Heckman Selection Model

Motivating Example: Wages and Female Labor Supply#

A labor economist wants to estimate the returns to education for married women. She has a large survey with information on education, age, number of children, husband's income, and — for women who are employed — their hourly wage.

Here is the problem: wages are only observed for women who work. Of the 753 women in her sample, only 428 are employed. The remaining 325 women have missing wages — not because of data errors, but because they chose not to participate in the labor market.

If she simply runs OLS on the 428 working women, she estimates wages conditional on employment. But employment itself is a choice that depends on potential wages. Women with very low potential wages may choose not to work (because the opportunity cost of leisure or home production exceeds their market wage). By restricting the sample to employed women, the researcher is selecting on an outcome that is correlated with the error term in the wage equation.

This problem is sample selection bias. It is not the same as omitted variable bias or measurement error — it arises because the sample is non-randomly drawn from the population. The observed wage distribution is truncated from below: women with the lowest potential wages are disproportionately absent from the sample.

The Heckman selection model (Heckman, 1979) solves this problem. It jointly models two processes: (1) the selection equation — whether a woman works — and (2) the outcome equation — what she earns if she works. By estimating the correlation between the errors in these two equations, the model corrects for the non-random selection into the observed sample. The key correction term is the inverse Mills ratio, which captures the expected value of the error in the wage equation conditional on the woman being selected into the sample.

This approach contributed to James Heckman's 2000 Nobel Prize in Economics (awarded for his broader development of theory and methods for analyzing selective samples) and remains one of the most widely used corrections for sample selection bias across the social sciences (Mroz, 1987).

What the Heckman Model Does#

The Heckman selection model corrects for bias that arises when the outcome variable is observed only for a non-random subset of the population. It does so by modeling the selection process explicitly and incorporating information about selection into the outcome equation.

The model consists of two equations:

Selection equation (who is observed):

where $D_i = 1$ if the outcome is observed (e.g., the woman works) and $D_i = 0$ otherwise. $Z_i$ is a vector of covariates that affect selection, and $u_i$ is the error term.

Outcome equation (what is the outcome, conditional on being observed):

where $Y_i$ is the outcome of interest (e.g., wages), $X_i$ is a vector of covariates, and $\varepsilon_i$ is the error term.

The key assumption is that the errors $(u_i, \varepsilon_i)$ are jointly normally distributed:

where $\rho$ is the correlation between the selection and outcome errors, and $\sigma$ is the standard deviation of $\varepsilon_i$. The variance of $u_i$ is normalized to 1 (as in a standard probit model).

The Selection Bias Problem#

If $\rho \neq 0$, then $E[\varepsilon_i | D_i = 1] \neq 0$ — the expected error in the wage equation is non-zero for the selected sample. Specifically:

The term $\lambda(Z_i^{\prime}\gamma) = \frac{\phi(Z_i^{\prime}\gamma)}{\Phi(Z_i^{\prime}\gamma)}$ is the inverse Mills ratio, where $\phi(\cdot)$ is the standard normal PDF and $\Phi(\cdot)$ is the standard normal CDF. Running OLS on the selected sample omits this term, producing biased estimates of $\beta$ whenever $\rho \neq 0$.

Two Estimation Approaches#

Heckman Two-Step (Heckit):

1. Estimate the selection equation as a probit: $P(D_i = 1 | Z_i) = \Phi(Z_i^{\prime}\gamma)$
2. Compute the inverse Mills ratio: $\hat{\lambda}_i = \phi(Z_i^{\prime}\hat{\gamma}) / \Phi(Z_i^{\prime}\hat{\gamma})$
3. Include $\hat{\lambda}_i$ as an additional regressor in the outcome equation and estimate by OLS:

$Y_i = X_i^{\prime}\beta + \rho\sigma \cdot \hat{\lambda}_i + \text{error}$

The coefficient on $\hat{\lambda}_i$ estimates $\rho\sigma$. Standard errors must be corrected because $\hat{\lambda}_i$ is a generated regressor (the standard OLS standard errors are too small).

Full Information Maximum Likelihood (FIML):

Estimate both equations simultaneously by maximizing the joint likelihood of the observed data. FIML is more efficient than the two-step estimator when the joint normality assumption holds, but it requires this assumption for the entire joint distribution, not just the conditional mean. With a weak exclusion restriction, both methods are fragile — MLE leverages the functional form more aggressively, which helps only if normality is correct (Puhani, 2000).

When to Use the Heckman Model#

• Your outcome is observed only for a selected subsample and you believe the selection is non-random
• You have a credible exclusion restriction — a variable that affects selection but not the outcome
• Joint normality of the error terms is a reasonable approximation
• You want to estimate causal effects or population-level relationships, not just conditional associations for the selected sample

When NOT to Use the Heckman Model#

• Your missing data are missing at random (MAR) — standard imputation or inverse probability weighting may suffice
• You have no credible exclusion restriction — without one, the model is identified only through functional form (normality), which is fragile (Bushway et al., 2007)
• The normality assumption is substantially violated (e.g., the outcome distribution is heavily skewed, bounded, or multimodal) — consider semiparametric alternatives or Lee bounds
• You have a randomized experiment with attrition — Lee bounds provide a more robust approach

Common Confusions#

When (Not) to Use This Method#

Use the Heckman Model When:

1. Your outcome is observed only for a non-randomly selected subset. Wages for workers, test scores for students who did not drop out, firm performance for firms that survived, analyst forecasts for covered firms. The key question is: would including the unobserved outcomes change your conclusions?

2. You have a credible exclusion restriction. A variable that plausibly affects selection but not the outcome. Without one, the model relies on functional form assumptions that are typically indefensible.

3. Joint normality is a reasonable approximation. If the outcome variable is continuous, approximately symmetric, and not heavily skewed or bounded, normality is more plausible.

4. You want to recover population-level parameters. If you only care about the effect of education on wages for women who work, OLS on the selected sample is fine. But if you want the effect of education on potential wages for all women (workers and non-workers), you need the selection correction.

Do NOT Use the Heckman Model When:

1. Your missing data are missing at random (MAR). If missingness depends only on observed covariates, standard methods (multiple imputation, inverse probability weighting) are sufficient and do not require the normality assumption.

2. You have no exclusion restriction and cannot defend one. Without an exclusion restriction, you are relying entirely on the nonlinearity of the normal CDF for identification. Certo et al. (2016) find that many management papers apply Heckman without a credible exclusion restriction, rendering the correction unreliable.

3. The normality assumption is doubtful. If the outcome variable is heavily skewed (e.g., firm value, patent counts) or bounded (e.g., proportions, ratings on a fixed scale), the joint normality assumption is suspect. Consider semiparametric alternatives or Lee bounds.

4. You have a randomized experiment with differential attrition. Lee bounds provide a nonparametric approach that does not require normality or an exclusion restriction. They give bounds on the treatment effect rather than a point estimate, but the bounds are valid under weaker assumptions.

5. Selection is on observables only. If you believe you have observed all the variables that drive selection, matching or inverse probability weighting can address selection without parametric distributional assumptions.

Connection to Other Methods#

The Heckman model relates to several other methods for handling selection and endogeneity:

• Logit/Probit: the selection equation in the Heckman model is a probit. If you are only interested in modeling the selection decision itself (e.g., labor force participation), a standalone probit is sufficient. The Heckman model adds the outcome equation and the correlation between the two.

• OLS: the outcome equation estimated on the selected sample is OLS with selection bias. The Heckman model adds the inverse Mills ratio to correct this bias. If $\rho = 0$ (no selection bias), the Heckman model reduces to OLS on the selected sample.

• IV/2SLS: conceptually similar in that both require an exclusion restriction. In IV, the instrument affects the endogenous regressor but not the outcome directly. In Heckman, the excluded variable affects selection but not the outcome. The Heckman two-step can be viewed as a control function approach — adding a correction term rather than instrumenting.

• Lee bounds: a nonparametric alternative that bounds the treatment effect under weaker assumptions (monotonicity of selection, no functional form). Lee bounds are popular in program evaluation when normality is questionable. The trade-off is that you get bounds rather than a point estimate.

• Matching: addresses selection on observables. Matching assumes that, conditional on observed covariates, selection is as good as random. The Heckman model addresses selection on unobservables — the case where selection depends on factors correlated with the outcome error term even after conditioning on observables.

• Control function approach: the Heckman two-step is a special case of the control function approach. In the general control function framework, the correction term can take forms other than the inverse Mills ratio (e.g., for non-normal error distributions). Rivers and Vuong (1988) extends this to simultaneous equations with endogenous regressors.

For the Heckman model to provide valid correction, two key conditions must hold.

Condition 1: Joint Normality#

Plain language: The unobserved factors affecting selection and the unobserved factors affecting the outcome must follow a bivariate normal distribution.

Formally: $(u_i, \varepsilon_i) \sim N(0, \Sigma)$ where $\Sigma = \begin{pmatrix} 1 & \rho\sigma \\ \rho\sigma & \sigma^2 \end{pmatrix}$.

This assumption is crucial because the inverse Mills ratio correction is derived from the properties of the bivariate normal distribution. If the errors are not jointly normal, the functional form of the correction term $\lambda(\cdot)$ is wrong, and the bias correction may itself be biased.

Condition 2: Exclusion Restriction#

Plain language: At least one variable must appear in the selection equation but not in the outcome equation. This variable affects whether the outcome is observed but does not directly affect the outcome itself.

Formally: There exists a variable $Z_k$ such that $\gamma_k \neq 0$ (it affects selection) but $Z_k$ does not appear in $X_i$ (it is excluded from the outcome equation).

Why it matters: Without an exclusion restriction, the Heckman model is identified only through the nonlinearity of the inverse Mills ratio — which comes from the normality assumption. If normality is even slightly wrong, this "identification through functional form" can produce wildly inaccurate corrections. Bushway et al. (2007) demonstrate that without a credible exclusion restriction, the Heckman correction provides no improvement over naive OLS and may increase bias.

Examples of exclusion restrictions in practice:

Adjust the selection correlation to see how naive OLS on the selected sample diverges from the true effect. When selection is strong, OLS is badly biased; the Heckman correction uses the inverse Mills ratio to recover the true coefficient.

Watch how non-random selection into the labor force distorts the observed wage-education relationship:

Derivation of the Inverse Mills Ratio Correction#

Heckman Selection Model with Diagnostics#

1# Requires: sampleSelection
2# sampleSelection: R package for Heckman selection models (Toomet & Henningsen)
3library(sampleSelection)
4
5# --- Step 1: Prepare the data ---
6# Outcome: log(wage), observed ONLY for working women (selected sample)
7# Selection: whether the woman works (lfp = 1)
8# Exclusion restriction: nwifeinc (non-wife income), kidslt6 (kids under 6)
9# These affect labor force participation but should not directly affect wage rate
10
11# --- Step 2: Heckman two-step estimator ---
12# heckit() estimates the selection and outcome equations jointly
13# selection: probit for participation (first stage)
14# outcome: OLS for wages, corrected by the inverse Mills ratio (second stage)
15heck_2step <- heckit(
16selection = lfp ~ age + I(age^2) + educ + nwifeinc + kidslt6 + kidsge6,
17outcome   = log(wage) ~ educ + exper + I(exper^2) + city,
18data = df,
19method = "2step"
20)
21summary(heck_2step)
22# Coefficient on lambda (inverse Mills ratio) estimates rho * sigma
23# If lambda is significant, this is consistent with selection bias under the model
24
25# --- Step 3: Full information MLE ---
26# MLE estimates the full joint model under bivariate normality
27# More efficient than two-step but more sensitive to normality violations
28heck_mle <- heckit(
29selection = lfp ~ age + I(age^2) + educ + nwifeinc + kidslt6 + kidsge6,
30outcome   = log(wage) ~ educ + exper + I(exper^2) + city,
31data = df,
32method = "ml"
33)
34summary(heck_mle)
35# MLE directly estimates rho and sigma (not just rho*sigma)
36# Compare two-step and MLE: large discrepancies suggest normality problems
37
38# --- Step 4: Diagnostics ---
39# Test H0: rho = 0 (no selection bias)
40# If not rejected, OLS on the selected sample is consistent
41
42# Compare with naive OLS (ignoring selection)
43ols_naive <- lm(log(wage) ~ educ + exper + I(exper^2) + city,
44              data = df[df$lfp == 1, ])
45summary(ols_naive)
46# If Heckman and OLS coefficients differ substantially, selection bias
47# is economically meaningful and the correction is needed

F.1 Significance of the Inverse Mills Ratio (Lambda)#

The most basic diagnostic: is $\hat{\lambda}$ (the coefficient on the inverse Mills ratio) statistically significant? Under the null hypothesis $H_0: \rho = 0$, there is no selection bias, and OLS on the selected sample is consistent. If $\hat{\lambda}$ is insignificant, either (a) there is genuinely no selection bias, or (b) the exclusion restriction is too weak to detect it.

• In the two-step estimator, test whether the coefficient on $\hat{\lambda}_i$ is significant (use corrected standard errors)
• In MLE, test $H_0: \rho = 0$ using the Wald test on $\hat{\rho}$ or a likelihood ratio test comparing the joint model to separate probit + OLS

F.2 Normality Tests#

Since the correction relies on joint normality, assess whether this assumption is plausible:

1. Residual normality: test the outcome equation residuals for normality (Shapiro-Wilk, Jarque-Bera, Q-Q plot). This checks marginal normality of $\varepsilon_i$, which is necessary but not sufficient for joint normality.

2. Polynomial Mills ratio test: add $\hat{\lambda}_i^2$ and $\hat{\lambda}_i^3$ to the outcome equation. If these higher-order terms are significant, the linear Mills ratio correction is insufficient — evidence against normality. The polynomial Mills ratio test is a specification test for the functional form of the correction.

3. Compare two-step and MLE: large discrepancies between the two estimators suggest normality violations (MLE relies more heavily on normality than the two-step estimator).

F.3 Exclusion Restriction Strength#

A weak exclusion restriction produces imprecise estimates and makes the model fragile:

1. Joint significance test: test whether the excluded variables are jointly significant in the probit selection equation ($\chi^2$ test). An insignificant exclusion restriction means the model is unidentified without functional form.

2. Magnitude of the first-stage coefficients: the excluded variables should have economically meaningful effects on selection, not just statistical significance.

3. Collinearity check: verify that the inverse Mills ratio is not highly collinear with the covariates in the outcome equation. High collinearity (variance inflation factor (VIF) > 10 for $\hat{\lambda}$) indicates that the model cannot distinguish the selection correction from the direct effects of the covariates.

F.4 Two-Step vs. MLE Comparison#

Estimate the model using both methods and compare:

• Similar results: reassuring. Both methods are estimating the same parameters, and the normality assumption is likely adequate.
• Different results: cause for concern. Possible explanations: (1) normality is violated (MLE is more sensitive), (2) the exclusion restriction is weak (MLE leverages different information than two-step), (3) the sample size is too small for MLE to converge properly.

1library(sampleSelection)
2
3# Fit two-step and MLE
4heck_2s <- heckit(
5selection = lfp ~ age + I(age^2) + educ + nwifeinc + kidslt6 + kidsge6,
6outcome   = log(wage) ~ educ + exper + I(exper^2) + city,
7data = df, method = "2step"
8)
9heck_ml <- heckit(
10selection = lfp ~ age + I(age^2) + educ + nwifeinc + kidslt6 + kidsge6,
11outcome   = log(wage) ~ educ + exper + I(exper^2) + city,
12data = df, method = "ml"
13)
14
15# F.1 Test significance of lambda (rho * sigma)
16summary(heck_2s)  # Check p-value on "Inverse Mills Ratio"
17
18# F.2 Polynomial Mills ratio test for normality
19# Compute Mills ratio from first-stage probit
20probit_fit <- glm(lfp ~ age + I(age^2) + educ + nwifeinc +
21                  kidslt6 + kidsge6,
22                family = binomial(link = "probit"), data = df)
23Zgamma <- predict(probit_fit, type = "link")
24mills <- dnorm(Zgamma) / pnorm(Zgamma)
25
26# Add squared and cubed Mills ratio to outcome equation
27selected <- df[df$lfp == 1, ]
28mills_sel <- mills[df$lfp == 1]
29selected$mills  <- mills_sel
30selected$mills2 <- mills_sel^2
31selected$mills3 <- mills_sel^3
32
33norm_test <- lm(log(wage) ~ educ + exper + I(exper^2) + city +
34                mills + mills2 + mills3,
35              data = selected)
36# Joint test of mills2 and mills3
37library(car)
38linearHypothesis(norm_test, c("mills2 = 0", "mills3 = 0"))
39
40# F.3 Exclusion restriction strength
41anova(
42glm(lfp ~ age + I(age^2) + educ + kidsge6,
43    family = binomial(link = "probit"), data = df),
44probit_fit, test = "Chisq"
45)
46
47# F.4 Compare two-step and MLE
48cbind(TwoStep = coef(heck_2s, part = "outcome"),
49    MLE = coef(heck_ml, part = "outcome"))

F.5 Interpreting Your Results#

What Rho Tells You

The parameter $\rho$ — the correlation between the selection and outcome errors — is the key parameter that drives the selection correction.

What Lambda ($\hat{\lambda}$ Coefficient) Captures

The coefficient on the inverse Mills ratio in the two-step estimator is $\hat{\rho}\hat{\sigma}$, sometimes denoted $\hat{\lambda}$ or $\hat{\delta}$. Its magnitude and significance determine:

1. Significance of $\hat{\lambda}$: if significant, this result is consistent with correlation between the selection and outcome disturbances under the maintained model — but significance alone does not prove selection bias is present (Certo et al., 2016). If insignificant, either selection bias is absent or the model lacks power to detect it (weak exclusion restriction).

2. Magnitude of $\hat{\lambda}$: the product $\rho\sigma$ determines the size of the bias correction. A large $|\hat{\lambda}|$ coefficient means the selection correction substantially changes the outcome equation coefficients.

3. Sign of $\hat{\lambda}$: reveals the direction of selection. Negative $\hat{\lambda}$ means negative selection ($\rho < 0$) — those who select in have lower unobserved outcome potential than a random draw from the population, so OLS on the selected sample underestimates the population mean.

What to Report in a Table

A well-reported Heckman model should include:

1. Both equations: report the full selection equation (probit coefficients) and the outcome equation (OLS coefficients with Mills ratio correction)
2. Lambda ($\rho\sigma$): the coefficient on the inverse Mills ratio, with its standard error and p-value
3. Rho ($\hat{\rho}$) and sigma ($\hat{\sigma}$): if using MLE, report these separately
4. Number of observations: total N, number selected (observed outcome), number not selected
5. Exclusion restriction: clearly identify which variables appear in the selection equation but not the outcome equation
6. Exclusion restriction strength: chi-squared test of the excluded variables in the selection equation
7. Estimation method: two-step or MLE
8. Naive OLS comparison: show how the coefficients change when selection is ignored

H.1 Concept Checks#

H.2 Guided Exercise#

H.3 Error Detective#

H.4 You Are the Referee#

• Lee bounds: when normality is doubtful or no credible exclusion restriction exists. Lee bounds require only a monotonicity assumption (treatment does not make anyone less likely to be selected). You get an interval rather than a point estimate, but it is valid under weaker assumptions. Recommended as a robustness check alongside the Heckman model.

• Control function approach: a generalization of the Heckman two-step. Instead of assuming joint normality (which implies the correction term is the inverse Mills ratio), you can use nonparametric or semiparametric estimates of the control function. This relaxes the distributional assumption at the cost of requiring a stronger exclusion restriction and a larger sample.

• Semiparametric selection models: methods by Powell (1987), Newey (1999), and others estimate the selection correction without assuming normality. These use kernel or series estimators for the conditional expectation of the error. They require larger samples and are computationally more complex.

• Inverse probability weighting (IPW): weight each observation by the inverse of its probability of being selected (estimated from the selection equation). Unlike Heckman, IPW does not require normality and can accommodate nonlinear outcome models. However, IPW only addresses selection on observables — it cannot correct for selection on unobservables.

• Bivariate probit: when both the selection and the outcome are binary. The Heckman model assumes a continuous outcome; bivariate probit handles two binary equations with correlated errors. Uses the same exclusion restriction logic.

• Bounds approaches (Manski bounds): when you want the weakest possible assumptions. Manski worst-case bounds assume nothing about the selection process and provide very wide intervals. Lee bounds tighten these by adding a monotonicity assumption.