution but are often limited to binary choices or rely on restrictive assumptions (e.g., fixed scale matrices) that hinder the modeling of substitution patterns in multinomial settings. The authors argue that imposing asymmetry directly on the conditional distribution of latent utilities offers a parsimonious alternative that does not require reference points or complex predictor transformations.

Methodology
The paper proposes the Skewed Multinomial Probit (SMNP) model, which replaces the standard multivariate normal distribution of latent utilities with a multivariate skew-normal distribution.

• Model Specification: The model defines observed discrete choices $Y_i$ based on latent utilities $Z_i = X_i\beta + \varepsilon_i$, where $\varepsilon_i$ follows a multivariate skew-normal distribution $SN_J(0, \Sigma, \alpha)$. The skewness is governed by a shape parameter vector $\alpha$, which is reparameterized into a skewness parameter vector $\delta$ for interpretability. The model nests the standard MNP when $\delta = 0$.
• Identification Challenges: Introducing skewness creates identification issues due to the interaction between the skewness parameters and the scale normalization required in MNP models (typically fixing a diagonal element of the covariance matrix to one). The authors address this by developing a novel reparameterization of the implied covariance matrix $\Sigma_u$ (where $\Sigma = \Sigma_u + \delta\delta^\top$). This parameterization fixes the scale and ensures the covariance matrix remains positive definite by construction, specifically by constraining the first element of $\Sigma_u$ to $1 - \delta_1^2$.
• Bayesian Inference: Estimation is performed via a Bayesian framework using Markov Chain Monte Carlo (MCMC). Because the skew-normal distribution does not yield a conjugate posterior for latent utilities, the authors employ a double data augmentation scheme:
  1. Augmentation with latent utilities $Z$.
  2. Augmentation with auxiliary variables $w$ based on the conditionally Gaussian representation of the multivariate skew-normal distribution.
     This strategy transforms the problem into a conditionally Gaussian regression model. The resulting sampler is a Metropolis-Hastings within Gibbs algorithm. All parameters are updated via Gibbs steps except for the first skewness parameter $\delta_1$, which requires a univariate Metropolis-Hastings step to respect the constraint $|\delta_1| < 1$.

Key Contributions

1. Model Extension: The paper extends the MNP framework to allow for alternative-specific skewness, capturing asymmetric choice responses without altering the observable utility function or requiring reference points.
2. Identification and Estimation: It provides a fully identified parameterization that resolves the interaction between skewness and scale normalization, ensuring positive definiteness of the covariance matrix.
3. Computational Algorithm: It develops a tractable Bayesian inference method using double data augmentation, enabling joint inference over all model parameters (including the skewness vector and covariance matrix) in a multinomial setting, a gap left by recent work that assumed fixed scale matrices.

Results
The paper validates the SMNP model through numerical experiments and two empirical applications (laundry detergent and ketchup purchases).

• Numerical Experiments:
  • In data generated from a skewed distribution, the SMNP successfully recovers the asymmetric choice responses and significantly outperforms the standard MNP in predictive accuracy (measured by hit rate, log score, and censored likelihood scores).
  • In data generated from a symmetric distribution, the SMNP shrinks toward the standard MNP (posterior mass of skewness parameters centers at zero) without sacrificing predictive performance, demonstrating that the added flexibility does not incur a penalty when symmetry holds.
• Empirical Applications:
  • Laundry Detergent: The SMNP detects significant skewness for specific brands (e.g., EraPlus). This asymmetry leads to economically meaningful differences in price elasticities; the standard MNP underestimates price sensitivity and the magnitude of demand loss/gain for specific brands compared to the SMNP.
  • Ketchup: The model identifies asymmetry for the Del Monte brand. The SMNP alters the predicted substitution patterns, suggesting that the MNP may misallocate substitution across competing brands (e.g., misestimating the cross-price elasticity between Hunts and Del Monte vs. store brands).
  • In both cases, the SMNP improves probabilistic prediction metrics (log scores and censored likelihoods) relative to the MNP.

Significance and Claims
The authors claim that the SMNP model offers a "parsimonious way" to capture asymmetric choice responses, avoiding the misspecification risks associated with symmetric latent utilities. The model is presented as a practical tool for practitioners (firms and governments) to make better pricing and policy decisions by providing more accurate estimates of price elasticities and substitution patterns. The paper emphasizes that while the model captures the phenomenon of asymmetry, it does not inherently identify the specific behavioral mechanism (e.g., loss aversion vs. switching costs) generating it. The authors modestly suggest that future research could explore approximate Bayesian methods for larger choice sets or models with random coefficients, noting that the current exact sampler may face mixing challenges in those complex settings.