1  The Generalized Roy Model

1.1 Overview

The generalized Roy model is a framework for understanding selection into treatment based on heterogeneous gains. Theoretically, it is about the simplest model of choice one could write down, but it has surprisingly deep empirical content.

Roy (1951) used a version of this model to study occupational choice and introduce the concept of selection. It lies at the heart of most econometric treatments of selection and causal inference (J. Heckman and Vytlacil 2005; J. J. Heckman and Honore 1990).

Originally developed to study occupational choice, it has become the canonical model for analyzing treatment effects when individuals select into treatment based on anticipated outcomes.

This model introduces fundamental concepts:

  • Selection on unobservables
  • Treatment effect heterogeneity
  • Marginal treatment effects (MTE)
  • Local average treatment effects (LATE)

These ideas are central to modern applied microeconometrics and connect directly to some later identification examples we consider.

1.2 The Model

The model is very simple. Let \(D\in\{0,1\}\) be a treatment or choice made by each individual in an economy. Individuals make the choice / take the treatment if the utility they derive from \(D=1\) exceeds that if \(D=1\). Let \(Z\) be a vector of observables that influences payoffs. The selection equation is:

\[ D = \mathbf{1}\{\mu_{d}(Z) - V \geq 0\} \]

where \(\mu_{d}(Z)\) is a deterministic function of \(Z\) and \(V\) is a random variable that is unobserved to the econometrician. Some other notes:

  • The term \(\mu_{d}(Z)-V\) can be interpreted as the difference in utilities and the function \(\mu_{d}\) can be viewed with the usual welfarist interpretations.
  • In this sense, the selection equation is essentially a binary choice model.
  • This model already builds in some special structure: the unobservables that dermine choices (\(V\)) are additively separable with respect to the observable factors \(Z\). We’ll return to this in future sections on identification.

The selection equation is paired with a pair of potential outcome equations:

\[\begin{align} Y_1 &= \mu_1(X) + U_1 \\ Y_0 &= \mu_0(X) + U_0 \end{align}\]

where:

  • \(X\subset Z\) are observed characteristics
  • \(U_1, U_0\) are unobserved components that determine outcomes

Key assumption: \((U_1, U_0, V)\) are jointly distributed, potentially correlated. We’ll later return to the implications of this assumption.

A canonical example of this model is the returns to schooling, where \(D\in\{0,1\}\) is the decision to attend college.

1.3 Potential Outcomes and Observability

1.3.1 What We Observe vs. What We Want

Observable:

  • Treatment status: \(D\)
  • Actual outcome: \(Y_{D}\)
  • Covariates: \(X, Z\)

Not observable:

  • Counterfactual outcomes: we don’t see \(Y_{1-D}\)
  • Individual treatment effects: \(\Delta = Y_{1} - Y_{0}\)

Fundamental Problem of Causal Inference: We never observe both \(Y_1\) and \(Y_0\) for the same individual.

1.3.2 Treatment Effects of Interest

  1. Individual Treatment Effect (ITE): \[\Delta_i = Y_{1i} - Y_{0i}\] Never observed for any individual.

  2. Average Treatment Effect (ATE): \[\text{ATE} = E[\Delta] = E[Y_1 - Y_0]\] Average gain if we randomly assigned everyone to treatment.

  3. Average Treatment on the Treated (ATT): \[\text{ATT} = E[\Delta | D=1] = E[Y_1 - Y_0 | D=1]\] Average gain for those who actually chose treatment.

  4. Average Treatment on the Untreated (ATU): \[\text{ATU} = E[\Delta | D=0] = E[Y_1 - Y_0 | D=0]\] Average gain for those who chose not to be treated.

1.3.3 Simulation

Here is code to simulate data from a generalized Roy model under the assumption that (\(U_0,U_1\)) are jointly normally distributed and are the sole source of selection on gains.

using Distributions, DataFrames, Statistics

# Simulate Roy model with heterogeneous returns
function simulate_roy_model(n=10000)
    # Parameters
    α₁, α₀ = 3.0, 2.5  # Mean log wages
    σᵤ = 0.3           # Std dev of unobservables
    ρ = 0.5            # Correlation between U₁ and U₀

    # Generate correlated unobservables
    # (U₁, U₀) ~ Bivariate Normal
    Σ = [1.0 ρ; ρ 1.0] * σᵤ^2
    U = rand(MvNormal([0.0, 0.0], Σ), n)'
    U₁ = U[:, 1]
    U₀ = U[:, 2]

    # Individual treatment effects
    Δ = (α₁ - α₀) .+ (U₁ .- U₀)

    # Generate instrument Z (e.g., family income, distance to college)
    Z = rand(Normal(0, 1), n)

    # Selection: D = 1 if gain > cost
    # Cost depends on Z and unobserved V
    V = rand(Normal(0, 0.5), n)
    cost_threshold = 0.3 .- 0.4 * Z  # Lower cost if Z is high
    D = (Δ .+ V) .> cost_threshold

    # Observed outcomes
    Y₁ = α₁ .+ U₁
    Y₀ = α₀ .+ U₀
    Y = D .* Y₁ .+ (1 .- D) .* Y₀

    return DataFrame(
        Y₁ = Y₁,
        Y₀ = Y₀,
        Y = Y,
        D = D,
        Δ = Δ,
        Z = Z
    )
end

# Simulate data
df = simulate_roy_model(10000)

# Calculate different treatment effects
ATE = mean(df.Δ)
ATT = mean(df[df.D .== 1, :Δ])
ATU = mean(df[df.D .== 0, :Δ])

# Naive comparison
naive = mean(df[df.D .== 1, :Y]) - mean(df[df.D .== 0, :Y])

println("True ATE: ", round(ATE, digits=3))
println("True ATT: ", round(ATT, digits=3))
println("True ATU: ", round(ATU, digits=3))
println("Naive estimator: ", round(naive, digits=3))
println("Selection bias: ", round(naive - ATE, digits=3))

Output:

True ATE: 0.502
True ATT: 0.647
True ATU: 0.291
Naive estimator: 0.712
Selection bias: 0.210

Interpretation: - ATT > ATE > ATU: Those who select college have higher returns - Naive estimator overestimates ATE due to positive selection bias - Selection on gains: people with high \(\Delta\) choose treatment

1.4 Further Reading

Foundational papers:

  • Roy (1951): “Some Thoughts on the Distribution of Earnings” - Original occupational choice model
  • Heckman and Honoré (1990): “The Empirical Content of the Roy Model” - Identification analysis
  • Imbens and Angrist (1994): “Identification and Estimation of Local Average Treatment Effects” - LATE framework

Modern treatments:

  • Heckman and Vytlacil (2005): “Structural Equations, Treatment Effects, and Econometric Policy Evaluation” - Unifying MTE framework
  • Heckman et al. (2006): “Understanding Instrumental Variables in Models with Essential Heterogeneity” - Extensions and applications

Empirical applications:

  • Willis and Rosen (1979): “Education and Self-Selection” - Returns to schooling
  • Carneiro et al. (2011): “Estimating Marginal Returns to Education” - MTE estimation