Selection Model

Definition

A selection model is a regression framework in which the sample or treatment assignment is not random but depends on variables correlated with the outcome. The canonical problem: unit $i$ is observed in state $j$ partly because of unobserved characteristics that also affect the outcome, so naive ordinary least squares (OLS) regression of outcome on state confounds the causal effect of state with the selection effect of those unobserved characteristics. Selection models correct for this by jointly modeling the assignment mechanism and the outcome, linking the two via shared unobservables and exploiting an instrument (a variable that affects assignment but not the outcome directly) for identification.

Key Ideas

Sample selection (Heckman 1979): The simplest case — outcome $y_i^*$ is only observed when a selection indicator $s_i=1$ ; if $\text{corr}(\varepsilon_i^s, \varepsilon_i^y) \neq 0$ then OLS on the selected sample is biased. The Heckman two-step estimator uses the inverse Mills ratio $\lambda(x_i'\hat\gamma)$ as a control function.
Endogenous treatment (Heckman 1978): When treatment is a discrete choice — e.g., which hospital a patient attends — and the choice depends on unobservables that also affect outcomes, the treatment variable is endogenous. Classical instrumental variables (IV) corrects this in linear models; selection models handle binary and multinomial outcomes.
Local average treatment effect (LATE): Angrist-Imbens-Rubin (1996) show that IV with a binary instrument identifies the average treatment effect (ATE) for compliers (units whose treatment status changes with the instrument), not the population ATE. The Bayesian simultaneous-equation approach in Geweke-Gowrisankaran-Town (2003) instead estimates structural quality parameters directly under a full distributional model.
Identification: Requires an exclusion restriction — a variable $z_i$ that shifts the assignment probability but has zero direct effect on the outcome. In health applications, patient-hospital distance is a natural instrument (affects which hospital is chosen; does not directly affect survival conditional on medical condition).
Bayesian treatment: Rather than two-step estimation, the full joint likelihood over latent utilities, latent outcomes, and structural parameters is specified; Markov Chain Monte Carlo (MCMC) with data augmentation draws from the joint posterior. This propagates uncertainty about the selection correction through to outcome parameter estimates, unlike classical two-step methods.

How It Works

Simultaneous-Equation Selection Model

The Geweke-Gowrisankaran-Town (2003) formulation for hospital selection and mortality:

Choice equation (multinomial probit): Patient $i$ chooses hospital $h$ if $y_{ih}^* = x_i'\gamma_h + \delta_h\varepsilon_i + \eta_{ih} > \max_{h'\neq h} y_{ih'}^*$ where $\varepsilon_i$ is the patient-specific mortality shock shared with the outcome equation and $\eta_{ih}$ are independent choice disturbances.

Outcome equation (binary probit): Mortality occurs if $m_i^* = x_i'\beta + q_{s(i)} + \varepsilon_i > 0$ where $q_j$ is the unobserved quality of hospital $j$ and $\varepsilon_i \sim \mathcal{N}(0,1)$ .

The vector $\delta = (\delta_1,\dots,\delta_J)'$ captures how the unobserved severity $\varepsilon_i$ loads onto each hospital's choice utility. If $\delta_j > 0$ , sicker patients (high $\varepsilon_i$ ) are more likely to choose hospital $j$ , so naive mortality rates at $j$ are inflated relative to quality $q_j$ .

Markov Chain Monte Carlo (MCMC) with Data Augmentation

Draw latent mortality utilities $m_i^*$ from truncated normals (Albert-Chib 1993b).
Draw latent hospital choice utilities $y_{ih}^*$ from truncated normals (Geweke-Keane-Runkle 1994, 1997).
Draw mortality shocks $\varepsilon_i$ from their full conditional (normal, given both sets of latent variables).
Draw hospital quality indicators $q_j$ from conjugate normal conditionals.
Draw severity loadings $\delta_j$ and structural coefficients $(\beta, \gamma_h)$ from their posteriors.
Draw hierarchical hyperparameters (quality group means and variances) from conjugate posteriors.

Why It Matters

Identification of causal effects: When units self-select into treatment (hospitals, schools, programs) based on unobservables, naive comparisons confound selection with treatment effects. Selection models provide a principled framework for disentangling the two.
Health economics: Hospital quality rankings from raw mortality rates or even risk-adjusted rates can be reversed once selection is accounted for. In Geweke-Gowrisankaran-Town (2003), the selection component of between-hospital mortality variance is 8 $\times$ larger than the independent component, making uncorrected rankings nearly uninformative about true quality.
Policy relevance: Public report cards, pay-for-performance schemes, and hospital closures based on unadjusted or poorly adjusted mortality rates can penalize high-quality hospitals that attract high-risk patients and reward low-quality hospitals that turn away the sick.
General principle: Selection bias applies whenever treatment assignment is endogenous — school choice, program participation, firm entry, occupational sorting. The Bayesian approach to selection models extends naturally to any setting where the outcome and assignment both involve latent Gaussian structures.

Open Questions

Validity of exclusion restrictions is untestable within the model — distance instruments assume that proximity affects quality choice but not survival conditional on treatment. Empirical plausibility depends on domain knowledge.
Weak instruments inflate posterior uncertainty but do not necessarily bias posterior means in the Bayesian framework the way they inflate IV standard errors in classical settings; however, posterior estimates remain sensitive to instrument relevance.
Extensions to dynamic settings (repeated hospitalizations, competing risks) and to continuous outcomes require further development of the data-augmentation strategies.