Heckman — Sample selection model (Heckit)

The Heckman model (Heckit) corrects sample selection bias — when whether an observation has an outcome value depends on factors related to that outcome. Example: we only observe wages of people who work; the working sample is non-random ⇒ OLS is biased.

When to use

Use Heckman when the outcome sample is endogenously selected (e.g. wage ↔ labor-force participation). You need an exclusion restriction: a variable that affects being selected but not the outcome directly.

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Two-equation structure

• Selection equation: $S_i = 1[Z_i \gamma + u_i > 0]$ (Probit).
• Outcome equation: $Y_i = X_i \beta + \rho \sigma_\varepsilon \, \lambda(Z_i \gamma) + \xi_i$, where $\lambda(\cdot)$ is the inverse Mills ratio (IMR).

A significant IMR coefficient ⇒ sample selection bias is present (and Heckman is warranted).

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Two estimation approaches

Approach	Description
Two-step (Heckit)	Step 1 Probit selection → compute IMR; step 2 OLS outcome with IMR
MLE	Estimate both equations jointly (more efficient)

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Running in EcoLab

1. Modeling module → Limited dependent variable family → Heckman.
2. Declare the outcome equation ($Y$, $X$) and the selection equation ($Z$, including the exclusion variable).
3. Choose two-step or MLE; run; read the IMR coefficient ($\rho$) to confirm bias; export the replication code.

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Replication code

Stata

* ===== Heckman Selection Model (Heckit) =====
* Two-step estimator
* Outcome eq.: lnwage = f(educ, exper)
* Selection eq.: working = f(married, kids)  ← exclusion restriction
heckman lnwage educ exper, select(working = married kids)

* MLE estimator (more efficient)
heckman lnwage educ exper, select(working = married kids) twostep

* Key output:
* - mills (lambda): significant → selection bias present
* - rho: correlation between selection and outcome errors
* - sigma: std. deviation of outcome error

R

# ===== Heckman Selection Model (Heckit) =====
library(sampleSelection)

# Two-step Heckit estimator
# Selection eq.: working ~ married + kids
# Outcome eq.:   lnwage ~ educ + exper
model <- heckit(
  selection = working ~ married + kids,
  outcome   = lnwage ~ educ + exper,
  data      = df,
  method    = "2step"
)

summary(model)

# MLE estimator
model_ml <- heckit(
  selection = working ~ married + kids,
  outcome   = lnwage ~ educ + exper,
  data      = df,
  method    = "ml"
)

summary(model_ml)

# Key: check the inverse Mills ratio (IMR) coefficient
# and rho (correlation between errors)

Python

# ===== Heckman Selection Model — Two-step manual =====
import numpy as np
import statsmodels.api as sm
from scipy.stats import norm

# Step 1: Probit selection equation
Z = sm.add_constant(df[["married", "kids"]])
sel_model = sm.Probit(df["working"], Z).fit()

# Compute the Inverse Mills Ratio (IMR / lambda)
Zg = sel_model.predict(Z)          # linear index P(working=1)
imr = norm.pdf(norm.ppf(Zg)) / Zg  # IMR = phi(Phi^{-1}(p)) / p

# Step 2: OLS outcome equation with IMR added
# Only for observations where working == 1
subset = df[df["working"] == 1].copy()
X = sm.add_constant(subset[["educ", "exper"]])
X["imr"] = imr[df["working"] == 1]

outcome_model = sm.OLS(subset["lnwage"], X).fit()
print(outcome_model.summary())

# Key: if the IMR coefficient is significant,
# sample selection bias is confirmed
print("IMR coefficient:", outcome_model.params["imr"])
print("IMR p-value:", outcome_model.pvalues["imr"])

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Limitations

• Heavily depends on a valid exclusion restriction; without it, the model is poorly identified (collinearity with the IMR).
• Sensitive to the bivariate-normal error assumption.

Video tutorial

Video Tutorial: Guide to running Heckman selection model in EcoLab

See also

• Probit · Tobit · Truncated · Catalog