The Heckman correction is a statistical technique to correct bias from non-randomly selected samples or otherwise incidentally truncated dependent variables, a pervasive issue in quantitative social sciences when using observational data.
[1] Conceptually, this is achieved by explicitly modelling the individual sampling probability of each observation (the so-called selection equation) together with the conditional expectation of the dependent variable (the so-called outcome equation).
The resulting likelihood function is mathematically similar to the tobit model for censored dependent variables, a connection first drawn by James Heckman in 1974.
[4] Heckman received the Nobel Memorial Prize in Economic Sciences in 2000 for his work in this field.
[5] Statistical analyses based on non-randomly selected samples can lead to erroneous conclusions.
Heckman discussed bias from using nonrandom selected samples to estimate behavioral relationships as a specification error.
In the first stage, the researcher formulates a model, based on economic theory, for the probability of working.
Estimation of the model yields results that can be used to predict this employment probability for each individual.
In the second stage, the researcher corrects for self-selection by incorporating a transformation of these predicted individual probabilities as an additional explanatory variable.
This equation demonstrates Heckman's insight that sample selection can be viewed as a form of omitted-variables bias, as conditional on both X and on
term, and including it as an additional explanatory variable in linear regression estimation of the wage equation.
Heckman's achievements have generated a large number of empirical applications in economics as well as in other social sciences.
[6] The Heckman correction is a two-step M-estimator where the covariance matrix generated by OLS estimation of the second stage is inconsistent.
[7] Correct standard errors and other statistics can be generated from an asymptotic approximation or by resampling, such as through a bootstrap.