Discrete Choice Models

Definition

A discrete choice model describes a decision-maker's selection from a finite set of mutually exclusive alternatives $\{1, \ldots, J\}$. The random utility model (McFadden 1974) specifies utility $U_{ij} = V_{ij} + \varepsilon_{ij}$ for alternative $j$, where $V_{ij}$ is the systematic component (function of observed attributes and individual characteristics) and $\varepsilon_{ij}$ is an unobserved idiosyncratic shock. The decision-maker chooses $j^* = \arg\max_j U_{ij}$.

Key Ideas

• Logit model: if $\varepsilon_{ij} \overset{\text{iid}}{\sim}$ Type-I Extreme Value, $\Pr(j^*=j) = \exp(V_{ij}) / \sum_k \exp(V_{ik})$ (McFadden 1973). Computationally tractable but exhibits the Independence of Irrelevant Alternatives (IIA) property.
• Probit model: if $(\varepsilon_{i1},\ldots,\varepsilon_{iJ}) \sim \mathcal{N}(0,\Sigma)$, the choice probability is a multivariate normal integral; IIA is avoided when $\Sigma$ has off-diagonal elements. See Multinomial Probit.
• Mixed logit / random-coefficient logit: allows parameters $\beta$ to vary randomly across individuals from a mixing distribution $p(\beta|\tau)$; nests logit and is IIA-free. Estimation by simulation (Geweke-Hajivassiliou-Keane (GHK) simulator) or Hierarchical Bayes (HB).
• IIA problem: in standard logit, the ratio of probabilities for any two alternatives depends only on those two alternatives, not on the rest of the choice set. This generates counter-intuitive predictions when alternatives are close substitutes.

How It Works

Estimation typically proceeds by maximum likelihood (probit/logit) or Markov Chain Monte Carlo (MCMC) for hierarchical Bayes specifications. Individual-specific parameters are recovered from the posterior $p(\beta_i | y_i, \tau)$ via Gibbs sampling.

Heterogeneity vs. State Dependence (Keane 1997)

In panel choice data (repeated purchases by the same consumer), brand choices are strongly persistent: buying A last period predicts buying A this period. This persistence has two observationally similar but behaviorally opposite explanations. Heterogeneity — stable unobserved preferences: some consumers just like A, so the correlation across their own choices is spurious state dependence. True state dependence — history changes preferences: buying A raises the utility of A next time (habit, loyalty, switching costs). Telling them apart is Heckman's (1981) classic identification problem, and it matters because only true state dependence implies that a one-off nudge (e.g., a promotion) has lasting effects. Keane (1997) models both simultaneously — a rich error structure (random effects + serial correlation) for heterogeneity plus a lagged-choice term for state dependence — in a panel multinomial probit estimated by method of simulated moments with the GHK simulator. On Nielsen ketchup scanner data he finds genuine state dependence survives controlling for rich heterogeneity, but the simulated long-run effect of a promotion-induced purchase on future choice is positive and small.

Why It Matters

Discrete choice models are the workhorse of applied microeconomics and marketing, enabling estimation of willingness to pay, elasticities, and counterfactual market shares from observed choices.

Related

• Multinomial Probit
• Binary Probit
• Conjoint Analysis
• Nonhomothetic Preferences
• Simulation-Based Estimation
• Count-Data Regression
• Consumer Heterogeneity
• GHK Simulator
• Keane (1997) Modeling Heterogeneity and State Dependence in Consumer Choice Behavior
• Michael P. Keane
• Amemiya (1975) Qualitative Response Models
• Takeshi Amemiya