Methods

This page walks through the full model in the order it is estimated, explaining both the intuition behind each choice and the computational methods used. The model follows Aakvik, Heckman, and Vytlacil (2005).

Step 1: Why a Roy Model?

Binary Roy Model

Model selection and outcomes jointly with a one-factor error structure.

✓ Corrects for observable and unobservable selection
✓ Allows children who gain more to also be more likely to be selected
✓ Identifies ATE, ATT, and MTE
✓ Natural for binary outcomes (aspiration is a target, not a scale)

The key additional insight of the Roy model over IV: children who are most likely to be selected into CI may also be those who benefit most from it. A motivated family that actively seeks sponsorship for their child may be exactly the type for whom the program works best. Standard IV treats this as a nuisance; the Roy model explicitly estimates it via the correlation between the selection error and the outcome errors.

Step 2: The Three-Equation System

The model has three latent variable equations.

(1) Selection Equation

Equation (1): Selection equation

\[\begin{aligned} S^*_i = \; &\gamma_0 + \gamma_1\,\text{Age6}_i + \gamma_2\,\text{Age7}_i + \gamma_3\,\text{Age8}_i \\ &+ \gamma_4\,\text{AssetIndex}_i + \gamma_5\,\text{Protestant}_i + \gamma_6\,\text{SiteCI}_i - U_{Si} \end{aligned}\] \[S_i = \mathbf{1}[S^*_i > 0]\]

\(S^*_i\) is the latent “propensity to be sponsored.” When it exceeds zero, the child is sponsored (\(S_i = 1\)). The vector \(Z_i\) contains all the covariates above; \(U_{Si}\) is an unobserved factor (motivation, family circumstances) that also drives selection.

Variables:

Regressor Role
Age6, Age7, Age8 Dummy = 1 if child was that age when CI arrived. Exclusion restrictions (see Step 3)
Asset index First principal component of household assets. Proxies for income.
Protestant Church attendance is positively correlated with CI access.
SiteCI Dummy = 1 if child lives in a community with a CI project.

(2) Outcome Equations

Equations (2)–(3): Potential outcome equations

\[Y^*_{1i} = \beta^1_0 + \rho_{HE,i}\,\beta^1_2 + \text{Dist}_i\,\beta^1_3 + \tilde{X}_i\,\beta^1_4 - U_{1i} \qquad (\text{sponsored})\] \[Y^*_{0i} = \beta^0_0 + \rho_{HE,i}\,\beta^0_2 + \text{Dist}_i\,\beta^0_3 + \tilde{X}_i\,\beta^0_4 - U_{0i} \qquad (\text{non-sponsored})\] \[Y_{ji} = \mathbf{1}[Y^*_{ji} > 0], \quad j \in \{0, 1\}\]

Each child has two potential outcomes: \(Y_{1i}\) (aspiration if sponsored) and \(Y_{0i}\) (aspiration if not). We only observe one of them, the one corresponding to actual sponsorship status:

\[Y_i = S_i \cdot Y_{1i} + (1 - S_i) \cdot Y_{0i}\]

Regressors in the outcome equations:

Variable Description
\(\rho_{HE,i}\) Perceived return to higher education (from subjective expectations). Only in Model 2.
\(\text{Dist}_i\) Distance in km to nearest university. Proxy for access constraints.
\(\tilde{X}_i\) Gender, asset index, parental education, Prospera participation dummy.
Note

Model 1 does not include \(\rho_{HE}\). Model 2 adds it, reducing the sample to 271 children who correctly interpreted the probability questions. Comparing the two lets us test whether subjective beliefs about returns shift aspirations.

Step 3: Identification: Exclusion Restrictions

For the Roy model to identify the causal effect of sponsorship, we need variables that affect whether a child is sponsored but do not directly affect their aspiration level. These are called exclusion restrictions.

The instrument: Age-at-arrival dummies (Age6, Age7, Age8).

A child who was 6, 7, or 8 years old when CI arrived in their village is significantly more likely to be sponsored than one who was 9 or older (the eligibility cutoff). The omitted category is age ≥ 9, too old to meet the eligibility criterion comfortably.

Validity argument: Current aspirations at ages 12–15 depend on current characteristics (income, parental education, access to schools, social environment), not on how old a child happened to be when a program arrived several years ago. Age-at-arrival shifts selection probability without having a direct channel into current aspiration.

The first-stage results confirm the instruments are strong: children who were 6, 7, or 8 years old at CI arrival are approximately 21–26 percentage points more likely to be sponsored than older children, all else equal.

Step 4: One-Factor Error Structure

The three error terms \(U_{Si}\), \(U_{1i}\), \(U_{0i}\) share a common latent factor \(\theta_i\):

One-factor error structure

\[U_{Si} = -\theta_i + \varepsilon_{Si}, \qquad U_{1i} = -\alpha_1\theta_i + \varepsilon_{1i}, \qquad U_{0i} = -\alpha_0\theta_i + \varepsilon_{0i}\]

Think of \(\theta_i\) as an unobserved characteristic of the child or family: motivation, perseverance, or the family’s belief in education. It affects all three equations.

What the factor loadings \(\alpha_1\) and \(\alpha_0\) capture:

  • \(\alpha_1 > 0\) means children with high \(\theta\) (high motivation) are both more likely to be selected and benefit more from CI when sponsored
  • \(\alpha_0 > 0\) means children with high \(\theta\) are also more ambitious even without CI
  • The difference \(\alpha_1 - \alpha_0\) measures selection on gains: whether those selected into CI are those who gain most from it

Under standard IV, we implicitly assume \(\alpha_1 = \alpha_0\), the treatment effect does not depend on unobserved characteristics. The Roy model relaxes this. If \(\alpha_1 \neq \alpha_0\), IV gives a biased estimate of ATT, and can even have the wrong sign.

Normalization: \(\text{Var}(\theta_i) = \text{Var}(\varepsilon_{Si}) = \text{Var}(\varepsilon_{1i}) = \text{Var}(\varepsilon_{0i}) = 1\). This pins down the scale and allows identification.

The covariances implied by the factor structure are: \[\text{Cov}(U_S, U_1) = \alpha_1, \qquad \text{Cov}(U_S, U_0) = \alpha_0, \qquad \text{Cov}(U_1, U_0) = \alpha_0\alpha_1\]

Step 5: Maximum Likelihood Estimation

Because \(\theta_i\) is unobserved, we cannot condition on it directly. Instead, we integrate it out when forming the likelihood.

The Likelihood Function

Conditional probabilities given \(\theta_i\)

\[\Pr(Y_i = 1 \mid S_i = 1, X_i, \theta_i) = \Phi(X_i\hat\beta_1 + \hat\alpha_1\theta_i)\] \[\Pr(Y_i = 1 \mid S_i = 0, X_i, \theta_i) = \Phi(X_i\hat\beta_0 + \hat\alpha_0\theta_i)\] \[\Pr(S_i = 1 \mid Z_i, \theta_i) = \Phi(Z_i\hat\gamma + \theta_i)\]

Multiplying across equations (conditional independence given \(\theta_i\)) and integrating over \(\theta_i \sim N(0,1)\):

Marginal likelihood

\[L = \prod_{i=1}^N \int \Pr(S_i, Y_i \mid X_i, Z_i, \theta_i)\,\phi(\theta_i)\,d\theta_i\]

This integral does not have a closed form, because \(\Phi(\cdot)\) is a nonlinear function of \(\theta_i\), we cannot analytically multiply out the normal density and integrate. We need a numerical method.

Gauss-Hermite Quadrature

Gauss-Hermite quadrature approximates integrals of the form \(\int f(\theta)\,\phi(\theta)\,d\theta\) with a weighted sum:

\[\int f(\theta)\,\phi(\theta)\,d\theta \;\approx\; \sum_{k=1}^{K} w_k \cdot f(\theta_k)\]

\(\theta_1,\ldots,\theta_K\) are quadrature nodes (roots of Hermite polynomials), chosen to span the support of the normal distribution efficiently. \(w_1,\ldots,w_K\) are weights that give each node its appropriate importance.

Intuition: Instead of integrating over infinitely many possible values of \(\theta\), we evaluate the likelihood at \(K = 10\) carefully selected points that together provide an excellent approximation to the normal distribution. With 10 nodes, the approximation is accurate to many decimal places for smooth integrands like ours.

The practical computation is:

\[L_i \approx \sum_{k=1}^{10} w_k \cdot \Pr(S_i, Y_i \mid X_i, Z_i, \theta_k)\]

Parameters \((\gamma, \beta_0, \beta_1, \alpha_0, \alpha_1)\) are estimated by maximizing \(\sum_i \ln L_i\).

An alternative is to draw random values of \(\theta\) from \(N(0,1)\) and average over them (Monte Carlo). Gauss-Hermite quadrature is preferred here because it is deterministic (no simulation noise), faster (10 nodes vs. hundreds of draws for equivalent accuracy), and reproducible. For a single factor with normal distribution, quadrature is essentially exact.

Standard Errors

Standard errors are computed by bootstrap (resampling with replacement at the child level).

Step 6: From Estimates to Treatment Effects

Once \((\hat\gamma, \hat\beta_0, \hat\beta_1, \hat\alpha_0, \hat\alpha_1)\) are estimated, three treatment parameters are computed.

Average Treatment Effect (ATE)

The expected effect of sponsorship for a randomly chosen child with characteristics \(x\):

Equation (4): Average Treatment Effect

\[ATE(x) = \Pr(Y_1 = 1 \mid X = x) - \Pr(Y_0 = 1 \mid X = x) = \Phi\!\left(\frac{x\hat\beta_1}{\sqrt{1 + \hat\alpha_1^2}}\right) - \Phi\!\left(\frac{x\hat\beta_0}{\sqrt{1 + \hat\alpha_0^2}}\right)\]

Average Treatment Effect on the Treated (ATT)

The expected effect for actually sponsored children:

Equation (5): Average Treatment Effect on the Treated

\[ATT(x,z) = \frac{1}{F_{U_S}(z\hat{\gamma})}\Bigl(F_{U_1,U_S}(x\hat{\beta}_1,\, z\hat{\gamma}) - F_{U_0,U_S}(x\hat{\beta}_0,\, z\hat{\gamma})\Bigr)\]

This conditions on \(S_i = 1\), so it averages only over the sponsored group. The bivariate normal distributions \(F_{U_S,U_1}\) and \(F_{U_S,U_0}\) are recoverable from the factor structure.

Marginal Treatment Effect (MTE)

The effect for children at the margin of selection, those who are just indifferent between being sponsored and not:

Equation (6): Marginal Treatment Effect

\[MTE(x, u_S) = \Pr(Y_1 = 1 \mid X = x,\, U_S = u_S) - \Pr(Y_0 = 1 \mid X = x,\, U_S = u_S)\]

\(u_S\) is the value of the selection error at the margin. For small values of \(u_S\) (the children most likely to be selected), the MTE gives the effect for children who would always be sponsored regardless of the instrument. For large \(u_S\), the MTE gives the effect for children who are very unlikely to be selected.

The MTE as a building block: Both ATE and ATT are weighted averages of the MTE curve. ATE uses equal weights across the \(u_S\) distribution; ATT over-weights low-\(u_S\) children (those who are likely to be selected). This is why ATE ≠ ATT when the MTE curve is not flat.

Practical computation of the MTE

\[MTE(x, u_S) = \frac{\int \left[\Phi(x\hat\beta_1 + \hat\alpha_1\theta) - \Phi(x\hat\beta_0 + \hat\alpha_0\theta)\right] \phi(u_S + \theta)\,\phi(\theta)\,d\theta}{\phi(u_S)/\sqrt{2}}\]

Again evaluated numerically using Gauss-Hermite quadrature.

Summary of Modeling Choices

Choice What it is Why it matters
Binary Roy model Jointly models selection and two potential outcomes Corrects for selection on both observables and unobservable gains
One-factor error structure Errors share a common latent factor \(\theta_i\) Parsimonious way to allow correlation across equations
Exclusion restriction Age-at-arrival dummies (Age6/7/8) Instrument that shifts P(sponsored) without directly affecting aspirations
MLE with Gauss-Hermite quadrature Numerically integrates out \(\theta\) Handles the non-closed-form integral efficiently and accurately
Bootstrap standard errors Resampling at the child level Robust to model complexity and small sample
Binary outcome Aspiration modeled as 0/1 Children have aspiration targets, not continuous ambition scales

The paper reports three additional robustness checks:

  1. Continuous Roy model using the same instruments — conclusions unchanged
  2. IV approach — conclusions unchanged (sponsorship effect not significant)
  3. Prospera subsample — restricts to children who benefit from Prospera in both groups; findings are stable

All three confirm that the main results are not driven by the binary specification, the specific exclusion restrictions, or the overlap with the Prospera program.