Maximum simulated likelihood estimation of the panel sample selection model

ABSTRACT

Heckman's (1976, 1979) sample selection model has been employed in many studies of linear and nonlinear regression applications. It is well known that ignoring the sample selectivity may result in inconsistency of the estimator due to the correlation between the statistical errors in the selection and main equations. In this article, we reconsider the maximum likelihood estimator for the panel sample selection model in Keane et al. (1988). Since the panel data model contains individual effects, such as fixed or random effects, the likelihood function is more complicated than that of the classical Heckman model. As an alternative to the existing derivation of the likelihood function in the literature, we show that the conditional distribution of the main equation follows a closed skew-normal (CSN) distribution, of which the linear transformation is still a CSN. Although the evaluation of the likelihood function involves high-dimensional integration, we show that the integration can be further simplified into a one-dimensional problem and can be evaluated by the simulated likelihood method. Moreover, we also conduct a Monte Carlo experiment to investigate the finite sample performance of the proposed estimator and find that our estimator provides reliable and quite satisfactory results.

KEYWORDS:

• Closed skew-normal
• maximum likelihood estimation
• panel data
• sample selection

JEL CLASSIFICATION:

• C33
• C40

Acknowledgments

The authors thank Herman Bierens, Peter Schmidt, and two anonymous referees for their comments. The usual disclaimer applies.

Notes

See Heckman (1979).

After the within transformation, the last period sample is not informative anymore, so we may drop it directly.

We thank an anonymous referee for pointing this out.

More discussions about the MSL can be found in the works of Train (2003) and Greene (2005).

See Appendix II for the proof.

We thank an anonymous referee for the suggestion of using the ordinary least squares (OLS) estimates as the initial values. Since the starting values also contain the values of λ, σ_v, and ρ, which are not available from the OLS approach, in order to have a unified way to set the initials in all simulations we draw the initials from the normal distribution. The initial values are chosen randomly, which is similar to the approach of Krief (2014) in spirit.

The worker's age is closely related to experience if we use the usual implicit experience formula to compute her experience. Therefore, the effect of age in the regression is interpreted as the effect of experience.