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Nowadays most people run wage regressions conditional on being employed rather than using Heckman selection models. So what are its drawbacks and why did this approach fall out of fashion?

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I think it is because Heckman selection models have disadvantages compared to more modern models.

The Heckman two-step estimator is a limited information maximum likelihood estimator. In asymptotic theory and in finite samples, the full information (FIML) estimator exhibits better statistical properties (source). FIML is more computationally difficult but that is no longer problem with modern PCs (nowadays its easy to get PC with high RAM).

The canonical model assumes the errors are jointly normal. If that assumption fails, the estimator is generally inconsistent and can provide misleading inference in small samples (source).

The model obtains formal identification from the normality assumption when the same covariates appear in the selection equation and the equation of interest, but identification will be tenuous unless there are many observations in the tails where there is substantial nonlinearity in the inverse Mills ratio. Generally, an exclusion restriction is required to generate credible estimates: there must be at least one variable which appears with a non-zero coefficient in the selection equation but does not appear in the equation of interest, essentially an instrument. If no such variable is available, it may be difficult to correct for sampling selectivity (source).

Nowadays you can estimate models that do not suffer from the problems above. Biggest advantage of Heckman model was that it was computationally easy to estimate but as hardware improved that became less of an advantage.

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