Elrod and Keane 1995 - Factor-Analytic Probit Model Specification

Summary

Brand utilities for household $i$ at occasion $t$ are ${\tilde{u}}_{it}=\alpha +{\tilde{\mu }}_i+{\widetilde{ϵ}}_{it}$ with ${\tilde{\mu }}_i\sim N(0,\Sigma )$, $\Sigma =L{L}^{\prime }+\kappa I$. Brand intercepts are tied to the common factors by $\alpha =Lc$, so a new brand’s share is predicted from its factor positions alone. Setting $L=0$ gives the equisimilar probit - the structureless, IIA-like benchmark within the same family.

Overview

Probit models are random-utility models: consumers buy the brand with the highest utility on each occasion, and unobserved (to the analyst) utility components are normal. Unlike logit, a general covariance for the errors relaxes independence of irrelevant alternatives.

Main Content

Choice rule (Eq. 1)

$d_{ijt}=1$ iff ${\tilde{u}}_{ijt}=\max \{{\tilde{u}}_{i1t},\dots ,{\tilde{u}}_{iJt}\}$, for household $i$, brand $j\in \{\mathrm{1..}J\}$, occasion $t$.

Factor-analytic probit (Eqs. 2-5)

${\tilde{u}}_{it}=\alpha +{\tilde{\mu }}_i+{\widetilde{ϵ}}_{it}$; ${\widetilde{ϵ}}_{it}\sim N(0,\kappa I)$ iid over households and occasions; ${\tilde{\mu }}_i\sim N(0,\Sigma )$ with $\Sigma =L{L}^{\prime }+\kappa I$, where $L$ is a $J\times M$ lower-triangular loadings matrix on $M$ common factors, and $\kappa >0$ is a scalar unique (brand-specific) variance shared by all brands. The brand intercepts satisfy $\alpha =Lc$ with $c$ an $M$-vector (the “mean importance vector”).

Reading. $\alpha$ is the average utility of each brand; ${\tilde{\mu }}_i$ is the household’s deviation from the population mean; $\kappa$ equal across brands makes unique variance a property of the product class, so repositioned or new brands inherit it. Equivalently (Eq. 6): ${\tilde{u}}_{ijt}=l_j^{\prime }{\tilde{w}}_i+{\tilde{\eta }}_{ijt}+{\widetilde{ϵ}}_{ijt}$ with $l_j$ brand $j$‘s values on the $M$ attributes, ${\tilde{w}}_i\sim N(c,I)$ the household’s importance weights, and ${\tilde{\eta }}_{ijt}$ iid $N(0,\kappa )$ - so the model predicts the share of a new brand once its position on the common factors is specified.

Identification

The model (2)-(4) is under-identified: a common positive affine transformation of all ${\tilde{u}}_{ijt}$ leaves choice probabilities unchanged. By convention the estimation algorithm fixes the utility of the last brand $J$ at zero mean and unit variance (so ${\alpha }_J=0$ and the $J$-th row of $L$ is zero, with the remaining variance constraint absorbing the scale). Hence only $J-1$ intercepts and $\kappa$ plus the free elements of $L$ are estimated.

Equisimilar probit - the structureless case (Eq. 8)

$L=0$, so $\Sigma =\kappa I$: there are no common factors. This is the nearest probit model to IIA: knowledge of a consumer’s utility for any subset of brands says nothing about the other brands. Only the $J-1$ intercepts and $\kappa$ are estimated. $\kappa \to 0$ means no heterogeneity (nobody is loyal to any brand); $\kappa \to \infty$ means every household is absolutely loyal to one brand. It is not a special case of market structure analysis, as brand intercepts cannot be a linear function of factors (this would force equal shares).

Dirichlet-multinomial benchmark (Eq. 7). The structureless stochastic brand choice model (Bass, Jeuland & Wright 1976) has $E(p_k\mid p_j)/E(p_{k^{\prime }}\mid p_j)=E(p_k\mid d_{ij})/E(p_{k^{\prime }}\mid d_{ij})=E(p_k)/E(p_{k^{\prime }})$: information about a household’s purchase of brand $j$ tells nothing about its relative preferences for other brands (IIA in the aggregate and for the individual). Elrod & Winer (1991) found it predicted best among their competing models.

Examples

Substitution under a one-factor probit versus IIA

Setup. $J=4$ brands, one factor with loadings $0.8\times (1,1,-1,-1)$, $\kappa =1$, equal intercepts $\alpha =0$. Solution. Simulate $u=\mu +ϵ$ with $\mu \sim N(0,L{L}^{\prime }+\kappa I)$; shares are 0.25 each. Remove brand 0 and re-simulate: shares become $(0.394,0.302,0.304)$ for brands 1-3, versus the IIA (logit-style) proportional prediction $(0.333,0.333,0.333)$. Interpretation. Brand 1, which loads like the removed brand, picks up 39% of the market rather than 33%: market structure appears as asymmetric substitution.

import numpy as np
rng = np.random.default_rng(0)
J = 4
L = 0.8*np.array([[1.], [1.], [-1.], [-1.]])
Sigma = L @ L.T + 1.0*np.eye(J)
u = rng.multivariate_normal(np.zeros(J), Sigma, size=200_000)
print(np.bincount(u.argmax(1), minlength=J)/len(u))          # [0.25 .. ]
u3 = rng.multivariate_normal(np.zeros(3), Sigma[1:, 1:], size=200_000)
print(np.bincount(u3.argmax(1), minlength=3)/len(u3))         # ~[0.394 0.302 0.304]

Connections

• Probit factor covariance vs nested logit and mixed logit: Market Structure Analysis - IIA, Nested Logit and Choice Maps (Concept Note), Random Coefficients Logit Model.
• Probit as random utility model: Discrete Choice Models.

See Also

Elrod and Keane 1995 - Simulated Moments and Panel Estimation