Notes on Control Function Method

Motivation

• Endogeneity: When an explanatory variable (treatment) is correlated with the unobserved error term (e.g., due to omitted variables, measurement error, or simultaneity).
• Consequence: Standard regression (e.g., OLS) yields biased estimates.
• Goal: The control function (CF) approach “purges” endogeneity by modeling the correlation between the treatment and unobservables.

The control-function (CF) approach tackles this by explicitly modelling the source of endogeneity and then partialling it out. For linear models that modelling step turns out to be algebraically equivalent to 2SLS, but the real power of CF is that it extends seamlessly to nonlinear or limited‐dependent‐variable settings where 2SLS cannot be applied directly.

How It Works

Outcome model: $Y={\beta }_0+{\beta }_{1}T+{\beta }_{2}X+U$

• $T$: Endogenous treatment
• $X$: Exogenous controls
• $U$: Unobservables (correlated with $T$).

First-stage model: $T={\gamma }_0+{\gamma }_{1}Z+{\gamma }_{2}X+V$

• $Z$: Instrumental variable (IV)
• $V$: First-stage error.

CF Insight:

If $U$ and $V$ are correlated (i.e., $\text{Cov}(U,V)\ne 0$), we can decompose $U$ into:
$$U=\rho V+ϵ$$ where $ϵ$ is uncorrelated with $T$ and $V$ (by construction).

Estimation

1. First Stage:

   • Regress $T$ on $Z$ and $X$:
     $$\hat{T}={\hat{\gamma }}_0+{\hat{\gamma }}_{1}Z+{\hat{\gamma }}_{2}X$$
   • Obtain the residual: $\hat{V}=T-\hat{T}$.

2. Second Stage:

   • Add $\hat{V}$ to the outcome model:
     $$Y={\beta }_0+{\beta }_{1}T+{\beta }_{2}X+\rho \hat{V}+ϵ$$
   • Estimate via OLS.

Why this works:

• $\hat{V}$ “controls for” the part of $U$ correlated with $T$.
• Once $\hat{V}$ is included, $T$ becomes exogenous in the modified model ($\text{Cov}(T,ϵ)=0$).
• ${\hat{\beta }}_1$ is consistent for the causal effect.

Key Assumptions

1. Instrument Validity:

   • $Z$ is relevant: $\text{Cov}(Z,T)\ne 0$ (strong first stage).
   • $Z$ is exogenous: $\text{Cov}(Z,U)=0$.

2. Correct Functional Form:

   • Linearity in the first stage and control function (e.g., $U=\rho V+ϵ$).

3. Exclusion Restriction: $Z$ affects $Y$ only through $T$.

CF vs. Other Methods

Advantage of CF:

• Directly models the endogeneity structure (via $\hat{V}$).
• Extends to non-additive errors, discrete outcomes, and heteroscedastic settings.

Example

Problem: Estimate returns to education ($T$) on wages ($Y$), where ability ($U$) is unobserved and correlated with education.

• IV: Distance to college ($Z$).

• Steps:

  1. Regress: education $\sim$ distance + controls → get residuals $\hat{V}$.

  2. Regress: wages $\sim$ education + controls + $\hat{V}$.

• Result: Coefficient on education is causal.

Generalization

The principle is identical: obtain a residual that captures unobserved heterogeneity driving $T$, then include it (or a flexible transformation) in the structural equation.

Why This Matters

• CF is essential when:

  • You have a valid IV and suspect omitted variable bias.
  • You work with nonlinear models (e.g., binary/duration outcomes).

• Software Implementation:

  • Stata: ivregress 2sls (equivalent to CF in linear cases) or cmp for nonlinear.
  • R: ivreg (linear), controlfunction package.

Critical Caveats

• Weak Instruments: If $Z$ is weak, $\hat{V}$ is noisy → bias.
• Functional Form Misspecification: If $U\ne \rho V+ϵ$, CF fails.
• No Magic Bullet: Validity of $Z$ is untestable and must be justified theoretically.

Bottom Line

The CF approach harnesses IV residuals to “control” for endogeneity, converting an endogenous variable into a conditionally exogenous one. It’s a blend of IV intuition and regression control — powerful when assumptions hold.