Common mistake 1 with Heckman selection

No exclusion restriction. If the selection and outcome equations contain the same variables, treat any correction with deep suspicion however good the p-values look.

Common mistake 2 with Heckman selection

Quoting the two-step standard errors. They are wrong in a known direction.

Common mistake 3 with Heckman selection

Believing the correction more than the assumption behind it. Heckman trades a selection problem for a normality assumption; whether that is a good trade is a matter of argument, not of output.

Common mistake 4 with Heckman selection

Applying the correction when the outcome is missing for reasons unrelated to the outcome. If selection really is ignorable, Heckman adds collinearity and noise for no gain, and the inverse Mills ratio is close to a linear function of the other regressors.

EconStatsCraft › Heckman selection

Heckman selection

For an outcome you only observe for a self-selected group - a wage only for people who work. Reports the two-step estimator so you can see the mechanism, then full maximum likelihood for inference, and tests whether selection was distorting anything at all.

What it needs

Outcome (observed for some rows only)
Outcome equation regressors
Observed indicator (1 = outcome seen) - Must be 0/1.
Selection equation regressors - Include at least one variable NOT in the outcome equation.

The intuition

You observe wages only for people who work, and people work partly because their wage would be high. So the wages you see are not a random sample of the wages that exist, and a regression on them answers a question about workers rather than about people. Heckman's insight is that if you can model who ends up in the sample, the expected error among those selected becomes a computable quantity - and once it is computable it can be put in the regression as a variable.

When to use it

The outcome is missing for a systematic, behavioural reason rather than at random, and you can name a variable that shifts whether it is observed without belonging in the outcome equation.

Where the estimator comes from

Two equations: selection d_i=\mathbf{1}[z_i'\gamma+u_i>0] and outcome y_i=x_i'\beta+\varepsilon_i, observed only when d_i=1, with (u,\varepsilon) jointly normal and \mathrm{Corr}(u,\varepsilon)=\rho. Then E[y_i\mid d_i=1]=x_i'\beta+\rho\sigma_\varepsilon\lambda(z_i'\gamma),\qquad \lambda(\cdot)=\frac{\phi(\cdot)}{\Phi(\cdot)}, the inverse Mills ratio. OLS on the selected sample omits that term, and because \lambda depends on z it is an omitted variable correlated with the regressors whenever x and z overlap. The two-step estimator fits a probit for selection, constructs \hat{\lambda}_i, and adds it to the outcome regression; its coefficient estimates \rho\sigma_\varepsilon, so testing it is a test of whether selection bias exists at all.

The model

E[y\mid \mathbf{x},\text{observed}]=\mathbf{x}^{\prime}\boldsymbol{\beta}+\rho\sigma\,\lambda(\mathbf{z}^{\prime}\boldsymbol{\gamma}),\qquad \lambda(a)=\frac{\phi(a)}{\Phi(a)}

What it assumes

Joint normality of the two errors. This is the load-bearing assumption and it is not testable from the data you have.
An exclusion restriction: at least one variable in the selection equation that does not belong in the outcome equation. Without it the correction rests entirely on the curvature of the inverse Mills ratio and is close to unidentified.
The selection indicator is measured without error.
The exclusion restriction is VALID, not merely present. The excluded variable has to move selection while having no direct effect on the outcome, and that is an economic argument rather than something the data can check - a variable can be highly significant in the selection equation and still belong in the outcome equation.

Statistical properties

Identification without an exclusion restriction rests entirely on the nonlinearity of the inverse Mills ratio, which is nearly linear over much of its range. That makes lambda nearly collinear with x and the estimates wildly unstable.
The exclusion restriction is the real requirement: a variable in the selection equation that is credibly absent from the outcome equation. Number of young children affects labour force participation and plausibly not the offered wage - and finding such a variable is as hard as finding an instrument.
The two-step standard errors must account for lambda being estimated. Naive OLS standard errors on the second step are wrong, and full maximum likelihood is more efficient when the normality assumption holds.
The whole thing hinges on joint normality. Unlike OLS, where normality only affects small-sample inference, here it determines the FORM of the correction term, so misspecification makes the estimator inconsistent.
A significant coefficient on lambda is evidence of selection; an insignificant one means OLS on the selected sample was adequate, which is a legitimate and common result worth reporting.
Selection on unobservables is what this addresses. Selection on observables is handled by controlling for them or by matching, and needs none of this machinery.

How to read the output

Read the test of rho = 0 first. If rho is indistinguishable from zero there was no detectable selection problem, and plain OLS on the observed rows was fine. Saying so is a finding, not a failure.
Use the maximum-likelihood table for inference. The two-step standard errors ignore that the inverse Mills ratio was itself estimated, so they are too small.
Compare the three columns. If uncorrected OLS sits well away from the two corrected estimates, selection was biasing your answer.
The coefficient on the inverse Mills ratio is rho times sigma, so its t-statistic is the selection test itself. A large coefficient with a wide standard error means the correction is making a substantial adjustment on thin evidence.

A worked reading

A wage equation on working women. OLS on the observed wages gives a return to schooling that is biased in a direction that depends on the correlation between the participation error and the wage error - if the women who work are those with unusually high unobserved earning capacity, rho is positive and OLS overstates the constant while the slope bias depends on how participation covaries with schooling. Adding lambda with number of young children excluded from the wage equation gives a corrected estimate, and the t-statistic on lambda is the test of whether the correction was needed. Without that exclusion, the same regression is identified only by the curvature of the Mills ratio, and its standard errors will say so.

Symptom → diagnosis → what to do

If you see this	It means	Do this
No exclusion restriction	Identification from functional form alone	Extremely fragile. Report OLS alongside and be honest
lambda coefficient insignificant	No detectable selection bias	OLS on the selected sample is defensible
Enormous standard errors	lambda nearly collinear with x	The classic symptom of a missing exclusion restriction
rho estimated at plus or minus 1	A boundary solution	The model is not identified in this sample
Two-step and MLE disagreeing sharply	Normality is doing heavy lifting	Distrust both
Selection driven by observables only	The wrong model	Control for them, or use matching

Common mistakes

No exclusion restriction. If the selection and outcome equations contain the same variables, treat any correction with deep suspicion however good the p-values look.
Quoting the two-step standard errors. They are wrong in a known direction.
Believing the correction more than the assumption behind it. Heckman trades a selection problem for a normality assumption; whether that is a good trade is a matter of argument, not of output.
Applying the correction when the outcome is missing for reasons unrelated to the outcome. If selection really is ignorable, Heckman adds collinearity and noise for no gain, and the inverse Mills ratio is close to a linear function of the other regressors.

Where to read this properly

Heckman (1979), "Sample Selection Bias as a Specification Error", Econometrica 47(1).
Puhani (2000), "The Heckman Correction for Sample Selection and Its Critique", Journal of Economic Surveys 14(1).
Wooldridge, Econometric Analysis of Cross Section and Panel Data (2010), ch. 19.
Angrist and Pischke, Mostly Harmless Econometrics (2009), ch. 3, for a sceptical view.

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Heckman selection

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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